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+The Project Gutenberg EBook of The Hindu-Arabic Numerals, by 
+David Eugene Smith and Louis Charles Karpinski
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: The Hindu-Arabic Numerals
+
+Author: David Eugene Smith
+        Louis Charles Karpinski
+
+Release Date: September 14, 2007 [EBook #22599]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE HINDU-ARABIC NUMERALS ***
+
+
+
+
+Produced by David Newman, Chuck Greif, Keith Edkins and
+the Online Distributed Proofreading Team at
+https://www.pgdp.net (This file was produced from images
+from the Cornell University Library: Historical Mathematics
+Monographs collection.)
+
+
+
+
+
+Transcriber's Note:
+
+The following codes are used for characters that are not present in the
+character set used for this version of the book.
+
+  [=a] a with macron (etc.)
+  [.g] g with dot above (etc.)
+  ['s] s with acute accent
+  [d.] d with dot below (etc.)
+  [d=] d with line below
+  [H)] H with breve below
+
+
+
+
+
+THE
+
+HINDU-ARABIC NUMERALS
+
+BY
+DAVID EUGENE SMITH
+AND
+LOUIS CHARLES KARPINSKI
+
+BOSTON AND LONDON
+GINN AND COMPANY, PUBLISHERS
+1911
+
+COPYRIGHT, 1911, BY DAVID EUGENE SMITH
+AND LOUIS CHARLES KARPINSKI
+ALL RIGHTS RESERVED
+811.7
+
+THE ATHENÆUM PRESS
+GINN AND COMPANY · PROPRIETORS
+BOSTON · U.S.A.
+
+       *       *       *       *       *
+
+
+{iii}
+
+PREFACE
+
+So familiar are we with the numerals that bear the misleading name of
+Arabic, and so extensive is their use in Europe and the Americas, that it
+is difficult for us to realize that their general acceptance in the
+transactions of commerce is a matter of only the last four centuries, and
+that they are unknown to a very large part of the human race to-day. It
+seems strange that such a labor-saving device should have struggled for
+nearly a thousand years after its system of place value was perfected
+before it replaced such crude notations as the one that the Roman conqueror
+made substantially universal in Europe. Such, however, is the case, and
+there is probably no one who has not at least some slight passing interest
+in the story of this struggle. To the mathematician and the student of
+civilization the interest is generally a deep one; to the teacher of the
+elements of knowledge the interest may be less marked, but nevertheless it
+is real; and even the business man who makes daily use of the curious
+symbols by which we express the numbers of commerce, cannot fail to have
+some appreciation for the story of the rise and progress of these tools of
+his trade.
+
+This story has often been told in part, but it is a long time since any
+effort has been made to bring together the fragmentary narrations and to
+set forth the general problem of the origin and development of these {iv}
+numerals. In this little work we have attempted to state the history of
+these forms in small compass, to place before the student materials for the
+investigation of the problems involved, and to express as clearly as
+possible the results of the labors of scholars who have studied the subject
+in different parts of the world. We have had no theory to exploit, for the
+history of mathematics has seen too much of this tendency already, but as
+far as possible we have weighed the testimony and have set forth what seem
+to be the reasonable conclusions from the evidence at hand.
+
+To facilitate the work of students an index has been prepared which we hope
+may be serviceable. In this the names of authors appear only when some use
+has been made of their opinions or when their works are first mentioned in
+full in a footnote.
+
+If this work shall show more clearly the value of our number system, and
+shall make the study of mathematics seem more real to the teacher and
+student, and shall offer material for interesting some pupil more fully in
+his work with numbers, the authors will feel that the considerable labor
+involved in its preparation has not been in vain.
+
+We desire to acknowledge our especial indebtedness to Professor Alexander
+Ziwet for reading all the proof, as well as for the digest of a Russian
+work, to Professor Clarence L. Meader for Sanskrit transliterations, and to
+Mr. Steven T. Byington for Arabic transliterations and the scheme of
+pronunciation of Oriental names, and also our indebtedness to other
+scholars in Oriental learning for information.
+
+DAVID EUGENE SMITH
+
+LOUIS CHARLES KARPINSKI
+
+       *       *       *       *       *
+
+
+{v}
+
+CONTENTS
+
+  CHAPTER
+
+        PRONUNCIATION OF ORIENTAL NAMES                              vi
+
+  I.    EARLY IDEAS OF THEIR ORIGIN                                   1
+
+  II.   EARLY HINDU FORMS WITH NO PLACE VALUE                        12
+
+  III.  LATER HINDU FORMS, WITH A PLACE VALUE                        38
+
+  IV.   THE SYMBOL ZERO                                              51
+
+  V.    THE QUESTION OF THE INTRODUCTION OF THE
+        NUMERALS INTO EUROPE BY BOETHIUS                             63
+
+  VI.   THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS              91
+
+  VII.  THE DEFINITE INTRODUCTION OF THE NUMERALS INTO EUROPE        99
+
+  VIII. THE SPREAD OF THE NUMERALS IN EUROPE                        128
+
+  INDEX                                                             153
+
+       *       *       *       *       *
+
+
+{vi}
+
+PRONUNCIATION OF ORIENTAL NAMES
+
+(S) = in Sanskrit names and words; (A) = in Arabic names and words.
+
+B, D, F, G, H, J, L, M, N, P, SH (A), T, TH (A), V, W, X, Z, as in English.
+
+A, (S) like _u_ in _but_: thus _pandit_, pronounced _pundit_. (A) like _a_
+in _ask_ or in _man_. [=A], as in _father_.
+
+C, (S) like _ch_ in _church_ (Italian _c_ in _cento_).
+
+[D.], [N.], [S.], [T.], (S) _d_, _n_, _sh_, _t_, made with the tip of the
+tongue turned up and back into the dome of the palate. [D.], [S.], [T.],
+[Z.], (A) _d_, _s_, _t_, _z_, made with the tongue spread so that the
+sounds are produced largely against the side teeth. Europeans commonly
+pronounce [D.], [N.], [S.], [T.], [Z.], both (S) and (A), as simple _d_,
+_n_, _sh_ (S) or _s_ (A), _t_, _z_. [D=] (A), like _th_ in _this_.
+
+E, (S) as in _they_. (A) as in _bed_.
+
+[.G], (A) a voiced consonant formed below the vocal cords; its sound is
+compared by some to a _g_, by others to a guttural _r_; in Arabic words
+adopted into English it is represented by _gh_ (e.g. _ghoul_), less often
+_r_ (e.g. _razzia_).
+
+H preceded by _b_, _c_, _t_, _[t.]_, etc. does not form a single sound with
+these letters, but is a more or less distinct _h_ sound following them; cf.
+the sounds in _abhor, boathook_, etc., or, more accurately for (S), the
+"bhoys" etc. of Irish brogue. H (A) retains its consonant sound at the end
+of a word. [H.], (A) an unvoiced consonant formed below the vocal cords;
+its sound is sometimes compared to German hard _ch_, and may be represented
+by an _h_ as strong as possible. In Arabic words adopted into English it is
+represented by _h_, e.g. in _sahib_, _hakeem_. [H.] (S) is final consonant
+_h_, like final _h_ (A).
+
+I, as in _pin_. [=I], as in _pique_.
+
+K, as in _kick_.
+
+KH, (A) the hard _ch_ of Scotch _loch_, German _ach_, especially of German
+as pronounced by the Swiss.
+
+[.M], [.N], (S) like French final _m_ or _n_, nasalizing the preceding
+vowel.
+
+[N.], see [D.]. Ñ, like _ng_ in _singing_.
+
+O, (S) as in _so_. (A) as in _obey_.
+
+Q, (A) like _k_ (or _c_) in _cook_; further back in the mouth than in
+_kick_.
+
+R, (S) English _r_, smooth and untrilled. (A) stronger. [R.], (S) r used as
+vowel, as in _apron_ when pronounced _aprn_ and not _apern_; modern Hindus
+say _ri_, hence our _amrita_, _Krishna_, for _a-m[r.]ta, K[r.][s.][n.]a_.
+
+S, as in _same_. [S.], see [D.]. ['S], (S) English _sh_ (German _sch_).
+
+[T.], see [D.].
+
+U, as in _put_. [=U], as in _rule_.
+
+Y, as in _you_.
+
+[Z.], see [D.].
+
+`, (A) a sound kindred to the spiritus lenis (that is, to our ears, the
+mere distinct separation of a vowel from the preceding sound, as at the
+beginning of a word in German) and to _[h.]_. The ` is a very distinct
+sound in Arabic, but is more nearly represented by the spiritus lenis than
+by any sound that we can produce without much special training. That is, it
+should be treated as silent, but the sounds that precede and follow it
+should not run together. In Arabic words adopted into English it is treated
+as silent, e.g. in _Arab_, _amber_, _Caaba_ (_`Arab_, _`anbar_, _ka`abah_).
+
+(A) A final long vowel is shortened before _al_ (_'l_) or _ibn_ (whose _i_
+is then silent).
+
+Accent: (S) as if Latin; in determining the place of the accent _[.m]_ and
+_[.n]_ count as consonants, but _h_ after another consonant does not. (A),
+on the last syllable that contains a long vowel or a vowel followed by two
+consonants, except that a final long vowel is not ordinarily accented; if
+there is no long vowel nor two consecutive consonants, the accent falls on
+the first syllable. The words _al_ and _ibn_ are never accented.
+
+       *       *       *       *       *
+
+
+{1}
+
+THE HINDU-ARABIC NUMERALS
+
+CHAPTER I
+
+EARLY IDEAS OF THEIR ORIGIN
+
+It has long been recognized that the common numerals used in daily life are
+of comparatively recent origin. The number of systems of notation employed
+before the Christian era was about the same as the number of written
+languages, and in some cases a single language had several systems. The
+Egyptians, for example, had three systems of writing, with a numerical
+notation for each; the Greeks had two well-defined sets of numerals, and
+the Roman symbols for number changed more or less from century to century.
+Even to-day the number of methods of expressing numerical concepts is much
+greater than one would believe before making a study of the subject, for
+the idea that our common numerals are universal is far from being correct.
+It will be well, then, to think of the numerals that we still commonly call
+Arabic, as only one of many systems in use just before the Christian era.
+As it then existed the system was no better than many others, it was of
+late origin, it contained no zero, it was cumbersome and little used, {2}
+and it had no particular promise. Not until centuries later did the system
+have any standing in the world of business and science; and had the place
+value which now characterizes it, and which requires a zero, been worked
+out in Greece, we might have been using Greek numerals to-day instead of
+the ones with which we are familiar.
+
+Of the first number forms that the world used this is not the place to
+speak. Many of them are interesting, but none had much scientific value. In
+Europe the invention of notation was generally assigned to the eastern
+shores of the Mediterranean until the critical period of about a century
+ago,--sometimes to the Hebrews, sometimes to the Egyptians, but more often
+to the early trading Phoenicians.[1]
+
+The idea that our common numerals are Arabic in origin is not an old one.
+The mediæval and Renaissance writers generally recognized them as Indian,
+and many of them expressly stated that they were of Hindu origin.[2] {3}
+Others argued that they were probably invented by the Chaldeans or the Jews
+because they increased in value from right to left, an argument that would
+apply quite as well to the Roman and Greek systems, or to any other. It
+was, indeed, to the general idea of notation that many of these writers
+referred, as is evident from the words of England's earliest arithmetical
+textbook-maker, Robert Recorde (c. 1542): "In that thinge all men do agree,
+that the Chaldays, whiche fyrste inuented thys arte, did set these figures
+as thei set all their letters. for they wryte backwarde as you tearme it,
+and so doo they reade. And that may appeare in all Hebrewe, Chaldaye and
+Arabike bookes ... where as the Greekes, Latines, and all nations of
+Europe, do wryte and reade from the lefte hand towarde the ryghte."[3]
+Others, and {4} among them such influential writers as Tartaglia[4] in
+Italy and Köbel[5] in Germany, asserted the Arabic origin of the numerals,
+while still others left the matter undecided[6] or simply dismissed them as
+"barbaric."[7] Of course the Arabs themselves never laid claim to the
+invention, always recognizing their indebtedness to the Hindus both for the
+numeral forms and for the distinguishing feature of place value. Foremost
+among these writers was the great master of the golden age of Bagdad, one
+of the first of the Arab writers to collect the mathematical classics of
+both the East and the West, preserving them and finally passing them on to
+awakening Europe. This man was Mo[h.]ammed the Son of Moses, from
+Khow[=a]rezm, or, more after the manner of the Arab, Mo[h.]ammed ibn
+M[=u]s[=a] al-Khow[=a]razm[=i],[8] a man of great {5} learning and one to
+whom the world is much indebted for its present knowledge of algebra[9] and
+of arithmetic. Of him there will often be occasion to speak; and in the
+arithmetic which he wrote, and of which Adelhard of Bath[10] (c. 1130) may
+have made the translation or paraphrase,[11] he stated distinctly that the
+numerals were due to the Hindus.[12] This is as plainly asserted by later
+Arab {6} writers, even to the present day.[13] Indeed the phrase _`ilm
+hind[=i]_, "Indian science," is used by them for arithmetic, as also the
+adjective _hind[=i]_ alone.[14]
+
+Probably the most striking testimony from Arabic sources is that given by
+the Arabic traveler and scholar Mohammed ibn A[h.]med, Ab[=u]
+'l-R[=i][h.][=a]n al-B[=i]r[=u]n[=i] (973-1048), who spent many years in
+Hindustan. He wrote a large work on India,[15] one on ancient
+chronology,[16] the "Book of the Ciphers," unfortunately lost, which
+treated doubtless of the Hindu art of calculating, and was the author of
+numerous other works. Al-B[=i]r[=u]n[=i] was a man of unusual attainments,
+being versed in Arabic, Persian, Sanskrit, Hebrew, and Syriac, as well as
+in astronomy, chronology, and mathematics. In his work on India he gives
+detailed information concerning the language and {7} customs of the people
+of that country, and states explicitly[17] that the Hindus of his time did
+not use the letters of their alphabet for numerical notation, as the Arabs
+did. He also states that the numeral signs called _a[.n]ka_[18] had
+different shapes in various parts of India, as was the case with the
+letters. In his _Chronology of Ancient Nations_ he gives the sum of a
+geometric progression and shows how, in order to avoid any possibility of
+error, the number may be expressed in three different systems: with Indian
+symbols, in sexagesimal notation, and by an alphabet system which will be
+touched upon later. He also speaks[19] of "179, 876, 755, expressed in
+Indian ciphers," thus again attributing these forms to Hindu sources.
+
+Preceding Al-B[=i]r[=u]n[=i] there was another Arabic writer of the tenth
+century, Mo[t.]ahhar ibn [T.][=a]hir,[20] author of the _Book of the
+Creation and of History_, who gave as a curiosity, in Indian (N[=a]gar[=i])
+symbols, a large number asserted by the people of India to represent the
+duration of the world. Huart feels positive that in Mo[t.]ahhar's time the
+present Arabic symbols had not yet come into use, and that the Indian
+symbols, although known to scholars, were not current. Unless this were the
+case, neither the author nor his readers would have found anything
+extraordinary in the appearance of the number which he cites.
+
+Mention should also be made of a widely-traveled student, Al-Mas`[=u]d[=i]
+(885?-956), whose journeys carried him from Bagdad to Persia, India,
+Ceylon, and even {8} across the China sea, and at other times to
+Madagascar, Syria, and Palestine.[21] He seems to have neglected no
+accessible sources of information, examining also the history of the
+Persians, the Hindus, and the Romans. Touching the period of the Caliphs
+his work entitled _Meadows of Gold_ furnishes a most entertaining fund of
+information. He states[22] that the wise men of India, assembled by the
+king, composed the _Sindhind_. Further on[23] he states, upon the authority
+of the historian Mo[h.]ammed ibn `Al[=i] `Abd[=i], that by order of
+Al-Man[s.][=u]r many works of science and astrology were translated into
+Arabic, notably the _Sindhind_ (_Siddh[=a]nta_). Concerning the meaning and
+spelling of this name there is considerable diversity of opinion.
+Colebrooke[24] first pointed out the connection between _Siddh[=a]nta_ and
+_Sindhind_. He ascribes to the word the meaning "the revolving ages."[25]
+Similar designations are collected by Sédillot,[26] who inclined to the
+Greek origin of the sciences commonly attributed to the Hindus.[27]
+Casiri,[28] citing the _T[=a]r[=i]kh al-[h.]okam[=a]_ or _Chronicles of the
+Learned_,[29] refers to the work {9} as the _Sindum-Indum_ with the meaning
+"perpetuum æternumque." The reference[30] in this ancient Arabic work to
+Al-Khow[=a]razm[=i] is worthy of note.
+
+This _Sindhind_ is the book, says Mas`[=u]d[=i],[31] which gives all that
+the Hindus know of the spheres, the stars, arithmetic,[32] and the other
+branches of science. He mentions also Al-Khow[=a]razm[=i] and [H.]abash[33]
+as translators of the tables of the _Sindhind_. Al-B[=i]r[=u]n[=i][34]
+refers to two other translations from a work furnished by a Hindu who came
+to Bagdad as a member of the political mission which Sindh sent to the
+caliph Al-Man[s.][=u]r, in the year of the Hejira 154 (A.D. 771).
+
+The oldest work, in any sense complete, on the history of Arabic literature
+and history is the _Kit[=a]b al-Fihrist_, written in the year 987 A.D., by
+Ibn Ab[=i] Ya`q[=u]b al-Nad[=i]m. It is of fundamental importance for the
+history of Arabic culture. Of the ten chief divisions of the work, the
+seventh demands attention in this discussion for the reason that its second
+subdivision treats of mathematicians and astronomers.[35]
+
+{10}
+
+The first of the Arabic writers mentioned is Al-Kind[=i] (800-870 A.D.),
+who wrote five books on arithmetic and four books on the use of the Indian
+method of reckoning. Sened ibn `Al[=i], the Jew, who was converted to Islam
+under the caliph Al-M[=a]m[=u]n, is also given as the author of a work on
+the Hindu method of reckoning. Nevertheless, there is a possibility[36]
+that some of the works ascribed to Sened ibn `Al[=i] are really works of
+Al-Khow[=a]razm[=i], whose name immediately precedes his. However, it is to
+be noted in this connection that Casiri[37] also mentions the same writer
+as the author of a most celebrated work on arithmetic.
+
+To Al-[S.][=u]f[=i], who died in 986 A.D., is also credited a large work on
+the same subject, and similar treatises by other writers are mentioned. We
+are therefore forced to the conclusion that the Arabs from the early ninth
+century on fully recognized the Hindu origin of the new numerals.
+
+Leonard of Pisa, of whom we shall speak at length in the chapter on the
+Introduction of the Numerals into Europe, wrote his _Liber Abbaci_[38] in
+1202. In this work he refers frequently to the nine Indian figures,[39]
+thus showing again the general consensus of opinion in the Middle Ages that
+the numerals were of Hindu origin.
+
+Some interest also attaches to the oldest documents on arithmetic in our
+own language. One of the earliest {11} treatises on algorism is a
+commentary[40] on a set of verses called the _Carmen de Algorismo_, written
+by Alexander de Villa Dei (Alexandra de Ville-Dieu), a Minorite monk of
+about 1240 A.D. The text of the first few lines is as follows:
+
+ "Hec algorism' ars p'sens dicit' in qua
+  Talib; indor[um] fruim bis quinq; figuris.[41]
+
+"This boke is called the boke of algorim or augrym after lewder use. And
+this boke tretys of the Craft of Nombryng, the quych crafte is called also
+Algorym. Ther was a kyng of Inde the quich heyth Algor & he made this
+craft.... Algorisms, in the quych we use teen figurys of Inde."
+
+       *       *       *       *       *
+
+
+{12}
+
+CHAPTER II
+
+EARLY HINDU FORMS WITH NO PLACE VALUE
+
+While it is generally conceded that the scientific development of astronomy
+among the Hindus towards the beginning of the Christian era rested upon
+Greek[42] or Chinese[43] sources, yet their ancient literature testifies to
+a high state of civilization, and to a considerable advance in sciences, in
+philosophy, and along literary lines, long before the golden age of Greece.
+From the earliest times even up to the present day the Hindu has been wont
+to put his thought into rhythmic form. The first of this poetry--it well
+deserves this name, being also worthy from a metaphysical point of
+view[44]--consists of the Vedas, hymns of praise and poems of worship,
+collected during the Vedic period which dates from approximately 2000 B.C.
+to 1400 B.C.[45] Following this work, or possibly contemporary with it, is
+the Brahmanic literature, which is partly ritualistic (the
+Br[=a]hma[n.]as), and partly philosophical (the Upanishads). Our especial
+interest is {13} in the S[=u]tras, versified abridgments of the ritual and
+of ceremonial rules, which contain considerable geometric material used in
+connection with altar construction, and also numerous examples of rational
+numbers the sum of whose squares is also a square, i.e. "Pythagorean
+numbers," although this was long before Pythagoras lived. Whitney[46]
+places the whole of the Veda literature, including the Vedas, the
+Br[=a]hma[n.]as, and the S[=u]tras, between 1500 B.C. and 800 B.C., thus
+agreeing with Bürk[47] who holds that the knowledge of the Pythagorean
+theorem revealed in the S[=u]tras goes back to the eighth century B.C.
+
+The importance of the S[=u]tras as showing an independent origin of Hindu
+geometry, contrary to the opinion long held by Cantor[48] of a Greek
+origin, has been repeatedly emphasized in recent literature,[49] especially
+since the appearance of the important work of Von Schroeder.[50] Further
+fundamental mathematical notions such as the conception of irrationals and
+the use of gnomons, as well as the philosophical doctrine of the
+transmigration of souls,--all of these having long been attributed to the
+Greeks,--are shown in these works to be native to India. Although this
+discussion does not bear directly upon the {14} origin of our numerals, yet
+it is highly pertinent as showing the aptitude of the Hindu for
+mathematical and mental work, a fact further attested by the independent
+development of the drama and of epic and lyric poetry.
+
+It should be stated definitely at the outset, however, that we are not at
+all sure that the most ancient forms of the numerals commonly known as
+Arabic had their origin in India. As will presently be seen, their forms
+may have been suggested by those used in Egypt, or in Eastern Persia, or in
+China, or on the plains of Mesopotamia. We are quite in the dark as to
+these early steps; but as to their development in India, the approximate
+period of the rise of their essential feature of place value, their
+introduction into the Arab civilization, and their spread to the West, we
+have more or less definite information. When, therefore, we consider the
+rise of the numerals in the land of the Sindhu,[51] it must be understood
+that it is only the large movement that is meant, and that there must
+further be considered the numerous possible sources outside of India itself
+and long anterior to the first prominent appearance of the number symbols.
+
+No one attempts to examine any detail in the history of ancient India
+without being struck with the great dearth of reliable material.[52] So
+little sympathy have the people with any save those of their own caste that
+a general literature is wholly lacking, and it is only in the observations
+of strangers that any all-round view of scientific progress is to be found.
+There is evidence that primary schools {15} existed in earliest times, and
+of the seventy-two recognized sciences writing and arithmetic were the most
+prized.[53] In the Vedic period, say from 2000 to 1400 B.C., there was the
+same attention to astronomy that was found in the earlier civilizations of
+Babylon, China, and Egypt, a fact attested by the Vedas themselves.[54]
+Such advance in science presupposes a fair knowledge of calculation, but of
+the manner of calculating we are quite ignorant and probably always shall
+be. One of the Buddhist sacred books, the _Lalitavistara_, relates that
+when the B[=o]dhisattva[55] was of age to marry, the father of Gopa, his
+intended bride, demanded an examination of the five hundred suitors, the
+subjects including arithmetic, writing, the lute, and archery. Having
+vanquished his rivals in all else, he is matched against Arjuna the great
+arithmetician and is asked to express numbers greater than 100 kotis.[56]
+In reply he gave a scheme of number names as high as 10^{53}, adding that
+he could proceed as far as 10^{421},[57] all of which suggests the system
+of Archimedes and the unsettled question of the indebtedness of the West to
+the East in the realm of ancient mathematics.[58] Sir Edwin Arnold, {16} in
+_The Light of Asia_, does not mention this part of the contest, but he
+speaks of Buddha's training at the hands of the learned Vi[s.]vamitra:
+
+  "And Viswamitra said, 'It is enough,
+  Let us to numbers. After me repeat
+  Your numeration till we reach the lakh,[59]
+  One, two, three, four, to ten, and then by tens
+  To hundreds, thousands.' After him the child
+  Named digits, decads, centuries, nor paused,
+  The round lakh reached, but softly murmured on,
+  Then comes the k[=o]ti, nahut, ninnahut,
+  Khamba, viskhamba, abab, attata,
+  To kumuds, gundhikas, and utpalas,
+  By pundar[=i]kas into padumas,
+  Which last is how you count the utmost grains
+  Of Hastagiri ground to finest dust;[60]
+  But beyond that a numeration is,
+  The K[=a]tha, used to count the stars of night,
+  The K[=o]ti-K[=a]tha, for the ocean drops;
+  Ingga, the calculus of circulars;
+  Sarvanikchepa, by the which you deal
+  With all the sands of Gunga, till we come
+  To Antah-Kalpas, where the unit is
+  The sands of the ten crore Gungas.  If one seeks
+  More comprehensive scale, th' arithmic mounts
+  By the Asankya, which is the tale
+  Of all the drops that in ten thousand years
+  Would fall on all the worlds by daily rain;
+  Thence unto Maha Kalpas, by the which
+  The gods compute their future and their past.'"
+
+{17}
+
+Thereupon Vi[s.]vamitra [=A]c[=a]rya[61] expresses his approval of the
+task, and asks to hear the "measure of the line" as far as y[=o]jana, the
+longest measure bearing name. This given, Buddha adds:
+
+  ... "'And master! if it please,
+  I shall recite how many sun-motes lie
+  From end to end within a y[=o]jana.'
+  Thereat, with instant skill, the little prince
+  Pronounced the total of the atoms true.
+  But Viswamitra heard it on his face
+  Prostrate before the boy; 'For thou,' he cried,
+  'Art Teacher of thy teachers--thou, not I,
+  Art G[=u]r[=u].'"
+
+It is needless to say that this is far from being history. And yet it puts
+in charming rhythm only what the ancient _Lalitavistara_ relates of the
+number-series of the Buddha's time. While it extends beyond all reason,
+nevertheless it reveals a condition that would have been impossible unless
+arithmetic had attained a considerable degree of advancement.
+
+To this pre-Christian period belong also the _Ved[=a][.n]gas_, or "limbs
+for supporting the Veda," part of that great branch of Hindu literature
+known as _Sm[r.]iti_ (recollection), that which was to be handed down by
+tradition. Of these the sixth is known as _Jyoti[s.]a_ (astronomy), a short
+treatise of only thirty-six verses, written not earlier than 300 B.C., and
+affording us some knowledge of the extent of number work in that
+period.[62] The Hindus {18} also speak of eighteen ancient Siddh[=a]ntas or
+astronomical works, which, though mostly lost, confirm this evidence.[63]
+
+As to authentic histories, however, there exist in India none relating to
+the period before the Mohammedan era (622 A.D.). About all that we know of
+the earlier civilization is what we glean from the two great epics, the
+Mah[=a]bh[=a]rata[64] and the R[=a]m[=a]yana, from coins, and from a few
+inscriptions.[65]
+
+It is with this unsatisfactory material, then, that we have to deal in
+searching for the early history of the Hindu-Arabic numerals, and the fact
+that many unsolved problems exist and will continue to exist is no longer
+strange when we consider the conditions. It is rather surprising that so
+much has been discovered within a century, than that we are so uncertain as
+to origins and dates and the early spread of the system. The probability
+being that writing was not introduced into India before the close of the
+fourth century B.C., and literature existing only in spoken form prior to
+that period,[66] the number work was doubtless that of all primitive
+peoples, palpable, merely a matter of placing sticks or cowries or pebbles
+on the ground, of marking a sand-covered board, or of cutting notches or
+tying cords as is still done in parts of Southern India to-day.[67]
+
+{19}
+
+The early Hindu numerals[68] may be classified into three great groups, (1)
+the Kharo[s.][t.]h[=i], (2) the Br[=a]hm[=i], and (3) the word and letter
+forms; and these will be considered in order.
+
+The Kharo[s.][t.]h[=i] numerals are found in inscriptions formerly known as
+Bactrian, Indo-Bactrian, and Aryan, and appearing in ancient Gandh[=a]ra,
+now eastern Afghanistan and northern Punjab. The alphabet of the language
+is found in inscriptions dating from the fourth century B.C. to the third
+century A.D., and from the fact that the words are written from right to
+left it is assumed to be of Semitic origin. No numerals, however, have been
+found in the earliest of these inscriptions, number-names probably having
+been written out in words as was the custom with many ancient peoples. Not
+until the time of the powerful King A['s]oka, in the third century B.C., do
+numerals appear in any inscriptions thus far discovered; and then only in
+the primitive form of marks, quite as they would be found in Egypt, Greece,
+Rome, or in {20} various other parts of the world. These A['s]oka[69]
+inscriptions, some thirty in all, are found in widely separated parts of
+India, often on columns, and are in the various vernaculars that were
+familiar to the people. Two are in the Kharo[s.][t.]h[=i] characters, and
+the rest in some form of Br[=a]hm[=i]. In the Kharo[s.][t.]h[=i]
+inscriptions only four numerals have been found, and these are merely
+vertical marks for one, two, four, and five, thus:
+
+  |     ||     |||     ||||
+
+In the so-called ['S]aka inscriptions, possibly of the first century B.C.,
+more numerals are found, and in more highly developed form, the
+right-to-left system appearing, together with evidences of three different
+scales of counting,--four, ten, and twenty. The numerals of this period are
+as follows:
+
+[Illustration]
+
+There are several noteworthy points to be observed in studying this system.
+In the first place, it is probably not as early as that shown in the
+N[=a]n[=a] Gh[=a]t forms hereafter given, although the inscriptions
+themselves at N[=a]n[=a] Gh[=a]t are later than those of the A['s]oka
+period. The {21} four is to this system what the X was to the Roman,
+probably a canceling of three marks as a workman does to-day for five, or a
+laying of one stick across three others. The ten has never been
+satisfactorily explained. It is similar to the A of the Kharo[s.][t.]h[=i]
+alphabet, but we have no knowledge as to why it was chosen. The twenty is
+evidently a ligature of two tens, and this in turn suggested a kind of
+radix, so that ninety was probably written in a way reminding one of the
+quatre-vingt-dix of the French. The hundred is unexplained, although it
+resembles the letter _ta_ or _tra_ of the Br[=a]hm[=i] alphabet with 1
+before (to the right of) it. The two hundred is only a variant of the
+symbol for hundred, with two vertical marks.[70]
+
+This system has many points of similarity with the Nabatean numerals[71] in
+use in the first centuries of the Christian era. The cross is here used for
+four, and the Kharo[s.][t.]h[=i] form is employed for twenty. In addition
+to this there is a trace of an analogous use of a scale of twenty. While
+the symbol for 100 is quite different, the method of forming the other
+hundreds is the same. The correspondence seems to be too marked to be
+wholly accidental.
+
+It is not in the Kharo[s.][t.]h[=i] numerals, therefore, that we can hope
+to find the origin of those used by us, and we turn to the second of the
+Indian types, the Br[=a]hm[=i] characters. The alphabet attributed to
+Brahm[=a] is the oldest of the several known in India, and was used from
+the earliest historic times. There are various theories of its origin, {22}
+none of which has as yet any wide acceptance,[72] although the problem
+offers hope of solution in due time. The numerals are not as old as the
+alphabet, or at least they have not as yet been found in inscriptions
+earlier than those in which the edicts of A['s]oka appear, some of these
+having been incised in Br[=a]hm[=i] as well as Kharo[s.][t.]h[=i]. As
+already stated, the older writers probably wrote the numbers in words, as
+seems to have been the case in the earliest Pali writings of Ceylon.[73]
+
+The following numerals are, as far as known, the only ones to appear in the
+A['s]oka edicts:[74]
+
+[Illustration]
+
+These fragments from the third century B.C., crude and unsatisfactory as
+they are, are the undoubted early forms from which our present system
+developed. They next appear in the second century B.C. in some inscriptions
+in the cave on the top of the N[=a]n[=a] Gh[=a]t hill, about seventy-five
+miles from Poona in central India. These inscriptions may be memorials of
+the early Andhra dynasty of southern India, but their chief interest lies
+in the numerals which they contain.
+
+The cave was made as a resting-place for travelers ascending the hill,
+which lies on the road from Kaly[=a]na to Junar. It seems to have been cut
+out by a descendant {23} of King ['S][=a]tav[=a]hana,[75] for inside the
+wall opposite the entrance are representations of the members of his
+family, much defaced, but with the names still legible. It would seem that
+the excavation was made by order of a king named Vedisiri, and "the
+inscription contains a list of gifts made on the occasion of the
+performance of several _yagnas_ or religious sacrifices," and numerals are
+to be seen in no less than thirty places.[76]
+
+There is considerable dispute as to what numerals are really found in these
+inscriptions, owing to the difficulty of deciphering them; but the
+following, which have been copied from a rubbing, are probably number
+forms:[77]
+
+[Illustration]
+
+The inscription itself, so important as containing the earliest
+considerable Hindu numeral system connected with our own, is of sufficient
+interest to warrant reproducing part of it in facsimile, as is done on page
+24.
+
+{24}
+
+[Illustration]
+
+The next very noteworthy evidence of the numerals, and this quite complete
+as will be seen, is found in certain other cave inscriptions dating back to
+the first or second century A.D. In these, the Nasik[78] cave inscriptions,
+the forms are as follows:
+
+[Illustration]
+
+From this time on, until the decimal system finally adopted the first nine
+characters and replaced the rest of the Br[=a]hm[=i] notation by adding the
+zero, the progress of these forms is well marked. It is therefore well to
+present synoptically the best-known specimens that have come down to us,
+and this is done in the table on page 25.[79]
+
+{25}
+
+TABLE SHOWING THE PROGRESS OF NUMBER FORMS IN INDIA
+
+  NUMERALS 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 1000
+  A['s]oka[80]                [Illustration]
+  ['S]aka[81]                 [Illustration]
+  A['s]oka[82]                [Illustration]
+  N[=a]gar[=i][83]            [Illustration]
+  Nasik[84]                   [Illustration]
+  K[s.]atrapa[85]             [Illustration]
+  Ku[s.]ana [86]              [Illustration]
+  Gupta[87]                   [Illustration]
+  Valhab[=i][88]              [Illustration]
+  Nepal [89]                  [Illustration]
+  Kali[.n]ga[90]              [Illustration]
+  V[=a]k[=a][t.]aka[91]       [Illustration]
+
+[Most of these numerals are given by Bühler, loc. cit., Tafel IX.]
+
+{26} With respect to these numerals it should first be noted that no zero
+appears in the table, and as a matter of fact none existed in any of the
+cases cited. It was therefore impossible to have any place value, and the
+numbers like twenty, thirty, and other multiples of ten, one hundred, and
+so on, required separate symbols except where they were written out in
+words. The ancient Hindus had no less than twenty of these symbols,[92] a
+number that was afterward greatly increased. The following are examples of
+their method of indicating certain numbers between one hundred and one
+thousand:
+
+  [93] [Numerals] for 174
+  [94] [Numerals] for 191
+  [95] [Numerals] for 269
+  [96] [Numerals] for 252
+  [97] [Numerals] for 400
+  [98] [Numerals] for 356
+
+{27}
+
+To these may be added the following numerals below one hundred, similar to
+those in the table:
+
+  [Numerals][99]    for 90
+  [Numerals][100]   for 70
+
+We have thus far spoken of the Kharo[s.][t.]h[=i] and Br[=a]hm[=i]
+numerals, and it remains to mention the third type, the word and letter
+forms. These are, however, so closely connected with the perfecting of the
+system by the invention of the zero that they are more appropriately
+considered in the next chapter, particularly as they have little relation
+to the problem of the origin of the forms known as the Arabic.
+
+Having now examined types of the early forms it is appropriate to turn our
+attention to the question of their origin. As to the first three there is
+no question. The [1 vertical stroke] or [1 horizontal stroke] is simply one
+stroke, or one stick laid down by the computer. The [2 vertical strokes] or
+[2 horizontal strokes] represents two strokes or two sticks, and so for the
+[3 vertical strokes] and [3 horizontal strokes]. From some primitive [2
+vertical strokes] came the two of Egypt, of Rome, of early Greece, and of
+various other civilizations. It appears in the three Egyptian numeral
+systems in the following forms:
+
+  Hieroglyphic [2 vertical strokes]
+  Hieratic     [Hieratic 2]
+  Demotic      [Demotic 2]
+
+The last of these is merely a cursive form as in the Arabic [Arabic 2],
+which becomes our 2 if tipped through a right angle. From some primitive [2
+horizontal strokes] came the Chinese {28} symbol, which is practically
+identical with the symbols found commonly in India from 150 B.C. to 700
+A.D. In the cursive form it becomes [2 horizontal strokes joined], and this
+was frequently used for two in Germany until the 18th century. It finally
+went into the modern form 2, and the [3 horizontal strokes] in the same way
+became our 3.
+
+There is, however, considerable ground for interesting speculation with
+respect to these first three numerals. The earliest Hindu forms were
+perpendicular. In the N[=a]n[=a] Gh[=a]t inscriptions they are vertical.
+But long before either the A['s]oka or the N[=a]n[=a] Gh[=a]t inscriptions
+the Chinese were using the horizontal forms for the first three numerals,
+but a vertical arrangement for four.[101] Now where did China get these
+forms? Surely not from India, for she had them, as her monuments and
+literature[102] show, long before the Hindus knew them. The tradition is
+that China brought her civilization around the north of Tibet, from
+Mongolia, the primitive habitat being Mesopotamia, or possibly the oases of
+Turkestan. Now what numerals did Mesopotamia use? The Babylonian system,
+simple in its general principles but very complicated in many of its
+details, is now well known.[103] In particular, one, two, and three were
+represented by vertical arrow-heads. Why, then, did the Chinese write {29}
+theirs horizontally? The problem now takes a new interest when we find that
+these Babylonian forms were not the primitive ones of this region, but that
+the early Sumerian forms were horizontal.[104]
+
+What interpretation shall be given to these facts? Shall we say that it was
+mere accident that one people wrote "one" vertically and that another wrote
+it horizontally? This may be the case; but it may also be the case that the
+tribal migrations that ended in the Mongol invasion of China started from
+the Euphrates while yet the Sumerian civilization was prominent, or from
+some common source in Turkestan, and that they carried to the East the
+primitive numerals of their ancient home, the first three, these being all
+that the people as a whole knew or needed. It is equally possible that
+these three horizontal forms represent primitive stick-laying, the most
+natural position of a stick placed in front of a calculator being the
+horizontal one. When, however, the cuneiform writing developed more fully,
+the vertical form may have been proved the easier to make, so that by the
+time the migrations to the West began these were in use, and from them came
+the upright forms of Egypt, Greece, Rome, and other Mediterranean lands,
+and those of A['s]oka's time in India. After A['s]oka, and perhaps among
+the merchants of earlier centuries, the horizontal forms may have come down
+into India from China, thus giving those of the N[=a]n[=a] Gh[=a]t cave and
+of later inscriptions. This is in the realm of speculation, but it is not
+improbable that further epigraphical studies may confirm the hypothesis.
+
+{30}
+
+As to the numerals above three there have been very many conjectures. The
+figure one of the Demotic looks like the one of the Sanskrit, the two
+(reversed) like that of the Arabic, the four has some resemblance to that
+in the Nasik caves, the five (reversed) to that on the K[s.]atrapa coins,
+the nine to that of the Ku[s.]ana inscriptions, and other points of
+similarity have been imagined. Some have traced resemblance between the
+Hieratic five and seven and those of the Indian inscriptions. There have
+not, therefore, been wanting those who asserted an Egyptian origin for
+these numerals.[105] There has already been mentioned the fact that the
+Kharo[s.][t.]h[=i] numerals were formerly known as Bactrian, Indo-Bactrian,
+and Aryan. Cunningham[106] was the first to suggest that these numerals
+were derived from the alphabet of the Bactrian civilization of Eastern
+Persia, perhaps a thousand years before our era, and in this he was
+supported by the scholarly work of Sir E. Clive Bayley,[107] who in turn
+was followed by Canon Taylor.[108] The resemblance has not proved
+convincing, however, and Bayley's drawings {31} have been criticized as
+being affected by his theory. The following is part of the hypothesis:[109]
+
+  _Numeral_      _Hindu_       _Bactrian_       _Sanskrit_
+     4           [Symbol]      [Symbol] = ch    chatur, Lat. quattuor
+     5           [Symbol]      [Symbol] = p     pancha, Gk. [Greek:p/ente]
+     6           [Symbol]      [Symbol] = s     [s.]a[s.]
+     7           [Symbol]      [Symbol] = [s.]  sapta
+     ( the s and [s.] are interchanged as occasionally in N. W. India)
+
+Bühler[110] rejects this hypothesis, stating that in four cases (four, six,
+seven, and ten) the facts are absolutely against it.
+
+While the relation to ancient Bactrian forms has been generally doubted, it
+is agreed that most of the numerals resemble Br[=a]hm[=i] letters, and we
+would naturally expect them to be initials.[111] But, knowing the ancient
+pronunciation of most of the number names,[112] we find this not to be the
+case. We next fall back upon the hypothesis {32} that they represent the
+order of letters[113] in the ancient alphabet. From what we know of this
+order, however, there seems also no basis for this assumption. We have,
+therefore, to confess that we are not certain that the numerals were
+alphabetic at all, and if they were alphabetic we have no evidence at
+present as to the basis of selection. The later forms may possibly have
+been alphabetical expressions of certain syllables called _ak[s.]aras_,
+which possessed in Sanskrit fixed numerical values,[114] but this is
+equally uncertain with the rest. Bayley also thought[115] that some of the
+forms were Phoenician, as notably the use of a circle for twenty, but the
+resemblance is in general too remote to be convincing.
+
+There is also some slight possibility that Chinese influence is to be seen
+in certain of the early forms of Hindu numerals.[116]
+
+{33}
+
+More absurd is the hypothesis of a Greek origin, supposedly supported by
+derivation of the current symbols from the first nine letters of the Greek
+alphabet.[117] This difficult feat is accomplished by twisting some of the
+letters, cutting off, adding on, and effecting other changes to make the
+letters fit the theory. This peculiar theory was first set up by
+Dasypodius[118] (Conrad Rauhfuss), and was later elaborated by Huet.[119]
+
+{34}
+
+A bizarre derivation based upon early Arabic (c. 1040 A.D.) sources is
+given by Kircher in his work[120] on number mysticism. He quotes from
+Abenragel,[121] giving the Arabic and a Latin translation[122] and stating
+that the ordinary Arabic forms are derived from sectors of a circle,
+[circle].
+
+Out of all these conflicting theories, and from all the resemblances seen
+or imagined between the numerals of the West and those of the East, what
+conclusions are we prepared to draw as the evidence now stands? Probably
+none that is satisfactory. Indeed, upon the evidence at {35} hand we might
+properly feel that everything points to the numerals as being substantially
+indigenous to India. And why should this not be the case? If the king
+Srong-tsan-Gampo (639 A.D.), the founder of Lh[=a]sa,[123] could have set
+about to devise a new alphabet for Tibet, and if the Siamese, and the
+Singhalese, and the Burmese, and other peoples in the East, could have
+created alphabets of their own, why should not the numerals also have been
+fashioned by some temple school, or some king, or some merchant guild? By
+way of illustration, there are shown in the table on page 36 certain
+systems of the East, and while a few resemblances are evident, it is also
+evident that the creators of each system endeavored to find original forms
+that should not be found in other systems. This, then, would seem to be a
+fair interpretation of the evidence. A human mind cannot readily create
+simple forms that are absolutely new; what it fashions will naturally
+resemble what other minds have fashioned, or what it has known through
+hearsay or through sight. A circle is one of the world's common stock of
+figures, and that it should mean twenty in Phoenicia and in India is hardly
+more surprising than that it signified ten at one time in Babylon.[124] It
+is therefore quite probable that an extraneous origin cannot be found for
+the very sufficient reason that none exists.
+
+Of absolute nonsense about the origin of the symbols which we use much has
+been written. Conjectures, {36} however, without any historical evidence
+for support, have no place in a serious discussion of the gradual evolution
+of the present numeral forms.[125]
+
+  TABLE OF CERTAIN EASTERN SYSTEMS
+  Siam            [Illustration: numerals]
+  Burma[126]      [Illustration: numerals]
+  Malabar[127]    [Illustration: numerals]
+  Tibet[128]      [Illustration: numerals]
+  Ceylon[129]     [Illustration: numerals]
+  Malayalam[129]  [Illustration: numerals]
+
+{37}
+
+We may summarize this chapter by saying that no one knows what suggested
+certain of the early numeral forms used in India. The origin of some is
+evident, but the origin of others will probably never be known. There is no
+reason why they should not have been invented by some priest or teacher or
+guild, by the order of some king, or as part of the mysticism of some
+temple. Whatever the origin, they were no better than scores of other
+ancient systems and no better than the present Chinese system when written
+without the zero, and there would never have been any chance of their
+triumphal progress westward had it not been for this relatively late
+symbol. There could hardly be demanded a stronger proof of the Hindu origin
+of the character for zero than this, and to it further reference will be
+made in Chapter IV.
+
+       *       *       *       *       *
+
+
+{38}
+
+CHAPTER III
+
+LATER HINDU FORMS, WITH A PLACE VALUE
+
+Before speaking of the perfected Hindu numerals with the zero and the place
+value, it is necessary to consider the third system mentioned on page
+19,--the word and letter forms. The use of words with place value began at
+least as early as the 6th century of the Christian era. In many of the
+manuals of astronomy and mathematics, and often in other works in
+mentioning dates, numbers are represented by the names of certain objects
+or ideas. For example, zero is represented by "the void" (_['s][=u]nya_),
+or "heaven-space" (_ambara [=a]k[=a]['s]a_); one by "stick" (_rupa_),
+"moon" (_indu ['s]a['s]in_), "earth" (_bh[=u]_), "beginning" (_[=a]di_),
+"Brahma," or, in general, by anything markedly unique; two by "the twins"
+(_yama_), "hands" (_kara_), "eyes" (_nayana_), etc.; four by "oceans," five
+by "senses" (_vi[s.]aya_) or "arrows" (the five arrows of K[=a]mad[=e]va);
+six by "seasons" or "flavors"; seven by "mountain" (_aga_), and so on.[130]
+These names, accommodating themselves to the verse in which scientific
+works were written, had the additional advantage of not admitting, as did
+the figures, easy alteration, since any change would tend to disturb the
+meter.
+
+{39}
+
+As an example of this system, the date "['S]aka Sa[m.]vat, 867" (A.D. 945
+or 946), is given by "_giri-ra[s.]a-vasu_," meaning "the mountains"
+(seven), "the flavors" (six), and the gods "_Vasu_" of which there were
+eight. In reading the date these are read from right to left.[131] The
+period of invention of this system is uncertain. The first trace seems to
+be in the _['S]rautas[=u]tra_ of K[=a]ty[=a]yana and
+L[=a][t.]y[=a]yana.[132] It was certainly known to Var[=a]ha-Mihira (d.
+587),[133] for he used it in the _B[r.]hat-Sa[m.]hit[=a]._[134] It has also
+been asserted[135] that [=A]ryabha[t.]a (c. 500 A.D.) was familiar with
+this system, but there is nothing to prove the statement.[136] The earliest
+epigraphical examples of the system are found in the Bayang (Cambodia)
+inscriptions of 604 and 624 A.D.[137]
+
+Mention should also be made, in this connection, of a curious system of
+alphabetic numerals that sprang up in southern India. In this we have the
+numerals represented by the letters as given in the following table:
+
+  1        2       3       4       5        6       7      8     9      0
+  k        kh      g       gh      [.n]     c       ch     j     jh     ñ
+  [t.]     [t.]h   [d.]    [d.]h   [n.]     t       th     d     th     n
+  p        ph      b       bh      m
+  y        r       l       v       ['s]     [s.]    s      h      l
+
+{40}
+
+By this plan a numeral might be represented by any one of several letters,
+as shown in the preceding table, and thus it could the more easily be
+formed into a word for mnemonic purposes. For example, the word
+
+    2     3      1     5    6       5       1
+  _kha_ _gont_ _yan_ _me_ _[s.]a_ _m[=a]_ _pa_
+
+has the value 1,565,132, reading from right to left.[138] This, the oldest
+specimen (1184 A.D.) known of this notation, is given in a commentary on
+the Rigveda, representing the number of days that had elapsed from the
+beginning of the Kaliyuga. Burnell[139] states that this system is even yet
+in use for remembering rules to calculate horoscopes, and for astronomical
+tables.
+
+A second system of this kind is still used in the pagination of manuscripts
+in Ceylon, Siam, and Burma, having also had its rise in southern India. In
+this the thirty-four consonants when followed by _a_ (as _ka_ ... _la_)
+designate the numbers 1-34; by _[=a]_ (as _k[=a]_ ... _l[=a]_), those from
+35 to 68; by _i_ (_ki_ ... _li_), those from 69 to 102, inclusive; and so
+on.[140]
+
+As already stated, however, the Hindu system as thus far described was no
+improvement upon many others of the ancients, such as those used by the
+Greeks and the Hebrews. Having no zero, it was impracticable to designate
+the tens, hundreds, and other units of higher order by the same symbols
+used for the units from one to nine. In other words, there was no
+possibility of place value without some further improvement. So the
+N[=a]n[=a] Gh[=a]t {41} symbols required the writing of "thousand seven
+twenty-four" about like T 7, tw, 4 in modern symbols, instead of 7024, in
+which the seven of the thousands, the two of the tens (concealed in the
+word twenty, being originally "twain of tens," the _-ty_ signifying ten),
+and the four of the units are given as spoken and the order of the unit
+(tens, hundreds, etc.) is given by the place. To complete the system only
+the zero was needed; but it was probably eight centuries after the
+N[=a]n[=a] Gh[=a]t inscriptions were cut, before this important symbol
+appeared; and not until a considerably later period did it become well
+known. Who it was to whom the invention is due, or where he lived, or even
+in what century, will probably always remain a mystery.[141] It is possible
+that one of the forms of ancient abacus suggested to some Hindu astronomer
+or mathematician the use of a symbol to stand for the vacant line when the
+counters were removed. It is well established that in different parts of
+India the names of the higher powers took different forms, even the order
+being interchanged.[142] Nevertheless, as the significance of the name of
+the unit was given by the order in reading, these variations did not lead
+to error. Indeed the variation itself may have necessitated the
+introduction of a word to signify a vacant place or lacking unit, with the
+ultimate introduction of a zero symbol for this word.
+
+To enable us to appreciate the force of this argument a large number,
+8,443,682,155, may be considered as the Hindus wrote and read it, and then,
+by way of contrast, as the Greeks and Arabs would have read it.
+
+{42}
+
+_Modern American reading_, 8 billion, 443 million, 682 thousand, 155.
+
+_Hindu_, 8 padmas, 4 vyarbudas, 4 k[=o][t.]is, 3 prayutas, 6 lak[s.]as, 8
+ayutas, 2 sahasra, 1 ['s]ata, 5 da['s]an, 5.
+
+_Arabic and early German_, eight thousand thousand thousand and four
+hundred thousand thousand and forty-three thousand thousand, and six
+hundred thousand and eighty-two thousand and one hundred fifty-five (or
+five and fifty).
+
+_Greek_, eighty-four myriads of myriads and four thousand three hundred
+sixty-eight myriads and two thousand and one hundred fifty-five.
+
+As Woepcke[143] pointed out, the reading of numbers of this kind shows that
+the notation adopted by the Hindus tended to bring out the place idea. No
+other language than the Sanskrit has made such consistent application, in
+numeration, of the decimal system of numbers. The introduction of myriads
+as in the Greek, and thousands as in Arabic and in modern numeration, is
+really a step away from a decimal scheme. So in the numbers below one
+hundred, in English, eleven and twelve are out of harmony with the rest of
+the -teens, while the naming of all the numbers between ten and twenty is
+not analogous to the naming of the numbers above twenty. To conform to our
+written system we should have ten-one, ten-two, ten-three, and so on, as we
+have twenty-one, twenty-two, and the like. The Sanskrit is consistent, the
+units, however, preceding the tens and hundreds. Nor did any other ancient
+people carry the numeration as far as did the Hindus.[144]
+
+{43}
+
+When the _a[.n]kapalli_,[145] the decimal-place system of writing numbers,
+was perfected, the tenth symbol was called the _['s][=u]nyabindu_,
+generally shortened to _['s][=u]nya_ (the void). Brockhaus[146] has well
+said that if there was any invention for which the Hindus, by all their
+philosophy and religion, were well fitted, it was the invention of a symbol
+for zero. This making of nothingness the crux of a tremendous achievement
+was a step in complete harmony with the genius of the Hindu.
+
+It is generally thought that this _['s][=u]nya_ as a symbol was not used
+before about 500 A.D., although some writers have placed it earlier.[147]
+Since [=A]ryabha[t.]a gives our common method of extracting roots, it would
+seem that he may have known a decimal notation,[148] although he did not
+use the characters from which our numerals are derived.[149] Moreover, he
+frequently speaks of the {44} void.[150] If he refers to a symbol this
+would put the zero as far back as 500 A.D., but of course he may have
+referred merely to the concept of nothingness.
+
+A little later, but also in the sixth century, Var[=a]ha-Mihira[151] wrote
+a work entitled _B[r.]hat Sa[m.]hit[=a]_[152] in which he frequently uses
+_['s][=u]nya_ in speaking of numerals, so that it has been thought that he
+was referring to a definite symbol. This, of course, would add to the
+probability that [=A]ryabha[t.]a was doing the same.
+
+It should also be mentioned as a matter of interest, and somewhat related
+to the question at issue, that Var[=a]ha-Mihira used the word-system with
+place value[153] as explained above.
+
+The first kind of alphabetic numerals and also the word-system (in both of
+which the place value is used) are plays upon, or variations of, position
+arithmetic, which would be most likely to occur in the country of its
+origin.[154]
+
+At the opening of the next century (c. 620 A.D.) B[=a][n.]a[155] wrote of
+Subandhus's _V[=a]savadatt[=a]_ as a celebrated work, {45} and mentioned
+that the stars dotting the sky are here compared with zeros, these being
+points as in the modern Arabic system. On the other hand, a strong argument
+against any Hindu knowledge of the symbol zero at this time is the fact
+that about 700 A.D. the Arabs overran the province of Sind and thus had an
+opportunity of knowing the common methods used there for writing numbers.
+And yet, when they received the complete system in 776 they looked upon it
+as something new.[156] Such evidence is not conclusive, but it tends to
+show that the complete system was probably not in common use in India at
+the beginning of the eighth century. On the other hand, we must bear in
+mind the fact that a traveler in Germany in the year 1700 would probably
+have heard or seen nothing of decimal fractions, although these were
+perfected a century before that date. The élite of the mathematicians may
+have known the zero even in [=A]ryabha[t.]a's time, while the merchants and
+the common people may not have grasped the significance of the novelty
+until a long time after. On the whole, the evidence seems to point to the
+west coast of India as the region where the complete system was first
+seen.[157] As mentioned above, traces of the numeral words with place
+value, which do not, however, absolutely require a decimal place-system of
+symbols, are found very early in Cambodia, as well as in India.
+
+Concerning the earliest epigraphical instances of the use of the nine
+symbols, plus the zero, with place value, there {46} is some question.
+Colebrooke[158] in 1807 warned against the possibility of forgery in many
+of the ancient copper-plate land grants. On this account Fleet, in the
+_Indian Antiquary_,[159] discusses at length this phase of the work of the
+epigraphists in India, holding that many of these forgeries were made about
+the end of the eleventh century. Colebrooke[160] takes a more rational view
+of these forgeries than does Kaye, who seems to hold that they tend to
+invalidate the whole Indian hypothesis. "But even where that may be
+suspected, the historical uses of a monument fabricated so much nearer to
+the times to which it assumes to belong, will not be entirely superseded.
+The necessity of rendering the forged grant credible would compel a
+fabricator to adhere to history, and conform to established notions: and
+the tradition, which prevailed in his time, and by which he must be guided,
+would probably be so much nearer to the truth, as it was less remote from
+the period which it concerned."[161] Bühler[162] gives the copper-plate
+Gurjara inscription of Cedi-sa[m.]vat 346 (595 A.D.) as the oldest
+epigraphical use of the numerals[163] "in which the symbols correspond to
+the alphabet numerals of the period and the place." Vincent A. Smith[164]
+quotes a stone inscription of 815 A.D., dated Sa[m.]vat 872. So F. Kielhorn
+in the _Epigraphia Indica_[165] gives a Pathari pillar inscription of
+Parabala, dated Vikrama-sa[m.]vat 917, which corresponds to 861 A.D., {47}
+and refers also to another copper-plate inscription dated Vikrama-sa[m.]vat
+813 (756 A.D.). The inscription quoted by V. A. Smith above is that given
+by D. R. Bhandarkar,[166] and another is given by the same writer as of
+date Saka-sa[m.]vat 715 (798 A.D.), being incised on a pilaster.
+Kielhorn[167] also gives two copper-plate inscriptions of the time of
+Mahendrapala of Kanauj, Valhab[=i]-sa[m.]vat 574 (893 A.D.) and
+Vikrama-sa[m.]vat 956 (899 A.D.). That there should be any inscriptions of
+date as early even as 750 A.D., would tend to show that the system was at
+least a century older. As will be shown in the further development, it was
+more than two centuries after the introduction of the numerals into Europe
+that they appeared there upon coins and inscriptions. While Thibaut[168]
+does not consider it necessary to quote any specific instances of the use
+of the numerals, he states that traces are found from 590 A.D. on. "That
+the system now in use by all civilized nations is of Hindu origin cannot be
+doubted; no other nation has any claim upon its discovery, especially since
+the references to the origin of the system which are found in the nations
+of western Asia point unanimously towards India."[169]
+
+The testimony and opinions of men like Bühler, Kielhorn, V. A. Smith,
+Bhandarkar, and Thibaut are entitled to the most serious consideration. As
+authorities on ancient Indian epigraphy no others rank higher. Their work
+is accepted by Indian scholars the world over, and their united judgment as
+to the rise of the system with a place value--that it took place in India
+as early as the {48} sixth century A.D.--must stand unless new evidence of
+great weight can be submitted to the contrary.
+
+Many early writers remarked upon the diversity of Indian numeral forms.
+Al-B[=i]r[=u]n[=i] was probably the first; noteworthy is also Johannes
+Hispalensis,[170] who gives the variant forms for seven and four. We insert
+on p. 49 a table of numerals used with place value. While the chief
+authority for this is Bühler,[171] several specimens are given which are
+not found in his work and which are of unusual interest.
+
+The ['S][=a]rad[=a] forms given in the table use the circle as a symbol for
+1 and the dot for zero. They are taken from the paging and text of _The
+Kashmirian Atharva-Veda_[172], of which the manuscript used is certainly
+four hundred years old. Similar forms are found in a manuscript belonging
+to the University of Tübingen. Two other series presented are from Tibetan
+books in the library of one of the authors.
+
+For purposes of comparison the modern Sanskrit and Arabic numeral forms are
+added.
+
+  Sanskrit, [Illustration]
+  Arabic, [Illustration]
+
+{49}
+
+NUMERALS USED WITH PLACE VALUE
+
+      1  2  3  4  5  6  7  8  9  0
+  a[173] [Illustration]
+  b[174] [Illustration]
+  c[175] [Illustration]
+  d[176] [Illustration]
+  e[177] [Illustration]
+  f[178] [Illustration]
+  g[179] [Illustration]
+  h[180] [Illustration]
+  i[180] [Illustration]
+  j[181] [Illustration]
+  k[181] [Illustration]
+  l[182] [Illustration]
+  m[183] [Illustration]
+  n[184] [Illustration]
+
+       *       *       *       *       *
+
+
+{51}
+
+CHAPTER IV
+
+THE SYMBOL ZERO
+
+What has been said of the improved Hindu system with a place value does not
+touch directly the origin of a symbol for zero, although it assumes that
+such a symbol exists. The importance of such a sign, the fact that it is a
+prerequisite to a place-value system, and the further fact that without it
+the Hindu-Arabic numerals would never have dominated the computation system
+of the western world, make it proper to devote a chapter to its origin and
+history.
+
+It was some centuries after the primitive Br[=a]hm[=i] and
+Kharo[s.][t.]h[=i] numerals had made their appearance in India that the
+zero first appeared there, although such a character was used by the
+Babylonians[185] in the centuries immediately preceding the Christian era.
+The symbol is [Babylonian zero symbol] or [Babylonian zero symbol], and
+apparently it was not used in calculation. Nor does it always occur when
+units of any order are lacking; thus 180 is written [Babylonian numerals
+180] with the meaning three sixties and no units, since 181 immediately
+following is [Babylonian numerals 181], three sixties and one unit.[186]
+The main {52} use of this Babylonian symbol seems to have been in the
+fractions, 60ths, 3600ths, etc., and somewhat similar to the Greek use of
+[Greek: o], for [Greek: ouden], with the meaning _vacant_.
+
+"The earliest undoubted occurrence of a zero in India is an inscription at
+Gwalior, dated Samvat 933 (876 A.D.). Where 50 garlands are mentioned (line
+20), 50 is written [Gwalior numerals 50]. 270 (line 4) is written [Gwalior
+numerals 270]."[187] The Bakh[s.][=a]l[=i] Manuscript[188] probably
+antedates this, using the point or dot as a zero symbol. Bayley mentions a
+grant of Jaika Rashtrakúta of Bharuj, found at Okamandel, of date 738 A.D.,
+which contains a zero, and also a coin with indistinct Gupta date 707 (897
+A.D.), but the reliability of Bayley's work is questioned. As has been
+noted, the appearance of the numerals in inscriptions and on coins would be
+of much later occurrence than the origin and written exposition of the
+system. From the period mentioned the spread was rapid over all of India,
+save the southern part, where the Tamil and Malayalam people retain the old
+system even to the present day.[189]
+
+Aside from its appearance in early inscriptions, there is still another
+indication of the Hindu origin of the symbol in the special treatment of
+the concept zero in the early works on arithmetic. Brahmagupta, who lived
+in Ujjain, the center of Indian astronomy,[190] in the early part {53} of
+the seventh century, gives in his arithmetic[191] a distinct treatment of
+the properties of zero. He does not discuss a symbol, but he shows by his
+treatment that in some way zero had acquired a special significance not
+found in the Greek or other ancient arithmetics. A still more scientific
+treatment is given by Bh[=a]skara,[192] although in one place he permits
+himself an unallowed liberty in dividing by zero. The most recently
+discovered work of ancient Indian mathematical lore, the
+Ganita-S[=a]ra-Sa[.n]graha[193] of Mah[=a]v[=i]r[=a]c[=a]rya (c. 830 A.D.),
+while it does not use the numerals with place value, has a similar
+discussion of the calculation with zero.
+
+What suggested the form for the zero is, of course, purely a matter of
+conjecture. The dot, which the Hindus used to fill up lacunæ in their
+manuscripts, much as we indicate a break in a sentence,[194] would have
+been a more natural symbol; and this is the one which the Hindus first
+used[195] and which most Arabs use to-day. There was also used for this
+purpose a cross, like our X, and this is occasionally found as a zero
+symbol.[196] In the Bakh[s.][=a]l[=i] manuscript above mentioned, the word
+_['s][=u]nya_, with the dot as its symbol, is used to denote the unknown
+quantity, as well as to denote zero. An analogous use of the {54} zero, for
+the unknown quantity in a proportion, appears in a Latin manuscript of some
+lectures by Gottfried Wolack in the University of Erfurt in 1467 and
+1468.[197] The usage was noted even as early as the eighteenth
+century.[198]
+
+The small circle was possibly suggested by the spurred circle which was
+used for ten.[199] It has also been thought that the omicron used by
+Ptolemy in his _Almagest_, to mark accidental blanks in the sexagesimal
+system which he employed, may have influenced the Indian writers.[200] This
+symbol was used quite generally in Europe and Asia, and the Arabic
+astronomer Al-Batt[=a]n[=i][201] (died 929 A.D.) used a similar symbol in
+connection with the alphabetic system of numerals. The occasional use by
+Al-Batt[=a]n[=i] of the Arabic negative, _l[=a]_, to indicate the absence
+of minutes {55} (or seconds), is noted by Nallino.[202] Noteworthy is also
+the use of the [Circle] for unity in the ['S][=a]rad[=a] characters of the
+Kashmirian Atharva-Veda, the writing being at least 400 years old.
+Bh[=a]skara (c. 1150) used a small circle above a number to indicate
+subtraction, and in the Tartar writing a redundant word is removed by
+drawing an oval around it. It would be interesting to know whether our
+score mark [score mark], read "four in the hole," could trace its pedigree
+to the same sources. O'Creat[203] (c. 1130), in a letter to his teacher,
+Adelhard of Bath, uses [Greek: t] for zero, being an abbreviation for the
+word _teca_ which we shall see was one of the names used for zero, although
+it could quite as well be from [Greek: tziphra]. More rarely O'Creat uses
+[circle with bar], applying the name _cyfra_ to both forms. Frater
+Sigsboto[204] (c. 1150) uses the same symbol. Other peculiar forms are
+noted by Heiberg[205] as being in use among the Byzantine Greeks in the
+fifteenth century. It is evident from the text that some of these writers
+did not understand the import of the new system.[206]
+
+Although the dot was used at first in India, as noted above, the small
+circle later replaced it and continues in use to this day. The Arabs,
+however, did not adopt the {56} circle, since it bore some resemblance to
+the letter which expressed the number five in the alphabet system.[207] The
+earliest Arabic zero known is the dot, used in a manuscript of 873
+A.D.[208] Sometimes both the dot and the circle are used in the same work,
+having the same meaning, which is the case in an Arabic MS., an abridged
+arithmetic of Jamshid,[209] 982 A.H. (1575 A.D.). As given in this work the
+numerals are [symbols]. The form for 5 varies, in some works becoming
+[symbol] or [symbol]; [symbol] is found in Egypt and [symbol] appears in
+some fonts of type. To-day the Arabs use the 0 only when, under European
+influence, they adopt the ordinary system. Among the Chinese the first
+definite trace of zero is in the work of Tsin[210] of 1247 A.D. The form is
+the circular one of the Hindus, and undoubtedly was brought to China by
+some traveler.
+
+The name of this all-important symbol also demands some attention,
+especially as we are even yet quite undecided as to what to call it. We
+speak of it to-day as _zero, naught_, and even _cipher_; the telephone
+operator often calls it _O_, and the illiterate or careless person calls it
+_aught_. In view of all this uncertainty we may well inquire what it has
+been called in the past.[211]
+
+{57}
+
+As already stated, the Hindus called it _['s][=u]nya_, "void."[212] This
+passed over into the Arabic as _a[s.]-[s.]ifr_ or _[s.]ifr_.[213] When
+Leonard of Pisa (1202) wrote upon the Hindu numerals he spoke of this
+character as _zephirum_.[214] Maximus Planudes (1330), writing under both
+the Greek and the Arabic influence, called it _tziphra_.[215] In a treatise
+on arithmetic written in the Italian language by Jacob of Florence[216]
+{58} (1307) it is called _zeuero_,[217] while in an arithmetic of Giovanni
+di Danti of Arezzo (1370) the word appears as _çeuero_.[218] Another form
+is _zepiro_,[219] which was also a step from _zephirum_ to zero.[220]
+
+Of course the English _cipher_, French _chiffre_, is derived from the same
+Arabic word, _a[s.]-[s.]ifr_, but in several languages it has come to mean
+the numeral figures in general. A trace of this appears in our word
+_ciphering_, meaning figuring or computing.[221] Johann Huswirt[222] uses
+the word with both meanings; he gives for the tenth character the four
+names _theca, circulus, cifra_, and _figura nihili_. In this statement
+Huswirt probably follows, as did many writers of that period, the
+_Algorismus_ of Johannes de Sacrobosco (c. 1250 A.D.), who was also known
+as John of Halifax or John of Holywood. The commentary of {59} Petrus de
+Dacia[223] (c. 1291 A.D.) on the _Algorismus vulgaris_ of Sacrobosco was
+also widely used. The widespread use of this Englishman's work on
+arithmetic in the universities of that time is attested by the large
+number[224] of MSS. from the thirteenth to the seventeenth century still
+extant, twenty in Munich, twelve in Vienna, thirteen in Erfurt, several in
+England given by Halliwell,[225] ten listed in Coxe's _Catalogue of the
+Oxford College Library_, one in the Plimpton collection,[226] one in the
+Columbia University Library, and, of course, many others.
+
+From _a[s.]-[s.]ifr _has come _zephyr, cipher,_ and finally the abridged
+form _zero_. The earliest printed work in which is found this final form
+appears to be Calandri's arithmetic of 1491,[227] while in manuscript it
+appears at least as early as the middle of the fourteenth century.[228] It
+also appears in a work, _Le Kadran des marchans_, by Jehan {60}
+Certain,[229] written in 1485. This word soon became fairly well known in
+Spain[230] and France.[231] The medieval writers also spoke of it as the
+_sipos_,[232] and occasionally as the _wheel_,[233] _circulus_[234] (in
+German _das Ringlein_[235]), _circular {61} note_,[236] _theca_,[237] long
+supposed to be from its resemblance to the Greek theta, but explained by
+Petrus de Dacia as being derived from the name of the iron[238] used to
+brand thieves and robbers with a circular mark placed on the forehead or on
+the cheek. It was also called _omicron_[239] (the Greek _o_), being
+sometimes written õ or [Greek: ph] to distinguish it from the letter _o_.
+It also went by the name _null_[240] (in the Latin books {62} _nihil_[241]
+or _nulla_,[242] and in the French _rien_[243]), and very commonly by the
+name _cipher_.[244] Wallis[245] gives one of the earliest extended
+discussions of the various forms of the word, giving certain other
+variations worthy of note, as _ziphra_, _zifera_, _siphra_, _ciphra_,
+_tsiphra_, _tziphra,_ and the Greek [Greek: tziphra].[246]
+
+       *       *       *       *       *
+
+
+{63}
+
+CHAPTER V
+
+THE QUESTION OF THE INTRODUCTION OF THE NUMERALS INTO EUROPE BY BOETHIUS
+
+Just as we were quite uncertain as to the origin of the numeral forms, so
+too are we uncertain as to the time and place of their introduction into
+Europe. There are two general theories as to this introduction. The first
+is that they were carried by the Moors to Spain in the eighth or ninth
+century, and thence were transmitted to Christian Europe, a theory which
+will be considered later. The second, advanced by Woepcke,[247] is that
+they were not brought to Spain by the Moors, but that they were already in
+Spain when the Arabs arrived there, having reached the West through the
+Neo-Pythagoreans. There are two facts to support this second theory: (1)
+the forms of these numerals are characteristic, differing materially from
+those which were brought by Leonardo of Pisa from Northern Africa early in
+the thirteenth century (before 1202 A.D.); (2) they are essentially those
+which {64} tradition has so persistently assigned to Boethius (c. 500
+A.D.), and which he would naturally have received, if at all, from these
+same Neo-Pythagoreans or from the sources from which they derived them.
+Furthermore, Woepcke points out that the Arabs on entering Spain (711 A.D.)
+would naturally have followed their custom of adopting for the computation
+of taxes the numerical systems of the countries they conquered,[248] so
+that the numerals brought from Spain to Italy, not having undergone the
+same modifications as those of the Eastern Arab empire, would have
+differed, as they certainly did, from those that came through Bagdad. The
+theory is that the Hindu system, without the zero, early reached Alexandria
+(say 450 A.D.), and that the Neo-Pythagorean love for the mysterious and
+especially for the Oriental led to its use as something bizarre and
+cabalistic; that it was then passed along the Mediterranean, reaching
+Boethius in Athens or in Rome, and to the schools of Spain, being
+discovered in Africa and Spain by the Arabs even before they themselves
+knew the improved system with the place value.
+
+{65}
+
+A recent theory set forth by Bubnov[249] also deserves mention, chiefly
+because of the seriousness of purpose shown by this well-known writer.
+Bubnov holds that the forms first found in Europe are derived from ancient
+symbols used on the abacus, but that the zero is of Hindu origin. This
+theory does not seem tenable, however, in the light of the evidence already
+set forth.
+
+Two questions are presented by Woepcke's theory: (1) What was the nature of
+these Spanish numerals, and how were they made known to Italy? (2) Did
+Boethius know them?
+
+The Spanish forms of the numerals were called the _[h.]ur[=u]f
+al-[.g]ob[=a]r_, the [.g]ob[=a]r or dust numerals, as distinguished from
+the _[h.]ur[=u]f al-jumal_ or alphabetic numerals. Probably the latter,
+under the influence of the Syrians or Jews,[250] were also used by the
+Arabs. The significance of the term [.g]ob[=a]r is doubtless that these
+numerals were written on the dust abacus, this plan being distinct from the
+counter method of representing numbers. It is also worthy of note that
+Al-B[=i]r[=u]n[=i] states that the Hindus often performed numerical
+computations in the sand. The term is found as early as c. 950, in the
+verses of an anonymous writer of Kairw[=a]n, in Tunis, in which the author
+speaks of one of his works on [.g]ob[=a]r calculation;[251] and, much
+later, the Arab writer Ab[=u] Bekr Mo[h.]ammed ibn `Abdall[=a]h, surnamed
+al-[H.]a[s.][s.][=a]r {66} (the arithmetician), wrote a work of which the
+second chapter was "On the dust figures."[252]
+
+The [.g]ob[=a]r numerals themselves were first made known to modern
+scholars by Silvestre de Sacy, who discovered them in an Arabic manuscript
+from the library of the ancient abbey of St.-Germain-des-Prés.[253] The
+system has nine characters, but no zero. A dot above a character indicates
+tens, two dots hundreds, and so on, [5 with dot] meaning 50, and [5 with 3
+dots] meaning 5000. It has been suggested that possibly these dots,
+sprinkled like dust above the numerals, gave rise to the word
+_[.g]ob[=a]r_,[254] but this is not at all probable. This system of dots is
+found in Persia at a much later date with numerals quite like the modern
+Arabic;[255] but that it was used at all is significant, for it is hardly
+likely that the western system would go back to Persia, when the perfected
+Hindu one was near at hand.
+
+At first sight there would seem to be some reason for believing that this
+feature of the [.g]ob[=a]r system was of {67} Arabic origin, and that the
+present zero of these people,[256] the dot, was derived from it. It was
+entirely natural that the Semitic people generally should have adopted such
+a scheme, since their diacritical marks would suggest it, not to speak of
+the possible influence of the Greek accents in the Hellenic number system.
+When we consider, however, that the dot is found for zero in the
+Bakh[s.][=a]l[=i] manuscript,[257] and that it was used in subscript form
+in the _Kit[=a]b al-Fihrist_[258] in the tenth century, and as late as the
+sixteenth century,[259] although in this case probably under Arabic
+influence, we are forced to believe that this form may also have been of
+Hindu origin.
+
+The fact seems to be that, as already stated,[260] the Arabs did not
+immediately adopt the Hindu zero, because it resembled their 5; they used
+the superscript dot as serving their purposes fairly well; they may,
+indeed, have carried this to the west and have added it to the [.g]ob[=a]r
+forms already there, just as they transmitted it to the Persians.
+Furthermore, the Arab and Hebrew scholars of Northern Africa in the tenth
+century knew these numerals as Indian forms, for a commentary on the
+_S[=e]fer Ye[s.][=i]r[=a]h_ by Ab[=u] Sahl ibn Tamim (probably composed at
+Kairw[=a]n, c. 950) speaks of "the Indian arithmetic known under the name
+of _[.g]ob[=a]r_ or dust calculation."[261] All this suggests that the
+Arabs may very {68} likely have known the [.g]ob[=a]r forms before the
+numerals reached them again in 773.[262] The term "[.g]ob[=a]r numerals"
+was also used without any reference to the peculiar use of dots.[263] In
+this connection it is worthy of mention that the Algerians employed two
+different forms of numerals in manuscripts even of the fourteenth
+century,[264] and that the Moroccans of to-day employ the European forms
+instead of the present Arabic.
+
+The Indian use of subscript dots to indicate the tens, hundreds, thousands,
+etc., is established by a passage in the _Kit[=a]b al-Fihrist_[265] (987
+A.D.) in which the writer discusses the written language of the people of
+India. Notwithstanding the importance of this reference for the early
+history of the numerals, it has not been mentioned by previous writers on
+this subject. The numeral forms given are those which have usually been
+called Indian,[266] in opposition to [.g]ob[=a]r. In this document the dots
+are placed below the characters, instead of being superposed as described
+above. The significance was the same.
+
+In form these [.g]ob[=a]r numerals resemble our own much more closely than
+the Arab numerals do. They varied more or less, but were substantially as
+follows:
+
+{69}
+
+  1[267][Illustration]
+  2[268][Illustration]
+  3[269][Illustration]
+  4[270][Illustration]
+  5[271][Illustration]
+  6[271][Illustration]
+
+The question of the possible influence of the Egyptian demotic and hieratic
+ordinal forms has been so often suggested that it seems well to introduce
+them at this point, for comparison with the [.g]ob[=a]r forms. They would
+as appropriately be used in connection with the Hindu forms, and the
+evidence of a relation of the first three with all these systems is
+apparent. The only further resemblance is in the Demotic 4 and in the 9, so
+that the statement that the Hindu forms in general came from {70} this
+source has no foundation. The first four Egyptian cardinal numerals[272]
+resemble more the modern Arabic.
+
+[Illustration: DEMOTIC AND HIERATIC ORDINALS]
+
+This theory of the very early introduction of the numerals into Europe
+fails in several points. In the first place the early Western forms are not
+known; in the second place some early Eastern forms are like the
+[.g]ob[=a]r, as is seen in the third line on p. 69, where the forms are
+from a manuscript written at Shiraz about 970 A.D., and in which some
+western Arabic forms, e.g. [symbol] for 2, are also used. Probably most
+significant of all is the fact that the [.g]ob[=a]r numerals as given by
+Sacy are all, with the exception of the symbol for eight, either single
+Arabic letters or combinations of letters. So much for the Woepcke theory
+and the meaning of the [.g]ob[=a]r numerals. We now have to consider the
+question as to whether Boethius knew these [.g]ob[=a]r forms, or forms akin
+to them.
+
+This large question[273] suggests several minor ones: (1) Who was Boethius?
+(2) Could he have known these numerals? (3) Is there any positive or strong
+circumstantial evidence that he did know them? (4) What are the
+probabilities in the case?
+
+{71}
+
+First, who was Boethius,--Divus[274] Boethius as he was called in the
+Middle Ages? Anicius Manlius Severinus Boethius[275] was born at Rome c.
+475. He was a member of the distinguished family of the Anicii,[276] which
+had for some time before his birth been Christian. Early left an orphan,
+the tradition is that he was taken to Athens at about the age of ten, and
+that he remained there eighteen years.[277] He married Rusticiana, daughter
+of the senator Symmachus, and this union of two such powerful families
+allowed him to move in the highest circles.[278] Standing strictly for the
+right, and against all iniquity at court, he became the object of hatred on
+the part of all the unscrupulous element near the throne, and his bold
+defense of the ex-consul Albinus, unjustly accused of treason, led to his
+imprisonment at Pavia[279] and his execution in 524.[280] Not many
+generations after his death, the period being one in which historical
+criticism was at its lowest ebb, the church found it profitable to look
+upon his execution as a martyrdom.[281] He was {72} accordingly looked upon
+as a saint,[282] his bones were enshrined,[283] and as a natural
+consequence his books were among the classics in the church schools for a
+thousand years.[284] It is pathetic, however, to think of the medieval
+student trying to extract mental nourishment from a work so abstract, so
+meaningless, so unnecessarily complicated, as the arithmetic of Boethius.
+
+He was looked upon by his contemporaries and immediate successors as a
+master, for Cassiodorus[285] (c. 490-c. 585 A.D.) says to him: "Through
+your translations the music of Pythagoras and the astronomy of Ptolemy are
+read by those of Italy, and the arithmetic of Nicomachus and the geometry
+of Euclid are known to those of the West."[286] Founder of the medieval
+scholasticism, {73} distinguishing the trivium and quadrivium,[287] writing
+the only classics of his time, Gibbon well called him "the last of the
+Romans whom Cato or Tully could have acknowledged for their
+countryman."[288]
+
+The second question relating to Boethius is this: Could he possibly have
+known the Hindu numerals? In view of the relations that will be shown to
+have existed between the East and the West, there can only be an
+affirmative answer to this question. The numerals had existed, without the
+zero, for several centuries; they had been well known in India; there had
+been a continued interchange of thought between the East and West; and
+warriors, ambassadors, scholars, and the restless trader, all had gone back
+and forth, by land or more frequently by sea, between the Mediterranean
+lands and the centers of Indian commerce and culture. Boethius could very
+well have learned one or more forms of Hindu numerals from some traveler or
+merchant.
+
+To justify this statement it is necessary to speak more fully of these
+relations between the Far East and Europe. It is true that we have no
+records of the interchange of learning, in any large way, between eastern
+Asia and central Europe in the century preceding the time of Boethius. But
+it is one of the mistakes of scholars to believe that they are the sole
+transmitters of knowledge. {74} As a matter of fact there is abundant
+reason for believing that Hindu numerals would naturally have been known to
+the Arabs, and even along every trade route to the remote west, long before
+the zero entered to make their place-value possible, and that the
+characters, the methods of calculating, the improvements that took place
+from time to time, the zero when it appeared, and the customs as to solving
+business problems, would all have been made known from generation to
+generation along these same trade routes from the Orient to the Occident.
+It must always be kept in mind that it was to the tradesman and the
+wandering scholar that the spread of such learning was due, rather than to
+the school man. Indeed, Avicenna[289] (980-1037 A.D.) in a short biography
+of himself relates that when his people were living at Bokh[=a]ra his
+father sent him to the house of a grocer to learn the Hindu art of
+reckoning, in which this grocer (oil dealer, possibly) was expert. Leonardo
+of Pisa, too, had a similar training.
+
+The whole question of this spread of mercantile knowledge along the trade
+routes is so connected with the [.g]ob[=a]r numerals, the Boethius
+question, Gerbert, Leonardo of Pisa, and other names and events, that a
+digression for its consideration now becomes necessary.[290]
+
+{75}
+
+Even in very remote times, before the Hindu numerals were sculptured in the
+cave of N[=a]n[=a] Gh[=a]t, there were trade relations between Arabia and
+India. Indeed, long before the Aryans went to India the great Turanian race
+had spread its civilization from the Mediterranean to the Indus.[291] At a
+much later period the Arabs were the intermediaries between Egypt and Syria
+on the west, and the farther Orient.[292] In the sixth century B.C.,
+Hecatæus,[293] the father of geography, was acquainted not only with the
+Mediterranean lands but with the countries as far as the Indus,[294] and in
+Biblical times there were regular triennial voyages to India. Indeed, the
+story of Joseph bears witness to the caravan trade from India, across
+Arabia, and on to the banks of the Nile. About the same time as Hecatæus,
+Scylax, a Persian admiral under Darius, from Caryanda on the coast of Asia
+Minor, traveled to {76} northwest India and wrote upon his ventures.[295]
+He induced the nations along the Indus to acknowledge the Persian
+supremacy, and such number systems as there were in these lands would
+naturally have been known to a man of his attainments.
+
+A century after Scylax, Herodotus showed considerable knowledge of India,
+speaking of its cotton and its gold,[296] telling how Sesostris[297] fitted
+out ships to sail to that country, and mentioning the routes to the east.
+These routes were generally by the Red Sea, and had been followed by the
+Phoenicians and the Sabæans, and later were taken by the Greeks and
+Romans.[298]
+
+In the fourth century B.C. the West and East came into very close
+relations. As early as 330, Pytheas of Massilia (Marseilles) had explored
+as far north as the northern end of the British Isles and the coasts of the
+German Sea, while Macedon, in close touch with southern France, was also
+sending her armies under Alexander[299] through Afghanistan as far east as
+the Punjab.[300] Pliny tells us that Alexander the Great employed surveyors
+to measure {77} the roads of India; and one of the great highways is
+described by Megasthenes, who in 295 B.C., as the ambassador of Seleucus,
+resided at P[=a]tal[=i]pu[t.]ra, the present Patna.[301]
+
+The Hindus also learned the art of coining from the Greeks, or possibly
+from the Chinese, and the stores of Greco-Hindu coins still found in
+northern India are a constant source of historical information.[302] The
+R[=a]m[=a]yana speaks of merchants traveling in great caravans and
+embarking by sea for foreign lands.[303] Ceylon traded with Malacca and
+Siam, and Java was colonized by Hindu traders, so that mercantile knowledge
+was being spread about the Indies during all the formative period of the
+numerals.
+
+Moreover the results of the early Greek invasion were embodied by
+Dicæarchus of Messana (about 320 B.C.) in a map that long remained a
+standard. Furthermore, Alexander did not allow his influence on the East to
+cease. He divided India into three satrapies,[304] placing Greek governors
+over two of them and leaving a Hindu ruler in charge of the third, and in
+Bactriana, a part of Ariana or ancient Persia, he left governors; and in
+these the western civilization was long in evidence. Some of the Greek and
+Roman metrical and astronomical terms {78} found their way, doubtless at
+this time, into the Sanskrit language.[305] Even as late as from the second
+to the fifth centuries A.D., Indian coins showed the Hellenic influence.
+The Hindu astronomical terminology reveals the same relationship to western
+thought, for Var[=a]ha-Mihira (6th century A.D.), a contemporary of
+[=A]ryabha[t.]a, entitled a work of his the _B[r.]hat-Sa[m.]hit[=a]_, a
+literal translation of [Greek: megalê suntaxis] of Ptolemy;[306] and in
+various ways is this interchange of ideas apparent.[307] It could not have
+been at all unusual for the ancient Greeks to go to India, for Strabo lays
+down the route, saying that all who make the journey start from Ephesus and
+traverse Phrygia and Cappadocia before taking the direct road.[308] The
+products of the East were always finding their way to the West, the Greeks
+getting their ginger[309] from Malabar, as the Phoenicians had long before
+brought gold from Malacca.
+
+Greece must also have had early relations with China, for there is a
+notable similarity between the Greek and Chinese life, as is shown in their
+houses, their domestic customs, their marriage ceremonies, the public
+story-tellers, the puppet shows which Herodotus says were introduced from
+Egypt, the street jugglers, the games of dice,[310] the game of
+finger-guessing,[311] the water clock, the {79} music system, the use of
+the myriad,[312] the calendars, and in many other ways.[313] In passing
+through the suburbs of Peking to-day, on the way to the Great Bell temple,
+one is constantly reminded of the semi-Greek architecture of Pompeii, so
+closely does modern China touch the old classical civilization of the
+Mediterranean. The Chinese historians tell us that about 200 B.C. their
+arms were successful in the far west, and that in 180 B.C. an ambassador
+went to Bactria, then a Greek city, and reported that Chinese products were
+on sale in the markets there.[314] There is also a noteworthy resemblance
+between certain Greek and Chinese words,[315] showing that in remote times
+there must have been more or less interchange of thought.
+
+The Romans also exchanged products with the East. Horace says, "A busy
+trader, you hasten to the farthest Indies, flying from poverty over sea,
+over crags, over fires."[316] The products of the Orient, spices and jewels
+from India, frankincense from Persia, and silks from China, being more in
+demand than the exports from the Mediterranean lands, the balance of trade
+was against the West, and thus Roman coin found its way eastward. In 1898,
+for example, a number of Roman coins dating from 114 B.C. to Hadrian's time
+were found at Pakl[=i], a part of the Haz[=a]ra district, sixteen miles
+north of Abbott[=a]b[=a]d,[317] and numerous similar discoveries have been
+made from time to time.
+
+{80}
+
+Augustus speaks of envoys received by him from India, a thing never before
+known,[318] and it is not improbable that he also received an embassy from
+China.[319] Suetonius (first century A.D.) speaks in his history of these
+relations,[320] as do several of his contemporaries,[321] and Vergil[322]
+tells of Augustus doing battle in Persia. In Pliny's time the trade of the
+Roman Empire with Asia amounted to a million and a quarter dollars a year,
+a sum far greater relatively then than now,[323] while by the time of
+Constantine Europe was in direct communication with the Far East.[324]
+
+In view of these relations it is not beyond the range of possibility that
+proof may sometime come to light to show that the Greeks and Romans knew
+something of the {81} number system of India, as several writers have
+maintained.[325]
+
+Returning to the East, there are many evidences of the spread of knowledge
+in and about India itself. In the third century B.C. Buddhism began to be a
+connecting medium of thought. It had already permeated the Himalaya
+territory, had reached eastern Turkestan, and had probably gone thence to
+China. Some centuries later (in 62 A.D.) the Chinese emperor sent an
+ambassador to India, and in 67 A.D. a Buddhist monk was invited to
+China.[326] Then, too, in India itself A['s]oka, whose name has already
+been mentioned in this work, extended the boundaries of his domains even
+into Afghanistan, so that it was entirely possible for the numerals of the
+Punjab to have worked their way north even at that early date.[327]
+
+Furthermore, the influence of Persia must not be forgotten in considering
+this transmission of knowledge. In the fifth century the Persian medical
+school at Jondi-Sapur admitted both the Hindu and the Greek doctrines, and
+Firdus[=i] tells us that during the brilliant reign of {82} Khosr[=u]
+I,[328] the golden age of Pahlav[=i] literature, the Hindu game of chess
+was introduced into Persia, at a time when wars with the Greeks were
+bringing prestige to the Sassanid dynasty.
+
+Again, not far from the time of Boethius, in the sixth century, the
+Egyptian monk Cosmas, in his earlier years as a trader, made journeys to
+Abyssinia and even to India and Ceylon, receiving the name _Indicopleustes_
+(the Indian traveler). His map (547 A.D.) shows some knowledge of the earth
+from the Atlantic to India. Such a man would, with hardly a doubt, have
+observed every numeral system used by the people with whom he
+sojourned,[329] and whether or not he recorded his studies in permanent
+form he would have transmitted such scraps of knowledge by word of mouth.
+
+As to the Arabs, it is a mistake to feel that their activities began with
+Mohammed. Commerce had always been held in honor by them, and the
+Qoreish[330] had annually for many generations sent caravans bearing the
+spices and textiles of Yemen to the shores of the Mediterranean. In the
+fifth century they traded by sea with India and even with China, and
+[H.]ira was an emporium for the wares of the East,[331] so that any numeral
+system of any part of the trading world could hardly have remained
+isolated.
+
+Long before the warlike activity of the Arabs, Alexandria had become the
+great market-place of the world. From this center caravans traversed Arabia
+to Hadramaut, where they met ships from India. Others went north to
+Damascus, while still others made their way {83} along the southern shores
+of the Mediterranean. Ships sailed from the isthmus of Suez to all the
+commercial ports of Southern Europe and up into the Black Sea. Hindus were
+found among the merchants[332] who frequented the bazaars of Alexandria,
+and Brahmins were reported even in Byzantium.
+
+Such is a very brief résumé of the evidence showing that the numerals of
+the Punjab and of other parts of India as well, and indeed those of China
+and farther Persia, of Ceylon and the Malay peninsula, might well have been
+known to the merchants of Alexandria, and even to those of any other
+seaport of the Mediterranean, in the time of Boethius. The Br[=a]hm[=i]
+numerals would not have attracted the attention of scholars, for they had
+no zero so far as we know, and therefore they were no better and no worse
+than those of dozens of other systems. If Boethius was attracted to them it
+was probably exactly as any one is naturally attracted to the bizarre or
+the mystic, and he would have mentioned them in his works only
+incidentally, as indeed they are mentioned in the manuscripts in which they
+occur.
+
+In answer therefore to the second question, Could Boethius have known the
+Hindu numerals? the reply must be, without the slightest doubt, that he
+could easily have known them, and that it would have been strange if a man
+of his inquiring mind did not pick up many curious bits of information of
+this kind even though he never thought of making use of them.
+
+Let us now consider the third question, Is there any positive or strong
+circumstantial evidence that Boethius did know these numerals? The question
+is not new, {84} nor is it much nearer being answered than it was over two
+centuries ago when Wallis (1693) expressed his doubts about it[333] soon
+after Vossius (1658) had called attention to the matter.[334] Stated
+briefly, there are three works on mathematics attributed to Boethius:[335]
+(1) the arithmetic, (2) a work on music, and (3) the geometry.[336]
+
+The genuineness of the arithmetic and the treatise on music is generally
+recognized, but the geometry, which contains the Hindu numerals with the
+zero, is under suspicion.[337] There are plenty of supporters of the idea
+that Boethius knew the numerals and included them in this book,[338] and on
+the other hand there are as many who {85} feel that the geometry, or at
+least the part mentioning the numerals, is spurious.[339] The argument of
+those who deny the authenticity of the particular passage in question may
+briefly be stated thus:
+
+1. The falsification of texts has always been the subject of complaint. It
+was so with the Romans,[340] it was common in the Middle Ages,[341] and it
+is much more prevalent {86} to-day than we commonly think. We have but to
+see how every hymn-book compiler feels himself authorized to change at will
+the classics of our language, and how unknown editors have mutilated
+Shakespeare, to see how much more easy it was for medieval scribes to
+insert or eliminate paragraphs without any protest from critics.[342]
+
+2. If Boethius had known these numerals he would have mentioned them in his
+arithmetic, but he does not do so.[343]
+
+3. If he had known them, and had mentioned them in any of his works, his
+contemporaries, disciples, and successors would have known and mentioned
+them. But neither Capella (c. 475)[344] nor any of the numerous medieval
+writers who knew the works of Boethius makes any reference to the
+system.[345]
+
+{87}
+
+4. The passage in question has all the appearance of an interpolation by
+some scribe. Boethius is speaking of angles, in his work on geometry, when
+the text suddenly changes to a discussion of classes of numbers.[346] This
+is followed by a chapter in explanation of the abacus,[347] in which are
+described those numeral forms which are called _apices_ or
+_caracteres_.[348] The forms[349] of these characters vary in different
+manuscripts, but in general are about as shown on page 88. They are
+commonly written with the 9 at the left, decreasing to the unit at the
+right, numerous writers stating that this was because they were derived
+from Semitic sources in which the direction of writing is the opposite of
+our own. This practice continued until the sixteenth century.[350] The
+writer then leaves the subject entirely, using the Roman numerals for the
+rest of his discussion, a proceeding so foreign to the method of Boethius
+as to be inexplicable on the hypothesis of authenticity. Why should such a
+scholarly writer have given them with no mention of their origin or use?
+Either he would have mentioned some historical interest attaching to them,
+or he would have used them in some discussion; he certainly would not have
+left the passage as it is.
+
+{88}
+
+FORMS OF THE NUMERALS, LARGELY FROM WORKS ON THE ABACUS[351]
+
+  a[352] [Illustration]
+  b[353] [Illustration]
+  c[354] [Illustration]
+  d[355] [Illustration]
+  e[356] [Illustration]
+  f[357] [Illustration]
+  g[358] [Illustration]
+  h[359] [Illustration]
+  i[360] [Illustration]
+
+{89}
+
+Sir E. Clive Bayley has added[361] a further reason for believing them
+spurious, namely that the 4 is not of the N[=a]n[=a] Gh[=a]t type, but of
+the Kabul form which the Arabs did not receive until 776;[362] so that it
+is not likely, even if the characters were known in Europe in the time of
+Boethius, that this particular form was recognized. It is worthy of
+mention, also, that in the six abacus forms from the chief manuscripts as
+given by Friedlein,[363] each contains some form of zero, which symbol
+probably originated in India about this time or later. It could hardly have
+reached Europe so soon.
+
+As to the fourth question, Did Boethius probably know the numerals? It
+seems to be a fair conclusion, according to our present evidence, that (1)
+Boethius might very easily have known these numerals without the zero, but,
+(2) there is no reliable evidence that he did know them. And just as
+Boethius might have come in contact with them, so any other inquiring mind
+might have done so either in his time or at any time before they definitely
+appeared in the tenth century. These centuries, five in number, represented
+the darkest of the Dark Ages, and even if these numerals were occasionally
+met and studied, no trace of them would be likely to show itself in the
+{90} literature of the period, unless by chance it should get into the
+writings of some man like Alcuin. As a matter of fact, it was not until the
+ninth or tenth century that there is any tangible evidence of their
+presence in Christendom. They were probably known to merchants here and
+there, but in their incomplete state they were not of sufficient importance
+to attract any considerable attention.
+
+As a result of this brief survey of the evidence several conclusions seem
+reasonable: (1) commerce, and travel for travel's sake, never died out
+between the East and the West; (2) merchants had every opportunity of
+knowing, and would have been unreasonably stupid if they had not known, the
+elementary number systems of the peoples with whom they were trading, but
+they would not have put this knowledge in permanent written form; (3)
+wandering scholars would have known many and strange things about the
+peoples they met, but they too were not, as a class, writers; (4) there is
+every reason a priori for believing that the [.g]ob[=a]r numerals would
+have been known to merchants, and probably to some of the wandering
+scholars, long before the Arabs conquered northern Africa; (5) the wonder
+is not that the Hindu-Arabic numerals were known about 1000 A.D., and that
+they were the subject of an elaborate work in 1202 by Fibonacci, but rather
+that more extended manuscript evidence of their appearance before that time
+has not been found. That they were more or less known early in the Middle
+Ages, certainly to many merchants of Christian Europe, and probably to
+several scholars, but without the zero, is hardly to be doubted. The lack
+of documentary evidence is not at all strange, in view of all of the
+circumstances.
+
+       *       *       *       *       *
+
+
+{91}
+
+CHAPTER VI
+
+THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS
+
+If the numerals had their origin in India, as seems most probable, when did
+the Arabs come to know of them? It is customary to say that it was due to
+the influence of Mohammedanism that learning spread through Persia and
+Arabia; and so it was, in part. But learning was already respected in these
+countries long before Mohammed appeared, and commerce flourished all
+through this region. In Persia, for example, the reign of Khosr[=u]
+Nu['s][=i]rw[=a]n,[364] the great contemporary of Justinian the law-maker,
+was characterized not only by an improvement in social and economic
+conditions, but by the cultivation of letters. Khosr[=u] fostered learning,
+inviting to his court scholars from Greece, and encouraging the
+introduction of culture from the West as well as from the East. At this
+time Aristotle and Plato were translated, and portions of the
+_Hito-pad[=e]['s]a_, or Fables of Pilpay, were rendered from the Sanskrit
+into Persian. All this means that some three centuries before the great
+intellectual ascendancy of Bagdad a similar fostering of learning was
+taking place in Persia, and under pre-Mohammedan influences.
+
+{92}
+
+The first definite trace that we have of the introduction of the Hindu
+system into Arabia dates from 773 A.D.,[365] when an Indian astronomer
+visited the court of the caliph, bringing with him astronomical tables
+which at the caliph's command were translated into Arabic by
+Al-Faz[=a]r[=i].[366] Al-Khow[=a]razm[=i] and [H.]abash (A[h.]med ibn
+`Abdall[=a]h, died c. 870) based their well-known tables upon the work of
+Al-F[=a]zar[=i]. It may be asserted as highly probable that the numerals
+came at the same time as the tables. They were certainly known a few
+decades later, and before 825 A.D., about which time the original of the
+_Algoritmi de numero Indorum_ was written, as that work makes no pretense
+of being the first work to treat of the Hindu numerals.
+
+The three writers mentioned cover the period from the end of the eighth to
+the end of the ninth century. While the historians Al-Ma['s]`[=u]d[=i] and
+Al-B[=i]r[=u]n[=i] follow quite closely upon the men mentioned, it is well
+to note again the Arab writers on Hindu arithmetic, contemporary with
+Al-Khow[=a]razm[=i], who were mentioned in chapter I, viz. Al-Kind[=i],
+Sened ibn `Al[=i], and Al-[S.][=u]f[=i].
+
+For over five hundred years Arabic writers and others continued to apply to
+works on arithmetic the name "Indian." In the tenth century such writers
+are `Abdall[=a]h ibn al-[H.]asan, Ab[=u] 'l-Q[=a]sim[367] (died 987 A.D.)
+of Antioch, and Mo[h.]ammed ibn `Abdall[=a]h, Ab[=u] Na[s.]r[368] (c. 982),
+of Kalw[=a]d[=a] near Bagdad. Others of the same period or {93} earlier
+(since they are mentioned in the _Fihrist_,[369] 987 A.D.), who explicitly
+use the word "Hindu" or "Indian," are Sin[=a]n ibn al-Fat[h.][370] of
+[H.]arr[=a]n, and Ahmed ibn `Omar, al-Kar[=a]b[=i]s[=i].[371] In the
+eleventh century come Al-B[=i]r[=u]n[=i][372] (973-1048) and `Ali ibn
+A[h.]med, Ab[=u] 'l-[H.]asan, Al-Nasaw[=i][373] (c. 1030). The following
+century brings similar works by Ish[=a]q ibn Y[=u]suf al-[S.]ardaf[=i][374]
+and Sam[=u]'[=i]l ibn Ya[h.]y[=a] ibn `Abb[=a]s al-Ma[.g]reb[=i]
+al-Andalus[=i][375] (c. 1174), and in the thirteenth century are
+`Abdallat[=i]f ibn Y[=u]suf ibn Mo[h.]ammed, Muwaffaq al-D[=i]n Ab[=u]
+Mo[h.]ammed al-Ba[.g]d[=a]d[=i][376] (c. 1231), and Ibn al-Bann[=a].[377]
+
+The Greek monk Maximus Planudes, writing in the first half of the
+fourteenth century, followed the Arabic usage in calling his work _Indian
+Arithmetic_.[378] There were numerous other Arabic writers upon arithmetic,
+as that subject occupied one of the high places among the sciences, but
+most of them did not feel it necessary to refer to the origin of the
+symbols, the knowledge of which might well have been taken for granted.
+
+{94}
+
+One document, cited by Woepcke,[379] is of special interest since it shows
+at an early period, 970 A.D., the use of the ordinary Arabic forms
+alongside the [.g]ob[=a]r. The title of the work is _Interesting and
+Beautiful Problems on Numbers_ copied by A[h.]med ibn Mo[h.]ammed ibn
+`Abdaljal[=i]l, Ab[=u] Sa`[=i]d, al-Sijz[=i],[380] (951-1024) from a work
+by a priest and physician, Na[z.][=i]f ibn Yumn,[381] al-Qass (died c.
+990). Suter does not mention this work of Na[z.][=i]f.
+
+The second reason for not ascribing too much credit to the purely Arab
+influence is that the Arab by himself never showed any intellectual
+strength. What took place after Mo[h.]ammed had lighted the fire in the
+hearts of his people was just what always takes place when different types
+of strong races blend,--a great renaissance in divers lines. It was seen in
+the blending of such types at Miletus in the time of Thales, at Rome in the
+days of the early invaders, at Alexandria when the Greek set firm foot on
+Egyptian soil, and we see it now when all the nations mingle their vitality
+in the New World. So when the Arab culture joined with the Persian, a new
+civilization rose and flourished.[382] The Arab influence came not from its
+purity, but from its intermingling with an influence more cultured if less
+virile.
+
+As a result of this interactivity among peoples of diverse interests and
+powers, Mohammedanism was to the world from the eighth to the thirteenth
+century what Rome and Athens and the Italo-Hellenic influence generally had
+{95} been to the ancient civilization. "If they did not possess the spirit
+of invention which distinguished the Greeks and the Hindus, if they did not
+show the perseverance in their observations that characterized the Chinese
+astronomers, they at least possessed the virility of a new and victorious
+people, with a desire to understand what others had accomplished, and a
+taste which led them with equal ardor to the study of algebra and of
+poetry, of philosophy and of language."[383]
+
+It was in 622 A.D. that Mo[h.]ammed fled from Mecca, and within a century
+from that time the crescent had replaced the cross in Christian Asia, in
+Northern Africa, and in a goodly portion of Spain. The Arab empire was an
+ellipse of learning with its foci at Bagdad and Cordova, and its rulers not
+infrequently took pride in demanding intellectual rather than commercial
+treasure as the result of conquest.[384]
+
+It was under these influences, either pre-Mohammedan or later, that the
+Hindu numerals found their way to the North. If they were known before
+Mo[h.]ammed's time, the proof of this fact is now lost. This much, however,
+is known, that in the eighth century they were taken to Bagdad. It was
+early in that century that the Mohammedans obtained their first foothold in
+northern India, thus foreshadowing an epoch of supremacy that endured with
+varied fortunes until after the golden age of Akbar the Great (1542-1605)
+and Shah Jehan. They also conquered Khorassan and Afghanistan, so that the
+learning and the commercial customs of India at once found easy {96} access
+to the newly-established schools and the bazaars of Mesopotamia and western
+Asia. The particular paths of conquest and of commerce were either by way
+of the Khyber Pass and through Kabul, Herat and Khorassan, or by sea
+through the strait of Ormuz to Basra (Busra) at the head of the Persian
+Gulf, and thence to Bagdad. As a matter of fact, one form of Arabic
+numerals, the one now in use by the Arabs, is attributed to the influence
+of Kabul, while the other, which eventually became our numerals, may very
+likely have reached Arabia by the other route. It is in Bagdad,[385] D[=a]r
+al-Sal[=a]m--"the Abode of Peace," that our special interest in the
+introduction of the numerals centers. Built upon the ruins of an ancient
+town by Al-Man[s.][=u]r[386] in the second half of the eighth century, it
+lies in one of those regions where the converging routes of trade give rise
+to large cities.[387] Quite as well of Bagdad as of Athens might Cardinal
+Newman have said:[388]
+
+"What it lost in conveniences of approach, it gained in its neighborhood to
+the traditions of the mysterious East, and in the loveliness of the region
+in which it lay. Hither, then, as to a sort of ideal land, where all
+archetypes of the great and the fair were found in substantial being, and
+all departments of truth explored, and all diversities of intellectual
+power exhibited, where taste and philosophy were majestically enthroned as
+in a royal court, where there was no sovereignty but that of mind, and no
+nobility but that of genius, where professors were {97} rulers, and princes
+did homage, thither flocked continually from the very corners of the _orbis
+terrarum_ the many-tongued generation, just rising, or just risen into
+manhood, in order to gain wisdom." For here it was that Al-Man[s.][=u]r and
+Al-M[=a]m[=u]n and H[=a]r[=u]n al-Rash[=i]d (Aaron the Just) made for a
+time the world's center of intellectual activity in general and in the
+domain of mathematics in particular.[389] It was just after the _Sindhind_
+was brought to Bagdad that Mo[h.]ammed ibn M[=u]s[=a] al-Khow[=a]razm[=i],
+whose name has already been mentioned,[390] was called to that city. He was
+the most celebrated mathematician of his time, either in the East or West,
+writing treatises on arithmetic, the sundial, the astrolabe, chronology,
+geometry, and algebra, and giving through the Latin transliteration of his
+name, _algoritmi_, the name of algorism to the early arithmetics using the
+new Hindu numerals.[391] Appreciating at once the value of the position
+system so recently brought from India, he wrote an arithmetic based upon
+these numerals, and this was translated into Latin in the time of Adelhard
+of Bath (c. 1180), although possibly by his contemporary countryman Robert
+Cestrensis.[392] This translation was found in Cambridge and was published
+by Boncompagni in 1857.[393]
+
+Contemporary with Al-Khow[=a]razm[=i], and working also under
+Al-M[=a]m[=u]n, was a Jewish astronomer, Ab[=u] 'l-[T.]eiyib, {98} Sened
+ibn `Al[=i], who is said to have adopted the Mohammedan religion at the
+caliph's request. He also wrote a work on Hindu arithmetic,[394] so that
+the subject must have been attracting considerable attention at that time.
+Indeed, the struggle to have the Hindu numerals replace the Arabic did not
+cease for a long time thereafter. `Al[=i] ibn A[h.]med al-Nasaw[=i], in his
+arithmetic of c. 1025, tells us that the symbolism of number was still
+unsettled in his day, although most people preferred the strictly Arabic
+forms.[395]
+
+We thus have the numerals in Arabia, in two forms: one the form now used
+there, and the other the one used by Al-Khow[=a]razm[=i]. The question then
+remains, how did this second form find its way into Europe? and this
+question will be considered in the next chapter.
+
+       *       *       *       *       *
+
+
+{99}
+
+CHAPTER VII
+
+THE DEFINITE INTRODUCTION OF THE NUMERALS INTO EUROPE
+
+It being doubtful whether Boethius ever knew the Hindu numeral forms,
+certainly without the zero in any case, it becomes necessary now to
+consider the question of their definite introduction into Europe. From what
+has been said of the trade relations between the East and the West, and of
+the probability that it was the trader rather than the scholar who carried
+these numerals from their original habitat to various commercial centers,
+it is evident that we shall never know when they first made their
+inconspicuous entrance into Europe. Curious customs from the East and from
+the tropics,--concerning games, social peculiarities, oddities of dress,
+and the like,--are continually being related by sailors and traders in
+their resorts in New York, London, Hamburg, and Rotterdam to-day, customs
+that no scholar has yet described in print and that may not become known
+for many years, if ever. And if this be so now, how much more would it have
+been true a thousand years before the invention of printing, when learning
+was at its lowest ebb. It was at this period of low esteem of culture that
+the Hindu numerals undoubtedly made their first appearance in Europe.
+
+There were many opportunities for such knowledge to reach Spain and Italy.
+In the first place the Moors went into Spain as helpers of a claimant of
+the throne, and {100} remained as conquerors. The power of the Goths, who
+had held Spain for three centuries, was shattered at the battle of Jerez de
+la Frontera in 711, and almost immediately the Moors became masters of
+Spain and so remained for five hundred years, and masters of Granada for a
+much longer period. Until 850 the Christians were absolutely free as to
+religion and as to holding political office, so that priests and monks were
+not infrequently skilled both in Latin and Arabic, acting as official
+translators, and naturally reporting directly or indirectly to Rome. There
+was indeed at this time a complaint that Christian youths cultivated too
+assiduously a love for the literature of the Saracen, and married too
+frequently the daughters of the infidel.[396] It is true that this happy
+state of affairs was not permanent, but while it lasted the learning and
+the customs of the East must have become more or less the property of
+Christian Spain. At this time the [.g]ob[=a]r numerals were probably in
+that country, and these may well have made their way into Europe from the
+schools of Cordova, Granada, and Toledo.
+
+Furthermore, there was abundant opportunity for the numerals of the East to
+reach Europe through the journeys of travelers and ambassadors. It was from
+the records of Suleim[=a]n the Merchant, a well-known Arab trader of the
+ninth century, that part of the story of Sindb[=a]d the Sailor was
+taken.[397] Such a merchant would have been particularly likely to know the
+numerals of the people whom he met, and he is a type of man that may well
+have taken such symbols to European markets. A little later, {101} Ab[=u]
+'l-[H.]asan `Al[=i] al-Mas`[=u]d[=i] (d. 956) of Bagdad traveled to the
+China Sea on the east, at least as far south as Zanzibar, and to the
+Atlantic on the west,[398] and he speaks of the nine figures with which the
+Hindus reckoned.[399]
+
+There was also a Bagdad merchant, one Ab[=u] 'l-Q[=a]sim `Obeidall[=a]h ibn
+A[h.]med, better known by his Persian name Ibn Khord[=a][d.]beh,[400] who
+wrote about 850 A.D. a work entitled _Book of Roads and Provinces_[401] in
+which the following graphic account appears:[402] "The Jewish merchants
+speak Persian, Roman (Greek and Latin), Arabic, French, Spanish, and
+Slavic. They travel from the West to the East, and from the East to the
+West, sometimes by land, sometimes by sea. They take ship from France on
+the Western Sea, and they voyage to Farama (near the ruins of the ancient
+Pelusium); there they transfer their goods to caravans and go by land to
+Colzom (on the Red Sea). They there reëmbark on the Oriental (Red) Sea and
+go to Hejaz and to Jiddah, and thence to the Sind, India, and China.
+Returning, they bring back the products of the oriental lands.... These
+journeys are also made by land. The merchants, leaving France and Spain,
+cross to Tangier and thence pass through the African provinces and Egypt.
+They then go to Ramleh, visit Damascus, Kufa, Bagdad, and Basra, penetrate
+into Ahwaz, Fars, Kerman, Sind, and thus reach India and China." Such
+travelers, about 900 A.D., must necessarily have spread abroad a knowledge
+of all number {102} systems used in recording prices or in the computations
+of the market. There is an interesting witness to this movement, a
+cruciform brooch now in the British Museum. It is English, certainly as
+early as the eleventh century, but it is inlaid with a piece of paste on
+which is the Mohammedan inscription, in Kufic characters, "There is no God
+but God." How did such an inscription find its way, perhaps in the time of
+Alcuin of York, to England? And if these Kufic characters reached there,
+then why not the numeral forms as well?
+
+Even in literature of the better class there appears now and then some
+stray proof of the important fact that the great trade routes to the far
+East were never closed for long, and that the customs and marks of trade
+endured from generation to generation. The _Gulist[=a]n_ of the Persian
+poet Sa`d[=i][403] contains such a passage:
+
+"I met a merchant who owned one hundred and forty camels, and fifty slaves
+and porters.... He answered to me: 'I want to carry sulphur of Persia to
+China, which in that country, as I hear, bears a high price; and thence to
+take Chinese ware to Roum; and from Roum to load up with brocades for Hind;
+and so to trade Indian steel (_pûlab_) to Halib. From Halib I will convey
+its glass to Yeman, and carry the painted cloths of Yeman back to
+Persia.'"[404] On the other hand, these men were not of the learned class,
+nor would they preserve in treatises any knowledge that they might have,
+although this knowledge would occasionally reach the ears of the learned as
+bits of curious information.
+
+{103}
+
+There were also ambassadors passing back and forth from time to time,
+between the East and the West, and in particular during the period when
+these numerals probably began to enter Europe. Thus Charlemagne (c. 800)
+sent emissaries to Bagdad just at the time of the opening of the
+mathematical activity there.[405] And with such ambassadors must have gone
+the adventurous scholar, inspired, as Alcuin says of Archbishop Albert of
+York (766-780),[406] to seek the learning of other lands. Furthermore, the
+Nestorian communities, established in Eastern Asia and in India at this
+time, were favored both by the Persians and by their Mohammedan conquerors.
+The Nestorian Patriarch of Syria, Timotheus (778-820), sent missionaries
+both to India and to China, and a bishop was appointed for the latter
+field. Ibn Wahab, who traveled to China in the ninth century, found images
+of Christ and the apostles in the Emperor's court.[407] Such a learned body
+of men, knowing intimately the countries in which they labored, could
+hardly have failed to make strange customs known as they returned to their
+home stations. Then, too, in Alfred's time (849-901) emissaries went {104}
+from England as far as India,[408] and generally in the Middle Ages
+groceries came to Europe from Asia as now they come from the colonies and
+from America. Syria, Asia Minor, and Cyprus furnished sugar and wool, and
+India yielded her perfumes and spices, while rich tapestries for the courts
+and the wealthy burghers came from Persia and from China.[409] Even in the
+time of Justinian (c. 550) there seems to have been a silk trade with
+China, which country in turn carried on commerce with Ceylon,[410] and
+reached out to Turkestan where other merchants transmitted the Eastern
+products westward. In the seventh century there was a well-defined commerce
+between Persia and India, as well as between Persia and
+Constantinople.[411] The Byzantine _commerciarii_ were stationed at the
+outposts not merely as customs officers but as government purchasing
+agents.[412]
+
+Occasionally there went along these routes of trade men of real learning,
+and such would surely have carried the knowledge of many customs back and
+forth. Thus at a period when the numerals are known to have been partly
+understood in Italy, at the opening of the eleventh century, one
+Constantine, an African, traveled from Italy through a great part of Africa
+and Asia, even on to India, for the purpose of learning the sciences of the
+Orient. He spent thirty-nine years in travel, having been hospitably
+received in Babylon, and upon his return he was welcomed with great honor
+at Salerno.[413]
+
+A very interesting illustration of this intercourse also appears in the
+tenth century, when the son of Otto I {105} (936-973) married a princess
+from Constantinople. This monarch was in touch with the Moors of Spain and
+invited to his court numerous scholars from abroad,[414] and his
+intercourse with the East as well as the West must have brought together
+much of the learning of each.
+
+Another powerful means for the circulation of mysticism and philosophy, and
+more or less of culture, took its start just before the conversion of
+Constantine (c. 312), in the form of Christian pilgrim travel. This was a
+feature peculiar to the zealots of early Christianity, found in only a
+slight degree among their Jewish predecessors in the annual pilgrimage to
+Jerusalem, and almost wholly wanting in other pre-Christian peoples. Chief
+among these early pilgrims were the two Placentians, John and Antonine the
+Elder (c. 303), who, in their wanderings to Jerusalem, seem to have started
+a movement which culminated centuries later in the crusades.[415] In 333 a
+Bordeaux pilgrim compiled the first Christian guide-book, the _Itinerary
+from Bordeaux to Jerusalem_,[416] and from this time on the holy pilgrimage
+never entirely ceased.
+
+Still another certain route for the entrance of the numerals into Christian
+Europe was through the pillaging and trading carried on by the Arabs on the
+northern shores of the Mediterranean. As early as 652 A.D., in the
+thirtieth year of the Hejira, the Mohammedans descended upon the shores of
+Sicily and took much spoil. Hardly had the wretched Constans given place to
+the {106} young Constantine IV when they again attacked the island and
+plundered ancient Syracuse. Again in 827, under Asad, they ravaged the
+coasts. Although at this time they failed to conquer Syracuse, they soon
+held a good part of the island, and a little later they successfully
+besieged the city. Before Syracuse fell, however, they had plundered the
+shores of Italy, even to the walls of Rome itself; and had not Leo IV, in
+849, repaired the neglected fortifications, the effects of the Moslem raid
+of that year might have been very far-reaching. Ibn Khord[=a][d.]beh, who
+left Bagdad in the latter part of the ninth century, gives a picture of the
+great commercial activity at that time in the Saracen city of Palermo. In
+this same century they had established themselves in Piedmont, and in 906
+they pillaged Turin.[417] On the Sorrento peninsula the traveler who climbs
+the hill to the beautiful Ravello sees still several traces of the Arab
+architecture, reminding him of the fact that about 900 A.D. Amalfi was a
+commercial center of the Moors.[418] Not only at this time, but even a
+century earlier, the artists of northern India sold their wares at such
+centers, and in the courts both of H[=a]r[=u]n al-Rash[=i]d and of
+Charlemagne.[419] Thus the Arabs dominated the Mediterranean Sea long
+before Venice
+
+              "held the gorgeous East in fee
+  And was the safeguard of the West,"
+
+and long before Genoa had become her powerful rival.[420]
+
+{107}
+
+Only a little later than this the brothers Nicolo and Maffeo Polo entered
+upon their famous wanderings.[421] Leaving Constantinople in 1260, they
+went by the Sea of Azov to Bokhara, and thence to the court of Kublai Khan,
+penetrating China, and returning by way of Acre in 1269 with a commission
+which required them to go back to China two years later. This time they
+took with them Nicolo's son Marco, the historian of the journey, and went
+across the plateau of Pamir; they spent about twenty years in China, and
+came back by sea from China to Persia.
+
+The ventures of the Poli were not long unique, however: the thirteenth
+century had not closed before Roman missionaries and the merchant Petrus de
+Lucolongo had penetrated China. Before 1350 the company of missionaries was
+large, converts were numerous, churches and Franciscan convents had been
+organized in the East, travelers were appealing for the truth of their
+accounts to the "many" persons in Venice who had been in China,
+Tsuan-chau-fu had a European merchant community, and Italian trade and
+travel to China was a thing that occupied two chapters of a commercial
+handbook.[422]
+
+{108}
+
+It is therefore reasonable to conclude that in the Middle Ages, as in the
+time of Boethius, it was a simple matter for any inquiring scholar to
+become acquainted with such numerals of the Orient as merchants may have
+used for warehouse or price marks. And the fact that Gerbert seems to have
+known only the forms of the simplest of these, not comprehending their full
+significance, seems to prove that he picked them up in just this way.
+
+Even if Gerbert did not bring his knowledge of the Oriental numerals from
+Spain, he may easily have obtained them from the marks on merchant's goods,
+had he been so inclined. Such knowledge was probably obtainable in various
+parts of Italy, though as parts of mere mercantile knowledge the forms
+might soon have been lost, it needing the pen of the scholar to preserve
+them. Trade at this time was not stagnant. During the eleventh and twelfth
+centuries the Slavs, for example, had very great commercial interests,
+their trade reaching to Kiev and Novgorod, and thence to the East.
+Constantinople was a great clearing-house of commerce with the Orient,[423]
+and the Byzantine merchants must have been entirely familiar with the
+various numerals of the Eastern peoples. In the eleventh century the
+Italian town of Amalfi established a factory[424] in Constantinople, and
+had trade relations with Antioch and Egypt. Venice, as early as the ninth
+century, had a valuable trade with Syria and Cairo.[425] Fifty years after
+Gerbert died, in the time of Cnut, the Dane and the Norwegian pushed their
+commerce far beyond the northern seas, both by caravans through Russia to
+the Orient, and by their venturesome barks which {109} sailed through the
+Strait of Gibraltar into the Mediterranean.[426] Only a little later,
+probably before 1200 A.D., a clerk in the service of Thomas à Becket,
+present at the latter's death, wrote a life of the martyr, to which
+(fortunately for our purposes) he prefixed a brief eulogy of the city of
+London.[427] This clerk, William Fitz Stephen by name, thus speaks of the
+British capital:
+
+  Aurum mittit Arabs: species et thura Sabæus:
+  Arma Sythes: oleum palmarum divite sylva
+  Pingue solum Babylon: Nilus lapides pretiosos:
+  Norwegi, Russi, varium grisum, sabdinas:
+  Seres, purpureas vestes: Galli, sua vina.
+
+Although, as a matter of fact, the Arabs had no gold to send, and the
+Scythians no arms, and Egypt no precious stones save only the turquoise,
+the Chinese (_Seres_) may have sent their purple vestments, and the north
+her sables and other furs, and France her wines. At any rate the verses
+show very clearly an extensive foreign trade.
+
+Then there were the Crusades, which in these times brought the East in
+touch with the West. The spirit of the Orient showed itself in the songs of
+the troubadours, and the _baudekin_,[428] the canopy of Bagdad,[429] became
+common in the churches of Italy. In Sicily and in Venice the textile
+industries of the East found place, and made their way even to the
+Scandinavian peninsula.[430]
+
+We therefore have this state of affairs: There was abundant intercourse
+between the East and West for {110} some centuries before the Hindu
+numerals appear in any manuscripts in Christian Europe. The numerals must
+of necessity have been known to many traders in a country like Italy at
+least as early as the ninth century, and probably even earlier, but there
+was no reason for preserving them in treatises. Therefore when a man like
+Gerbert made them known to the scholarly circles, he was merely describing
+what had been familiar in a small way to many people in a different walk of
+life.
+
+Since Gerbert[431] was for a long time thought to have been the one to
+introduce the numerals into Italy,[432] a brief sketch of this unique
+character is proper. Born of humble parents,[433] this remarkable man
+became the counselor and companion of kings, and finally wore the papal
+tiara as Sylvester II, from 999 until his death in 1003.[434] He was early
+brought under the influence of the monks at Aurillac, and particularly of
+Raimund, who had been a pupil of Odo of Cluny, and there in due time he
+himself took holy orders. He visited Spain in about 967 in company with
+Count Borel,[435] remaining there three years, {111} and studying under
+Bishop Hatto of Vich,[436] a city in the province of Barcelona,[437] then
+entirely under Christian rule. Indeed, all of Gerbert's testimony is as to
+the influence of the Christian civilization upon his education. Thus he
+speaks often of his study of Boethius,[438] so that if the latter knew the
+numerals Gerbert would have learned them from him.[439] If Gerbert had
+studied in any Moorish schools he would, under the decree of the emir
+Hish[=a]m (787-822), have been obliged to know Arabic, which would have
+taken most of his three years in Spain, and of which study we have not the
+slightest hint in any of his letters.[440] On the other hand, Barcelona was
+the only Christian province in immediate touch with the Moorish
+civilization at that time.[441] Furthermore we know that earlier in the
+same century King Alonzo of Asturias (d. 910) confided the education of his
+son Ordoño to the Arab scholars of the court of the {112} w[=a]l[=i] of
+Saragossa,[442] so that there was more or less of friendly relation between
+Christian and Moor.
+
+After his three years in Spain, Gerbert went to Italy, about 970, where he
+met Pope John XIII, being by him presented to the emperor Otto I. Two years
+later (972), at the emperor's request, he went to Rheims, where he studied
+philosophy, assisting to make of that place an educational center; and in
+983 he became abbot at Bobbio. The next year he returned to Rheims, and
+became archbishop of that diocese in 991. For political reasons he returned
+to Italy in 996, became archbishop of Ravenna in 998, and the following
+year was elected to the papal chair. Far ahead of his age in wisdom, he
+suffered as many such scholars have even in times not so remote by being
+accused of heresy and witchcraft. As late as 1522, in a biography published
+at Venice, it is related that by black art he attained the papacy, after
+having given his soul to the devil.[443] Gerbert was, however, interested
+in astrology,[444] although this was merely the astronomy of that time and
+was such a science as any learned man would wish to know, even as to-day we
+wish to be reasonably familiar with physics and chemistry.
+
+That Gerbert and his pupils knew the [.g]ob[=a]r numerals is a fact no
+longer open to controversy.[445] Bernelinus and Richer[446] call them by
+the well-known name of {113} "caracteres," a word used by Radulph of Laon
+in the same sense a century later.[447] It is probable that Gerbert was the
+first to describe these [.g]ob[=a]r numerals in any scientific way in
+Christian Europe, but without the zero. If he knew the latter he certainly
+did not understand its use.[448]
+
+The question still to be settled is as to where he found these numerals.
+That he did not bring them from Spain is the opinion of a number of careful
+investigators.[449] This is thought to be the more probable because most of
+the men who made Spain famous for learning lived after Gerbert was there.
+Such were Ibn S[=i]n[=a] (Avicenna) who lived at the beginning, and Gerber
+of Seville who flourished in the middle, of the eleventh century, and
+Ab[=u] Roshd (Averroës) who lived at the end of the twelfth.[450] Others
+hold that his proximity to {114} the Arabs for three years makes it
+probable that he assimilated some of their learning, in spite of the fact
+that the lines between Christian and Moor at that time were sharply
+drawn.[451] Writers fail, however, to recognize that a commercial numeral
+system would have been more likely to be made known by merchants than by
+scholars. The itinerant peddler knew no forbidden pale in Spain, any more
+than he has known one in other lands. If the [.g]ob[=a]r numerals were used
+for marking wares or keeping simple accounts, it was he who would have
+known them, and who would have been the one rather than any Arab scholar to
+bring them to the inquiring mind of the young French monk. The facts that
+Gerbert knew them only imperfectly, that he used them solely for
+calculations, and that the forms are evidently like the Spanish
+[.g]ob[=a]r, make it all the more probable that it was through the small
+tradesman of the Moors that this versatile scholar derived his knowledge.
+Moreover the part of the geometry bearing his name, and that seems
+unquestionably his, shows the Arab influence, proving that he at least came
+into contact with the transplanted Oriental learning, even though
+imperfectly.[452] There was also the persistent Jewish merchant trading
+with both peoples then as now, always alive to the acquiring of useful
+knowledge, and it would be very natural for a man like Gerbert to welcome
+learning from such a source.
+
+On the other hand, the two leading sources of information as to the life of
+Gerbert reveal practically nothing to show that he came within the Moorish
+sphere of influence during his sojourn in Spain. These sources {115} are
+his letters and the history written by Richer. Gerbert was a master of the
+epistolary art, and his exalted position led to the preservation of his
+letters to a degree that would not have been vouchsafed even by their
+classic excellence.[453] Richer was a monk at St. Remi de Rheims, and was
+doubtless a pupil of Gerbert. The latter, when archbishop of Rheims, asked
+Richer to write a history of his times, and this was done. The work lay in
+manuscript, entirely forgotten until Pertz discovered it at Bamberg in
+1833.[454] The work is dedicated to Gerbert as archbishop of Rheims,[455]
+and would assuredly have testified to such efforts as he may have made to
+secure the learning of the Moors.
+
+Now it is a fact that neither the letters nor this history makes any
+statement as to Gerbert's contact with the Saracens. The letters do not
+speak of the Moors, of the Arab numerals, nor of Cordova. Spain is not
+referred to by that name, and only one Spanish scholar is mentioned. In one
+of his letters he speaks of Joseph Ispanus,[456] or Joseph Sapiens, but who
+this Joseph the Wise of Spain may have been we do not know. Possibly {116}
+it was he who contributed the morsel of knowledge so imperfectly
+assimilated by the young French monk.[457] Within a few years after
+Gerbert's visit two young Spanish monks of lesser fame, and doubtless with
+not that keen interest in mathematical matters which Gerbert had, regarded
+the apparently slight knowledge which they had of the Hindu numeral forms
+as worthy of somewhat permanent record[458] in manuscripts which they were
+transcribing. The fact that such knowledge had penetrated to their modest
+cloisters in northern Spain--the one Albelda or Albaida--indicates that it
+was rather widely diffused.
+
+Gerbert's treatise _Libellus de numerorum divisione_[459] is characterized
+by Chasles as "one of the most obscure documents in the history of
+science."[460] The most complete information in regard to this and the
+other mathematical works of Gerbert is given by Bubnov,[461] who considers
+this work to be genuine.[462]
+
+{117}
+
+So little did Gerbert appreciate these numerals that in his works known as
+the _Regula de abaco computi_ and the _Libellus_ he makes no use of them at
+all, employing only the Roman forms.[463] Nevertheless Bernelinus[464]
+refers to the nine [.g]ob[=a]r characters.[465] These Gerbert had marked on
+a thousand _jetons_ or counters,[466] using the latter on an abacus which
+he had a sign-maker prepare for him.[467] Instead of putting eight counters
+in say the tens' column, Gerbert would put a single counter marked 8, and
+so for the other places, leaving the column empty where we would place a
+zero, but where he, lacking the zero, had no counter to place. These
+counters he possibly called _caracteres_, a name which adhered also to the
+figures themselves. It is an interesting speculation to consider whether
+these _apices_, as they are called in the Boethius interpolations, were in
+any way suggested by those Roman jetons generally known in numismatics as
+_tesserae_, and bearing the figures I-XVI, the sixteen referring to the
+number of _assi_ in a _sestertius_.[468] The {118} name _apices_ adhered to
+the Hindu-Arabic numerals until the sixteenth century.[469]
+
+To the figures on the _apices_ were given the names Igin, andras, ormis,
+arbas, quimas, calctis or caltis, zenis, temenias, celentis, sipos,[470]
+the origin and meaning of which still remain a mystery. The Semitic origin
+of several of the words seems probable. _Wahud_, _thaneine_, {119}
+_thalata_, _arba_, _kumsa_, _setta_, _sebba_, _timinia_, _taseud_ are given
+by the Rev. R. Patrick[471] as the names, in an Arabic dialect used in
+Morocco, for the numerals from one to nine. Of these the words for four,
+five, and eight are strikingly like those given above.
+
+The name _apices_ was not, however, a common one in later times. _Notae_
+was more often used, and it finally gave the name to notation.[472] Still
+more common were the names _figures_, _ciphers_, _signs_, _elements_, and
+_characters_.[473]
+
+So little effect did the teachings of Gerbert have in making known the new
+numerals, that O'Creat, who lived a century later, a friend and pupil of
+Adelhard {120} of Bath, used the zero with the Roman characters, in
+contrast to Gerbert's use of the [.g]ob[=a]r forms without the zero.[474]
+O'Creat uses three forms for zero, o, [=o], and [Greek: t], as in Maximus
+Planudes. With this use of the zero goes, naturally, a place value, for he
+writes III III for 33, ICCOO and I. II. [tau]. [tau] for 1200,
+I. O. VIII. IX for 1089, and I. IIII. IIII. [tau][tau][tau][tau] for the
+square of 1200.
+
+The period from the time of Gerbert until after the appearance of
+Leonardo's monumental work may be called the period of the abacists. Even
+for many years after the appearance early in the twelfth century of the
+books explaining the Hindu art of reckoning, there was strife between the
+abacists, the advocates of the abacus, and the algorists, those who favored
+the new numerals. The words _cifra_ and _algorismus cifra_ were used with a
+somewhat derisive significance, indicative of absolute uselessness, as
+indeed the zero is useless on an abacus in which the value of any unit is
+given by the column which it occupies.[475] So Gautier de Coincy
+(1177-1236) in a work on the miracles of Mary says:
+
+  A horned beast, a sheep,
+  An algorismus-cipher,
+  Is a priest, who on such a feast day
+  Does not celebrate the holy Mother.[476]
+
+So the abacus held the field for a long time, even against the new algorism
+employing the new numerals. {121} Geoffrey Chaucer[477] describes in _The
+Miller's Tale_ the clerk with
+
+ "His Almageste and bokes grete and smale,
+  His astrelabie, longinge for his art,
+  His augrim-stones layen faire apart
+  On shelves couched at his beddes heed."
+
+So, too, in Chaucer's explanation of the astrolabe,[478] written for his
+son Lewis, the number of degrees is expressed on the instrument in
+Hindu-Arabic numerals: "Over the whiche degrees ther ben noumbres of
+augrim, that devyden thilke same degrees fro fyve to fyve," and "... the
+nombres ... ben writen in augrim," meaning in the way of the algorism.
+Thomas Usk about 1387 writes:[479] "a sypher in augrim have no might in
+signification of it-selve, yet he yeveth power in signification to other."
+So slow and so painful is the assimilation of new ideas.
+
+Bernelinus[480] states that the abacus is a well-polished board (or table),
+which is covered with blue sand and used by geometers in drawing
+geometrical figures. We have previously mentioned the fact that the Hindus
+also performed mathematical computations in the sand, although there is no
+evidence to show that they had any column abacus.[481] For the purposes of
+computation, Bernelinus continues, the board is divided into thirty
+vertical columns, three of which are reserved for fractions. Beginning with
+the units columns, each set of {122} three columns (_lineae_ is the word
+which Bernelinus uses) is grouped together by a semicircular arc placed
+above them, while a smaller arc is placed over the units column and another
+joins the tens and hundreds columns. Thus arose the designation _arcus
+pictagore_[482] or sometimes simply _arcus_.[483] The operations of
+addition, subtraction, and multiplication upon this form of the abacus
+required little explanation, although they were rather extensively treated,
+especially the multiplication of different orders of numbers. But the
+operation of division was effected with some difficulty. For the
+explanation of the method of division by the use of the complementary
+difference,[484] long the stumbling-block in the way of the medieval
+arithmetician, the reader is referred to works on the history of
+mathematics[485] and to works relating particularly to the abacus.[486]
+
+Among the writers on the subject may be mentioned Abbo[487] of Fleury (c.
+970), Heriger[488] of Lobbes or Laubach {123} (c. 950-1007), and Hermannus
+Contractus[489] (1013-1054), all of whom employed only the Roman numerals.
+Similarly Adelhard of Bath (c. 1130), in his work _Regulae Abaci_,[490]
+gives no reference to the new numerals, although it is certain that he knew
+them. Other writers on the abacus who used some form of Hindu numerals were
+Gerland[491] (first half of twelfth century) and Turchill[492] (c. 1200).
+For the forms used at this period the reader is referred to the plate on
+page 88.
+
+After Gerbert's death, little by little the scholars of Europe came to know
+the new figures, chiefly through the introduction of Arab learning. The
+Dark Ages had passed, although arithmetic did not find another advocate as
+prominent as Gerbert for two centuries. Speaking of this great revival,
+Raoul Glaber[493] (985-c. 1046), a monk of the great Benedictine abbey of
+Cluny, of the eleventh century, says: "It was as though the world had
+arisen and tossed aside the worn-out garments of ancient time, and wished
+to apparel itself in a white robe of churches." And with this activity in
+religion came a corresponding interest in other lines. Algorisms began to
+appear, and knowledge from the outside world found {124} interested
+listeners. Another Raoul, or Radulph, to whom we have referred as Radulph
+of Laon,[494] a teacher in the cloister school of his city, and the brother
+of Anselm of Laon[495] the celebrated theologian, wrote a treatise on
+music, extant but unpublished, and an arithmetic which Nagl first published
+in 1890.[496] The latter work, preserved to us in a parchment manuscript of
+seventy-seven leaves, contains a curious mixture of Roman and [.g]ob[=a]r
+numerals, the former for expressing large results, the latter for practical
+calculation. These [.g]ob[=a]r "caracteres" include the sipos (zero),
+[Symbol], of which, however, Radulph did not know the full significance;
+showing that at the opening of the twelfth century the system was still
+uncertain in its status in the church schools of central France.
+
+At the same time the words _algorismus_ and _cifra_ were coming into
+general use even in non-mathematical literature. Jordan [497] cites
+numerous instances of such use from the works of Alanus ab Insulis[498]
+(Alain de Lille), Gautier de Coincy (1177-1236), and others.
+
+Another contributor to arithmetic during this interesting period was a
+prominent Spanish Jew called variously John of Luna, John of Seville,
+Johannes Hispalensis, Johannes Toletanus, and Johannes Hispanensis de
+Luna.[499] {125} His date is rather closely fixed by the fact that he
+dedicated a work to Raimund who was archbishop of Toledo between 1130 and
+1150.[500] His interests were chiefly in the translation of Arabic works,
+especially such as bore upon the Aristotelian philosophy. From the
+standpoint of arithmetic, however, the chief interest centers about a
+manuscript entitled _Joannis Hispalensis liber Algorismi de Practica
+Arismetrice_ which Boncompagni found in what is now the _Bibliothèque
+nationale_ at Paris. Although this distinctly lays claim to being
+Al-Khow[=a]razm[=i]'s work,[501] the evidence is altogether against the
+statement,[502] but the book is quite as valuable, since it represents the
+knowledge of the time in which it was written. It relates to the operations
+with integers and sexagesimal fractions, including roots, and contains no
+applications.[503]
+
+Contemporary with John of Luna, and also living in Toledo, was Gherard of
+Cremona,[504] who has sometimes been identified, but erroneously, with
+Gernardus,[505] the {126} author of a work on algorism. He was a physician,
+an astronomer, and a mathematician, translating from the Arabic both in
+Italy and in Spain. In arithmetic he was influential in spreading the ideas
+of algorism.
+
+Four Englishmen--Adelhard of Bath (c. 1130), Robert of Chester (Robertus
+Cestrensis, c. 1143), William Shelley, and Daniel Morley (1180)--are
+known[506] to have journeyed to Spain in the twelfth century for the
+purpose of studying mathematics and Arabic. Adelhard of Bath made
+translations from Arabic into Latin of Al-Khow[=a]razm[=i]'s astronomical
+tables[507] and of Euclid's Elements,[508] while Robert of Chester is known
+as the translator of Al-Khow[=a]razm[=i]'s algebra.[509] There is no reason
+to doubt that all of these men, and others, were familiar with the numerals
+which the Arabs were using.
+
+The earliest trace we have of computation with Hindu numerals in Germany is
+in an Algorismus of 1143, now in the Hofbibliothek in Vienna.[510] It is
+bound in with a {127} _Computus_ by the same author and bearing the date
+given. It contains chapters "De additione," "De diminutione," "De
+mediatione," "De divisione," and part of a chapter on multiplication. The
+numerals are in the usual medieval forms except the 2 which, as will be
+seen from the illustration,[511] is somewhat different, and the 3, which
+takes the peculiar shape [Symbol], a form characteristic of the twelfth
+century.
+
+It was about the same time that the _Sefer ha-Mispar_,[512] the Book of
+Number, appeared in the Hebrew language. The author, Rabbi Abraham ibn Meïr
+ibn Ezra,[513] was born in Toledo (c. 1092). In 1139 he went to Egypt,
+Palestine, and the Orient, spending also some years in Italy. Later he
+lived in southern France and in England. He died in 1167. The probability
+is that he acquired his knowledge of the Hindu arithmetic[514] in his
+native town of Toledo, but it is also likely that the knowledge of other
+systems which he acquired on travels increased his appreciation of this
+one. We have mentioned the fact that he used the first letters of the
+Hebrew alphabet, [Hebrew: A B G D H W Z CH T`], for the numerals 9 8 7 6 5
+4 3 2 1, and a circle for the zero. The quotation in the note given below
+shows that he knew of the Hindu origin; but in his manuscript, although he
+set down the Hindu forms, he used the above nine Hebrew letters with place
+value for all computations.
+
+       *       *       *       *       *
+
+
+{128}
+
+CHAPTER VIII
+
+THE SPREAD OF THE NUMERALS IN EUROPE
+
+Of all the medieval writers, probably the one most influential in
+introducing the new numerals to the scholars of Europe was Leonardo
+Fibonacci, of Pisa.[515] This remarkable man, the most noteworthy
+mathematical genius of the Middle Ages, was born at Pisa about 1175.[516]
+
+The traveler of to-day may cross the Via Fibonacci on his way to the Campo
+Santo, and there he may see at the end of the long corridor, across the
+quadrangle, the statue of Leonardo in scholars garb. Few towns have honored
+a mathematician more, and few mathematicians have so distinctly honored
+their birthplace. Leonardo was born in the golden age of this city, the
+period of its commercial, religious, and intellectual prosperity.[517]
+{129} Situated practically at the mouth of the Arno, Pisa formed with Genoa
+and Venice the trio of the greatest commercial centers of Italy at the
+opening of the thirteenth century. Even before Venice had captured the
+Levantine trade, Pisa had close relations with the East. An old Latin
+chronicle relates that in 1005 "Pisa was captured by the Saracens," that in
+the following year "the Pisans overthrew the Saracens at Reggio," and that
+in 1012 "the Saracens came to Pisa and destroyed it." The city soon
+recovered, however, sending no fewer than a hundred and twenty ships to
+Syria in 1099,[518] founding a merchant colony in Constantinople a few
+years later,[519] and meanwhile carrying on an interurban warfare in Italy
+that seemed to stimulate it to great activity.[520] A writer of 1114 tells
+us that at that time there were many heathen people--Turks, Libyans,
+Parthians, and Chaldeans--to be found in Pisa. It was in the midst of such
+wars, in a cosmopolitan and commercial town, in a center where literary
+work was not appreciated,[521] that the genius of Leonardo appears as one
+of the surprises of history, warning us again that "we should draw no
+horoscope; that we should expect little, for what we expect will not come
+to pass."[522]
+
+Leonardo's father was one William,[523] and he had a brother named
+Bonaccingus,[524] but nothing further is {130} known of his family. As to
+Fibonacci, most writers[525] have assumed that his father's name was
+Bonaccio,[526] whence _filius Bonaccii_, or Fibonacci. Others[527] believe
+that the name, even in the Latin form of _filius Bonaccii_ as used in
+Leonardo's work, was simply a general one, like our Johnson or Bronson
+(Brown's son); and the only contemporary evidence that we have bears out
+this view. As to the name Bigollo, used by Leonardo, some have thought it a
+self-assumed one meaning blockhead, a term that had been applied to him by
+the commercial world or possibly by the university circle, and taken by him
+that he might prove what a blockhead could do. Milanesi,[528] however, has
+shown that the word Bigollo (or Pigollo) was used in Tuscany to mean a
+traveler, and was naturally assumed by one who had studied, as Leonardo
+had, in foreign lands.
+
+Leonardo's father was a commercial agent at Bugia, the modern Bougie,[529]
+the ancient Saldae on the coast of Barbary,[530] a royal capital under the
+Vandals and again, a century before Leonardo, under the Beni Hammad. It had
+one of the best harbors on the coast, sheltered as it is by Mt. Lalla
+Guraia,[531] and at the close of the twelfth century it was a center of
+African commerce. It was here that Leonardo was taken as a child, and here
+he went to school to a Moorish master. When he reached the years of young
+manhood he started on a tour of the Mediterranean Sea, and visited Egypt,
+Syria, Greece, Sicily, and Provence, meeting with scholars as well as with
+{131} merchants, and imbibing a knowledge of the various systems of numbers
+in use in the centers of trade. All these systems, however, he says he
+counted almost as errors compared with that of the Hindus.[532] Returning
+to Pisa, he wrote his _Liber Abaci_[533] in 1202, rewriting it in
+1228.[534] In this work the numerals are explained and are used in the
+usual computations of business. Such a treatise was not destined to be
+popular, however, because it was too advanced for the mercantile class, and
+too novel for the conservative university circles. Indeed, at this time
+mathematics had only slight place in the newly established universities, as
+witness the oldest known statute of the Sorbonne at Paris, dated 1215,
+where the subject is referred to only in an incidental way.[535] The period
+was one of great commercial activity, and on this very {132} account such a
+book would attract even less attention than usual.[536]
+
+It would now be thought that the western world would at once adopt the new
+numerals which Leonardo had made known, and which were so much superior to
+anything that had been in use in Christian Europe. The antagonism of the
+universities would avail but little, it would seem, against such an
+improvement. It must be remembered, however, that there was great
+difficulty in spreading knowledge at this time, some two hundred and fifty
+years before printing was invented. "Popes and princes and even great
+religious institutions possessed far fewer books than many farmers of the
+present age. The library belonging to the Cathedral Church of San Martino
+at Lucca in the ninth century contained only nineteen volumes of
+abridgments from ecclesiastical commentaries."[537] Indeed, it was not
+until the early part of the fifteenth century that Palla degli Strozzi took
+steps to carry out the project that had been in the mind of Petrarch, the
+founding of a public library. It was largely by word of mouth, therefore,
+that this early knowledge had to be transmitted. Fortunately the presence
+of foreign students in Italy at this time made this transmission feasible.
+(If human nature was the same then as now, it is not impossible that the
+very opposition of the faculties to the works of Leonardo led the students
+to investigate {133} them the more zealously.) At Vicenza in 1209, for
+example, there were Bohemians, Poles, Frenchmen, Burgundians, Germans, and
+Spaniards, not to speak of representatives of divers towns of Italy; and
+what was true there was also true of other intellectual centers. The
+knowledge could not fail to spread, therefore, and as a matter of fact we
+find numerous bits of evidence that this was the case. Although the bankers
+of Florence were forbidden to use these numerals in 1299, and the statutes
+of the university of Padua required stationers to keep the price lists of
+books "non per cifras, sed per literas claros,"[538] the numerals really
+made much headway from about 1275 on.
+
+It was, however, rather exceptional for the common people of Germany to use
+the Arabic numerals before the sixteenth century, a good witness to this
+fact being the popular almanacs. Calendars of 1457-1496[539] have generally
+the Roman numerals, while Köbel's calendar of 1518 gives the Arabic forms
+as subordinate to the Roman. In the register of the Kreuzschule at Dresden
+the Roman forms were used even until 1539.
+
+While not minimizing the importance of the scientific work of Leonardo of
+Pisa, we may note that the more popular treatises by Alexander de Villa Dei
+(c. 1240 A.D.) and John of Halifax (Sacrobosco, c. 1250 A.D.) were much
+more widely used, and doubtless contributed more to the spread of the
+numerals among the common people.
+
+{134}
+
+The _Carmen de Algorismo_[540] of Alexander de Villa Dei was written in
+verse, as indeed were many other textbooks of that time. That it was widely
+used is evidenced by the large number of manuscripts[541] extant in
+European libraries. Sacrobosco's _Algorismus_,[542] in which some lines
+from the Carmen are quoted, enjoyed a wide popularity as a textbook for
+university instruction.[543] The work was evidently written with this end
+in view, as numerous commentaries by university lecturers are found.
+Probably the most widely used of these was that of Petrus de Dacia[544]
+written in 1291. These works throw an interesting light upon the method of
+instruction in mathematics in use in the universities from the thirteenth
+even to the sixteenth century. Evidently the text was first read and copied
+by students.[545] Following this came line by line an exposition of the
+text, such as is given in Petrus de Dacia's commentary.
+
+Sacrobosco's work is of interest also because it was probably due to the
+extended use of this work that the {135} term _Arabic numerals_ became
+common. In two places there is mention of the inventors of this system. In
+the introduction it is stated that this science of reckoning was due to a
+philosopher named Algus, whence the name _algorismus_,[546] and in the
+section on numeration reference is made to the Arabs as the inventors of
+this science.[547] While some of the commentators, Petrus de Dacia[548]
+among them, knew of the Hindu origin, most of them undoubtedly took the
+text as it stood; and so the Arabs were credited with the invention of the
+system.
+
+The first definite trace that we have of an algorism in the French language
+is found in a manuscript written about 1275.[549] This interesting leaf,
+for the part on algorism consists of a single folio, was noticed by the
+Abbé Leboeuf as early as 1741,[550] and by Daunou in 1824.[551] It then
+seems to have been lost in the multitude of Paris manuscripts; for although
+Chasles[552] relates his vain search for it, it was not rediscovered until
+1882. In that year M. Ch. Henry found it, and to his care we owe our
+knowledge of the interesting manuscript. The work is anonymous and is
+devoted almost entirely to geometry, only {136} two pages (one folio)
+relating to arithmetic. In these the forms of the numerals are given, and a
+very brief statement as to the operations, it being evident that the writer
+himself had only the slightest understanding of the subject.
+
+Once the new system was known in France, even thus superficially, it would
+be passed across the Channel to England. Higden,[553] writing soon after
+the opening of the fourteenth century, speaks of the French influence at
+that time and for some generations preceding:[554] "For two hundred years
+children in scole, agenst the usage and manir of all other nations beeth
+compelled for to leave hire own language, and for to construe hir lessons
+and hire thynges in Frensche.... Gentilmen children beeth taught to speke
+Frensche from the tyme that they bith rokked in hir cradell; and
+uplondissche men will likne himself to gentylmen, and fondeth with greet
+besynesse for to speke Frensche."
+
+The question is often asked, why did not these new numerals attract more
+immediate attention? Why did they have to wait until the sixteenth century
+to be generally used in business and in the schools? In reply it may be
+said that in their elementary work the schools always wait upon the demands
+of trade. That work which pretends to touch the life of the people must
+come reasonably near doing so. Now the computations of business until about
+1500 did not demand the new figures, for two reasons: First, cheap paper
+was not known. Paper-making of any kind was not introduced into Europe
+until {137} the twelfth century, and cheap paper is a product of the
+nineteenth. Pencils, too, of the modern type, date only from the sixteenth
+century. In the second place, modern methods of operating, particularly of
+multiplying and dividing (operations of relatively greater importance when
+all measures were in compound numbers requiring reductions at every step),
+were not yet invented. The old plan required the erasing of figures after
+they had served their purpose, an operation very simple with counters,
+since they could be removed. The new plan did not as easily permit this.
+Hence we find the new numerals very tardily admitted to the counting-house,
+and not welcomed with any enthusiasm by teachers.[555]
+
+Aside from their use in the early treatises on the new art of reckoning,
+the numerals appeared from time to time in the dating of manuscripts and
+upon monuments. The oldest definitely dated European document known {138}
+to contain the numerals is a Latin manuscript,[556] the Codex Vigilanus,
+written in the Albelda Cloister not far from Logroño in Spain, in 976 A.D.
+The nine characters (of [.g]ob[=a]r type), without the zero, are given as
+an addition to the first chapters of the third book of the _Origines_ by
+Isidorus of Seville, in which the Roman numerals are under discussion.
+Another Spanish copy of the same work, of 992 A.D., contains the numerals
+in the corresponding section. The writer ascribes an Indian origin to them
+in the following words: "Item de figuris arithmetic[e,]. Scire debemus in
+Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et
+geometria et ceteris liberalibus disciplinis concedere. Et hoc manifestum
+est in nobem figuris, quibus designant unumquemque gradum cuiuslibet
+gradus. Quarum hec sunt forma." The nine [.g]ob[=a]r characters follow.
+Some of the abacus forms[557] previously given are doubtless also of the
+tenth century. The earliest Arabic documents containing the numerals are
+two manuscripts of 874 and 888 A.D.[558] They appear about a century later
+in a work[559] written at Shiraz in 970 A.D. There is also an early trace
+of their use on a pillar recently discovered in a church apparently
+destroyed as early as the tenth century, not far from the Jeremias
+Monastery, in Egypt. {139} A graffito in Arabic on this pillar has the date
+349 A.H., which corresponds to 961 A.D.[560] For the dating of Latin
+documents the Arabic forms were used as early as the thirteenth
+century.[561]
+
+On the early use of these numerals in Europe the only scientific study
+worthy the name is that made by Mr. G. F. Hill of the British Museum.[562]
+From his investigations it appears that the earliest occurrence of a date
+in these numerals on a coin is found in the reign of Roger of Sicily in
+1138.[563] Until recently it was thought that the earliest such date was
+1217 A.D. for an Arabic piece and 1388 for a Turkish one.[564] Most of the
+seals and medals containing dates that were at one time thought to be very
+early have been shown by Mr. Hill to be of relatively late workmanship.
+There are, however, in European manuscripts, numerous instances of the use
+of these numerals before the twelfth century. Besides the example in the
+Codex Vigilanus, another of the tenth century has been found in the St.
+Gall MS. now in the University Library at Zürich, the forms differing
+materially from those in the Spanish codex.
+
+The third specimen in point of time in Mr. Hill's list is from a Vatican
+MS. of 1077. The fourth and fifth specimens are from the Erlangen MS. of
+Boethius, of the same {140} (eleventh) century, and the sixth and seventh
+are also from an eleventh-century MS. of Boethius at Chartres. These and
+other early forms are given by Mr. Hill in this table, which is reproduced
+with his kind permission.
+
+EARLIEST MANUSCRIPT FORMS
+
+[Illustration]
+
+This is one of more than fifty tables given in Mr. Hill's valuable paper,
+and to this monograph students {141} are referred for details as to the
+development of number-forms in Europe from the tenth to the sixteenth
+century. It is of interest to add that he has found that among the earliest
+dates of European coins or medals in these numerals, after the Sicilian one
+already mentioned, are the following: Austria, 1484; Germany, 1489
+(Cologne); Switzerland, 1424 (St. Gall); Netherlands, 1474; France, 1485;
+Italy, 1390.[565]
+
+The earliest English coin dated in these numerals was struck in 1551,[566]
+although there is a Scotch piece of 1539.[567] In numbering pages of a
+printed book these numerals were first used in a work of Petrarch's
+published at Cologne in 1471.[568] The date is given in the following form
+in the _Biblia Pauperum_,[569] a block-book of 1470,
+
+[Illustration]
+
+while in another block-book which possibly goes back to c. 1430[570] the
+numerals appear in several illustrations, with forms as follows:
+
+[Illustration]
+
+Many printed works anterior to 1471 have pages or chapters numbered by
+hand, but many of these numerals are {142} of date much later than the
+printing of the work. Other works were probably numbered directly after
+printing. Thus the chapters 2, 3, 4, 5, 6 in a book of 1470[571] are
+numbered as follows: Capitulem [Symbol 2]m.,... [Symbol 3]m.,... 4m.,...
+v,... vi, and followed by Roman numerals. This appears in the body of the
+text, in spaces left by the printer to be filled in by hand. Another
+book[572] of 1470 has pages numbered by hand with a mixture of Roman and
+Hindu numerals, thus,
+
+  [Illustration] for 125      [Illustration] for 150
+  [Illustration] for 147      [Illustration] for 202
+
+As to monumental inscriptions,[573] there was once thought to be a
+gravestone at Katharein, near Troppau, with the date 1007, and one at
+Biebrich of 1299. There is no doubt, however, of one at Pforzheim of 1371
+and one at Ulm of 1388.[574] Certain numerals on Wells Cathedral have been
+assigned to the thirteenth century, but they are undoubtedly considerably
+later.[575]
+
+The table on page 143 will serve to supplement that from Mr. Hill's
+work.[576]
+
+{143}
+
+EARLY MANUSCRIPT FORMS
+
+  [577] [Illustration] Twelfth century A.D.
+  [578] [Illustration] 1197 A.D.
+  [579] [Illustration] 1275 A.D.
+  [580] [Illustration] c. 1294 A.D.
+  [581] [Illustration] c. 1303 A.D.
+  [582] [Illustration] c. 1360 A.D.
+  [583] [Illustration] c. 1442 A.D.
+
+{144}
+
+[Illustration]
+
+For the sake of further comparison, three illustrations from works in Mr.
+Plimpton's library, reproduced from the _Rara Arithmetica_, may be
+considered. The first is from a Latin manuscript on arithmetic,[584] of
+which the original was written at Paris in 1424 by Rollandus, a Portuguese
+physician, who prepared the work at the command of John of Lancaster, Duke
+of Bedford, at one time Protector of England and Regent of France, to whom
+the work is dedicated. The figures show the successive powers of 2. The
+second illustration is from Luca da Firenze's _Inprencipio darte
+dabacho_,[585] c. 1475, and the third is from an anonymous manuscript[586]
+of about 1500.
+
+[Illustration]
+
+As to the forms of the numerals, fashion played a leading part until
+printing was invented. This tended to fix these forms, although in writing
+there is still a great variation, as witness the French 5 and the German 7
+and 9. Even in printing there is not complete uniformity, {145} and it is
+often difficult for a foreigner to distinguish between the 3 and 5 of the
+French types.
+
+[Illustration]
+
+As to the particular numerals, the following are some of the forms to be
+found in the later manuscripts and in the early printed books.
+
+1. In the early printed books "one" was often i, perhaps to save types,
+just as some modern typewriters use the same character for l and 1.[587] In
+the manuscripts the "one" appears in such forms as[588]
+
+[Illustration]
+
+2. "Two" often appears as z in the early printed books, 12 appearing as
+iz.[589] In the medieval manuscripts the following forms are common:[590]
+
+[Illustration]
+
+{146}
+
+It is evident, from the early traces, that it is merely a cursive form for
+the primitive [2 horizontal strokes], just as 3 comes from [3 horizontal
+strokes], as in the N[=a]n[=a] Gh[=a]t inscriptions.
+
+3. "Three" usually had a special type in the first printed books, although
+occasionally it appears as [Symbol].[591] In the medieval manuscripts it
+varied rather less than most of the others. The following are common
+forms:[592]
+
+[Illustration]
+
+4. "Four" has changed greatly; and one of the first tests as to the age of
+a manuscript on arithmetic, and the place where it was written, is the
+examination of this numeral. Until the time of printing the most common
+form was [Symbol], although the Florentine manuscript of Leonard of Pisa's
+work has the form [Symbol];[593] but the manuscripts show that the
+Florentine arithmeticians and astronomers rather early began to straighten
+the first of these forms up to forms like [Symbol][594] and [Symbol][594]
+or [Symbol],[595] more closely resembling our own. The first printed books
+generally used our present form[596] with the closed top [Symbol], the open
+top used in writing ( [Symbol]) being {147} purely modern. The following
+are other forms of the four, from various manuscripts:[597]
+
+[Illustration]
+
+5. "Five" also varied greatly before the time of printing. The following
+are some of the forms:[598]
+
+[Illustration]
+
+6. "Six" has changed rather less than most of the others. The chief
+variation has been in the slope of the top, as will be seen in the
+following:[599]
+
+[Illustration]
+
+7. "Seven," like "four," has assumed its present erect form only since the
+fifteenth century. In medieval times it appeared as follows:[600]
+
+[Illustration]
+
+{148}
+
+8. "Eight," like "six," has changed but little. In medieval times there are
+a few variants of interest as follows:[601]
+
+[Illustration]
+
+In the sixteenth century, however, there was manifested a tendency to write
+it [Symbol].[602]
+
+9. "Nine" has not varied as much as most of the others. Among the medieval
+forms are the following:[603]
+
+[Illustration]
+
+0. The shape of the zero also had a varied history. The following are
+common medieval forms:[604]
+
+[Illustration]
+
+The explanation of the place value was a serious matter to most of the
+early writers. If they had been using an abacus constructed like the
+Russian chotü, and had placed this before all learners of the positional
+system, there would have been little trouble. But the medieval {149}
+line-reckoning, where the lines stood for powers of 10 and the spaces for
+half of such powers, did not lend itself to this comparison. Accordingly we
+find such labored explanations as the following, from _The Crafte of
+Nombrynge_:
+
+"Euery of these figuris bitokens hym selfe & no more, yf he stonde in the
+first place of the rewele....
+
+"If it stonde in the secunde place of the rewle, he betokens ten tymes hym
+selfe, as this figure 2 here 20 tokens ten tyme hym selfe, that is twenty,
+for he hym selfe betokens tweyne, & ten tymes twene is twenty. And for he
+stondis on the lyft side & in the secunde place, he betokens ten tyme hym
+selfe. And so go forth....
+
+"Nil cifra significat sed dat signare sequenti. Expone this verse. A cifre
+tokens no[gh]t, bot he makes the figure to betoken that comes after hym
+more than he shuld & he were away, as thus 10. here the figure of one
+tokens ten, & yf the cifre were away & no figure byfore hym he schuld token
+bot one, for than he schuld stonde in the first place...."[605]
+
+It would seem that a system that was thus used for dating documents, coins,
+and monuments, would have been generally adopted much earlier than it was,
+particularly in those countries north of Italy where it did not come into
+general use until the sixteenth century. This, however, has been the fate
+of many inventions, as witness our neglect of logarithms and of contracted
+processes to-day.
+
+As to Germany, the fifteenth century saw the rise of the new symbolism; the
+sixteenth century saw it slowly {150} gain the mastery; the seventeenth
+century saw it finally conquer the system that for two thousand years had
+dominated the arithmetic of business. Not a little of the success of the
+new plan was due to Luther's demand that all learning should go into the
+vernacular.[606]
+
+During the transition period from the Roman to the Arabic numerals, various
+anomalous forms found place. For example, we have in the fourteenth century
+c[alpha] for 104;[607] 1000. 300. 80 et 4 for 1384;[608] and in a
+manuscript of the fifteenth century 12901 for 1291.[609] In the same
+century m. cccc. 8II appears for 1482,[610] while M^oCCCC^o50 (1450) and
+MCCCCXL6 (1446) are used by Theodoricus Ruffi about the same time.[611] To
+the next century belongs the form 1vojj for 1502. Even in Sfortunati's
+_Nuovo lume_[612] the use of ordinals is quite confused, the propositions
+on a single page being numbered "tertia," "4," and "V."
+
+Although not connected with the Arabic numerals in any direct way, the
+medieval astrological numerals may here be mentioned. These are given by
+several early writers, but notably by Noviomagus (1539),[613] as
+follows[614]:
+
+[Illustration]
+
+{151}
+
+Thus we find the numerals gradually replacing the Roman forms all over
+Europe, from the time of Leonardo of Pisa until the seventeenth century.
+But in the Far East to-day they are quite unknown in many countries, and
+they still have their way to make. In many parts of India, among the common
+people of Japan and China, in Siam and generally about the Malay Peninsula,
+in Tibet, and among the East India islands, the natives still adhere to
+their own numeral forms. Only as Western civilization is making its way
+into the commercial life of the East do the numerals as used by us find
+place, save as the Sanskrit forms appear in parts of India. It is therefore
+with surprise that the student of mathematics comes to realize how modern
+are these forms so common in the West, how limited is their use even at the
+present time, and how slow the world has been and is in adopting such a
+simple device as the Hindu-Arabic numerals.
+
+       *       *       *       *       *
+
+
+{153}
+
+INDEX
+
+_Transcriber's note: many of the entries refer to footnotes linked from the
+page numbers given._
+
+  Abbo of Fleury, 122
+  `Abdall[=a]h ibn al-[H.]asan, 92
+  `Abdallat[=i]f ibn Y[=u]suf, 93
+  `Abdalq[=a]dir ibn `Al[=i] al-Sakh[=a]w[=i], 6
+  Abenragel, 34
+  Abraham ibn Meïr ibn Ezra, _see_ Rabbi ben Ezra
+  Ab[=u] `Al[=i] al-[H.]osein ibn S[=i]n[=a], 74
+  Ab[=u] 'l-[H.]asan, 93, 100
+  Ab[=u] 'l-Q[=a]sim, 92
+  Ab[=u] 'l-[T.]eiyib, 97
+  Ab[=u] Na[s.]r, 92
+  Ab[=u] Roshd, 113
+  Abu Sahl Dunash ibn Tamim, 65, 67
+  Adelhard of Bath, 5, 55, 97, 119, 123, 126
+  Adhemar of Chabanois, 111
+  A[h.]med al-Nasaw[=i], 98
+  A[h.]med ibn `Abdall[=a]h, 9, 92
+  A[h.]med ibn Mo[h.]ammed, 94
+  A[h.]med ibn `Omar, 93
+  Ak[s.]aras, 32
+  Alanus ab Insulis, 124
+  Al-Ba[.g]d[=a]d[=i], 93
+  Al-Batt[=a]n[=i], 54
+  Albelda (Albaida) MS., 116
+  Albert, J., 62
+  Albert of York, 103
+  Al-B[=i]r[=u]n[=i], 6, 41, 49, 65, 92, 93
+  Alcuin, 103
+  Alexander the Great, 76
+  Alexander de Villa Dei, 11, 133
+  Alexandria, 64, 82
+  Al-Faz[=a]r[=i], 92
+  Alfred, 103
+  Algebra, etymology, 5
+  Algerian numerals, 68
+  Algorism, 97
+  Algorismus, 124, 126, 135
+  Algorismus cifra, 120
+  Al-[H.]a[s.][s.][=a]r, 65
+  `Al[=i] ibn Ab[=i] Bekr, 6
+  `Al[=i] ibn A[h.]med, 93, 98
+  Al-Kar[=a]b[=i]s[=i], 93
+  Al-Khow[=a]razm[=i], 4, 9, 10, 92, 97, 98, 125, 126
+  Al-Kind[=i], 10, 92
+  Almagest, 54
+  Al-Ma[.g]reb[=i], 93
+  Al-Ma[h.]all[=i], 6
+  Al-M[=a]m[=u]n, 10, 97
+  Al-Man[s.][=u]r, 96, 97
+  Al-Mas`[=u]d[=i], 7, 92
+  Al-Nad[=i]m, 9
+  Al-Nasaw[=i], 93, 98
+  Alphabetic numerals, 39, 40, 43
+  Al-Q[=a]sim, 92
+  Al-Qass, 94
+  Al-Sakh[=a]w[=i], 6
+  Al-[S.]ardaf[=i], 93
+  Al-Sijz[=i], 94
+  Al-S[=u]f[=i], 10, 92
+  Ambrosoli, 118
+  A[.n]kapalli, 43
+  Apices, 87, 117, 118
+  Arabs, 91-98
+  Arbuthnot, 141
+  {154}
+  Archimedes, 15, 16
+  Arcus Pictagore, 122
+  Arjuna, 15
+  Arnold, E., 15, 102
+  Ars memorandi, 141
+  [=A]ryabha[t.]a, 39, 43, 44
+  Aryan numerals, 19
+  Aschbach, 134
+  Ashmole, 134
+  A['s]oka, 19, 20, 22, 81
+  A[s.]-[s.]ifr, 57, 58
+  Astrological numerals, 150
+  Atharva-Veda, 48, 49, 55
+  Augustus, 80
+  Averroës, 113
+  Avicenna, 58, 74, 113
+
+  Babylonian numerals, 28
+  Babylonian zero, 51
+  Bacon, R., 131
+  Bactrian numerals, 19, 30
+  Bæda, 2, 72
+  Bagdad, 4, 96
+  Bakh[s.][=a]l[=i] manuscript, 43, 49, 52, 53
+  Ball, C. J., 35
+  Ball, W. W. R., 36, 131
+  B[=a][n.]a, 44
+  Barth, A., 39
+  Bayang inscriptions, 39
+  Bayer, 33
+  Bayley, E. C., 19, 23, 30, 32, 52, 89
+  Beazley, 75
+  Bede, _see_ Bæda
+  Beldomandi, 137
+  Beloch, J., 77
+  Bendall, 25, 52
+  Benfey, T., 26
+  Bernelinus, 88, 112, 117, 121
+  Besagne, 128
+  Besant, W., 109
+  Bettino, 36
+  Bhandarkar, 18, 47, 49
+  Bh[=a]skara, 53, 55
+  Biernatzki, 32
+  Biot, 32
+  Björnbo, A. A., 125, 126
+  Blassière, 119
+  Bloomfield, 48
+  Blume, 85
+  Boeckh, 62
+  Boehmer, 143
+  Boeschenstein, 119
+  Boethius, 63, 70-73, 83-90
+  Boissière, 63
+  Bombelli, 81
+  Bonaini, 128
+  Boncompagni, 5, 6, 10, 48, 49, 123, 125
+  Borghi, 59
+  Borgo, 119
+  Bougie, 130
+  Bowring, J., 56
+  Brahmagupta, 52
+  Br[=a]hma[n.]as, 12, 13
+  Br[=a]hm[=i], 19, 20, 31, 83
+  Brandis, J., 54
+  B[r.]hat-Sa[m.]hita, 39, 44, 78
+  Brockhaus, 43
+  Bubnov, 65, 84, 110, 116
+  Buddha, education of, 15, 16
+  Büdinger, 110
+  Bugia, 130
+  Bühler, G., 15, 19, 22, 31, 44, 49
+  Burgess, 25
+  Bürk, 13
+  Burmese numerals, 36
+  Burnell, A. C., 18, 40
+  Buteo, 61
+
+  Calandri, 59, 81
+  Caldwell, R., 19
+  Calendars, 133
+  Calmet, 34
+  Cantor, M., 5, 13, 30, 43, 84
+  {155}
+  Capella, 86
+  Cappelli, 143
+  Caracteres, 87, 113, 117, 119
+  Cardan, 119
+  Carmen de Algorismo, 11, 134
+  Casagrandi, 132
+  Casiri, 8, 10
+  Cassiodorus, 72
+  Cataldi, 62
+  Cataneo, 3
+  Caxton, 143, 146
+  Ceretti, 32
+  Ceylon numerals, 36
+  Chalfont, F. H., 28
+  Champenois, 60
+  Characters, _see_ Caracteres
+  Charlemagne, 103
+  Chasles, 54, 60, 85, 116, 122, 135
+  Chassant, L. A., 142
+  Chaucer, 121
+  Chiarini, 145, 146
+  Chiffre, 58
+  Chinese numerals, 28, 56
+  Chinese zero, 56
+  Cifra, 120, 124
+  Cipher, 58
+  Circulus, 58, 60
+  Clichtoveus, 61, 119, 145
+  Codex Vigilanus, 138
+  Codrington, O., 139
+  Coins dated, 141
+  Colebrooke, 8, 26, 46, 53
+  Constantine, 104, 105
+  Cosmas, 82
+  Cossali, 5
+  Counters, 117
+  Courteille, 8
+  Coxe, 59
+  Crafte of Nombrynge, 11, 87, 149
+  Crusades, 109
+  Cunningham, A., 30, 75
+  Curtze, 55, 59, 126, 134
+  Cyfra, 55
+
+  Dagomari, 146
+  D'Alviella, 15
+  Dante, 72
+  Dasypodius, 33, 67, 63
+  Daunou, 135
+  Delambre, 54
+  Devan[=a]gar[=i], 7
+  Devoulx, A., 68
+  Dhruva, 49
+  Dicæarchus of Messana, 77
+  Digits, 119
+  Diodorus Siculus, 76
+  Du Cange, 62
+  Dumesnil, 36
+  Dutt, R. C., 12, 15, 18, 75
+  Dvived[=i], 44
+
+  East and West, relations, 73-81, 100-109
+  Egyptian numerals, 27
+  Eisenlohr, 28
+  Elia Misrachi, 57
+  Enchiridion Algorismi, 58
+  Eneström, 5, 48, 59, 97, 125, 128
+  Europe, numerals in, 63, 99, 128, 136
+  Eusebius Caesariensis, 142
+  Euting, 21
+  Ewald, P., 116
+
+  Fazzari, 53, 54
+  Fibonacci, _see_ Leonardo of Pisa
+  Figura nihili, 58
+  Figures, 119. _See_ numerals.
+  Fihrist, 67, 68, 93
+  Finaeus, 57
+  Firdus[=i], 81
+  Fitz Stephen, W., 109
+  Fleet, J. C., 19, 20, 49
+  {156}
+  Florus, 80
+  Flügel, G., 68
+  Francisco de Retza, 142
+  François, 58
+  Friedlein, G., 84, 113, 116, 122
+  Froude, J. A., 129
+
+  Gandh[=a]ra, 19
+  Garbe, 48
+  Gasbarri, 58
+  Gautier de Coincy, 120, 124
+  Gemma Frisius, 2, 3, 119
+  Gerber, 113
+  Gerbert, 108, 110-120, 122
+  Gerhardt, C. I., 43, 56, 93, 118
+  Gerland, 88, 123
+  Gherard of Cremona, 125
+  Gibbon, 72
+  Giles, H. A., 79
+  Ginanni, 81
+  Giovanni di Danti, 58
+  Glareanus, 4, 119
+  Gnecchi, 71, 117
+  [.G]ob[=a]r numerals, 65, 100, 112, 124, 138
+  Gow, J., 81
+  Grammateus, 61
+  Greek origin, 33
+  Green, J. R., 109
+  Greenwood, I., 62, 119
+  Guglielmini, 128
+  Gulist[=a]n, 102
+  Günther, S., 131
+  Guyard, S., 82
+
+  [H.]abash, 9, 92
+  Hager, J. (G.), 28, 32
+  Halliwell, 59, 85
+  Hankel, 93
+  H[=a]r[=u]n al-Rash[=i]d, 97, 106
+  Havet, 110
+  Heath, T. L., 125
+  Hebrew numerals, 127
+  Hecatæus, 75
+  Heiberg, J. L., 55, 85, 148
+  Heilbronner, 5
+  Henry, C., 5, 31, 55, 87, 120, 135
+  Heriger, 122
+  Hermannus Contractus, 123
+  Herodotus, 76, 78
+  Heyd, 75
+  Higden, 136
+  Hill, G. F., 52, 139, 142
+  Hillebrandt, A., 15, 74
+  Hilprecht, H. V., 28
+  Hindu forms, early, 12
+  Hindu number names, 42
+  Hodder, 62
+  Hoernle, 43, 49
+  Holywood, _see_ Sacrobosco
+  Hopkins, E. W., 12
+  Horace, 79, 80
+  [H.]osein ibn Mo[h.]ammed al-Ma[h.]all[=i], 6
+  Hostus, M., 56
+  Howard, H. H., 29
+  Hrabanus Maurus, 72
+  Huart, 7
+  Huet, 33
+  Hugo, H., 57
+  Humboldt, A. von, 62
+  Huswirt, 58
+
+  Iamblichus, 81
+  Ibn Ab[=i] Ya`q[=u]b, 9
+  Ibn al-Adam[=i], 92
+  Ibn al-Bann[=a], 93
+  Ibn Khord[=a][d.]beh, 101, 106
+  Ibn Wahab, 103
+  India, history of, 14
+    writing in, 18
+  Indicopleustes, 83
+  Indo-Bactrian numerals, 19
+  {157}
+  Indr[=a]j[=i], 23
+  Is[h.][=a]q ibn Y[=u]suf al-[S.]ardaf[=i], 93
+
+  Jacob of Florence, 57
+  Jacquet, E., 38
+  Jamshid, 56
+  Jehan Certain, 59
+  Jetons, 58, 117
+  Jevons, F. B., 76
+  Johannes Hispalensis, 48, 88, 124
+  John of Halifax, _see_ Sacrobosco
+  John of Luna, _see_ Johannes Hispalensis
+  Jordan, L., 58, 124
+  Joseph Ispanus (Joseph Sapiens), 115
+  Justinian, 104
+
+  Kále, M. R., 26
+  Karabacek, 56
+  Karpinski, L. C., 126, 134, 138
+  K[=a]ty[=a]yana, 39
+  Kaye, C. R., 6, 16, 43, 46, 121
+  Keane, J., 75, 82
+  Keene, H. G., 15
+  Kern, 44
+  Kharo[s.][t.]h[=i], 19, 20
+  Khosr[=u], 82, 91
+  Kielhorn, F., 46, 47
+  Kircher, A., 34
+  Kit[=a]b al-Fihrist, _see_ Fihrist
+  Kleinwächter, 32
+  K[=l]os, 62
+  Köbel, 4, 58, 60, 119, 123
+  Krumbacher, K., 57
+  Kuckuck, 62, 133
+  Kugler, F. X., 51
+
+  Lachmann, 85
+  Lacouperie, 33, 35
+  Lalitavistara, 15, 17
+  Lami, G., 57
+  La Roche, 61
+  Lassen, 39
+  L[=a][t.]y[=a]yana, 39
+  Leboeuf, 135
+  Leonardo of Pisa, 5, 10, 57, 64, 74, 120, 128-133
+  Lethaby, W. R., 142
+  Levi, B., 13
+  Levias, 3
+  Libri, 73, 85, 95
+  Light of Asia, 16
+  Luca da Firenze, 144
+  Lucas, 128
+
+  Mah[=a]bh[=a]rata, 18
+  Mah[=a]v[=i]r[=a]c[=a]rya, 53
+  Malabar numerals, 36
+  Malayalam numerals, 36
+  Mannert, 81
+  Margarita Philosophica, 146
+  Marie, 78
+  Marquardt, J., 85
+  Marshman, J. C., 17
+  Martin, T. H., 30, 62, 85, 113
+  Martines, D. C., 58
+  M[=a]sh[=a]ll[=a]h, 3
+  Maspero, 28
+  Mauch, 142
+  Maximus Planudes, 2, 57, 66, 93, 120
+  Megasthenes, 77
+  Merchants, 114
+  Meynard, 8
+  Migne, 87
+  Mikami, Y., 56
+  Milanesi, 128
+  Mo[h.]ammed ibn `Abdall[=a]h, 92
+  Mo[h.]ammed ibn A[h.]med, 6
+  Mo[h.]ammed ibn `Al[=i] `Abd[=i], 8
+  Mo[h.]ammed ibn M[=u]s[=a], _see_ Al-Khow[=a]razm[=i]
+  Molinier, 123
+  Monier-Williams, 17
+  {158}
+  Morley, D., 126
+  Moroccan numerals, 68, 119
+  Mortet, V., 11
+  Moseley, C. B., 33
+  Mo[t.]ahhar ibn [T.][=a]hir, 7
+  Mueller, A., 68
+  Mumford, J. K., 109
+  Muwaffaq al-D[=i]n, 93
+
+  Nabatean forms, 21
+  Nallino, 4, 54, 55
+  Nagl, A., 55, 110, 113, 126
+  N[=a]n[=a] Gh[=a]t inscriptions, 20, 22, 23, 40
+  Narducci, 123
+  Nasik cave inscriptions, 24
+  Na[z.][=i]f ibn Yumn, 94
+  Neander, A., 75
+  Neophytos, 57, 62
+  Neo-Pythagoreans, 64
+  Nesselmann, 58
+  Newman, Cardinal, 96
+  Newman, F. W., 131
+  Nöldeke, Th., 91
+  Notation, 61
+  Note, 61, 119
+  Noviomagus, 45, 61, 119, 150
+  Null, 61
+  Numerals,
+    Algerian, 68
+    astrological, 150
+    Br[=a]hm[=i], 19-22, 83
+    early ideas of origin, 1
+    Hindu, 26
+    Hindu, classified, 19, 38
+    Kharo[s.][t.]h[=i], 19-22
+    Moroccan, 68
+    Nabatean, 21
+    origin, 27, 30, 31, 37
+    supposed Arabic origin, 2
+    supposed Babylonian origin, 28
+    supposed Chaldean and Jewish origin, 3
+    supposed Chinese origin, 28, 32
+    supposed Egyptian origin, 27, 30, 69, 70
+    supposed Greek origin, 33
+    supposed Phoenician origin, 32
+    tables of, 22-27, 36, 48, 49, 69, 88, 140, 143, 145-148
+
+  O'Creat, 5, 55, 119, 120
+  Olleris, 110, 113
+  Oppert, G., 14, 75
+
+  Pali, 22
+  Pañcasiddh[=a]ntik[=a], 44
+  Paravey, 32, 57
+  P[=a]tal[=i]pu[t.]ra, 77
+  Patna, 77
+  Patrick, R., 119
+  Payne, E. J., 106
+  Pegolotti, 107
+  Peletier, 2, 62
+  Perrot, 80
+  Persia, 66, 91, 107
+  Pertz, 115
+  Petrus de Dacia, 59, 61, 62
+  Pez, P. B., 117
+  "Philalethes," 75
+  Phillips, G., 107
+  Picavet, 105
+  Pichler, F., 141
+  Pihan, A. P., 36
+  Pisa, 128
+  Place value, 26, 42, 46, 48
+  Planudes, _see_ Maximus Planudes
+  Plimpton, G. A., 56, 59, 85, 143, 144, 145, 148
+  Pliny, 76
+  Polo, N. and M., 107
+  {159}
+  Prändel, J. G., 54
+  Prinsep, J., 20, 31
+  Propertius, 80
+  Prosdocimo de' Beldomandi, 137
+  Prou, 143
+  Ptolemy, 54, 78
+  Putnam, 103
+  Pythagoras, 63
+  Pythagorean numbers, 13
+  Pytheas of Massilia, 76
+
+  Rabbi ben Ezra, 60, 127
+  Radulph of Laon, 60, 113, 118, 124
+  Raets, 62
+  Rainer, _see_ Gemma Frisius
+  R[=a]m[=a]yana, 18
+  Ramus, 2, 41, 60, 61
+  Raoul Glaber, 123
+  Rapson, 77
+  Rauhfuss, _see_ Dasypodius
+  Raumer, K. von, 111
+  Reclus, E., 14, 96, 130
+  Recorde, 3, 58
+  Reinaud, 67, 74, 80
+  Reveillaud, 36
+  Richer, 110, 112, 115
+  Riese, A., 119
+  Robertson, 81
+  Robertus Cestrensis, 97, 126
+  Rodet, 5, 44
+  Roediger, J., 68
+  Rollandus, 144
+  Romagnosi, 81
+  Rosen, F., 5
+  Rotula, 60
+  Rudolff, 85
+  Rudolph, 62, 67
+  Ruffi, 150
+
+  Sachau, 6
+  Sacrobosco, 3, 58, 133
+  Sacy, S. de, 66, 70
+  Sa`d[=i], 102
+  ['S]aka inscriptions, 20
+  Sam[=u]'[=i]l ibn Ya[h.]y[=a], 93
+  ['S][=a]rad[=a] characters, 55
+  Savonne, 60
+  Scaliger, J. C., 73
+  Scheubel, 62
+  Schlegel, 12
+  Schmidt, 133
+  Schonerus, 87, 119
+  Schroeder, L. von, 13
+  Scylax, 75
+  Sedillot, 8, 34
+  Senart, 20, 24, 25
+  Sened ibn `Al[=i], 10, 98
+  Sfortunati, 62, 150
+  Shelley, W., 126
+  Siamese numerals, 36
+  Siddh[=a]nta, 8, 18
+  [S.]ifr, 57
+  Sigsboto, 55
+  Sih[=a]b al-D[=i]n, 67
+  Silberberg, 60
+  Simon, 13
+  Sin[=a]n ibn al-Fat[h.], 93
+  Sindbad, 100
+  Sindhind, 97
+  Sipos, 60
+  Sirr, H. C., 75
+  Skeel, C. A., 74
+  Smith, D. E., 11, 17, 53, 86, 141, 143
+  Smith, V. A., 20, 35, 46, 47
+  Smith, Wm., 75
+  Sm[r.]ti, 17
+  Spain, 64, 65, 100
+  Spitta-Bey, 5
+  Sprenger, 94
+  ['S]rautas[=u]tra, 39
+  Steffens, F., 116
+  Steinschneider, 5, 57, 65, 66, 98, 126
+  Stifel, 62
+  {160}
+  Subandhus, 44
+  Suetonius, 80
+  Suleim[=a]n, 100
+  ['S][=u]nya, 43, 53, 57
+  Suter, 5, 9, 68, 69, 93, 116, 131
+  S[=u]tras, 13
+  Sykes, P. M., 75
+  Sylvester II, _see_ Gerbert
+  Symonds, J. A., 129
+
+  Tannery, P., 62, 84, 85
+  Tartaglia, 4, 61
+  Taylor, I., 19, 30
+  Teca, 55, 61
+  Tennent, J. E., 75
+  Texada, 60
+  Theca, 58, 61
+  Theophanes, 64
+  Thibaut, G., 12, 13, 16, 44, 47
+  Tibetan numerals, 36
+  Timotheus, 103
+  Tonstall, C., 3, 61
+  Trenchant, 60
+  Treutlein, 5, 63, 123
+  Trevisa, 136
+  Treviso arithmetic, 145
+  Trivium and quadrivium, 73
+  Tsin, 56
+  Tunis, 65
+  Turchill, 88, 118, 123
+  Turnour, G., 75
+  Tziphra, 57, 62
+  [Greek: tziphra], 55, 57, 62
+  Tzwivel, 61, 118, 145
+
+  Ujjain, 32
+  Unger, 133
+  Upanishads, 12
+  Usk, 121
+
+  Valla, G., 61
+  Van der Schuere, 62
+  Var[=a]ha-Mihira, 39, 44, 78
+  V[=a]savadatt[=a], 44
+  Vaux, Carra de, 9, 74
+  Vaux, W. S. W., 91
+  Ved[=a][.n]gas, 17
+  Vedas, 12, 15, 17
+  Vergil, 80
+  Vincent, A. J. H., 57
+  Vogt, 13
+  Voizot, P., 36
+  Vossius, 4, 76, 81, 84
+
+  Wallis, 3, 62, 84, 116
+  Wappler, E., 54, 126
+  Wäschke, H., 2, 93
+  Wattenbach, 143
+  Weber, A., 31
+  Weidler, I. F., 34, 66
+  Weidler, I. F. and G. I., 63, 66
+  Weissenborn, 85, 110
+  Wertheim, G., 57, 61
+  Whitney, W. D., 13
+  Wilford, F., 75
+  Wilkens, 62
+  Wilkinson, J. G., 70
+  Willichius, 3
+  Woepcke, 3, 6, 42, 63, 64, 65, 67, 69, 70, 94, 113, 138
+  Wolack, G., 54
+  Woodruff, C. E., 32
+  Word and letter numerals, 38, 44
+  Wüstenfeld, 74
+
+  Yule, H., 107
+
+  Zephirum, 57, 58
+  Zephyr, 59
+  Zepiro, 58
+  Zero, 26, 38, 40, 43, 45, 49, 51-62, 67
+  Zeuero, 58
+
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+chapters have been rewritten in part and greatly improved. The order of
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+Notes
+
+al-Mekk[=i] on a treatise on [.g]ob[=a]r arithmetic (explained later)
+called _Al-murshidah_, found by Woepcke in Paris (_Propagation_, p. 66),
+there is mentioned the fact that there are "nine Indian figures" and "a
+second kind of Indian figures ... although these are the figures of the
+[.g]ob[=a]r writing." So in a commentary by [H.]osein ibn Mo[h.]ammed
+al-Ma[h.]all[=i] (died in 1756) on the _Mokhta[s.]ar f[=i]`ilm
+el-[h.]is[=a]b_ (Extract from Arithmetic) by `Abdalq[=a]dir ibn `Al[=i]
+al-Sakh[=a]w[=i] (died c. 1000) it is related that "the preface treats of
+the forms of the figures of Hindu signs, such as were established by the
+Hindu nation." [Woepcke, _Propagation_, p. 63.]]
+
+which, of course, are interpolations. An interesting example of a forgery
+in ecclesiastical matters is in the charter said to have been given by St.
+Patrick, granting indulgences to the benefactors of Glastonbury, dated "In
+nomine domini nostri Jhesu Christi Ego Patricius humilis servunculus Dei
+anno incarnationis ejusdem ccccxxx." Now if the Benedictines are right in
+saying that Dionysius Exiguus, a Scythian monk, first arranged the
+Christian chronology c. 532 A.D., this can hardly be other than spurious.
+See Arbuthnot, loc. cit., p. 38.
+
+[1] "_Discipulus._ Quis primus invenit numerum apud Hebræos et Ægyptios?
+_Magister._ Abraham primus invenit numerum apud Hebræos, deinde Moses; et
+Abraham tradidit istam scientiam numeri ad Ægyptios, et docuit eos: deinde
+Josephus." [Bede, _De computo dialogus_ (doubtfully assigned to him),
+_Opera omnia_, Paris, 1862, Vol. I, p. 650.]
+
+"Alii referunt ad Phoenices inventores arithmeticæ, propter eandem
+commerciorum caussam: Alii ad Indos: Ioannes de Sacrobosco, cujus
+sepulchrum est Lutetiæ in comitio Maturinensi, refert ad Arabes." [Ramus,
+_Arithmeticæ libri dvo_, Basel, 1569, p. 112.]
+
+Similar notes are given by Peletarius in his commentary on the arithmetic
+of Gemma Frisius (1563 ed., fol. 77), and in his own work (1570 Lyons ed.,
+p. 14): "La valeur des Figures commence au coste dextre tirant vers le
+coste senestre: au rebours de notre maniere d'escrire par ce que la
+premiere prattique est venue des Chaldees: ou des Pheniciens, qui ont été
+les premiers traffiquers de marchandise."
+
+[2] Maximus Planudes (c. 1330) states that "the nine symbols come from the
+Indians." [Wäschke's German translation, Halle, 1878, p. 3.] Willichius
+speaks of the "Zyphræ Indicæ," in his _Arithmeticæ libri tres_ (Strasburg,
+1540, p. 93), and Cataneo of "le noue figure de gli Indi," in his _Le
+pratiche delle dve prime mathematiche_ (Venice, 1546, fol. 1). Woepcke is
+not correct, therefore, in saying ("Mémoire sur la propagation des chiffres
+indiens," hereafter referred to as _Propagation_ [_Journal Asiatique_, Vol.
+I (6), 1863, p. 34]) that Wallis (_A Treatise on Algebra, both historical
+and practical_, London, 1685, p. 13, and _De algebra tractatus_, Latin
+edition in his _Opera omnia_, 1693, Vol. II, p. 10) was one of the first to
+give the Hindu origin.
+
+[3] From the 1558 edition of _The Grovnd of Artes_, fol. C, 5. Similarly
+Bishop Tonstall writes: "Qui a Chaldeis primum in finitimos, deinde in
+omnes pene gentes fluxit.... Numerandi artem a Chaldeis esse profectam: qui
+dum scribunt, a dextra incipiunt, et in leuam progrediuntur." [_De arte
+supputandi_, London, 1522, fol. B, 3.] Gemma Frisius, the great continental
+rival of Recorde, had the same idea: "Primùm autem appellamus dexterum
+locum, eo quòd haec ars vel à Chaldæis, vel ab Hebræis ortum habere
+credatur, qui etiam eo ordine scribunt"; but this refers more evidently to
+the Arabic numerals. [_Arithmeticæ practicæ methodvs facilis_, Antwerp,
+1540, fol. 4 of the 1563 ed.] Sacrobosco (c. 1225) mentions the same thing.
+Even the modern Jewish writers claim that one of their scholars,
+M[=a]sh[=a]ll[=a]h (c. 800), introduced them to the Mohammedan world. [C.
+Levias, _The Jewish Encyclopedia_, New York, 1905, Vol. IX, p. 348.]
+
+[4] "... & que esto fu trouato di fare da gli Arabi con diece figure." [_La
+prima parte del general trattato di nvmeri, et misvre_, Venice, 1556, fol.
+9 of the 1592 edition.]
+
+[5] "Vom welchen Arabischen auch disz Kunst entsprungen ist." [_Ain nerv
+geordnet Rechenbiechlin_, Augsburg, 1514, fol. 13 of the 1531 edition. The
+printer used the letters _rv_ for _w_ in "new" in the first edition, as he
+had no _w_ of the proper font.]
+
+[6] Among them Glareanus: "Characteres simplices sunt nouem significatiui,
+ab Indis usque, siue Chaldæis asciti .1.2.3.4.5.6.7.8.9. Est item unus .0
+circulus, qui nihil significat." [_De VI. Arithmeticae practicae
+speciebvs_, Paris, 1539, fol. 9 of the 1543 edition.]
+
+[7] "Barbarische oder gemeine Ziffern." [Anonymous, _Das Einmahl Eins cum
+notis variorum_, Dresden, 1703, p. 3.] So Vossius (_De universae matheseos
+natura et constitutione liber_, Amsterdam, 1650, p. 34) calls them
+"Barbaras numeri notas." The word at that time was possibly synonymous with
+Arabic.
+
+[8] His full name was `Ab[=u] `Abdall[=a]h Mo[h.]ammed ibn M[=u]s[=a]
+al-Khow[=a]razm[=i]. He was born in Khow[=a]rezm, "the lowlands," the
+country about the present Khiva and bordering on the Oxus, and lived at
+Bagdad under the caliph al-M[=a]m[=u]n. He died probably between 220 and
+230 of the Mohammedan era, that is, between 835 and 845 A.D., although some
+put the date as early as 812. The best account of this great scholar may be
+found in an article by C. Nallino, "Al-[H)]uw[=a]rizm[=i]" in the _Atti
+della R. Accad. dei Lincei_, Rome, 1896. See also _Verhandlungen des 5.
+Congresses der Orientalisten_, Berlin, 1882, Vol. II, p. 19; W. Spitta-Bey
+in the _Zeitschrift der deutschen Morgenländ. Gesellschaft_, Vol. XXXIII,
+p. 224; Steinschneider in the _Zeitschrift der deutschen Morgenländ.
+Gesellschaft_, Vol. L, p. 214; Treutlein in the _Abhandlungen zur
+Geschichte der Mathematik_, Vol. I, p. 5; Suter, "Die Mathematiker und
+Astronomen der Araber und ihre Werke," _Abhandlungen zur Geschichte der
+Mathematik_, Vol. X, Leipzig, 1900, p. 10, and "Nachträge," in Vol. XIV, p.
+158; Cantor, _Geschichte der Mathematik_, Vol. I, 3d ed., pp. 712-733 etc.;
+F. Woepcke in _Propagation_, p. 489. So recently has he become known that
+Heilbronner, writing in 1742, merely mentions him as "Ben-Musa, inter
+Arabes celebris Geometra, scripsit de figuris planis & sphericis."
+[_Historia matheseos universæ_, Leipzig, 1742, p. 438.]
+
+In this work most of the Arabic names will be transliterated substantially
+as laid down by Suter in his work _Die Mathematiker_ etc., except where
+this violates English pronunciation. The scheme of pronunciation of
+oriental names is set forth in the preface.
+
+[9] Our word _algebra_ is from the title of one of his works, Al-jabr
+wa'l-muq[=a]balah, Completion and Comparison. The work was translated into
+English by F. Rosen, London, 1831, and treated in _L'Algèbre
+d'al-Kh[=a]rizmi et les méthodes indienne et grecque_, Léon Rodet, Paris,
+1878, extract from the _Journal Asiatique_. For the derivation of the word
+_algebra_, see Cossali, _Scritti Inediti_, pp. 381-383, Rome, 1857;
+Leonardo's _Liber Abbaci_ (1202), p. 410, Rome, 1857; both published by B.
+Boncompagni. "Almuchabala" also was used as a name for algebra.
+
+[10] This learned scholar, teacher of O'Creat who wrote the _Helceph_
+("_Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum_"),
+studied in Toledo, learned Arabic, traveled as far east as Egypt, and
+brought from the Levant numerous manuscripts for study and translation. See
+Henry in the _Abhandlungen zur Geschichte der Mathematik_, Vol. III, p.
+131; Woepcke in _Propagation_, p. 518.
+
+[11] The title is _Algoritmi de numero Indorum_. That he did not make this
+translation is asserted by Eneström in the _Bibliotheca Mathematica_, Vol.
+I (3), p. 520.
+
+[12] Thus he speaks "de numero indorum per .IX. literas," and proceeds:
+"Dixit algoritmi: Cum uidissem yndos constituisse .IX. literas in uniuerso
+numero suo, propter dispositionem suam quam posuerunt, uolui patefacere de
+opera quod fit per eas aliquid quod esset leuius discentibus, si deus
+uoluerit." [Boncompagni, _Trattati d'Aritmetica_,  Rome, 1857.] Discussed
+by F. Woepcke, _Sur l'introduction de l'arithmétique indienne en Occident_,
+Rome, 1859.
+
+[13] Thus in a commentary by `Al[=i] ibn Ab[=i] Bekr ibn al-Jam[=a]l
+al-An[s.][=a]r[=i
+
+[14] See also Woepcke, _Propagation_, p. 505. The origin is discussed at
+much length by G. R. Kaye, "Notes on Indian Mathematics.--Arithmetical
+Notation," _Journ. and Proc. of the Asiatic Soc. of Bengal_, Vol. III,
+1907, p. 489.
+
+[15] _Alberuni's India_, Arabic version, London, 1887; English translation,
+ibid., 1888.
+
+[16] _Chronology of Ancient Nations_, London, 1879. Arabic and English
+versions, by C. E. Sachau.
+
+[17] _India_, Vol. I, chap. xvi.
+
+[18] The Hindu name for the symbols of the decimal place system.
+
+[19] Sachau's English edition of the _Chronology_, p. 64.
+
+[20] _Littérature arabe_, Cl. Huart, Paris, 1902.
+
+[21] Huart, _History of Arabic Literature_, English ed., New York, 1903, p.
+182 seq.
+
+[22] Al-Mas`[=u]d[=i]'s _Meadows of Gold_, translated in part by Aloys
+Sprenger, London, 1841; _Les prairies d'or_, trad. par C. Barbier de
+Meynard et Pavet de Courteille, Vols. I to IX, Paris, 1861-1877.
+
+[23] _Les prairies d'or_, Vol. VIII, p. 289 seq.
+
+[24] _Essays_, Vol. II, p. 428.
+
+[25] Loc. cit., p. 504.
+
+[26] _Matériaux pour servir à l'histoire comparée des sciences
+mathématiques chez les Grecs et les Orientaux_, 2 vols., Paris, 1845-1849,
+pp. 438-439.
+
+[27] He made an exception, however, in favor of the numerals, loc. cit.,
+Vol. II, p. 503.
+
+[28] _Bibliotheca Arabico-Hispana Escurialensis_, Madrid, 1760-1770, pp.
+426-427.
+
+[29] The author, Ibn al-Qif[t.][=i], flourished A.D. 1198 [Colebrooke, loc.
+cit., note Vol. II, p. 510].
+
+[30] "Liber Artis Logisticae à Mohamado Ben Musa _Alkhuarezmita_ exornatus,
+qui ceteros omnes brevitate methodi ac facilitate praestat, Indorum que in
+praeclarissimis inventis ingenium & acumen ostendit." [Casiri, loc. cit.,
+p. 427.]
+
+[31] Maçoudi, _Le livre de l'avertissement et de la révision_. Translation
+by B. Carra de Vaux, Paris, 1896.
+
+[32] Verifying the hypothesis of Woepcke, _Propagation_, that the Sindhind
+included a treatment of arithmetic.
+
+[33] A[h.]med ibn `Abdall[=a]h, Suter, _Die Mathematiker_, etc., p. 12.
+
+[34] _India_, Vol. II, p. 15.
+
+[35] See H. Suter, "Das Mathematiker-Verzeichniss im Fihrist,"
+_Abhandlungen zur Geschichte der Mathematik_, Vol. VI, Leipzig, 1892. For
+further references to early Arabic writers the reader is referred to H.
+Suter, _Die Mathematiker und Astronomen der Araber und ihre Werke_. Also
+"Nachträge und Berichtigungen" to the same (_Abhandlungen_,  Vol. XIV,
+1902, pp. 155-186).
+
+[36] Suter, loc. cit., note 165, pp. 62-63.
+
+[37] "Send Ben Ali,... tùm arithmetica scripta maximè celebrata, quae
+publici juris fecit." [Loc. cit., p. 440.]
+
+[38] _Scritti di Leonardo Pisano_, Vol. I, _Liber Abbaci_ (1857); Vol. II,
+_Scritti_ (1862); published by Baldassarre Boncompagni, Rome. Also _Tre
+Scritti Inediti_, and _Intorno ad Opere di Leonardo Pisano_, Rome, 1854.
+
+[39] "Ubi ex mirabili magisterio in arte per novem figuras indorum
+introductus" etc. In another place, as a heading to a separate division, he
+writes, "De cognitione novem figurarum yndorum" etc. "Novem figure indorum
+he sunt 9 8 7 6 5 4 3 2 1."
+
+[40] See _An Ancient English Algorism_, by David Eugene Smith, in
+_Festschrift Moritz Cantor_, Leipzig, 1909. See also Victor Mortet, "Le
+plus ancien traité francais d'algorisme," _Bibliotheca Mathematica_, Vol.
+IX (3), pp. 55-64.
+
+[41] These are the two opening lines of the _Carmen de Algorismo_ that the
+anonymous author is explaining. They should read as follows:
+
+  Haec algorismus ars praesens dicitur, in qua
+  Talibus Indorum fruimur bis quinque figuris.
+
+What follows is the translation.
+
+[42] Thibaut, _Astronomie, Astrologie und Mathematik_, Strassburg, 1899.
+
+[43] Gustave Schlegel, _Uranographie chinoise ou preuves directes que
+l'astronomie primitive est originaire de la Chine, et qu'elle a été
+empruntée par les anciens peuples occidentaux à la sphère chinoise; ouvrage
+accompagné d'un atlas céleste chinois et grec_, The Hague and Leyden, 1875.
+
+[44] E. W. Hopkins, _The Religions of India_, Boston, 1898, p. 7.
+
+[45] R. C. Dutt, _History of India_, London, 1906.
+
+[46] W. D. Whitney, _Sanskrit Grammar_, 3d ed., Leipzig, 1896.
+
+[47] "Das [=A]pastamba-['S]ulba-S[=u]tra," _Zeitschrift der deutschen
+Morgenländischen Gesellschaft_, Vol. LV, p. 543, and Vol. LVI, p. 327.
+
+[48] _Geschichte der Math._, Vol. I, 2d ed., p. 595.
+
+[49] L. von Schroeder, _Pythagoras und die Inder_, Leipzig, 1884; H. Vogt,
+"Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale
+gekannt?" _Bibliotheca Mathematica_, Vol. VII (3), pp. 6-20; A. Bürk, loc.
+cit.; Max Simon, _Geschichte der Mathematik im Altertum_, Berlin, 1909, pp.
+137-165; three S[=u]tras are translated in part by Thibaut, _Journal of the
+Asiatic Society of Bengal_, 1875, and one appeared in _The Pandit_, 1875;
+Beppo Levi, "Osservazioni e congetture sopra la geometria degli indiani,"
+_Bibliotheca Mathematica_, Vol. IX (3), 1908, pp. 97-105.
+
+[50] Loc. cit.; also _Indiens Literatur und Cultur_, Leipzig, 1887.
+
+[51] It is generally agreed that the name of the river Sindhu, corrupted by
+western peoples to Hindhu, Indos, Indus, is the root of Hindustan and of
+India. Reclus, _Asia_, English ed., Vol. III, p. 14.
+
+[52] See the comments of Oppert, _On the Original Inhabitants of
+Bharatavar[s.]a or India_, London, 1893, p. 1.
+
+[53] A. Hillebrandt, _Alt-Indien_, Breslau, 1899, p. 111. Fragmentary
+records relate that Kh[=a]ravela, king of Kali[.n]ga, learned as a boy
+_lekh[=a]_ (writing), _ga[n.]an[=a]_ (reckoning), and _r[=u]pa_ (arithmetic
+applied to monetary affairs and mensuration), probably in the 5th century
+B.C. [Bühler, _Indische Palaeographie_, Strassburg, 1896, p. 5.]
+
+[54] R. C. Dutt, _A History of Civilization in Ancient India_, London,
+1893, Vol. I, p. 174.
+
+[55] The Buddha. The date of his birth is uncertain. Sir Edwin Arnold put
+it c. 620 B.C.
+
+[56] I.e. 100·10^7.
+
+[57] There is some uncertainty about this limit.
+
+[58] This problem deserves more study than has yet been given it. A
+beginning may be made with Comte Goblet d'Alviella, _Ce que l'Inde doit à
+la Grèce_, Paris, 1897, and H. G. Keene's review, "The Greeks in India," in
+the _Calcutta Review_, Vol. CXIV, 1902, p. 1. See also F. Woepeke,
+_Propagation_, p. 253; G. R. Kaye, loc. cit., p. 475 seq., and "The Source
+of Hindu Mathematics," _Journal of the Royal Asiatic Society_, July, 1910,
+pp. 749-760; G. Thibaut, _Astronomie, Astrologie und Mathematik_, pp. 43-50
+and 76-79. It will be discussed more fully in Chapter VI.
+
+[59] I.e. to 100,000. The lakh is still the common large unit in India,
+like the myriad in ancient Greece and the million in the West.
+
+[60] This again suggests the _Psammites_, or _De harenae numero_ as it is
+called in the 1544 edition of the _Opera_ of Archimedes, a work in which
+the great Syracusan proposes to show to the king "by geometric proofs which
+you can follow, that the numbers which have been named by us ... are
+sufficient to exceed not only the number of a sand-heap as large as the
+whole earth, but one as large as the universe." For a list of early
+editions of this work see D. E. Smith, _Rara Arithmetica_, Boston, 1909, p.
+227.
+
+[61] I.e. the Wise.
+
+[62] Sir Monier Monier-Williams, _Indian Wisdom_, 4th ed., London, 1893,
+pp. 144, 177. See also J. C. Marshman, _Abridgment of the History of
+India_, London, 1893, p. 2.
+
+[63] For a list and for some description of these works see R. C. Dutt, _A
+History of Civilization in Ancient India_, Vol. II, p. 121.
+
+[64] Professor Ramkrishna Gopal Bhandarkar fixes the date as the fifth
+century B.C. ["Consideration of the Date of the Mah[=a]bh[=a]rata," in the
+_Journal of the Bombay Branch of the R. A. Soc._, Bombay, 1873, Vol. X, p.
+2.].
+
+[65] Marshman, loc. cit., p. 2.
+
+[66] A. C. Burnell, _South Indian Palæography_, 2d ed., London, 1878, p. 1,
+seq.
+
+[67] This extensive subject of palpable arithmetic, essentially the history
+of the abacus, deserves to be treated in a work by itself.
+
+[68] The following are the leading sources of information upon this
+subject: G. Bühler, _Indische Palaeographie_, particularly chap. vi; A. C.
+Burnell, _South Indian Palæography_, 2d ed., London, 1878, where tables of
+the various Indian numerals are given in Plate XXIII; E. C. Bayley, "On the
+Genealogy of Modern Numerals," _Journal of the Royal Asiatic Society_, Vol.
+XIV, part 3, and Vol. XV, part 1, and reprint, London, 1882; I. Taylor, in
+_The Academy_, January 28, 1882, with a repetition of his argument in his
+work _The Alphabet_, London, 1883, Vol. II, p. 265, based on Bayley; G. R.
+Kaye, loc. cit., in some respects one of the most critical articles thus
+far published; J. C. Fleet, _Corpus inscriptionum Indicarum_, London, 1888,
+Vol. III, with facsimiles of many Indian inscriptions, and _Indian
+Epigraphy_, Oxford, 1907, reprinted from the _Imperial Gazetteer of India_,
+Vol. II, pp. 1-88, 1907; G. Thibaut, loc. cit., _Astronomie_ etc.; R.
+Caldwell, _Comparative Grammar of the Dravidian Languages_, London, 1856,
+p. 262 seq.; and _Epigraphia Indica_ (official publication of the
+government of India), Vols. I-IX. Another work of Bühler's, _On the Origin
+of the Indian Br[=a]hma Alphabet_, is also of value.
+
+[69] The earliest work on the subject was by James Prinsep, "On the
+Inscriptions of Piyadasi or A['s]oka," etc., _Journal of the Asiatic
+Society of Bengal_, 1838, following a preliminary suggestion in the same
+journal in 1837. See also "A['s]oka Notes," by V. A. Smith, _The Indian
+Antiquary_, Vol. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, pp. 151-159, June,
+1909; _The Early History of India_, 2d ed., Oxford, 1908, p. 154; J. F.
+Fleet, "The Last Words of A['s]oka," _Journal of the Royal Asiatic
+Society_, October, 1909, pp. 981-1016; E. Senart, _Les inscriptions de
+Piyadasi_, 2 vols., Paris, 1887.
+
+[70] For a discussion of the minor details of this system, see Bühler, loc.
+cit., p. 73.
+
+[71] Julius Euting, _Nabatäische Inschriften aus Arabien_, Berlin, 1885,
+pp. 96-97, with a table of numerals.
+
+[72] For the five principal theories see Bühler, loc. cit., p. 10.
+
+[73] Bayley, loc. cit., reprint p. 3.
+
+[74] Bühler, loc. cit.; _Epigraphia Indica_, Vol. III, p. 134; _Indian
+Antiquary_, Vol. VI, p. 155 seq., and Vol. X, p. 107.
+
+[75] Pandit Bhagav[=a]nl[=a]l Indr[=a]j[=i], "On Ancient N[=a]g[=a]ri
+Numeration; from an Inscription at N[=a]negh[=a]t," _Journal of the Bombay
+Branch of the Royal Asiatic Society_, 1876, Vol. XII, p. 404.
+
+[76] Ib., p. 405. He gives also a plate and an interpretation of each
+numeral.
+
+[77] These may be compared with Bühler's drawings, loc. cit.; with Bayley,
+loc. cit., p. 337 and plates; and with Bayley's article in the
+_Encyclopædia Britannica_, 9th ed., art. "Numerals."
+
+[78] E. Senart, "The Inscriptions in the Caves at Nasik," _Epigraphia
+Indica_, Vol. VIII, pp. 59-96; "The Inscriptions in the Cave at Karle,"
+_Epigraphia Indica_, Vol. VII, pp. 47-74; Bühler, _Palaeographie_, Tafel
+IX.
+
+[79] See Fleet, loc. cit. See also T. Benfey, _Sanskrit Grammar_, London,
+1863, p. 217; M. R. Kále, _Higher Sanskrit Grammar_, 2d ed., Bombay, 1898,
+p. 110, and other authorities as cited.
+
+[80] Kharo[s.][t.]h[=i] numerals, A['s]oka inscriptions, c. 250 B.C.
+Senart, _Notes d'épigraphie indienne_. Given by Bühler, loc. cit., Tafel I.
+
+[81] Same, ['S]aka inscriptions, probably of the first century B.C. Senart,
+loc. cit.; Bühler, loc. cit.
+
+[82] Br[=a]hm[=i] numerals, A['s]oka inscriptions, c. 250 B.C. _Indian
+Antiquary_, Vol. VI, p. 155 seq.
+
+[83] Same, N[=a]n[=a] Gh[=a]t inscriptions, c. 150 B.C. Bhagav[=a]nl[=a]l
+Indr[=a]j[=i], _On Ancient N[=a]gar[=i] Numeration_, loc. cit. Copied from
+a squeeze of the original.
+
+[84] Same, Nasik inscription, c. 100 B.C. Burgess, _Archeological Survey
+Report, Western India_; Senart, _Epigraphia Indica_, Vol. VII, pp. 47-79,
+and Vol. VIII, pp. 59-96.
+
+[85] K[s.]atrapa coins, c. 200 A.D. _Journal of the Royal Asiatic Society_,
+1890, p. 639.
+
+[86] Ku[s.]ana inscriptions, c. 150 A.D. _Epigraphia Indica_, Vol. I, p.
+381, and Vol. II, p. 201.
+
+[87] Gupta Inscriptions, c. 300 A.D. to 450 A.D. Fleet, loc. cit., Vol.
+III.
+
+[88] Valhab[=i], c. 600 A.D. _Corpus_, Vol. III.
+
+[89] Bendall's Table of Numerals, in _Cat. Sansk. Budd. MSS._, British
+Museum.
+
+[90] _Indian Antiquary_, Vol. XIII, 120; _Epigraphia Indica_, Vol. III, 127
+ff.
+
+[91] Fleet, loc. cit.
+
+[92] Bayley, loc. cit., p. 335.
+
+[93] From a copper plate of 493 A.D., found at K[=a]r[=i]tal[=a][=i],
+Central India. [Fleet, loc. cit., Plate XVI.] It should be stated, however,
+that many of these copper plates, being deeds of property, have forged
+dates so as to give the appearance of antiquity of title. On the other
+hand, as Colebrooke long ago pointed out, a successful forgery has to
+imitate the writing of the period in question, so that it becomes evidence
+well worth considering, as shown in Chapter III.
+
+[94] From a copper plate of 510 A.D., found at Majhgaw[=a]in, Central
+India. [Fleet, loc. cit., Plate XIV.]
+
+[95] From an inscription of 588 A.D., found at B[=o]dh-Gay[=a], Bengal
+Presidency. [Fleet, loc. cit., Plate XXIV.]
+
+[96] From a copper plate of 571 A.D., found at M[=a]liy[=a], Bombay
+Presidency. [Fleet, loc. cit., Plate XXIV.]
+
+[97] From a Bijayaga[d.]h pillar inscription of 372 A.D. [Fleet, loc. cit.,
+Plate XXXVI, C.]
+
+[98] From a copper plate of 434 A.D. [_Indian Antiquary_, Vol. I, p. 60.]
+
+[99] Gadhwa inscription, c. 417 A.D. [Fleet, loc. cit., Plate IV, D.]
+
+[100] K[=a]r[=i]tal[=a][=i] plate of 493 A.D., referred to above.
+
+[101] It seems evident that the Chinese four, curiously enough called
+"eight in the mouth," is only a cursive [4 vertical strokes].
+
+[102] Chalfont, F. H., _Memoirs of the Carnegie Museum_, Vol. IV, no. 1; J.
+Hager, _An Explanation of the Elementary Characters of the Chinese_,
+London, 1801.
+
+[103] H. V. Hilprecht, _Mathematical, Metrological and Chronological
+Tablets from the Temple Library at Nippur_, Vol. XX, part I, of Series A,
+Cuneiform Texts Published by the Babylonian Expedition of the University of
+Pennsylvania, 1906; A. Eisenlohr, _Ein altbabylonischer Felderplan_,
+Leipzig, 1906; Maspero, _Dawn of Civilization_, p. 773.
+
+[104] Sir H. H. Howard, "On the Earliest Inscriptions from Chaldea,"
+_Proceedings of the Society of Biblical Archæology_, XXI, p. 301, London,
+1899.
+
+[105] For a bibliography of the principal hypotheses of this nature see
+Bühler, loc. cit., p. 77. Bühler (p. 78) feels that of all these hypotheses
+that which connects the Br[=a]hm[=i] with the Egyptian numerals is the most
+plausible, although he does not adduce any convincing proof. Th. Henri
+Martin, "Les signes numéraux et l'arithmétique chez les peuples de
+l'antiquité et du moyen âge" (being an examination of Cantor's
+_Mathematische Beiträge zum Culturleben der Völker_), _Annali di matematica
+pura ed applicata_, Vol. V, Rome, 1864, pp. 8, 70. Also, same author,
+"Recherches nouvelles sur l'origine de notre système de numération écrite,"
+_Revue Archéologique_, 1857, pp. 36, 55. See also the tables given later in
+this work.
+
+[106] _Journal of the Royal Asiatic Society, Bombay Branch_, Vol. XXIII.
+
+[107] Loc. cit., reprint, Part I, pp. 12, 17. Bayley's deductions are
+generally regarded as unwarranted.
+
+[108] _The Alphabet_; London, 1883, Vol. II, pp. 265, 266, and _The
+Academy_ of Jan. 28, 1882.
+
+[109] Taylor, _The Alphabet_, loc. cit., table on p. 266.
+
+[110] Bühler, _On the Origin of the Indian Br[=a]hma Alphabet_, Strassburg,
+1898, footnote, pp. 52, 53.
+
+[111] Albrecht Weber, _History of Indian Literature_, English ed., Boston,
+1878, p. 256: "The Indian figures from 1-9 are abbreviated forms of the
+initial letters of the numerals themselves...: the zero, too, has arisen
+out of the first letter of the word _[s.]unya_ (empty) (it occurs even in
+Piñgala). It is the decimal place value of these figures which gives them
+significance." C. Henry, "Sur l'origine de quelques notations
+mathématiques," _Revue Archéologique_, June and July, 1879, attempts to
+derive the Boethian forms from the initials of Latin words. See also J.
+Prinsep, "Examination of the Inscriptions from Girnar in Gujerat, and
+Dhauli in Cuttach," _Journal of the Asiatic Society of Bengal_, 1838,
+especially Plate XX, p. 348; this was the first work on the subject.
+
+[112] Bühler, _Palaeographie_, p. 75, gives the list, with the list of
+letters (p. 76) corresponding to the number symbols.
+
+[113] For a general discussion of the connection between the numerals and
+the different kinds of alphabets, see the articles by U. Ceretti, "Sulla
+origine delle cifre numerali moderne," _Rivista di fisica, matematica e
+scienze naturali_, Pisa and Pavia, 1909, anno X, numbers 114, 118, 119, and
+120, and continuation in 1910.
+
+[114] This is one of Bühler's hypotheses. See Bayley, loc. cit., reprint p.
+4; a good bibliography of original sources is given in this work, p. 38.
+
+[115] Loc. cit., reprint, part I, pp. 12, 17. See also Burnell, loc. cit.,
+p. 64, and tables in plate XXIII.
+
+[116] This was asserted by G. Hager (_Memoria sulle cifre arabiche_, Milan,
+1813, also published in _Fundgruben des Orients_, Vienna, 1811, and in
+_Bibliothèque Britannique_, Geneva, 1812). See also the recent article by
+Major Charles E. Woodruff, "The Evolution of Modern Numerals from Tally
+Marks," _American Mathematical Monthly_, August-September, 1909.
+Biernatzki, "Die Arithmetik der Chinesen," _Crelle's Journal für die reine
+und angewandte Mathematik_, Vol. LII, 1857, pp. 59-96, also asserts the
+priority of the Chinese claim for a place system and the zero, but upon the
+flimsiest authority. Ch. de Paravey, _Essai sur l'origine unique et
+hiéroglyphique des chiffres et des lettres de tous les peuples_, Paris,
+1826; G. Kleinwächter, "The Origin of the Arabic Numerals," _China Review_,
+Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp. 28-30; Biot, "Note sur la
+connaissance que les Chinois ont eue de la valeur de position des
+chiffres," _Journal Asiatique_, 1839, pp. 497-502. A. Terrien de
+Lacouperie, "The Old Numerals, the Counting-Rods and the Swan-Pan in
+China," _Numismatic Chronicle_, Vol. III (3), pp. 297-340, and Crowder B.
+Moseley, "Numeral Characters: Theory of Origin and Development," _American
+Antiquarian_, Vol. XXII, pp. 279-284, both propose to derive our numerals
+from Chinese characters, in much the same way as is done by Major Woodruff,
+in the article above cited.
+
+[117] The Greeks, probably following the Semitic custom, used nine letters
+of the alphabet for the numerals from 1 to 9, then nine others for 10 to
+90, and further letters to represent 100 to 900. As the ordinary Greek
+alphabet was insufficient, containing only twenty-four letters, an alphabet
+of twenty-seven letters was used.
+
+[118] _Institutiones mathematicae_, 2 vols., Strassburg, 1593-1596, a
+somewhat rare work from which the following quotation is taken:
+
+"_Quis est harum Cyphrarum autor?_
+
+"A quibus hae usitatae syphrarum notae sint inventae: hactenus incertum
+fuit: meo tamen iudicio, quod exiguum esse fateor: a graecis librarijs
+(quorum olim magna fuit copia) literae Graecorum quibus veteres Graeci
+tamquam numerorum notis sunt usi: fuerunt corruptae. vt ex his licet
+videre.
+
+"Graecorum Literae corruptae.
+
+[Illustration]
+
+_"Sed qua ratione graecorum literae ita fuerunt corruptae?_
+
+"Finxerunt has corruptas Graecorum literarum notas: vel abiectione vt in
+nota binarij numeri, vel additione vt in ternarij, vel inuersione vt in
+septenarij, numeri nota, nostrae notae, quibus hodie utimur: ab his sola
+differunt elegantia, vt apparet."
+
+See also Bayer, _Historia regni Graecorum Bactriani_, St. Petersburg, 1788,
+pp. 129-130, quoted by Martin, _Recherches nouvelles_, etc., loc. cit.
+
+[119] P. D. Huet, _Demonstratio evangelica_, Paris, 1769, note to p. 139 on
+p. 647: "Ab Arabibus vel ab Indis inventas esse, non vulgus eruditorum
+modo, sed doctissimi quique ad hanc diem arbitrati sunt. Ego vero falsum id
+esse, merosque esse Graecorum characteres aio; à librariis Graecae linguae
+ignaris interpolatos, et diuturna scribendi consuetudine corruptos. Nam
+primum 1 apex fuit, seu virgula, nota [Greek: monados]. 2, est ipsum [beta]
+extremis suis truncatum. [gamma], si in sinistram partem inclinaveris &
+cauda mutilaveris & sinistrum cornu sinistrorsum flexeris, fiet 3. Res ipsa
+loquitur 4 ipsissimum esse [Delta], cujus crus sinistrum erigitur [Greek:
+kata katheton], & infra basim descendit; basis vero ipsa ultra crus
+producta eminet. Vides quam 5 simile sit [Greek: tôi] [epsilon]; infimo
+tantum semicirculo, qui sinistrorsum patebat, dextrorsum converso. [Greek:
+episêmon bau] quod ita notabatur [digamma], rotundato ventre, pede
+detracto, peperit [Greek: to] 6. Ex [Zeta] basi sua mutilato, ortum est
+[Greek: to] 7. Si [Eta] inflexis introrsum apicibus in rotundiorem &
+commodiorem formam mutaveris, exurget [Greek: to] 8. At 9 ipsissimum est
+[alt theta]."
+
+I. Weidler, _Spicilegium observationum ad historiam notarum numeralium_,
+Wittenberg, 1755, derives them from the Hebrew letters; Dom Augustin
+Calmet, "Recherches sur l'origine des chiffres d'arithmétique," _Mémoires
+pour l'histoire des sciences et des beaux arts_, Trévoux, 1707 (pp.
+1620-1635, with two plates), derives the current symbols from the Romans,
+stating that they are relics of the ancient "Notae Tironianae." These
+"notes" were part of a system of shorthand invented, or at least perfected,
+by Tiro, a slave who was freed by Cicero. L. A. Sedillot, "Sur l'origine de
+nos chiffres," _Atti dell' Accademia pontificia dei nuovi Lincei_, Vol.
+XVIII, 1864-1865, pp. 316-322, derives the Arabic forms from the Roman
+numerals.
+
+[120] Athanasius Kircher, _Arithmologia sive De abditis Numerorum,
+mysterijs qua origo, antiquitas & fabrica Numerorum exponitur_, Rome, 1665.
+
+[121] See Suter, _Die Mathematiker und Astronomen der Araber_, p. 100.
+
+[122] "Et hi numeri sunt numeri Indiani, a Brachmanis Indiae Sapientibus ex
+figura circuli secti inuenti."
+
+[123] V. A. Smith, _The Early History of India_, Oxford, 2d ed., 1908, p.
+333.
+
+[124] C. J. Ball, "An Inscribed Limestone Tablet from Sippara,"
+_Proceedings of the Society of Biblical Archæology_, Vol. XX, p. 25
+(London, 1898). Terrien de Lacouperie states that the Chinese used the
+circle for 10 before the beginning of the Christian era. [_Catalogue of
+Chinese Coins_, London, 1892, p. xl.]
+
+[125] For a purely fanciful derivation from the corresponding number of
+strokes, see W. W. R. Ball, _A Short Account of the History of
+Mathematics_, 1st ed., London, 1888, p. 147; similarly J. B. Reveillaud,
+_Essai sur les chiffres arabes_, Paris, 1883; P. Voizot, "Les chiffres
+arabes et leur origine," _La Nature_, 1899, p. 222; G. Dumesnil, "De la
+forme des chiffres usuels," _Annales de l'université de Grenoble_, 1907,
+Vol. XIX, pp. 657-674, also a note in _Revue Archéologique_, 1890, Vol. XVI
+(3), pp. 342-348; one of the earliest references to a possible derivation
+from points is in a work by Bettino entitled _Apiaria universae
+philosophiae mathematicae in quibus paradoxa et noua machinamenta ad usus
+eximios traducta, et facillimis demonstrationibus confirmata_, Bologna,
+1545, Vol. II, Apiarium XI, p. 5.
+
+[126] _Alphabetum Barmanum_, Romae, MDCCLXXVI, p. 50. The 1 is evidently
+Sanskrit, and the 4, 7, and possibly 9 are from India.
+
+[127] _Alphabetum Grandonico-Malabaricum_, Romae, MDCCLXXII, p. 90. The
+zero is not used, but the symbols for 10, 100, and so on, are joined to the
+units to make the higher numbers.
+
+[128] _Alphabetum Tangutanum_, Romae, MDCCLXXIII, p. 107. In a Tibetan MS.
+in the library of Professor Smith, probably of the eighteenth century,
+substantially these forms are given.
+
+[129] Bayley, loc. cit., plate II. Similar forms to these here shown, and
+numerous other forms found in India, as well as those of other oriental
+countries, are given by A. P. Pihan, _Exposé des signes de numération
+usités chez les peuples orientaux anciens et modernes_, Paris, 1860.
+
+[130] Bühler, loc. cit., p. 80; J. F. Fleet, _Corpus inscriptionum
+Indicarum_, Vol. III, Calcutta, 1888. Lists of such words are given also by
+Al-B[=i]r[=u]n[=i] in his work _India_; by Burnell, loc. cit.; by E.
+Jacquet, "Mode d'expression symbolique des nombres employé par les Indiens,
+les Tibétains et les Javanais," _Journal Asiatique_, Vol. XVI, Paris, 1835.
+
+[131] This date is given by Fleet, loc. cit., Vol. III, p. 73, as the
+earliest epigraphical instance of this usage in India proper.
+
+[132] Weber, _Indische Studien_, Vol. VIII, p. 166 seq.
+
+[133] _Journal of the Royal Asiatic Society_, Vol. I (N.S.), p. 407.
+
+[134] VIII, 20, 21.
+
+[135] Th. H. Martin, _Les signes numéraux_ ..., Rome, 1864; Lassen,
+_Indische Alterthumskunde_, Vol. II, 2d ed., Leipzig and London, 1874, p.
+1153.
+
+[136] But see Burnell, loc. cit., and Thibaut, _Astronomie, Astrologie und
+Mathematik_, p. 71.
+
+[137] A. Barth, "Inscriptions Sanscrites du Cambodge," in the _Notices et
+extraits des Mss. de la Bibliothèque nationale_, Vol. XXVII, Part I, pp.
+1-180, 1885; see also numerous articles in _Journal Asiatique_, by
+Aymonier.
+
+[138] Bühler, loc. cit., p. 82.
+
+[139] Loc. cit., p. 79.
+
+[140] Bühler, loc. cit., p. 83. The Hindu astrologers still use an
+alphabetical system of numerals. [Burnell, loc. cit., p. 79.]
+
+[141] Well could Ramus say, "Quicunq; autem fuerit inventor decem notarum
+laudem magnam meruit."
+
+[142] Al-B[=i]r[=u]n[=i] gives lists.
+
+[143] _Propagation_, loc. cit., p. 443.
+
+[144] See the quotation from _The Light of Asia_ in Chapter II, p. 16.
+
+[145] The nine ciphers were called _a[.n]ka_.
+
+[146] "Zur Geschichte des indischen Ziffernsystems," _Zeitschrift für die
+Kunde des Morgenlandes_, Vol. IV, 1842, pp. 74-83.
+
+[147] It is found in the Bakh[s.][=a]l[=i] MS. of an elementary arithmetic
+which Hoernle placed, at first, about the beginning of our era, but the
+date is much in question. G. Thibaut, loc. cit., places it between 700 and
+900 A.D.; Cantor places the body of the work about the third or fourth
+century A.D., _Geschichte der Mathematik_, Vol. I (3), p. 598.
+
+[148] For the opposite side of the case see G. R. Kaye, "Notes on Indian
+Mathematics, No. 2.--[=A]ryabha[t.]a," _Journ. and Proc. of the Asiatic
+Soc. of Bengal_, Vol. IV, 1908, pp. 111-141.
+
+[149] He used one of the alphabetic systems explained above. This ran up to
+10^{18} and was not difficult, beginning as follows:
+
+[Illustration]
+
+the same letter (_ka_) appearing in the successive consonant forms, _ka_,
+_kha_, _ga_, _gha_, etc. See C. I. Gerhardt, _Über die Entstehung und
+Ausbreitung des dekadischen Zahlensystems_, Programm, p. 17, Salzwedel,
+1853, and _Études historiques sur l'arithmétique de position_, Programm, p.
+24, Berlin, 1856; E. Jacquet, _Mode d'expression symbolique des nombres_,
+loc. cit., p. 97; L. Rodet, "Sur la véritable signification de la notation
+numérique inventée par [=A]ryabhata," _Journal Asiatique_, Vol. XVI (7),
+pp. 440-485. On the two [=A]ryabha[t.]as see Kaye, _Bibl. Math._, Vol. X
+(3), p. 289.
+
+[150] Using _kha_, a synonym of _['s][=u]nya_. [Bayley, loc. cit., p. 22,
+and L. Rodet, _Journal Asiatique_, Vol. XVI (7), p. 443.]
+
+[151] Var[=a]ha-Mihira, _Pañcasiddh[=a]ntik[=a]_, translated by G. Thibaut
+and M. S. Dvived[=i], Benares, 1889; see Bühler, loc. cit., p. 78; Bayley,
+loc. cit., p. 23.
+
+[152] _B[r.]hat Sa[m.]hit[=a]_, translated by Kern, _Journal of the Royal
+Asiatic Society_, 1870-1875.
+
+[153] It is stated by Bühler in a personal letter to Bayley (loc. cit., p.
+65) that there are hundreds of instances of this usage in the _B[r.]hat
+Sa[m.]hit[=a]_. The system was also used in the _Pañcasiddh[=a]ntik[=a]_ as
+early as 505 A.D. [Bühler, _Palaeographie_, p. 80, and Fleet, _Journal of
+the Royal Asiatic Society_, 1910, p. 819.]
+
+[154] Cantor, _Geschichte der Mathematik_, Vol. I (3), p. 608.
+
+[155] Bühler, loc. cit., p. 78.
+
+[156] Bayley, p. 38.
+
+[157] Noviomagus, in his _De numeris libri duo_, Paris, 1539, confesses his
+ignorance as to the origin of the zero, but says: "D. Henricus Grauius, vir
+Graecè & Hebraicè eximè doctus, Hebraicam originem ostendit," adding that
+Valla "Indis Orientalibus gentibus inventionem tribuit."
+
+[158] See _Essays_, Vol. II, pp. 287 and 288.
+
+[159] Vol. XXX, p. 205 seqq.
+
+[160] Loc. cit., p. 284 seqq.
+
+[161] Colebrooke, loc. cit., p. 288.
+
+[162] Loc. cit., p. 78.
+
+[163] Hereafter, unless expressly stated to the contrary, we shall use the
+word "numerals" to mean numerals with place value.
+
+[164] "The Gurjaras of R[=a]jput[=a]na and Kanauj," in _Journal of the
+Royal Asiatic Society_, January and April, 1909.
+
+[165] Vol. IX, 1908, p. 248.
+
+[166] _Epigraphia Indica_, Vol. IX, pp. 193 and 198.
+
+[167] _Epigraphia Indica_, Vol. IX, p. 1.
+
+[168] Loc. cit., p. 71.
+
+[169] Thibaut, p. 71.
+
+[170] "Est autem in aliquibus figurarum istaram apud multos diuersitas.
+Quidam enim septimam hanc figuram representant," etc. [Boncompagni,
+_Trattati_, p. 28.] Eneström has shown that very likely this work is
+incorrectly attributed to Johannes Hispalensis. [_Bibliotheca Mathematica_,
+Vol. IX (3), p. 2.]
+
+[171] _Indische Palaeographie_, Tafel IX.
+
+[172] Edited by Bloomfield and Garbe, Baltimore, 1901, containing
+photographic reproductions of the manuscript.
+
+[173] Bakh[s.][=a]l[=i] MS. See page 43; Hoernle, R., _The Indian
+Antiquary_, Vol. XVII, pp. 33-48, 1 plate; Hoernle, _Verhandlungen des VII.
+Internationalen Orientalisten-Congresses, Arische Section_, Vienna, 1888,
+"On the Baksh[=a]l[=i] Manuscript," pp. 127-147, 3 plates; Bühler, loc.
+cit.
+
+[174] 3, 4, 6, from H. H. Dhruva, "Three Land-Grants from Sankheda,"
+_Epigraphia Indica_, Vol. II, pp. 19-24 with plates; date 595 A.D. 7, 1, 5,
+from Bhandarkar, "Daulatabad Plates," _Epigraphia Indica_, Vol. IX, part V;
+date c. 798 A.D.
+
+[175] 8, 7, 2, from "Buckhala Inscription of Nagabhatta," Bhandarkar,
+_Epigraphia Indica_, Vol. IX, part V; date 815 A.D. 5 from "The Morbi
+Copper-Plate," Bhandarkar, _The Indian Antiquary_, Vol. II, pp. 257-258,
+with plate; date 804 A.D. See Bühler, loc. cit.
+
+[176] 8 from the above Morbi Copper-Plate. 4, 5, 7, 9, and 0, from "Asni
+Inscription of Mahipala," _The Indian Antiquary_, Vol. XVI, pp. 174-175;
+inscription is on red sandstone, date 917 A.D. See Bühler.
+
+[177] 8, 9, 4, from "Rashtrakuta Grant of Amoghavarsha," J. F. Fleet, _The
+Indian Antiquary_, Vol. XII, pp. 263-272; copper-plate grant of date c. 972
+A.D. See Bühler. 7, 3, 5, from "Torkhede Copper-Plate Grant of the Time of
+Govindaraja of Gujerat," Fleet, _Epigraphia Indica_, Vol. III, pp. 53-58.
+See Bühler.
+
+[178] From "A Copper-Plate Grant of King Tritochanapâla Chanlukya of
+L[=a][t.]ade['s]a," H.H. Dhruva, _Indian Antiquary_, Vol. XII, pp. 196-205;
+date 1050 A.D. See Bühler.
+
+[179] Burnell, A. C., _South Indian Palæography_, plate XXIII,
+Telugu-Canarese numerals of the eleventh century. See Bühler.
+
+[180] From a manuscript of the second half of the thirteenth century,
+reproduced in "Della vita e delle opere di Leonardo Pisano," Baldassare
+Boncompagni, Rome, 1852, in _Atti dell' Accademia Pontificia dei nuovi
+Lincei_, anno V.
+
+[181] From a fourteenth-century manuscript, as reproduced in _Della vita_
+etc., Boncompagni, loc. cit.
+
+[182] From a Tibetan MS. in the library of D. E. Smith.
+
+[183] From a Tibetan block-book in the library of D. E. Smith.
+
+[184] ['S][=a]rad[=a] numerals from _The Kashmirian Atharva-Veda,
+reproduced by chromophotography from the manuscript in the University
+Library at Tübingen_, Bloomfield and Garbe, Baltimore, 1901. Somewhat
+similar forms are given under "Numération Cachemirienne," by Pihan,
+_Exposé_ etc., p. 84.
+
+[185] Franz X. Kugler, _Die Babylonische Mondrechnung_, Freiburg i. Br.,
+1900, in the numerous plates at the end of the book; practically all of
+these contain the symbol to which reference is made. Cantor, _Geschichte_,
+Vol. I, p. 31.
+
+[186] F. X. Kugler, _Sternkunde und Sterndienst in Babel_, I. Buch, from
+the beginnings to the time of Christ, Münster i. Westfalen, 1907. It also
+has numerous tables containing the above zero.
+
+[187] From a letter to D. E. Smith, from G. F. Hill of the British Museum.
+See also his monograph "On the Early Use of Arabic Numerals in Europe," in
+_Archæologia_, Vol. LXII (1910), p. 137.
+
+[188] R. Hoernle, "The Baksh[=a]l[=i] Manuscript," _Indian Antiquary_, Vol.
+XVII, pp. 33-48 and 275-279, 1888; Thibaut, _Astronomie, Astrologie und
+Mathematik_, p. 75; Hoernle, _Verhandlungen_, loc. cit., p. 132.
+
+[189] Bayley, loc. cit., Vol. XV, p. 29. Also Bendall, "On a System of
+Numerals used in South India," _Journal of the Royal Asiatic Society_,
+1896, pp. 789-792.
+
+[190] V. A. Smith, _The Early History of India_, 2d ed., Oxford, 1908, p.
+14.
+
+[191] Colebrooke, _Algebra, with Arithmetic and Mensuration, from the
+Sanskrit of Brahmegupta and Bháscara_, London, 1817, pp. 339-340.
+
+[192] Ibid., p. 138.
+
+[193] D. E. Smith, in the _Bibliotheca Mathematica_, Vol. IX (3), pp.
+106-110.
+
+[194] As when we use three dots (...).
+
+[195] "The Hindus call the nought explicitly _['s][=u]nyabindu_ 'the dot
+marking a blank,' and about 500 A.D. they marked it by a simple dot, which
+latter is commonly used in inscriptions and MSS. in order to mark a blank,
+and which was later converted into a small circle." [Bühler, _On the Origin
+of the Indian Alphabet_, p. 53, note.]
+
+[196] Fazzari, _Dell' origine delle parole zero e cifra_, Naples, 1903.
+
+[197] E. Wappler, "Zur Geschichte der Mathematik im 15. Jahrhundert," in
+the _Zeitschrift für Mathematik und Physik_, Vol. XLV, _Hist.-lit. Abt._,
+p. 47. The manuscript is No. C. 80, in the Dresden library.
+
+[198] J. G. Prändel, _Algebra nebst ihrer literarischen Geschichte_, p.
+572, Munich, 1795.
+
+[199] See the table, p. 23. Does the fact that the early European
+arithmetics, following the Arab custom, always put the 0 after the 9,
+suggest that the 0 was derived from the old Hindu symbol for 10?
+
+[200] Bayley, loc. cit., p. 48. From this fact Delambre (_Histoire de
+l'astronomie ancienne_) inferred that Ptolemy knew the zero, a theory
+accepted by Chasles, _Aperçu historique sur l'origine et le développement
+des méthodes en géométrie_, 1875 ed., p. 476; Nesselmann, however, showed
+(_Algebra der Griechen_, 1842, p. 138), that Ptolemy merely used [Greek: o]
+for [Greek: ouden], with no notion of zero. See also G. Fazzari, "Dell'
+origine delle parole zero e cifra," _Ateneo_, Anno I, No. 11, reprinted at
+Naples in 1903, where the use of the point and the small cross for zero is
+also mentioned. Th. H. Martin, _Les signes numéraux_ etc., reprint p. 30,
+and J. Brandis, _Das Münz-, Mass- und Gewichtswesen in Vorderasien bis auf
+Alexander den Grossen_, Berlin, 1866, p. 10, also discuss this usage of
+[Greek: o], without the notion of place value, by the Greeks.
+
+[201] _Al-Batt[=a]n[=i] sive Albatenii opus astronomicum_. Ad fidem codicis
+escurialensis arabice editum, latine versum, adnotationibus instructum a
+Carolo Alphonso Nallino, 1899-1907. Publicazioni del R. Osservatorio di
+Brera in Milano, No. XL.
+
+[202] Loc. cit., Vol. II, p. 271.
+
+[203] C. Henry, "Prologus N. Ocreati in Helceph ad Adelardum Batensem
+magistrum suum," _Abhandlungen zur Geschichte der Mathematik_, Vol. III,
+1880.
+
+[204] Max. Curtze, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts,"
+_Abhandlungen zur Geschichte der Mathematik_, Vol. VIII, 1898, pp. 1-27;
+Alfred Nagl, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und über
+die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im
+christl. Abendlande," _Zeitschrift für Mathematik und Physik, Hist.-lit.
+Abth._, Vol. XXXIV, pp. 129-146 and 161-170, with one plate.
+
+[205] "Byzantinische Analekten," _Abhandlungen zur Geschichte der
+Mathematik_, Vol. IX, pp. 161-189.
+
+[206] [symbol] or [symbol] for 0. [symbol] also used for 5. [symbols] for
+13. [Heiberg, loc. cit.]
+
+[207] Gerhardt, _Études historiques sur l'arithmétique de position_,
+Berlin, 1856, p. 12; J. Bowring, _The Decimal System in Numbers, Coins, &
+Accounts_, London, 1854, p. 33.
+
+[208] Karabacek, _Wiener Zeitschrift für die Kunde des Morgenlandes_, Vol.
+XI, p. 13; _Führer durch die Papyrus-Ausstellung Erzherzog Rainer_, Vienna,
+1894, p. 216.
+
+[209] In the library of G. A. Plimpton, Esq.
+
+[210] Cantor, _Geschichte_, Vol. I (3), p. 674; Y. Mikami, "A Remark on the
+Chinese Mathematics in Cantor's Geschichte der Mathematik," _Archiv der
+Mathematik und Physik_, Vol. XV (3), pp. 68-70.
+
+[211] Of course the earlier historians made innumerable guesses as to the
+origin of the word _cipher_. E.g. Matthew Hostus, _De numeratione
+emendata_, Antwerp, 1582, p. 10, says: "Siphra vox Hebræam originem sapit
+refértque: & ut docti arbitrantur, à verbo saphar, quod Ordine numerauit
+significat. Unde Sephar numerus est: hinc Siphra (vulgo corruptius). Etsi
+verò gens Iudaica his notis, quæ hodie Siphræ vocantur, usa non fuit:
+mansit tamen rei appellatio apud multas gentes." Dasypodius, _Institutiones
+mathematicae_, Vol. I, 1593, gives a large part of this quotation word for
+word, without any mention of the source. Hermannus Hugo, _De prima
+scribendi origine_, Trajecti ad Rhenum, 1738, pp. 304-305, and note, p.
+305; Karl Krumbacher, "Woher stammt das Wort Ziffer (Chiffre)?", _Études de
+philologie néo-grecque_, Paris, 1892.
+
+[212] Bühler, loc. cit., p. 78 and p. 86.
+
+[213] Fazzari, loc. cit., p. 4. So Elia Misrachi (1455-1526) in his
+posthumous _Book of Number_, Constantinople, 1534, explains _sifra_ as
+being Arabic. See also Steinschneider, _Bibliotheca Mathematica_, 1893, p.
+69, and G. Wertheim, _Die Arithmetik des Elia Misrachi_, Programm,
+Frankfurt, 1893.
+
+[214] "Cum his novem figuris, et cum hoc signo 0, quod arabice zephirum
+appellatur, scribitur quilibet numerus."
+
+[215] [Greek: tziphra], a form also used by Neophytos (date unknown,
+probably c. 1330). It is curious that Finaeus (1555 ed., f. 2) used the
+form _tziphra_ throughout. A. J. H. Vincent ["Sur l'origine de nos
+chiffres," _Notices et Extraits des MSS._, Paris, 1847, pp. 143-150] says:
+"Ce cercle fut nommé par les uns, _sipos, rota, galgal_ ...; par les autres
+_tsiphra_ (de [Hebrew: TSPR], _couronne_ ou _diadème_) ou _ciphra_ (de
+[Hebrew: SPR], _numération_)." Ch. de Paravey, _Essai sur l'origine unique
+et hiéroglyphique des chiffres et des lettres de tous les peuples_, Paris,
+1826, p. 165, a rather fanciful work, gives "vase, vase arrondi et fermé
+par un couvercle, qui est le symbole de la 10^e Heure, [symbol]," among the
+Chinese; also "Tsiphron Zéron, ou tout à fait vide en arabe, [Greek:
+tziphra] en grec ... d'où chiffre (qui dérive plutôt, suivant nous, de
+l'Hébreu _Sepher_, compter.")
+
+[216] "Compilatus a Magistro Jacobo de Florentia apud montem pesalanum,"
+and described by G. Lami in his _Catalogus codicum manuscriptorum qui in
+bibliotheca Riccardiana Florentiæ adservantur_. See Fazzari, loc. cit., p.
+5.
+
+[217] "Et doveto sapere chel zeuero per se solo non significa nulla ma è
+potentia di fare significare, ... Et decina o centinaia o migliaia non si
+puote scrivere senza questo segno 0. la quale si chiama zeuero." [Fazzari,
+loc. cit., p. 5.]
+
+[218] Ibid., p. 6.
+
+[219] Avicenna (980-1036), translation by Gasbarri et François, "più il
+punto (gli Arabi adoperavano il punto in vece dello zero il cui segno 0 in
+arabo si chiama _zepiro_ donde il vocabolo zero), che per sè stesso non
+esprime nessun numero." This quotation is taken from D. C. Martines,
+_Origine e progressi dell' aritmetica_, Messina, 1865.
+
+[220] Leo Jordan, "Materialien zur Geschichte der arabischen Zahlzeichen in
+Frankreich," _Archiv für Kulturgeschichte_, Berlin, 1905, pp. 155-195,
+gives the following two schemes of derivation, (1) "zefiro, zeviro, zeiro,
+zero," (2) "zefiro, zefro, zevro, zero."
+
+[221] Köbel (1518 ed., f. A_4) speaks of the numerals in general as "die
+der gemain man Zyfer nendt." Recorde (_Grounde of Artes_, 1558 ed., f. B_6)
+says that the zero is "called priuatly a Cyphar, though all the other
+sometimes be likewise named."
+
+[222] "Decimo X 0 theca, circul[us] cifra sive figura nihili appelat'."
+[_Enchiridion Algorismi_, Cologne, 1501.] Later, "quoniam de integris tam
+in cifris quam in proiectilibus,"--the word _proiectilibus_ referring to
+markers "thrown" and used on an abacus, whence the French _jetons_ and the
+English expression "to _cast_ an account."
+
+[223] "Decima vero o dicitur teca, circulus, vel cyfra vel figura nichili."
+[Maximilian Curtze, _Petri Philomeni de Dacia in Algorismum Vulgarem
+Johannis de Sacrobosco commentarius, una cum Algorismo ipso_, Copenhagen,
+1897, p. 2.] Curtze cites five manuscripts (fourteenth and fifteenth
+centuries) of Dacia's commentary in the libraries at Erfurt, Leipzig, and
+Salzburg, in addition to those given by Eneström, _Öfversigt af Kongl.
+Vetenskaps-Akademiens Förhandlingar_, 1885, pp. 15-27, 65-70; 1886, pp.
+57-60.
+
+[224] Curtze, loc. cit., p. VI.
+
+[225] _Rara Mathematica_, London, 1841, chap, i, "Joannis de Sacro-Bosco
+Tractatus de Arte Numerandi."
+
+[226] Smith, _Rara Arithmetica_, Boston, 1909.
+
+[227] In the 1484 edition, Borghi uses the form "çefiro: ouero nulla:"
+while in the 1488 edition he uses "zefiro: ouero nulla," and in the 1540
+edition, f. 3, appears "Chiamata zero, ouero nulla." Woepcke asserted that
+it first appeared in Calandri (1491) in this sentence: "Sono dieci le
+figure con le quali ciascuno numero si può significare: delle quali n'è una
+che si chiama zero: et per se sola nulla significa." (f. 4). [See
+_Propagation_, p. 522.]
+
+[228] Boncompagni _Bulletino_, Vol. XVI, pp. 673-685.
+
+[229] Leo Jordan, loc. cit. In the _Catalogue of MSS., Bibl. de l'Arsenal_,
+Vol. III, pp. 154-156, this work is No. 2904 (184 S.A.F.), Bibl. Nat., and
+is also called _Petit traicté de algorisme_.
+
+[230] Texada (1546) says that there are "nueue letros yvn zero o cifra" (f.
+3).
+
+[231] Savonne (1563, 1751 ed., f. 1): "Vne ansi formee (o) qui s'appelle
+nulle, & entre marchans zero," showing the influence of Italian names on
+French mercantile customs. Trenchant (Lyons, 1566, 1578 ed., p. 12) also
+says: "La derniere qui s'apele nulle, ou zero;" but Champenois, his
+contemporary, writing in Paris in 1577 (although the work was not published
+until 1578), uses "cipher," the Italian influence showing itself less in
+this center of university culture than in the commercial atmosphere of
+Lyons.
+
+[232] Thus Radulph of Laon (c. 1100): "Inscribitur in ultimo ordine et
+figura [symbol] sipos nomine, quae, licet numerum nullum signitet, tantum
+ad alia quaedam utilis, ut insequentibus declarabitur." ["Der Arithmetische
+Tractat des Radulph von Laon," _Abhandlungen zur Geschichte der
+Mathematik_, Vol. V, p. 97, from a manuscript of the thirteenth century.]
+Chasles (_Comptes rendus_, t. 16, 1843, pp. 1393, 1408) calls attention to
+the fact that Radulph did not know how to use the zero, and he doubts if
+the sipos was really identical with it. Radulph says: "... figuram, cui
+sipos nomen est [symbol] in motum rotulae formatam nullius numeri
+significatione inscribi solere praediximus," and thereafter uses _rotula_.
+He uses the sipos simply as a kind of marker on the abacus.
+
+[233] Rabbi ben Ezra (1092-1168) used both [Hebrew: GLGL], _galgal_ (the
+Hebrew for _wheel_), and [Hebrew: SPR'], _sifra_. See M. Steinschneider,
+"Die Mathematik bei den Juden," in _Bibliotheca Mathematica_, 1893, p. 69,
+and Silberberg, _Das Buch der Zahl des R. Abraham ibn Esra_, Frankfurt a.
+M., 1895, p. 96, note 23; in this work the Hebrew letters are used for
+numerals with place value, having the zero.
+
+[234] E.g., in the twelfth-century _Liber aligorismi_ (see Boncompagni's
+_Trattati_, II, p. 28). So Ramus (_Libri II_, 1569 ed., p. 1) says:
+"Circulus quæ nota est ultima: nil per se significat." (See also the
+Schonerus ed. of Ramus, 1586, p. 1.)
+
+[235] "Und wirt das ringlein o. die Ziffer genant die nichts bedeut."
+[Köbel's _Rechenbuch_, 1549 ed., f. 10, and other editions.]
+
+[236] I.e. "circular figure," our word _notation_ having come from the
+medieval _nota_. Thus Tzwivel (1507, f. 2) says: "Nota autem circularis .o.
+per se sumpta nihil vsus habet. alijs tamen adiuncta earum significantiam
+et auget et ordinem permutat quantum quo ponit ordinem. vt adiuncta note
+binarij hoc modo 20 facit eam significare bis decem etc." Also (ibid., f.
+4), "figura circularis," "circularis nota." Clichtoveus (1503 ed., f.
+XXXVII) calls it "nota aut circularis o," "circularis nota," and "figura
+circularis." Tonstall (1522, f. B_3) says of it: "Decimo uero nota ad
+formam [symbol] litteræ circulari figura est: quam alij circulum, uulgus
+cyphram uocat," and later (f. C_4) speaks of the "circulos." Grammateus, in
+his _Algorismus de integris_ (Erfurt, 1523, f. A_2), speaking of the nine
+significant figures, remarks: "His autem superadditur decima figura
+circularis ut 0 existens que ratione sua nihil significat." Noviomagus (_De
+Numeris libri II_, Paris, 1539, chap. xvi, "De notis numerorum, quas
+zyphras vocant") calls it "circularis nota, quam ex his solam, alij
+sipheram, Georgius Valla zyphram."
+
+[237] Huswirt, as above. Ramus (_Scholae mathematicae_, 1569 ed., p. 112)
+discusses the name interestingly, saying: "Circulum appellamus cum multis,
+quam alii thecam, alii figuram nihili, alii figuram privationis, seu
+figuram nullam vocant, alii ciphram, cùm tamen hodie omnes hæ notæ vulgò
+ciphræ nominentur, & his notis numerare idem sit quod ciphrare." Tartaglia
+(1592 ed., f. 9) says: "si chiama da alcuni tecca, da alcuni circolo, da
+altri cifra, da altri zero, & da alcuni altri nulla."
+
+[238] "Quare autem aliis nominibus vocetur, non dicit auctor, quia omnia
+alia nomina habent rationem suae lineationis sive figurationis. Quia
+rotunda est, dicitur haec figura teca ad similitudinem tecae. Teca enim est
+ferrum figurae rotundae, quod ignitum solet in quibusdam regionibus imprimi
+fronti vel maxillae furis seu latronum." [Loc. cit., p. 26.] But in Greek
+_theca_ ([THEKE], [Greek: thêkê]) is a place to put something, a
+receptacle. If a vacant column, e.g. in the abacus, was so called, the
+initial might have given the early forms [symbol] and [symbol] for the
+zero.
+
+[239] Buteo, _Logistica_, Lyons, 1559. See also Wertheim in the
+_Bibliotheca Mathematica_, 1901, p. 214.
+
+[240] "0 est appellee chiffre ou nulle ou figure de nulle valeur." [La
+Roche, _L'arithmétique_, Lyons, 1520.]
+
+[241] "Decima autem figura nihil uocata," "figura nihili (quam etiam cifram
+uocant)." [Stifel, _Arithmetica integra_, 1544, f. 1.]
+
+[242] "Zifra, & Nulla uel figura Nihili." [Scheubel, 1545, p. 1 of ch. 1.]
+_Nulla_ is also used by Italian writers. Thus Sfortunati (1545 ed., f. 4)
+says: "et la decima nulla & e chiamata questa decima zero;" Cataldi (1602,
+p. 1): "La prima, che è o, si chiama nulla, ouero zero, ouero niente." It
+also found its way into the Dutch arithmetics, e.g. Raets (1576, 1580 ed.,
+f. A_3): "Nullo dat ist niet;" Van der Schuere (1600, 1624 ed., f. 7);
+Wilkens (1669 ed., p. 1). In Germany Johann Albert (Wittenberg, 1534) and
+Rudolff (1526) both adopted the Italian _nulla_ and popularized it. (See
+also Kuckuck, _Die Rechenkunst im sechzehnten Jahrhundert_, Berlin, 1874,
+p. 7; Günther, _Geschichte_, p. 316.)
+
+[243] "La dixième s'appelle chifre vulgairement: les vns l'appellant zero:
+nous la pourrons appeller vn Rien." [Peletier, 1607 ed., p. 14.]
+
+[244] It appears in the Polish arithmetic of K[=l]os (1538) as _cyfra_.
+"The Ciphra 0 augmenteth places, but of himselfe signifieth not," Digges,
+1579, p. 1. Hodder (10th ed., 1672, p. 2) uses only this word (cypher or
+cipher), and the same is true of the first native American arithmetic,
+written by Isaac Greenwood (1729, p. 1). Petrus de Dacia derives _cyfra_
+from circumference. "Vocatur etiam cyfra, quasi circumfacta vel
+circumferenda, quod idem est, quod circulus non habito respectu ad
+centrum." [Loc. cit., p. 26.]
+
+[245] _Opera mathematica_, 1695, Oxford, Vol. I, chap. ix, _Mathesis
+universalis_, "De figuris numeralibus," pp. 46-49; Vol. II, _Algebra_, p.
+10.
+
+[246] Martin, _Origine de notre système de numération écrite_, note 149, p.
+36 of reprint, spells [Greek: tsiphra] from Maximus Planudes, citing Wallis
+as an authority. This is an error, for Wallis gives the correct form as
+above.
+
+Alexander von Humboldt, "Über die bei verschiedenen Völkern üblichen
+Systeme von Zahlzeichen und über den Ursprung des Stellenwerthes in den
+indischen Zahlen," Crelle's _Journal für reine und angewandte Mathematik_,
+Vol. IV, 1829, called attention to the work [Greek: arithmoi Indikoi] of
+the monk Neophytos, supposed to be of the fourteenth century. In this work
+the forms [Greek: tzuphra] and [Greek: tzumphra] appear. See also Boeckh,
+_De abaco Graecorum_, Berlin, 1841, and Tannery, "Le Scholie du moine
+Néophytos," _Revue Archéologique_, 1885, pp. 99-102. Jordan, loc. cit.,
+gives from twelfth and thirteenth century manuscripts the forms _cifra_,
+_ciffre_, _chifras_, and _cifrus_. Du Cange, _Glossarium mediae et infimae
+Latinitatis_, Paris, 1842, gives also _chilerae_. Dasypodius,
+_Institutiones Mathematicae_, Strassburg, 1593-1596, adds the forms
+_zyphra_ and _syphra_. Boissière, _L'art d'arythmetique contenant toute
+dimention, tres-singulier et commode, tant pour l'art militaire que autres
+calculations_, Paris, 1554: "Puis y en a vn autre dict zero lequel ne
+designe nulle quantité par soy, ains seulement les loges vuides."
+
+[247] _Propagation_, pp. 27, 234, 442. Treutlein, "Das Rechnen im 16.
+Jahrhundert," _Abhandlungen zur Geschichte der Mathematik_, Vol. I, p. 5,
+favors the same view. It is combated by many writers, e.g. A. C. Burnell,
+loc. cit., p. 59. Long before Woepcke, I. F. and G. I. Weidler, _De
+characteribus numerorum vulgaribus et eorum aetatibus_, Wittenberg, 1727,
+asserted the possibility of their introduction into Greece by Pythagoras or
+one of his followers: "Potuerunt autem ex oriente, uel ex phoenicia, ad
+graecos traduci, uel Pythagorae, uel eius discipulorum auxilio, cum aliquis
+eo, proficiendi in literis causa, iter faceret, et hoc quoque inuentum
+addisceret."
+
+[248] E.g., they adopted the Greek numerals in use in Damascus and Syria,
+and the Coptic in Egypt. Theophanes (758-818 A.D.), _Chronographia_,
+Scriptores Historiae Byzantinae, Vol. XXXIX, Bonnae, 1839, p. 575, relates
+that in 699 A.D. the caliph Wal[=i]d forbade the use of the Greek language
+in the bookkeeping of the treasury of the caliphate, but permitted the use
+of the Greek alphabetic numerals, since the Arabs had no convenient number
+notation: [Greek: kai ekôluse graphesthai Hellênisti tous dêmosious tôn
+logothesiôn kôdikas, all' Arabiois auta parasêmainesthai, chôris tôn
+psêphôn, epeidê adunaton têi ekeinôn glôssêi monada ê duada ê triada ê oktô
+hêmisu ê tria graphesthai; dio kai heôs sêmeron eisin sun autois notarioi
+Christianoi.] The importance of this contemporaneous document was pointed
+out by Martin, loc. cit. Karabacek, "Die Involutio im arabischen
+Schriftwesen," Vol. CXXXV of _Sitzungsberichte d. phil.-hist. Classe d. k.
+Akad. d. Wiss._, Vienna, 1896, p. 25, gives an Arabic date of 868 A.D. in
+Greek letters.
+
+[249] _The Origin and History of Our Numerals_ (in Russian), Kiev, 1908;
+_The Independence of European Arithmetic_ (in Russian), Kiev.
+
+[250] Woepcke, loc. cit., pp. 462, 262.
+
+[251] Woepcke, loc. cit., p. 240. _[H.]is[=a]b-al-[.G]ob[=a]r_, by an
+anonymous author, probably Ab[=u] Sahl Dunash ibn Tamim, is given by
+Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca Mathematica_,
+1896, p. 26.
+
+[252] Steinschneider in the _Abhandlungen_, Vol. III, p. 110.
+
+[253] See his _Grammaire arabe_, Vol. I, Paris, 1810, plate VIII; Gerhardt,
+_Études_, pp. 9-11, and _Entstehung_ etc., p. 8; I. F. Weidler,
+_Spicilegium observationum ad historiam notarum numeralium pertinentium_,
+Wittenberg, 1755, speaks of the "figura cifrarum Saracenicarum" as being
+different from that of the "characterum Boethianorum," which are similar to
+the "vulgar" or common numerals; see also Humboldt, loc. cit.
+
+[254] Gerhardt mentions it in his _Entstehung_ etc., p. 8; Woepcke,
+_Propagation_, states that these numerals were used not for calculation,
+but very much as we use Roman numerals. These superposed dots are found
+with both forms of numerals (_Propagation_, pp. 244-246).
+
+[255] Gerhardt (_Études_, p. 9) from a manuscript in the Bibliothèque
+Nationale. The numeral forms are [symbols], 20 being indicated by [symbol
+with dot] and 200 by [symbol with 2 dots]. This scheme of zero dots was
+also adopted by the Byzantine Greeks, for a manuscript of Planudes in the
+Bibliothèque Nationale has numbers like [pi alpha with 4 dots] for
+8,100,000,000. See Gerhardt, _Études_, p. 19. Pihan, _Exposé_ etc., p. 208,
+gives two forms, Asiatic and Maghrebian, of "Ghob[=a]r" numerals.
+
+[256] See Chap. IV.
+
+[257] Possibly as early as the third century A.D., but probably of the
+eighth or ninth. See Cantor, I (3), p. 598.
+
+[258] Ascribed by the Arabic writer to India.
+
+[259] See Woepcke's description of a manuscript in the Chasles library,
+"Recherches sur l'histoire des sciences mathématiques chez les orientaux,"
+_Journal Asiatique_, IV (5), 1859, p. 358, note.
+
+[260] P. 56.
+
+[261] Reinaud, _Mémoire sur l'Inde_, p. 399. In the fourteenth century one
+Sih[=a]b al-D[=i]n wrote a work on which, a scholiast to the Bodleian
+manuscript remarks: "The science is called Algobar because the inventor had
+the habit of writing the figures on a tablet covered with sand." [Gerhardt,
+_Études, _p. 11, note.]
+
+[262] Gerhardt, _Entstehung _etc., p. 20.
+
+[263] H. Suter, "Das Rechenbuch des Ab[=u] Zakar[=i]j[=a]
+el-[H.]a[s.][s.][=a]r," _Bibliotheca Mathematica_, Vol. II (3), p. 15.
+
+[264] A. Devoulx, "Les chiffres arabes," _Revue Africaine_, Vol. XVI, pp.
+455-458.
+
+[265] _Kit[=a]b al-Fihrist_, G. Flügel, Leipzig, Vol. I, 1871, and Vol. II,
+1872. This work was published after Professor Flügel's death by J. Roediger
+and A. Mueller. The first volume contains the Arabic text and the second
+volume contains critical notes upon it.
+
+[266] Like those of line 5 in the illustration on page 69.
+
+[267] Woepcke, _Recherches sur l'histoire des sciences mathématiques chez
+les orientaux_, loc. cit.; _Propagation, _p. 57.
+
+[268] Al-[H.]a[s.][s.][=a]r's forms, Suter, _Bibliotheca Mathematica_, Vol.
+II (3), p. 15.
+
+[269] Woepcke, _Sur une donnée historique_, etc., loc. cit. The name
+_[.g]ob[=a]r_ is not used in the text. The manuscript from which these are
+taken is the oldest (970 A.D.) Arabic document known to contain all of the
+numerals.
+
+[270] Silvestre de Sacy, loc. cit. He gives the ordinary modern Arabic
+forms, calling them _Indien_.
+
+[271] Woepcke, "Introduction au calcul Gob[=a]r[=i] et Haw[=a][=i]," _Atti
+dell' accademia pontificia dei nuovi Lincei_, Vol. XIX. The adjective
+applied to the forms in 5 is _gob[=a]r[=i]_ and to those in 6 _indienne_.
+This is the direct opposite of Woepcke's use of these adjectives in the
+_Recherches sur l'histoire_ cited above, in which the ordinary Arabic forms
+(like those in row 5) are called _indiens_.
+
+These forms are usually written from right to left.
+
+[272] J. G. Wilkinson, _The Manners and Customs of the Ancient Egyptians_,
+revised by S. Birch, London, 1878, Vol. II, p. 493, plate XVI.
+
+[273] There is an extensive literature on this "Boethius-Frage." The reader
+who cares to go fully into it should consult the various volumes of the
+_Jahrbuch über die Fortschritte der Mathematik_.
+
+[274] This title was first applied to Roman emperors in posthumous coins of
+Julius Cæsar. Subsequently the emperors assumed it during their own
+lifetimes, thus deifying themselves. See F. Gnecchi, _Monete romane_, 2d
+ed., Milan, 1900, p. 299.
+
+[275] This is the common spelling of the name, although the more correct
+Latin form is Boëtius. See Harper's _Dict. of Class. Lit. and Antiq._, New
+York, 1897, Vol. I, p. 213. There is much uncertainty as to his life. A
+good summary of the evidence is given in the last two editions of the
+_Encyclopædia Britannica_.
+
+[276] His father, Flavius Manlius Boethius, was consul in 487.
+
+[277] There is, however, no good historic evidence of this sojourn in
+Athens.
+
+[278] His arithmetic is dedicated to Symmachus: "Domino suo patricio
+Symmacho Boetius." [Friedlein ed., p. 3.]
+
+[279] It was while here that he wrote _De consolatione philosophiae_.
+
+[280] It is sometimes given as 525.
+
+[281] There was a medieval tradition that he was executed because of a work
+on the Trinity.
+
+[282] Hence the _Divus_ in his name.
+
+[283] Thus Dante, speaking of his burial place in the monastery of St.
+Pietro in Ciel d'Oro, at Pavia, says:
+
+      "The saintly soul, that shows
+  The world's deceitfulness, to all who hear him,
+  Is, with the sight of all the good that is,
+  Blest there. The limbs, whence it was driven, lie
+  Down in Cieldauro; and from martyrdom
+  And exile came it here."--_Paradiso_, Canto X.
+
+[284] Not, however, in the mercantile schools. The arithmetic of Boethius
+would have been about the last book to be thought of in such institutions.
+While referred to by Bæda (672-735) and Hrabanus Maurus (c. 776-856), it
+was only after Gerbert's time that the _Boëtii de institutione arithmetica
+libri duo_ was really a common work.
+
+[285] Also spelled Cassiodorius.
+
+[286] As a matter of fact, Boethius could not have translated any work by
+Pythagoras on music, because there was no such work, but he did make the
+theories of the Pythagoreans known. Neither did he translate Nicomachus,
+although he embodied many of the ideas of the Greek writer in his own
+arithmetic. Gibbon follows Cassiodorus in these statements in his _Decline
+and Fall of the Roman Empire_, chap. xxxix. Martin pointed out with
+positiveness the similarity of the first book of Boethius to the first five
+books of Nicomachus. [_Les signes numéraux_ etc., reprint, p. 4.]
+
+[287] The general idea goes back to Pythagoras, however.
+
+[288] J. C. Scaliger in his _Poëtice_ also said of him: "Boethii Severini
+ingenium, eruditio, ars, sapientia facile provocat omnes auctores, sive
+illi Graeci sint, sive Latini" [Heilbronner, _Hist. math. univ._, p. 387].
+Libri, speaking of the time of Boethius, remarks: "Nous voyons du temps de
+Théodoric, les lettres reprendre une nouvelle vie en Italie, les écoles
+florissantes et les savans honorés. Et certes les ouvrages de Boëce, de
+Cassiodore, de Symmaque, surpassent de beaucoup toutes les productions du
+siècle précédent." [_Histoire des mathématiques_, Vol. I, p. 78.]
+
+[289] Carra de Vaux, _Avicenne_, Paris, 1900; Woepcke, _Sur
+l'introduction_, etc.; Gerhardt, _Entstehung_ etc., p. 20. Avicenna is a
+corruption from Ibn S[=i]n[=a], as pointed out by Wüstenfeld, _Geschichte
+der arabischen Aerzte und Naturforscher_, Göttingen, 1840. His full name is
+Ab[=u] `Al[=i] al-[H.]osein ibn S[=i]n[=a]. For notes on Avicenna's
+arithmetic, see Woepcke, _Propagation_, p. 502.
+
+[290] On the early travel between the East and the West the following works
+may be consulted: A. Hillebrandt, _Alt-Indien_, containing "Chinesische
+Reisende in Indien," Breslau, 1899, p. 179; C. A. Skeel, _Travel in the
+First Century after Christ_, Cambridge, 1901, p. 142; M. Reinaud,
+"Relations politiques et commerciales de l'empire romain avec l'Asie
+orientale," in the _Journal Asiatique_, Mars-Avril, 1863, Vol. I (6), p.
+93; Beazley, _Dawn of Modern Geography, a History of Exploration and
+Geographical Science from the Conversion of the Roman Empire to A.D. 1420_,
+London, 1897-1906, 3 vols.; Heyd, _Geschichte des Levanthandels im
+Mittelalter_, Stuttgart, 1897; J. Keane, _The Evolution of Geography_,
+London, 1899, p. 38; A. Cunningham, _Corpus inscriptionum Indicarum_,
+Calcutta, 1877, Vol. I; A. Neander, _General History of the Christian
+Religion and Church_, 5th American ed., Boston, 1855, Vol. III, p. 89; R.
+C. Dutt, _A History of Civilization in Ancient India_, Vol. II, Bk. V,
+chap, ii; E. C. Bayley, loc. cit., p. 28 et seq.; A. C. Burnell, loc. cit.,
+p. 3; J. E. Tennent, _Ceylon_, London, 1859, Vol. I, p. 159; Geo. Turnour,
+_Epitome of the History of Ceylon_, London, n.d., preface; "Philalethes,"
+_History of Ceylon_, London, 1816, chap, i; H. C. Sirr, _Ceylon and the
+Cingalese_, London, 1850, Vol. I, chap. ix. On the Hindu knowledge of the
+Nile see F. Wilford, _Asiatick Researches_, Vol. III, p. 295, Calcutta,
+1792.
+
+[291] G. Oppert, _On the Ancient Commerce of India_, Madras, 1879, p. 8.
+
+[292] Gerhardt, _Études_ etc., pp. 8, 11.
+
+[293] See Smith's _Dictionary of Greek and Roman Biography and Mythology_.
+
+[294] P. M. Sykes, _Ten Thousand Miles in Persia, or Eight Years in Irán_,
+London, 1902, p. 167. Sykes was the first European to follow the course of
+Alexander's army across eastern Persia.
+
+[295] Bühler, _Indian Br[=a]hma Alphabet_, note, p. 27; _Palaeographie_, p.
+2; _Herodoti Halicarnassei historia_, Amsterdam, 1763, Bk. IV, p. 300;
+Isaac Vossius, _Periplus Scylacis Caryandensis_, 1639. It is doubtful
+whether the work attributed to Scylax was written by him, but in any case
+the work dates back to the fourth century B.C. See Smith's _Dictionary of
+Greek and Roman Biography_.
+
+[296] Herodotus, Bk. III.
+
+[297] Rameses II(?), the _Sesoosis_ of Diodorus Siculus.
+
+[298] _Indian Antiquary_, Vol. I, p. 229; F. B. Jevons, _Manual of Greek
+Antiquities_, London, 1895, p. 386. On the relations, political and
+commercial, between India and Egypt c. 72 B.C., under Ptolemy Auletes, see
+the _Journal Asiatique_, 1863, p. 297.
+
+[299] Sikandar, as the name still remains in northern India.
+
+[300] _Harper's Classical Dict._, New York, 1897, Vol. I, p. 724; F. B.
+Jevons, loc. cit., p. 389; J. C. Marshman, _Abridgment of the History of
+India_, chaps. i and ii.
+
+[301] Oppert, loc. cit., p. 11. It was at or near this place that the first
+great Indian mathematician, [=A]ryabha[t.]a, was born in 476 A.D.
+
+[302] Bühler, _Palaeographie_, p. 2, speaks of Greek coins of a period
+anterior to Alexander, found in northern India. More complete information
+may be found in _Indian Coins_, by E. J. Rapson, Strassburg, 1898, pp. 3-7.
+
+[303] Oppert, loc. cit., p. 14; and to him is due other similar
+information.
+
+[304] J. Beloch, _Griechische Geschichte_, Vol. III, Strassburg, 1904, pp.
+30-31.
+
+[305] E.g., the denarius, the words for hour and minute ([Greek: hôra,
+lepton]), and possibly the signs of the zodiac. [R. Caldwell, _Comparative
+Grammar of the Dravidian Languages_, London, 1856, p. 438.] On the probable
+Chinese origin of the zodiac see Schlegel, loc. cit.
+
+[306] Marie, Vol. II, p. 73; R. Caldwell, loc. cit.
+
+[307] A. Cunningham, loc. cit., p. 50.
+
+[308] C. A. J. Skeel, _Travel_, loc. cit., p. 14.
+
+[309] _Inchiver_, from _inchi_, "the green root." [_Indian Antiquary_, Vol.
+I, p. 352.]
+
+[310] In China dating only from the second century A.D., however.
+
+[311] The Italian _morra_.
+
+[312] J. Bowring, _The Decimal System_, London, 1854, p. 2.
+
+[313] H. A. Giles, lecture at Columbia University, March 12, 1902, on
+"China and Ancient Greece."
+
+[314] Giles, loc. cit.
+
+[315] E.g., the names for grape, radish (_la-po_, [Greek: rhaphê]),
+water-lily (_si-kua_,  "west gourds"; [Greek: sikua], "gourds"), are much
+alike. [Giles, loc. cit.]
+
+[316] _Epistles_, I, 1, 45-46. On the Roman trade routes, see Beazley, loc.
+cit., Vol. I, p. 179.
+
+[317] _Am. Journ. of Archeol._, Vol. IV, p. 366.
+
+[318] M. Perrot gives this conjectural restoration of his words: "Ad me ex
+India regum legationes saepe missi sunt numquam antea visae apud quemquam
+principem Romanorum." [M. Reinaud, "Relations politiques et commerciales de
+l'empire romain avec l'Asie orientale," _Journ. Asiat._, Vol. I (6), p.
+93.]
+
+[319] Reinaud, loc. cit., p. 189. Florus, II, 34 (IV, 12), refers to it:
+"Seres etiam habitantesque sub ipso sole Indi, cum gemmis et margaritis
+elephantes quoque inter munera trahentes nihil magis quam longinquitatem
+viae imputabant." Horace shows his geographical knowledge by saying: "Not
+those who drink of the deep Danube shall now break the Julian edicts; not
+the Getae, not the Seres, nor the perfidious Persians, nor those born on
+the river Tanaïs." [_Odes_, Bk. IV, Ode 15, 21-24.]
+
+[320] "Qua virtutis moderationisque fama Indos etiam ac Scythas auditu modo
+cognitos pellexit ad amicitiam suam populique Romani ultro per legatos
+petendam." [Reinaud, loc. cit., p. 180.]
+
+[321] Reinaud, loc. cit., p. 180.
+
+[322] _Georgics_, II, 170-172. So Propertius (_Elegies_, III, 4):
+
+  Arma deus Caesar dites meditatur ad Indos
+    Et freta gemmiferi findere classe maris.
+
+"The divine Cæsar meditated carrying arms against opulent India, and with
+his ships to cut the gem-bearing seas."
+
+[323] Heyd, loc. cit., Vol. I, p. 4.
+
+[324] Reinaud, loc. cit., p. 393.
+
+[325] The title page of Calandri (1491), for example, represents Pythagoras
+with these numerals before him. [Smith, _Rara Arithmetica_, p. 46.] Isaacus
+Vossius, _Observationes ad Pomponium Melam de situ orbis_, 1658, maintained
+that the Arabs derived these numerals from the west. A learned dissertation
+to this effect, but deriving them from the Romans instead of the Greeks,
+was written by Ginanni in 1753 (_Dissertatio mathematica critica de
+numeralium notarum minuscularum origine_, Venice, 1753). See also Mannert,
+_De numerorum quos arabicos vocant vera origine Pythagorica_, Nürnberg,
+1801. Even as late as 1827 Romagnosi (in his supplement to _Ricerche
+storiche sull' India_ etc., by Robertson, Vol. II, p. 580, 1827) asserted
+that Pythagoras originated them. [R. Bombelli, _L'antica numerazione
+italica_, Rome, 1876, p. 59.] Gow (_Hist. of Greek Math._, p. 98) thinks
+that Iamblichus must have known a similar system in order to have worked
+out certain of his theorems, but this is an unwarranted deduction from the
+passage given.
+
+[326] A. Hillebrandt, _Alt-Indien_, p. 179.
+
+[327] J. C. Marshman, loc. cit., chaps. i and ii.
+
+[328] He reigned 631-579 A.D.; called Nu['s][=i]rw[=a]n, _the holy one_.
+
+[329] J. Keane, _The Evolution of Geography_, London, 1899, p. 38.
+
+[330] The Arabs who lived in and about Mecca.
+
+[331] S. Guyard, in _Encyc. Brit._, 9th ed., Vol. XVI, p. 597.
+
+[332] Oppert, loc. cit., p. 29.
+
+[333] "At non credendum est id in Autographis contigisse, aut vetustioribus
+Codd. MSS." [Wallis, _Opera omnia_, Vol. II, p. 11.]
+
+[334] In _Observationes ad Pomponium Melam de situ orbis_. The question was
+next taken up in a large way by Weidler, loc. cit., _De characteribus_
+etc., 1727, and in _Spicilegium_ etc., 1755.
+
+[335] The best edition of these works is that of G. Friedlein, _Anicii
+Manlii Torquati Severini Boetii de institutione arithmetica libri duo, de
+institutione musica libri quinque. Accedit geometria quae fertur
+Boetii_.... Leipzig.... MDCCCLXVII.
+
+[336] See also P. Tannery, "Notes sur la pseudo-géometrie de Boèce," in
+_Bibliotheca Mathematica_, Vol. I (3), p. 39. This is not the geometry in
+two books in which are mentioned the numerals. There is a manuscript of
+this pseudo-geometry of the ninth century, but the earliest one of the
+other work is of the eleventh century (Tannery), unless the Vatican codex
+is of the tenth century as Friedlein (p. 372) asserts.
+
+[337] Friedlein feels that it is partly spurious, but he says: "Eorum
+librorum, quos Boetius de geometria scripsisse dicitur, investigare veram
+inscriptionem nihil aliud esset nisi operam et tempus perdere." [Preface,
+p. v.] N. Bubnov in the Russian _Journal of the Ministry of Public
+Instruction_, 1907, in an article of which a synopsis is given in the
+_Jahrbuch über die Fortschritte der Mathematik_ for 1907, asserts that the
+geometry was written in the eleventh century.
+
+[338] The most noteworthy of these was for a long time Cantor
+(_Geschichte_, Vol. I., 3d ed., pp. 587-588), who in his earlier days even
+believed that Pythagoras had known them. Cantor says (_Die römischen
+Agrimensoren_, Leipzig, 1875, p. 130): "Uns also, wir wiederholen es, ist
+die Geometrie des Boetius echt, dieselbe Schrift, welche er nach Euklid
+bearbeitete, von welcher ein Codex bereits in Jahre 821 im  Kloster
+Reichenau vorhanden war, von welcher ein anderes Exemplar im Jahre 982 zu
+Mantua in die Hände Gerbert's gelangte, von welcher mannigfache
+Handschriften noch heute vorhanden sind." But against this opinion of the
+antiquity of MSS. containing these numerals is the important statement of
+P. Tannery, perhaps the most critical of modern historians of mathematics,
+that none exists earlier than the eleventh century. See also J. L. Heiberg
+in _Philologus, Zeitschrift f. d. klass. Altertum_, Vol. XLIII, p. 508.
+
+Of Cantor's predecessors, Th. H. Martin was one of the most prominent, his
+argument for authenticity appearing in the _Revue Archéologique_  for
+1856-1857, and in his treatise _Les signes numéraux_ etc. See also M.
+Chasles, "De la connaissance qu'ont eu les anciens d'une numération
+décimale écrite qui fait usage de neuf chiffres prenant les valeurs de
+position," _Comptes rendus_, Vol. VI, pp. 678-680; "Sur l'origine de notre
+système de numération," _Comptes rendus_, Vol. VIII, pp. 72-81; and note
+"Sur le passage du premier livre de la géométrie de Boèce, relatif à un
+nouveau système de numération," in his work _Aperçu historique sur
+l'origine et le devéloppement des méthodes en géométrie_, of which the
+first edition appeared in 1837.
+
+[339] J. L. Heiberg places the book in the eleventh century on philological
+grounds, _Philologus_, loc. cit.; Woepcke, in _Propagation_, p. 44; Blume,
+Lachmann, and Rudorff, _Die Schriften der römischen Feldmesser_, Berlin,
+1848; Boeckh, _De abaco graecorum_, Berlin, 1841; Friedlein, in his Leipzig
+edition of 1867; Weissenborn, _Abhandlungen_, Vol. II, p. 185, his
+_Gerbert_, pp. 1, 247, and his _Geschichte der Einführung der jetzigen
+Ziffern in Europa durch Gerbert_, Berlin, 1892, p. 11; Bayley, loc. cit.,
+p. 59; Gerhardt, _Études_, p. 17, _Entstehung und Ausbreitung_, p. 14;
+Nagl, _Gerbert_, p. 57; Bubnov, loc. cit. See also the discussion by
+Chasles, Halliwell, and Libri, in the _Comptes rendus_, 1839, Vol. IX, p.
+447, and in Vols. VIII, XVI, XVII of the same journal.
+
+[340] J. Marquardt, _La vie privée des Romains_, Vol. II (French trans.),
+p. 505, Paris, 1893.
+
+[341] In a Plimpton manuscript of the arithmetic of Boethius of the
+thirteenth century, for example, the Roman numerals are all replaced by the
+Arabic, and the same is true in the first printed edition of the book. (See
+Smith's _Rara Arithmetica_, pp. 434, 25-27.) D. E. Smith also copied from a
+manuscript of the arithmetic in the Laurentian library at Florence, of
+1370, the following forms, [Forged numerals
+
+[342] Halliwell, in his _Rara Mathematica, _p. 107, states that the
+disputed passage is not in a manuscript belonging to Mr. Ames, nor in one
+at Trinity College. See also Woepcke, in _Propagation_, pp. 37 and 42. It
+was the evident corruption of the texts in such editions of Boethius as
+those of Venice, 1499, Basel, 1546 and 1570, that led Woepcke to publish
+his work _Sur l'introduction de l'arithmétique indienne en Occident_.
+
+[343] They are found in none of the very ancient manuscripts, as, for
+example, in the ninth-century (?) codex in the Laurentian library which one
+of the authors has examined. It should be said, however, that the disputed
+passage was written after the arithmetic, for it contains a reference to
+that work. See the Friedlein ed., p. 397.
+
+[344] Smith, _Rara Arithmetica_, p. 66.
+
+[345] J. L. Heiberg, _Philologus_, Vol. XLIII, p. 507.
+
+[346] "Nosse autem huius artis dispicientem, quid sint digiti, quid
+articuli, quid compositi, quid incompositi numeri." [Friedlein ed., p.
+395.]
+
+[347] _De ratione abaci._ In this he describes "quandam formulam, quam ob
+honorem sui praeceptoris mensam Pythagoream nominabant ... a posterioribus
+appellabatur abacus." This, as pictured in the text, is the common Gerbert
+abacus. In the edition in Migne's _Patrologia Latina_, Vol. LXIII, an
+ordinary multiplication table (sometimes called Pythagorean abacus) is
+given in the illustration.
+
+[348] "Habebant enim diverse formatos apices vel caracteres." See the
+reference to Gerbert on p. 117.
+
+[349] C. Henry, "Sur l'origine de quelques notations mathématiques," _Revue
+Archéologique_, 1879, derives these from the initial letters used as
+abbreviations for the names of the numerals, a theory that finds few
+supporters.
+
+[350] E.g., it appears in Schonerus, _Algorithmus Demonstratus_, Nürnberg,
+1534, f. A4. In England it appeared in the earliest English arithmetical
+manuscript known, _The Crafte of Nombrynge_: "¶ fforthermore ye most
+vndirstonde that in this craft ben vsid teen figurys, as here bene writen
+for ensampul, [Numerals] ... in the quych we vse teen figurys of Inde.
+Questio. ¶ why ten fyguris of Inde? Solucio. for as I have sayd afore thei
+were fonde fyrst in Inde of a kynge of that Cuntre, that was called Algor."
+See Smith, _An Early English Algorism_, loc. cit.
+
+[351] Friedlein ed., p. 397.
+
+[352] Carlsruhe codex of Gerlando.
+
+[353] Munich codex of Gerlando.
+
+[354] Carlsruhe codex of Bernelinus.
+
+[355] Munich codex of Bernelinus.
+
+[356] Turchill, c. 1200.
+
+[357] Anon. MS., thirteenth century, Alexandrian Library, Rome.
+
+[358] Twelfth-century Boethius, Friedlein, p. 396.
+
+[359] Vatican codex, tenth century, Boethius.
+
+[360] a, h, i, are from the Friedlein ed.; the original in the manuscript
+from which a is taken contains a zero symbol, as do all of the six plates
+given by Friedlein. b-e from the Boncompagni _Bulletino_, Vol. X, p. 596; f
+ibid., Vol. XV, p. 186; g _Memorie della classe di sci., Reale Acc. dei
+Lincei_, An. CCLXXIV (1876-1877), April, 1877. A twelfth-century
+arithmetician, possibly John of Luna (Hispalensis, of Seville, c. 1150),
+speaks of the great diversity of these forms even in his day, saying: "Est
+autem in aliquibus figuram istarum apud multos diuersitas. Quidam enim
+septimam hanc figuram representant [Symbol] alii autem sic [Symbol], uel
+sic [Symbol]. Quidam vero quartam sic [Symbol]." [Boncompagni, _Trattati_,
+Vol. II, p. 28.]
+
+[361] Loc. cit., p. 59.
+
+[362] Ibid., p. 101.
+
+[363] Loc. cit., p. 396.
+
+[364] Khosr[=u] I, who began to reign in 531 A.D. See W. S. W Vaux,
+_Persia, _London, 1875, p. 169; Th. Nöldeke, _Aufsätze zur persichen
+Geschichte_, Leipzig, 1887, p. 113, and his article in the ninth edition of
+the _Encyclopædia Britannica_.
+
+[365] Colebrooke, _Essays_, Vol. II, p. 504, on the authority of Ibn
+al-Adam[=i], astronomer, in a work published by his continuator Al-Q[=a]sim
+in 920 A.D.; Al-B[=i]r[=u]n[=i], _India, _Vol. II, p. 15.
+
+[366] H. Suter, _Die Mathematiker_ etc., pp. 4-5, states that
+Al-Faz[=a]r[=i] died between 796 and 806.
+
+[367] Suter, loc. cit., p. 63.
+
+[368] Suter, loc. cit., p. 74.
+
+[369] Suter, _Das Mathematiker-Verzeichniss im Fihrist_. The references to
+Suter, unless otherwise stated, are to his later work _Die Mathematiker und
+Astronomen der Araber_ etc.
+
+[370] Suter, _Fihrist_, p. 37, no date.
+
+[371] Suter, _Fihrist_, p. 38, no date.
+
+[372] Possibly late tenth, since he refers to one arithmetical work which
+is entitled _Book of the Cyphers_ in his _Chronology_, English ed., p. 132.
+Suter, _Die Mathematiker_ etc., pp. 98-100, does not mention this work; see
+the _Nachträge und Berichtigungen_, pp. 170-172.
+
+[373] Suter, pp. 96-97.
+
+[374] Suter, p. 111.
+
+[375] Suter, p. 124. As the name shows, he came from the West.
+
+[376] Suter, p. 138.
+
+[377] Hankel, _Zur Geschichte der Mathematik_, p. 256, refers to him as
+writing on the Hindu art of reckoning; Suter, p. 162.
+
+[378] [Greek: Psêphophoria kat' Indous], Greek ed., C. I. Gerhardt, Halle,
+1865; and German translation, _Das Rechenbuch des Maximus Planudes_, H.
+Wäschke, Halle, 1878.
+
+[379] "Sur une donnée historique relative à l'emploi des chiffres indiens
+par les Arabes," Tortolini's _Annali di scienze mat. e fis._, 1855.
+
+[380] Suter, p. 80.
+
+[381] Suter, p. 68.
+
+[382] Sprenger also calls attention to this fact, in the _Zeitschrift d.
+deutschen morgenländ. Gesellschaft_, Vol. XLV, p. 367.
+
+[383] Libri, _Histoire des mathématiques_, Vol. I, p. 147.
+
+[384] "Dictant la paix à l'empereur de Constantinople, l'Arabe victorieux
+demandait des manuscrits et des savans." [Libri, loc. cit., p. 108.]
+
+[385] Persian _bagadata_, "God-given."
+
+[386] One of the Abbassides, the (at least pretended) descendants of
+`Al-Abb[=a]s, uncle and adviser of Mo[h.]ammed.
+
+[387] E. Reclus, _Asia_, American ed., N. Y., 1891, Vol. IV, p. 227.
+
+[388] _Historical Sketches_, Vol. III, chap. iii.
+
+[389] On its prominence at that period see Villicus, p. 70.
+
+[390] See pp. 4-5.
+
+[391] Smith, D. E., in the _Cantor Festschrift_, 1909, note pp. 10-11. See
+also F. Woepcke, _Propagation_.
+
+[392] Eneström, in _Bibliotheca Mathematica_, Vol. I (3), p. 499; Cantor,
+_Geschichte_, Vol. I (3), p. 671.
+
+[393] Cited in Chapter I. It begins: "Dixit algoritmi: laudes deo rectori
+nostro atque defensori dicamus dignas." It is devoted entirely to the
+fundamental operations and contains no applications.
+
+[394] M. Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca
+Mathematica_, Vol. VIII (2), p. 99. See also the reference to this writer
+in Chapter I.
+
+[395] Part of this work has been translated from a Leyden MS. by F.
+Woepcke, _Propagation_, and more recently by H. Suter, _Bibliotheca
+Mathematica_, Vol. VII (3), pp. 113-119.
+
+[396] A. Neander, _General History of the Christian Religion and Church_,
+5th American ed., Boston, 1855, Vol. III, p. 335.
+
+[397] Beazley, loc. cit., Vol. I, p. 49.
+
+[398] Beazley, loc. cit., Vol. I, pp. 50, 460.
+
+[399] See pp. 7-8.
+
+[400] The name also appears as Mo[h.]ammed Ab[=u]'l-Q[=a]sim, and Ibn
+Hauqal. Beazley, loc. cit., Vol. I, p. 45.
+
+[401] _Kit[=a]b al-mas[=a]lik wa'l-mam[=a]lik._
+
+[402] Reinaud, _Mém. sur l'Inde_; in Gerhardt, _Études_, p. 18.
+
+[403] Born at Shiraz in 1193. He himself had traveled from India to Europe.
+
+[404] _Gulistan_ (_Rose Garden_), Gateway the third, XXII. Sir Edwin
+Arnold's translation, N. Y., 1899, p. 177.
+
+[405] Cunningham, loc. cit., p. 81.
+
+[406] Putnam, _Books_, Vol. I, p. 227:
+
+ "Non semel externas peregrino tramite terras
+  Jam peragravit ovans, sophiae deductus amore,
+  Si quid forte novi librorum seu studiorum
+  Quod secum ferret, terris reperiret in illis.
+  Hic quoque Romuleum venit devotus ad urbem."
+
+("More than once he has traveled joyfully through remote regions and by
+strange roads, led on by his zeal for knowledge and seeking to discover in
+foreign lands novelties in books or in studies which he could take back
+with him. And this zealous student journeyed to the city of Romulus.")
+
+[407] A. Neander, _General History of the Christian Religion and Church_,
+5th American ed., Boston, 1855, Vol. III, p. 89, note 4; Libri, _Histoire_,
+Vol. I, p. 143.
+
+[408] Cunningham, loc. cit., p. 81.
+
+[409] Heyd, loc. cit., Vol. I, p. 4.
+
+[410] Ibid., p. 5.
+
+[411] Ibid., p. 21.
+
+[412] Ibid., p. 23.
+
+[413] Libri, _Histoire_, Vol. I, p. 167.
+
+[414] Picavet, _Gerbert, un pape philosophe, d'après l'histoire et d'après
+la légende_, Paris, 1897, p. 19.
+
+[415] Beazley, loc. cit., Vol. I, chap, i, and p. 54 seq.
+
+[416] Ibid., p. 57.
+
+[417] Libri, _Histoire_, Vol. I, p. 110, n., citing authorities, and p.
+152.
+
+[418] Possibly the old tradition, "Prima dedit nautis usum magnetis
+Amalphis," is true so far as it means the modern form of compass card. See
+Beazley, loc. cit., Vol. II, p. 398.
+
+[419] R. C. Dutt, loc. cit., Vol. II, p. 312.
+
+[420] E. J. Payne, in _The Cambridge Modern History_, London, 1902, Vol. I,
+chap. i.
+
+[421] Geo. Phillips, "The Identity of Marco Polo's Zaitun with Changchau,
+in T'oung pao," _Archives pour servir à l'étude de l'histoire de l'Asie
+orientale_, Leyden, 1890, Vol. I, p. 218. W. Heyd, _Geschichte des
+Levanthandels im Mittelalter_, Vol. II, p. 216.
+
+The Palazzo dei Poli, where Marco was born and died, still stands in the
+Corte del Milione, in Venice. The best description of the Polo travels, and
+of other travels of the later Middle Ages, is found in C. R. Beazley's
+_Dawn of Modern Geography_, Vol. III, chap, ii, and Part II.
+
+[422] Heyd, loc. cit., Vol. II, p. 220; H. Yule, in _Encyclopædia
+Britannica_, 9th (10th) or 11th ed., article "China." The handbook cited is
+Pegolotti's _Libro di divisamenti di paesi_, chapters i-ii, where it is
+implied that $60,000 would be a likely amount for a merchant going to China
+to invest in his trip.
+
+[423] Cunningham, loc. cit., p. 194.
+
+[424] I.e. a commission house.
+
+[425] Cunningham, loc. cit., p. 186.
+
+[426] J. R. Green, _Short History of the English People_, New York, 1890,
+p. 66.
+
+[427] W. Besant, _London_, New York, 1892, p. 43.
+
+[428] _Baldakin_, _baldekin_, _baldachino_.
+
+[429] Italian _Baldacco_.
+
+[430] J. K. Mumford, _Oriental Rugs_, New York, 1901, p. 18.
+
+[431] Or Girbert, the Latin forms _Gerbertus_ and _Girbertus_ appearing
+indifferently in the documents of his time.
+
+[432] See, for example, J. C. Heilbronner, _Historia matheseos universæ_,
+p. 740.
+
+[433] "Obscuro loco natum," as an old chronicle of Aurillac has it.
+
+[434] N. Bubnov, _Gerberti postea Silvestri II papae opera mathematica_,
+Berlin, 1899, is the most complete and reliable source of information;
+Picavet, loc. cit., _Gerbert_ etc.; Olleris, _Oeuvres de Gerbert_, Paris,
+1867; Havet, _Lettres de Gerbert_, Paris, 1889 ; H. Weissenborn, _Gerbert;
+Beiträge zur Kenntnis der Mathematik des Mittelalters_, Berlin, 1888, and
+_Zur Geschichte der Einführung der jetzigen Ziffern in Europa durch
+Gerbert_, Berlin, 1892; Büdinger, _Ueber Gerberts wissenschaftliche und
+politische Stellung_, Cassel, 1851; Richer, "Historiarum liber III," in
+Bubnov, loc. cit., pp. 376-381; Nagl, _Gerbert und die Rechenkunst des 10.
+Jahrhunderts_, Vienna, 1888.
+
+[435] Richer tells of the visit to Aurillac by Borel, a Spanish nobleman,
+just as Gerbert was entering into young manhood. He relates how
+affectionately the abbot received him, asking if there were men in Spain
+well versed in the arts. Upon Borel's reply in the affirmative, the abbot
+asked that one of his young men might accompany him upon his return, that
+he might carry on his studies there.
+
+[436] Vicus Ausona. Hatto also appears as Atton and Hatton.
+
+[437] This is all that we know of his sojourn in Spain, and this comes from
+his pupil Richer. The stories told by Adhemar of Chabanois, an apparently
+ignorant and certainly untrustworthy contemporary, of his going to Cordova,
+are unsupported. (See e.g. Picavet, p. 34.) Nevertheless this testimony is
+still accepted: K. von Raumer, for example (_Geschichte der Pädagogik_, 6th
+ed., 1890, Vol. I, p. 6), says "Mathematik studierte man im Mittelalter bei
+den Arabern in Spanien. Zu ihnen gieng Gerbert, nachmaliger Pabst Sylvester
+II."
+
+[438] Thus in a letter to Aldaberon he says: "Quos post repperimus
+speretis, id est VIII volumina Boeti de astrologia, praeclarissima quoque
+figurarum geometriæ, aliaque non minus admiranda" (Epist. 8). Also in a
+letter to Rainard (Epist. 130), he says: "Ex tuis sumptibus fac ut michi
+scribantur M. Manlius (Manilius in one MS.) de astrologia."
+
+[439] Picavet, loc. cit., p. 31.
+
+[440] Picavet, loc. cit., p. 36.
+
+[441] Havet, loc. cit., p. vii.
+
+[442] Picavet, loc. cit., p. 37.
+
+[443] "Con sinistre arti conseguri la dignita del Pontificato.... Lasciato
+poi l' abito, e 'l monasterio, e datosi tutto in potere del diavolo."
+[Quoted in Bombelli, _L'antica numerazione Italica_, Rome, 1876, p. 41 n.]
+
+[444] He writes from Rheims in 984 to one Lupitus, in Barcelona, saying:
+"Itaque librum de astrologia translatum a te michi petenti dirige,"
+presumably referring to some Arabic treatise. [Epist. no. 24 of the Havet
+collection, p. 19.]
+
+[445] See Bubnov, loc. cit., p. x.
+
+[446] Olleris, loc. cit., p. 361, l. 15, for Bernelinus; and Bubnov, loc.
+cit., p. 381, l. 4, for Richer.
+
+[447] Woepcke found this in a Paris MS. of Radulph of Laon, c. 1100.
+[_Propagation_, p. 246.] "Et prima quidem trium spaciorum superductio
+unitatis caractere inscribitur, qui chaldeo nomine dicitur igin." See also
+Alfred Nagl, "Der arithmetische Tractat des Radulph von Laon"
+(_Abhandlungen zur Geschichte der Mathematik_, Vol. V, pp. 85-133), p. 97.
+
+[448] Weissenborn, loc. cit., p. 239. When Olleris (_Oeuvres de Gerbert_,
+Paris, 1867, p. cci) says, "C'est à lui et non point aux Arabes, que
+l'Europe doit son système et ses signes de numération," he exaggerates,
+since the evidence is all against his knowing the place value. Friedlein
+emphasizes this in the _Zeitschrift für Mathematik und Physik_, Vol. XII
+(1867), _Literaturzeitung_, p. 70: "Für das _System_ unserer Numeration ist
+die _Null_ das wesentlichste Merkmal, und diese kannte Gerbert nicht. Er
+selbst schrieb alle Zahlen mit den römischen Zahlzeichen und man kann ihm
+also nicht verdanken, was er selbst nicht kannte."
+
+[449] E.g., Chasles, Büdinger, Gerhardt, and Richer. So Martin (_Recherches
+nouvelles_ etc.) believes that Gerbert received them from Boethius or his
+followers. See Woepcke, _Propagation_, p. 41.
+
+[450] Büdinger, loc. cit., p. 10. Nevertheless, in Gerbert's time one
+Al-Man[s.][=u]r, governing Spain under the name of Hish[=a]m (976-1002),
+called from the Orient Al-Be[.g][=a]n[=i] to teach his son, so that
+scholars were recognized. [Picavet, p. 36.]
+
+[451] Weissenborn, loc. cit., p. 235.
+
+[452] Ibid., p. 234.
+
+[453] These letters, of the period 983-997, were edited by Havet, loc.
+cit., and, less completely, by Olleris, loc. cit. Those touching
+mathematical topics were edited by Bubnov, loc. cit., pp. 98-106.
+
+[454] He published it in the _Monumenta Germaniae historica_, "Scriptores,"
+Vol. III, and at least three other editions have since appeared, viz. those
+by Guadet in 1845, by Poinsignon in 1855, and by Waitz in 1877.
+
+[455] Domino ac beatissimo Patri Gerberto, Remorum archiepiscopo, Richerus
+Monchus, Gallorum congressibus in volumine regerendis, imperii tui, pater
+sanctissime Gerberte, auctoritas seminarium dedit.
+
+[456] In epistle 17 (Havet collection) he speaks of the "De multiplicatione
+et divisione numerorum libellum a Joseph Ispano editum abbas Warnerius" (a
+person otherwise unknown). In epistle 25 he says: "De multiplicatione et
+divisione numerorum, Joseph Sapiens sententias quasdam edidit."
+
+[457] H. Suter, "Zur Frage über den Josephus Sapiens," _Bibliotheca
+Mathematica_, Vol. VIII (2), p. 84; Weissenborn, _Einführung_, p. 14; also
+his _Gerbert_; M. Steinschneider, in _Bibliotheca Mathematica_, 1893, p.
+68. Wallis (_Algebra_, 1685, chap. 14) went over the list of Spanish
+Josephs very carefully, but could find nothing save that "Josephus Hispanus
+seu Josephus sapiens videtur aut Maurus fuisse aut alius quis in Hispania."
+
+[458] P. Ewald, _Mittheilungen, Neues Archiv d. Gesellschaft für ältere
+deutsche Geschichtskunde_, Vol. VIII, 1883, pp. 354-364. One of the
+manuscripts is of 976 A.D. and the other of 992 A.D. See also Franz
+Steffens, _Lateinische Paläographie_, Freiburg (Schweiz), 1903, pp.
+xxxix-xl. The forms are reproduced in the plate on page 140.
+
+[459] It is entitled _Constantino suo Gerbertus scolasticus_, because it
+was addressed to Constantine, a monk of the Abbey of Fleury. The text of
+the letter to Constantine, preceding the treatise on the Abacus, is given
+in the _Comptes rendus_, Vol. XVI (1843), p. 295. This book seems to have
+been written c. 980 A.D. [Bubnov, loc. cit., p. 6.]
+
+[460] "Histoire de l'Arithmétique," _Comptes rendus_, Vol. XVI (1843), pp.
+156, 281.
+
+[461] Loc. cit., _Gerberti Opera_ etc.
+
+[462] Friedlein thought it spurious. See _Zeitschrift für Mathematik und
+Physik_, Vol. XII (1867), Hist.-lit. suppl., p. 74. It was discovered in
+the library of the Benedictine monastry of St. Peter, at Salzburg, and was
+published by Peter Bernhard Pez in 1721. Doubt was first cast upon it in
+the Olleris edition (_Oeuvres de Gerbert_). See Weissenborn, _Gerbert_, pp.
+2, 6, 168, and Picavet, p. 81. Hock, Cantor, and Th. Martin place the
+composition of the work at c. 996 when Gerbert was in Germany, while
+Olleris and Picavet refer it to the period when he was at Rheims.
+
+[463] Picavet, loc. cit., p. 182.
+
+[464] Who wrote after Gerbert became pope, for he uses, in his preface, the
+words, "a domino pape Gerberto." He was quite certainly not later than the
+eleventh century; we do not have exact information about the time in which
+he lived.
+
+[465] Picavet, loc. cit., p. 182. Weissenborn, _Gerbert_, p. 227. In
+Olleris, _Liber Abaci_ (of Bernelinus), p. 361.
+
+[466] Richer, in Bubnov, loc. cit., p. 381.
+
+[467] Weissenborn, _Gerbert_, p. 241.
+
+[468] Writers on numismatics are quite uncertain as to their use. See F.
+Gnecchi, _Monete Romane_, 2d ed., Milan, 1900, cap. XXXVII. For pictures of
+old Greek tesserae of Sarmatia, see S. Ambrosoli, _Monete Greche_, Milan,
+1899, p. 202.
+
+[469] Thus Tzwivel's arithmetic of 1507, fol. 2, v., speaks of the ten
+figures as "characteres sive numerorum apices a diuo Seuerino Boetio."
+
+[470] Weissenborn uses _sipos_ for 0. It is not given by Bernelinus, and
+appears in Radulph of Laon, in the twelfth century. See Günther's
+_Geschichte_, p. 98, n.; Weissenborn, p. 11; Pihan, _Exposé_ etc., pp.
+xvi-xxii.
+
+In Friedlein's _Boetius_, p. 396, the plate shows that all of the six
+important manuscripts from which the illustrations are taken contain the
+symbol, while four out of five which give the words use the word _sipos_
+for 0. The names appear in a twelfth-century anonymous manuscript in the
+Vatican, in a passage beginning
+
+  Ordine primigeno sibi nomen possidet igin.
+  Andras ecce locum mox uendicat ipse secundum
+  Ormis post numeros incompositus sibi primus.
+
+[Boncompagni _Buttetino_, XV, p. 132.] Turchill (twelfth century) gives the
+names Igin, andras, hormis, arbas, quimas, caletis, zenis, temenias,
+celentis, saying: "Has autem figuras, ut donnus [dominus] Gvillelmus Rx
+testatur, a pytagoricis habemus, nomina uero ab arabibus." (Who the William
+R. was is not known. Boncompagni _Bulletino_ XV, p. 136.) Radulph of Laon
+(d. 1131) asserted that they were Chaldean (_Propagation_, p. 48 n.). A
+discussion of the whole question is also given in E. C. Bayley, loc. cit.
+Huet, writing in 1679, asserted that they were of Semitic origin, as did
+Nesselmann in spite of his despair over ormis, calctis, and celentis; see
+Woepcke, _Propagation_, p. 48. The names were used as late as the fifteenth
+century, without the zero, but with the superscript dot for 10's, two dots
+for 100's, etc., as among the early Arabs. Gerhardt mentions having seen a
+fourteenth or fifteenth century manuscript in the Bibliotheca Amploniana
+with the names "Ingnin, andras, armis, arbas, quinas, calctis, zencis,
+zemenias, zcelentis," and the statement "Si unum punctum super ingnin
+ponitur, X significat.... Si duo puncta super ... figuras superponunter,
+fiet decuplim illius quod cum uno puncto significabatur," in
+_Monatsberichte der K. P. Akad. d. Wiss._, Berlin, 1867, p. 40.
+
+[471] _A chart of ten numerals in 200 tongues_, by Rev. R. Patrick, London,
+1812.
+
+[472] "Numeratio figuralis est cuiusuis numeri per notas, et figuras
+numerates descriptio." [Clichtoveus, edition of c. 1507, fol. C ii, v.]
+"Aristoteles enim uoces rerum [Greek: sumbola] uocat: id translatum, sonat
+notas." [Noviomagus, _De Numeris Libri II_, cap. vi.] "Alphabetum decem
+notarum." [Schonerus, notes to Ramus, 1586, p. 3 seq.] Richer says: "novem
+numero notas omnem numerum significantes." [Bubnov, loc. cit., p. 381.]
+
+[473] "Il y a dix Characteres, autrement Figures, Notes, ou Elements."
+[Peletier, edition of 1607, p. 13.] "Numerorum notas alij figuras, alij
+signa, alij characteres uocant." [Glareanus, 1545 edition, f. 9, r.] "Per
+figuras (quas zyphras uocant) assignationem, quales sunt hæ notulæ, 1. 2.
+3. 4...." [Noviomagus, _De Numeris Libri II_, cap. vi.] Gemma Frisius also
+uses _elementa_ and Cardan uses _literae_. In the first arithmetic by an
+American (Greenwood, 1729) the author speaks of "a few Arabian _Charecters_
+or Numeral Figures, called _Digits_" (p. 1), and as late as 1790, in the
+third edition of J. J. Blassière's arithmetic (1st ed. 1769), the name
+_characters_ is still in use, both for "de Latynsche en de Arabische" (p.
+4), as is also the term "Cyfferletters" (p. 6, n.). _Ziffer_, the modern
+German form of cipher, was commonly used to designate any of the nine
+figures, as by Boeschenstein and Riese, although others, like Köbel, used
+it only for the zero. So _zifre_ appears in the arithmetic by Borgo, 1550
+ed. In a Munich codex of the twelfth century, attributed to Gerland, they
+are called _characters_ only: "Usque ad VIIII. enim porrigitur omnis
+numerus et qui supercrescit eisdem designator Karacteribus." [Boncompagni
+_Bulletino_, Vol. X. p. 607.]
+
+[474] The title of his work is _Prologus N. Ocreati in Helceph_ (Arabic
+_al-qeif_, investigation or memoir) _ad Adelardum Batensem magistrum suum_.
+The work was made known by C. Henry, in the _Zeitschrift für Mathematik und
+Physik_, Vol. XXV, p. 129, and in the _Abhandlungen zur Geschichte der
+Mathematik_, Vol. III; Weissenborn, _Gerbert_, p. 188.
+
+[475] The zero is indicated by a vacant column.
+
+[476] Leo Jordan, loc. cit., p. 170. "Chifre en augorisme" is the
+expression used, while a century later "giffre en argorisme" and "cyffres
+d'augorisme" are similarly used.
+
+[477] _The Works of Geoffrey Chaucer_, edited by W. W. Skeat, Vol. IV,
+Oxford, 1894, p. 92.
+
+[478] Loc. cit., Vol. III, pp. 179 and 180.
+
+[479] In Book II, chap, vii, of _The Testament of Love_, printed with
+Chaucer's Works, loc. cit., Vol. VII, London, 1897.
+
+[480] _Liber Abacci_, published in Olleris, _Oeuvres de Gerbert_, pp.
+357-400.
+
+[481] G. R. Kaye, "The Use of the Abacus in Ancient India," _Journal and
+Proceedings of the Asiatic Society of Bengal_, 1908, pp. 293-297.
+
+[482] _Liber Abbaci_, by Leonardo Pisano, loc. cit., p. 1.
+
+[483] Friedlein, "Die Entwickelung des Rechnens mit Columnen," _Zeitschrift
+für Mathematik und Physik_, Vol. X, p. 247.
+
+[484] The divisor 6 or 16 being increased by the difference 4, to 10 or 20
+respectively.
+
+[485] E.g. Cantor, Vol. I, p. 882.
+
+[486] Friedlein, loc. cit.; Friedlein, "Gerbert's Regeln der Division" and
+"Das Rechnen mit Columnen vor dem 10. Jahrhundert," _Zeitschrift für
+Mathematik und Physik_, Vol. IX; Bubnov, loc. cit., pp. 197-245; M.
+Chasles, "Histoire de l'arithmétique. Recherches des traces du système de
+l'abacus, après que cette méthode a pris le nom d'Algorisme.--Preuves qu'à
+toutes les époques, jusq'au XVI^e siècle, on a su que l'arithmétique
+vulgaire avait pour origine cette méthode ancienne," _Comptes rendus_, Vol.
+XVII, pp. 143-154, also "Règles de l'abacus," _Comptes rendus_, Vol. XVI,
+pp. 218-246, and "Analyse et explication du traité de Gerbert," _Comptes
+rendus_, Vol. XVI, pp. 281-299.
+
+[487] Bubnov, loc. cit., pp. 203-204, "Abbonis abacus."
+
+[488] "Regulae de numerorum abaci rationibus," in Bubnov, loc. cit., pp.
+205-225.
+
+[489] P. Treutlein, "Intorno ad alcuni scritti inediti relativi al calcolo
+dell' abaco," _Bulletino di bibliografia e di storia delle scienze
+matematiche e fisiche_, Vol. X, pp. 589-647.
+
+[490] "Intorno ad uno scritto inedito di Adelhardo di Bath intitolato
+'Regulae Abaci,'" B. Boncompagni, in his _Bulletino_, Vol. XIV, pp. 1-134.
+
+[491] Treutlein, loc. cit.; Boncompagni, "Intorno al Tractatus de Abaco di
+Gerlando," _Bulletino_, Vol. X, pp. 648-656.
+
+[492] E. Narducci, "Intorno a due trattati inediti d'abaco contenuti in due
+codici Vaticani del secolo XII," Boncompagni _Bulletino_, Vol. XV, pp.
+111-162.
+
+[493] See Molinier, _Les sources de l'histoire de France_, Vol. II, Paris,
+1902, pp. 2, 3.
+
+[494] Cantor, _Geschichte_, Vol. I, p. 762. A. Nagl in the _Abhandlungen
+zur Geschichte der Mathematik_, Vol. V, p. 85.
+
+[495] 1030-1117.
+
+[496] _Abhandlungen zur Geschichte der Mathematik_, Vol. V, pp. 85-133. The
+work begins "Incipit Liber Radulfi laudunensis de abaco."
+
+[497] _Materialien zur Geschichte der arabischen Zahlzeichen in
+Frankreich_, loc. cit.
+
+[498] Who died in 1202.
+
+[499] Cantor, _Geschichte_, Vol. I (3), pp. 800-803; Boncompagni,
+_Trattati_, Part II. M. Steinschneider ("Die Mathematik bei den Juden,"
+_Bibliotheca Mathematica_, Vol. X (2), p. 79) ingeniously derives another
+name by which he is called (Abendeuth) from Ibn Da[=u]d (Son of David). See
+also _Abhandlungen_, Vol. III, p. 110.
+
+[500] John is said to have died in 1157.
+
+[501] For it says, "Incipit prologus in libro alghoarismi de practica
+arismetrice. Qui editus est a magistro Johanne yspalensi." It is published
+in full in the second part of Boncompagni's _Trattati d'aritmetica_.
+
+[502] Possibly, indeed, the meaning of "libro alghoarismi" is not "to
+Al-Khow[=a]razm[=i]'s book," but "to a book of algorism." John of Luna says
+of it: "Hoc idem est illud etiam quod ... alcorismus dicere videtur."
+[_Trattati_, p. 68.]
+
+[503] For a résumé, see Cantor, Vol. I (3), pp. 800-803. As to the author,
+see Eneström in the _Bibliotheca Mathematica_, Vol. VI (3), p. 114, and
+Vol. IX (3), p. 2.
+
+[504] Born at Cremona (although some have asserted at Carmona, in
+Andalusia) in 1114; died at Toledo in 1187. Cantor, loc. cit.; Boncompagni,
+_Atti d. R. Accad. d. n. Lincei_, 1851.
+
+[505] See _Abhandlungen zur Geschichte der Mathematik_, Vol. XIV, p. 149;
+_Bibliotheca Mathematica_, Vol. IV (3), p. 206. Boncompagni had a
+fourteenth-century manuscript of his work, _Gerardi Cremonensis artis
+metrice practice_. See also T. L. Heath, _The Thirteen Books of Euclid's
+Elements_, 3 vols., Cambridge, 1908, Vol. I, pp. 92-94 ; A. A. Björnbo,
+"Gerhard von Cremonas Übersetzung von Alkwarizmis Algebra und von Euklids
+Elementen," _Bibliotheca Mathematica_, Vol. VI (3), pp. 239-248.
+
+[506] Wallis, _Algebra_, 1685, p. 12 seq.
+
+[507] Cantor, _Geschichte_, Vol. I (3), p. 906; A. A. Björnbo,
+"Al-Chw[=a]rizm[=i]'s trigonometriske Tavler," _Festskrift til H. G.
+Zeuthen_, Copenhagen, 1909, pp. 1-17.
+
+[508] Heath, loc. cit., pp. 93-96.
+
+[509] M. Steinschneider, _Zeitschrift der deutschen morgenländischen
+Gesellschaft_, Vol. XXV, 1871, p. 104, and _Zeitschrift für Mathematik und
+Physik_, Vol. XVI, 1871, pp. 392-393; M. Curtze, _Centralblatt für
+Bibliothekswesen_, 1899, p. 289; E. Wappler, _Zur Geschichte der deutschen
+Algebra im 15. Jahrhundert_, Programm, Zwickau, 1887; L. C. Karpinski,
+"Robert of Chester's Translation of the Algebra of Al-Khow[=a]razm[=i],"
+_Bibliotheca Mathematica_, Vol. XI (3), p. 125. He is also known as
+Robertus Retinensis, or Robert of Reading.
+
+[510] Nagl, A., "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und
+über die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im
+christl. Abendlande," in the _Zeitschrift für Mathematik und Physik,
+Hist.-lit. Abth._, Vol. XXXIV, p. 129. Curtze, _Abhandlungen zur Geschichte
+der Mathematik_, Vol. VIII, pp. 1-27.
+
+[511] See line _a_ in the plate on p. 143.
+
+[512] _Sefer ha-Mispar, Das Buch der Zahl, ein hebräisch-arithmetisches
+Werk des R. Abraham ibn Esra_, Moritz Silberberg, Frankfurt a. M., 1895.
+
+[513] Browning's "Rabbi ben Ezra."
+
+[514] "Darum haben auch die Weisen Indiens all ihre Zahlen durch neun
+bezeichnet und Formen für die 9 Ziffern gebildet." [_Sefer ha-Mispar_, loc.
+cit., p. 2.]
+
+[515] F. Bonaini, "Memoria unica sincrona di Leonardo Fibonacci," Pisa,
+1858, republished in 1867, and appearing in the _Giornale Arcadico_, Vol.
+CXCVII (N.S. LII); Gaetano Milanesi, _Documento inedito e sconosciuto a
+Lionardo Fibonacci_, Roma, 1867; Guglielmini, _Elogio di Lionardo Pisano_,
+Bologna, 1812, p. 35; Libri, _Histoire des sciences mathématiques_, Vol.
+II, p. 25; D. Martines, _Origine e progressi dell' aritmetica_, Messina,
+1865, p. 47; Lucas, in Boncompagni _Bulletino_, Vol. X, pp. 129, 239;
+Besagne, ibid., Vol. IX, p. 583; Boncompagni, three works as cited in Chap.
+I; G. Eneström, "Ueber zwei angebliche mathematische Schulen im
+christlichen Mittelalter," _Bibliotheca Mathematica_, Vol. VIII (3), pp.
+252-262; Boncompagni, "Della vita e delle opere di Leonardo Pisano," loc.
+cit.
+
+[516] The date is purely conjectural. See the _Bibliotheca Mathematica_,
+Vol. IV (3), p. 215.
+
+[517] An old chronicle relates that in 1063 Pisa fought a great battle with
+the Saracens at Palermo, capturing six ships, one being "full of wondrous
+treasure," and this was devoted to building the cathedral.
+
+[518] Heyd, loc. cit., Vol. I, p. 149.
+
+[519] Ibid., p. 211.
+
+[520] J. A. Symonds, _Renaissance in Italy. The Age of Despots._ New York,
+1883, p. 62.
+
+[521] Symonds, loc. cit., p. 79.
+
+[522] J. A. Froude, _The Science of History_, London, 1864. "Un brevet
+d'apothicaire n'empêcha pas Dante d'être le plus grand poète de l'Italie,
+et ce fut un petit marchand de Pise qui donna l'algèbre aux Chrétiens."
+[Libri, _Histoire_, Vol. I, p. xvi.]
+
+[523] A document of 1226, found and published in 1858, reads: "Leonardo
+bigollo quondam Guilielmi."
+
+[524] "Bonaccingo germano suo."
+
+[525] E.g. Libri, Guglielmini, Tiraboschi.
+
+[526] Latin, _Bonaccius_.
+
+[527] Boncompagni and Milanesi.
+
+[528] Reprint, p. 5.
+
+[529] Whence the French name for candle.
+
+[530] Now part of Algiers.
+
+[531] E. Reclus, _Africa_, New York, 1893, Vol. II, p. 253.
+
+[532] "Sed hoc totum et algorismum atque arcus pictagore quasi errorem
+computavi respectu modi indorum." Woepcke, _Propagation_ etc., regards this
+as referring to two different systems, but the expression may very well
+mean algorism as performed upon the Pythagorean arcs (or table).
+
+[533] "Book of the Abacus," this term then being used, and long afterwards
+in Italy, to mean merely the arithmetic of computation.
+
+[534] "Incipit liber Abaci a Leonardo filio Bonacci compositus anno 1202 et
+correctus ab eodem anno 1228." Three MSS. of the thirteenth century are
+known, viz. at Milan, at Siena, and in the Vatican library. The work was
+first printed by Boncompagni in 1857.
+
+[535] I.e. in relation to the quadrivium. "Non legant in festivis diebus,
+nisi Philosophos et rhetoricas et quadrivalia et barbarismum et ethicam, si
+placet." Suter, _Die Mathematik auf den Universitäten des Mittelalters_,
+Zürich, 1887, p. 56. Roger Bacon gives a still more gloomy view of Oxford
+in his time in his _Opus minus_, in the _Rerum Britannicarum medii aevi
+scriptores_, London, 1859, Vol. I, p. 327. For a picture of Cambridge at
+this time consult F. W. Newman, _The English Universities, translated from
+the German of V. A. Huber_, London, 1843, Vol. I, p. 61; W. W. R. Ball,
+_History of Mathematics at Cambridge_, 1889; S. Günther, _Geschichte des
+mathematischen Unterrichts im deutschen Mittelalter bis zum Jahre 1525_,
+Berlin, 1887, being Vol. III of _Monumenta Germaniae paedagogica_.
+
+[536] On the commercial activity of the period, it is known that bills of
+exchange passed between Messina and Constantinople in 1161, and that a bank
+was founded at Venice in 1170, the Bank of San Marco being established in
+the following year. The activity of Pisa was very manifest at this time.
+Heyd, loc. cit., Vol. II, p. 5; V. Casagrandi, _Storia e cronologia_, 3d
+ed., Milan, 1901, p. 56.
+
+[537] J. A. Symonds, loc. cit., Vol. II, p. 127.
+
+[538] I. Taylor, _The Alphabet_, London, 1883, Vol. II, p. 263.
+
+[539] Cited by Unger's History, p. 15. The Arabic numerals appear in a
+Regensburg chronicle of 1167 and in Silesia in 1340. See Schmidt's
+_Encyclopädie der Erziehung_, Vol. VI, p. 726; A. Kuckuk, "Die Rechenkunst
+im sechzehnten Jahrhundert," _Festschrift zur dritten Säcularfeier des
+Berlinischen Gymnasiums zum grauen Kloster_, Berlin, 1874, p. 4.
+
+[540] The text is given in Halliwell, _Rara Mathematica_, London, 1839.
+
+[541] Seven are given in Ashmole's _Catalogue of Manuscripts in the Oxford
+Library_, 1845.
+
+[542] Maximilian Curtze, _Petri Philomeni de Dacia in Algorismum Vulgarem
+Johannis de Sacrobosco commentarius, una cum Algorismo ipso_, Copenhagen,
+1897; L. C. Karpinski, "Jordanus Nemorarius and John of Halifax," _American
+Mathematical Monthly_, Vol. XVII, pp. 108-113.
+
+[543] J. Aschbach, _Geschichte der Wiener Universität im ersten
+Jahrhunderte ihres Bestehens_, Wien, 1865, p. 93.
+
+[544] Curtze, loc. cit., gives the text.
+
+[545] Curtze, loc. cit., found some forty-five copies of the _Algorismus_
+in three libraries of Munich, Venice, and Erfurt (Amploniana). Examination
+of two manuscripts from the Plimpton collection and the Columbia library
+shows such marked divergence from each other and from the text published by
+Curtze that the conclusion seems legitimate that these were students'
+lecture notes. The shorthand character of the writing further confirms this
+view, as it shows that they were written largely for the personal use of
+the writers.
+
+[546] "Quidam philosophus edidit nomine Algus, unde et Algorismus
+nuncupatur." [Curtze, loc. cit., p. 1.]
+
+[547] "Sinistrorsum autera scribimus in hac arte more arabico sive iudaico,
+huius scientiae inventorum." [Curtze, loc. cit., p. 7.] The Plimpton
+manuscript omits the words "sive iudaico."
+
+[548] "Non enim omnis numerus per quascumque figuras Indorum
+repraesentatur, sed tantum determinatus per determinatam, ut 4 non per
+5,..." [Curtze, loc. cit., p. 25.]
+
+[549] C. Henry, "Sur les deux plus anciens traités français d'algorisme et
+de géométrie," Boncompagni _Bulletino_, Vol. XV, p. 49; Victor Mortet, "Le
+plus ancien traité français d'algorisme," loc. cit.
+
+[550] _L'État des sciences en France, depute la mort du Roy Robert, arrivée
+en 1031, jusqu'à celle de Philippe le Bel, arrivée en 1314_, Paris, 1741.
+
+[551] _Discours sur l'état des lettres en France au XIII^e siecle_, Paris,
+1824.
+
+[552] _Aperçu historique_, Paris, 1876 ed., p. 464.
+
+[553] Ranulf Higden, a native of the west of England, entered St.
+Werburgh's monastery at Chester in 1299. He was a Benedictine monk and
+chronicler, and died in 1364. His _Polychronicon_, a history in seven
+books, was printed by Caxton in 1480.
+
+[554] Trevisa's translation, Higden having written in Latin.
+
+[555] An illustration of this feeling is seen in the writings of Prosdocimo
+de' Beldomandi (b. c. 1370-1380, d. 1428): "Inveni in quam pluribus libris
+algorismi nuncupatis mores circa numeros operandi satis varios atque
+diversos, qui licet boni existerent atque veri erant, tamen fastidiosi, tum
+propter ipsarum regularum multitudinem, tum propter earum deleationes, tum
+etiam propter ipsarum operationum probationes, utrum si bone fuerint vel
+ne. Erant et etiam isti modi interim fastidiosi, quod si in aliquo calculo
+astroloico error contigisset, calculatorem operationem suam a capite
+incipere oportebat, dato quod error suus adhuc satis propinquus existeret;
+et hoc propter figuras in sua operatione deletas. Indigebat etiam
+calculator semper aliquo lapide vel sibi conformi, super quo scribere atque
+faciliter delere posset figuras cum quibus operabatur in calculo suo. Et
+quia haec omnia satis fastidiosa atque laboriosa mihi visa sunt, disposui
+libellum edere in quo omnia ista abicerentur: qui etiam algorismus sive
+liber de numeris denominari poterit. Scias tamen quod in hoc libello ponere
+non intendo nisi ea quae ad calculum necessaria sunt, alia quae in aliis
+libris practice arismetrice tanguntur, ad calculum non necessaria, propter
+brevitatem dimitendo." [Quoted by A. Nagl, _Zeitschrift für Mathematik und
+Physik, Hist.-lit. Abth._, Vol. XXXIV, p. 143; Smith, _Rara Arithmetica_,
+p. 14, in facsimile.]
+
+[556] P. Ewald, loc. cit.; Franz Steffens, _Lateinische Paläographie_, pp.
+xxxix-xl. We are indebted to Professor J. M. Burnam for a photograph of
+this rare manuscript.
+
+[557] See the plate of forms on p. 88.
+
+[558] Karabacek, loc. cit., p. 56; Karpinski, "Hindu Numerals in the
+Fihrist," _Bibliotheca Mathematica_, Vol. XI (3), p. 121.
+
+[559] Woepcke, "Sur une donnée historique," etc., loc. cit., and "Essai
+d'une restitution de travaux perdus d'Apollonius sur les quantités
+irrationnelles, d'après des indications tirées d'un manuscrit arabe," _Tome
+XIV des Mémoires présentés par divers savants à l'Académie des sciences_,
+Paris, 1856, note, pp. 6-14.
+
+[560] _Archeological Report of the Egypt Exploration Fund for 1908-1909_,
+London, 1910, p. 18.
+
+[561] There was a set of astronomical tables in Boncompagni's library
+bearing this date: "Nota quod anno d[=n]i [=n]ri ihû x[=p]i. 1264.
+perfecto." See Narducci's _Catalogo_, p. 130.
+
+[562] "On the Early use of Arabic Numerals in Europe," read before the
+Society of Antiquaries April 14, 1910, and published in _Archæologia_ in
+the same year.
+
+[563] Ibid., p. 8, n. The date is part of an Arabic inscription.
+
+[564] O. Codrington, _A Manual of Musalman Numismatics_, London, 1904.
+
+[565] See Arbuthnot, _The Mysteries of Chronology_, London, 1900, pp. 75,
+78, 98; F. Pichler, _Repertorium der steierischen Münzkunde_, Grätz, 1875,
+where the claim is made of an Austrian coin of 1458; _Bibliotheca
+Mathematica_, Vol. X (2), p. 120, and Vol. XII (2), p. 120. There is a
+Brabant piece of 1478 in the collection of D. E. Smith.
+
+[566] A specimen is in the British Museum. [Arbuthnot, p. 79.]
+
+[567] Ibid., p. 79.
+
+[568] _Liber de Remediis utriusque fortunae Coloniae._
+
+[569] Fr. Walthern et Hans Hurning, Nördlingen.
+
+[570] _Ars Memorandi_, one of the oldest European block-books.
+
+[571] Eusebius Caesariensis, _De praeparatione evangelica_, Venice, Jenson,
+1470. The above statement holds for copies in the Astor Library and in the
+Harvard University Library.
+
+[572] Francisco de Retza, _Comestorium vitiorum_, Nürnberg, 1470. The copy
+referred to is in the Astor Library.
+
+[573] See Mauch, "Ueber den Gebrauch arabischer Ziffern und die
+Veränderungen derselben," _Anzeiger für Kunde der deutschen Vorzeit_, 1861,
+columns 46, 81, 116, 151, 189, 229, and 268; Calmet, _Recherches sur
+l'origine des chiffres d'arithmétique_, plate, loc. cit.
+
+[574] Günther, _Geschichte_, p. 175, n.; Mauch, loc. cit.
+
+[575] These are given by W. R. Lethaby, from drawings by J. T. Irvine, in
+the _Proceedings of the Society of Antiquaries_, 1906, p. 200.
+
+[576] There are some ill-tabulated forms to be found in J. Bowring, _The
+Decimal System_, London, 1854, pp. 23, 25, and in L. A. Chassant,
+_Dictionnaire des abréviations latines et françaises ... du moyen âge_,
+Paris, MDCCCLXVI, p. 113. The best sources we have at present, aside from
+the Hill monograph, are P. Treutlein, _Geschichte unserer Zahlzeichen_,
+Karlsruhe, 1875; Cantor's _Geschichte_, Vol. I, table; M. Prou, _Manuel de
+paléographie latine et française_, 2d ed., Paris, 1892, p. 164; A.
+Cappelli, _Dizionario di abbreviature latine ed italiane_, Milan, 1899. An
+interesting early source is found in the rare Caxton work of 1480, _The
+Myrrour of the World_. In Chap. X is a cut with the various numerals, the
+chapter beginning "The fourth scyence is called arsmetrique." Two of the
+fifteen extant copies of this work are at present in the library of Mr. J.
+P. Morgan, in New York.
+
+[577] From the twelfth-century manuscript on arithmetic, Curtze, loc. cit.,
+_Abhandlungen_, and Nagl, loc. cit. The forms are copied from Plate VII in
+_Zeitschrift für Mathematik und Physik_, Vol. XXXIV.
+
+[578] From the Regensburg chronicle. Plate containing some of these
+numerals in _Monumenta Germaniae historica_, "Scriptores" Vol. XVII, plate
+to p. 184; Wattenbach, _Anleitung zur lateinischen Palaeographie_, Leipzig,
+1886, p. 102; Boehmer, _Fontes rerum Germanicarum_, Vol. III, Stuttgart,
+1852, p. lxv.
+
+[579] French Algorismus of 1275; from an unpublished photograph of the
+original, in the possession of D. E. Smith. See also p. 135.
+
+[580] From a manuscript of Boethius c. 1294, in Mr. Plimpton's library.
+Smith, _Rara Arithmetica_, Plate I.
+
+[581] Numerals in a 1303 manuscript in Sigmaringen, copied from Wattenbach,
+loc. cit., p. 102.
+
+[582] From a manuscript, Add. Manuscript 27,589, British Museum, 1360 A.D.
+The work is a computus in which the date 1360 appears, assigned in the
+British Museum catalogue to the thirteenth century.
+
+[583] From the copy of Sacrabosco's _Algorismus_ in Mr. Plimpton's library.
+Date c. 1442. See Smith, _Rara Arithmetica_, p. 450.
+
+[584] See _Rara Arithmetica_, pp. 446-447.
+
+[585] Ibid., pp. 469-470.
+
+[586] Ibid., pp. 477-478.
+
+[587] The i is used for "one" in the Treviso arithmetic (1478), Clichtoveus
+(c. 1507 ed., where both i and j are so used), Chiarini (1481), Sacrobosco
+(1488 ed.), and Tzwivel (1507 ed., where jj and jz are used for 11 and 12).
+This was not universal, however, for the _Algorithmus linealis_ of c. 1488
+has a special type for 1. In a student's notebook of lectures taken at the
+University of Würzburg in 1660, in Mr. Plimpton's library, the ones are all
+in the form of i.
+
+[588] Thus the date [Numerals 1580], for 1580, appears in a MS. in the
+Laurentian library at Florence. The second and the following five
+characters are taken from Cappelli's _Dizionario_, p. 380, and are from
+manuscripts of the twelfth, thirteenth, fourteenth, sixteenth, seventeenth,
+and eighteenth centuries, respectively.
+
+[589] E.g. Chiarini's work of 1481; Clichtoveus (c. 1507).
+
+[590] The first is from an algorismus of the thirteenth century, in the
+Hannover Library. [See Gerhardt, "Ueber die Entstehung und Ausbreitung des
+dekadischen Zahlensystems," loc. cit., p. 28.] The second character is from
+a French algorismus, c. 1275. [Boncompagni _Bulletino_, Vol. XV, p. 51.]
+The third and the following sixteen characters are given by Cappelli, loc.
+cit., and are from manuscripts of the twelfth (1), thirteenth (2),
+fourteenth (7), fifteenth (3), sixteenth (1), seventeenth (2), and
+eighteenth (1) centuries, respectively.
+
+[591] Thus Chiarini (1481) has [Symbol] for 23.
+
+[592] The first of these is from a French algorismus, c. 1275. The second
+and the following eight characters are given by Cappelli, loc. cit., and
+are from manuscripts of the twelfth (2), thirteenth, fourteenth, fifteenth
+(3), seventeenth, and eighteenth centuries, respectively.
+
+[593] See Nagl, loc. cit.
+
+[594] Hannover algorismus, thirteenth century.
+
+[595] See the Dagomari manuscript, in _Rara Arithmetica_, pp. 435, 437-440.
+
+[596] But in the woodcuts of the _Margarita Philosophica_ (1503) the old
+forms are used, although the new ones appear in the text. In Caxton's
+_Myrrour of the World_ (1480) the old form is used.
+
+[597] Cappelli, loc. cit. They are partly from manuscripts of the tenth,
+twelfth, thirteenth (3), fourteenth (7), fifteenth (6), and eighteenth
+centuries, respectively. Those in the third line are from Chassant's
+_Dictionnaire_, p. 113, without mention of dates.
+
+[598] The first is from the Hannover algorismus, thirteenth century. The
+second is taken from the Rollandus manuscript, 1424. The others in the
+first two lines are from Cappelli, twelfth (3), fourteenth (6), fifteenth
+(13) centuries, respectively. The third line is from Chassant, loc. cit.,
+p. 113, no mention of dates.
+
+[599] The first of these forms is from the Hannover algorismus, thirteenth
+century. The following are from Cappelli, fourteenth (3), fifteenth,
+sixteenth (2), and eighteenth centuries, respectively.
+
+[600] The first of these is taken from the Hannover algorismus, thirteenth
+century. The following forms are from Cappelli, twelfth, thirteenth,
+fourteenth (5), fifteenth (2), seventeenth, and eighteenth centuries,
+respectively.
+
+[601] All of these are given by Cappelli, thirteenth, fourteenth, fifteenth
+(2), and sixteenth centuries, respectively.
+
+[602] Smith, _Rara Arithmetica_, p. 489. This is also seen in several of
+the Plimpton manuscripts, as in one written at Ancona in 1684. See also
+Cappelli, loc. cit.
+
+[603] French algorismus, c. 1275, for the first of these forms. Cappelli,
+thirteenth, fourteenth, fifteenth (3), and seventeenth centuries,
+respectively. The last three are taken from _Byzantinische Analekten_, J.
+L. Heiberg, being forms of the fifteenth century, but not at all common.
+[Symbol: Qoppa] was the old Greek symbol for 90.
+
+[604] For the first of these the reader is referred to the forms ascribed
+to Boethius, in the illustration on p. 88; for the second, to Radulph of
+Laon, see p. 60. The third is used occasionally in the Rollandus (1424)
+manuscript, in Mr. Plimpton's library. The remaining three are from
+Cappelli, fourteenth (2) and seventeenth centuries.
+
+[605] Smith, _An Early English Algorism_.
+
+[606] Kuckuck, p. 5.
+
+[607] A. Cappelli, loc. cit., p. 372.
+
+[608] Smith, _Rara Arithmetica_, p. 443.
+
+[609] Curtze, _Petri Philomeni de Dacia_ etc., p. IX.
+
+[610] Cappelli, loc. cit., p. 376.
+
+[611] Curtze, loc. cit., pp. VIII-IX, note.
+
+[612] Edition of 1544-1545, f. 52.
+
+[613] _De numeris libri II_, 1544 ed., cap. XV. Heilbronner, loc. cit., p.
+736, also gives them, and compares this with other systems.
+
+[614] Noviomagus says of them: "De quibusdam Astrologicis, sive Chaldaicis
+numerorum notis.... Sunt & aliæ quædam notæ, quibus Chaldaei & Astrologii
+quemlibet numerum artificiose & arguté describunt, scitu periucundae, quas
+nobis communicauit Rodolphus Paludanus Nouiomagus."
+
+
+
+
+
+
+End of the Project Gutenberg EBook of The Hindu-Arabic Numerals, by 
+David Eugene Smith and Louis Charles Karpinski
+
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+<pre>
+
+The Project Gutenberg EBook of The Hindu-Arabic Numerals, by 
+David Eugene Smith and Louis Charles Karpinski
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: The Hindu-Arabic Numerals
+
+Author: David Eugene Smith
+        Louis Charles Karpinski
+
+Release Date: September 14, 2007 [EBook #22599]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE HINDU-ARABIC NUMERALS ***
+
+
+
+
+Produced by David Newman, Chuck Greif, Keith Edkins and
+the Online Distributed Proofreading Team at
+https://www.pgdp.net (This file was produced from images
+from the Cornell University Library: Historical Mathematics
+Monographs collection.)
+
+
+
+
+
+
+</pre>
+
+
+<table border="0" cellpadding="10" style="background-color: #ccccff;">
+<tr>
+<td style="width:25%; vertical-align:top">
+Transcriber's note:
+</td>
+<td>
+Sections in Greek or Hebrew will yield a transliteration
+when the pointer is moved over them, and words using diacritic characters in the
+Latin Extended Additional block, which may not display in some fonts or browsers, will
+display an unaccented version.
+</td>
+</tr>
+</table>
+
+<h3>THE</h3>
+
+<h2>HINDU-ARABIC NUMERALS</h2>
+
+<p class="cenhead">BY<br />
+DAVID EUGENE SMITH<br />
+AND<br />
+LOUIS CHARLES KARPINSKI</p>
+
+<p class="cenhead">BOSTON AND LONDON<br />
+GINN AND COMPANY, PUBLISHERS<br />
+1911</p>
+
+<p class="cenhead">COPYRIGHT, 1911, BY DAVID EUGENE SMITH<br />
+AND LOUIS CHARLES KARPINSKI<br />
+ALL RIGHTS RESERVED<br />
+811.7</p>
+
+<p class="cenhead"><b>The Athenæum Press</b><br />
+GINN AND COMPANY · PROPRIETORS<br />
+BOSTON · U.S.A.</p>
+
+<hr class="full" >
+
+<p><!-- Page iii --><span class="pagenum"><a name="pageiii"></a>[iii]</span></p>
+
+<h3>PREFACE</h3>
+
+  <p>So familiar are we with the numerals that bear the misleading name of
+  Arabic, and so extensive is their use in Europe and the Americas, that it
+  is difficult for us to realize that their general acceptance in the
+  transactions of commerce is a matter of only the last four centuries, and
+  that they are unknown to a very large part of the human race to-day. It
+  seems strange that such a labor-saving device should have struggled for
+  nearly a thousand years after its system of place value was perfected
+  before it replaced such crude notations as the one that the Roman
+  conqueror made substantially universal in Europe. Such, however, is the
+  case, and there is probably no one who has not at least some slight
+  passing interest in the story of this struggle. To the mathematician and
+  the student of civilization the interest is generally a deep one; to the
+  teacher of the elements of knowledge the interest may be less marked, but
+  nevertheless it is real; and even the business man who makes daily use of
+  the curious symbols by which we express the numbers of commerce, cannot
+  fail to have some appreciation for the story of the rise and progress of
+  these tools of his trade.</p>
+
+  <p>This story has often been told in part, but it is a long time since
+  any effort has been made to bring together the fragmentary narrations and
+  to set forth the general problem of the origin and development of these
+  <!-- Page iv --><span class="pagenum"><a
+  name="pageiv"></a>[iv]</span>numerals. In this little work we have
+  attempted to state the history of these forms in small compass, to place
+  before the student materials for the investigation of the problems
+  involved, and to express as clearly as possible the results of the labors
+  of scholars who have studied the subject in different parts of the world.
+  We have had no theory to exploit, for the history of mathematics has seen
+  too much of this tendency already, but as far as possible we have weighed
+  the testimony and have set forth what seem to be the reasonable
+  conclusions from the evidence at hand.</p>
+
+  <p>To facilitate the work of students an index has been prepared which we
+  hope may be serviceable. In this the names of authors appear only when
+  some use has been made of their opinions or when their works are first
+  mentioned in full in a footnote.</p>
+
+  <p>If this work shall show more clearly the value of our number system,
+  and shall make the study of mathematics seem more real to the teacher and
+  student, and shall offer material for interesting some pupil more fully
+  in his work with numbers, the authors will feel that the considerable
+  labor involved in its preparation has not been in vain.</p>
+
+  <p>We desire to acknowledge our especial indebtedness to Professor
+  Alexander Ziwet for reading all the proof, as well as for the digest of a
+  Russian work, to Professor Clarence L. Meader for Sanskrit
+  transliterations, and to Mr. Steven T. Byington for Arabic
+  transliterations and the scheme of pronunciation of Oriental names, and
+  also our indebtedness to other scholars in Oriental learning for
+  information.</p>
+
+  <p class="author">DAVID EUGENE SMITH
+
+  <p class="author">LOUIS CHARLES KARPINSKI
+
+<hr class="full" >
+
+<p><!-- Page v --><span class="pagenum"><a name="pagev"></a>[v]</span></p>
+
+<h3>CONTENTS</h3>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>CHAPTER</p>
+    </div>
+
+    <div class="stanza">
+      <p class="i6">PRONUNCIATION OF ORIENTAL NAMES                              <a href="#pagevi">vi</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>I.    EARLY IDEAS OF THEIR ORIGIN                                   <a href="#page1">1</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>II.   EARLY HINDU FORMS WITH NO PLACE VALUE                        <a href="#page12">12</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>III.  LATER HINDU FORMS, WITH A PLACE VALUE                        <a href="#page38">38</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>IV.   THE SYMBOL ZERO                                              <a href="#page51">51</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>V.    THE QUESTION OF THE INTRODUCTION OF THE</p>
+      <p class="i6">NUMERALS INTO EUROPE BY BOETHIUS                             <a href="#page63">63</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>VI.   THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS              <a href="#page91">91</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>VII.  THE DEFINITE INTRODUCTION OF THE NUMERALS INTO EUROPE        <a href="#page99">99</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>VIII. THE SPREAD OF THE NUMERALS IN EUROPE                        <a href="#page128">128</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>INDEX                                                             <a href="#page153">153</a></p>
+    </div>
+  </div>
+<hr class="full" >
+
+<p><!-- Page vi --><span class="pagenum"><a name="pagevi"></a>[vi]</span></p>
+
+<h3>PRONUNCIATION OF ORIENTAL NAMES</h3>
+
+  <p>(S) = in Sanskrit names and words; (A) = in Arabic names and
+  words.</p>
+
+  <p><b>b</b>, <b>d</b>, <b>f</b>, <b>g</b>, <b>h</b>, <b>j</b>, <b>l</b>,
+  <b>m</b>, <b>n</b>, <b>p</b>, <b>sh</b> (A), <b>t</b>, <b>th</b> (A),
+  <b>v</b>, <b>w</b>, <b>x</b>, <b>z</b>, as in English.</p>
+
+  <p><b>a</b>, (S) like <i>u</i> in <i>but</i>: thus <i>pandit</i>,
+  pronounced <i>pundit</i>. (A) like <i>a</i> in <i>ask</i> or in
+  <i>man</i>. <b>&#x101;</b>, as in <i>father</i>.</p>
+
+  <p><b>c</b>, (S) like <i>ch</i> in <i>church</i> (Italian <i>c</i> in
+  <i>cento</i>).</p>
+
+  <p><b><span class="special" title="d-dot-below">&#x1E0D;</span></b>,
+  <b><span class="special" title="n-dot-below">&#x1E47;</span></b>,
+  <b><span class="special" title="s-dot-below">&#x1E63;</span></b>,
+  <b><span class="special" title="t-dot-below">&#x1E6D;</span></b>, (S)
+  <i>d</i>, <i>n</i>, <i>sh</i>, <i>t</i>, made with the tip of the tongue
+  turned up and back into the dome of the palate. <b><span class="special"
+  title="d-dot-below">&#x1E0D;</span></b>, <b><span class="special"
+  title="s-dot-below">&#x1E63;</span></b>, <b><span class="special"
+  title="t-dot-below">&#x1E6D;</span></b>, <b><span class="special"
+  title="z-dot-below">&#x1E93;</span></b>, (A) <i>d</i>, <i>s</i>,
+  <i>t</i>, <i>z</i>, made with the tongue spread so that the sounds are
+  produced largely against the side teeth. Europeans commonly pronounce
+  <b><span class="special" title="d-dot-below">&#x1E0D;</span></b>,
+  <b><span class="special" title="n-dot-below">&#x1E47;</span></b>,
+  <b><span class="special" title="s-dot-below">&#x1E63;</span></b>,
+  <b><span class="special" title="t-dot-below">&#x1E6D;</span></b>,
+  <b><span class="special" title="z-dot-below">&#x1E93;</span></b>, both
+  (S) and (A), as simple <i>d</i>, <i>n</i>, <i>sh</i> (S) or <i>s</i> (A),
+  <i>t</i>, <i>z</i>. <b><span class="special"
+  title="d-line-below">&#x1E0F;</span></b> (A), like <i>th</i> in
+  <i>this</i>.</p>
+
+  <p><b>e</b>, (S) as in <i>they</i>. (A) as in <i>bed</i>.</p>
+
+  <p><b>&#x121;</b>, (A) a voiced consonant formed below the vocal cords;
+  its sound is compared by some to a <i>g</i>, by others to a guttural
+  <i>r</i>; in Arabic words adopted into English it is represented by
+  <i>gh</i> (e.g. <i>ghoul</i>), less often <i>r</i> (e.g.
+  <i>razzia</i>).</p>
+
+  <p><b>h</b> preceded by <i>b</i>, <i>c</i>, <i>t</i>, <i><span
+  class="special" title="t-dot-below">&#x1E6D;</span></i>, etc. does not
+  form a single sound with these letters, but is a more or less distinct
+  <i>h</i> sound following them; cf. the sounds in <i>abhor, boathook</i>,
+  etc., or, more accurately for (S), the "bhoys" etc. of Irish brogue.
+  <b>h</b> (A) retains its consonant sound at the end of a word. <b><span
+  class="special" title="h-dot-below">&#x1E25;</span></b>, (A) an unvoiced
+  consonant formed below the vocal cords; its sound is sometimes compared
+  to German hard <i>ch</i>, and may be represented by an <i>h</i> as strong
+  as possible. In Arabic words adopted into English it is represented by
+  <i>h</i>, e.g. in <i>sahib</i>, <i>hakeem</i>. <b><span class="special"
+  title="h-dot-below">&#x1E25;</span></b> (S) is final consonant <i>h</i>,
+  like final <i>h</i> (A).</p>
+
+  <p><b>i</b>, as in <i>pin</i>. <b>&#x12B;</b>, as in <i>pique</i>.</p>
+
+  <p><b>k</b>, as in <i>kick</i>.</p>
+
+  <p><b>kh</b>, (A) the hard <i>ch</i> of Scotch <i>loch</i>, German
+  <i>ach</i>, especially of German as pronounced by the Swiss.</p>
+
+  <p><b><span class="special" title="m-dot-above">&#x1E41;</span></b>,
+  <b><span class="special" title="n-dot-above">&#x1E45;</span></b>, (S)
+  like French final <i>m</i> or <i>n</i>, nasalizing the preceding
+  vowel.</p>
+
+  <p><b><span class="special" title="n-dot-below">&#x1E47;</span></b>, see
+  <b><span class="special" title="d-dot-below">&#x1E0D;</span></b>.
+  <b>ñ</b>, like <i>ng</i> in <i>singing</i>.</p>
+
+  <p><b>o</b>, (S) as in <i>so</i>. (A) as in <i>obey</i>.</p>
+
+  <p><b>q</b>, (A) like <i>k</i> (or <i>c</i>) in <i>cook</i>; further back
+  in the mouth than in <i>kick</i>.</p>
+
+  <p><b>r</b>, (S) English <i>r</i>, smooth and untrilled. (A) stronger.
+  <b><span class="special" title="r-dot-below">&#x1E5B;</span></b>, (S) r
+  used as vowel, as in <i>apron</i> when pronounced <i>aprn</i> and not
+  <i>apern</i>; modern Hindus say <i>ri</i>, hence our <i>amrita</i>,
+  <i>Krishna</i>, for <i><span class="special"
+  title="a-mrta">a-m&#x1E5B;ta</span>, <span class="special"
+  title="Krsna">K&#x1E5B;&#x1E63;&#x1E47;a</span></i>.</p>
+
+  <p><b>s</b>, as in <i>same</i>. <b><span class="special"
+  title="s-dot-below">&#x1E63;</span></b>, see <b><span class="special"
+  title="d-dot-below">&#x1E0D;</span></b>. <b>&#x15B;</b>, (S) English
+  <i>sh</i> (German <i>sch</i>).</p>
+
+  <p><b><span class="special" title="t-dot-below">&#x1E6D;</span></b>, see
+  <b><span class="special" title="d-dot-below">&#x1E0D;</span></b>.</p>
+
+  <p><b>u</b>, as in <i>put</i>. <b>&#x16B;</b>, as in <i>rule</i>.</p>
+
+  <p><b>y</b>, as in <i>you</i>.</p>
+
+  <p><b><span class="special" title="z-dot-below">&#x1E93;</span></b>, see
+  <b><span class="special" title="d-dot-below">&#x1E0D;</span></b>.</p>
+
+  <p><b>&#x201B;</b>, (A) a sound kindred to the spiritus lenis (that is,
+  to our ears, the mere distinct separation of a vowel from the preceding
+  sound, as at the beginning of a word in German) and to <i><span
+  class="special" title="h-dot-below">&#x1E25;</span></i>. The &#x201B; is
+  a very distinct sound in Arabic, but is more nearly represented by the
+  spiritus lenis than by any sound that we can produce without much special
+  training. That is, it should be treated as silent, but the sounds that
+  precede and follow it should not run together. In Arabic words adopted
+  into English it is treated as silent, e.g. in <i>Arab</i>, <i>amber</i>,
+  <i>Caaba</i> (<i>&#x201B;Arab</i>, <i>&#x201B;anbar</i>,
+  <i>ka&#x201B;abah</i>).</p>
+
+  <p>(A) A final long vowel is shortened before <i>al</i> (<i>'l</i>) or
+  <i>ibn</i> (whose <i>i</i> is then silent).</p>
+
+  <p>Accent: (S) as if Latin; in determining the place of the accent
+  <i><span class="special" title="m-dot-above">&#x1E41;</span></i> and
+  <i><span class="special" title="n-dot-above">&#x1E45;</span></i> count as
+  consonants, but <i>h</i> after another consonant does not. (A), on the
+  last syllable that contains a long vowel or a vowel followed by two
+  consonants, except that a final long vowel is not ordinarily accented; if
+  there is no long vowel nor two consecutive consonants, the accent falls
+  on the first syllable. The words <i>al</i> and <i>ibn</i> are never
+  accented.</p>
+
+<hr class="full" >
+
+<p><!-- Page 1 --><span class="pagenum"><a name="page1"></a>[1]</span></p>
+
+<h2>THE HINDU-ARABIC NUMERALS</h2>
+
+<h3>CHAPTER I</h3>
+
+<p class="cenhead">EARLY IDEAS OF THEIR ORIGIN</p>
+
+  <p>It has long been recognized that the common numerals used in daily
+  life are of comparatively recent origin. The number of systems of
+  notation employed before the Christian era was about the same as the
+  number of written languages, and in some cases a single language had
+  several systems. The Egyptians, for example, had three systems of
+  writing, with a numerical notation for each; the Greeks had two
+  well-defined sets of numerals, and the Roman symbols for number changed
+  more or less from century to century. Even to-day the number of methods
+  of expressing numerical concepts is much greater than one would believe
+  before making a study of the subject, for the idea that our common
+  numerals are universal is far from being correct. It will be well, then,
+  to think of the numerals that we still commonly call Arabic, as only one
+  of many systems in use just before the Christian era. As it then existed
+  the system was no better than many others, it was of late origin, it
+  contained no zero, it was cumbersome and little used, <!-- Page 2
+  --><span class="pagenum"><a name="page2"></a>[2]</span>and it had no
+  particular promise. Not until centuries later did the system have any
+  standing in the world of business and science; and had the place value
+  which now characterizes it, and which requires a zero, been worked out in
+  Greece, we might have been using Greek numerals to-day instead of the
+  ones with which we are familiar.</p>
+
+  <p>Of the first number forms that the world used this is not the place to
+  speak. Many of them are interesting, but none had much scientific value.
+  In Europe the invention of notation was generally assigned to the eastern
+  shores of the Mediterranean until the critical period of about a century
+  ago,&mdash;sometimes to the Hebrews, sometimes to the Egyptians, but more
+  often to the early trading Ph&oelig;nicians.<a name="NtA_1"
+  href="#Nt_1"><sup>[1]</sup></a></p>
+
+  <p>The idea that our common numerals are Arabic in origin is not an old
+  one. The mediæval and Renaissance writers generally recognized them as
+  Indian, and many of them expressly stated that they were of Hindu
+  origin.<a name="NtA_2" href="#Nt_2"><sup>[2]</sup></a> <!-- Page 3
+  --><span class="pagenum"><a name="page3"></a>[3]</span>Others argued that
+  they were probably invented by the Chaldeans or the Jews because they
+  increased in value from right to left, an argument that would apply quite
+  as well to the Roman and Greek systems, or to any other. It was, indeed,
+  to the general idea of notation that many of these writers referred, as
+  is evident from the words of England's earliest arithmetical
+  textbook-maker, Robert Recorde (c. 1542): "In that thinge all men do
+  agree, that the Chaldays, whiche fyrste inuented thys arte, did set these
+  figures as thei set all their letters. for they wryte backwarde as you
+  tearme it, and so doo they reade. And that may appeare in all Hebrewe,
+  Chaldaye and Arabike bookes ... where as the Greekes, Latines, and all
+  nations of Europe, do wryte and reade from the lefte hand towarde the
+  ryghte."<a name="NtA_3" href="#Nt_3"><sup>[3]</sup></a> Others, and <!--
+  Page 4 --><span class="pagenum"><a name="page4"></a>[4]</span>among them
+  such influential writers as Tartaglia<a name="NtA_4"
+  href="#Nt_4"><sup>[4]</sup></a> in Italy and Köbel<a name="NtA_5"
+  href="#Nt_5"><sup>[5]</sup></a> in Germany, asserted the Arabic origin of
+  the numerals, while still others left the matter undecided<a name="NtA_6"
+  href="#Nt_6"><sup>[6]</sup></a> or simply dismissed them as "barbaric."<a
+  name="NtA_7" href="#Nt_7"><sup>[7]</sup></a> Of course the Arabs
+  themselves never laid claim to the invention, always recognizing their
+  indebtedness to the Hindus both for the numeral forms and for the
+  distinguishing feature of place value. Foremost among these writers was
+  the great master of the golden age of Bagdad, one of the first of the
+  Arab writers to collect the mathematical classics of both the East and
+  the West, preserving them and finally passing them on to awakening
+  Europe. This man was <span class="special"
+  title="Mohammed">Mo&#x1E25;ammed</span> the Son of Moses, from
+  Khow&#x101;rezm, or, more after the manner of the Arab, <span
+  class="special" title="Mohammed">Mo&#x1E25;ammed</span> ibn
+  M&#x16B;s&#x101; al-Khow&#x101;razm&#x12B;,<a name="NtA_8"
+  href="#Nt_8"><sup>[8]</sup></a> a man of great <!-- Page 5 --><span
+  class="pagenum"><a name="page5"></a>[5]</span>learning and one to whom
+  the world is much indebted for its present knowledge of algebra<a
+  name="NtA_9" href="#Nt_9"><sup>[9]</sup></a> and of arithmetic. Of him
+  there will often be occasion to speak; and in the arithmetic which he
+  wrote, and of which Adelhard of Bath<a name="NtA_10"
+  href="#Nt_10"><sup>[10]</sup></a> (c. 1130) may have made the translation
+  or paraphrase,<a name="NtA_11" href="#Nt_11"><sup>[11]</sup></a> he
+  stated distinctly that the numerals were due to the Hindus.<a
+  name="NtA_12" href="#Nt_12"><sup>[12]</sup></a> This is as plainly
+  asserted by later Arab <!-- Page 6 --><span class="pagenum"><a
+  name="page6"></a>[6]</span>writers, even to the present day.<a
+  name="NtA_13" href="#Nt_13"><sup>[13]</sup></a> Indeed the phrase
+  <i>&#x201B;ilm hind&#x12B;</i>, "Indian science," is used by them for
+  arithmetic, as also the adjective <i>hind&#x12B;</i> alone.<a
+  name="NtA_14" href="#Nt_14"><sup>[14]</sup></a></p>
+
+  <p>Probably the most striking testimony from Arabic sources is that given
+  by the Arabic traveler and scholar <span class="special" title="Mohammed ibn Ahmed"
+  >Mohammed ibn A&#x1E25;med</span>, <span class="special" title="Abu 'l-Rihan al-Biruni"
+  >Ab&#x16B; 'l-R&#x12B;&#x1E25;&#x101;n al-B&#x12B;r&#x16B;n&#x12B;</span>
+  (973-1048), who spent many years in Hindustan. He wrote a large work on
+  India,<a name="NtA_15" href="#Nt_15"><sup>[15]</sup></a> one on ancient
+  chronology,<a name="NtA_16" href="#Nt_16"><sup>[16]</sup></a> the "Book
+  of the Ciphers," unfortunately lost, which treated doubtless of the Hindu
+  art of calculating, and was the author of numerous other works.
+  Al-B&#x12B;r&#x16B;n&#x12B; was a man of unusual attainments, being
+  versed in Arabic, Persian, Sanskrit, Hebrew, and Syriac, as well as in
+  astronomy, chronology, and mathematics. In his work on India he gives
+  detailed information concerning the language and <!-- Page 7 --><span
+  class="pagenum"><a name="page7"></a>[7]</span>customs of the people of
+  that country, and states explicitly<a name="NtA_17"
+  href="#Nt_17"><sup>[17]</sup></a> that the Hindus of his time did not use
+  the letters of their alphabet for numerical notation, as the Arabs did.
+  He also states that the numeral signs called <i><span class="special"
+  title="anka">a&#x1E45;ka</span></i><a name="NtA_18"
+  href="#Nt_18"><sup>[18]</sup></a> had different shapes in various parts
+  of India, as was the case with the letters. In his <i>Chronology of
+  Ancient Nations</i> he gives the sum of a geometric progression and shows
+  how, in order to avoid any possibility of error, the number may be
+  expressed in three different systems: with Indian symbols, in sexagesimal
+  notation, and by an alphabet system which will be touched upon later. He
+  also speaks<a name="NtA_19" href="#Nt_19"><sup>[19]</sup></a> of "179,
+  876, 755, expressed in Indian ciphers," thus again attributing these
+  forms to Hindu sources.</p>
+
+  <p>Preceding Al-B&#x12B;r&#x16B;n&#x12B; there was another Arabic writer
+  of the tenth century, <span class="special" title="Motahhar ibn Tahir"
+  >Mo&#x1E6D;ahhar ibn &#x1E6C;&#x101;hir</span>,<a name="NtA_20"
+  href="#Nt_20"><sup>[20]</sup></a> author of the <i>Book of the Creation
+  and of History</i>, who gave as a curiosity, in Indian
+  (N&#x101;gar&#x12B;) symbols, a large number asserted by the people of
+  India to represent the duration of the world. Huart feels positive that
+  in <span class="special" title="Motahhar's">Mo&#x1E6D;ahhar's</span> time
+  the present Arabic symbols had not yet come into use, and that the Indian
+  symbols, although known to scholars, were not current. Unless this were
+  the case, neither the author nor his readers would have found anything
+  extraordinary in the appearance of the number which he cites.</p>
+
+  <p>Mention should also be made of a widely-traveled student,
+  Al-Mas&#x201B;&#x16B;d&#x12B; (885?-956), whose journeys carried him from
+  Bagdad to Persia, India, Ceylon, and even <!-- Page 8 --><span
+  class="pagenum"><a name="page8"></a>[8]</span>across the China sea, and
+  at other times to Madagascar, Syria, and Palestine.<a name="NtA_21"
+  href="#Nt_21"><sup>[21]</sup></a> He seems to have neglected no
+  accessible sources of information, examining also the history of the
+  Persians, the Hindus, and the Romans. Touching the period of the Caliphs
+  his work entitled <i>Meadows of Gold</i> furnishes a most entertaining
+  fund of information. He states<a name="NtA_22"
+  href="#Nt_22"><sup>[22]</sup></a> that the wise men of India, assembled
+  by the king, composed the <i>Sindhind</i>. Further on<a name="NtA_23"
+  href="#Nt_23"><sup>[23]</sup></a> he states, upon the authority of the
+  historian <span class="special" title="Mohammed">Mo&#x1E25;ammed</span>
+  ibn &#x201B;Al&#x12B; &#x201B;Abd&#x12B;, that by order of <span
+  class="special" title="Al-Mansur">Al-Man&#x1E63;&#x16B;r</span> many
+  works of science and astrology were translated into Arabic, notably the
+  <i>Sindhind</i> (<i>Siddh&#x101;nta</i>). Concerning the meaning and
+  spelling of this name there is considerable diversity of opinion.
+  Colebrooke<a name="NtA_24" href="#Nt_24"><sup>[24]</sup></a> first
+  pointed out the connection between <i>Siddh&#x101;nta</i> and
+  <i>Sindhind</i>. He ascribes to the word the meaning "the revolving
+  ages."<a name="NtA_25" href="#Nt_25"><sup>[25]</sup></a> Similar
+  designations are collected by Sédillot,<a name="NtA_26"
+  href="#Nt_26"><sup>[26]</sup></a> who inclined to the Greek origin of the
+  sciences commonly attributed to the Hindus.<a name="NtA_27"
+  href="#Nt_27"><sup>[27]</sup></a> Casiri,<a name="NtA_28"
+  href="#Nt_28"><sup>[28]</sup></a> citing the <i><span class="special"
+  title="Tarikh al-hokama">T&#x101;r&#x12B;kh
+  al-&#x1E25;okam&#x101;</span></i> or <i>Chronicles of the Learned</i>,<a
+  name="NtA_29" href="#Nt_29"><sup>[29]</sup></a> refers to the work <!--
+  Page 9 --><span class="pagenum"><a name="page9"></a>[9]</span>as the
+  <i>Sindum-Indum</i> with the meaning "perpetuum æternumque." The
+  reference<a name="NtA_30" href="#Nt_30"><sup>[30]</sup></a> in this
+  ancient Arabic work to Al-Khow&#x101;razm&#x12B; is worthy of note.</p>
+
+  <p>This <i>Sindhind</i> is the book, says Mas&#x201B;&#x16B;d&#x12B;,<a
+  name="NtA_31" href="#Nt_31"><sup>[31]</sup></a> which gives all that the
+  Hindus know of the spheres, the stars, arithmetic,<a name="NtA_32"
+  href="#Nt_32"><sup>[32]</sup></a> and the other branches of science. He
+  mentions also Al-Khow&#x101;razm&#x12B; and <span class="special"
+  title="Habash">&#x1E24;abash</span><a name="NtA_33"
+  href="#Nt_33"><sup>[33]</sup></a> as translators of the tables of the
+  <i>Sindhind</i>. Al-B&#x12B;r&#x16B;n&#x12B;<a name="NtA_34"
+  href="#Nt_34"><sup>[34]</sup></a> refers to two other translations from a
+  work furnished by a Hindu who came to Bagdad as a member of the political
+  mission which Sindh sent to the caliph <span class="special"
+  title="Al-Mansur">Al-Man&#x1E63;&#x16B;r</span>, in the year of the
+  Hejira 154 (<span class="scac">A.D.</span> 771).</p>
+
+  <p>The oldest work, in any sense complete, on the history of Arabic
+  literature and history is the <i>Kit&#x101;b al-Fihrist</i>, written in
+  the year 987 <span class="scac">A.D.</span>, by Ibn Ab&#x12B;
+  Ya&#x201B;q&#x16B;b al-Nad&#x12B;m. It is of fundamental importance for
+  the history of Arabic culture. Of the ten chief divisions of the work,
+  the seventh demands attention in this discussion for the reason that its
+  second subdivision treats of mathematicians and astronomers.<a
+  name="NtA_35" href="#Nt_35"><sup>[35]</sup></a></p>
+
+<p><!-- Page 10 --><span class="pagenum"><a name="page10"></a>[10]</span></p>
+
+  <p>The first of the Arabic writers mentioned is Al-Kind&#x12B; (800-870
+  <span class="scac">A.D.</span>), who wrote five books on arithmetic and
+  four books on the use of the Indian method of reckoning. Sened ibn
+  &#x201B;Al&#x12B;, the Jew, who was converted to Islam under the caliph
+  Al-M&#x101;m&#x16B;n, is also given as the author of a work on the Hindu
+  method of reckoning. Nevertheless, there is a possibility<a name="NtA_36"
+  href="#Nt_36"><sup>[36]</sup></a> that some of the works ascribed to
+  Sened ibn &#x201B;Al&#x12B; are really works of
+  Al-Khow&#x101;razm&#x12B;, whose name immediately precedes his. However,
+  it is to be noted in this connection that Casiri<a name="NtA_37"
+  href="#Nt_37"><sup>[37]</sup></a> also mentions the same writer as the
+  author of a most celebrated work on arithmetic.</p>
+
+  <p>To <span class="special"
+  title="Al-Sufi">Al-&#x1E62;&#x16B;f&#x12B;</span>, who died in 986 <span
+  class="scac">A.D.</span>, is also credited a large work on the same
+  subject, and similar treatises by other writers are mentioned. We are
+  therefore forced to the conclusion that the Arabs from the early ninth
+  century on fully recognized the Hindu origin of the new numerals.</p>
+
+  <p>Leonard of Pisa, of whom we shall speak at length in the chapter on
+  the Introduction of the Numerals into Europe, wrote his <i>Liber
+  Abbaci</i><a name="NtA_38" href="#Nt_38"><sup>[38]</sup></a> in 1202. In
+  this work he refers frequently to the nine Indian figures,<a
+  name="NtA_39" href="#Nt_39"><sup>[39]</sup></a> thus showing again the
+  general consensus of opinion in the Middle Ages that the numerals were of
+  Hindu origin.</p>
+
+  <p>Some interest also attaches to the oldest documents on arithmetic in
+  our own language. One of the earliest <!-- Page 11 --><span
+  class="pagenum"><a name="page11"></a>[11]</span>treatises on algorism is
+  a commentary<a name="NtA_40" href="#Nt_40"><sup>[40]</sup></a> on a set
+  of verses called the <i>Carmen de Algorismo</i>, written by Alexander de
+  Villa Dei (Alexandra de Ville-Dieu), a Minorite monk of about 1240 <span
+  class="scac">A.D.</span> The text of the first few lines is as
+  follows:</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p class="hg3">"Hec algorism' ars p'sens dicit' in qua</p>
+      <p>Talib; indor<a href="images/017a.png"><img src="images/017a.png" class="middle" style="height:1.2ex" alt="um" /></a> fruim bis quinq; figuris.<a name="NtA_41" href="#Nt_41"><sup>[41]</sup></a></p>
+    </div>
+  </div>
+  <p>"This boke is called the boke of algorim or augrym after lewder use.
+  And this boke tretys of the Craft of Nombryng, the quych crafte is called
+  also Algorym. Ther was a kyng of Inde the quich heyth Algor &amp; he made
+  this craft.... Algorisms, in the quych we use teen figurys of Inde."</p>
+
+<hr class="full" >
+
+<p><!-- Page 12 --><span class="pagenum"><a name="page12"></a>[12]</span></p>
+
+<h3>CHAPTER II</h3>
+
+<p class="cenhead">EARLY HINDU FORMS WITH NO PLACE VALUE</p>
+
+  <p>While it is generally conceded that the scientific development of
+  astronomy among the Hindus towards the beginning of the Christian era
+  rested upon Greek<a name="NtA_42" href="#Nt_42"><sup>[42]</sup></a> or
+  Chinese<a name="NtA_43" href="#Nt_43"><sup>[43]</sup></a> sources, yet
+  their ancient literature testifies to a high state of civilization, and
+  to a considerable advance in sciences, in philosophy, and along literary
+  lines, long before the golden age of Greece. From the earliest times even
+  up to the present day the Hindu has been wont to put his thought into
+  rhythmic form. The first of this poetry&mdash;it well deserves this name,
+  being also worthy from a metaphysical point of view<a name="NtA_44"
+  href="#Nt_44"><sup>[44]</sup></a>&mdash;consists of the Vedas, hymns of
+  praise and poems of worship, collected during the Vedic period which
+  dates from approximately 2000 <span class="scac">B.C.</span> to 1400
+  <span class="scac">B.C.</span><a name="NtA_45"
+  href="#Nt_45"><sup>[45]</sup></a> Following this work, or possibly
+  contemporary with it, is the Brahmanic literature, which is partly
+  ritualistic (the <span class="special"
+  title="Brahmanas">Br&#x101;hma&#x1E47;as</span>), and partly
+  philosophical (the Upanishads). Our especial interest is <!-- Page 13
+  --><span class="pagenum"><a name="page13"></a>[13]</span>in the
+  S&#x16B;tras, versified abridgments of the ritual and of ceremonial
+  rules, which contain considerable geometric material used in connection
+  with altar construction, and also numerous examples of rational numbers
+  the sum of whose squares is also a square, i.e. "Pythagorean numbers,"
+  although this was long before Pythagoras lived. Whitney<a name="NtA_46"
+  href="#Nt_46"><sup>[46]</sup></a> places the whole of the Veda
+  literature, including the Vedas, the <span class="special"
+  title="Brahmanas">Br&#x101;hma&#x1E47;as</span>, and the S&#x16B;tras,
+  between 1500 <span class="scac">B.C.</span> and 800 <span
+  class="scac">B.C.</span>, thus agreeing with Bürk<a name="NtA_47"
+  href="#Nt_47"><sup>[47]</sup></a> who holds that the knowledge of the
+  Pythagorean theorem revealed in the S&#x16B;tras goes back to the eighth
+  century <span class="scac">B.C.</span></p>
+
+  <p>The importance of the S&#x16B;tras as showing an independent origin of
+  Hindu geometry, contrary to the opinion long held by Cantor<a
+  name="NtA_48" href="#Nt_48"><sup>[48]</sup></a> of a Greek origin, has
+  been repeatedly emphasized in recent literature,<a name="NtA_49"
+  href="#Nt_49"><sup>[49]</sup></a> especially since the appearance of the
+  important work of Von Schroeder.<a name="NtA_50"
+  href="#Nt_50"><sup>[50]</sup></a> Further fundamental mathematical
+  notions such as the conception of irrationals and the use of gnomons, as
+  well as the philosophical doctrine of the transmigration of
+  souls,&mdash;all of these having long been attributed to the
+  Greeks,&mdash;are shown in these works to be native to India. Although
+  this discussion does not bear directly upon the <!-- Page 14 --><span
+  class="pagenum"><a name="page14"></a>[14]</span>origin of our numerals,
+  yet it is highly pertinent as showing the aptitude of the Hindu for
+  mathematical and mental work, a fact further attested by the independent
+  development of the drama and of epic and lyric poetry.</p>
+
+  <p>It should be stated definitely at the outset, however, that we are not
+  at all sure that the most ancient forms of the numerals commonly known as
+  Arabic had their origin in India. As will presently be seen, their forms
+  may have been suggested by those used in Egypt, or in Eastern Persia, or
+  in China, or on the plains of Mesopotamia. We are quite in the dark as to
+  these early steps; but as to their development in India, the approximate
+  period of the rise of their essential feature of place value, their
+  introduction into the Arab civilization, and their spread to the West, we
+  have more or less definite information. When, therefore, we consider the
+  rise of the numerals in the land of the Sindhu,<a name="NtA_51"
+  href="#Nt_51"><sup>[51]</sup></a> it must be understood that it is only
+  the large movement that is meant, and that there must further be
+  considered the numerous possible sources outside of India itself and long
+  anterior to the first prominent appearance of the number symbols.</p>
+
+  <p>No one attempts to examine any detail in the history of ancient India
+  without being struck with the great dearth of reliable material.<a
+  name="NtA_52" href="#Nt_52"><sup>[52]</sup></a> So little sympathy have
+  the people with any save those of their own caste that a general
+  literature is wholly lacking, and it is only in the observations of
+  strangers that any all-round view of scientific progress is to be found.
+  There is evidence that primary schools <!-- Page 15 --><span
+  class="pagenum"><a name="page15"></a>[15]</span>existed in earliest
+  times, and of the seventy-two recognized sciences writing and arithmetic
+  were the most prized.<a name="NtA_53" href="#Nt_53"><sup>[53]</sup></a>
+  In the Vedic period, say from 2000 to 1400 <span
+  class="scac">B.C.</span>, there was the same attention to astronomy that
+  was found in the earlier civilizations of Babylon, China, and Egypt, a
+  fact attested by the Vedas themselves.<a name="NtA_54"
+  href="#Nt_54"><sup>[54]</sup></a> Such advance in science presupposes a
+  fair knowledge of calculation, but of the manner of calculating we are
+  quite ignorant and probably always shall be. One of the Buddhist sacred
+  books, the <i>Lalitavistara</i>, relates that when the
+  B&#x14D;dhisattva<a name="NtA_55" href="#Nt_55"><sup>[55]</sup></a> was
+  of age to marry, the father of Gopa, his intended bride, demanded an
+  examination of the five hundred suitors, the subjects including
+  arithmetic, writing, the lute, and archery. Having vanquished his rivals
+  in all else, he is matched against Arjuna the great arithmetician and is
+  asked to express numbers greater than 100 kotis.<a name="NtA_56"
+  href="#Nt_56"><sup>[56]</sup></a> In reply he gave a scheme of number
+  names as high as 10<sup>53</sup>, adding that he could proceed as far as
+  10<sup>421</sup>,<a name="NtA_57" href="#Nt_57"><sup>[57]</sup></a> all
+  of which suggests the system of Archimedes and the unsettled question of
+  the indebtedness of the West to the East in the realm of ancient
+  mathematics.<a name="NtA_58" href="#Nt_58"><sup>[58]</sup></a> Sir Edwin
+  Arnold, <!-- Page 16 --><span class="pagenum"><a
+  name="page16"></a>[16]</span>in <i>The Light of Asia</i>, does not
+  mention this part of the contest, but he speaks of Buddha's training at
+  the hands of the learned <span class="special"
+  title="Visvamitra">Vi&#x1E63;vamitra</span>:</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>"And Viswamitra said, 'It is enough,</p>
+      <p>Let us to numbers. After me repeat</p>
+      <p>Your numeration till we reach the lakh,<a name="NtA_59" href="#Nt_59"><sup>[59]</sup></a></p>
+      <p>One, two, three, four, to ten, and then by tens</p>
+      <p>To hundreds, thousands.' After him the child</p>
+      <p>Named digits, decads, centuries, nor paused,</p>
+      <p>The round lakh reached, but softly murmured on,</p>
+      <p>Then comes the k&#x14D;ti, nahut, ninnahut,</p>
+      <p>Khamba, viskhamba, abab, attata,</p>
+      <p>To kumuds, gundhikas, and utpalas,</p>
+      <p>By pundar&#x12B;kas into padumas,</p>
+      <p>Which last is how you count the utmost grains</p>
+      <p>Of Hastagiri ground to finest dust;<a name="NtA_60" href="#Nt_60"><sup>[60]</sup></a></p>
+      <p>But beyond that a numeration is,</p>
+      <p>The K&#x101;tha, used to count the stars of night,</p>
+      <p>The K&#x14D;ti-K&#x101;tha, for the ocean drops;</p>
+      <p>Ingga, the calculus of circulars;</p>
+      <p>Sarvanikchepa, by the which you deal</p>
+      <p>With all the sands of Gunga, till we come</p>
+      <p>To Antah-Kalpas, where the unit is</p>
+      <p>The sands of the ten crore Gungas.  If one seeks</p>
+      <p>More comprehensive scale, th' arithmic mounts</p>
+      <p>By the Asankya, which is the tale</p>
+      <p>Of all the drops that in ten thousand years</p>
+      <p>Would fall on all the worlds by daily rain;</p>
+      <p>Thence unto Maha Kalpas, by the which</p>
+      <p>The gods compute their future and their past.'"</p>
+    </div>
+  </div>
+<p><!-- Page 17 --><span class="pagenum"><a name="page17"></a>[17]</span></p>
+
+  <p>Thereupon <span class="special" title="Visvamitra Acarya"
+  >Vi&#x1E63;vamitra &#x100;c&#x101;rya</span><a name="NtA_61"
+  href="#Nt_61"><sup>[61]</sup></a> expresses his approval of the task, and
+  asks to hear the "measure of the line" as far as y&#x14D;jana, the
+  longest measure bearing name. This given, Buddha adds:</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>... "'And master! if it please,</p>
+      <p>I shall recite how many sun-motes lie</p>
+      <p>From end to end within a y&#x14D;jana.'</p>
+      <p>Thereat, with instant skill, the little prince</p>
+      <p>Pronounced the total of the atoms true.</p>
+      <p>But Viswamitra heard it on his face</p>
+      <p>Prostrate before the boy; 'For thou,' he cried,</p>
+      <p>'Art Teacher of thy teachers&mdash;thou, not I,</p>
+      <p>Art G&#x16B;r&#x16B;.'"</p>
+    </div>
+  </div>
+  <p>It is needless to say that this is far from being history. And yet it
+  puts in charming rhythm only what the ancient <i>Lalitavistara</i>
+  relates of the number-series of the Buddha's time. While it extends
+  beyond all reason, nevertheless it reveals a condition that would have
+  been impossible unless arithmetic had attained a considerable degree of
+  advancement.</p>
+
+  <p>To this pre-Christian period belong also the <i><span class="special"
+  title="Vedangas">Ved&#x101;&#x1E45;gas</span></i>, or "limbs for
+  supporting the Veda," part of that great branch of Hindu literature known
+  as <i><span class="special" title="Smriti">Sm&#x1E5B;iti</span></i>
+  (recollection), that which was to be handed down by tradition. Of these
+  the sixth is known as <i><span class="special"
+  title="Jyotisa">Jyoti&#x1E63;a</span></i> (astronomy), a short treatise
+  of only thirty-six verses, written not earlier than 300 <span
+  class="scac">B.C.</span>, and affording us some knowledge of the extent
+  of number work in that period.<a name="NtA_62"
+  href="#Nt_62"><sup>[62]</sup></a> The Hindus <!-- Page 18 --><span
+  class="pagenum"><a name="page18"></a>[18]</span>also speak of eighteen
+  ancient Siddh&#x101;ntas or astronomical works, which, though mostly
+  lost, confirm this evidence.<a name="NtA_63"
+  href="#Nt_63"><sup>[63]</sup></a></p>
+
+  <p>As to authentic histories, however, there exist in India none relating
+  to the period before the Mohammedan era (622 <span
+  class="scac">A.D.</span>). About all that we know of the earlier
+  civilization is what we glean from the two great epics, the
+  Mah&#x101;bh&#x101;rata<a name="NtA_64" href="#Nt_64"><sup>[64]</sup></a>
+  and the R&#x101;m&#x101;yana, from coins, and from a few inscriptions.<a
+  name="NtA_65" href="#Nt_65"><sup>[65]</sup></a></p>
+
+  <p>It is with this unsatisfactory material, then, that we have to deal in
+  searching for the early history of the Hindu-Arabic numerals, and the
+  fact that many unsolved problems exist and will continue to exist is no
+  longer strange when we consider the conditions. It is rather surprising
+  that so much has been discovered within a century, than that we are so
+  uncertain as to origins and dates and the early spread of the system. The
+  probability being that writing was not introduced into India before the
+  close of the fourth century <span class="scac">B.C.</span>, and
+  literature existing only in spoken form prior to that period,<a
+  name="NtA_66" href="#Nt_66"><sup>[66]</sup></a> the number work was
+  doubtless that of all primitive peoples, palpable, merely a matter of
+  placing sticks or cowries or pebbles on the ground, of marking a
+  sand-covered board, or of cutting notches or tying cords as is still done
+  in parts of Southern India to-day.<a name="NtA_67"
+  href="#Nt_67"><sup>[67]</sup></a></p>
+
+<p><!-- Page 19 --><span class="pagenum"><a name="page19"></a>[19]</span></p>
+
+  <p>The early Hindu numerals<a name="NtA_68"
+  href="#Nt_68"><sup>[68]</sup></a> may be classified into three great
+  groups, (1) the <span class="special"
+  title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span>, (2) the
+  Br&#x101;hm&#x12B;, and (3) the word and letter forms; and these will be
+  considered in order.</p>
+
+  <p>The <span class="special"
+  title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span> numerals are found
+  in inscriptions formerly known as Bactrian, Indo-Bactrian, and Aryan, and
+  appearing in ancient Gandh&#x101;ra, now eastern Afghanistan and northern
+  Punjab. The alphabet of the language is found in inscriptions dating from
+  the fourth century <span class="scac">B.C.</span> to the third century
+  <span class="scac">A.D.</span>, and from the fact that the words are
+  written from right to left it is assumed to be of Semitic origin. No
+  numerals, however, have been found in the earliest of these inscriptions,
+  number-names probably having been written out in words as was the custom
+  with many ancient peoples. Not until the time of the powerful King
+  A&#x15B;oka, in the third century <span class="scac">B.C.</span>, do
+  numerals appear in any inscriptions thus far discovered; and then only in
+  the primitive form of marks, quite as they would be found in Egypt,
+  Greece, Rome, or in <!-- Page 20 --><span class="pagenum"><a
+  name="page20"></a>[20]</span>various other parts of the world. These
+  A&#x15B;oka<a name="NtA_69" href="#Nt_69"><sup>[69]</sup></a>
+  inscriptions, some thirty in all, are found in widely separated parts of
+  India, often on columns, and are in the various vernaculars that were
+  familiar to the people. Two are in the <span class="special"
+  title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span> characters, and
+  the rest in some form of Br&#x101;hm&#x12B;. In the <span class="special"
+  title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span> inscriptions only
+  four numerals have been found, and these are merely vertical marks for
+  one, two, four, and five, thus:</p>
+
+  <div class="figcenter" style="width:25%;">
+      <a href="images/026a.png"><img style="width:100%" src="images/026a.png"
+      alt="Numerals in Kharosthi inscriptions." title="Numerals in Kharosthi inscriptions." /></a>
+  </div>
+  <p>In the so-called &#x15A;aka inscriptions, possibly of the first
+  century <span class="scac">B.C.</span>, more numerals are found, and in
+  more highly developed form, the right-to-left system appearing, together
+  with evidences of three different scales of counting,&mdash;four, ten,
+  and twenty. The numerals of this period are as follows:</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/026b.png"><img style="width:100%" src="images/026b.png"
+      alt="Numerals in Saka inscriptions." title="Numerals in Saka inscriptions." /></a>
+  </div>
+  <p>There are several noteworthy points to be observed in studying this
+  system. In the first place, it is probably not as early as that shown in
+  the N&#x101;n&#x101; Gh&#x101;t forms hereafter given, although the
+  inscriptions themselves at N&#x101;n&#x101; Gh&#x101;t are later than
+  those of the A&#x15B;oka period. The <!-- Page 21 --><span
+  class="pagenum"><a name="page21"></a>[21]</span>four is to this system
+  what the X was to the Roman, probably a canceling of three marks as a
+  workman does to-day for five, or a laying of one stick across three
+  others. The ten has never been satisfactorily explained. It is similar to
+  the A of the <span class="special"
+  title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span> alphabet, but we
+  have no knowledge as to why it was chosen. The twenty is evidently a
+  ligature of two tens, and this in turn suggested a kind of radix, so that
+  ninety was probably written in a way reminding one of the
+  quatre-vingt-dix of the French. The hundred is unexplained, although it
+  resembles the letter <i>ta</i> or <i>tra</i> of the Br&#x101;hm&#x12B;
+  alphabet with 1 before (to the right of) it. The two hundred is only a
+  variant of the symbol for hundred, with two vertical marks.<a
+  name="NtA_70" href="#Nt_70"><sup>[70]</sup></a></p>
+
+  <p>This system has many points of similarity with the Nabatean numerals<a
+  name="NtA_71" href="#Nt_71"><sup>[71]</sup></a> in use in the first
+  centuries of the Christian era. The cross is here used for four, and the
+  <span class="special"
+  title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span> form is employed
+  for twenty. In addition to this there is a trace of an analogous use of a
+  scale of twenty. While the symbol for 100 is quite different, the method
+  of forming the other hundreds is the same. The correspondence seems to be
+  too marked to be wholly accidental.</p>
+
+  <p>It is not in the <span class="special"
+  title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span> numerals,
+  therefore, that we can hope to find the origin of those used by us, and
+  we turn to the second of the Indian types, the Br&#x101;hm&#x12B;
+  characters. The alphabet attributed to Brahm&#x101; is the oldest of the
+  several known in India, and was used from the earliest historic times.
+  There are various theories of its origin, <!-- Page 22 --><span
+  class="pagenum"><a name="page22"></a>[22]</span>none of which has as yet
+  any wide acceptance,<a name="NtA_72" href="#Nt_72"><sup>[72]</sup></a>
+  although the problem offers hope of solution in due time. The numerals
+  are not as old as the alphabet, or at least they have not as yet been
+  found in inscriptions earlier than those in which the edicts of
+  A&#x15B;oka appear, some of these having been incised in
+  Br&#x101;hm&#x12B; as well as <span class="special"
+  title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span>. As already
+  stated, the older writers probably wrote the numbers in words, as seems
+  to have been the case in the earliest Pali writings of Ceylon.<a
+  name="NtA_73" href="#Nt_73"><sup>[73]</sup></a></p>
+
+  <p>The following numerals are, as far as known, the only ones to appear
+  in the A&#x15B;oka edicts:<a name="NtA_74"
+  href="#Nt_74"><sup>[74]</sup></a></p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/028a.png"><img style="width:100%" src="images/028a.png"
+      alt="Numerals in Asoka edicts." title="Numerals in Asoka edicts." /></a>
+  </div>
+  <p>These fragments from the third century <span class="scac">B.C.</span>,
+  crude and unsatisfactory as they are, are the undoubted early forms from
+  which our present system developed. They next appear in the second
+  century <span class="scac">B.C.</span> in some inscriptions in the cave
+  on the top of the N&#x101;n&#x101; Gh&#x101;t hill, about seventy-five
+  miles from Poona in central India. These inscriptions may be memorials of
+  the early Andhra dynasty of southern India, but their chief interest lies
+  in the numerals which they contain.</p>
+
+  <p>The cave was made as a resting-place for travelers ascending the hill,
+  which lies on the road from Kaly&#x101;na to Junar. It seems to have been
+  cut out by a descendant <!-- Page 23 --><span class="pagenum"><a
+  name="page23"></a>[23]</span>of King &#x15A;&#x101;tav&#x101;hana,<a
+  name="NtA_75" href="#Nt_75"><sup>[75]</sup></a> for inside the wall
+  opposite the entrance are representations of the members of his family,
+  much defaced, but with the names still legible. It would seem that the
+  excavation was made by order of a king named Vedisiri, and "the
+  inscription contains a list of gifts made on the occasion of the
+  performance of several <i>yagnas</i> or religious sacrifices," and
+  numerals are to be seen in no less than thirty places.<a name="NtA_76"
+  href="#Nt_76"><sup>[76]</sup></a></p>
+
+  <p>There is considerable dispute as to what numerals are really found in
+  these inscriptions, owing to the difficulty of deciphering them; but the
+  following, which have been copied from a rubbing, are probably number
+  forms:<a name="NtA_77" href="#Nt_77"><sup>[77]</sup></a></p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/029a.png"><img style="width:100%" src="images/029a.png"
+      alt="Numerals from Nana Ghat inscriptions." title="Numerals from Nana Ghat inscriptions." /></a>
+  </div>
+  <p>The inscription itself, so important as containing the earliest
+  considerable Hindu numeral system connected with our own, is of
+  sufficient interest to warrant reproducing part of it in facsimile, as is
+  done on page 24.</p>
+
+<p><!-- Page 24 --><span class="pagenum"><a name="page24"></a>[24]</span></p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/030a.png"><img style="width:100%" src="images/030a.png"
+      alt="Nânâghât Inscriptions." title="Nânâghât Inscriptions." /></a>
+  </div>
+  <p>The next very noteworthy evidence of the numerals, and this quite
+  complete as will be seen, is found in certain other cave inscriptions
+  dating back to the first or second century <span class="scac">A.D.</span>
+  In these, the Nasik<a name="NtA_78" href="#Nt_78"><sup>[78]</sup></a>
+  cave inscriptions, the forms are as follows:</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/030b.png"><img style="width:100%" src="images/030b.png"
+      alt="Numerals from Nasik cave inscriptions." title="Numerals from Nasik cave inscriptions." /></a>
+  </div>
+  <p>From this time on, until the decimal system finally adopted the first
+  nine characters and replaced the rest of the Br&#x101;hm&#x12B; notation
+  by adding the zero, the progress of these forms is well marked. It is
+  therefore well to present synoptically the best-known specimens that have
+  come down to us, and this is done in the table on page 25.<a
+  name="NtA_79" href="#Nt_79"><sup>[79]</sup></a></p>
+
+<p><!-- Page 25 --><span class="pagenum"><a name="page25"></a>[25]</span></p>
+
+<h3><span class="sc">Table showing the Progress of Number Forms in India</span></h3>
+
+<table class="nobctr">
+<tr><td><span class="sc">Numerals</span></td><td><a href="images/031.png"><img src="images/031.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr>
+<tr><td>A&#x15B;oka<a name="NtA_80" href="#Nt_80"><sup>[80]</sup></a></td><td><a href="images/031a.png"><img src="images/031a.png" class="middle" style="height:4.5ex" alt="Asoka" /></a></td></tr>
+<tr><td>&#x15A;aka<a name="NtA_81" href="#Nt_81"><sup>[81]</sup></a></td><td><a href="images/031b.png"><img src="images/031b.png" class="middle" style="height:4.5ex" alt="Saka" /></a></td></tr>
+<tr><td>A&#x15B;oka<a name="NtA_82" href="#Nt_82"><sup>[82]</sup></a></td><td><a href="images/031c.png"><img src="images/031c.png" class="middle" style="height:4.5ex" alt="Asoka" /></a></td></tr>
+<tr><td>N&#x101;gar&#x12B;<a name="NtA_83" href="#Nt_83"><sup>[83]</sup></a></td><td><a href="images/031d.png"><img src="images/031d.png" class="middle" style="height:4.5ex" alt="Nagari" /></a></td></tr>
+<tr><td>Nasik<a name="NtA_84" href="#Nt_84"><sup>[84]</sup></a></td><td><a href="images/031e.png"><img src="images/031e.png" class="middle" style="height:4.5ex" alt="Nasik" /></a></td></tr>
+<tr><td><span class="special" title="Ksatrapa">K&#x1E63;atrapa</span><a name="NtA_85" href="#Nt_85"><sup>[85]</sup></a></td><td><a href="images/031f.png"><img src="images/031f.png" class="middle" style="height:4.5ex" alt="Ksatrapa" /></a></td></tr>
+<tr><td><span class="special" title="Kusana">Ku&#x1E63;ana</span><a name="NtA_86" href="#Nt_86"><sup>[86]</sup></a></td><td><a href="images/031g.png"><img src="images/031g.png" class="middle" style="height:4.5ex" alt="Kusana" /></a></td></tr>
+<tr><td>Gupta<a name="NtA_87" href="#Nt_87"><sup>[87]</sup></a></td><td><a href="images/031h.png"><img src="images/031h.png" class="middle" style="height:4.5ex" alt="Gupta" /></a></td></tr>
+<tr><td>Valhab&#x12B;<a name="NtA_88" href="#Nt_88"><sup>[88]</sup></a></td><td><a href="images/031i.png"><img src="images/031i.png" class="middle" style="height:4.5ex" alt="Valhabi" /></a></td></tr>
+<tr><td>Nepal<a name="NtA_89" href="#Nt_89"><sup>[89]</sup></a></td><td><a href="images/031j.png"><img src="images/031j.png" class="middle" style="height:4.5ex" alt="Nepal" /></a></td></tr>
+<tr><td><span class="special" title="Kalinga">Kali&#x1E45;ga</span><a name="NtA_90" href="#Nt_90"><sup>[90]</sup></a></td><td><a href="images/031k.png"><img src="images/031k.png" class="middle" style="height:4.5ex" alt="Kalinga" /></a></td></tr>
+<tr><td><span class="special" title="Vakataka">V&#x101;k&#x101;&#x1E6D;aka</span><a name="NtA_91" href="#Nt_91"><sup>[91]</sup></a></td><td><a href="images/031l.png"><img src="images/031l.png" class="middle" style="height:4.5ex" alt="Vakataka" /></a></td></tr>
+</table>
+
+  <p>[Most of these numerals are given by Bühler, loc. cit., Tafel IX.]</p>
+
+<p><!-- Page 26 --><span class="pagenum"><a name="page26"></a>[26]</span></p>
+
+  <p>With respect to these numerals it should first be noted that no zero
+  appears in the table, and as a matter of fact none existed in any of the
+  cases cited. It was therefore impossible to have any place value, and the
+  numbers like twenty, thirty, and other multiples of ten, one hundred, and
+  so on, required separate symbols except where they were written out in
+  words. The ancient Hindus had no less than twenty of these symbols,<a
+  name="NtA_92" href="#Nt_92"><sup>[92]</sup></a> a number that was
+  afterward greatly increased. The following are examples of their method
+  of indicating certain numbers between one hundred and one thousand:</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p><a name="NtA_93" href="#Nt_93"><sup>[93]</sup></a> <a href="images/032a.png"><img src="images/032a.png" class="middle" style="height:3ex" alt="Numerals" /></a> for 174</p>
+      <p><a name="NtA_94" href="#Nt_94"><sup>[94]</sup></a> <a href="images/032b.png"><img src="images/032b.png" class="middle" style="height:3ex" alt="Numerals" /></a> for 191</p>
+      <p><a name="NtA_95" href="#Nt_95"><sup>[95]</sup></a> <a href="images/032c.png"><img src="images/032c.png" class="middle" style="height:3ex" alt="Numerals" /></a> for 269</p>
+      <p><a name="NtA_96" href="#Nt_96"><sup>[96]</sup></a> <a href="images/032d.png"><img src="images/032d.png" class="middle" style="height:3ex" alt="Numerals" /></a> for 252</p>
+      <p><a name="NtA_97" href="#Nt_97"><sup>[97]</sup></a> <a href="images/032e.png"><img src="images/032e.png" class="middle" style="height:3ex" alt="Numerals" /></a> for 400</p>
+      <p><a name="NtA_98" href="#Nt_98"><sup>[98]</sup></a> <a href="images/032f.png"><img src="images/032f.png" class="middle" style="height:3.5ex" alt="Numerals" /></a> for 356</p>
+    </div>
+  </div>
+<p><!-- Page 27 --><span class="pagenum"><a name="page27"></a>[27]</span></p>
+
+  <p>To these may be added the following numerals below one hundred,
+  similar to those in the table:</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p><a href="images/033a.png"><img src="images/033a.png" class="middle" style="height:3ex" alt="Numerals" /></a><a name="NtA_99" href="#Nt_99"><sup>[99]</sup></a>    for 90</p>
+      <p><a href="images/033b.png"><img src="images/033b.png" class="middle" style="height:3ex" alt="Numerals" /></a><a name="NtA_100" href="#Nt_100"><sup>[100]</sup></a>   for 70</p>
+    </div>
+  </div>
+  <p>We have thus far spoken of the <span class="special"
+  title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span> and
+  Br&#x101;hm&#x12B; numerals, and it remains to mention the third type,
+  the word and letter forms. These are, however, so closely connected with
+  the perfecting of the system by the invention of the zero that they are
+  more appropriately considered in the next chapter, particularly as they
+  have little relation to the problem of the origin of the forms known as
+  the Arabic.</p>
+
+  <p>Having now examined types of the early forms it is appropriate to turn
+  our attention to the question of their origin. As to the first three
+  there is no question. The <a href="images/033c.png"><img
+  src="images/033c.png" class="middle" style="height:1.5ex" alt="1 vertical
+  stroke" /></a> or <a href="images/033d.png"><img src="images/033d.png"
+  class="middle" style="height:1.5ex" alt="1 horizontal stroke" /></a> is
+  simply one stroke, or one stick laid down by the computer. The <a
+  href="images/033e.png"><img src="images/033e.png" class="middle"
+  style="height:1.5ex" alt="2 vertical strokes" /></a> or <a
+  href="images/033f.png"><img src="images/033f.png" class="middle"
+  style="height:1.5ex" alt="2 horizontal strokes" /></a> represents two
+  strokes or two sticks, and so for the <a href="images/033g.png"><img
+  src="images/033g.png" class="middle" style="height:1.5ex" alt="3 vertical
+  strokes" /></a> and <a href="images/033h.png"><img src="images/033h.png"
+  class="middle" style="height:1.5ex" alt="3 horizontal strokes" /></a>.
+  From some primitive <a href="images/033e.png"><img src="images/033e.png"
+  class="middle" style="height:1.5ex" alt="2 vertical strokes" /></a> came
+  the two of Egypt, of Rome, of early Greece, and of various other
+  civilizations. It appears in the three Egyptian numeral systems in the
+  following forms:</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Hieroglyphic <a href="images/033e.png"><img src="images/033e.png" class="middle" style="height:1.5ex" alt="2 vertical strokes" /></a></p>
+      <p>Hieratic     <a href="images/033i.png"><img src="images/033i.png" class="middle" style="height:1.7ex" alt="Hieratic 2" /></a></p>
+      <p>Demotic      <a href="images/033j.png"><img src="images/033j.png" class="middle" style="height:1.7ex" alt="Demotic 2" /></a></p>
+    </div>
+  </div>
+  <p>The last of these is merely a cursive form as in the Arabic <a
+  href="images/033k.png"><img src="images/033k.png" class="middle"
+  style="height:1.7ex" alt="Arabic 2" /></a>, which becomes our 2 if tipped
+  through a right angle. From some primitive <a href="images/033f.png"><img
+  src="images/033f.png" class="middle" style="height:1.5ex" alt="2
+  horizontal strokes" /></a> came the Chinese <!-- Page 28 --><span
+  class="pagenum"><a name="page28"></a>[28]</span>symbol, which is
+  practically identical with the symbols found commonly in India from 150
+  <span class="scac">B.C.</span> to 700 <span class="scac">A.D.</span> In
+  the cursive form it becomes <a href="images/034a.png"><img
+  src="images/034a.png" class="middle" style="height:1.5ex" alt="2
+  horizontal strokes joined" /></a>, and this was frequently used for two
+  in Germany until the 18th century. It finally went into the modern form
+  2, and the <a href="images/033h.png"><img src="images/033h.png"
+  class="middle" style="height:1.5ex" alt="3 horizontal strokes" /></a> in
+  the same way became our 3.</p>
+
+  <p>There is, however, considerable ground for interesting speculation
+  with respect to these first three numerals. The earliest Hindu forms were
+  perpendicular. In the N&#x101;n&#x101; Gh&#x101;t inscriptions they are
+  vertical. But long before either the A&#x15B;oka or the N&#x101;n&#x101;
+  Gh&#x101;t inscriptions the Chinese were using the horizontal forms for
+  the first three numerals, but a vertical arrangement for four.<a
+  name="NtA_101" href="#Nt_101"><sup>[101]</sup></a> Now where did China
+  get these forms? Surely not from India, for she had them, as her
+  monuments and literature<a name="NtA_102"
+  href="#Nt_102"><sup>[102]</sup></a> show, long before the Hindus knew
+  them. The tradition is that China brought her civilization around the
+  north of Tibet, from Mongolia, the primitive habitat being Mesopotamia,
+  or possibly the oases of Turkestan. Now what numerals did Mesopotamia
+  use? The Babylonian system, simple in its general principles but very
+  complicated in many of its details, is now well known.<a name="NtA_103"
+  href="#Nt_103"><sup>[103]</sup></a> In particular, one, two, and three
+  were represented by vertical arrow-heads. Why, then, did the Chinese
+  write <!-- Page 29 --><span class="pagenum"><a
+  name="page29"></a>[29]</span>theirs horizontally? The problem now takes a
+  new interest when we find that these Babylonian forms were not the
+  primitive ones of this region, but that the early Sumerian forms were
+  horizontal.<a name="NtA_104" href="#Nt_104"><sup>[104]</sup></a></p>
+
+  <p>What interpretation shall be given to these facts? Shall we say that
+  it was mere accident that one people wrote "one" vertically and that
+  another wrote it horizontally? This may be the case; but it may also be
+  the case that the tribal migrations that ended in the Mongol invasion of
+  China started from the Euphrates while yet the Sumerian civilization was
+  prominent, or from some common source in Turkestan, and that they carried
+  to the East the primitive numerals of their ancient home, the first
+  three, these being all that the people as a whole knew or needed. It is
+  equally possible that these three horizontal forms represent primitive
+  stick-laying, the most natural position of a stick placed in front of a
+  calculator being the horizontal one. When, however, the cuneiform writing
+  developed more fully, the vertical form may have been proved the easier
+  to make, so that by the time the migrations to the West began these were
+  in use, and from them came the upright forms of Egypt, Greece, Rome, and
+  other Mediterranean lands, and those of A&#x15B;oka's time in India.
+  After A&#x15B;oka, and perhaps among the merchants of earlier centuries,
+  the horizontal forms may have come down into India from China, thus
+  giving those of the N&#x101;n&#x101; Gh&#x101;t cave and of later
+  inscriptions. This is in the realm of speculation, but it is not
+  improbable that further epigraphical studies may confirm the
+  hypothesis.</p>
+
+<p><!-- Page 30 --><span class="pagenum"><a name="page30"></a>[30]</span></p>
+
+  <p>As to the numerals above three there have been very many conjectures.
+  The figure one of the Demotic looks like the one of the Sanskrit, the two
+  (reversed) like that of the Arabic, the four has some resemblance to that
+  in the Nasik caves, the five (reversed) to that on the <span
+  class="special" title="Ksatrapa">K&#x1E63;atrapa</span> coins, the nine
+  to that of the <span class="special" title="Kusana">Ku&#x1E63;ana</span>
+  inscriptions, and other points of similarity have been imagined. Some
+  have traced resemblance between the Hieratic five and seven and those of
+  the Indian inscriptions. There have not, therefore, been wanting those
+  who asserted an Egyptian origin for these numerals.<a name="NtA_105"
+  href="#Nt_105"><sup>[105]</sup></a> There has already been mentioned the
+  fact that the <span class="special"
+  title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span> numerals were
+  formerly known as Bactrian, Indo-Bactrian, and Aryan. Cunningham<a
+  name="NtA_106" href="#Nt_106"><sup>[106]</sup></a> was the first to
+  suggest that these numerals were derived from the alphabet of the
+  Bactrian civilization of Eastern Persia, perhaps a thousand years before
+  our era, and in this he was supported by the scholarly work of Sir E.
+  Clive Bayley,<a name="NtA_107" href="#Nt_107"><sup>[107]</sup></a> who in
+  turn was followed by Canon Taylor.<a name="NtA_108"
+  href="#Nt_108"><sup>[108]</sup></a> The resemblance has not proved
+  convincing, however, and Bayley's drawings <!-- Page 31 --><span
+  class="pagenum"><a name="page31"></a>[31]</span>have been criticized as
+  being affected by his theory. The following is part of the hypothesis:<a
+  name="NtA_109" href="#Nt_109"><sup>[109]</sup></a></p>
+
+<table class="nobctr">
+<tr><td><i>Numeral</i></td><td><i>Hindu</i></td><td><i>Bactrian</i></td><td><i>Sanskrit</i></td></tr>
+<tr><td align="center">4</td><td align="center"><a href="images/037a.png"><img src="images/037a.png" class="middle" style="height:3.6ex" alt="Symbol" /></a></td><td align="center"><a href="images/037b.png"><img src="images/037b.png" class="middle" style="height:3.6ex" alt="Symbol" /></a> = ch</td><td>chatur, Lat. quattuor</td></tr>
+<tr><td align="center">5</td><td align="center"><a href="images/037c.png"><img src="images/037c.png" class="middle" style="height:3.6ex" alt="Symbol" /></a></td><td align="center"><a href="images/037d.png"><img src="images/037d.png" class="middle" style="height:3.6ex" alt="Symbol" /></a> = p</td><td>pancha, Gk. <span title="pente" class="grk">&pi;&#x1F73;&nu;&tau;&epsilon;</span></td></tr>
+<tr><td align="center">6</td><td align="center"><a href="images/037e.png"><img src="images/037e.png" class="middle" style="height:3.6ex" alt="Symbol" /></a></td><td align="center"><a href="images/037f.png"><img src="images/037f.png" class="middle" style="height:3.6ex" alt="Symbol" /></a> = s</td><td><span class="special" title="sas">&#x1E63;a&#x1E63;</span></td></tr>
+<tr><td align="center">7</td><td align="center"><a href="images/037g.png"><img src="images/037g.png" class="middle" style="height:3.6ex" alt="Symbol" /></a></td><td align="center"><a href="images/037h.png"><img src="images/037h.png" class="middle" style="height:3.6ex" alt="Symbol" /></a> = <span class="special" title="s-dot-below">&#x1E63;</span></td><td>sapta</td></tr>
+<tr><td align="center" colspan="4">(the s and <span class="special" title="s-dot-below">&#x1E63;</span> are interchanged as occasionally in N. W. India)</td></tr>
+</table>
+
+  <p>Bühler<a name="NtA_110" href="#Nt_110"><sup>[110]</sup></a> rejects
+  this hypothesis, stating that in four cases (four, six, seven, and ten)
+  the facts are absolutely against it.</p>
+
+  <p>While the relation to ancient Bactrian forms has been generally
+  doubted, it is agreed that most of the numerals resemble
+  Br&#x101;hm&#x12B; letters, and we would naturally expect them to be
+  initials.<a name="NtA_111" href="#Nt_111"><sup>[111]</sup></a> But,
+  knowing the ancient pronunciation of most of the number names,<a
+  name="NtA_112" href="#Nt_112"><sup>[112]</sup></a> we find this not to be
+  the case. We next fall back upon the hypothesis <!-- Page 32 --><span
+  class="pagenum"><a name="page32"></a>[32]</span>that they represent the
+  order of letters<a name="NtA_113" href="#Nt_113"><sup>[113]</sup></a> in
+  the ancient alphabet. From what we know of this order, however, there
+  seems also no basis for this assumption. We have, therefore, to confess
+  that we are not certain that the numerals were alphabetic at all, and if
+  they were alphabetic we have no evidence at present as to the basis of
+  selection. The later forms may possibly have been alphabetical
+  expressions of certain syllables called <i><span class="special"
+  title="aksaras">ak&#x1E63;aras</span></i>, which possessed in Sanskrit
+  fixed numerical values,<a name="NtA_114"
+  href="#Nt_114"><sup>[114]</sup></a> but this is equally uncertain with
+  the rest. Bayley also thought<a name="NtA_115"
+  href="#Nt_115"><sup>[115]</sup></a> that some of the forms were
+  Ph&oelig;nician, as notably the use of a circle for twenty, but the
+  resemblance is in general too remote to be convincing.</p>
+
+  <p>There is also some slight possibility that Chinese influence is to be
+  seen in certain of the early forms of Hindu numerals.<a name="NtA_116"
+  href="#Nt_116"><sup>[116]</sup></a></p>
+
+<p><!-- Page 33 --><span class="pagenum"><a name="page33"></a>[33]</span></p>
+
+  <p>More absurd is the hypothesis of a Greek origin, supposedly supported
+  by derivation of the current symbols from the first nine letters of the
+  Greek alphabet.<a name="NtA_117" href="#Nt_117"><sup>[117]</sup></a> This
+  difficult feat is accomplished by twisting some of the letters, cutting
+  off, adding on, and effecting other changes to make the letters fit the
+  theory. This peculiar theory was first set up by Dasypodius<a
+  name="NtA_118" href="#Nt_118"><sup>[118]</sup></a> (Conrad Rauhfuss), and
+  was later elaborated by Huet.<a name="NtA_119"
+  href="#Nt_119"><sup>[119]</sup></a></p>
+
+<p><!-- Page 34 --><span class="pagenum"><a name="page34"></a>[34]</span></p>
+
+  <p>A bizarre derivation based upon early Arabic (c. 1040 <span
+  class="scac">A.D.</span>) sources is given by Kircher in his work<a
+  name="NtA_120" href="#Nt_120"><sup>[120]</sup></a> on number mysticism.
+  He quotes from Abenragel,<a name="NtA_121"
+  href="#Nt_121"><sup>[121]</sup></a> giving the Arabic and a Latin
+  translation<a name="NtA_122" href="#Nt_122"><sup>[122]</sup></a> and
+  stating that the ordinary Arabic forms are derived from sectors of a
+  circle, <a href="images/040d.png"><img src="images/040d.png"
+  class="middle" style="height:2ex" alt="circle" /></a>.</p>
+
+  <p>Out of all these conflicting theories, and from all the resemblances
+  seen or imagined between the numerals of the West and those of the East,
+  what conclusions are we prepared to draw as the evidence now stands?
+  Probably none that is satisfactory. Indeed, upon the evidence at <!--
+  Page 35 --><span class="pagenum"><a name="page35"></a>[35]</span>hand we
+  might properly feel that everything points to the numerals as being
+  substantially indigenous to India. And why should this not be the case?
+  If the king Srong-tsan-Gampo (639 <span class="scac">A.D.</span>), the
+  founder of Lh&#x101;sa,<a name="NtA_123"
+  href="#Nt_123"><sup>[123]</sup></a> could have set about to devise a new
+  alphabet for Tibet, and if the Siamese, and the Singhalese, and the
+  Burmese, and other peoples in the East, could have created alphabets of
+  their own, why should not the numerals also have been fashioned by some
+  temple school, or some king, or some merchant guild? By way of
+  illustration, there are shown in the table on page 36 certain systems of
+  the East, and while a few resemblances are evident, it is also evident
+  that the creators of each system endeavored to find original forms that
+  should not be found in other systems. This, then, would seem to be a fair
+  interpretation of the evidence. A human mind cannot readily create simple
+  forms that are absolutely new; what it fashions will naturally resemble
+  what other minds have fashioned, or what it has known through hearsay or
+  through sight. A circle is one of the world's common stock of figures,
+  and that it should mean twenty in Ph&oelig;nicia and in India is hardly
+  more surprising than that it signified ten at one time in Babylon.<a
+  name="NtA_124" href="#Nt_124"><sup>[124]</sup></a> It is therefore quite
+  probable that an extraneous origin cannot be found for the very
+  sufficient reason that none exists.</p>
+
+  <p>Of absolute nonsense about the origin of the symbols which we use much
+  has been written. Conjectures, <!-- Page 36 --><span class="pagenum"><a
+  name="page36"></a>[36]</span>however, without any historical evidence for
+  support, have no place in a serious discussion of the gradual evolution
+  of the present numeral forms.<a name="NtA_125"
+  href="#Nt_125"><sup>[125]</sup></a></p>
+
+<h3><span class="sc">Table of Certain Eastern Systems</span></h3>
+
+<table class="nobctr">
+<tr><td>&nbsp;</td><td><a href="images/042.png"><img src="images/042.png" class="middle" style="height:3.6ex" alt="0 12 3 4 5 6 7 8 9 10" /></a></td></tr>
+<tr><td valign="middle">Siam</td><td><a href="images/042a.png"><img src="images/042a.png" class="middle" style="height:5.2ex" alt="Siamese numerals" /></a></td></tr>
+<tr><td valign="middle">Burma<a name="NtA_126" href="#Nt_126"><sup>[126]</sup></a></td><td><a href="images/042b.png"><img src="images/042b.png" class="middle" style="height:5.2ex" alt="Burmese numerals" /></a></td></tr>
+<tr><td valign="middle">Malabar<a name="NtA_127" href="#Nt_127"><sup>[127]</sup></a></td><td><a href="images/042c.png"><img src="images/042c.png" class="middle" style="height:5.2ex" alt="Malaberese numerals" /></a></td></tr>
+<tr><td valign="middle">Tibet<a name="NtA_128" href="#Nt_128"><sup>[128]</sup></a></td><td><a href="images/042d.png"><img src="images/042d.png" class="middle" style="height:5.2ex" alt="Tibetan numerals" /></a></td></tr>
+<tr><td valign="middle">Ceylon<a name="NtA_129" href="#Nt_129"><sup>[129]</sup></a></td><td><a href="images/042e.png"><img src="images/042e.png" class="middle" style="height:5.2ex" alt="Celanese numerals" /></a></td></tr>
+<tr><td valign="middle">Malayalam<a href="#Nt_129"><sup>[129]</sup></a></td><td><a href="images/042f.png"><img src="images/042f.png" class="middle" style="height:5.2ex" alt="Malayalam numerals" /></a></td></tr>
+</table>
+
+<p><!-- Page 37 --><span class="pagenum"><a name="page37"></a>[37]</span></p>
+
+  <p>We may summarize this chapter by saying that no one knows what
+  suggested certain of the early numeral forms used in India. The origin of
+  some is evident, but the origin of others will probably never be known.
+  There is no reason why they should not have been invented by some priest
+  or teacher or guild, by the order of some king, or as part of the
+  mysticism of some temple. Whatever the origin, they were no better than
+  scores of other ancient systems and no better than the present Chinese
+  system when written without the zero, and there would never have been any
+  chance of their triumphal progress westward had it not been for this
+  relatively late symbol. There could hardly be demanded a stronger proof
+  of the Hindu origin of the character for zero than this, and to it
+  further reference will be made in Chapter IV.</p>
+
+<hr class="full" >
+
+<p><!-- Page 38 --><span class="pagenum"><a name="page38"></a>[38]</span></p>
+
+<h3>CHAPTER III</h3>
+
+<p class="cenhead">LATER HINDU FORMS, WITH A PLACE VALUE</p>
+
+  <p>Before speaking of the perfected Hindu numerals with the zero and the
+  place value, it is necessary to consider the third system mentioned on
+  page 19,&mdash;the word and letter forms. The use of words with place
+  value began at least as early as the 6th century of the Christian era. In
+  many of the manuals of astronomy and mathematics, and often in other
+  works in mentioning dates, numbers are represented by the names of
+  certain objects or ideas. For example, zero is represented by "the void"
+  (<i>&#x15B;&#x16B;nya</i>), or "heaven-space" (<i>ambara
+  &#x101;k&#x101;&#x15B;a</i>); one by "stick" (<i>rupa</i>), "moon"
+  (<i>indu &#x15B;a&#x15B;in</i>), "earth" (<i>bh&#x16B;</i>), "beginning"
+  (<i>&#x101;di</i>), "Brahma," or, in general, by anything markedly
+  unique; two by "the twins" (<i>yama</i>), "hands" (<i>kara</i>), "eyes"
+  (<i>nayana</i>), etc.; four by "oceans," five by "senses" (<i><span
+  class="special" title="visaya">vi&#x1E63;aya</span></i>) or "arrows" (the
+  five arrows of K&#x101;mad&#x113;va); six by "seasons" or "flavors";
+  seven by "mountain" (<i>aga</i>), and so on.<a name="NtA_130"
+  href="#Nt_130"><sup>[130]</sup></a> These names, accommodating themselves
+  to the verse in which scientific works were written, had the additional
+  advantage of not admitting, as did the figures, easy alteration, since
+  any change would tend to disturb the meter.</p>
+
+<p><!-- Page 39 --><span class="pagenum"><a name="page39"></a>[39]</span></p>
+
+  <p>As an example of this system, the date "<span class="special"
+  title="Saka Samvat">&#x15A;aka Sa&#x1E43;vat</span>, 867" (<span
+  class="scac">A.D.</span> 945 or 946), is given by "<i><span
+  class="special" title="giri-rasa-vasu">giri-ra&#x1E63;a-vasu</span></i>,"
+  meaning "the mountains" (seven), "the flavors" (six), and the gods
+  "<i>Vasu</i>" of which there were eight. In reading the date these are
+  read from right to left.<a name="NtA_131"
+  href="#Nt_131"><sup>[131]</sup></a> The period of invention of this
+  system is uncertain. The first trace seems to be in the
+  <i>&#x15A;rautas&#x16B;tra</i> of K&#x101;ty&#x101;yana and <span
+  class="special" title="Latyayana">L&#x101;&#x1E6D;y&#x101;yana</span>.<a
+  name="NtA_132" href="#Nt_132"><sup>[132]</sup></a> It was certainly known
+  to Var&#x101;ha-Mihira (d. 587),<a name="NtA_133"
+  href="#Nt_133"><sup>[133]</sup></a> for he used it in the <i><span
+  class="special"
+  title="Brhat-Samhita">B&#x1E5B;hat-Sa&#x1E43;hit&#x101;</span>.</i><a
+  name="NtA_134" href="#Nt_134"><sup>[134]</sup></a> It has also been
+  asserted<a name="NtA_135" href="#Nt_135"><sup>[135]</sup></a> that <span
+  class="special" title="Aryabhata">&#x100;ryabha&#x1E6D;a</span> (c. 500
+  <span class="scac">A.D.</span>) was familiar with this system, but there
+  is nothing to prove the statement.<a name="NtA_136"
+  href="#Nt_136"><sup>[136]</sup></a> The earliest epigraphical examples of
+  the system are found in the Bayang (Cambodia) inscriptions of 604 and 624
+  <span class="scac">A.D.</span><a name="NtA_137"
+  href="#Nt_137"><sup>[137]</sup></a></p>
+
+  <p>Mention should also be made, in this connection, of a curious system
+  of alphabetic numerals that sprang up in southern India. In this we have
+  the numerals represented by the letters as given in the following
+  table:</p>
+
+<table class="nobctr">
+<tr><td style="width:6%">1</td><td style="width:6%">2</td><td style="width:6%">3</td><td style="width:6%">4</td><td style="width:6%">5</td><td style="width:6%">6</td><td style="width:6%">7</td><td style="width:6%">8</td><td style="width:6%">9</td><td style="width:6%">0</td></tr>
+<tr><td>k</td><td>kh</td><td>g</td><td>gh</td><td> <span class="special" title="n-dot-above">&#x1E45;</span></td><td>c</td><td>ch</td><td>j</td><td>jh</td><td>ñ</td></tr>
+<tr><td><span class="special" title="t-dot-below">&#x1E6D;</span></td><td><span class="special" title="t-dot-below h">&#x1E6D;h</span></td><td><span class="special" title="d-dot-below">&#x1E0D;</span></td><td><span class="special" title="d-dot-below h">&#x1E0D;h</span></td><td><span class="special" title="n-dot-below">&#x1E47;</span></td><td>t</td><td>th</td><td>d</td><td>th</td><td>n</td></tr>
+<tr><td>p</td><td>ph</td><td>b</td><td>bh</td><td>m</td></tr>
+<tr><td>y</td><td>r</td><td>l</td><td>v</td><td>&#x15B;</td><td><span class="special" title="s-dot-below">&#x1E63;</span></td><td>s</td><td>h</td><td>l</td></tr>
+</table>
+
+<p><!-- Page 40 --><span class="pagenum"><a name="page40"></a>[40]</span></p>
+
+  <p>By this plan a numeral might be represented by any one of several
+  letters, as shown in the preceding table, and thus it could the more
+  easily be formed into a word for mnemonic purposes. For example, the
+  word</p>
+
+<table class="nobctr">
+<tr><td>2</td><td>3</td><td>1</td><td>5</td><td>6</td><td>5</td><td>1</td></tr>
+<tr><td><i>kha</i></td><td><i>gont</i></td><td><i>yan</i></td><td><i>me</i></td><td><i><span class="special" title="s-dot-under a">&#x1E63;a</span></i></td><td><i>m&#x101;</i></td><td><i>pa</i></td></tr>
+</table>
+
+  <p>has the value 1,565,132, reading from right to left.<a name="NtA_138"
+  href="#Nt_138"><sup>[138]</sup></a> This, the oldest specimen (1184 <span
+  class="scac">A.D.</span>) known of this notation, is given in a
+  commentary on the Rigveda, representing the number of days that had
+  elapsed from the beginning of the Kaliyuga. Burnell<a name="NtA_139"
+  href="#Nt_139"><sup>[139]</sup></a> states that this system is even yet
+  in use for remembering rules to calculate horoscopes, and for
+  astronomical tables.</p>
+
+  <p>A second system of this kind is still used in the pagination of
+  manuscripts in Ceylon, Siam, and Burma, having also had its rise in
+  southern India. In this the thirty-four consonants when followed by
+  <i>a</i> (as <i>ka</i> ... <i>la</i>) designate the numbers 1-34; by
+  <i>&#x101;</i> (as <i>k&#x101;</i> ... <i>l&#x101;</i>), those from 35 to
+  68; by <i>i</i> (<i>ki</i> ... <i>li</i>), those from 69 to 102,
+  inclusive; and so on.<a name="NtA_140"
+  href="#Nt_140"><sup>[140]</sup></a></p>
+
+  <p>As already stated, however, the Hindu system as thus far described was
+  no improvement upon many others of the ancients, such as those used by
+  the Greeks and the Hebrews. Having no zero, it was impracticable to
+  designate the tens, hundreds, and other units of higher order by the same
+  symbols used for the units from one to nine. In other words, there was no
+  possibility of place value without some further improvement. So the
+  N&#x101;n&#x101; Gh&#x101;t <!-- Page 41 --><span class="pagenum"><a
+  name="page41"></a>[41]</span>symbols required the writing of "thousand
+  seven twenty-four" about like T 7, tw, 4 in modern symbols, instead of
+  7024, in which the seven of the thousands, the two of the tens (concealed
+  in the word twenty, being originally "twain of tens," the <i>-ty</i>
+  signifying ten), and the four of the units are given as spoken and the
+  order of the unit (tens, hundreds, etc.) is given by the place. To
+  complete the system only the zero was needed; but it was probably eight
+  centuries after the N&#x101;n&#x101; Gh&#x101;t inscriptions were cut,
+  before this important symbol appeared; and not until a considerably later
+  period did it become well known. Who it was to whom the invention is due,
+  or where he lived, or even in what century, will probably always remain a
+  mystery.<a name="NtA_141" href="#Nt_141"><sup>[141]</sup></a> It is
+  possible that one of the forms of ancient abacus suggested to some Hindu
+  astronomer or mathematician the use of a symbol to stand for the vacant
+  line when the counters were removed. It is well established that in
+  different parts of India the names of the higher powers took different
+  forms, even the order being interchanged.<a name="NtA_142"
+  href="#Nt_142"><sup>[142]</sup></a> Nevertheless, as the significance of
+  the name of the unit was given by the order in reading, these variations
+  did not lead to error. Indeed the variation itself may have necessitated
+  the introduction of a word to signify a vacant place or lacking unit,
+  with the ultimate introduction of a zero symbol for this word.</p>
+
+  <p>To enable us to appreciate the force of this argument a large number,
+  8,443,682,155, may be considered as the Hindus wrote and read it, and
+  then, by way of contrast, as the Greeks and Arabs would have read it.</p>
+
+<p><!-- Page 42 --><span class="pagenum"><a name="page42"></a>[42]</span></p>
+
+  <p><i>Modern American reading</i>, 8 billion, 443 million, 682 thousand,
+  155.</p>
+
+  <p><i>Hindu</i>, 8 padmas, 4 vyarbudas, 4 <span class="special"
+  title="kotis">k&#x14D;&#x1E6D;is</span>, 3 prayutas, 6 <span
+  class="special" title="laksas">lak&#x1E63;as</span>, 8 ayutas, 2 sahasra,
+  1 &#x15B;ata, 5 da&#x15B;an, 5.</p>
+
+  <p><i>Arabic and early German</i>, eight thousand thousand thousand and
+  four hundred thousand thousand and forty-three thousand thousand, and six
+  hundred thousand and eighty-two thousand and one hundred fifty-five (or
+  five and fifty).</p>
+
+  <p><i>Greek</i>, eighty-four myriads of myriads and four thousand three
+  hundred sixty-eight myriads and two thousand and one hundred
+  fifty-five.</p>
+
+  <p>As Woepcke<a name="NtA_143" href="#Nt_143"><sup>[143]</sup></a>
+  pointed out, the reading of numbers of this kind shows that the notation
+  adopted by the Hindus tended to bring out the place idea. No other
+  language than the Sanskrit has made such consistent application, in
+  numeration, of the decimal system of numbers. The introduction of myriads
+  as in the Greek, and thousands as in Arabic and in modern numeration, is
+  really a step away from a decimal scheme. So in the numbers below one
+  hundred, in English, eleven and twelve are out of harmony with the rest
+  of the -teens, while the naming of all the numbers between ten and twenty
+  is not analogous to the naming of the numbers above twenty. To conform to
+  our written system we should have ten-one, ten-two, ten-three, and so on,
+  as we have twenty-one, twenty-two, and the like. The Sanskrit is
+  consistent, the units, however, preceding the tens and hundreds. Nor did
+  any other ancient people carry the numeration as far as did the Hindus.<a
+  name="NtA_144" href="#Nt_144"><sup>[144]</sup></a></p>
+
+<p><!-- Page 43 --><span class="pagenum"><a name="page43"></a>[43]</span></p>
+
+  <p>When the <i><span class="special"
+  title="ankapalli">a&#x1E45;kapalli</span></i>,<a name="NtA_145"
+  href="#Nt_145"><sup>[145]</sup></a> the decimal-place system of writing
+  numbers, was perfected, the tenth symbol was called the
+  <i>&#x15B;&#x16B;nyabindu</i>, generally shortened to
+  <i>&#x15B;&#x16B;nya</i> (the void). Brockhaus<a name="NtA_146"
+  href="#Nt_146"><sup>[146]</sup></a> has well said that if there was any
+  invention for which the Hindus, by all their philosophy and religion,
+  were well fitted, it was the invention of a symbol for zero. This making
+  of nothingness the crux of a tremendous achievement was a step in
+  complete harmony with the genius of the Hindu.</p>
+
+  <p>It is generally thought that this <i>&#x15B;&#x16B;nya</i> as a symbol
+  was not used before about 500 <span class="scac">A.D.</span>, although
+  some writers have placed it earlier.<a name="NtA_147"
+  href="#Nt_147"><sup>[147]</sup></a> Since <span class="special"
+  title="Aryabhata">&#x100;ryabha&#x1E6D;a</span> gives our common method
+  of extracting roots, it would seem that he may have known a decimal
+  notation,<a name="NtA_148" href="#Nt_148"><sup>[148]</sup></a> although
+  he did not use the characters from which our numerals are derived.<a
+  name="NtA_149" href="#Nt_149"><sup>[149]</sup></a> Moreover, he
+  frequently speaks of the <!-- Page 44 --><span class="pagenum"><a
+  name="page44"></a>[44]</span>void.<a name="NtA_150"
+  href="#Nt_150"><sup>[150]</sup></a> If he refers to a symbol this would
+  put the zero as far back as 500 <span class="scac">A.D.</span>, but of
+  course he may have referred merely to the concept of nothingness.</p>
+
+  <p>A little later, but also in the sixth century, Var&#x101;ha-Mihira<a
+  name="NtA_151" href="#Nt_151"><sup>[151]</sup></a> wrote a work entitled
+  <i><span class="special" title="Brhat Samhita">B&#x1E5B;hat
+  Sa&#x1E43;hit&#x101;</span></i><a name="NtA_152"
+  href="#Nt_152"><sup>[152]</sup></a> in which he frequently uses
+  <i>&#x15B;&#x16B;nya</i> in speaking of numerals, so that it has been
+  thought that he was referring to a definite symbol. This, of course,
+  would add to the probability that <span class="special"
+  title="Aryabhata">&#x100;ryabha&#x1E6D;a</span> was doing the same.</p>
+
+  <p>It should also be mentioned as a matter of interest, and somewhat
+  related to the question at issue, that Var&#x101;ha-Mihira used the
+  word-system with place value<a name="NtA_153"
+  href="#Nt_153"><sup>[153]</sup></a> as explained above.</p>
+
+  <p>The first kind of alphabetic numerals and also the word-system (in
+  both of which the place value is used) are plays upon, or variations of,
+  position arithmetic, which would be most likely to occur in the country
+  of its origin.<a name="NtA_154" href="#Nt_154"><sup>[154]</sup></a></p>
+
+  <p>At the opening of the next century (c. 620 <span
+  class="scac">A.D.</span>) <span class="special"
+  title="Bana">B&#x101;&#x1E47;a</span><a name="NtA_155"
+  href="#Nt_155"><sup>[155]</sup></a> wrote of Subandhus's
+  <i>V&#x101;savadatt&#x101;</i> as a celebrated work, <!-- Page 45
+  --><span class="pagenum"><a name="page45"></a>[45]</span>and mentioned
+  that the stars dotting the sky are here compared with zeros, these being
+  points as in the modern Arabic system. On the other hand, a strong
+  argument against any Hindu knowledge of the symbol zero at this time is
+  the fact that about 700 <span class="scac">A.D.</span> the Arabs overran
+  the province of Sind and thus had an opportunity of knowing the common
+  methods used there for writing numbers. And yet, when they received the
+  complete system in 776 they looked upon it as something new.<a
+  name="NtA_156" href="#Nt_156"><sup>[156]</sup></a> Such evidence is not
+  conclusive, but it tends to show that the complete system was probably
+  not in common use in India at the beginning of the eighth century. On the
+  other hand, we must bear in mind the fact that a traveler in Germany in
+  the year 1700 would probably have heard or seen nothing of decimal
+  fractions, although these were perfected a century before that date. The
+  élite of the mathematicians may have known the zero even in <span
+  class="special" title="Aryabhata">&#x100;ryabha&#x1E6D;a</span>'s time,
+  while the merchants and the common people may not have grasped the
+  significance of the novelty until a long time after. On the whole, the
+  evidence seems to point to the west coast of India as the region where
+  the complete system was first seen.<a name="NtA_157"
+  href="#Nt_157"><sup>[157]</sup></a> As mentioned above, traces of the
+  numeral words with place value, which do not, however, absolutely require
+  a decimal place-system of symbols, are found very early in Cambodia, as
+  well as in India.</p>
+
+  <p>Concerning the earliest epigraphical instances of the use of the nine
+  symbols, plus the zero, with place value, there <!-- Page 46 --><span
+  class="pagenum"><a name="page46"></a>[46]</span>is some question.
+  Colebrooke<a name="NtA_158" href="#Nt_158"><sup>[158]</sup></a> in 1807
+  warned against the possibility of forgery in many of the ancient
+  copper-plate land grants. On this account Fleet, in the <i>Indian
+  Antiquary</i>,<a name="NtA_159" href="#Nt_159"><sup>[159]</sup></a>
+  discusses at length this phase of the work of the epigraphists in India,
+  holding that many of these forgeries were made about the end of the
+  eleventh century. Colebrooke<a name="NtA_160"
+  href="#Nt_160"><sup>[160]</sup></a> takes a more rational view of these
+  forgeries than does Kaye, who seems to hold that they tend to invalidate
+  the whole Indian hypothesis. "But even where that may be suspected, the
+  historical uses of a monument fabricated so much nearer to the times to
+  which it assumes to belong, will not be entirely superseded. The
+  necessity of rendering the forged grant credible would compel a
+  fabricator to adhere to history, and conform to established notions: and
+  the tradition, which prevailed in his time, and by which he must be
+  guided, would probably be so much nearer to the truth, as it was less
+  remote from the period which it concerned."<a name="NtA_161"
+  href="#Nt_161"><sup>[161]</sup></a> Bühler<a name="NtA_162"
+  href="#Nt_162"><sup>[162]</sup></a> gives the copper-plate Gurjara
+  inscription of <span class="special"
+  title="Cedi-samvat">Cedi-sa&#x1E43;vat</span> 346 (595 <span
+  class="scac">A.D.</span>) as the oldest epigraphical use of the
+  numerals<a name="NtA_163" href="#Nt_163"><sup>[163]</sup></a> "in which
+  the symbols correspond to the alphabet numerals of the period and the
+  place." Vincent A. Smith<a name="NtA_164"
+  href="#Nt_164"><sup>[164]</sup></a> quotes a stone inscription of 815
+  <span class="scac">A.D.</span>, dated <span class="special"
+  title="Samvat">Sa&#x1E43;vat</span> 872. So F. Kielhorn in the
+  <i>Epigraphia Indica</i><a name="NtA_165"
+  href="#Nt_165"><sup>[165]</sup></a> gives a Pathari pillar inscription of
+  Parabala, dated <span class="special"
+  title="Vikrama-samvat">Vikrama-sa&#x1E43;vat</span> 917, which
+  corresponds to 861 <span class="scac">A.D.</span>, <!-- Page 47 --><span
+  class="pagenum"><a name="page47"></a>[47]</span>and refers also to
+  another copper-plate inscription dated <span class="special"
+  title="Vikrama-samvat">Vikrama-sa&#x1E43;vat</span> 813 (756 <span
+  class="scac">A.D.</span>). The inscription quoted by V. A. Smith above is
+  that given by D. R. Bhandarkar,<a name="NtA_166"
+  href="#Nt_166"><sup>[166]</sup></a> and another is given by the same
+  writer as of date <span class="special"
+  title="Saka-samvat">Saka-sa&#x1E43;vat</span> 715 (798 <span
+  class="scac">A.D.</span>), being incised on a pilaster. Kielhorn<a
+  name="NtA_167" href="#Nt_167"><sup>[167]</sup></a> also gives two
+  copper-plate inscriptions of the time of Mahendrapala of Kanauj, <span
+  class="special" title="Valhabi-samvat">Valhab&#x12B;-sa&#x1E43;vat</span>
+  574 (893 <span class="scac">A.D.</span>) and <span class="special"
+  title="Vikrama-samvat">Vikrama-sa&#x1E43;vat</span> 956 (899 <span
+  class="scac">A.D.</span>). That there should be any inscriptions of date
+  as early even as 750 <span class="scac">A.D.</span>, would tend to show
+  that the system was at least a century older. As will be shown in the
+  further development, it was more than two centuries after the
+  introduction of the numerals into Europe that they appeared there upon
+  coins and inscriptions. While Thibaut<a name="NtA_168"
+  href="#Nt_168"><sup>[168]</sup></a> does not consider it necessary to
+  quote any specific instances of the use of the numerals, he states that
+  traces are found from 590 <span class="scac">A.D.</span> on. "That the
+  system now in use by all civilized nations is of Hindu origin cannot be
+  doubted; no other nation has any claim upon its discovery, especially
+  since the references to the origin of the system which are found in the
+  nations of western Asia point unanimously towards India."<a
+  name="NtA_169" href="#Nt_169"><sup>[169]</sup></a></p>
+
+  <p>The testimony and opinions of men like Bühler, Kielhorn, V. A. Smith,
+  Bhandarkar, and Thibaut are entitled to the most serious consideration.
+  As authorities on ancient Indian epigraphy no others rank higher. Their
+  work is accepted by Indian scholars the world over, and their united
+  judgment as to the rise of the system with a place value&mdash;that it
+  took place in India as early as the <!-- Page 48 --><span
+  class="pagenum"><a name="page48"></a>[48]</span>sixth century <span
+  class="scac">A.D.</span>&mdash;must stand unless new evidence of great
+  weight can be submitted to the contrary.</p>
+
+  <p>Many early writers remarked upon the diversity of Indian numeral
+  forms. Al-B&#x12B;r&#x16B;n&#x12B; was probably the first; noteworthy is
+  also Johannes Hispalensis,<a name="NtA_170"
+  href="#Nt_170"><sup>[170]</sup></a> who gives the variant forms for seven
+  and four. We insert on p. 49 a table of numerals used with place value.
+  While the chief authority for this is Bühler,<a name="NtA_171"
+  href="#Nt_171"><sup>[171]</sup></a> several specimens are given which are
+  not found in his work and which are of unusual interest.</p>
+
+  <p>The &#x15A;&#x101;rad&#x101; forms given in the table use the circle
+  as a symbol for 1 and the dot for zero. They are taken from the paging
+  and text of <i>The Kashmirian Atharva-Veda</i><a name="NtA_172"
+  href="#Nt_172"><sup>[172]</sup></a>, of which the manuscript used is
+  certainly four hundred years old. Similar forms are found in a manuscript
+  belonging to the University of Tübingen. Two other series presented are
+  from Tibetan books in the library of one of the authors.</p>
+
+  <p>For purposes of comparison the modern Sanskrit and Arabic numeral
+  forms are added.</p>
+
+<table class="nobctr">
+<tr><td valign="middle">Sanskrit,</td><td><a href="images/054a.png"><img src="images/054a.png" class="middle" style="height:6ex" alt="Sanskrit" /></a></td></tr>
+<tr><td valign="middle">Arabic,</td><td><a href="images/054b.png"><img src="images/054b.png" class="middle" style="height:6ex" alt="Sanskrit" /></a></td></tr>
+</table>
+
+<p><!-- Page 49 --><span class="pagenum"><a name="page49"></a>[49]</span></p>
+
+<h3><span class="sc">Numerals used with Place Value</span></h3>
+
+<table class="nobctr">
+<tr><td>&nbsp;</td><td><a href="images/055.png"><img src="images/055.png" class="middle" style="height:5ex" alt="1 2 3 4 5 6 7 8 9 0" /></a></td></tr>
+<tr><td valign="middle">a <a name="NtA_173" href="#Nt_173"><sup>[173]</sup></a></td><td><a href="images/055a.png"><img src="images/055a.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">b <a name="NtA_174" href="#Nt_174"><sup>[174]</sup></a></td><td><a href="images/055b.png"><img src="images/055b.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">c <a name="NtA_175" href="#Nt_175"><sup>[175]</sup></a></td><td><a href="images/055c.png"><img src="images/055c.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">d <a name="NtA_176" href="#Nt_176"><sup>[176]</sup></a></td><td><a href="images/055d.png"><img src="images/055d.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">e <a name="NtA_177" href="#Nt_177"><sup>[177]</sup></a></td><td><a href="images/055e.png"><img src="images/055e.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">f <a name="NtA_178" href="#Nt_178"><sup>[178]</sup></a></td><td><a href="images/055f.png"><img src="images/055f.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">g <a name="NtA_179" href="#Nt_179"><sup>[179]</sup></a></td><td><a href="images/055g.png"><img src="images/055g.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">h <a name="NtA_180" href="#Nt_180"><sup>[180]</sup></a></td><td><a href="images/055h.png"><img src="images/055h.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">i <a href="#Nt_180"><sup>[180]</sup></a></td><td><a href="images/055i.png"><img src="images/055i.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">j <a name="NtA_181" href="#Nt_181"><sup>[181]</sup></a></td><td><a href="images/055j.png"><img src="images/055j.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">k <a href="#Nt_181"><sup>[181]</sup></a></td><td><a href="images/055k.png"><img src="images/055k.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">l <a name="NtA_182" href="#Nt_182"><sup>[182]</sup></a></td><td><a href="images/055l.png"><img src="images/055l.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">m <a name="NtA_183" href="#Nt_183"><sup>[183]</sup></a></td><td><a href="images/055m.png"><img src="images/055m.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">n <a name="NtA_184" href="#Nt_184"><sup>[184]</sup></a></td><td><a href="images/055n.png"><img src="images/055n.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr>
+</table>
+
+<hr class="full" >
+
+<p><!-- Page 51 --><span class="pagenum"><a name="page51"></a>[51]</span></p>
+
+<h3>CHAPTER IV</h3>
+
+<p class="cenhead">THE SYMBOL ZERO</p>
+
+  <p>What has been said of the improved Hindu system with a place value
+  does not touch directly the origin of a symbol for zero, although it
+  assumes that such a symbol exists. The importance of such a sign, the
+  fact that it is a prerequisite to a place-value system, and the further
+  fact that without it the Hindu-Arabic numerals would never have dominated
+  the computation system of the western world, make it proper to devote a
+  chapter to its origin and history.</p>
+
+  <p>It was some centuries after the primitive Br&#x101;hm&#x12B; and <span
+  class="special" title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span>
+  numerals had made their appearance in India that the zero first appeared
+  there, although such a character was used by the Babylonians<a
+  name="NtA_185" href="#Nt_185"><sup>[185]</sup></a> in the centuries
+  immediately preceding the Christian era. The symbol is <a
+  href="images/057a.png"><img src="images/057a.png" class="middle"
+  style="height:2ex" alt="Babylonian zero symbol" /></a> or <a
+  href="images/057b.png"><img src="images/057b.png" class="middle"
+  style="height:2ex" alt="Babylonian zero symbol" /></a>, and apparently it
+  was not used in calculation. Nor does it always occur when units of any
+  order are lacking; thus 180 is written <a href="images/057c.png"><img
+  src="images/057c.png" class="middle" style="height:2.2ex" alt="Babylonian
+  numerals 180" /></a> with the meaning three sixties and no units, since
+  181 immediately following is <a href="images/057d.png"><img
+  src="images/057d.png" class="middle" style="height:2.2ex" alt="Babylonian
+  numerals 181" /></a>, three sixties and one unit.<a name="NtA_186"
+  href="#Nt_186"><sup>[186]</sup></a> The main <!-- Page 52 --><span
+  class="pagenum"><a name="page52"></a>[52]</span>use of this Babylonian
+  symbol seems to have been in the fractions, 60ths, 3600ths, etc., and
+  somewhat similar to the Greek use of <span title="o" class="grk"
+  >&omicron;</span>, for <span title="ouden" class="grk"
+  >&omicron;&#x1F50;&delta;&#x1F73;&nu;</span>, with the meaning
+  <i>vacant</i>.</p>
+
+  <p>"The earliest undoubted occurrence of a zero in India is an
+  inscription at Gwalior, dated Samvat 933 (876 <span
+  class="scac">A.D.</span>). Where 50 garlands are mentioned (line 20), 50
+  is written <a href="images/058a.png"><img src="images/058a.png"
+  class="middle" style="height:2ex" alt="Gwalior numerals 50" /></a>. 270
+  (line 4) is written <a href="images/058b.png"><img src="images/058b.png"
+  class="middle" style="height:2ex" alt="Gwalior numerals 270" /></a>."<a
+  name="NtA_187" href="#Nt_187"><sup>[187]</sup></a> The <span
+  class="special" title="Bakhsali">Bakh&#x1E63;&#x101;l&#x12B;</span>
+  Manuscript<a name="NtA_188" href="#Nt_188"><sup>[188]</sup></a> probably
+  antedates this, using the point or dot as a zero symbol. Bayley mentions
+  a grant of Jaika Rashtrakúta of Bharuj, found at Okamandel, of date 738
+  <span class="scac">A.D.</span>, which contains a zero, and also a coin
+  with indistinct Gupta date 707 (897 <span class="scac">A.D.</span>), but
+  the reliability of Bayley's work is questioned. As has been noted, the
+  appearance of the numerals in inscriptions and on coins would be of much
+  later occurrence than the origin and written exposition of the system.
+  From the period mentioned the spread was rapid over all of India, save
+  the southern part, where the Tamil and Malayalam people retain the old
+  system even to the present day.<a name="NtA_189"
+  href="#Nt_189"><sup>[189]</sup></a></p>
+
+  <p>Aside from its appearance in early inscriptions, there is still
+  another indication of the Hindu origin of the symbol in the special
+  treatment of the concept zero in the early works on arithmetic.
+  Brahmagupta, who lived in Ujjain, the center of Indian astronomy,<a
+  name="NtA_190" href="#Nt_190"><sup>[190]</sup></a> in the early part <!--
+  Page 53 --><span class="pagenum"><a name="page53"></a>[53]</span>of the
+  seventh century, gives in his arithmetic<a name="NtA_191"
+  href="#Nt_191"><sup>[191]</sup></a> a distinct treatment of the
+  properties of zero. He does not discuss a symbol, but he shows by his
+  treatment that in some way zero had acquired a special significance not
+  found in the Greek or other ancient arithmetics. A still more scientific
+  treatment is given by Bh&#x101;skara,<a name="NtA_192"
+  href="#Nt_192"><sup>[192]</sup></a> although in one place he permits
+  himself an unallowed liberty in dividing by zero. The most recently
+  discovered work of ancient Indian mathematical lore, the <span
+  class="special"
+  title="Ganita-Sara-Sangraha">Ganita-S&#x101;ra-Sa&#x1E45;graha</span><a
+  name="NtA_193" href="#Nt_193"><sup>[193]</sup></a> of
+  Mah&#x101;v&#x12B;r&#x101;c&#x101;rya (c. 830 <span
+  class="scac">A.D.</span>), while it does not use the numerals with place
+  value, has a similar discussion of the calculation with zero.</p>
+
+  <p>What suggested the form for the zero is, of course, purely a matter of
+  conjecture. The dot, which the Hindus used to fill up lacunæ in their
+  manuscripts, much as we indicate a break in a sentence,<a name="NtA_194"
+  href="#Nt_194"><sup>[194]</sup></a> would have been a more natural
+  symbol; and this is the one which the Hindus first used<a name="NtA_195"
+  href="#Nt_195"><sup>[195]</sup></a> and which most Arabs use to-day.
+  There was also used for this purpose a cross, like our X, and this is
+  occasionally found as a zero symbol.<a name="NtA_196"
+  href="#Nt_196"><sup>[196]</sup></a> In the <span class="special"
+  title="Bakhsali">Bakh&#x1E63;&#x101;l&#x12B;</span> manuscript above
+  mentioned, the word <i>&#x15B;&#x16B;nya</i>, with the dot as its symbol,
+  is used to denote the unknown quantity, as well as to denote zero. An
+  analogous use of the <!-- Page 54 --><span class="pagenum"><a
+  name="page54"></a>[54]</span>zero, for the unknown quantity in a
+  proportion, appears in a Latin manuscript of some lectures by Gottfried
+  Wolack in the University of Erfurt in 1467 and 1468.<a name="NtA_197"
+  href="#Nt_197"><sup>[197]</sup></a> The usage was noted even as early as
+  the eighteenth century.<a name="NtA_198"
+  href="#Nt_198"><sup>[198]</sup></a></p>
+
+  <p>The small circle was possibly suggested by the spurred circle which
+  was used for ten.<a name="NtA_199" href="#Nt_199"><sup>[199]</sup></a> It
+  has also been thought that the omicron used by Ptolemy in his
+  <i>Almagest</i>, to mark accidental blanks in the sexagesimal system
+  which he employed, may have influenced the Indian writers.<a
+  name="NtA_200" href="#Nt_200"><sup>[200]</sup></a> This symbol was used
+  quite generally in Europe and Asia, and the Arabic astronomer
+  Al-Batt&#x101;n&#x12B;<a name="NtA_201"
+  href="#Nt_201"><sup>[201]</sup></a> (died 929 <span
+  class="scac">A.D.</span>) used a similar symbol in connection with the
+  alphabetic system of numerals. The occasional use by
+  Al-Batt&#x101;n&#x12B; of the Arabic negative, <i>l&#x101;</i>, to
+  indicate the absence of minutes <!-- Page 55 --><span class="pagenum"><a
+  name="page55"></a>[55]</span>(or seconds), is noted by Nallino.<a
+  name="NtA_202" href="#Nt_202"><sup>[202]</sup></a> Noteworthy is also the
+  use of the <a href="images/061a.png"><img src="images/061a.png"
+  class="middle" style="height:1.5ex" alt="Circle" /></a> for unity in the
+  &#x15A;&#x101;rad&#x101; characters of the Kashmirian Atharva-Veda, the
+  writing being at least 400 years old. Bh&#x101;skara (c. 1150) used a
+  small circle above a number to indicate subtraction, and in the Tartar
+  writing a redundant word is removed by drawing an oval around it. It
+  would be interesting to know whether our score mark <a
+  href="images/061b.png"><img src="images/061b.png" class="middle"
+  style="height:2.2ex" alt="score mark" /></a>, read "four in the hole,"
+  could trace its pedigree to the same sources. O'Creat<a name="NtA_203"
+  href="#Nt_203"><sup>[203]</sup></a> (c. 1130), in a letter to his
+  teacher, Adelhard of Bath, uses <span title="t" class="grk">&tau;</span>
+  for zero, being an abbreviation for the word <i>teca</i> which we shall
+  see was one of the names used for zero, although it could quite as well
+  be from <span title="tziphra" class="grk"
+  >&tau;&zeta;&#x1F77;&phi;&rho;&alpha;</span>. More rarely O'Creat uses <a
+  href="images/061c.png"><img src="images/061c.png" class="middle"
+  style="height:1.8ex" alt="circle with bar" /></a>, applying the name
+  <i>cyfra</i> to both forms. Frater Sigsboto<a name="NtA_204"
+  href="#Nt_204"><sup>[204]</sup></a> (c. 1150) uses the same symbol. Other
+  peculiar forms are noted by Heiberg<a name="NtA_205"
+  href="#Nt_205"><sup>[205]</sup></a> as being in use among the Byzantine
+  Greeks in the fifteenth century. It is evident from the text that some of
+  these writers did not understand the import of the new system.<a
+  name="NtA_206" href="#Nt_206"><sup>[206]</sup></a></p>
+
+  <p>Although the dot was used at first in India, as noted above, the small
+  circle later replaced it and continues in use to this day. The Arabs,
+  however, did not adopt the <!-- Page 56 --><span class="pagenum"><a
+  name="page56"></a>[56]</span>circle, since it bore some resemblance to
+  the letter which expressed the number five in the alphabet system.<a
+  name="NtA_207" href="#Nt_207"><sup>[207]</sup></a> The earliest Arabic
+  zero known is the dot, used in a manuscript of 873 <span
+  class="scac">A.D.</span><a name="NtA_208"
+  href="#Nt_208"><sup>[208]</sup></a> Sometimes both the dot and the circle
+  are used in the same work, having the same meaning, which is the case in
+  an Arabic MS., an abridged arithmetic of Jamshid,<a name="NtA_209"
+  href="#Nt_209"><sup>[209]</sup></a> 982 A.H. (1575 <span
+  class="scac">A.D.</span>). As given in this work the numerals are <a
+  href="images/062a.png"><img src="images/062a.png" class="middle"
+  style="height:2.8ex" alt="symbols" /></a>. The form for 5 varies, in some
+  works becoming <a href="images/062b.png"><img src="images/062b.png"
+  class="middle" style="height:2ex" alt="symbol" /></a> or <a
+  href="images/062c.png"><img src="images/062c.png" class="middle"
+  style="height:2ex" alt="symbol" /></a>; <a href="images/062d.png"><img
+  src="images/062d.png" class="middle" style="height:2ex" alt="symbol"
+  /></a> is found in Egypt and <a href="images/062e.png"><img
+  src="images/062e.png" class="middle" style="height:2ex" alt="symbol"
+  /></a> appears in some fonts of type. To-day the Arabs use the 0 only
+  when, under European influence, they adopt the ordinary system. Among the
+  Chinese the first definite trace of zero is in the work of Tsin<a
+  name="NtA_210" href="#Nt_210"><sup>[210]</sup></a> of 1247 <span
+  class="scac">A.D.</span> The form is the circular one of the Hindus, and
+  undoubtedly was brought to China by some traveler.</p>
+
+  <p>The name of this all-important symbol also demands some attention,
+  especially as we are even yet quite undecided as to what to call it. We
+  speak of it to-day as <i>zero, naught</i>, and even <i>cipher</i>; the
+  telephone operator often calls it <i>O</i>, and the illiterate or
+  careless person calls it <i>aught</i>. In view of all this uncertainty we
+  may well inquire what it has been called in the past.<a name="NtA_211"
+  href="#Nt_211"><sup>[211]</sup></a></p>
+
+<p><!-- Page 57 --><span class="pagenum"><a name="page57"></a>[57]</span></p>
+
+  <p>As already stated, the Hindus called it <i>&#x15B;&#x16B;nya</i>,
+  "void."<a name="NtA_212" href="#Nt_212"><sup>[212]</sup></a> This passed
+  over into the Arabic as <i><span class="special"
+  title="as-sifr">a&#x1E63;-&#x1E63;ifr</span></i> or <i><span
+  class="special" title="sifr">&#x1E63;ifr</span></i>.<a name="NtA_213"
+  href="#Nt_213"><sup>[213]</sup></a> When Leonard of Pisa (1202) wrote
+  upon the Hindu numerals he spoke of this character as <i>zephirum</i>.<a
+  name="NtA_214" href="#Nt_214"><sup>[214]</sup></a> Maximus Planudes
+  (1330), writing under both the Greek and the Arabic influence, called it
+  <i>tziphra</i>.<a name="NtA_215" href="#Nt_215"><sup>[215]</sup></a> In a
+  treatise on arithmetic written in the Italian language by Jacob of
+  Florence<a name="NtA_216" href="#Nt_216"><sup>[216]</sup></a> <!-- Page
+  58 --><span class="pagenum"><a name="page58"></a>[58]</span>(1307) it is
+  called <i>zeuero</i>,<a name="NtA_217"
+  href="#Nt_217"><sup>[217]</sup></a> while in an arithmetic of Giovanni di
+  Danti of Arezzo (1370) the word appears as <i>çeuero</i>.<a
+  name="NtA_218" href="#Nt_218"><sup>[218]</sup></a> Another form is
+  <i>zepiro</i>,<a name="NtA_219" href="#Nt_219"><sup>[219]</sup></a> which
+  was also a step from <i>zephirum</i> to zero.<a name="NtA_220"
+  href="#Nt_220"><sup>[220]</sup></a></p>
+
+  <p>Of course the English <i>cipher</i>, French <i>chiffre</i>, is derived
+  from the same Arabic word, <i><span class="special"
+  title="as-sifr">a&#x1E63;-&#x1E63;ifr</span></i>, but in several
+  languages it has come to mean the numeral figures in general. A trace of
+  this appears in our word <i>ciphering</i>, meaning figuring or
+  computing.<a name="NtA_221" href="#Nt_221"><sup>[221]</sup></a> Johann
+  Huswirt<a name="NtA_222" href="#Nt_222"><sup>[222]</sup></a> uses the
+  word with both meanings; he gives for the tenth character the four names
+  <i>theca, circulus, cifra</i>, and <i>figura nihili</i>. In this
+  statement Huswirt probably follows, as did many writers of that period,
+  the <i>Algorismus</i> of Johannes de Sacrobosco (c. 1250 <span
+  class="scac">A.D.</span>), who was also known as John of Halifax or John
+  of Holywood. The commentary of <!-- Page 59 --><span class="pagenum"><a
+  name="page59"></a>[59]</span>Petrus de Dacia<a name="NtA_223"
+  href="#Nt_223"><sup>[223]</sup></a> (c. 1291 <span
+  class="scac">A.D.</span>) on the <i>Algorismus vulgaris</i> of Sacrobosco
+  was also widely used. The widespread use of this Englishman's work on
+  arithmetic in the universities of that time is attested by the large
+  number<a name="NtA_224" href="#Nt_224"><sup>[224]</sup></a> of MSS. from
+  the thirteenth to the seventeenth century still extant, twenty in Munich,
+  twelve in Vienna, thirteen in Erfurt, several in England given by
+  Halliwell,<a name="NtA_225" href="#Nt_225"><sup>[225]</sup></a> ten
+  listed in Coxe's <i>Catalogue of the Oxford College Library</i>, one in
+  the Plimpton collection,<a name="NtA_226"
+  href="#Nt_226"><sup>[226]</sup></a> one in the Columbia University
+  Library, and, of course, many others.</p>
+
+  <p>From <i><span class="special"
+  title="as-sifr">a&#x1E63;-&#x1E63;ifr</span> </i>has come <i>zephyr,
+  cipher,</i> and finally the abridged form <i>zero</i>. The earliest
+  printed work in which is found this final form appears to be Calandri's
+  arithmetic of 1491,<a name="NtA_227" href="#Nt_227"><sup>[227]</sup></a>
+  while in manuscript it appears at least as early as the middle of the
+  fourteenth century.<a name="NtA_228" href="#Nt_228"><sup>[228]</sup></a>
+  It also appears in a work, <i>Le Kadran des marchans</i>, by Jehan <!--
+  Page 60 --><span class="pagenum"><a
+  name="page60"></a>[60]</span>Certain,<a name="NtA_229"
+  href="#Nt_229"><sup>[229]</sup></a> written in 1485. This word soon
+  became fairly well known in Spain<a name="NtA_230"
+  href="#Nt_230"><sup>[230]</sup></a> and France.<a name="NtA_231"
+  href="#Nt_231"><sup>[231]</sup></a> The medieval writers also spoke of it
+  as the <i>sipos</i>,<a name="NtA_232" href="#Nt_232"><sup>[232]</sup></a>
+  and occasionally as the <i>wheel</i>,<a name="NtA_233"
+  href="#Nt_233"><sup>[233]</sup></a> <i>circulus</i><a name="NtA_234"
+  href="#Nt_234"><sup>[234]</sup></a> (in German <i>das Ringlein</i><a
+  name="NtA_235" href="#Nt_235"><sup>[235]</sup></a>), <i>circular <!--
+  Page 61 --><span class="pagenum"><a
+  name="page61"></a>[61]</span>note</i>,<a name="NtA_236"
+  href="#Nt_236"><sup>[236]</sup></a> <i>theca</i>,<a name="NtA_237"
+  href="#Nt_237"><sup>[237]</sup></a> long supposed to be from its
+  resemblance to the Greek theta, but explained by Petrus de Dacia as being
+  derived from the name of the iron<a name="NtA_238"
+  href="#Nt_238"><sup>[238]</sup></a> used to brand thieves and robbers
+  with a circular mark placed on the forehead or on the cheek. It was also
+  called <i>omicron</i><a name="NtA_239"
+  href="#Nt_239"><sup>[239]</sup></a> (the Greek <i>o</i>), being sometimes
+  written õ or <span title="ph" class="grk">&phi;</span> to distinguish it
+  from the letter <i>o</i>. It also went by the name <i>null</i><a
+  name="NtA_240" href="#Nt_240"><sup>[240]</sup></a> (in the Latin books
+  <!-- Page 62 --><span class="pagenum"><a
+  name="page62"></a>[62]</span><i>nihil</i><a name="NtA_241"
+  href="#Nt_241"><sup>[241]</sup></a> or <i>nulla</i>,<a name="NtA_242"
+  href="#Nt_242"><sup>[242]</sup></a> and in the French <i>rien</i><a
+  name="NtA_243" href="#Nt_243"><sup>[243]</sup></a>), and very commonly by
+  the name <i>cipher</i>.<a name="NtA_244"
+  href="#Nt_244"><sup>[244]</sup></a> Wallis<a name="NtA_245"
+  href="#Nt_245"><sup>[245]</sup></a> gives one of the earliest extended
+  discussions of the various forms of the word, giving certain other
+  variations worthy of note, as <i>ziphra</i>, <i>zifera</i>,
+  <i>siphra</i>, <i>ciphra</i>, <i>tsiphra</i>, <i>tziphra,</i> and the
+  Greek <span title="tziphra" class="grk"
+  >&tau;&zeta;&#x1F77;&phi;&rho;&alpha;</span>.<a name="NtA_246"
+  href="#Nt_246"><sup>[246]</sup></a></p>
+
+<hr class="full" >
+
+<p><!-- Page 63 --><span class="pagenum"><a name="page63"></a>[63]</span></p>
+
+<h3>CHAPTER V</h3>
+
+<p class="cenhead">THE QUESTION OF THE INTRODUCTION OF THE
+NUMERALS INTO EUROPE BY BOETHIUS</p>
+
+  <p>Just as we were quite uncertain as to the origin of the numeral forms,
+  so too are we uncertain as to the time and place of their introduction
+  into Europe. There are two general theories as to this introduction. The
+  first is that they were carried by the Moors to Spain in the eighth or
+  ninth century, and thence were transmitted to Christian Europe, a theory
+  which will be considered later. The second, advanced by Woepcke,<a
+  name="NtA_247" href="#Nt_247"><sup>[247]</sup></a> is that they were not
+  brought to Spain by the Moors, but that they were already in Spain when
+  the Arabs arrived there, having reached the West through the
+  Neo-Pythagoreans. There are two facts to support this second theory: (1)
+  the forms of these numerals are characteristic, differing materially from
+  those which were brought by Leonardo of Pisa from Northern Africa early
+  in the thirteenth century (before 1202 <span class="scac">A.D.</span>);
+  (2) they are essentially those which <!-- Page 64 --><span
+  class="pagenum"><a name="page64"></a>[64]</span>tradition has so
+  persistently assigned to Boethius (c. 500 <span
+  class="scac">A.D.</span>), and which he would naturally have received, if
+  at all, from these same Neo-Pythagoreans or from the sources from which
+  they derived them. Furthermore, Woepcke points out that the Arabs on
+  entering Spain (711 <span class="scac">A.D.</span>) would naturally have
+  followed their custom of adopting for the computation of taxes the
+  numerical systems of the countries they conquered,<a name="NtA_248"
+  href="#Nt_248"><sup>[248]</sup></a> so that the numerals brought from
+  Spain to Italy, not having undergone the same modifications as those of
+  the Eastern Arab empire, would have differed, as they certainly did, from
+  those that came through Bagdad. The theory is that the Hindu system,
+  without the zero, early reached Alexandria (say 450 <span
+  class="scac">A.D.</span>), and that the Neo-Pythagorean love for the
+  mysterious and especially for the Oriental led to its use as something
+  bizarre and cabalistic; that it was then passed along the Mediterranean,
+  reaching Boethius in Athens or in Rome, and to the schools of Spain,
+  being discovered in Africa and Spain by the Arabs even before they
+  themselves knew the improved system with the place value.</p>
+
+<p><!-- Page 65 --><span class="pagenum"><a name="page65"></a>[65]</span></p>
+
+  <p>A recent theory set forth by Bubnov<a name="NtA_249"
+  href="#Nt_249"><sup>[249]</sup></a> also deserves mention, chiefly
+  because of the seriousness of purpose shown by this well-known writer.
+  Bubnov holds that the forms first found in Europe are derived from
+  ancient symbols used on the abacus, but that the zero is of Hindu origin.
+  This theory does not seem tenable, however, in the light of the evidence
+  already set forth.</p>
+
+  <p>Two questions are presented by Woepcke's theory: (1) What was the
+  nature of these Spanish numerals, and how were they made known to Italy?
+  (2) Did Boethius know them?</p>
+
+  <p>The Spanish forms of the numerals were called the <i><span
+  class="special" title="huruf al-gobar">&#x1E25;ur&#x16B;f
+  al-&#x121;ob&#x101;r</span></i>, the &#x121;ob&#x101;r or dust numerals,
+  as distinguished from the <i><span class="special" title="huruf al-jumal"
+  >&#x1E25;ur&#x16B;f al-jumal</span></i> or alphabetic numerals. Probably
+  the latter, under the influence of the Syrians or Jews,<a name="NtA_250"
+  href="#Nt_250"><sup>[250]</sup></a> were also used by the Arabs. The
+  significance of the term &#x121;ob&#x101;r is doubtless that these
+  numerals were written on the dust abacus, this plan being distinct from
+  the counter method of representing numbers. It is also worthy of note
+  that Al-B&#x12B;r&#x16B;n&#x12B; states that the Hindus often performed
+  numerical computations in the sand. The term is found as early as c. 950,
+  in the verses of an anonymous writer of Kairw&#x101;n, in Tunis, in which
+  the author speaks of one of his works on &#x121;ob&#x101;r calculation;<a
+  name="NtA_251" href="#Nt_251"><sup>[251]</sup></a> and, much later, the
+  Arab writer <span class="special" title="Abu Bekr Mohammed ibn `Abdallah"
+  >Ab&#x16B; Bekr Mo&#x1E25;ammed ibn &#x201B;Abdall&#x101;h</span>,
+  surnamed <span class="special"
+  title="al-Hassar">al-&#x1E24;a&#x1E63;&#x1E63;&#x101;r</span> <!-- Page
+  66 --><span class="pagenum"><a name="page66"></a>[66]</span>(the
+  arithmetician), wrote a work of which the second chapter was "On the dust
+  figures."<a name="NtA_252" href="#Nt_252"><sup>[252]</sup></a></p>
+
+  <p>The &#x121;ob&#x101;r numerals themselves were first made known to
+  modern scholars by Silvestre de Sacy, who discovered them in an Arabic
+  manuscript from the library of the ancient abbey of
+  St.-Germain-des-Prés.<a name="NtA_253"
+  href="#Nt_253"><sup>[253]</sup></a> The system has nine characters, but
+  no zero. A dot above a character indicates tens, two dots hundreds, and
+  so on, <a href="images/072a.png"><img src="images/072a.png"
+  class="middle" style="height:2ex" alt="5 with dot" /></a> meaning 50, and
+  <a href="images/072b.png"><img src="images/072b.png" class="middle"
+  style="height:2ex" alt="5 with 3 dots" /></a> meaning 5000. It has been
+  suggested that possibly these dots, sprinkled like dust above the
+  numerals, gave rise to the word <i>&#x121;ob&#x101;r</i>,<a
+  name="NtA_254" href="#Nt_254"><sup>[254]</sup></a> but this is not at all
+  probable. This system of dots is found in Persia at a much later date
+  with numerals quite like the modern Arabic;<a name="NtA_255"
+  href="#Nt_255"><sup>[255]</sup></a> but that it was used at all is
+  significant, for it is hardly likely that the western system would go
+  back to Persia, when the perfected Hindu one was near at hand.</p>
+
+  <p>At first sight there would seem to be some reason for believing that
+  this feature of the &#x121;ob&#x101;r system was of <!-- Page 67 --><span
+  class="pagenum"><a name="page67"></a>[67]</span>Arabic origin, and that
+  the present zero of these people,<a name="NtA_256"
+  href="#Nt_256"><sup>[256]</sup></a> the dot, was derived from it. It was
+  entirely natural that the Semitic people generally should have adopted
+  such a scheme, since their diacritical marks would suggest it, not to
+  speak of the possible influence of the Greek accents in the Hellenic
+  number system. When we consider, however, that the dot is found for zero
+  in the <span class="special"
+  title="Bakhsali">Bakh&#x1E63;&#x101;l&#x12B;</span> manuscript,<a
+  name="NtA_257" href="#Nt_257"><sup>[257]</sup></a> and that it was used
+  in subscript form in the <i>Kit&#x101;b al-Fihrist</i><a name="NtA_258"
+  href="#Nt_258"><sup>[258]</sup></a> in the tenth century, and as late as
+  the sixteenth century,<a name="NtA_259"
+  href="#Nt_259"><sup>[259]</sup></a> although in this case probably under
+  Arabic influence, we are forced to believe that this form may also have
+  been of Hindu origin.</p>
+
+  <p>The fact seems to be that, as already stated,<a name="NtA_260"
+  href="#Nt_260"><sup>[260]</sup></a> the Arabs did not immediately adopt
+  the Hindu zero, because it resembled their 5; they used the superscript
+  dot as serving their purposes fairly well; they may, indeed, have carried
+  this to the west and have added it to the &#x121;ob&#x101;r forms already
+  there, just as they transmitted it to the Persians. Furthermore, the Arab
+  and Hebrew scholars of Northern Africa in the tenth century knew these
+  numerals as Indian forms, for a commentary on the <i><span
+  class="special" title="Sefer Yesirah">S&#x113;fer
+  Ye&#x1E63;&#x12B;r&#x101;h</span></i> by Ab&#x16B; Sahl ibn Tamim
+  (probably composed at Kairw&#x101;n, c. 950) speaks of "the Indian
+  arithmetic known under the name of <i>&#x121;ob&#x101;r</i> or dust
+  calculation."<a name="NtA_261" href="#Nt_261"><sup>[261]</sup></a> All
+  this suggests that the Arabs may very <!-- Page 68 --><span
+  class="pagenum"><a name="page68"></a>[68]</span>likely have known the
+  &#x121;ob&#x101;r forms before the numerals reached them again in 773.<a
+  name="NtA_262" href="#Nt_262"><sup>[262]</sup></a> The term
+  "&#x121;ob&#x101;r numerals" was also used without any reference to the
+  peculiar use of dots.<a name="NtA_263"
+  href="#Nt_263"><sup>[263]</sup></a> In this connection it is worthy of
+  mention that the Algerians employed two different forms of numerals in
+  manuscripts even of the fourteenth century,<a name="NtA_264"
+  href="#Nt_264"><sup>[264]</sup></a> and that the Moroccans of to-day
+  employ the European forms instead of the present Arabic.</p>
+
+  <p>The Indian use of subscript dots to indicate the tens, hundreds,
+  thousands, etc., is established by a passage in the <i>Kit&#x101;b
+  al-Fihrist</i><a name="NtA_265" href="#Nt_265"><sup>[265]</sup></a> (987
+  <span class="scac">A.D.</span>) in which the writer discusses the written
+  language of the people of India. Notwithstanding the importance of this
+  reference for the early history of the numerals, it has not been
+  mentioned by previous writers on this subject. The numeral forms given
+  are those which have usually been called Indian,<a name="NtA_266"
+  href="#Nt_266"><sup>[266]</sup></a> in opposition to &#x121;ob&#x101;r.
+  In this document the dots are placed below the characters, instead of
+  being superposed as described above. The significance was the same.</p>
+
+  <p>In form these &#x121;ob&#x101;r numerals resemble our own much more
+  closely than the Arab numerals do. They varied more or less, but were
+  substantially as follows:</p>
+
+<p><!-- Page 69 --><span class="pagenum"><a name="page69"></a>[69]</span></p>
+
+<table class="nobctr">
+<tr><td>1 <a name="NtA_267" href="#Nt_267"><sup>[267]</sup></a></td><td><a href="images/075a.png"><img src="images/075a.png" class="middle" style="height:4.5ex" alt="numerals" /></a></td></tr>
+<tr><td>2 <a name="NtA_268" href="#Nt_268"><sup>[268]</sup></a></td><td><a href="images/075b.png"><img src="images/075b.png" class="middle" style="height:4.5ex" alt="numerals" /></a></td></tr>
+<tr><td>3 <a name="NtA_269" href="#Nt_269"><sup>[269]</sup></a></td><td><a href="images/075c.png"><img src="images/075c.png" class="middle" style="height:4.5ex" alt="numerals" /></a></td></tr>
+<tr><td>4 <a name="NtA_270" href="#Nt_270"><sup>[270]</sup></a></td><td><a href="images/075d.png"><img src="images/075d.png" class="middle" style="height:4.5ex" alt="numerals" /></a></td></tr>
+<tr><td>5 <a name="NtA_271" href="#Nt_271"><sup>[271]</sup></a></td><td><a href="images/075e.png"><img src="images/075e.png" class="middle" style="height:4.5ex" alt="numerals" /></a></td></tr>
+<tr><td>6 <a href="#Nt_271"><sup>[271]</sup></a></td><td><a href="images/075f.png"><img src="images/075f.png" class="middle" style="height:4.5ex" alt="numerals" /></a></td></tr>
+</table>
+
+  <p>The question of the possible influence of the Egyptian demotic and
+  hieratic ordinal forms has been so often suggested that it seems well to
+  introduce them at this point, for comparison with the &#x121;ob&#x101;r
+  forms. They would as appropriately be used in connection with the Hindu
+  forms, and the evidence of a relation of the first three with all these
+  systems is apparent. The only further resemblance is in the Demotic 4 and
+  in the 9, so that the statement that the Hindu forms in general came from
+  <!-- Page 70 --><span class="pagenum"><a
+  name="page70"></a>[70]</span>this source has no foundation. The first
+  four Egyptian cardinal numerals<a name="NtA_272"
+  href="#Nt_272"><sup>[272]</sup></a> resemble more the modern Arabic.</p>
+
+  <div class="figleft" style="width:20%;">
+      <a href="images/076a.png"><img style="width:100%" src="images/076a.png"
+      alt="Demotic and Hieratic Ordinals" title="Demotic and Hieratic Ordinals" /></a>
+    <span class="sc">Demotic and Hieratic Ordinals</span>
+  </div>
+  <p>This theory of the very early introduction of the numerals into Europe
+  fails in several points. In the first place the early Western forms are
+  not known; in the second place some early Eastern forms are like the
+  &#x121;ob&#x101;r, as is seen in the third line on p. <a
+  href="#page69">69</a>, where the forms are from a manuscript written at
+  Shiraz about 970 <span class="scac">A.D.</span>, and in which some
+  western Arabic forms, e.g. <a href="images/076b.png"><img
+  src="images/076b.png" class="middle" style="height:2ex" alt="symbol"
+  /></a> for 2, are also used. Probably most significant of all is the fact
+  that the &#x121;ob&#x101;r numerals as given by Sacy are all, with the
+  exception of the symbol for eight, either single Arabic letters or
+  combinations of letters. So much for the Woepcke theory and the meaning
+  of the &#x121;ob&#x101;r numerals. We now have to consider the question
+  as to whether Boethius knew these &#x121;ob&#x101;r forms, or forms akin
+  to them.</p>
+
+  <p>This large question<a name="NtA_273"
+  href="#Nt_273"><sup>[273]</sup></a> suggests several minor ones: (1) Who
+  was Boethius? (2) Could he have known these numerals? (3) Is there any
+  positive or strong circumstantial evidence that he did know them? (4)
+  What are the probabilities in the case?</p>
+
+<p><!-- Page 71 --><span class="pagenum"><a name="page71"></a>[71]</span></p>
+
+  <p>First, who was Boethius,&mdash;Divus<a name="NtA_274"
+  href="#Nt_274"><sup>[274]</sup></a> Boethius as he was called in the
+  Middle Ages? Anicius Manlius Severinus Boethius<a name="NtA_275"
+  href="#Nt_275"><sup>[275]</sup></a> was born at Rome c. 475. He was a
+  member of the distinguished family of the Anicii,<a name="NtA_276"
+  href="#Nt_276"><sup>[276]</sup></a> which had for some time before his
+  birth been Christian. Early left an orphan, the tradition is that he was
+  taken to Athens at about the age of ten, and that he remained there
+  eighteen years.<a name="NtA_277" href="#Nt_277"><sup>[277]</sup></a> He
+  married Rusticiana, daughter of the senator Symmachus, and this union of
+  two such powerful families allowed him to move in the highest circles.<a
+  name="NtA_278" href="#Nt_278"><sup>[278]</sup></a> Standing strictly for
+  the right, and against all iniquity at court, he became the object of
+  hatred on the part of all the unscrupulous element near the throne, and
+  his bold defense of the ex-consul Albinus, unjustly accused of treason,
+  led to his imprisonment at Pavia<a name="NtA_279"
+  href="#Nt_279"><sup>[279]</sup></a> and his execution in 524.<a
+  name="NtA_280" href="#Nt_280"><sup>[280]</sup></a> Not many generations
+  after his death, the period being one in which historical criticism was
+  at its lowest ebb, the church found it profitable to look upon his
+  execution as a martyrdom.<a name="NtA_281"
+  href="#Nt_281"><sup>[281]</sup></a> He was <!-- Page 72 --><span
+  class="pagenum"><a name="page72"></a>[72]</span>accordingly looked upon
+  as a saint,<a name="NtA_282" href="#Nt_282"><sup>[282]</sup></a> his
+  bones were enshrined,<a name="NtA_283"
+  href="#Nt_283"><sup>[283]</sup></a> and as a natural consequence his
+  books were among the classics in the church schools for a thousand
+  years.<a name="NtA_284" href="#Nt_284"><sup>[284]</sup></a> It is
+  pathetic, however, to think of the medieval student trying to extract
+  mental nourishment from a work so abstract, so meaningless, so
+  unnecessarily complicated, as the arithmetic of Boethius.</p>
+
+  <p>He was looked upon by his contemporaries and immediate successors as a
+  master, for Cassiodorus<a name="NtA_285"
+  href="#Nt_285"><sup>[285]</sup></a> (c. 490-c. 585 <span
+  class="scac">A.D.</span>) says to him: "Through your translations the
+  music of Pythagoras and the astronomy of Ptolemy are read by those of
+  Italy, and the arithmetic of Nicomachus and the geometry of Euclid are
+  known to those of the West."<a name="NtA_286"
+  href="#Nt_286"><sup>[286]</sup></a> Founder of the medieval
+  scholasticism, <!-- Page 73 --><span class="pagenum"><a
+  name="page73"></a>[73]</span>distinguishing the trivium and quadrivium,<a
+  name="NtA_287" href="#Nt_287"><sup>[287]</sup></a> writing the only
+  classics of his time, Gibbon well called him "the last of the Romans whom
+  Cato or Tully could have acknowledged for their countryman."<a
+  name="NtA_288" href="#Nt_288"><sup>[288]</sup></a></p>
+
+  <p>The second question relating to Boethius is this: Could he possibly
+  have known the Hindu numerals? In view of the relations that will be
+  shown to have existed between the East and the West, there can only be an
+  affirmative answer to this question. The numerals had existed, without
+  the zero, for several centuries; they had been well known in India; there
+  had been a continued interchange of thought between the East and West;
+  and warriors, ambassadors, scholars, and the restless trader, all had
+  gone back and forth, by land or more frequently by sea, between the
+  Mediterranean lands and the centers of Indian commerce and culture.
+  Boethius could very well have learned one or more forms of Hindu numerals
+  from some traveler or merchant.</p>
+
+  <p>To justify this statement it is necessary to speak more fully of these
+  relations between the Far East and Europe. It is true that we have no
+  records of the interchange of learning, in any large way, between eastern
+  Asia and central Europe in the century preceding the time of Boethius.
+  But it is one of the mistakes of scholars to believe that they are the
+  sole transmitters of knowledge. <!-- Page 74 --><span class="pagenum"><a
+  name="page74"></a>[74]</span>As a matter of fact there is abundant reason
+  for believing that Hindu numerals would naturally have been known to the
+  Arabs, and even along every trade route to the remote west, long before
+  the zero entered to make their place-value possible, and that the
+  characters, the methods of calculating, the improvements that took place
+  from time to time, the zero when it appeared, and the customs as to
+  solving business problems, would all have been made known from generation
+  to generation along these same trade routes from the Orient to the
+  Occident. It must always be kept in mind that it was to the tradesman and
+  the wandering scholar that the spread of such learning was due, rather
+  than to the school man. Indeed, Avicenna<a name="NtA_289"
+  href="#Nt_289"><sup>[289]</sup></a> (980-1037 <span
+  class="scac">A.D.</span>) in a short biography of himself relates that
+  when his people were living at Bokh&#x101;ra his father sent him to the
+  house of a grocer to learn the Hindu art of reckoning, in which this
+  grocer (oil dealer, possibly) was expert. Leonardo of Pisa, too, had a
+  similar training.</p>
+
+  <p>The whole question of this spread of mercantile knowledge along the
+  trade routes is so connected with the &#x121;ob&#x101;r numerals, the
+  Boethius question, Gerbert, Leonardo of Pisa, and other names and events,
+  that a digression for its consideration now becomes necessary.<a
+  name="NtA_290" href="#Nt_290"><sup>[290]</sup></a></p>
+
+<p><!-- Page 75 --><span class="pagenum"><a name="page75"></a>[75]</span></p>
+
+  <p>Even in very remote times, before the Hindu numerals were sculptured
+  in the cave of N&#x101;n&#x101; Gh&#x101;t, there were trade relations
+  between Arabia and India. Indeed, long before the Aryans went to India
+  the great Turanian race had spread its civilization from the
+  Mediterranean to the Indus.<a name="NtA_291"
+  href="#Nt_291"><sup>[291]</sup></a> At a much later period the Arabs were
+  the intermediaries between Egypt and Syria on the west, and the farther
+  Orient.<a name="NtA_292" href="#Nt_292"><sup>[292]</sup></a> In the sixth
+  century <span class="scac">B.C.</span>, Hecatæus,<a name="NtA_293"
+  href="#Nt_293"><sup>[293]</sup></a> the father of geography, was
+  acquainted not only with the Mediterranean lands but with the countries
+  as far as the Indus,<a name="NtA_294" href="#Nt_294"><sup>[294]</sup></a>
+  and in Biblical times there were regular triennial voyages to India.
+  Indeed, the story of Joseph bears witness to the caravan trade from
+  India, across Arabia, and on to the banks of the Nile. About the same
+  time as Hecatæus, Scylax, a Persian admiral under Darius, from Caryanda
+  on the coast of Asia Minor, traveled to <!-- Page 76 --><span
+  class="pagenum"><a name="page76"></a>[76]</span>northwest India and wrote
+  upon his ventures.<a name="NtA_295" href="#Nt_295"><sup>[295]</sup></a>
+  He induced the nations along the Indus to acknowledge the Persian
+  supremacy, and such number systems as there were in these lands would
+  naturally have been known to a man of his attainments.</p>
+
+  <p>A century after Scylax, Herodotus showed considerable knowledge of
+  India, speaking of its cotton and its gold,<a name="NtA_296"
+  href="#Nt_296"><sup>[296]</sup></a> telling how Sesostris<a
+  name="NtA_297" href="#Nt_297"><sup>[297]</sup></a> fitted out ships to
+  sail to that country, and mentioning the routes to the east. These routes
+  were generally by the Red Sea, and had been followed by the
+  Ph&oelig;nicians and the Sabæans, and later were taken by the Greeks and
+  Romans.<a name="NtA_298" href="#Nt_298"><sup>[298]</sup></a></p>
+
+  <p>In the fourth century <span class="scac">B.C.</span> the West and East
+  came into very close relations. As early as 330, Pytheas of Massilia
+  (Marseilles) had explored as far north as the northern end of the British
+  Isles and the coasts of the German Sea, while Macedon, in close touch
+  with southern France, was also sending her armies under Alexander<a
+  name="NtA_299" href="#Nt_299"><sup>[299]</sup></a> through Afghanistan as
+  far east as the Punjab.<a name="NtA_300"
+  href="#Nt_300"><sup>[300]</sup></a> Pliny tells us that Alexander the
+  Great employed surveyors to measure <!-- Page 77 --><span
+  class="pagenum"><a name="page77"></a>[77]</span>the roads of India; and
+  one of the great highways is described by Megasthenes, who in 295 <span
+  class="scac">B.C.</span>, as the ambassador of Seleucus, resided at <span
+  class="special"
+  title="Pataliputra">P&#x101;tal&#x12B;pu&#x1E6D;ra</span>, the present
+  Patna.<a name="NtA_301" href="#Nt_301"><sup>[301]</sup></a></p>
+
+  <p>The Hindus also learned the art of coining from the Greeks, or
+  possibly from the Chinese, and the stores of Greco-Hindu coins still
+  found in northern India are a constant source of historical
+  information.<a name="NtA_302" href="#Nt_302"><sup>[302]</sup></a> The
+  R&#x101;m&#x101;yana speaks of merchants traveling in great caravans and
+  embarking by sea for foreign lands.<a name="NtA_303"
+  href="#Nt_303"><sup>[303]</sup></a> Ceylon traded with Malacca and Siam,
+  and Java was colonized by Hindu traders, so that mercantile knowledge was
+  being spread about the Indies during all the formative period of the
+  numerals.</p>
+
+  <p>Moreover the results of the early Greek invasion were embodied by
+  Dicæarchus of Messana (about 320 <span class="scac">B.C.</span>) in a map
+  that long remained a standard. Furthermore, Alexander did not allow his
+  influence on the East to cease. He divided India into three satrapies,<a
+  name="NtA_304" href="#Nt_304"><sup>[304]</sup></a> placing Greek
+  governors over two of them and leaving a Hindu ruler in charge of the
+  third, and in Bactriana, a part of Ariana or ancient Persia, he left
+  governors; and in these the western civilization was long in evidence.
+  Some of the Greek and Roman metrical and astronomical terms <!-- Page 78
+  --><span class="pagenum"><a name="page78"></a>[78]</span>found their way,
+  doubtless at this time, into the Sanskrit language.<a name="NtA_305"
+  href="#Nt_305"><sup>[305]</sup></a> Even as late as from the second to
+  the fifth centuries <span class="scac">A.D.</span>, Indian coins showed
+  the Hellenic influence. The Hindu astronomical terminology reveals the
+  same relationship to western thought, for Var&#x101;ha-Mihira (6th
+  century <span class="scac">A.D.</span>), a contemporary of <span
+  class="special" title="Aryabhata">&#x100;ryabha&#x1E6D;a</span>, entitled
+  a work of his the <i><span class="special"
+  title="Brhat-Samhita">B&#x1E5B;hat-Sa&#x1E43;hit&#x101;</span></i>, a
+  literal translation of <span title="megalê suntaxis" class="grk"
+  >&mu;&epsilon;&gamma;&#x1F71;&lambda;&eta;
+  &sigma;&#x1F7B;&nu;&tau;&alpha;&xi;&iota;&sigmaf;</span> of Ptolemy;<a
+  name="NtA_306" href="#Nt_306"><sup>[306]</sup></a> and in various ways is
+  this interchange of ideas apparent.<a name="NtA_307"
+  href="#Nt_307"><sup>[307]</sup></a> It could not have been at all unusual
+  for the ancient Greeks to go to India, for Strabo lays down the route,
+  saying that all who make the journey start from Ephesus and traverse
+  Phrygia and Cappadocia before taking the direct road.<a name="NtA_308"
+  href="#Nt_308"><sup>[308]</sup></a> The products of the East were always
+  finding their way to the West, the Greeks getting their ginger<a
+  name="NtA_309" href="#Nt_309"><sup>[309]</sup></a> from Malabar, as the
+  Ph&oelig;nicians had long before brought gold from Malacca.</p>
+
+  <p>Greece must also have had early relations with China, for there is a
+  notable similarity between the Greek and Chinese life, as is shown in
+  their houses, their domestic customs, their marriage ceremonies, the
+  public story-tellers, the puppet shows which Herodotus says were
+  introduced from Egypt, the street jugglers, the games of dice,<a
+  name="NtA_310" href="#Nt_310"><sup>[310]</sup></a> the game of
+  finger-guessing,<a name="NtA_311" href="#Nt_311"><sup>[311]</sup></a> the
+  water clock, the <!-- Page 79 --><span class="pagenum"><a
+  name="page79"></a>[79]</span>music system, the use of the myriad,<a
+  name="NtA_312" href="#Nt_312"><sup>[312]</sup></a> the calendars, and in
+  many other ways.<a name="NtA_313" href="#Nt_313"><sup>[313]</sup></a> In
+  passing through the suburbs of Peking to-day, on the way to the Great
+  Bell temple, one is constantly reminded of the semi-Greek architecture of
+  Pompeii, so closely does modern China touch the old classical
+  civilization of the Mediterranean. The Chinese historians tell us that
+  about 200 <span class="scac">B.C.</span> their arms were successful in
+  the far west, and that in 180 <span class="scac">B.C.</span> an
+  ambassador went to Bactria, then a Greek city, and reported that Chinese
+  products were on sale in the markets there.<a name="NtA_314"
+  href="#Nt_314"><sup>[314]</sup></a> There is also a noteworthy
+  resemblance between certain Greek and Chinese words,<a name="NtA_315"
+  href="#Nt_315"><sup>[315]</sup></a> showing that in remote times there
+  must have been more or less interchange of thought.</p>
+
+  <p>The Romans also exchanged products with the East. Horace says, "A busy
+  trader, you hasten to the farthest Indies, flying from poverty over sea,
+  over crags, over fires."<a name="NtA_316"
+  href="#Nt_316"><sup>[316]</sup></a> The products of the Orient, spices
+  and jewels from India, frankincense from Persia, and silks from China,
+  being more in demand than the exports from the Mediterranean lands, the
+  balance of trade was against the West, and thus Roman coin found its way
+  eastward. In 1898, for example, a number of Roman coins dating from 114
+  <span class="scac">B.C.</span> to Hadrian's time were found at
+  Pakl&#x12B;, a part of the Haz&#x101;ra district, sixteen miles north of
+  Abbott&#x101;b&#x101;d,<a name="NtA_317"
+  href="#Nt_317"><sup>[317]</sup></a> and numerous similar discoveries have
+  been made from time to time.</p>
+
+<p><!-- Page 80 --><span class="pagenum"><a name="page80"></a>[80]</span></p>
+
+  <p>Augustus speaks of envoys received by him from India, a thing never
+  before known,<a name="NtA_318" href="#Nt_318"><sup>[318]</sup></a> and it
+  is not improbable that he also received an embassy from China.<a
+  name="NtA_319" href="#Nt_319"><sup>[319]</sup></a> Suetonius (first
+  century <span class="scac">A.D.</span>) speaks in his history of these
+  relations,<a name="NtA_320" href="#Nt_320"><sup>[320]</sup></a> as do
+  several of his contemporaries,<a name="NtA_321"
+  href="#Nt_321"><sup>[321]</sup></a> and Vergil<a name="NtA_322"
+  href="#Nt_322"><sup>[322]</sup></a> tells of Augustus doing battle in
+  Persia. In Pliny's time the trade of the Roman Empire with Asia amounted
+  to a million and a quarter dollars a year, a sum far greater relatively
+  then than now,<a name="NtA_323" href="#Nt_323"><sup>[323]</sup></a> while
+  by the time of Constantine Europe was in direct communication with the
+  Far East.<a name="NtA_324" href="#Nt_324"><sup>[324]</sup></a></p>
+
+  <p>In view of these relations it is not beyond the range of possibility
+  that proof may sometime come to light to show that the Greeks and Romans
+  knew something of the <!-- Page 81 --><span class="pagenum"><a
+  name="page81"></a>[81]</span>number system of India, as several writers
+  have maintained.<a name="NtA_325" href="#Nt_325"><sup>[325]</sup></a></p>
+
+  <p>Returning to the East, there are many evidences of the spread of
+  knowledge in and about India itself. In the third century <span
+  class="scac">B.C.</span> Buddhism began to be a connecting medium of
+  thought. It had already permeated the Himalaya territory, had reached
+  eastern Turkestan, and had probably gone thence to China. Some centuries
+  later (in 62 <span class="scac">A.D.</span>) the Chinese emperor sent an
+  ambassador to India, and in 67 <span class="scac">A.D.</span> a Buddhist
+  monk was invited to China.<a name="NtA_326"
+  href="#Nt_326"><sup>[326]</sup></a> Then, too, in India itself
+  A&#x15B;oka, whose name has already been mentioned in this work, extended
+  the boundaries of his domains even into Afghanistan, so that it was
+  entirely possible for the numerals of the Punjab to have worked their way
+  north even at that early date.<a name="NtA_327"
+  href="#Nt_327"><sup>[327]</sup></a></p>
+
+  <p>Furthermore, the influence of Persia must not be forgotten in
+  considering this transmission of knowledge. In the fifth century the
+  Persian medical school at Jondi-Sapur admitted both the Hindu and the
+  Greek doctrines, and Firdus&#x12B; tells us that during the brilliant
+  reign of <!-- Page 82 --><span class="pagenum"><a
+  name="page82"></a>[82]</span>Khosr&#x16B; I,<a name="NtA_328"
+  href="#Nt_328"><sup>[328]</sup></a> the golden age of Pahlav&#x12B;
+  literature, the Hindu game of chess was introduced into Persia, at a time
+  when wars with the Greeks were bringing prestige to the Sassanid
+  dynasty.</p>
+
+  <p>Again, not far from the time of Boethius, in the sixth century, the
+  Egyptian monk Cosmas, in his earlier years as a trader, made journeys to
+  Abyssinia and even to India and Ceylon, receiving the name
+  <i>Indicopleustes</i> (the Indian traveler). His map (547 <span
+  class="scac">A.D.</span>) shows some knowledge of the earth from the
+  Atlantic to India. Such a man would, with hardly a doubt, have observed
+  every numeral system used by the people with whom he sojourned,<a
+  name="NtA_329" href="#Nt_329"><sup>[329]</sup></a> and whether or not he
+  recorded his studies in permanent form he would have transmitted such
+  scraps of knowledge by word of mouth.</p>
+
+  <p>As to the Arabs, it is a mistake to feel that their activities began
+  with Mohammed. Commerce had always been held in honor by them, and the
+  Qoreish<a name="NtA_330" href="#Nt_330"><sup>[330]</sup></a> had annually
+  for many generations sent caravans bearing the spices and textiles of
+  Yemen to the shores of the Mediterranean. In the fifth century they
+  traded by sea with India and even with China, and <span class="special"
+  title="Hira">&#x1E24;ira</span> was an emporium for the wares of the
+  East,<a name="NtA_331" href="#Nt_331"><sup>[331]</sup></a> so that any
+  numeral system of any part of the trading world could hardly have
+  remained isolated.</p>
+
+  <p>Long before the warlike activity of the Arabs, Alexandria had become
+  the great market-place of the world. From this center caravans traversed
+  Arabia to Hadramaut, where they met ships from India. Others went north
+  to Damascus, while still others made their way <!-- Page 83 --><span
+  class="pagenum"><a name="page83"></a>[83]</span>along the southern shores
+  of the Mediterranean. Ships sailed from the isthmus of Suez to all the
+  commercial ports of Southern Europe and up into the Black Sea. Hindus
+  were found among the merchants<a name="NtA_332"
+  href="#Nt_332"><sup>[332]</sup></a> who frequented the bazaars of
+  Alexandria, and Brahmins were reported even in Byzantium.</p>
+
+  <p>Such is a very brief résumé of the evidence showing that the numerals
+  of the Punjab and of other parts of India as well, and indeed those of
+  China and farther Persia, of Ceylon and the Malay peninsula, might well
+  have been known to the merchants of Alexandria, and even to those of any
+  other seaport of the Mediterranean, in the time of Boethius. The
+  Br&#x101;hm&#x12B; numerals would not have attracted the attention of
+  scholars, for they had no zero so far as we know, and therefore they were
+  no better and no worse than those of dozens of other systems. If Boethius
+  was attracted to them it was probably exactly as any one is naturally
+  attracted to the bizarre or the mystic, and he would have mentioned them
+  in his works only incidentally, as indeed they are mentioned in the
+  manuscripts in which they occur.</p>
+
+  <p>In answer therefore to the second question, Could Boethius have known
+  the Hindu numerals? the reply must be, without the slightest doubt, that
+  he could easily have known them, and that it would have been strange if a
+  man of his inquiring mind did not pick up many curious bits of
+  information of this kind even though he never thought of making use of
+  them.</p>
+
+  <p>Let us now consider the third question, Is there any positive or
+  strong circumstantial evidence that Boethius did know these numerals? The
+  question is not new, <!-- Page 84 --><span class="pagenum"><a
+  name="page84"></a>[84]</span>nor is it much nearer being answered than it
+  was over two centuries ago when Wallis (1693) expressed his doubts about
+  it<a name="NtA_333" href="#Nt_333"><sup>[333]</sup></a> soon after
+  Vossius (1658) had called attention to the matter.<a name="NtA_334"
+  href="#Nt_334"><sup>[334]</sup></a> Stated briefly, there are three works
+  on mathematics attributed to Boethius:<a name="NtA_335"
+  href="#Nt_335"><sup>[335]</sup></a> (1) the arithmetic, (2) a work on
+  music, and (3) the geometry.<a name="NtA_336"
+  href="#Nt_336"><sup>[336]</sup></a></p>
+
+  <p>The genuineness of the arithmetic and the treatise on music is
+  generally recognized, but the geometry, which contains the Hindu numerals
+  with the zero, is under suspicion.<a name="NtA_337"
+  href="#Nt_337"><sup>[337]</sup></a> There are plenty of supporters of the
+  idea that Boethius knew the numerals and included them in this book,<a
+  name="NtA_338" href="#Nt_338"><sup>[338]</sup></a> and on the other hand
+  there are as many who <!-- Page 85 --><span class="pagenum"><a
+  name="page85"></a>[85]</span>feel that the geometry, or at least the part
+  mentioning the numerals, is spurious.<a name="NtA_339"
+  href="#Nt_339"><sup>[339]</sup></a> The argument of those who deny the
+  authenticity of the particular passage in question may briefly be stated
+  thus:</p>
+
+  <p>1. The falsification of texts has always been the subject of
+  complaint. It was so with the Romans,<a name="NtA_340"
+  href="#Nt_340"><sup>[340]</sup></a> it was common in the Middle Ages,<a
+  name="NtA_341" href="#Nt_341"><sup>[341]</sup></a> and it is much more
+  prevalent <!-- Page 86 --><span class="pagenum"><a
+  name="page86"></a>[86]</span>to-day than we commonly think. We have but
+  to see how every hymn-book compiler feels himself authorized to change at
+  will the classics of our language, and how unknown editors have mutilated
+  Shakespeare, to see how much more easy it was for medieval scribes to
+  insert or eliminate paragraphs without any protest from critics.<a
+  name="NtA_342" href="#Nt_342"><sup>[342]</sup></a></p>
+
+  <p>2. If Boethius had known these numerals he would have mentioned them
+  in his arithmetic, but he does not do so.<a name="NtA_343"
+  href="#Nt_343"><sup>[343]</sup></a></p>
+
+  <p>3. If he had known them, and had mentioned them in any of his works,
+  his contemporaries, disciples, and successors would have known and
+  mentioned them. But neither Capella (c. 475)<a name="NtA_344"
+  href="#Nt_344"><sup>[344]</sup></a> nor any of the numerous medieval
+  writers who knew the works of Boethius makes any reference to the
+  system.<a name="NtA_345" href="#Nt_345"><sup>[345]</sup></a></p>
+
+<p><!-- Page 87 --><span class="pagenum"><a name="page87"></a>[87]</span></p>
+
+  <p>4. The passage in question has all the appearance of an interpolation
+  by some scribe. Boethius is speaking of angles, in his work on geometry,
+  when the text suddenly changes to a discussion of classes of numbers.<a
+  name="NtA_346" href="#Nt_346"><sup>[346]</sup></a> This is followed by a
+  chapter in explanation of the abacus,<a name="NtA_347"
+  href="#Nt_347"><sup>[347]</sup></a> in which are described those numeral
+  forms which are called <i>apices</i> or <i>caracteres</i>.<a
+  name="NtA_348" href="#Nt_348"><sup>[348]</sup></a> The forms<a
+  name="NtA_349" href="#Nt_349"><sup>[349]</sup></a> of these characters
+  vary in different manuscripts, but in general are about as shown on page
+  <a href="#page88">88</a>. They are commonly written with the 9 at the
+  left, decreasing to the unit at the right, numerous writers stating that
+  this was because they were derived from Semitic sources in which the
+  direction of writing is the opposite of our own. This practice continued
+  until the sixteenth century.<a name="NtA_350"
+  href="#Nt_350"><sup>[350]</sup></a> The writer then leaves the subject
+  entirely, using the Roman numerals for the rest of his discussion, a
+  proceeding so foreign to the method of Boethius as to be inexplicable on
+  the hypothesis of authenticity. Why should such a scholarly writer have
+  given them with no mention of their origin or use? Either he would have
+  mentioned some historical interest attaching to them, or he would have
+  used them in some discussion; he certainly would not have left the
+  passage as it is.</p>
+
+<p><!-- Page 88 --><span class="pagenum"><a name="page88"></a>[88]</span></p>
+
+<h3><span class="sc">Forms of the Numerals, Largely from Works on the Abacus</span><a name="NtA_351" href="#Nt_351"><sup>[351]</sup></a></h3>
+
+<table class="nobctr">
+<tr><td>&nbsp;</td><td><a href="images/094.png"><img src="images/094.png" class="middle" style="height:2.25ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">a <a name="NtA_352" href="#Nt_352"><sup>[352]</sup></a></td><td><a href="images/094a.png"><img src="images/094a.png" class="middle" style="height:5.4ex" alt="1 2 3 4 5 6 7 8 9 0" /></a></td></tr>
+<tr><td valign="middle">b <a name="NtA_353" href="#Nt_353"><sup>[353]</sup></a></td><td><a href="images/094b.png"><img src="images/094b.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">c <a name="NtA_354" href="#Nt_354"><sup>[354]</sup></a></td><td><a href="images/094c.png"><img src="images/094c.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">d <a name="NtA_355" href="#Nt_355"><sup>[355]</sup></a></td><td><a href="images/094d.png"><img src="images/094d.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">e <a name="NtA_356" href="#Nt_356"><sup>[356]</sup></a></td><td><a href="images/094e.png"><img src="images/094e.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">f <a name="NtA_357" href="#Nt_357"><sup>[357]</sup></a></td><td><a href="images/094f.png"><img src="images/094f.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">g <a name="NtA_358" href="#Nt_358"><sup>[358]</sup></a></td><td><a href="images/094g.png"><img src="images/094g.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">h <a name="NtA_359" href="#Nt_359"><sup>[359]</sup></a></td><td><a href="images/094h.png"><img src="images/094h.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr>
+<tr><td valign="middle">i <a name="NtA_360" href="#Nt_360"><sup>[360]</sup></a></td><td><a href="images/094i.png"><img src="images/094i.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr>
+</table>
+
+<p><!-- Page 89 --><span class="pagenum"><a name="page89"></a>[89]</span></p>
+
+  <p>Sir E. Clive Bayley has added<a name="NtA_361"
+  href="#Nt_361"><sup>[361]</sup></a> a further reason for believing them
+  spurious, namely that the 4 is not of the N&#x101;n&#x101; Gh&#x101;t
+  type, but of the Kabul form which the Arabs did not receive until 776;<a
+  name="NtA_362" href="#Nt_362"><sup>[362]</sup></a> so that it is not
+  likely, even if the characters were known in Europe in the time of
+  Boethius, that this particular form was recognized. It is worthy of
+  mention, also, that in the six abacus forms from the chief manuscripts as
+  given by Friedlein,<a name="NtA_363" href="#Nt_363"><sup>[363]</sup></a>
+  each contains some form of zero, which symbol probably originated in
+  India about this time or later. It could hardly have reached Europe so
+  soon.</p>
+
+  <p>As to the fourth question, Did Boethius probably know the numerals? It
+  seems to be a fair conclusion, according to our present evidence, that
+  (1) Boethius might very easily have known these numerals without the
+  zero, but, (2) there is no reliable evidence that he did know them. And
+  just as Boethius might have come in contact with them, so any other
+  inquiring mind might have done so either in his time or at any time
+  before they definitely appeared in the tenth century. These centuries,
+  five in number, represented the darkest of the Dark Ages, and even if
+  these numerals were occasionally met and studied, no trace of them would
+  be likely to show itself in the <!-- Page 90 --><span class="pagenum"><a
+  name="page90"></a>[90]</span>literature of the period, unless by chance
+  it should get into the writings of some man like Alcuin. As a matter of
+  fact, it was not until the ninth or tenth century that there is any
+  tangible evidence of their presence in Christendom. They were probably
+  known to merchants here and there, but in their incomplete state they
+  were not of sufficient importance to attract any considerable
+  attention.</p>
+
+  <p>As a result of this brief survey of the evidence several conclusions
+  seem reasonable: (1) commerce, and travel for travel's sake, never died
+  out between the East and the West; (2) merchants had every opportunity of
+  knowing, and would have been unreasonably stupid if they had not known,
+  the elementary number systems of the peoples with whom they were trading,
+  but they would not have put this knowledge in permanent written form; (3)
+  wandering scholars would have known many and strange things about the
+  peoples they met, but they too were not, as a class, writers; (4) there
+  is every reason a priori for believing that the &#x121;ob&#x101;r
+  numerals would have been known to merchants, and probably to some of the
+  wandering scholars, long before the Arabs conquered northern Africa; (5)
+  the wonder is not that the Hindu-Arabic numerals were known about 1000
+  <span class="scac">A.D.</span>, and that they were the subject of an
+  elaborate work in 1202 by Fibonacci, but rather that more extended
+  manuscript evidence of their appearance before that time has not been
+  found. That they were more or less known early in the Middle Ages,
+  certainly to many merchants of Christian Europe, and probably to several
+  scholars, but without the zero, is hardly to be doubted. The lack of
+  documentary evidence is not at all strange, in view of all of the
+  circumstances.</p>
+
+<hr class="full" >
+
+<p><!-- Page 91 --><span class="pagenum"><a name="page91"></a>[91]</span></p>
+
+<h3>CHAPTER VI</h3>
+
+<p class="cenhead">THE DEVELOPMENT OF THE NUMERALS
+AMONG THE ARABS</p>
+
+  <p>If the numerals had their origin in India, as seems most probable,
+  when did the Arabs come to know of them? It is customary to say that it
+  was due to the influence of Mohammedanism that learning spread through
+  Persia and Arabia; and so it was, in part. But learning was already
+  respected in these countries long before Mohammed appeared, and commerce
+  flourished all through this region. In Persia, for example, the reign of
+  Khosr&#x16B; Nu&#x15B;&#x12B;rw&#x101;n,<a name="NtA_364"
+  href="#Nt_364"><sup>[364]</sup></a> the great contemporary of Justinian
+  the law-maker, was characterized not only by an improvement in social and
+  economic conditions, but by the cultivation of letters. Khosr&#x16B;
+  fostered learning, inviting to his court scholars from Greece, and
+  encouraging the introduction of culture from the West as well as from the
+  East. At this time Aristotle and Plato were translated, and portions of
+  the <i>Hito-pad&#x113;&#x15B;a</i>, or Fables of Pilpay, were rendered
+  from the Sanskrit into Persian. All this means that some three centuries
+  before the great intellectual ascendancy of Bagdad a similar fostering of
+  learning was taking place in Persia, and under pre-Mohammedan
+  influences.</p>
+
+<p><!-- Page 92 --><span class="pagenum"><a name="page92"></a>[92]</span></p>
+
+  <p>The first definite trace that we have of the introduction of the Hindu
+  system into Arabia dates from 773 <span class="scac">A.D.</span>,<a
+  name="NtA_365" href="#Nt_365"><sup>[365]</sup></a> when an Indian
+  astronomer visited the court of the caliph, bringing with him
+  astronomical tables which at the caliph's command were translated into
+  Arabic by Al-Faz&#x101;r&#x12B;.<a name="NtA_366"
+  href="#Nt_366"><sup>[366]</sup></a> Al-Khow&#x101;razm&#x12B; and <span
+  class="special" title="Habash">&#x1E24;abash</span> (<span
+  class="special" title="Ahmed ibn `Abdallah">A&#x1E25;med ibn
+  &#x201B;Abdall&#x101;h</span>, died c. 870) based their well-known tables
+  upon the work of Al-F&#x101;zar&#x12B;. It may be asserted as highly
+  probable that the numerals came at the same time as the tables. They were
+  certainly known a few decades later, and before 825 <span
+  class="scac">A.D.</span>, about which time the original of the
+  <i>Algoritmi de numero Indorum</i> was written, as that work makes no
+  pretense of being the first work to treat of the Hindu numerals.</p>
+
+  <p>The three writers mentioned cover the period from the end of the
+  eighth to the end of the ninth century. While the historians
+  Al-Ma&#x15B;&#x201B;&#x16B;d&#x12B; and Al-B&#x12B;r&#x16B;n&#x12B;
+  follow quite closely upon the men mentioned, it is well to note again the
+  Arab writers on Hindu arithmetic, contemporary with
+  Al-Khow&#x101;razm&#x12B;, who were mentioned in chapter I, viz.
+  Al-Kind&#x12B;, Sened ibn &#x201B;Al&#x12B;, and <span class="special"
+  title="Al-Sufi">Al-&#x1E62;&#x16B;f&#x12B;</span>.</p>
+
+  <p>For over five hundred years Arabic writers and others continued to
+  apply to works on arithmetic the name "Indian." In the tenth century such
+  writers are <span class="special" title="`Abdallah ibn al-Hasan"
+  >&#x201B;Abdall&#x101;h ibn al-&#x1E24;asan</span>, Ab&#x16B;
+  'l-Q&#x101;sim<a name="NtA_367" href="#Nt_367"><sup>[367]</sup></a> (died
+  987 <span class="scac">A.D.</span>) of Antioch, and <span class="special"
+  title="Mohammed ibn `Abdallah, Abu Nasr">Mo&#x1E25;ammed ibn
+  &#x201B;Abdall&#x101;h, Ab&#x16B; Na&#x1E63;r</span><a name="NtA_368"
+  href="#Nt_368"><sup>[368]</sup></a> (c. 982), of Kalw&#x101;d&#x101; near
+  Bagdad. Others of the same period or <!-- Page 93 --><span
+  class="pagenum"><a name="page93"></a>[93]</span>earlier (since they are
+  mentioned in the <i>Fihrist</i>,<a name="NtA_369"
+  href="#Nt_369"><sup>[369]</sup></a> 987 <span class="scac">A.D.</span>),
+  who explicitly use the word "Hindu" or "Indian," are <span
+  class="special" title="Sinan ibn al-Fath">Sin&#x101;n ibn
+  al-Fat&#x1E25;</span><a name="NtA_370"
+  href="#Nt_370"><sup>[370]</sup></a> of <span class="special"
+  title="Harran">&#x1E24;arr&#x101;n</span>, and Ahmed ibn &#x201B;Omar,
+  al-Kar&#x101;b&#x12B;s&#x12B;.<a name="NtA_371"
+  href="#Nt_371"><sup>[371]</sup></a> In the eleventh century come
+  Al-B&#x12B;r&#x16B;n&#x12B;<a name="NtA_372"
+  href="#Nt_372"><sup>[372]</sup></a> (973-1048) and <span class="special"
+  title="`Ali ibn Ahmed, Abu 'l-Hasan">&#x201B;Ali ibn A&#x1E25;med,
+  Ab&#x16B; 'l-&#x1E24;asan</span>, Al-Nasaw&#x12B;<a name="NtA_373"
+  href="#Nt_373"><sup>[373]</sup></a> (c. 1030). The following century
+  brings similar works by <span class="special" title="Ishaq ibn Yusuf al-Sardafi"
+  >Ish&#x101;q ibn Y&#x16B;suf al-&#x1E62;ardaf&#x12B;</span><a
+  name="NtA_374" href="#Nt_374"><sup>[374]</sup></a> and
+  Sam&#x16B;'&#x12B;l ibn <span class="special"
+  title="Yahya">Ya&#x1E25;y&#x101;</span> ibn &#x201B;Abb&#x101;s
+  al-Ma&#x121;reb&#x12B; al-Andalus&#x12B;<a name="NtA_375"
+  href="#Nt_375"><sup>[375]</sup></a> (c. 1174), and in the thirteenth
+  century are &#x201B;Abdallat&#x12B;f ibn Y&#x16B;suf ibn <span
+  class="special" title="Mohammed">Mo&#x1E25;ammed</span>, Muwaffaq
+  al-D&#x12B;n Ab&#x16B; <span class="special"
+  title="Mohammed">Mo&#x1E25;ammed</span> al-Ba&#x121;d&#x101;d&#x12B;<a
+  name="NtA_376" href="#Nt_376"><sup>[376]</sup></a> (c. 1231), and Ibn
+  al-Bann&#x101;.<a name="NtA_377" href="#Nt_377"><sup>[377]</sup></a></p>
+
+  <p>The Greek monk Maximus Planudes, writing in the first half of the
+  fourteenth century, followed the Arabic usage in calling his work
+  <i>Indian Arithmetic</i>.<a name="NtA_378"
+  href="#Nt_378"><sup>[378]</sup></a> There were numerous other Arabic
+  writers upon arithmetic, as that subject occupied one of the high places
+  among the sciences, but most of them did not feel it necessary to refer
+  to the origin of the symbols, the knowledge of which might well have been
+  taken for granted.</p>
+
+<p><!-- Page 94 --><span class="pagenum"><a name="page94"></a>[94]</span></p>
+
+  <p>One document, cited by Woepcke,<a name="NtA_379"
+  href="#Nt_379"><sup>[379]</sup></a> is of special interest since it shows
+  at an early period, 970 <span class="scac">A.D.</span>, the use of the
+  ordinary Arabic forms alongside the &#x121;ob&#x101;r. The title of the
+  work is <i>Interesting and Beautiful Problems on Numbers</i> copied by
+  <span class="special" title="Ahmed ibn Mohammed ibn `Abdaljalil"
+  >A&#x1E25;med ibn Mo&#x1E25;ammed ibn &#x201B;Abdaljal&#x12B;l</span>,
+  Ab&#x16B; Sa&#x201B;&#x12B;d, al-Sijz&#x12B;,<a name="NtA_380"
+  href="#Nt_380"><sup>[380]</sup></a> (951-1024) from a work by a priest
+  and physician, <span class="special"
+  title="Nazif">Na&#x1E93;&#x12B;f</span> ibn Yumn,<a name="NtA_381"
+  href="#Nt_381"><sup>[381]</sup></a> al-Qass (died c. 990). Suter does not
+  mention this work of <span class="special"
+  title="Nazif">Na&#x1E93;&#x12B;f</span>.</p>
+
+  <p>The second reason for not ascribing too much credit to the purely Arab
+  influence is that the Arab by himself never showed any intellectual
+  strength. What took place after <span class="special"
+  title="Mohammed">Mo&#x1E25;ammed</span> had lighted the fire in the
+  hearts of his people was just what always takes place when different
+  types of strong races blend,&mdash;a great renaissance in divers lines.
+  It was seen in the blending of such types at Miletus in the time of
+  Thales, at Rome in the days of the early invaders, at Alexandria when the
+  Greek set firm foot on Egyptian soil, and we see it now when all the
+  nations mingle their vitality in the New World. So when the Arab culture
+  joined with the Persian, a new civilization rose and flourished.<a
+  name="NtA_382" href="#Nt_382"><sup>[382]</sup></a> The Arab influence
+  came not from its purity, but from its intermingling with an influence
+  more cultured if less virile.</p>
+
+  <p>As a result of this interactivity among peoples of diverse interests
+  and powers, Mohammedanism was to the world from the eighth to the
+  thirteenth century what Rome and Athens and the Italo-Hellenic influence
+  generally had <!-- Page 95 --><span class="pagenum"><a
+  name="page95"></a>[95]</span>been to the ancient civilization. "If they
+  did not possess the spirit of invention which distinguished the Greeks
+  and the Hindus, if they did not show the perseverance in their
+  observations that characterized the Chinese astronomers, they at least
+  possessed the virility of a new and victorious people, with a desire to
+  understand what others had accomplished, and a taste which led them with
+  equal ardor to the study of algebra and of poetry, of philosophy and of
+  language."<a name="NtA_383" href="#Nt_383"><sup>[383]</sup></a></p>
+
+  <p>It was in 622 <span class="scac">A.D.</span> that <span
+  class="special" title="Mohammed">Mo&#x1E25;ammed</span> fled from Mecca,
+  and within a century from that time the crescent had replaced the cross
+  in Christian Asia, in Northern Africa, and in a goodly portion of Spain.
+  The Arab empire was an ellipse of learning with its foci at Bagdad and
+  Cordova, and its rulers not infrequently took pride in demanding
+  intellectual rather than commercial treasure as the result of conquest.<a
+  name="NtA_384" href="#Nt_384"><sup>[384]</sup></a></p>
+
+  <p>It was under these influences, either pre-Mohammedan or later, that
+  the Hindu numerals found their way to the North. If they were known
+  before <span class="special" title="Mohammed">Mo&#x1E25;ammed</span>'s
+  time, the proof of this fact is now lost. This much, however, is known,
+  that in the eighth century they were taken to Bagdad. It was early in
+  that century that the Mohammedans obtained their first foothold in
+  northern India, thus foreshadowing an epoch of supremacy that endured
+  with varied fortunes until after the golden age of Akbar the Great
+  (1542-1605) and Shah Jehan. They also conquered Khorassan and
+  Afghanistan, so that the learning and the commercial customs of India at
+  once found easy <!-- Page 96 --><span class="pagenum"><a
+  name="page96"></a>[96]</span>access to the newly-established schools and
+  the bazaars of Mesopotamia and western Asia. The particular paths of
+  conquest and of commerce were either by way of the Khyber Pass and
+  through Kabul, Herat and Khorassan, or by sea through the strait of Ormuz
+  to Basra (Busra) at the head of the Persian Gulf, and thence to Bagdad.
+  As a matter of fact, one form of Arabic numerals, the one now in use by
+  the Arabs, is attributed to the influence of Kabul, while the other,
+  which eventually became our numerals, may very likely have reached Arabia
+  by the other route. It is in Bagdad,<a name="NtA_385"
+  href="#Nt_385"><sup>[385]</sup></a> D&#x101;r al-Sal&#x101;m&mdash;"the
+  Abode of Peace," that our special interest in the introduction of the
+  numerals centers. Built upon the ruins of an ancient town by <span
+  class="special" title="Al-Mansur">Al-Man&#x1E63;&#x16B;r</span><a
+  name="NtA_386" href="#Nt_386"><sup>[386]</sup></a> in the second half of
+  the eighth century, it lies in one of those regions where the converging
+  routes of trade give rise to large cities.<a name="NtA_387"
+  href="#Nt_387"><sup>[387]</sup></a> Quite as well of Bagdad as of Athens
+  might Cardinal Newman have said:<a name="NtA_388"
+  href="#Nt_388"><sup>[388]</sup></a></p>
+
+  <p>"What it lost in conveniences of approach, it gained in its
+  neighborhood to the traditions of the mysterious East, and in the
+  loveliness of the region in which it lay. Hither, then, as to a sort of
+  ideal land, where all archetypes of the great and the fair were found in
+  substantial being, and all departments of truth explored, and all
+  diversities of intellectual power exhibited, where taste and philosophy
+  were majestically enthroned as in a royal court, where there was no
+  sovereignty but that of mind, and no nobility but that of genius, where
+  professors were <!-- Page 97 --><span class="pagenum"><a
+  name="page97"></a>[97]</span>rulers, and princes did homage, thither
+  flocked continually from the very corners of the <i>orbis terrarum</i>
+  the many-tongued generation, just rising, or just risen into manhood, in
+  order to gain wisdom." For here it was that <span class="special"
+  title="Al-Mansur">Al-Man&#x1E63;&#x16B;r</span> and Al-M&#x101;m&#x16B;n
+  and H&#x101;r&#x16B;n al-Rash&#x12B;d (Aaron the Just) made for a time
+  the world's center of intellectual activity in general and in the domain
+  of mathematics in particular.<a name="NtA_389"
+  href="#Nt_389"><sup>[389]</sup></a> It was just after the <i>Sindhind</i>
+  was brought to Bagdad that <span class="special"
+  title="Mohammed">Mo&#x1E25;ammed</span> ibn M&#x16B;s&#x101;
+  al-Khow&#x101;razm&#x12B;, whose name has already been mentioned,<a
+  name="NtA_390" href="#Nt_390"><sup>[390]</sup></a> was called to that
+  city. He was the most celebrated mathematician of his time, either in the
+  East or West, writing treatises on arithmetic, the sundial, the
+  astrolabe, chronology, geometry, and algebra, and giving through the
+  Latin transliteration of his name, <i>algoritmi</i>, the name of algorism
+  to the early arithmetics using the new Hindu numerals.<a name="NtA_391"
+  href="#Nt_391"><sup>[391]</sup></a> Appreciating at once the value of the
+  position system so recently brought from India, he wrote an arithmetic
+  based upon these numerals, and this was translated into Latin in the time
+  of Adelhard of Bath (c. 1180), although possibly by his contemporary
+  countryman Robert Cestrensis.<a name="NtA_392"
+  href="#Nt_392"><sup>[392]</sup></a> This translation was found in
+  Cambridge and was published by Boncompagni in 1857.<a name="NtA_393"
+  href="#Nt_393"><sup>[393]</sup></a></p>
+
+  <p>Contemporary with Al-Khow&#x101;razm&#x12B;, and working also under
+  Al-M&#x101;m&#x16B;n, was a Jewish astronomer, <span class="special"
+  title="Abu 'l-Teiyib">Ab&#x16B; 'l-&#x1E6C;eiyib</span>, <!-- Page 98
+  --><span class="pagenum"><a name="page98"></a>[98]</span>Sened ibn
+  &#x201B;Al&#x12B;, who is said to have adopted the Mohammedan religion at
+  the caliph's request. He also wrote a work on Hindu arithmetic,<a
+  name="NtA_394" href="#Nt_394"><sup>[394]</sup></a> so that the subject
+  must have been attracting considerable attention at that time. Indeed,
+  the struggle to have the Hindu numerals replace the Arabic did not cease
+  for a long time thereafter. &#x201B;Al&#x12B; ibn <span class="special"
+  title="Ahmed">A&#x1E25;med</span> al-Nasaw&#x12B;, in his arithmetic of
+  c. 1025, tells us that the symbolism of number was still unsettled in his
+  day, although most people preferred the strictly Arabic forms.<a
+  name="NtA_395" href="#Nt_395"><sup>[395]</sup></a></p>
+
+  <p>We thus have the numerals in Arabia, in two forms: one the form now
+  used there, and the other the one used by Al-Khow&#x101;razm&#x12B;. The
+  question then remains, how did this second form find its way into Europe?
+  and this question will be considered in the next chapter.</p>
+
+<hr class="full" >
+
+<p><!-- Page 99 --><span class="pagenum"><a name="page99"></a>[99]</span></p>
+
+<h3>CHAPTER VII</h3>
+
+<p class="cenhead">THE DEFINITE INTRODUCTION OF THE NUMERALS
+INTO EUROPE</p>
+
+  <p>It being doubtful whether Boethius ever knew the Hindu numeral forms,
+  certainly without the zero in any case, it becomes necessary now to
+  consider the question of their definite introduction into Europe. From
+  what has been said of the trade relations between the East and the West,
+  and of the probability that it was the trader rather than the scholar who
+  carried these numerals from their original habitat to various commercial
+  centers, it is evident that we shall never know when they first made
+  their inconspicuous entrance into Europe. Curious customs from the East
+  and from the tropics,&mdash;concerning games, social peculiarities,
+  oddities of dress, and the like,&mdash;are continually being related by
+  sailors and traders in their resorts in New York, London, Hamburg, and
+  Rotterdam to-day, customs that no scholar has yet described in print and
+  that may not become known for many years, if ever. And if this be so now,
+  how much more would it have been true a thousand years before the
+  invention of printing, when learning was at its lowest ebb. It was at
+  this period of low esteem of culture that the Hindu numerals undoubtedly
+  made their first appearance in Europe.</p>
+
+  <p>There were many opportunities for such knowledge to reach Spain and
+  Italy. In the first place the Moors went into Spain as helpers of a
+  claimant of the throne, and <!-- Page 100 --><span class="pagenum"><a
+  name="page100"></a>[100]</span>remained as conquerors. The power of the
+  Goths, who had held Spain for three centuries, was shattered at the
+  battle of Jerez de la Frontera in 711, and almost immediately the Moors
+  became masters of Spain and so remained for five hundred years, and
+  masters of Granada for a much longer period. Until 850 the Christians
+  were absolutely free as to religion and as to holding political office,
+  so that priests and monks were not infrequently skilled both in Latin and
+  Arabic, acting as official translators, and naturally reporting directly
+  or indirectly to Rome. There was indeed at this time a complaint that
+  Christian youths cultivated too assiduously a love for the literature of
+  the Saracen, and married too frequently the daughters of the infidel.<a
+  name="NtA_396" href="#Nt_396"><sup>[396]</sup></a> It is true that this
+  happy state of affairs was not permanent, but while it lasted the
+  learning and the customs of the East must have become more or less the
+  property of Christian Spain. At this time the &#x121;ob&#x101;r numerals
+  were probably in that country, and these may well have made their way
+  into Europe from the schools of Cordova, Granada, and Toledo.</p>
+
+  <p>Furthermore, there was abundant opportunity for the numerals of the
+  East to reach Europe through the journeys of travelers and ambassadors.
+  It was from the records of Suleim&#x101;n the Merchant, a well-known Arab
+  trader of the ninth century, that part of the story of Sindb&#x101;d the
+  Sailor was taken.<a name="NtA_397" href="#Nt_397"><sup>[397]</sup></a>
+  Such a merchant would have been particularly likely to know the numerals
+  of the people whom he met, and he is a type of man that may well have
+  taken such symbols to European markets. A little later, <!-- Page 101
+  --><span class="pagenum"><a name="page101"></a>[101]</span><span
+  class="special" title="Abu 'l-Hasan">Ab&#x16B; 'l-&#x1E24;asan</span>
+  &#x201B;Al&#x12B; al-Mas&#x201B;&#x16B;d&#x12B; (d. 956) of Bagdad
+  traveled to the China Sea on the east, at least as far south as Zanzibar,
+  and to the Atlantic on the west,<a name="NtA_398"
+  href="#Nt_398"><sup>[398]</sup></a> and he speaks of the nine figures
+  with which the Hindus reckoned.<a name="NtA_399"
+  href="#Nt_399"><sup>[399]</sup></a></p>
+
+  <p>There was also a Bagdad merchant, one Ab&#x16B; 'l-Q&#x101;sim
+  &#x201B;Obeidall&#x101;h ibn <span class="special"
+  title="Ahmed">A&#x1E25;med</span>, better known by his Persian name <span
+  class="special" title="Ibn Khordadbeh">Ibn
+  Khord&#x101;&#x1E0D;beh</span>,<a name="NtA_400"
+  href="#Nt_400"><sup>[400]</sup></a> who wrote about 850 <span
+  class="scac">A.D.</span> a work entitled <i>Book of Roads and
+  Provinces</i><a name="NtA_401" href="#Nt_401"><sup>[401]</sup></a> in
+  which the following graphic account appears:<a name="NtA_402"
+  href="#Nt_402"><sup>[402]</sup></a> "The Jewish merchants speak Persian,
+  Roman (Greek and Latin), Arabic, French, Spanish, and Slavic. They travel
+  from the West to the East, and from the East to the West, sometimes by
+  land, sometimes by sea. They take ship from France on the Western Sea,
+  and they voyage to Farama (near the ruins of the ancient Pelusium); there
+  they transfer their goods to caravans and go by land to Colzom (on the
+  Red Sea). They there reëmbark on the Oriental (Red) Sea and go to Hejaz
+  and to Jiddah, and thence to the Sind, India, and China. Returning, they
+  bring back the products of the oriental lands.... These journeys are also
+  made by land. The merchants, leaving France and Spain, cross to Tangier
+  and thence pass through the African provinces and Egypt. They then go to
+  Ramleh, visit Damascus, Kufa, Bagdad, and Basra, penetrate into Ahwaz,
+  Fars, Kerman, Sind, and thus reach India and China." Such travelers,
+  about 900 <span class="scac">A.D.</span>, must necessarily have spread
+  abroad a knowledge of all number <!-- Page 102 --><span
+  class="pagenum"><a name="page102"></a>[102]</span>systems used in
+  recording prices or in the computations of the market. There is an
+  interesting witness to this movement, a cruciform brooch now in the
+  British Museum. It is English, certainly as early as the eleventh
+  century, but it is inlaid with a piece of paste on which is the
+  Mohammedan inscription, in Kufic characters, "There is no God but God."
+  How did such an inscription find its way, perhaps in the time of Alcuin
+  of York, to England? And if these Kufic characters reached there, then
+  why not the numeral forms as well?</p>
+
+  <p>Even in literature of the better class there appears now and then some
+  stray proof of the important fact that the great trade routes to the far
+  East were never closed for long, and that the customs and marks of trade
+  endured from generation to generation. The <i>Gulist&#x101;n</i> of the
+  Persian poet Sa&#x201B;d&#x12B;<a name="NtA_403"
+  href="#Nt_403"><sup>[403]</sup></a> contains such a passage:</p>
+
+  <p>"I met a merchant who owned one hundred and forty camels, and fifty
+  slaves and porters.... He answered to me: 'I want to carry sulphur of
+  Persia to China, which in that country, as I hear, bears a high price;
+  and thence to take Chinese ware to Roum; and from Roum to load up with
+  brocades for Hind; and so to trade Indian steel (<i>pûlab</i>) to Halib.
+  From Halib I will convey its glass to Yeman, and carry the painted cloths
+  of Yeman back to Persia.'"<a name="NtA_404"
+  href="#Nt_404"><sup>[404]</sup></a> On the other hand, these men were not
+  of the learned class, nor would they preserve in treatises any knowledge
+  that they might have, although this knowledge would occasionally reach
+  the ears of the learned as bits of curious information.</p>
+
+<p><!-- Page 103 --><span class="pagenum"><a name="page103"></a>[103]</span></p>
+
+  <p>There were also ambassadors passing back and forth from time to time,
+  between the East and the West, and in particular during the period when
+  these numerals probably began to enter Europe. Thus Charlemagne (c. 800)
+  sent emissaries to Bagdad just at the time of the opening of the
+  mathematical activity there.<a name="NtA_405"
+  href="#Nt_405"><sup>[405]</sup></a> And with such ambassadors must have
+  gone the adventurous scholar, inspired, as Alcuin says of Archbishop
+  Albert of York (766-780),<a name="NtA_406"
+  href="#Nt_406"><sup>[406]</sup></a> to seek the learning of other lands.
+  Furthermore, the Nestorian communities, established in Eastern Asia and
+  in India at this time, were favored both by the Persians and by their
+  Mohammedan conquerors. The Nestorian Patriarch of Syria, Timotheus
+  (778-820), sent missionaries both to India and to China, and a bishop was
+  appointed for the latter field. Ibn Wahab, who traveled to China in the
+  ninth century, found images of Christ and the apostles in the Emperor's
+  court.<a name="NtA_407" href="#Nt_407"><sup>[407]</sup></a> Such a
+  learned body of men, knowing intimately the countries in which they
+  labored, could hardly have failed to make strange customs known as they
+  returned to their home stations. Then, too, in Alfred's time (849-901)
+  emissaries went <!-- Page 104 --><span class="pagenum"><a
+  name="page104"></a>[104]</span>from England as far as India,<a
+  name="NtA_408" href="#Nt_408"><sup>[408]</sup></a> and generally in the
+  Middle Ages groceries came to Europe from Asia as now they come from the
+  colonies and from America. Syria, Asia Minor, and Cyprus furnished sugar
+  and wool, and India yielded her perfumes and spices, while rich
+  tapestries for the courts and the wealthy burghers came from Persia and
+  from China.<a name="NtA_409" href="#Nt_409"><sup>[409]</sup></a> Even in
+  the time of Justinian (c. 550) there seems to have been a silk trade with
+  China, which country in turn carried on commerce with Ceylon,<a
+  name="NtA_410" href="#Nt_410"><sup>[410]</sup></a> and reached out to
+  Turkestan where other merchants transmitted the Eastern products
+  westward. In the seventh century there was a well-defined commerce
+  between Persia and India, as well as between Persia and Constantinople.<a
+  name="NtA_411" href="#Nt_411"><sup>[411]</sup></a> The Byzantine
+  <i>commerciarii</i> were stationed at the outposts not merely as customs
+  officers but as government purchasing agents.<a name="NtA_412"
+  href="#Nt_412"><sup>[412]</sup></a></p>
+
+  <p>Occasionally there went along these routes of trade men of real
+  learning, and such would surely have carried the knowledge of many
+  customs back and forth. Thus at a period when the numerals are known to
+  have been partly understood in Italy, at the opening of the eleventh
+  century, one Constantine, an African, traveled from Italy through a great
+  part of Africa and Asia, even on to India, for the purpose of learning
+  the sciences of the Orient. He spent thirty-nine years in travel, having
+  been hospitably received in Babylon, and upon his return he was welcomed
+  with great honor at Salerno.<a name="NtA_413"
+  href="#Nt_413"><sup>[413]</sup></a></p>
+
+  <p>A very interesting illustration of this intercourse also appears in
+  the tenth century, when the son of Otto I <!-- Page 105 --><span
+  class="pagenum"><a name="page105"></a>[105]</span>(936-973) married a
+  princess from Constantinople. This monarch was in touch with the Moors of
+  Spain and invited to his court numerous scholars from abroad,<a
+  name="NtA_414" href="#Nt_414"><sup>[414]</sup></a> and his intercourse
+  with the East as well as the West must have brought together much of the
+  learning of each.</p>
+
+  <p>Another powerful means for the circulation of mysticism and
+  philosophy, and more or less of culture, took its start just before the
+  conversion of Constantine (c. 312), in the form of Christian pilgrim
+  travel. This was a feature peculiar to the zealots of early Christianity,
+  found in only a slight degree among their Jewish predecessors in the
+  annual pilgrimage to Jerusalem, and almost wholly wanting in other
+  pre-Christian peoples. Chief among these early pilgrims were the two
+  Placentians, John and Antonine the Elder (c. 303), who, in their
+  wanderings to Jerusalem, seem to have started a movement which culminated
+  centuries later in the crusades.<a name="NtA_415"
+  href="#Nt_415"><sup>[415]</sup></a> In 333 a Bordeaux pilgrim compiled
+  the first Christian guide-book, the <i>Itinerary from Bordeaux to
+  Jerusalem</i>,<a name="NtA_416" href="#Nt_416"><sup>[416]</sup></a> and
+  from this time on the holy pilgrimage never entirely ceased.</p>
+
+  <p>Still another certain route for the entrance of the numerals into
+  Christian Europe was through the pillaging and trading carried on by the
+  Arabs on the northern shores of the Mediterranean. As early as 652 <span
+  class="scac">A.D.</span>, in the thirtieth year of the Hejira, the
+  Mohammedans descended upon the shores of Sicily and took much spoil.
+  Hardly had the wretched Constans given place to the <!-- Page 106
+  --><span class="pagenum"><a name="page106"></a>[106]</span>young
+  Constantine IV when they again attacked the island and plundered ancient
+  Syracuse. Again in 827, under Asad, they ravaged the coasts. Although at
+  this time they failed to conquer Syracuse, they soon held a good part of
+  the island, and a little later they successfully besieged the city.
+  Before Syracuse fell, however, they had plundered the shores of Italy,
+  even to the walls of Rome itself; and had not Leo IV, in 849, repaired
+  the neglected fortifications, the effects of the Moslem raid of that year
+  might have been very far-reaching. <span class="special" title="Ibn Khordadbeh"
+  >Ibn Khord&#x101;&#x1E0D;beh</span>, who left Bagdad in the latter part
+  of the ninth century, gives a picture of the great commercial activity at
+  that time in the Saracen city of Palermo. In this same century they had
+  established themselves in Piedmont, and in 906 they pillaged Turin.<a
+  name="NtA_417" href="#Nt_417"><sup>[417]</sup></a> On the Sorrento
+  peninsula the traveler who climbs the hill to the beautiful Ravello sees
+  still several traces of the Arab architecture, reminding him of the fact
+  that about 900 <span class="scac">A.D.</span> Amalfi was a commercial
+  center of the Moors.<a name="NtA_418" href="#Nt_418"><sup>[418]</sup></a>
+  Not only at this time, but even a century earlier, the artists of
+  northern India sold their wares at such centers, and in the courts both
+  of H&#x101;r&#x16B;n al-Rash&#x12B;d and of Charlemagne.<a name="NtA_419"
+  href="#Nt_419"><sup>[419]</sup></a> Thus the Arabs dominated the
+  Mediterranean Sea long before Venice</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p class="i12">"held the gorgeous East in fee</p>
+      <p>And was the safeguard of the West,"</p>
+    </div>
+  </div>
+  <p>and long before Genoa had become her powerful rival.<a name="NtA_420"
+  href="#Nt_420"><sup>[420]</sup></a></p>
+
+<p><!-- Page 107 --><span class="pagenum"><a name="page107"></a>[107]</span></p>
+
+  <p>Only a little later than this the brothers Nicolo and Maffeo Polo
+  entered upon their famous wanderings.<a name="NtA_421"
+  href="#Nt_421"><sup>[421]</sup></a> Leaving Constantinople in 1260, they
+  went by the Sea of Azov to Bokhara, and thence to the court of Kublai
+  Khan, penetrating China, and returning by way of Acre in 1269 with a
+  commission which required them to go back to China two years later. This
+  time they took with them Nicolo's son Marco, the historian of the
+  journey, and went across the plateau of Pamir; they spent about twenty
+  years in China, and came back by sea from China to Persia.</p>
+
+  <p>The ventures of the Poli were not long unique, however: the thirteenth
+  century had not closed before Roman missionaries and the merchant Petrus
+  de Lucolongo had penetrated China. Before 1350 the company of
+  missionaries was large, converts were numerous, churches and Franciscan
+  convents had been organized in the East, travelers were appealing for the
+  truth of their accounts to the "many" persons in Venice who had been in
+  China, Tsuan-chau-fu had a European merchant community, and Italian trade
+  and travel to China was a thing that occupied two chapters of a
+  commercial handbook.<a name="NtA_422"
+  href="#Nt_422"><sup>[422]</sup></a></p>
+
+<p><!-- Page 108 --><span class="pagenum"><a name="page108"></a>[108]</span></p>
+
+  <p>It is therefore reasonable to conclude that in the Middle Ages, as in
+  the time of Boethius, it was a simple matter for any inquiring scholar to
+  become acquainted with such numerals of the Orient as merchants may have
+  used for warehouse or price marks. And the fact that Gerbert seems to
+  have known only the forms of the simplest of these, not comprehending
+  their full significance, seems to prove that he picked them up in just
+  this way.</p>
+
+  <p>Even if Gerbert did not bring his knowledge of the Oriental numerals
+  from Spain, he may easily have obtained them from the marks on merchant's
+  goods, had he been so inclined. Such knowledge was probably obtainable in
+  various parts of Italy, though as parts of mere mercantile knowledge the
+  forms might soon have been lost, it needing the pen of the scholar to
+  preserve them. Trade at this time was not stagnant. During the eleventh
+  and twelfth centuries the Slavs, for example, had very great commercial
+  interests, their trade reaching to Kiev and Novgorod, and thence to the
+  East. Constantinople was a great clearing-house of commerce with the
+  Orient,<a name="NtA_423" href="#Nt_423"><sup>[423]</sup></a> and the
+  Byzantine merchants must have been entirely familiar with the various
+  numerals of the Eastern peoples. In the eleventh century the Italian town
+  of Amalfi established a factory<a name="NtA_424"
+  href="#Nt_424"><sup>[424]</sup></a> in Constantinople, and had trade
+  relations with Antioch and Egypt. Venice, as early as the ninth century,
+  had a valuable trade with Syria and Cairo.<a name="NtA_425"
+  href="#Nt_425"><sup>[425]</sup></a> Fifty years after Gerbert died, in
+  the time of Cnut, the Dane and the Norwegian pushed their commerce far
+  beyond the northern seas, both by caravans through Russia to the Orient,
+  and by their venturesome barks which <!-- Page 109 --><span
+  class="pagenum"><a name="page109"></a>[109]</span>sailed through the
+  Strait of Gibraltar into the Mediterranean.<a name="NtA_426"
+  href="#Nt_426"><sup>[426]</sup></a> Only a little later, probably before
+  1200 <span class="scac">A.D.</span>, a clerk in the service of Thomas à
+  Becket, present at the latter's death, wrote a life of the martyr, to
+  which (fortunately for our purposes) he prefixed a brief eulogy of the
+  city of London.<a name="NtA_427" href="#Nt_427"><sup>[427]</sup></a> This
+  clerk, William Fitz Stephen by name, thus speaks of the British
+  capital:</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Aurum mittit Arabs: species et thura Sabæus:</p>
+      <p>Arma Sythes: oleum palmarum divite sylva</p>
+      <p>Pingue solum Babylon: Nilus lapides pretiosos:</p>
+      <p>Norwegi, Russi, varium grisum, sabdinas:</p>
+      <p>Seres, purpureas vestes: Galli, sua vina.</p>
+    </div>
+  </div>
+  <p>Although, as a matter of fact, the Arabs had no gold to send, and the
+  Scythians no arms, and Egypt no precious stones save only the turquoise,
+  the Chinese (<i>Seres</i>) may have sent their purple vestments, and the
+  north her sables and other furs, and France her wines. At any rate the
+  verses show very clearly an extensive foreign trade.</p>
+
+  <p>Then there were the Crusades, which in these times brought the East in
+  touch with the West. The spirit of the Orient showed itself in the songs
+  of the troubadours, and the <i>baudekin</i>,<a name="NtA_428"
+  href="#Nt_428"><sup>[428]</sup></a> the canopy of Bagdad,<a
+  name="NtA_429" href="#Nt_429"><sup>[429]</sup></a> became common in the
+  churches of Italy. In Sicily and in Venice the textile industries of the
+  East found place, and made their way even to the Scandinavian
+  peninsula.<a name="NtA_430" href="#Nt_430"><sup>[430]</sup></a></p>
+
+  <p>We therefore have this state of affairs: There was abundant
+  intercourse between the East and West for <!-- Page 110 --><span
+  class="pagenum"><a name="page110"></a>[110]</span>some centuries before
+  the Hindu numerals appear in any manuscripts in Christian Europe. The
+  numerals must of necessity have been known to many traders in a country
+  like Italy at least as early as the ninth century, and probably even
+  earlier, but there was no reason for preserving them in treatises.
+  Therefore when a man like Gerbert made them known to the scholarly
+  circles, he was merely describing what had been familiar in a small way
+  to many people in a different walk of life.</p>
+
+  <p>Since Gerbert<a name="NtA_431" href="#Nt_431"><sup>[431]</sup></a> was
+  for a long time thought to have been the one to introduce the numerals
+  into Italy,<a name="NtA_432" href="#Nt_432"><sup>[432]</sup></a> a brief
+  sketch of this unique character is proper. Born of humble parents,<a
+  name="NtA_433" href="#Nt_433"><sup>[433]</sup></a> this remarkable man
+  became the counselor and companion of kings, and finally wore the papal
+  tiara as Sylvester II, from 999 until his death in 1003.<a name="NtA_434"
+  href="#Nt_434"><sup>[434]</sup></a> He was early brought under the
+  influence of the monks at Aurillac, and particularly of Raimund, who had
+  been a pupil of Odo of Cluny, and there in due time he himself took holy
+  orders. He visited Spain in about 967 in company with Count Borel,<a
+  name="NtA_435" href="#Nt_435"><sup>[435]</sup></a> remaining there three
+  years, <!-- Page 111 --><span class="pagenum"><a
+  name="page111"></a>[111]</span>and studying under Bishop Hatto of Vich,<a
+  name="NtA_436" href="#Nt_436"><sup>[436]</sup></a> a city in the province
+  of Barcelona,<a name="NtA_437" href="#Nt_437"><sup>[437]</sup></a> then
+  entirely under Christian rule. Indeed, all of Gerbert's testimony is as
+  to the influence of the Christian civilization upon his education. Thus
+  he speaks often of his study of Boethius,<a name="NtA_438"
+  href="#Nt_438"><sup>[438]</sup></a> so that if the latter knew the
+  numerals Gerbert would have learned them from him.<a name="NtA_439"
+  href="#Nt_439"><sup>[439]</sup></a> If Gerbert had studied in any Moorish
+  schools he would, under the decree of the emir Hish&#x101;m (787-822),
+  have been obliged to know Arabic, which would have taken most of his
+  three years in Spain, and of which study we have not the slightest hint
+  in any of his letters.<a name="NtA_440"
+  href="#Nt_440"><sup>[440]</sup></a> On the other hand, Barcelona was the
+  only Christian province in immediate touch with the Moorish civilization
+  at that time.<a name="NtA_441" href="#Nt_441"><sup>[441]</sup></a>
+  Furthermore we know that earlier in the same century King Alonzo of
+  Asturias (d. 910) confided the education of his son Ordoño to the Arab
+  scholars of the court of the <!-- Page 112 --><span class="pagenum"><a
+  name="page112"></a>[112]</span>w&#x101;l&#x12B; of Saragossa,<a
+  name="NtA_442" href="#Nt_442"><sup>[442]</sup></a> so that there was more
+  or less of friendly relation between Christian and Moor.</p>
+
+  <p>After his three years in Spain, Gerbert went to Italy, about 970,
+  where he met Pope John XIII, being by him presented to the emperor Otto
+  I. Two years later (972), at the emperor's request, he went to Rheims,
+  where he studied philosophy, assisting to make of that place an
+  educational center; and in 983 he became abbot at Bobbio. The next year
+  he returned to Rheims, and became archbishop of that diocese in 991. For
+  political reasons he returned to Italy in 996, became archbishop of
+  Ravenna in 998, and the following year was elected to the papal chair.
+  Far ahead of his age in wisdom, he suffered as many such scholars have
+  even in times not so remote by being accused of heresy and witchcraft. As
+  late as 1522, in a biography published at Venice, it is related that by
+  black art he attained the papacy, after having given his soul to the
+  devil.<a name="NtA_443" href="#Nt_443"><sup>[443]</sup></a> Gerbert was,
+  however, interested in astrology,<a name="NtA_444"
+  href="#Nt_444"><sup>[444]</sup></a> although this was merely the
+  astronomy of that time and was such a science as any learned man would
+  wish to know, even as to-day we wish to be reasonably familiar with
+  physics and chemistry.</p>
+
+  <p>That Gerbert and his pupils knew the &#x121;ob&#x101;r numerals is a
+  fact no longer open to controversy.<a name="NtA_445"
+  href="#Nt_445"><sup>[445]</sup></a> Bernelinus and Richer<a
+  name="NtA_446" href="#Nt_446"><sup>[446]</sup></a> call them by the
+  well-known name of <!-- Page 113 --><span class="pagenum"><a
+  name="page113"></a>[113]</span>"caracteres," a word used by Radulph of
+  Laon in the same sense a century later.<a name="NtA_447"
+  href="#Nt_447"><sup>[447]</sup></a> It is probable that Gerbert was the
+  first to describe these &#x121;ob&#x101;r numerals in any scientific way
+  in Christian Europe, but without the zero. If he knew the latter he
+  certainly did not understand its use.<a name="NtA_448"
+  href="#Nt_448"><sup>[448]</sup></a></p>
+
+  <p>The question still to be settled is as to where he found these
+  numerals. That he did not bring them from Spain is the opinion of a
+  number of careful investigators.<a name="NtA_449"
+  href="#Nt_449"><sup>[449]</sup></a> This is thought to be the more
+  probable because most of the men who made Spain famous for learning lived
+  after Gerbert was there. Such were Ibn S&#x12B;n&#x101; (Avicenna) who
+  lived at the beginning, and Gerber of Seville who flourished in the
+  middle, of the eleventh century, and Ab&#x16B; Roshd (Averroës) who lived
+  at the end of the twelfth.<a name="NtA_450"
+  href="#Nt_450"><sup>[450]</sup></a> Others hold that his proximity to
+  <!-- Page 114 --><span class="pagenum"><a
+  name="page114"></a>[114]</span>the Arabs for three years makes it
+  probable that he assimilated some of their learning, in spite of the fact
+  that the lines between Christian and Moor at that time were sharply
+  drawn.<a name="NtA_451" href="#Nt_451"><sup>[451]</sup></a> Writers fail,
+  however, to recognize that a commercial numeral system would have been
+  more likely to be made known by merchants than by scholars. The itinerant
+  peddler knew no forbidden pale in Spain, any more than he has known one
+  in other lands. If the &#x121;ob&#x101;r numerals were used for marking
+  wares or keeping simple accounts, it was he who would have known them,
+  and who would have been the one rather than any Arab scholar to bring
+  them to the inquiring mind of the young French monk. The facts that
+  Gerbert knew them only imperfectly, that he used them solely for
+  calculations, and that the forms are evidently like the Spanish
+  &#x121;ob&#x101;r, make it all the more probable that it was through the
+  small tradesman of the Moors that this versatile scholar derived his
+  knowledge. Moreover the part of the geometry bearing his name, and that
+  seems unquestionably his, shows the Arab influence, proving that he at
+  least came into contact with the transplanted Oriental learning, even
+  though imperfectly.<a name="NtA_452" href="#Nt_452"><sup>[452]</sup></a>
+  There was also the persistent Jewish merchant trading with both peoples
+  then as now, always alive to the acquiring of useful knowledge, and it
+  would be very natural for a man like Gerbert to welcome learning from
+  such a source.</p>
+
+  <p>On the other hand, the two leading sources of information as to the
+  life of Gerbert reveal practically nothing to show that he came within
+  the Moorish sphere of influence during his sojourn in Spain. These
+  sources <!-- Page 115 --><span class="pagenum"><a
+  name="page115"></a>[115]</span>are his letters and the history written by
+  Richer. Gerbert was a master of the epistolary art, and his exalted
+  position led to the preservation of his letters to a degree that would
+  not have been vouchsafed even by their classic excellence.<a
+  name="NtA_453" href="#Nt_453"><sup>[453]</sup></a> Richer was a monk at
+  St. Remi de Rheims, and was doubtless a pupil of Gerbert. The latter,
+  when archbishop of Rheims, asked Richer to write a history of his times,
+  and this was done. The work lay in manuscript, entirely forgotten until
+  Pertz discovered it at Bamberg in 1833.<a name="NtA_454"
+  href="#Nt_454"><sup>[454]</sup></a> The work is dedicated to Gerbert as
+  archbishop of Rheims,<a name="NtA_455"
+  href="#Nt_455"><sup>[455]</sup></a> and would assuredly have testified to
+  such efforts as he may have made to secure the learning of the Moors.</p>
+
+  <p>Now it is a fact that neither the letters nor this history makes any
+  statement as to Gerbert's contact with the Saracens. The letters do not
+  speak of the Moors, of the Arab numerals, nor of Cordova. Spain is not
+  referred to by that name, and only one Spanish scholar is mentioned. In
+  one of his letters he speaks of Joseph Ispanus,<a name="NtA_456"
+  href="#Nt_456"><sup>[456]</sup></a> or Joseph Sapiens, but who this
+  Joseph the Wise of Spain may have been we do not know. Possibly <!-- Page
+  116 --><span class="pagenum"><a name="page116"></a>[116]</span>it was he
+  who contributed the morsel of knowledge so imperfectly assimilated by the
+  young French monk.<a name="NtA_457" href="#Nt_457"><sup>[457]</sup></a>
+  Within a few years after Gerbert's visit two young Spanish monks of
+  lesser fame, and doubtless with not that keen interest in mathematical
+  matters which Gerbert had, regarded the apparently slight knowledge which
+  they had of the Hindu numeral forms as worthy of somewhat permanent
+  record<a name="NtA_458" href="#Nt_458"><sup>[458]</sup></a> in
+  manuscripts which they were transcribing. The fact that such knowledge
+  had penetrated to their modest cloisters in northern Spain&mdash;the one
+  Albelda or Albaida&mdash;indicates that it was rather widely
+  diffused.</p>
+
+  <p>Gerbert's treatise <i>Libellus de numerorum divisione</i><a
+  name="NtA_459" href="#Nt_459"><sup>[459]</sup></a> is characterized by
+  Chasles as "one of the most obscure documents in the history of
+  science."<a name="NtA_460" href="#Nt_460"><sup>[460]</sup></a> The most
+  complete information in regard to this and the other mathematical works
+  of Gerbert is given by Bubnov,<a name="NtA_461"
+  href="#Nt_461"><sup>[461]</sup></a> who considers this work to be
+  genuine.<a name="NtA_462" href="#Nt_462"><sup>[462]</sup></a></p>
+
+<p><!-- Page 117 --><span class="pagenum"><a name="page117"></a>[117]</span></p>
+
+  <p>So little did Gerbert appreciate these numerals that in his works
+  known as the <i>Regula de abaco computi</i> and the <i>Libellus</i> he
+  makes no use of them at all, employing only the Roman forms.<a
+  name="NtA_463" href="#Nt_463"><sup>[463]</sup></a> Nevertheless
+  Bernelinus<a name="NtA_464" href="#Nt_464"><sup>[464]</sup></a> refers to
+  the nine &#x121;ob&#x101;r characters.<a name="NtA_465"
+  href="#Nt_465"><sup>[465]</sup></a> These Gerbert had marked on a
+  thousand <i>jetons</i> or counters,<a name="NtA_466"
+  href="#Nt_466"><sup>[466]</sup></a> using the latter on an abacus which
+  he had a sign-maker prepare for him.<a name="NtA_467"
+  href="#Nt_467"><sup>[467]</sup></a> Instead of putting eight counters in
+  say the tens' column, Gerbert would put a single counter marked 8, and so
+  for the other places, leaving the column empty where we would place a
+  zero, but where he, lacking the zero, had no counter to place. These
+  counters he possibly called <i>caracteres</i>, a name which adhered also
+  to the figures themselves. It is an interesting speculation to consider
+  whether these <i>apices</i>, as they are called in the Boethius
+  interpolations, were in any way suggested by those Roman jetons generally
+  known in numismatics as <i>tesserae</i>, and bearing the figures I-XVI,
+  the sixteen referring to the number of <i>assi</i> in a
+  <i>sestertius</i>.<a name="NtA_468" href="#Nt_468"><sup>[468]</sup></a>
+  The <!-- Page 118 --><span class="pagenum"><a
+  name="page118"></a>[118]</span>name <i>apices</i> adhered to the
+  Hindu-Arabic numerals until the sixteenth century.<a name="NtA_469"
+  href="#Nt_469"><sup>[469]</sup></a></p>
+
+  <p>To the figures on the <i>apices</i> were given the names Igin, andras,
+  ormis, arbas, quimas, calctis or caltis, zenis, temenias, celentis,
+  sipos,<a name="NtA_470" href="#Nt_470"><sup>[470]</sup></a> the origin
+  and meaning of which still remain a mystery. The Semitic origin of
+  several of the words seems probable. <i>Wahud</i>, <i>thaneine</i>, <!--
+  Page 119 --><span class="pagenum"><a
+  name="page119"></a>[119]</span><i>thalata</i>, <i>arba</i>, <i>kumsa</i>,
+  <i>setta</i>, <i>sebba</i>, <i>timinia</i>, <i>taseud</i> are given by
+  the Rev. R. Patrick<a name="NtA_471" href="#Nt_471"><sup>[471]</sup></a>
+  as the names, in an Arabic dialect used in Morocco, for the numerals from
+  one to nine. Of these the words for four, five, and eight are strikingly
+  like those given above.</p>
+
+  <p>The name <i>apices</i> was not, however, a common one in later times.
+  <i>Notae</i> was more often used, and it finally gave the name to
+  notation.<a name="NtA_472" href="#Nt_472"><sup>[472]</sup></a> Still more
+  common were the names <i>figures</i>, <i>ciphers</i>, <i>signs</i>,
+  <i>elements</i>, and <i>characters</i>.<a name="NtA_473"
+  href="#Nt_473"><sup>[473]</sup></a></p>
+
+  <p>So little effect did the teachings of Gerbert have in making known the
+  new numerals, that O'Creat, who lived a century later, a friend and pupil
+  of Adelhard <!-- Page 120 --><span class="pagenum"><a
+  name="page120"></a>[120]</span>of Bath, used the zero with the Roman
+  characters, in contrast to Gerbert's use of the &#x121;ob&#x101;r forms
+  without the zero.<a name="NtA_474" href="#Nt_474"><sup>[474]</sup></a>
+  O'Creat uses three forms for zero, o, &#x14D;, and <span title="t" class="grk"
+  >&tau;</span>, as in Maximus Planudes. With this use of the zero goes,
+  naturally, a place value, for he writes III&nbsp;III for 33, ICCOO and
+  I.&nbsp;II.&nbsp;<span class="grk">&tau;</span>.&nbsp;<span class="grk">&tau;</span> for
+  1200, I.&nbsp;O.&nbsp;VIII.&nbsp;IX for 1089, and I.&nbsp;IIII.&nbsp;IIII.&nbsp;<span
+  class="grk">&tau;</span><span class="grk">&tau;</span><span
+  class="grk">&tau;</span><span class="grk">&tau;</span> for the square of
+  1200.</p>
+
+  <p>The period from the time of Gerbert until after the appearance of
+  Leonardo's monumental work may be called the period of the abacists. Even
+  for many years after the appearance early in the twelfth century of the
+  books explaining the Hindu art of reckoning, there was strife between the
+  abacists, the advocates of the abacus, and the algorists, those who
+  favored the new numerals. The words <i>cifra</i> and <i>algorismus
+  cifra</i> were used with a somewhat derisive significance, indicative of
+  absolute uselessness, as indeed the zero is useless on an abacus in which
+  the value of any unit is given by the column which it occupies.<a
+  name="NtA_475" href="#Nt_475"><sup>[475]</sup></a> So Gautier de Coincy
+  (1177-1236) in a work on the miracles of Mary says:</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>A horned beast, a sheep,</p>
+      <p>An algorismus-cipher,</p>
+      <p>Is a priest, who on such a feast day</p>
+      <p>Does not celebrate the holy Mother.<a name="NtA_476" href="#Nt_476"><sup>[476]</sup></a></p>
+    </div>
+  </div>
+  <p>So the abacus held the field for a long time, even against the new
+  algorism employing the new numerals. <!-- Page 121 --><span
+  class="pagenum"><a name="page121"></a>[121]</span>Geoffrey Chaucer<a
+  name="NtA_477" href="#Nt_477"><sup>[477]</sup></a> describes in <i>The
+  Miller's Tale</i> the clerk with</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p class="hg3">"His Almageste and bokes grete and smale,</p>
+      <p>His astrelabie, longinge for his art,</p>
+      <p>His augrim-stones layen faire apart</p>
+      <p>On shelves couched at his beddes heed."</p>
+    </div>
+  </div>
+  <p>So, too, in Chaucer's explanation of the astrolabe,<a name="NtA_478"
+  href="#Nt_478"><sup>[478]</sup></a> written for his son Lewis, the number
+  of degrees is expressed on the instrument in Hindu-Arabic numerals: "Over
+  the whiche degrees ther ben noumbres of augrim, that devyden thilke same
+  degrees fro fyve to fyve," and "... the nombres ... ben writen in
+  augrim," meaning in the way of the algorism. Thomas Usk about 1387
+  writes:<a name="NtA_479" href="#Nt_479"><sup>[479]</sup></a> "a sypher in
+  augrim have no might in signification of it-selve, yet he yeveth power in
+  signification to other." So slow and so painful is the assimilation of
+  new ideas.</p>
+
+  <p>Bernelinus<a name="NtA_480" href="#Nt_480"><sup>[480]</sup></a> states
+  that the abacus is a well-polished board (or table), which is covered
+  with blue sand and used by geometers in drawing geometrical figures. We
+  have previously mentioned the fact that the Hindus also performed
+  mathematical computations in the sand, although there is no evidence to
+  show that they had any column abacus.<a name="NtA_481"
+  href="#Nt_481"><sup>[481]</sup></a> For the purposes of computation,
+  Bernelinus continues, the board is divided into thirty vertical columns,
+  three of which are reserved for fractions. Beginning with the units
+  columns, each set of <!-- Page 122 --><span class="pagenum"><a
+  name="page122"></a>[122]</span>three columns (<i>lineae</i> is the word
+  which Bernelinus uses) is grouped together by a semicircular arc placed
+  above them, while a smaller arc is placed over the units column and
+  another joins the tens and hundreds columns. Thus arose the designation
+  <i>arcus pictagore</i><a name="NtA_482"
+  href="#Nt_482"><sup>[482]</sup></a> or sometimes simply <i>arcus</i>.<a
+  name="NtA_483" href="#Nt_483"><sup>[483]</sup></a> The operations of
+  addition, subtraction, and multiplication upon this form of the abacus
+  required little explanation, although they were rather extensively
+  treated, especially the multiplication of different orders of numbers.
+  But the operation of division was effected with some difficulty. For the
+  explanation of the method of division by the use of the complementary
+  difference,<a name="NtA_484" href="#Nt_484"><sup>[484]</sup></a> long the
+  stumbling-block in the way of the medieval arithmetician, the reader is
+  referred to works on the history of mathematics<a name="NtA_485"
+  href="#Nt_485"><sup>[485]</sup></a> and to works relating particularly to
+  the abacus.<a name="NtA_486" href="#Nt_486"><sup>[486]</sup></a></p>
+
+  <p>Among the writers on the subject may be mentioned Abbo<a
+  name="NtA_487" href="#Nt_487"><sup>[487]</sup></a> of Fleury (c. 970),
+  Heriger<a name="NtA_488" href="#Nt_488"><sup>[488]</sup></a> of Lobbes or
+  Laubach <!-- Page 123 --><span class="pagenum"><a
+  name="page123"></a>[123]</span>(c. 950-1007), and Hermannus Contractus<a
+  name="NtA_489" href="#Nt_489"><sup>[489]</sup></a> (1013-1054), all of
+  whom employed only the Roman numerals. Similarly Adelhard of Bath (c.
+  1130), in his work <i>Regulae Abaci</i>,<a name="NtA_490"
+  href="#Nt_490"><sup>[490]</sup></a> gives no reference to the new
+  numerals, although it is certain that he knew them. Other writers on the
+  abacus who used some form of Hindu numerals were Gerland<a name="NtA_491"
+  href="#Nt_491"><sup>[491]</sup></a> (first half of twelfth century) and
+  Turchill<a name="NtA_492" href="#Nt_492"><sup>[492]</sup></a> (c. 1200).
+  For the forms used at this period the reader is referred to the plate on
+  page <a href="#page88">88</a>.</p>
+
+  <p>After Gerbert's death, little by little the scholars of Europe came to
+  know the new figures, chiefly through the introduction of Arab learning.
+  The Dark Ages had passed, although arithmetic did not find another
+  advocate as prominent as Gerbert for two centuries. Speaking of this
+  great revival, Raoul Glaber<a name="NtA_493"
+  href="#Nt_493"><sup>[493]</sup></a> (985-c. 1046), a monk of the great
+  Benedictine abbey of Cluny, of the eleventh century, says: "It was as
+  though the world had arisen and tossed aside the worn-out garments of
+  ancient time, and wished to apparel itself in a white robe of churches."
+  And with this activity in religion came a corresponding interest in other
+  lines. Algorisms began to appear, and knowledge from the outside world
+  found <!-- Page 124 --><span class="pagenum"><a
+  name="page124"></a>[124]</span>interested listeners. Another Raoul, or
+  Radulph, to whom we have referred as Radulph of Laon,<a name="NtA_494"
+  href="#Nt_494"><sup>[494]</sup></a> a teacher in the cloister school of
+  his city, and the brother of Anselm of Laon<a name="NtA_495"
+  href="#Nt_495"><sup>[495]</sup></a> the celebrated theologian, wrote a
+  treatise on music, extant but unpublished, and an arithmetic which Nagl
+  first published in 1890.<a name="NtA_496"
+  href="#Nt_496"><sup>[496]</sup></a> The latter work, preserved to us in a
+  parchment manuscript of seventy-seven leaves, contains a curious mixture
+  of Roman and &#x121;ob&#x101;r numerals, the former for expressing large
+  results, the latter for practical calculation. These &#x121;ob&#x101;r
+  "caracteres" include the sipos (zero), <a href="images/130a.png"><img
+  src="images/130a.png" class="middle" style="height:2ex" alt="Symbol"
+  /></a>, of which, however, Radulph did not know the full significance;
+  showing that at the opening of the twelfth century the system was still
+  uncertain in its status in the church schools of central France.</p>
+
+  <p>At the same time the words <i>algorismus</i> and <i>cifra</i> were
+  coming into general use even in non-mathematical literature. Jordan <a
+  name="NtA_497" href="#Nt_497"><sup>[497]</sup></a> cites numerous
+  instances of such use from the works of Alanus ab Insulis<a
+  name="NtA_498" href="#Nt_498"><sup>[498]</sup></a> (Alain de Lille),
+  Gautier de Coincy (1177-1236), and others.</p>
+
+  <p>Another contributor to arithmetic during this interesting period was a
+  prominent Spanish Jew called variously John of Luna, John of Seville,
+  Johannes Hispalensis, Johannes Toletanus, and Johannes Hispanensis de
+  Luna.<a name="NtA_499" href="#Nt_499"><sup>[499]</sup></a> <!-- Page 125
+  --><span class="pagenum"><a name="page125"></a>[125]</span>His date is
+  rather closely fixed by the fact that he dedicated a work to Raimund who
+  was archbishop of Toledo between 1130 and 1150.<a name="NtA_500"
+  href="#Nt_500"><sup>[500]</sup></a> His interests were chiefly in the
+  translation of Arabic works, especially such as bore upon the
+  Aristotelian philosophy. From the standpoint of arithmetic, however, the
+  chief interest centers about a manuscript entitled <i>Joannis Hispalensis
+  liber Algorismi de Practica Arismetrice</i> which Boncompagni found in
+  what is now the <i>Bibliothèque nationale</i> at Paris. Although this
+  distinctly lays claim to being Al-Khow&#x101;razm&#x12B;'s work,<a
+  name="NtA_501" href="#Nt_501"><sup>[501]</sup></a> the evidence is
+  altogether against the statement,<a name="NtA_502"
+  href="#Nt_502"><sup>[502]</sup></a> but the book is quite as valuable,
+  since it represents the knowledge of the time in which it was written. It
+  relates to the operations with integers and sexagesimal fractions,
+  including roots, and contains no applications.<a name="NtA_503"
+  href="#Nt_503"><sup>[503]</sup></a></p>
+
+  <p>Contemporary with John of Luna, and also living in Toledo, was Gherard
+  of Cremona,<a name="NtA_504" href="#Nt_504"><sup>[504]</sup></a> who has
+  sometimes been identified, but erroneously, with Gernardus,<a
+  name="NtA_505" href="#Nt_505"><sup>[505]</sup></a> the <!-- Page 126
+  --><span class="pagenum"><a name="page126"></a>[126]</span>author of a
+  work on algorism. He was a physician, an astronomer, and a mathematician,
+  translating from the Arabic both in Italy and in Spain. In arithmetic he
+  was influential in spreading the ideas of algorism.</p>
+
+  <p>Four Englishmen&mdash;Adelhard of Bath (c. 1130), Robert of Chester
+  (Robertus Cestrensis, c. 1143), William Shelley, and Daniel Morley
+  (1180)&mdash;are known<a name="NtA_506"
+  href="#Nt_506"><sup>[506]</sup></a> to have journeyed to Spain in the
+  twelfth century for the purpose of studying mathematics and Arabic.
+  Adelhard of Bath made translations from Arabic into Latin of
+  Al-Khow&#x101;razm&#x12B;'s astronomical tables<a name="NtA_507"
+  href="#Nt_507"><sup>[507]</sup></a> and of Euclid's Elements,<a
+  name="NtA_508" href="#Nt_508"><sup>[508]</sup></a> while Robert of
+  Chester is known as the translator of Al-Khow&#x101;razm&#x12B;'s
+  algebra.<a name="NtA_509" href="#Nt_509"><sup>[509]</sup></a> There is no
+  reason to doubt that all of these men, and others, were familiar with the
+  numerals which the Arabs were using.</p>
+
+  <p>The earliest trace we have of computation with Hindu numerals in
+  Germany is in an Algorismus of 1143, now in the Hofbibliothek in
+  Vienna.<a name="NtA_510" href="#Nt_510"><sup>[510]</sup></a> It is bound
+  in with a <!-- Page 127 --><span class="pagenum"><a
+  name="page127"></a>[127]</span><i>Computus</i> by the same author and
+  bearing the date given. It contains chapters "De additione," "De
+  diminutione," "De mediatione," "De divisione," and part of a chapter on
+  multiplication. The numerals are in the usual medieval forms except the 2
+  which, as will be seen from the illustration,<a name="NtA_511"
+  href="#Nt_511"><sup>[511]</sup></a> is somewhat different, and the 3,
+  which takes the peculiar shape <a href="images/133a.png"><img
+  src="images/133a.png" class="middle" style="height:2ex" alt="Symbol"
+  /></a>, a form characteristic of the twelfth century.</p>
+
+  <p>It was about the same time that the <i>Sefer ha-Mispar</i>,<a
+  name="NtA_512" href="#Nt_512"><sup>[512]</sup></a> the Book of Number,
+  appeared in the Hebrew language. The author, Rabbi Abraham ibn Meïr ibn
+  Ezra,<a name="NtA_513" href="#Nt_513"><sup>[513]</sup></a> was born in
+  Toledo (c. 1092). In 1139 he went to Egypt, Palestine, and the Orient,
+  spending also some years in Italy. Later he lived in southern France and
+  in England. He died in 1167. The probability is that he acquired his
+  knowledge of the Hindu arithmetic<a name="NtA_514"
+  href="#Nt_514"><sup>[514]</sup></a> in his native town of Toledo, but it
+  is also likely that the knowledge of other systems which he acquired on
+  travels increased his appreciation of this one. We have mentioned the
+  fact that he used the first letters of the Hebrew alphabet, <span
+  lang="he" class="heb" title="A B G D H W Z CH T`" ><bdo dir="rtl">&#x5D0;
+  &#x5D1; &#x5D2; &#x5D3; &#x5D4; &#x5D5; &#x5D6; &#x5D7;
+  &#x5D8;</bdo></span>, for the numerals 9 8 7 6 5 4 3 2 1, and a circle
+  for the zero. The quotation in the note given below shows that he knew of
+  the Hindu origin; but in his manuscript, although he set down the Hindu
+  forms, he used the above nine Hebrew letters with place value for all
+  computations.</p>
+
+<hr class="full" >
+
+<p><!-- Page 128 --><span class="pagenum"><a name="page128"></a>[128]</span></p>
+
+<h3>CHAPTER VIII</h3>
+
+<p class="cenhead">THE SPREAD OF THE NUMERALS IN EUROPE</p>
+
+  <p>Of all the medieval writers, probably the one most influential in
+  introducing the new numerals to the scholars of Europe was Leonardo
+  Fibonacci, of Pisa.<a name="NtA_515" href="#Nt_515"><sup>[515]</sup></a>
+  This remarkable man, the most noteworthy mathematical genius of the
+  Middle Ages, was born at Pisa about 1175.<a name="NtA_516"
+  href="#Nt_516"><sup>[516]</sup></a></p>
+
+  <p>The traveler of to-day may cross the Via Fibonacci on his way to the
+  Campo Santo, and there he may see at the end of the long corridor, across
+  the quadrangle, the statue of Leonardo in scholars garb. Few towns have
+  honored a mathematician more, and few mathematicians have so distinctly
+  honored their birthplace. Leonardo was born in the golden age of this
+  city, the period of its commercial, religious, and intellectual
+  prosperity.<a name="NtA_517" href="#Nt_517"><sup>[517]</sup></a> <!--
+  Page 129 --><span class="pagenum"><a
+  name="page129"></a>[129]</span>Situated practically at the mouth of the
+  Arno, Pisa formed with Genoa and Venice the trio of the greatest
+  commercial centers of Italy at the opening of the thirteenth century.
+  Even before Venice had captured the Levantine trade, Pisa had close
+  relations with the East. An old Latin chronicle relates that in 1005
+  "Pisa was captured by the Saracens," that in the following year "the
+  Pisans overthrew the Saracens at Reggio," and that in 1012 "the Saracens
+  came to Pisa and destroyed it." The city soon recovered, however, sending
+  no fewer than a hundred and twenty ships to Syria in 1099,<a
+  name="NtA_518" href="#Nt_518"><sup>[518]</sup></a> founding a merchant
+  colony in Constantinople a few years later,<a name="NtA_519"
+  href="#Nt_519"><sup>[519]</sup></a> and meanwhile carrying on an
+  interurban warfare in Italy that seemed to stimulate it to great
+  activity.<a name="NtA_520" href="#Nt_520"><sup>[520]</sup></a> A writer
+  of 1114 tells us that at that time there were many heathen
+  people&mdash;Turks, Libyans, Parthians, and Chaldeans&mdash;to be found
+  in Pisa. It was in the midst of such wars, in a cosmopolitan and
+  commercial town, in a center where literary work was not appreciated,<a
+  name="NtA_521" href="#Nt_521"><sup>[521]</sup></a> that the genius of
+  Leonardo appears as one of the surprises of history, warning us again
+  that "we should draw no horoscope; that we should expect little, for what
+  we expect will not come to pass."<a name="NtA_522"
+  href="#Nt_522"><sup>[522]</sup></a></p>
+
+  <p>Leonardo's father was one William,<a name="NtA_523"
+  href="#Nt_523"><sup>[523]</sup></a> and he had a brother named
+  Bonaccingus,<a name="NtA_524" href="#Nt_524"><sup>[524]</sup></a> but
+  nothing further is <!-- Page 130 --><span class="pagenum"><a
+  name="page130"></a>[130]</span>known of his family. As to Fibonacci, most
+  writers<a name="NtA_525" href="#Nt_525"><sup>[525]</sup></a> have assumed
+  that his father's name was Bonaccio,<a name="NtA_526"
+  href="#Nt_526"><sup>[526]</sup></a> whence <i>filius Bonaccii</i>, or
+  Fibonacci. Others<a name="NtA_527" href="#Nt_527"><sup>[527]</sup></a>
+  believe that the name, even in the Latin form of <i>filius Bonaccii</i>
+  as used in Leonardo's work, was simply a general one, like our Johnson or
+  Bronson (Brown's son); and the only contemporary evidence that we have
+  bears out this view. As to the name Bigollo, used by Leonardo, some have
+  thought it a self-assumed one meaning blockhead, a term that had been
+  applied to him by the commercial world or possibly by the university
+  circle, and taken by him that he might prove what a blockhead could do.
+  Milanesi,<a name="NtA_528" href="#Nt_528"><sup>[528]</sup></a> however,
+  has shown that the word Bigollo (or Pigollo) was used in Tuscany to mean
+  a traveler, and was naturally assumed by one who had studied, as Leonardo
+  had, in foreign lands.</p>
+
+  <p>Leonardo's father was a commercial agent at Bugia, the modern
+  Bougie,<a name="NtA_529" href="#Nt_529"><sup>[529]</sup></a> the ancient
+  Saldae on the coast of Barbary,<a name="NtA_530"
+  href="#Nt_530"><sup>[530]</sup></a> a royal capital under the Vandals and
+  again, a century before Leonardo, under the Beni Hammad. It had one of
+  the best harbors on the coast, sheltered as it is by Mt. Lalla Guraia,<a
+  name="NtA_531" href="#Nt_531"><sup>[531]</sup></a> and at the close of
+  the twelfth century it was a center of African commerce. It was here that
+  Leonardo was taken as a child, and here he went to school to a Moorish
+  master. When he reached the years of young manhood he started on a tour
+  of the Mediterranean Sea, and visited Egypt, Syria, Greece, Sicily, and
+  Provence, meeting with scholars as well as with <!-- Page 131 --><span
+  class="pagenum"><a name="page131"></a>[131]</span>merchants, and imbibing
+  a knowledge of the various systems of numbers in use in the centers of
+  trade. All these systems, however, he says he counted almost as errors
+  compared with that of the Hindus.<a name="NtA_532"
+  href="#Nt_532"><sup>[532]</sup></a> Returning to Pisa, he wrote his
+  <i>Liber Abaci</i><a name="NtA_533" href="#Nt_533"><sup>[533]</sup></a>
+  in 1202, rewriting it in 1228.<a name="NtA_534"
+  href="#Nt_534"><sup>[534]</sup></a> In this work the numerals are
+  explained and are used in the usual computations of business. Such a
+  treatise was not destined to be popular, however, because it was too
+  advanced for the mercantile class, and too novel for the conservative
+  university circles. Indeed, at this time mathematics had only slight
+  place in the newly established universities, as witness the oldest known
+  statute of the Sorbonne at Paris, dated 1215, where the subject is
+  referred to only in an incidental way.<a name="NtA_535"
+  href="#Nt_535"><sup>[535]</sup></a> The period was one of great
+  commercial activity, and on this very <!-- Page 132 --><span
+  class="pagenum"><a name="page132"></a>[132]</span>account such a book
+  would attract even less attention than usual.<a name="NtA_536"
+  href="#Nt_536"><sup>[536]</sup></a></p>
+
+  <p>It would now be thought that the western world would at once adopt the
+  new numerals which Leonardo had made known, and which were so much
+  superior to anything that had been in use in Christian Europe. The
+  antagonism of the universities would avail but little, it would seem,
+  against such an improvement. It must be remembered, however, that there
+  was great difficulty in spreading knowledge at this time, some two
+  hundred and fifty years before printing was invented. "Popes and princes
+  and even great religious institutions possessed far fewer books than many
+  farmers of the present age. The library belonging to the Cathedral Church
+  of San Martino at Lucca in the ninth century contained only nineteen
+  volumes of abridgments from ecclesiastical commentaries."<a
+  name="NtA_537" href="#Nt_537"><sup>[537]</sup></a> Indeed, it was not
+  until the early part of the fifteenth century that Palla degli Strozzi
+  took steps to carry out the project that had been in the mind of
+  Petrarch, the founding of a public library. It was largely by word of
+  mouth, therefore, that this early knowledge had to be transmitted.
+  Fortunately the presence of foreign students in Italy at this time made
+  this transmission feasible. (If human nature was the same then as now, it
+  is not impossible that the very opposition of the faculties to the works
+  of Leonardo led the students to investigate <!-- Page 133 --><span
+  class="pagenum"><a name="page133"></a>[133]</span>them the more
+  zealously.) At Vicenza in 1209, for example, there were Bohemians, Poles,
+  Frenchmen, Burgundians, Germans, and Spaniards, not to speak of
+  representatives of divers towns of Italy; and what was true there was
+  also true of other intellectual centers. The knowledge could not fail to
+  spread, therefore, and as a matter of fact we find numerous bits of
+  evidence that this was the case. Although the bankers of Florence were
+  forbidden to use these numerals in 1299, and the statutes of the
+  university of Padua required stationers to keep the price lists of books
+  "non per cifras, sed per literas claros,"<a name="NtA_538"
+  href="#Nt_538"><sup>[538]</sup></a> the numerals really made much headway
+  from about 1275 on.</p>
+
+  <p>It was, however, rather exceptional for the common people of Germany
+  to use the Arabic numerals before the sixteenth century, a good witness
+  to this fact being the popular almanacs. Calendars of 1457-1496<a
+  name="NtA_539" href="#Nt_539"><sup>[539]</sup></a> have generally the
+  Roman numerals, while Köbel's calendar of 1518 gives the Arabic forms as
+  subordinate to the Roman. In the register of the Kreuzschule at Dresden
+  the Roman forms were used even until 1539.</p>
+
+  <p>While not minimizing the importance of the scientific work of Leonardo
+  of Pisa, we may note that the more popular treatises by Alexander de
+  Villa Dei (c. 1240 <span class="scac">A.D.</span>) and John of Halifax
+  (Sacrobosco, c. 1250 <span class="scac">A.D.</span>) were much more
+  widely used, and doubtless contributed more to the spread of the numerals
+  among the common people.</p>
+
+<p><!-- Page 134 --><span class="pagenum"><a name="page134"></a>[134]</span></p>
+
+  <p>The <i>Carmen de Algorismo</i><a name="NtA_540"
+  href="#Nt_540"><sup>[540]</sup></a> of Alexander de Villa Dei was written
+  in verse, as indeed were many other textbooks of that time. That it was
+  widely used is evidenced by the large number of manuscripts<a
+  name="NtA_541" href="#Nt_541"><sup>[541]</sup></a> extant in European
+  libraries. Sacrobosco's <i>Algorismus</i>,<a name="NtA_542"
+  href="#Nt_542"><sup>[542]</sup></a> in which some lines from the Carmen
+  are quoted, enjoyed a wide popularity as a textbook for university
+  instruction.<a name="NtA_543" href="#Nt_543"><sup>[543]</sup></a> The
+  work was evidently written with this end in view, as numerous
+  commentaries by university lecturers are found. Probably the most widely
+  used of these was that of Petrus de Dacia<a name="NtA_544"
+  href="#Nt_544"><sup>[544]</sup></a> written in 1291. These works throw an
+  interesting light upon the method of instruction in mathematics in use in
+  the universities from the thirteenth even to the sixteenth century.
+  Evidently the text was first read and copied by students.<a
+  name="NtA_545" href="#Nt_545"><sup>[545]</sup></a> Following this came
+  line by line an exposition of the text, such as is given in Petrus de
+  Dacia's commentary.</p>
+
+  <p>Sacrobosco's work is of interest also because it was probably due to
+  the extended use of this work that the <!-- Page 135 --><span
+  class="pagenum"><a name="page135"></a>[135]</span>term <i>Arabic
+  numerals</i> became common. In two places there is mention of the
+  inventors of this system. In the introduction it is stated that this
+  science of reckoning was due to a philosopher named Algus, whence the
+  name <i>algorismus</i>,<a name="NtA_546"
+  href="#Nt_546"><sup>[546]</sup></a> and in the section on numeration
+  reference is made to the Arabs as the inventors of this science.<a
+  name="NtA_547" href="#Nt_547"><sup>[547]</sup></a> While some of the
+  commentators, Petrus de Dacia<a name="NtA_548"
+  href="#Nt_548"><sup>[548]</sup></a> among them, knew of the Hindu origin,
+  most of them undoubtedly took the text as it stood; and so the Arabs were
+  credited with the invention of the system.</p>
+
+  <p>The first definite trace that we have of an algorism in the French
+  language is found in a manuscript written about 1275.<a name="NtA_549"
+  href="#Nt_549"><sup>[549]</sup></a> This interesting leaf, for the part
+  on algorism consists of a single folio, was noticed by the Abbé
+  Leb&oelig;uf as early as 1741,<a name="NtA_550"
+  href="#Nt_550"><sup>[550]</sup></a> and by Daunou in 1824.<a
+  name="NtA_551" href="#Nt_551"><sup>[551]</sup></a> It then seems to have
+  been lost in the multitude of Paris manuscripts; for although Chasles<a
+  name="NtA_552" href="#Nt_552"><sup>[552]</sup></a> relates his vain
+  search for it, it was not rediscovered until 1882. In that year M. Ch.
+  Henry found it, and to his care we owe our knowledge of the interesting
+  manuscript. The work is anonymous and is devoted almost entirely to
+  geometry, only <!-- Page 136 --><span class="pagenum"><a
+  name="page136"></a>[136]</span>two pages (one folio) relating to
+  arithmetic. In these the forms of the numerals are given, and a very
+  brief statement as to the operations, it being evident that the writer
+  himself had only the slightest understanding of the subject.</p>
+
+  <p>Once the new system was known in France, even thus superficially, it
+  would be passed across the Channel to England. Higden,<a name="NtA_553"
+  href="#Nt_553"><sup>[553]</sup></a> writing soon after the opening of the
+  fourteenth century, speaks of the French influence at that time and for
+  some generations preceding:<a name="NtA_554"
+  href="#Nt_554"><sup>[554]</sup></a> "For two hundred years children in
+  scole, agenst the usage and manir of all other nations beeth compelled
+  for to leave hire own language, and for to construe hir lessons and hire
+  thynges in Frensche.... Gentilmen children beeth taught to speke Frensche
+  from the tyme that they bith rokked in hir cradell; and uplondissche men
+  will likne himself to gentylmen, and fondeth with greet besynesse for to
+  speke Frensche."</p>
+
+  <p>The question is often asked, why did not these new numerals attract
+  more immediate attention? Why did they have to wait until the sixteenth
+  century to be generally used in business and in the schools? In reply it
+  may be said that in their elementary work the schools always wait upon
+  the demands of trade. That work which pretends to touch the life of the
+  people must come reasonably near doing so. Now the computations of
+  business until about 1500 did not demand the new figures, for two
+  reasons: First, cheap paper was not known. Paper-making of any kind was
+  not introduced into Europe until <!-- Page 137 --><span
+  class="pagenum"><a name="page137"></a>[137]</span>the twelfth century,
+  and cheap paper is a product of the nineteenth. Pencils, too, of the
+  modern type, date only from the sixteenth century. In the second place,
+  modern methods of operating, particularly of multiplying and dividing
+  (operations of relatively greater importance when all measures were in
+  compound numbers requiring reductions at every step), were not yet
+  invented. The old plan required the erasing of figures after they had
+  served their purpose, an operation very simple with counters, since they
+  could be removed. The new plan did not as easily permit this. Hence we
+  find the new numerals very tardily admitted to the counting-house, and
+  not welcomed with any enthusiasm by teachers.<a name="NtA_555"
+  href="#Nt_555"><sup>[555]</sup></a></p>
+
+  <p>Aside from their use in the early treatises on the new art of
+  reckoning, the numerals appeared from time to time in the dating of
+  manuscripts and upon monuments. The oldest definitely dated European
+  document known <!-- Page 138 --><span class="pagenum"><a
+  name="page138"></a>[138]</span>to contain the numerals is a Latin
+  manuscript,<a name="NtA_556" href="#Nt_556"><sup>[556]</sup></a> the
+  Codex Vigilanus, written in the Albelda Cloister not far from Logroño in
+  Spain, in 976 <span class="scac">A.D.</span> The nine characters (of
+  &#x121;ob&#x101;r type), without the zero, are given as an addition to
+  the first chapters of the third book of the <i>Origines</i> by Isidorus
+  of Seville, in which the Roman numerals are under discussion. Another
+  Spanish copy of the same work, of 992 <span class="scac">A.D.</span>,
+  contains the numerals in the corresponding section. The writer ascribes
+  an Indian origin to them in the following words: "Item de figuris
+  arithmetic&#x119;. Scire debemus in Indos subtilissimum ingenium habere
+  et ceteras gentes eis in arithmetica et geometria et ceteris liberalibus
+  disciplinis concedere. Et hoc manifestum est in nobem figuris, quibus
+  designant unumquemque gradum cuiuslibet gradus. Quarum hec sunt forma."
+  The nine &#x121;ob&#x101;r characters follow. Some of the abacus forms<a
+  name="NtA_557" href="#Nt_557"><sup>[557]</sup></a> previously given are
+  doubtless also of the tenth century. The earliest Arabic documents
+  containing the numerals are two manuscripts of 874 and 888 <span
+  class="scac">A.D.</span><a name="NtA_558"
+  href="#Nt_558"><sup>[558]</sup></a> They appear about a century later in
+  a work<a name="NtA_559" href="#Nt_559"><sup>[559]</sup></a> written at
+  Shiraz in 970 <span class="scac">A.D.</span> There is also an early trace
+  of their use on a pillar recently discovered in a church apparently
+  destroyed as early as the tenth century, not far from the Jeremias
+  Monastery, in Egypt. <!-- Page 139 --><span class="pagenum"><a
+  name="page139"></a>[139]</span>A graffito in Arabic on this pillar has
+  the date 349 <span class="scac">A.H.</span>, which corresponds to 961
+  <span class="scac">A.D.</span><a name="NtA_560"
+  href="#Nt_560"><sup>[560]</sup></a> For the dating of Latin documents the
+  Arabic forms were used as early as the thirteenth century.<a
+  name="NtA_561" href="#Nt_561"><sup>[561]</sup></a></p>
+
+  <p>On the early use of these numerals in Europe the only scientific study
+  worthy the name is that made by Mr. G. F. Hill of the British Museum.<a
+  name="NtA_562" href="#Nt_562"><sup>[562]</sup></a> From his
+  investigations it appears that the earliest occurrence of a date in these
+  numerals on a coin is found in the reign of Roger of Sicily in 1138.<a
+  name="NtA_563" href="#Nt_563"><sup>[563]</sup></a> Until recently it was
+  thought that the earliest such date was 1217 <span
+  class="scac">A.D.</span> for an Arabic piece and 1388 for a Turkish
+  one.<a name="NtA_564" href="#Nt_564"><sup>[564]</sup></a> Most of the
+  seals and medals containing dates that were at one time thought to be
+  very early have been shown by Mr. Hill to be of relatively late
+  workmanship. There are, however, in European manuscripts, numerous
+  instances of the use of these numerals before the twelfth century.
+  Besides the example in the Codex Vigilanus, another of the tenth century
+  has been found in the St. Gall MS. now in the University Library at
+  Zürich, the forms differing materially from those in the Spanish
+  codex.</p>
+
+  <p>The third specimen in point of time in Mr. Hill's list is from a
+  Vatican MS. of 1077. The fourth and fifth specimens are from the Erlangen
+  MS. of Boethius, of the same <!-- Page 140 --><span class="pagenum"><a
+  name="page140"></a>[140]</span>(eleventh) century, and the sixth and
+  seventh are also from an eleventh-century MS. of Boethius at Chartres.
+  These and other early forms are given by Mr. Hill in this table, which is
+  reproduced with his kind permission.</p>
+
+<h3><span class="sc">Earliest Manuscript Forms</span></h3>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/146a.png"><img style="width:100%" src="images/146a.png"
+      alt="Earliest Manuscript Forms" title="Earliest Manuscript Forms" /></a>
+  </div>
+  <p>This is one of more than fifty tables given in Mr. Hill's valuable
+  paper, and to this monograph students <!-- Page 141 --><span
+  class="pagenum"><a name="page141"></a>[141]</span>are referred for
+  details as to the development of number-forms in Europe from the tenth to
+  the sixteenth century. It is of interest to add that he has found that
+  among the earliest dates of European coins or medals in these numerals,
+  after the Sicilian one already mentioned, are the following: Austria,
+  1484; Germany, 1489 (Cologne); Switzerland, 1424 (St. Gall); Netherlands,
+  1474; France, 1485; Italy, 1390.<a name="NtA_565"
+  href="#Nt_565"><sup>[565]</sup></a></p>
+
+  <p>The earliest English coin dated in these numerals was struck in
+  1551,<a name="NtA_566" href="#Nt_566"><sup>[566]</sup></a> although there
+  is a Scotch piece of 1539.<a name="NtA_567"
+  href="#Nt_567"><sup>[567]</sup></a> In numbering pages of a printed book
+  these numerals were first used in a work of Petrarch's published at
+  Cologne in 1471.<a name="NtA_568" href="#Nt_568"><sup>[568]</sup></a> The
+  date is given in the following form in the <i>Biblia Pauperum</i>,<a
+  name="NtA_569" href="#Nt_569"><sup>[569]</sup></a> a block-book of
+  1470,</p>
+
+  <div class="figcenter" style="width:15%;">
+      <a href="images/147a.png"><img style="width:100%" src="images/147a.png"
+      alt="Numerals 1470" title="Numerals 1470" /></a>
+  </div>
+  <p>while in another block-book which possibly goes back to c. 1430<a
+  name="NtA_570" href="#Nt_570"><sup>[570]</sup></a> the numerals appear in
+  several illustrations, with forms as follows:</p>
+
+  <div class="figcenter" style="width:36%;">
+      <a href="images/147b.png"><img style="width:100%" src="images/147b.png"
+      alt="Numerals" title="Numerals" /></a>
+  </div>
+  <p>Many printed works anterior to 1471 have pages or chapters numbered by
+  hand, but many of these numerals are <!-- Page 142 --><span
+  class="pagenum"><a name="page142"></a>[142]</span>of date much later than
+  the printing of the work. Other works were probably numbered directly
+  after printing. Thus the chapters 2, 3, 4, 5, 6 in a book of 1470<a
+  name="NtA_571" href="#Nt_571"><sup>[571]</sup></a> are numbered as
+  follows: Capitulem <a href="images/148a.png"><img src="images/148a.png"
+  class="middle" style="height:1.5ex" alt="Symbol 2" /></a>m.,... <a
+  href="images/148b.png"><img src="images/148b.png" class="middle"
+  style="height:1.5ex" alt="Symbol 3" /></a>m.,... 4m.,... v,... vi, and
+  followed by Roman numerals. This appears in the body of the text, in
+  spaces left by the printer to be filled in by hand. Another book<a
+  name="NtA_572" href="#Nt_572"><sup>[572]</sup></a> of 1470 has pages
+  numbered by hand with a mixture of Roman and Hindu numerals, thus,</p>
+
+<table class="nobctr">
+<tr><td valign="middle"><a href="images/148c.png"><img src="images/148c.png" class="middle" style="height:3.5ex" alt="Symbols" /></a></td><td valign="middle">for 125</td>
+<td valign="middle"><a href="images/148e.png"><img src="images/148e.png" class="middle" style="height:3.5ex" alt="Symbols" /></a></td><td valign="middle">for 150</td></tr>
+<tr><td valign="middle"><a href="images/148d.png"><img src="images/148d.png" class="middle" style="height:3.5ex" alt="Symbols" /></a></td><td valign="middle">for 147</td>
+<td valign="middle"><a href="images/148f.png"><img src="images/148f.png" class="middle" style="height:3.5ex" alt="Symbols" /></a></td><td valign="middle">for 202</td></tr>
+</table>
+
+  <p>As to monumental inscriptions,<a name="NtA_573"
+  href="#Nt_573"><sup>[573]</sup></a> there was once thought to be a
+  gravestone at Katharein, near Troppau, with the date 1007, and one at
+  Biebrich of 1299. There is no doubt, however, of one at Pforzheim of 1371
+  and one at Ulm of 1388.<a name="NtA_574"
+  href="#Nt_574"><sup>[574]</sup></a> Certain numerals on Wells Cathedral
+  have been assigned to the thirteenth century, but they are undoubtedly
+  considerably later.<a name="NtA_575"
+  href="#Nt_575"><sup>[575]</sup></a></p>
+
+  <p>The table on page 143 will serve to supplement that from Mr. Hill's
+  work.<a name="NtA_576" href="#Nt_576"><sup>[576]</sup></a></p>
+
+<p><!-- Page 143 --><span class="pagenum"><a name="page143"></a>[143]</span></p>
+
+<h3><span class="sc">Early Manuscript Forms</span></h3>
+
+<table class="nobctr">
+<tr><td>&nbsp;</td><td><a href="images/149.png"><img src="images/149.png" class="middle" style="height:4.5ex" alt="1 2 3 4 5 6 7 8 9 0" /></a></td><td>&nbsp;</td></tr>
+<tr><td valign="middle">a <a name="NtA_577" href="#Nt_577"><sup>[577]</sup></a></td><td><a href="images/149a.png"><img src="images/149a.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> Twelfth century <span class="scac">A.D.</span></td></tr>
+<tr><td valign="middle">b <a name="NtA_578" href="#Nt_578"><sup>[578]</sup></a></td><td><a href="images/149b.png"><img src="images/149b.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> 1197 <span class="scac">A.D.</span></td></tr>
+<tr><td valign="middle">c <a name="NtA_579" href="#Nt_579"><sup>[579]</sup></a></td><td><a href="images/149c.png"><img src="images/149c.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> 1275 <span class="scac">A.D.</span></td></tr>
+<tr><td valign="middle">d <a name="NtA_580" href="#Nt_580"><sup>[580]</sup></a></td><td><a href="images/149d.png"><img src="images/149d.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> c. 1294 <span class="scac">A.D.</span></td></tr>
+<tr><td valign="middle">e <a name="NtA_581" href="#Nt_581"><sup>[581]</sup></a></td><td><a href="images/149e.png"><img src="images/149e.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> c. 1303 <span class="scac">A.D.</span></td></tr>
+<tr><td valign="middle">f <a name="NtA_582" href="#Nt_582"><sup>[582]</sup></a></td><td><a href="images/149f.png"><img src="images/149f.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> c. 1360 <span class="scac">A.D.</span></td></tr>
+<tr><td valign="middle">g <a name="NtA_583" href="#Nt_583"><sup>[583]</sup></a></td><td><a href="images/149g.png"><img src="images/149g.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> c. 1442 <span class="scac">A.D.</span></td></tr>
+</table>
+
+<p><!-- Page 144 --><span class="pagenum"><a name="page144"></a>[144]</span></p>
+
+  <div class="figleft" style="width:15%;">
+      <a href="images/150a.png"><img style="width:100%" src="images/150a.png"
+      alt="Powers of 2." title="Powers of 2." /></a>
+  </div>
+  <p>For the sake of further comparison, three illustrations from works in
+  Mr. Plimpton's library, reproduced from the <i>Rara Arithmetica</i>, may
+  be considered. The first is from a Latin manuscript on arithmetic,<a
+  name="NtA_584" href="#Nt_584"><sup>[584]</sup></a> of which the original
+  was written at Paris in 1424 by Rollandus, a Portuguese physician, who
+  prepared the work at the command of John of Lancaster, Duke of Bedford,
+  at one time Protector of England and Regent of France, to whom the work
+  is dedicated. The figures show the successive powers of 2. The second
+  illustration is from Luca da Firenze's <i>Inprencipio darte
+  dabacho</i>,<a name="NtA_585" href="#Nt_585"><sup>[585]</sup></a> c.
+  1475, and the third is from an anonymous manuscript<a name="NtA_586"
+  href="#Nt_586"><sup>[586]</sup></a> of about 1500.</p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/150b.png"><img style="width:100%" src="images/150b.png"
+      alt="Numerals." title="Numerals." /></a>
+  </div>
+  <p>As to the forms of the numerals, fashion played a leading part until
+  printing was invented. This tended to fix these forms, although in
+  writing there is still a great variation, as witness the French 5 and the
+  German 7 and 9. Even in printing there is not complete uniformity, <!--
+  Page 145 --><span class="pagenum"><a name="page145"></a>[145]</span>and
+  it is often difficult for a foreigner to distinguish between the 3 and 5
+  of the French types.</p>
+
+  <div class="figcenter" style="width:60%;">
+      <a href="images/150c.png"><img style="width:100%" src="images/150c.png"
+      alt="Numerals." title="Numerals." /></a>
+  </div>
+  <p>As to the particular numerals, the following are some of the forms to
+  be found in the later manuscripts and in the early printed books.</p>
+
+  <p>1. In the early printed books "one" was often i, perhaps to save
+  types, just as some modern typewriters use the same character for l and
+  1.<a name="NtA_587" href="#Nt_587"><sup>[587]</sup></a> In the
+  manuscripts the "one" appears in such forms as<a name="NtA_588"
+  href="#Nt_588"><sup>[588]</sup></a></p>
+
+  <div class="figcenter" style="width:36%;">
+      <a href="images/151a.png"><img style="width:100%" src="images/151a.png"
+      alt="Variations of 1." title="Variations of 1." /></a>
+  </div>
+  <p>2. "Two" often appears as z in the early printed books, 12 appearing
+  as iz.<a name="NtA_589" href="#Nt_589"><sup>[589]</sup></a> In the
+  medieval manuscripts the following forms are common:<a name="NtA_590"
+  href="#Nt_590"><sup>[590]</sup></a></p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/151b.png"><img style="width:100%" src="images/151b.png"
+      alt="Variations of 2." title="Variations of 2." /></a>
+  </div>
+<p><!-- Page 146 --><span class="pagenum"><a name="page146"></a>[146]</span></p>
+
+  <p>It is evident, from the early traces, that it is merely a cursive form
+  for the primitive <a href="images/033f.png"><img src="images/033f.png"
+  class="middle" style="height:1.5ex" alt="2 horizontal strokes" /></a>,
+  just as 3 comes from <a href="images/033h.png"><img src="images/033h.png"
+  class="middle" style="height:1.5ex" alt="3 horizontal strokes" /></a>, as
+  in the N&#x101;n&#x101; Gh&#x101;t inscriptions.</p>
+
+  <p>3. "Three" usually had a special type in the first printed books,
+  although occasionally it appears as <a href="images/152a.png"><img
+  src="images/152a.png" class="middle" style="height:2ex" alt="Symbol"
+  /></a>.<a name="NtA_591" href="#Nt_591"><sup>[591]</sup></a> In the
+  medieval manuscripts it varied rather less than most of the others. The
+  following are common forms:<a name="NtA_592"
+  href="#Nt_592"><sup>[592]</sup></a></p>
+
+  <div class="figcenter" style="width:48%;">
+      <a href="images/152b.png"><img style="width:100%" src="images/152b.png"
+      alt="Variations of 3." title="Variations of 3." /></a>
+  </div>
+  <p>4. "Four" has changed greatly; and one of the first tests as to the
+  age of a manuscript on arithmetic, and the place where it was written, is
+  the examination of this numeral. Until the time of printing the most
+  common form was <a href="images/152c.png"><img src="images/152c.png"
+  class="middle" style="height:2ex" alt="Symbol" /></a>, although the
+  Florentine manuscript of Leonard of Pisa's work has the form <a
+  href="images/152d.png"><img src="images/152d.png" class="middle"
+  style="height:2ex" alt="Symbol" /></a>;<a name="NtA_593"
+  href="#Nt_593"><sup>[593]</sup></a> but the manuscripts show that the
+  Florentine arithmeticians and astronomers rather early began to
+  straighten the first of these forms up to forms like <a
+  href="images/152e.png"><img src="images/152e.png" class="middle"
+  style="height:2ex" alt="Symbol" /></a><a name="NtA_594"
+  href="#Nt_594"><sup>[594]</sup></a> and <a href="images/152f.png"><img
+  src="images/152f.png" class="middle" style="height:2ex" alt="Symbol"
+  /></a><a href="#Nt_594"><sup>[594]</sup></a> or <a
+  href="images/152g.png"><img src="images/152g.png" class="middle"
+  style="height:2ex" alt="Symbol" /></a>,<a name="NtA_595"
+  href="#Nt_595"><sup>[595]</sup></a> more closely resembling our own. The
+  first printed books generally used our present form<a name="NtA_596"
+  href="#Nt_596"><sup>[596]</sup></a> with the closed top <a
+  href="images/152h.png"><img src="images/152h.png" class="middle"
+  style="height:2ex" alt="Symbol" /></a>, the open top used in writing ( <a
+  href="images/152i.png"><img src="images/152i.png" class="middle"
+  style="height:2ex" alt="Symbol" /></a>) being <!-- Page 147 --><span
+  class="pagenum"><a name="page147"></a>[147]</span>purely modern. The
+  following are other forms of the four, from various manuscripts:<a
+  name="NtA_597" href="#Nt_597"><sup>[597]</sup></a></p>
+
+  <div class="figcenter" style="width:50%;">
+      <a href="images/153a.png"><img style="width:100%" src="images/153a.png"
+      alt="Variations of 4." title="Variations of 4." /></a>
+  </div>
+  <p>5. "Five" also varied greatly before the time of printing. The
+  following are some of the forms:<a name="NtA_598"
+  href="#Nt_598"><sup>[598]</sup></a></p>
+
+  <div class="figcenter" style="width:42%;">
+      <a href="images/153b.png"><img style="width:100%" src="images/153b.png"
+      alt="Variations of 5." title="Variations of 5." /></a>
+  </div>
+  <p>6. "Six" has changed rather less than most of the others. The chief
+  variation has been in the slope of the top, as will be seen in the
+  following:<a name="NtA_599" href="#Nt_599"><sup>[599]</sup></a></p>
+
+  <div class="figcenter" style="width:30%;">
+      <a href="images/153c.png"><img style="width:100%" src="images/153c.png"
+      alt="Variations of 6." title="Variations of 6." /></a>
+  </div>
+  <p>7. "Seven," like "four," has assumed its present erect form only since
+  the fifteenth century. In medieval times it appeared as follows:<a
+  name="NtA_600" href="#Nt_600"><sup>[600]</sup></a></p>
+
+  <div class="figcenter" style="width:52%;">
+      <a href="images/153d.png"><img style="width:100%" src="images/153d.png"
+      alt="Variations of 7." title="Variations of 7." /></a>
+  </div>
+<p><!-- Page 148 --><span class="pagenum"><a name="page148"></a>[148]</span></p>
+
+  <p>8. "Eight," like "six," has changed but little. In medieval times
+  there are a few variants of interest as follows:<a name="NtA_601"
+  href="#Nt_601"><sup>[601]</sup></a></p>
+
+  <div class="figcenter" style="width:20%;">
+      <a href="images/154a.png"><img style="width:100%" src="images/154a.png"
+      alt="Variations of 8." title="Variations of 8." /></a>
+  </div>
+  <p>In the sixteenth century, however, there was manifested a tendency to
+  write it <a href="images/154b.png"><img src="images/154b.png"
+  class="middle" style="height:1.5ex" alt="Symbol" /></a>.<a name="NtA_602"
+  href="#Nt_602"><sup>[602]</sup></a></p>
+
+  <p>9. "Nine" has not varied as much as most of the others. Among the
+  medieval forms are the following:<a name="NtA_603"
+  href="#Nt_603"><sup>[603]</sup></a></p>
+
+  <div class="figcenter" style="width:45%;">
+      <a href="images/154c.png"><img style="width:100%" src="images/154c.png"
+      alt="Variations of 9." title="Variations of 9." /></a>
+  </div>
+  <p>0. The shape of the zero also had a varied history. The following are
+  common medieval forms:<a name="NtA_604"
+  href="#Nt_604"><sup>[604]</sup></a></p>
+
+  <div class="figcenter" style="width:30%;">
+      <a href="images/154d.png"><img style="width:100%" src="images/154d.png"
+      alt="Variations of 0." title="Variations of 0." /></a>
+  </div>
+  <p>The explanation of the place value was a serious matter to most of the
+  early writers. If they had been using an abacus constructed like the
+  Russian chotü, and had placed this before all learners of the positional
+  system, there would have been little trouble. But the medieval <!-- Page
+  149 --><span class="pagenum"><a
+  name="page149"></a>[149]</span>line-reckoning, where the lines stood for
+  powers of 10 and the spaces for half of such powers, did not lend itself
+  to this comparison. Accordingly we find such labored explanations as the
+  following, from <i>The Crafte of Nombrynge</i>:</p>
+
+  <p>"Euery of these figuris bitokens hym selfe &amp; no more, yf he stonde
+  in the first place of the rewele....</p>
+
+  <p>"If it stonde in the secunde place of the rewle, he betokens ten tymes
+  hym selfe, as this figure 2 here 20 tokens ten tyme hym selfe, that is
+  twenty, for he hym selfe betokens tweyne, &amp; ten tymes twene is
+  twenty. And for he stondis on the lyft side &amp; in the secunde place,
+  he betokens ten tyme hym selfe. And so go forth....</p>
+
+  <p>"Nil cifra significat sed dat signare sequenti. Expone this verse. A
+  cifre tokens no&#x21D;t, bot he makes the figure to betoken that comes
+  after hym more than he shuld &amp; he were away, as thus 10. here the
+  figure of one tokens ten, &amp; yf the cifre were away &amp; no figure
+  byfore hym he schuld token bot one, for than he schuld stonde in the
+  first place...."<a name="NtA_605" href="#Nt_605"><sup>[605]</sup></a></p>
+
+  <p>It would seem that a system that was thus used for dating documents,
+  coins, and monuments, would have been generally adopted much earlier than
+  it was, particularly in those countries north of Italy where it did not
+  come into general use until the sixteenth century. This, however, has
+  been the fate of many inventions, as witness our neglect of logarithms
+  and of contracted processes to-day.</p>
+
+  <p>As to Germany, the fifteenth century saw the rise of the new
+  symbolism; the sixteenth century saw it slowly <!-- Page 150 --><span
+  class="pagenum"><a name="page150"></a>[150]</span>gain the mastery; the
+  seventeenth century saw it finally conquer the system that for two
+  thousand years had dominated the arithmetic of business. Not a little of
+  the success of the new plan was due to Luther's demand that all learning
+  should go into the vernacular.<a name="NtA_606"
+  href="#Nt_606"><sup>[606]</sup></a></p>
+
+  <p>During the transition period from the Roman to the Arabic numerals,
+  various anomalous forms found place. For example, we have in the
+  fourteenth century c<span class="grk">&alpha;</span> for 104;<a
+  name="NtA_607" href="#Nt_607"><sup>[607]</sup></a> 1000. 300. 80 et 4 for
+  1384;<a name="NtA_608" href="#Nt_608"><sup>[608]</sup></a> and in a
+  manuscript of the fifteenth century 12901 for 1291.<a name="NtA_609"
+  href="#Nt_609"><sup>[609]</sup></a> In the same century m. cccc. 8II
+  appears for 1482,<a name="NtA_610" href="#Nt_610"><sup>[610]</sup></a>
+  while M<sup>o</sup>CCCC<sup>o</sup>50 (1450) and MCCCCXL6 (1446) are used
+  by Theodoricus Ruffi about the same time.<a name="NtA_611"
+  href="#Nt_611"><sup>[611]</sup></a> To the next century belongs the form
+  1vojj for 1502. Even in Sfortunati's <i>Nuovo lume</i><a name="NtA_612"
+  href="#Nt_612"><sup>[612]</sup></a> the use of ordinals is quite
+  confused, the propositions on a single page being numbered "tertia," "4,"
+  and "V."</p>
+
+  <p>Although not connected with the Arabic numerals in any direct way, the
+  medieval astrological numerals may here be mentioned. These are given by
+  several early writers, but notably by Noviomagus (1539),<a name="NtA_613"
+  href="#Nt_613"><sup>[613]</sup></a> as follows<a name="NtA_614"
+  href="#Nt_614"><sup>[614]</sup></a>:</p>
+
+  <div class="figcenter" style="width:52%;">
+      <a href="images/156a.png"><img style="width:100%" src="images/156a.png"
+      alt="Astrological numerals." title="Astrological numerals." /></a>
+  </div>
+<p><!-- Page 151 --><span class="pagenum"><a name="page151"></a>[151]</span></p>
+
+  <p>Thus we find the numerals gradually replacing the Roman forms all over
+  Europe, from the time of Leonardo of Pisa until the seventeenth century.
+  But in the Far East to-day they are quite unknown in many countries, and
+  they still have their way to make. In many parts of India, among the
+  common people of Japan and China, in Siam and generally about the Malay
+  Peninsula, in Tibet, and among the East India islands, the natives still
+  adhere to their own numeral forms. Only as Western civilization is making
+  its way into the commercial life of the East do the numerals as used by
+  us find place, save as the Sanskrit forms appear in parts of India. It is
+  therefore with surprise that the student of mathematics comes to realize
+  how modern are these forms so common in the West, how limited is their
+  use even at the present time, and how slow the world has been and is in
+  adopting such a simple device as the Hindu-Arabic numerals.</p>
+
+<hr class="full" >
+
+<p><!-- Page 153 --><span class="pagenum"><a name="page153"></a>[153]</span></p>
+
+<h3>INDEX</h3>
+
+  <p><i>Transcriber's note: many of the entries refer to footnotes linked
+  from the page numbers given.</i></p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Abbo of Fleury, <a href="#page122">122</a></p>
+      <p><span class="special" title="`Abdallah ibn al-Hasan">&#x201B;Abdall&#x101;h ibn al-&#x1E24;asan</span>, <a href="#page92">92</a></p>
+      <p>&#x201B;Abdallat&#x12B;f ibn Y&#x16B;suf, <a href="#page93">93</a></p>
+      <p>&#x201B;Abdalq&#x101;dir ibn &#x201B;Al&#x12B; al-Sakh&#x101;w&#x12B;, <a href="#page6">6</a></p>
+      <p>Abenragel, <a href="#page34">34</a></p>
+      <p>Abraham ibn Meïr ibn Ezra, <i>see</i> Rabbi ben Ezra</p>
+      <p><span class="special" title="Abu `Ali al-Hosein ibn Sina">Ab&#x16B; &#x201B;Al&#x12B; al-&#x1E24;osein ibn S&#x12B;n&#x101;</span>, <a href="#page74">74</a></p>
+      <p><span class="special" title="Abu 'l-Hasan">Ab&#x16B; 'l-&#x1E24;asan</span>, <a href="#page93">93</a>, <a href="#page100">100</a></p>
+      <p>Ab&#x16B; 'l-Q&#x101;sim, <a href="#page92">92</a></p>
+      <p><span class="special" title="Abu 'l-Teiyib">Ab&#x16B; 'l-&#x1E6C;eiyib</span>, <a href="#page97">97</a></p>
+      <p><span class="special" title="Abu Nasr">Ab&#x16B; Na&#x1E63;r</span>, <a href="#page92">92</a></p>
+      <p>Ab&#x16B; Roshd, <a href="#page113">113</a></p>
+      <p>Abu Sahl Dunash ibn Tamim, <a href="#page65">65</a>, <a href="#page67">67</a></p>
+      <p>Adelhard of Bath, <a href="#page5">5</a>, <a href="#page55">55</a>, <a href="#page97">97</a>, <a href="#page119">119</a>, <a href="#page123">123</a>, <a href="#page126">126</a></p>
+      <p>Adhemar of Chabanois, <a href="#page111">111</a></p>
+      <p><span class="special" title="Ahmed al-Nasawi">A&#x1E25;med al-Nasaw&#x12B;</span>, <a href="#page98">98</a></p>
+      <p><span class="special" title="Ahmed ibn `Abdallah">A&#x1E25;med ibn &#x201B;Abdall&#x101;h</span>, <a href="#page9">9</a>, <a href="#page92">92</a></p>
+      <p><span class="special" title="Ahmed ibn Mohammed">A&#x1E25;med ibn Mo&#x1E25;ammed</span>, <a href="#page94">94</a></p>
+      <p><span class="special" title="Ahmed ibn `Omar">A&#x1E25;med ibn &#x201B;Omar</span>, <a href="#page93">93</a></p>
+      <p><span class="special" title="Aksaras">Ak&#x1E63;aras</span>, <a href="#page32">32</a></p>
+      <p>Alanus ab Insulis, <a href="#page124">124</a></p>
+      <p>Al-Ba&#x121;d&#x101;d&#x12B;, <a href="#page93">93</a></p>
+      <p>Al-Batt&#x101;n&#x12B;, <a href="#page54">54</a></p>
+      <p>Albelda (Albaida) MS., <a href="#page116">116</a></p>
+      <p>Albert, J., <a href="#page62">62</a></p>
+      <p>Albert of York, <a href="#page103">103</a></p>
+      <p>Al-B&#x12B;r&#x16B;n&#x12B;, <a href="#page6">6</a>, <a href="#page41">41</a>, <a href="#page49">49</a>, <a href="#page65">65</a>, <a href="#page92">92</a>, <a href="#page93">93</a></p>
+      <p>Alcuin, <a href="#page103">103</a></p>
+      <p>Alexander the Great, <a href="#page76">76</a></p>
+      <p>Alexander de Villa Dei, <a href="#page11">11</a>, <a href="#page133">133</a></p>
+      <p>Alexandria, <a href="#page64">64</a>, <a href="#page82">82</a></p>
+      <p>Al-Faz&#x101;r&#x12B;, <a href="#page92">92</a></p>
+      <p>Alfred, <a href="#page103">103</a></p>
+      <p>Algebra, etymology, <a href="#page5">5</a></p>
+      <p>Algerian numerals, <a href="#page68">68</a></p>
+      <p>Algorism, <a href="#page97">97</a></p>
+      <p>Algorismus, <a href="#page124">124</a>, <a href="#page126">126</a>, <a href="#page135">135</a></p>
+      <p>Algorismus cifra, <a href="#page120">120</a></p>
+      <p><span class="special" title="Al-Hassar">Al-&#x1E24;a&#x1E63;&#x1E63;&#x101;r</span>, <a href="#page65">65</a></p>
+      <p>&#x201B;Al&#x12B; ibn Ab&#x12B; Bekr, <a href="#page6">6</a></p>
+      <p><span class="special" title="`Ali ibn Ahmed">&#x201B;Al&#x12B; ibn A&#x1E25;med</span>, <a href="#page93">93</a>, <a href="#page98">98</a></p>
+      <p>Al-Kar&#x101;b&#x12B;s&#x12B;, <a href="#page93">93</a></p>
+      <p>Al-Khow&#x101;razm&#x12B;, <a href="#page4">4</a>, <a href="#page9">9</a>, <a href="#page10">10</a>, <a href="#page92">92</a>, <a href="#page97">97</a>, <a href="#page98">98</a>, <a href="#page125">125</a>, <a href="#page126">126</a></p>
+      <p>Al-Kind&#x12B;, <a href="#page10">10</a>, <a href="#page92">92</a></p>
+      <p>Almagest, <a href="#page54">54</a></p>
+      <p>Al-Ma&#x121;reb&#x12B;, <a href="#page93">93</a></p>
+      <p><span class="special" title="Al-Mahalli">Al-Ma&#x1E25;all&#x12B;</span>, <a href="#page6">6</a></p>
+      <p>Al-M&#x101;m&#x16B;n, <a href="#page10">10</a>, <a href="#page97">97</a></p>
+      <p><span class="special" title="Al-Mansur">Al-Man&#x1E63;&#x16B;r</span>, <a href="#page96">96</a>, <a href="#page97">97</a></p>
+      <p>Al-Mas&#x201B;&#x16B;d&#x12B;, <a href="#page7">7</a>, <a href="#page92">92</a></p>
+      <p>Al-Nad&#x12B;m, <a href="#page9">9</a></p>
+      <p>Al-Nasaw&#x12B;, <a href="#page93">93</a>, <a href="#page98">98</a></p>
+      <p>Alphabetic numerals, <a href="#page39">39</a>, <a href="#page40">40</a>, <a href="#page43">43</a></p>
+      <p>Al-Q&#x101;sim, <a href="#page92">92</a></p>
+      <p>Al-Qass, <a href="#page94">94</a></p>
+      <p>Al-Sakh&#x101;w&#x12B;, <a href="#page6">6</a></p>
+      <p><span class="special" title="Al-Sardafi">Al-&#x1E62;ardaf&#x12B;</span>, <a href="#page93">93</a></p>
+      <p>Al-Sijz&#x12B;, <a href="#page94">94</a></p>
+      <p>Al-S&#x16B;f&#x12B;, <a href="#page10">10</a>, <a href="#page92">92</a></p>
+      <p>Ambrosoli, <a href="#page118">118</a></p>
+      <p><span class="special" title="Ankapalli">A&#x1E45;kapalli</span>, <a href="#page43">43</a></p>
+      <p>Apices, <a href="#page87">87</a>, <a href="#page117">117</a>, <a href="#page118">118</a></p>
+      <p>Arabs, <a href="#page91">91</a>-<a href="#page98">98</a></p>
+      <p>Arbuthnot, <a href="#page141">141</a></p>
+<!-- Page 154 --><span class="pagenum"><a name="page154"></a>[154]</span>
+      <p>Archimedes, <a href="#page15">15</a>, <a href="#page16">16</a></p>
+      <p>Arcus Pictagore, <a href="#page122">122</a></p>
+      <p>Arjuna, <a href="#page15">15</a></p>
+      <p>Arnold, E., <a href="#page15">15</a>, <a href="#page102">102</a></p>
+      <p>Ars memorandi, <a href="#page141">141</a></p>
+      <p><span class="special" title="Aryabhata">&#x100;ryabha&#x1E6D;a</span>, <a href="#page39">39</a>, <a href="#page43">43</a>, <a href="#page44">44</a></p>
+      <p>Aryan numerals, <a href="#page19">19</a></p>
+      <p>Aschbach, <a href="#page134">134</a></p>
+      <p>Ashmole, <a href="#page134">134</a></p>
+      <p>A&#x15B;oka, <a href="#page19">19</a>, <a href="#page20">20</a>, <a href="#page22">22</a>, <a href="#page81">81</a></p>
+      <p><span class="special" title="As-sifr">A&#x1E63;-&#x1E63;ifr</span>, <a href="#page57">57</a>, <a href="#page58">58</a></p>
+      <p>Astrological numerals, <a href="#page150">150</a></p>
+      <p>Atharva-Veda, <a href="#page48">48</a>, <a href="#page49">49</a>, <a href="#page55">55</a></p>
+      <p>Augustus, <a href="#page80">80</a></p>
+      <p>Averroës, <a href="#page113">113</a></p>
+      <p>Avicenna, <a href="#page58">58</a>, <a href="#page74">74</a>, <a href="#page113">113</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Babylonian numerals, <a href="#page28">28</a></p>
+      <p>Babylonian zero, <a href="#page51">51</a></p>
+      <p>Bacon, R., <a href="#page131">131</a></p>
+      <p>Bactrian numerals, <a href="#page19">19</a>, <a href="#page30">30</a></p>
+      <p>Bæda, <a href="#page2">2</a>, <a href="#page72">72</a></p>
+      <p>Bagdad, <a href="#page4">4</a>, <a href="#page96">96</a></p>
+      <p><span class="special" title="Bakhsali">Bakh&#x1E63;&#x101;l&#x12B;</span> manuscript, <a href="#page43">43</a>, <a href="#page49">49</a>, <a href="#page52">52</a>, <a href="#page53">53</a></p>
+      <p>Ball, C. J., <a href="#page35">35</a></p>
+      <p>Ball, W. W. R., <a href="#page36">36</a>, <a href="#page131">131</a></p>
+      <p><span class="special" title="Bana">B&#x101;&#x1E47;a</span>, <a href="#page44">44</a></p>
+      <p>Barth, A., <a href="#page39">39</a></p>
+      <p>Bayang inscriptions, <a href="#page39">39</a></p>
+      <p>Bayer, <a href="#page33">33</a></p>
+      <p>Bayley, E. C., <a href="#page19">19</a>, <a href="#page23">23</a>, <a href="#page30">30</a>, <a href="#page32">32</a>, <a href="#page52">52</a>, <a href="#page89">89</a></p>
+      <p>Beazley, <a href="#page75">75</a></p>
+      <p>Bede, <i>see</i> Bæda</p>
+      <p>Beldomandi, <a href="#page137">137</a></p>
+      <p>Beloch, J., <a href="#page77">77</a></p>
+      <p>Bendall, <a href="#page25">25</a>, <a href="#page52">52</a></p>
+      <p>Benfey, T., <a href="#page26">26</a></p>
+      <p>Bernelinus, <a href="#page88">88</a>, <a href="#page112">112</a>, <a href="#page117">117</a>, <a href="#page121">121</a></p>
+      <p>Besagne, <a href="#page128">128</a></p>
+      <p>Besant, W., <a href="#page109">109</a></p>
+      <p>Bettino, <a href="#page36">36</a></p>
+      <p>Bhandarkar, <a href="#page18">18</a>, <a href="#page47">47</a>, <a href="#page49">49</a></p>
+      <p>Bh&#x101;skara, <a href="#page53">53</a>, <a href="#page55">55</a></p>
+      <p>Biernatzki, <a href="#page32">32</a></p>
+      <p>Biot, <a href="#page32">32</a></p>
+      <p>Björnbo, A. A., <a href="#page125">125</a>, <a href="#page126">126</a></p>
+      <p>Blassière, <a href="#page119">119</a></p>
+      <p>Bloomfield, <a href="#page48">48</a></p>
+      <p>Blume, <a href="#page85">85</a></p>
+      <p>Boeckh, <a href="#page62">62</a></p>
+      <p>Boehmer, <a href="#page143">143</a></p>
+      <p>Boeschenstein, <a href="#page119">119</a></p>
+      <p>Boethius, <a href="#page63">63</a>, <a href="#page70">70</a>-<a href="#page73">73</a>, <a href="#page83">83</a>-<a href="#page90">90</a></p>
+      <p>Boissière, <a href="#page63">63</a></p>
+      <p>Bombelli, <a href="#page81">81</a></p>
+      <p>Bonaini, <a href="#page128">128</a></p>
+      <p>Boncompagni, <a href="#page5">5</a>, <a href="#page6">6</a>, <a href="#page10">10</a>, <a href="#page48">48</a>, <a href="#page49">49</a>, <a href="#page123">123</a>, <a href="#page125">125</a></p>
+      <p>Borghi, <a href="#page59">59</a></p>
+      <p>Borgo, <a href="#page119">119</a></p>
+      <p>Bougie, <a href="#page130">130</a></p>
+      <p>Bowring, J., <a href="#page56">56</a></p>
+      <p>Brahmagupta, <a href="#page52">52</a></p>
+      <p><span class="special" title="Brahmanas">Br&#x101;hma&#x1E47;as</span>, <a href="#page12">12</a>, <a href="#page13">13</a></p>
+      <p>Br&#x101;hm&#x12B;, <a href="#page19">19</a>, <a href="#page20">20</a>, <a href="#page31">31</a>, <a href="#page83">83</a></p>
+      <p>Brandis, J., <a href="#page54">54</a></p>
+      <p><span class="special" title="Brhat-Samhita">B&#x1E5B;hat-Sa&#x1E43;hita</span>, <a href="#page39">39</a>, <a href="#page44">44</a>, <a href="#page78">78</a></p>
+      <p>Brockhaus, <a href="#page43">43</a></p>
+      <p>Bubnov, <a href="#page65">65</a>, <a href="#page84">84</a>, <a href="#page110">110</a>, <a href="#page116">116</a></p>
+      <p>Buddha, education of, <a href="#page15">15</a>, <a href="#page16">16</a></p>
+      <p>Büdinger, <a href="#page110">110</a></p>
+      <p>Bugia, <a href="#page130">130</a></p>
+      <p>Bühler, G., <a href="#page15">15</a>, <a href="#page19">19</a>, <a href="#page22">22</a>, <a href="#page31">31</a>, <a href="#page44">44</a>, <a href="#page49">49</a></p>
+      <p>Burgess, <a href="#page25">25</a></p>
+      <p>Bürk, <a href="#page13">13</a></p>
+      <p>Burmese numerals, <a href="#page36">36</a></p>
+      <p>Burnell, A. C., <a href="#page18">18</a>, <a href="#page40">40</a></p>
+      <p>Buteo, <a href="#page61">61</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Calandri, <a href="#page59">59</a>, <a href="#page81">81</a></p>
+      <p>Caldwell, R., <a href="#page19">19</a></p>
+      <p>Calendars, <a href="#page133">133</a></p>
+      <p>Calmet, <a href="#page34">34</a></p>
+      <p>Cantor, M., <a href="#page5">5</a>, <a href="#page13">13</a>, <a href="#page30">30</a>, <a href="#page43">43</a>, <a href="#page84">84</a></p>
+<!-- Page 155 --><span class="pagenum"><a name="page155"></a>[155]</span>
+      <p>Capella, <a href="#page86">86</a></p>
+      <p>Cappelli, <a href="#page143">143</a></p>
+      <p>Caracteres, <a href="#page87">87</a>, <a href="#page113">113</a>, <a href="#page117">117</a>, <a href="#page119">119</a></p>
+      <p>Cardan, <a href="#page119">119</a></p>
+      <p>Carmen de Algorismo, <a href="#page11">11</a>, <a href="#page134">134</a></p>
+      <p>Casagrandi, <a href="#page132">132</a></p>
+      <p>Casiri, <a href="#page8">8</a>, <a href="#page10">10</a></p>
+      <p>Cassiodorus, <a href="#page72">72</a></p>
+      <p>Cataldi, <a href="#page62">62</a></p>
+      <p>Cataneo, <a href="#page3">3</a></p>
+      <p>Caxton, <a href="#page143">143</a>, <a href="#page146">146</a></p>
+      <p>Ceretti, <a href="#page32">32</a></p>
+      <p>Ceylon numerals, <a href="#page36">36</a></p>
+      <p>Chalfont, F. H., <a href="#page28">28</a></p>
+      <p>Champenois, <a href="#page60">60</a></p>
+      <p>Characters, <i>see</i> Caracteres</p>
+      <p>Charlemagne, <a href="#page103">103</a></p>
+      <p>Chasles, <a href="#page54">54</a>, <a href="#page60">60</a>, <a href="#page85">85</a>, <a href="#page116">116</a>, <a href="#page122">122</a>, <a href="#page135">135</a></p>
+      <p>Chassant, L. A., <a href="#page142">142</a></p>
+      <p>Chaucer, <a href="#page121">121</a></p>
+      <p>Chiarini, <a href="#page145">145</a>, <a href="#page146">146</a></p>
+      <p>Chiffre, <a href="#page58">58</a></p>
+      <p>Chinese numerals, <a href="#page28">28</a>, <a href="#page56">56</a></p>
+      <p>Chinese zero, <a href="#page56">56</a></p>
+      <p>Cifra, <a href="#page120">120</a>, <a href="#page124">124</a></p>
+      <p>Cipher, <a href="#page58">58</a></p>
+      <p>Circulus, <a href="#page58">58</a>, <a href="#page60">60</a></p>
+      <p>Clichtoveus, <a href="#page61">61</a>, <a href="#page119">119</a>, <a href="#page145">145</a></p>
+      <p>Codex Vigilanus, <a href="#page138">138</a></p>
+      <p>Codrington, O., <a href="#page139">139</a></p>
+      <p>Coins dated, <a href="#page141">141</a></p>
+      <p>Colebrooke, <a href="#page8">8</a>, <a href="#page26">26</a>, <a href="#page46">46</a>, <a href="#page53">53</a></p>
+      <p>Constantine, <a href="#page104">104</a>, <a href="#page105">105</a></p>
+      <p>Cosmas, <a href="#page82">82</a></p>
+      <p>Cossali, <a href="#page5">5</a></p>
+      <p>Counters, <a href="#page117">117</a></p>
+      <p>Courteille, <a href="#page8">8</a></p>
+      <p>Coxe, <a href="#page59">59</a></p>
+      <p>Crafte of Nombrynge, <a href="#page11">11</a>, <a href="#page87">87</a>, <a href="#page149">149</a></p>
+      <p>Crusades, <a href="#page109">109</a></p>
+      <p>Cunningham, A., <a href="#page30">30</a>, <a href="#page75">75</a></p>
+      <p>Curtze, <a href="#page55">55</a>, <a href="#page59">59</a>, <a href="#page126">126</a>, <a href="#page134">134</a></p>
+      <p>Cyfra, <a href="#page55">55</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Dagomari, <a href="#page146">146</a></p>
+      <p>D'Alviella, <a href="#page15">15</a></p>
+      <p>Dante, <a href="#page72">72</a></p>
+      <p>Dasypodius, <a href="#page33">33</a>, <a href="#page67">67</a>, <a href="#page63">63</a></p>
+      <p>Daunou, <a href="#page135">135</a></p>
+      <p>Delambre, <a href="#page54">54</a></p>
+      <p>Devan&#x101;gar&#x12B;, <a href="#page7">7</a></p>
+      <p>Devoulx, A., <a href="#page68">68</a></p>
+      <p>Dhruva, <a href="#page49">49</a></p>
+      <p>Dicæarchus of Messana, <a href="#page77">77</a></p>
+      <p>Digits, <a href="#page119">119</a></p>
+      <p>Diodorus Siculus, <a href="#page76">76</a></p>
+      <p>Du Cange, <a href="#page62">62</a></p>
+      <p>Dumesnil, <a href="#page36">36</a></p>
+      <p>Dutt, R. C., <a href="#page12">12</a>, <a href="#page15">15</a>, <a href="#page18">18</a>, <a href="#page75">75</a></p>
+      <p>Dvived&#x12B;, <a href="#page44">44</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>East and West, relations, <a href="#page73">73</a>-<a href="#page81">81</a>, <a href="#page100">100</a>-<a href="#page109">109</a></p>
+      <p>Egyptian numerals, <a href="#page27">27</a></p>
+      <p>Eisenlohr, <a href="#page28">28</a></p>
+      <p>Elia Misrachi, <a href="#page57">57</a></p>
+      <p>Enchiridion Algorismi, <a href="#page58">58</a></p>
+      <p>Eneström, <a href="#page5">5</a>, <a href="#page48">48</a>, <a href="#page59">59</a>, <a href="#page97">97</a>, <a href="#page125">125</a>, <a href="#page128">128</a></p>
+      <p>Europe, numerals in, <a href="#page63">63</a>, <a href="#page99">99</a>, <a href="#page128">128</a>, <a href="#page136">136</a></p>
+      <p>Eusebius Caesariensis, <a href="#page142">142</a></p>
+      <p>Euting, <a href="#page21">21</a></p>
+      <p>Ewald, P., <a href="#page116">116</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Fazzari, <a href="#page53">53</a>, <a href="#page54">54</a></p>
+      <p>Fibonacci, <i>see</i> Leonardo of Pisa</p>
+      <p>Figura nihili, <a href="#page58">58</a></p>
+      <p>Figures, <a href="#page119">119</a>. <i>See</i> numerals.</p>
+      <p>Fihrist, <a href="#page67">67</a>, <a href="#page68">68</a>, <a href="#page93">93</a></p>
+      <p>Finaeus, <a href="#page57">57</a></p>
+      <p>Firdus&#x12B;, <a href="#page81">81</a></p>
+      <p>Fitz Stephen, W., <a href="#page109">109</a></p>
+      <p>Fleet, J. C., <a href="#page19">19</a>, <a href="#page20">20</a>, <a href="#page49">49</a></p>
+<!-- Page 156 --><span class="pagenum"><a name="page156"></a>[156]</span>
+      <p>Florus, <a href="#page80">80</a></p>
+      <p>Flügel, G., <a href="#page68">68</a></p>
+      <p>Francisco de Retza, <a href="#page142">142</a></p>
+      <p>François, <a href="#page58">58</a></p>
+      <p>Friedlein, G., <a href="#page84">84</a>, <a href="#page113">113</a>, <a href="#page116">116</a>, <a href="#page122">122</a></p>
+      <p>Froude, J. A., <a href="#page129">129</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Gandh&#x101;ra, <a href="#page19">19</a></p>
+      <p>Garbe, <a href="#page48">48</a></p>
+      <p>Gasbarri, <a href="#page58">58</a></p>
+      <p>Gautier de Coincy, <a href="#page120">120</a>, <a href="#page124">124</a></p>
+      <p>Gemma Frisius, <a href="#page2">2</a>, <a href="#page3">3</a>, <a href="#page119">119</a></p>
+      <p>Gerber, <a href="#page113">113</a></p>
+      <p>Gerbert, <a href="#page108">108</a>, <a href="#page110">110</a>-<a href="#page120">120</a>, <a href="#page122">122</a></p>
+      <p>Gerhardt, C. I., <a href="#page43">43</a>, <a href="#page56">56</a>, <a href="#page93">93</a>, <a href="#page118">118</a></p>
+      <p>Gerland, <a href="#page88">88</a>, <a href="#page123">123</a></p>
+      <p>Gherard of Cremona, <a href="#page125">125</a></p>
+      <p>Gibbon, <a href="#page72">72</a></p>
+      <p>Giles, H. A., <a href="#page79">79</a></p>
+      <p>Ginanni, <a href="#page81">81</a></p>
+      <p>Giovanni di Danti, <a href="#page58">58</a></p>
+      <p>Glareanus, <a href="#page4">4</a>, <a href="#page119">119</a></p>
+      <p>Gnecchi, <a href="#page71">71</a>, <a href="#page117">117</a></p>
+      <p>&#x120;ob&#x101;r numerals, <a href="#page65">65</a>, <a href="#page100">100</a>, <a href="#page112">112</a>, <a href="#page124">124</a>, <a href="#page138">138</a></p>
+      <p>Gow, J., <a href="#page81">81</a></p>
+      <p>Grammateus, <a href="#page61">61</a></p>
+      <p>Greek origin, <a href="#page33">33</a></p>
+      <p>Green, J. R., <a href="#page109">109</a></p>
+      <p>Greenwood, I., <a href="#page62">62</a>, <a href="#page119">119</a></p>
+      <p>Guglielmini, <a href="#page128">128</a></p>
+      <p>Gulist&#x101;n, <a href="#page102">102</a></p>
+      <p>Günther, S., <a href="#page131">131</a></p>
+      <p>Guyard, S., <a href="#page82">82</a></p>
+    </div>
+
+    <div class="stanza">
+      <p><span class="special" title="Habash">&#x1E24;abash</span>, <a href="#page9">9</a>, <a href="#page92">92</a></p>
+      <p>Hager, J. (G.), <a href="#page28">28</a>, <a href="#page32">32</a></p>
+      <p>Halliwell, <a href="#page59">59</a>, <a href="#page85">85</a></p>
+      <p>Hankel, <a href="#page93">93</a></p>
+      <p>H&#x101;r&#x16B;n al-Rash&#x12B;d, <a href="#page97">97</a>, <a href="#page106">106</a></p>
+      <p>Havet, <a href="#page110">110</a></p>
+      <p>Heath, T. L., <a href="#page125">125</a></p>
+      <p>Hebrew numerals, <a href="#page127">127</a></p>
+      <p>Hecatæus, <a href="#page75">75</a></p>
+      <p>Heiberg, J. L., <a href="#page55">55</a>, <a href="#page85">85</a>, <a href="#page148">148</a></p>
+      <p>Heilbronner, <a href="#page5">5</a></p>
+      <p>Henry, C., <a href="#page5">5</a>, <a href="#page31">31</a>, <a href="#page55">55</a>, <a href="#page87">87</a>, <a href="#page120">120</a>, <a href="#page135">135</a></p>
+      <p>Heriger, <a href="#page122">122</a></p>
+      <p>Hermannus Contractus, <a href="#page123">123</a></p>
+      <p>Herodotus, <a href="#page76">76</a>, <a href="#page78">78</a></p>
+      <p>Heyd, <a href="#page75">75</a></p>
+      <p>Higden, <a href="#page136">136</a></p>
+      <p>Hill, G. F., <a href="#page52">52</a>, <a href="#page139">139</a>, <a href="#page142">142</a></p>
+      <p>Hillebrandt, A., <a href="#page15">15</a>, <a href="#page74">74</a></p>
+      <p>Hilprecht, H. V., <a href="#page28">28</a></p>
+      <p>Hindu forms, early, <a href="#page12">12</a></p>
+      <p>Hindu number names, <a href="#page42">42</a></p>
+      <p>Hodder, <a href="#page62">62</a></p>
+      <p>Hoernle, <a href="#page43">43</a>, <a href="#page49">49</a></p>
+      <p>Holywood, <i>see</i> Sacrobosco</p>
+      <p>Hopkins, E. W., <a href="#page12">12</a></p>
+      <p>Horace, <a href="#page79">79</a>, <a href="#page80">80</a></p>
+      <p><span class="special" title="Hosein ibn Mohammed al-Mahalli">&#x1E24;osein ibn Mo&#x1E25;ammed al-Ma&#x1E25;all&#x12B;</span>, <a href="#page6">6</a></p>
+      <p>Hostus, M., <a href="#page56">56</a></p>
+      <p>Howard, H. H., <a href="#page29">29</a></p>
+      <p>Hrabanus Maurus, <a href="#page72">72</a></p>
+      <p>Huart, <a href="#page7">7</a></p>
+      <p>Huet, <a href="#page33">33</a></p>
+      <p>Hugo, H., <a href="#page57">57</a></p>
+      <p>Humboldt, A. von, <a href="#page62">62</a></p>
+      <p>Huswirt, <a href="#page58">58</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Iamblichus, <a href="#page81">81</a></p>
+      <p>Ibn Ab&#x12B; Ya&#x201B;q&#x16B;b, <a href="#page9">9</a></p>
+      <p>Ibn al-Adam&#x12B;, <a href="#page92">92</a></p>
+      <p>Ibn al-Bann&#x101;, <a href="#page93">93</a></p>
+      <p><span class="special" title="Ibn Khordadbeh">Ibn Khord&#x101;&#x1E0D;beh</span>, <a href="#page101">101</a>, <a href="#page106">106</a></p>
+      <p>Ibn Wahab, <a href="#page103">103</a></p>
+      <p>India, history of, <a href="#page14">14</a></p>
+      <p class="i2">writing in, <a href="#page18">18</a></p>
+      <p>Indicopleustes, <a href="#page83">83</a></p>
+      <p>Indo-Bactrian numerals, <a href="#page19">19</a></p>
+<!-- Page 157 --><span class="pagenum"><a name="page157"></a>[157]</span>
+      <p>Indr&#x101;j&#x12B;, <a href="#page23">23</a></p>
+      <p><span class="special" title="Ishaq ibn Yusuf al-Sardafi">Is&#x1E25;&#x101;q ibn Y&#x16B;suf al-&#x1E62;ardaf&#x12B;</span>, <a href="#page93">93</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Jacob of Florence, <a href="#page57">57</a></p>
+      <p>Jacquet, E., <a href="#page38">38</a></p>
+      <p>Jamshid, <a href="#page56">56</a></p>
+      <p>Jehan Certain, <a href="#page59">59</a></p>
+      <p>Jetons, <a href="#page58">58</a>, <a href="#page117">117</a></p>
+      <p>Jevons, F. B., <a href="#page76">76</a></p>
+      <p>Johannes Hispalensis, <a href="#page48">48</a>, <a href="#page88">88</a>, <a href="#page124">124</a></p>
+      <p>John of Halifax, <i>see</i> Sacrobosco</p>
+      <p>John of Luna, <i>see</i> Johannes Hispalensis</p>
+      <p>Jordan, L., <a href="#page58">58</a>, <a href="#page124">124</a></p>
+      <p>Joseph Ispanus (Joseph Sapiens), <a href="#page115">115</a></p>
+      <p>Justinian, <a href="#page104">104</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Kále, M. R., <a href="#page26">26</a></p>
+      <p>Karabacek, <a href="#page56">56</a></p>
+      <p>Karpinski, L. C., <a href="#page126">126</a>, <a href="#page134">134</a>, <a href="#page138">138</a></p>
+      <p>K&#x101;ty&#x101;yana, <a href="#page39">39</a></p>
+      <p>Kaye, C. R., <a href="#page6">6</a>, <a href="#page16">16</a>, <a href="#page43">43</a>, <a href="#page46">46</a>, <a href="#page121">121</a></p>
+      <p>Keane, J., <a href="#page75">75</a>, <a href="#page82">82</a></p>
+      <p>Keene, H. G., <a href="#page15">15</a></p>
+      <p>Kern, <a href="#page44">44</a></p>
+      <p><span class="special" title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span>, <a href="#page19">19</a>, <a href="#page20">20</a></p>
+      <p>Khosr&#x16B;, <a href="#page82">82</a>, <a href="#page91">91</a></p>
+      <p>Kielhorn, F., <a href="#page46">46</a>, <a href="#page47">47</a></p>
+      <p>Kircher, A., <a href="#page34">34</a></p>
+      <p>Kit&#x101;b al-Fihrist, <i>see</i> Fihrist</p>
+      <p>Kleinwächter, <a href="#page32">32</a></p>
+      <p>K<span class="over">l</span>os, <a href="#page62">62</a></p>
+      <p>Köbel, <a href="#page4">4</a>, <a href="#page58">58</a>, <a href="#page60">60</a>, <a href="#page119">119</a>, <a href="#page123">123</a></p>
+      <p>Krumbacher, K., <a href="#page57">57</a></p>
+      <p>Kuckuck, <a href="#page62">62</a>, <a href="#page133">133</a></p>
+      <p>Kugler, F. X., <a href="#page51">51</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Lachmann, <a href="#page85">85</a></p>
+      <p>Lacouperie, <a href="#page33">33</a>, <a href="#page35">35</a></p>
+      <p>Lalitavistara, <a href="#page15">15</a>, <a href="#page17">17</a></p>
+      <p>Lami, G., <a href="#page57">57</a></p>
+      <p>La Roche, <a href="#page61">61</a></p>
+      <p>Lassen, <a href="#page39">39</a></p>
+      <p><span class="special" title="Latyayana">L&#x101;&#x1E6D;y&#x101;yana</span>, <a href="#page39">39</a></p>
+      <p>Leb&oelig;uf, <a href="#page135">135</a></p>
+      <p>Leonardo of Pisa, <a href="#page5">5</a>, <a href="#page10">10</a>, <a href="#page57">57</a>, <a href="#page64">64</a>, <a href="#page74">74</a>, <a href="#page120">120</a>, <a href="#page128">128</a>-<a href="#page133">133</a></p>
+      <p>Lethaby, W. R., <a href="#page142">142</a></p>
+      <p>Levi, B., <a href="#page13">13</a></p>
+      <p>Levias, <a href="#page3">3</a></p>
+      <p>Libri, <a href="#page73">73</a>, <a href="#page85">85</a>, <a href="#page95">95</a></p>
+      <p>Light of Asia, <a href="#page16">16</a></p>
+      <p>Luca da Firenze, <a href="#page144">144</a></p>
+      <p>Lucas, <a href="#page128">128</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Mah&#x101;bh&#x101;rata, <a href="#page18">18</a></p>
+      <p>Mah&#x101;v&#x12B;r&#x101;c&#x101;rya, <a href="#page53">53</a></p>
+      <p>Malabar numerals, <a href="#page36">36</a></p>
+      <p>Malayalam numerals, <a href="#page36">36</a></p>
+      <p>Mannert, <a href="#page81">81</a></p>
+      <p>Margarita Philosophica, <a href="#page146">146</a></p>
+      <p>Marie, <a href="#page78">78</a></p>
+      <p>Marquardt, J., <a href="#page85">85</a></p>
+      <p>Marshman, J. C., <a href="#page17">17</a></p>
+      <p>Martin, T. H., <a href="#page30">30</a>, <a href="#page62">62</a>, <a href="#page85">85</a>, <a href="#page113">113</a></p>
+      <p>Martines, D. C., <a href="#page58">58</a></p>
+      <p>M&#x101;sh&#x101;ll&#x101;h, <a href="#page3">3</a></p>
+      <p>Maspero, <a href="#page28">28</a></p>
+      <p>Mauch, <a href="#page142">142</a></p>
+      <p>Maximus Planudes, <a href="#page2">2</a>, <a href="#page57">57</a>, <a href="#page66">66</a>, <a href="#page93">93</a>, <a href="#page120">120</a></p>
+      <p>Megasthenes, <a href="#page77">77</a></p>
+      <p>Merchants, <a href="#page114">114</a></p>
+      <p>Meynard, <a href="#page8">8</a></p>
+      <p>Migne, <a href="#page87">87</a></p>
+      <p>Mikami, Y., <a href="#page56">56</a></p>
+      <p>Milanesi, <a href="#page128">128</a></p>
+      <p><span class="special" title="Mohammed ibn `Abdallah">Mo&#x1E25;ammed ibn &#x201B;Abdall&#x101;h</span>, <a href="#page92">92</a></p>
+      <p><span class="special" title="Mohammed ibn Ahmed">Mo&#x1E25;ammed ibn A&#x1E25;med</span>, <a href="#page6">6</a></p>
+      <p><span class="special" title="Mohammed ibn `Ali `Abdi">Mo&#x1E25;ammed ibn &#x201B;Al&#x12B; &#x201B;Abd&#x12B;</span>, <a href="#page8">8</a></p>
+      <p><span class="special" title="Mohammed ibn Musa">Mo&#x1E25;ammed ibn M&#x16B;s&#x101;</span>, <i>see</i> Al-Khow&#x101;razm&#x12B;</p>
+      <p>Molinier, <a href="#page123">123</a></p>
+      <p>Monier-Williams, <a href="#page17">17</a></p>
+<!-- Page 158 --><span class="pagenum"><a name="page158"></a>[158]</span>
+      <p>Morley, D., <a href="#page126">126</a></p>
+      <p>Moroccan numerals, <a href="#page68">68</a>, <a href="#page119">119</a></p>
+      <p>Mortet, V., <a href="#page11">11</a></p>
+      <p>Moseley, C. B., <a href="#page33">33</a></p>
+      <p><span class="special" title="Motahhar ibn Tahir">Mo&#x1E6D;ahhar ibn &#x1E6C;&#x101;hir</span>, <a href="#page7">7</a></p>
+      <p>Mueller, A., <a href="#page68">68</a></p>
+      <p>Mumford, J. K., <a href="#page109">109</a></p>
+      <p>Muwaffaq al-D&#x12B;n, <a href="#page93">93</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Nabatean forms, <a href="#page21">21</a></p>
+      <p>Nallino, <a href="#page4">4</a>, <a href="#page54">54</a>, <a href="#page55">55</a></p>
+      <p>Nagl, A., <a href="#page55">55</a>, <a href="#page110">110</a>, <a href="#page113">113</a>, <a href="#page126">126</a></p>
+      <p>N&#x101;n&#x101; Gh&#x101;t inscriptions, <a href="#page20">20</a>, <a href="#page22">22</a>, <a href="#page23">23</a>, <a href="#page40">40</a></p>
+      <p>Narducci, <a href="#page123">123</a></p>
+      <p>Nasik cave inscriptions, <a href="#page24">24</a></p>
+      <p><span class="special" title="Nazif ibn Yumn">Na&#x1E93;&#x12B;f ibn Yumn</span>, <a href="#page94">94</a></p>
+      <p>Neander, A., <a href="#page75">75</a></p>
+      <p>Neophytos, <a href="#page57">57</a>, <a href="#page62">62</a></p>
+      <p>Neo-Pythagoreans, <a href="#page64">64</a></p>
+      <p>Nesselmann, <a href="#page58">58</a></p>
+      <p>Newman, Cardinal, <a href="#page96">96</a></p>
+      <p>Newman, F. W., <a href="#page131">131</a></p>
+      <p>Nöldeke, Th., <a href="#page91">91</a></p>
+      <p>Notation, <a href="#page61">61</a></p>
+      <p>Note, <a href="#page61">61</a>, <a href="#page119">119</a></p>
+      <p>Noviomagus, <a href="#page45">45</a>, <a href="#page61">61</a>, <a href="#page119">119</a>, <a href="#page150">150</a></p>
+      <p>Null, <a href="#page61">61</a></p>
+      <p>Numerals,</p>
+      <p class="i2">Algerian, <a href="#page68">68</a></p>
+      <p class="i2">astrological, <a href="#page150">150</a></p>
+      <p class="i2">Br&#x101;hm&#x12B;, <a href="#page19">19</a>-<a href="#page22">22</a>, <a href="#page83">83</a></p>
+      <p class="i2">early ideas of origin, <a href="#page1">1</a></p>
+      <p class="i2">Hindu, <a href="#page26">26</a></p>
+      <p class="i2">Hindu, classified, <a href="#page19">19</a>, <a href="#page38">38</a></p>
+      <p class="i2"><span class="special" title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span>, <a href="#page19">19</a>-<a href="#page22">22</a></p>
+      <p class="i2">Moroccan, <a href="#page68">68</a></p>
+      <p class="i2">Nabatean, <a href="#page21">21</a></p>
+      <p class="i2">origin, <a href="#page27">27</a>, <a href="#page30">30</a>, <a href="#page31">31</a>, <a href="#page37">37</a></p>
+      <p class="i2">supposed Arabic origin, <a href="#page2">2</a></p>
+      <p class="i2">supposed Babylonian origin, <a href="#page28">28</a></p>
+      <p class="i2">supposed Chaldean and Jewish origin, <a href="#page3">3</a></p>
+      <p class="i2">supposed Chinese origin, <a href="#page28">28</a>, <a href="#page32">32</a></p>
+      <p class="i2">supposed Egyptian origin, <a href="#page27">27</a>, <a href="#page30">30</a>, <a href="#page69">69</a>, <a href="#page70">70</a></p>
+      <p class="i2">supposed Greek origin, <a href="#page33">33</a></p>
+      <p class="i2">supposed Ph&oelig;nician origin, <a href="#page32">32</a></p>
+      <p class="i2">tables of, <a href="#page22">22</a>-<a href="#page27">27</a>, <a href="#page36">36</a>, <a href="#page48">48</a>, <a href="#page49">49</a>, <a href="#page69">69</a>, <a href="#page88">88</a>, <a href="#page140">140</a>, <a href="#page143">143</a>, <a href="#page145">145</a>-<a href="#page148">148</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>O'Creat, <a href="#page5">5</a>, <a href="#page55">55</a>, <a href="#page119">119</a>, <a href="#page120">120</a></p>
+      <p>Olleris, <a href="#page110">110</a>, <a href="#page113">113</a></p>
+      <p>Oppert, G., <a href="#page14">14</a>, <a href="#page75">75</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Pali, <a href="#page22">22</a></p>
+      <p>Pañcasiddh&#x101;ntik&#x101;, <a href="#page44">44</a></p>
+      <p>Paravey, <a href="#page32">32</a>, <a href="#page57">57</a></p>
+      <p><span class="special" title="Pataliputra">P&#x101;tal&#x12B;pu&#x1E6D;ra</span>, <a href="#page77">77</a></p>
+      <p>Patna, <a href="#page77">77</a></p>
+      <p>Patrick, R., <a href="#page119">119</a></p>
+      <p>Payne, E. J., <a href="#page106">106</a></p>
+      <p>Pegolotti, <a href="#page107">107</a></p>
+      <p>Peletier, <a href="#page2">2</a>, <a href="#page62">62</a></p>
+      <p>Perrot, <a href="#page80">80</a></p>
+      <p>Persia, <a href="#page66">66</a>, <a href="#page91">91</a>, <a href="#page107">107</a></p>
+      <p>Pertz, <a href="#page115">115</a></p>
+      <p>Petrus de Dacia, <a href="#page59">59</a>, <a href="#page61">61</a>, <a href="#page62">62</a></p>
+      <p>Pez, P. B., <a href="#page117">117</a></p>
+      <p>"Philalethes," <a href="#page75">75</a></p>
+      <p>Phillips, G., <a href="#page107">107</a></p>
+      <p>Picavet, <a href="#page105">105</a></p>
+      <p>Pichler, F., <a href="#page141">141</a></p>
+      <p>Pihan, A. P., <a href="#page36">36</a></p>
+      <p>Pisa, <a href="#page128">128</a></p>
+      <p>Place value, <a href="#page26">26</a>, <a href="#page42">42</a>, <a href="#page46">46</a>, <a href="#page48">48</a></p>
+      <p>Planudes, <i>see</i> Maximus Planudes</p>
+      <p>Plimpton, G. A., <a href="#page56">56</a>, <a href="#page59">59</a>, <a href="#page85">85</a>, <a href="#page143">143</a>, <a href="#page144">144</a>, <a href="#page145">145</a>, <a href="#page148">148</a></p>
+      <p>Pliny, <a href="#page76">76</a></p>
+      <p>Polo, N. and M., <a href="#page107">107</a></p>
+<!-- Page 159 --><span class="pagenum"><a name="page159"></a>[159]</span>
+      <p>Prändel, J. G., <a href="#page54">54</a></p>
+      <p>Prinsep, J., <a href="#page20">20</a>, <a href="#page31">31</a></p>
+      <p>Propertius, <a href="#page80">80</a></p>
+      <p>Prosdocimo de' Beldomandi, <a href="#page137">137</a></p>
+      <p>Prou, <a href="#page143">143</a></p>
+      <p>Ptolemy, <a href="#page54">54</a>, <a href="#page78">78</a></p>
+      <p>Putnam, <a href="#page103">103</a></p>
+      <p>Pythagoras, <a href="#page63">63</a></p>
+      <p>Pythagorean numbers, <a href="#page13">13</a></p>
+      <p>Pytheas of Massilia, <a href="#page76">76</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Rabbi ben Ezra, <a href="#page60">60</a>, <a href="#page127">127</a></p>
+      <p>Radulph of Laon, <a href="#page60">60</a>, <a href="#page113">113</a>, <a href="#page118">118</a>, <a href="#page124">124</a></p>
+      <p>Raets, <a href="#page62">62</a></p>
+      <p>Rainer, <i>see</i> Gemma Frisius</p>
+      <p>R&#x101;m&#x101;yana, <a href="#page18">18</a></p>
+      <p>Ramus, <a href="#page2">2</a>, <a href="#page41">41</a>, <a href="#page60">60</a>, <a href="#page61">61</a></p>
+      <p>Raoul Glaber, <a href="#page123">123</a></p>
+      <p>Rapson, <a href="#page77">77</a></p>
+      <p>Rauhfuss, <i>see</i> Dasypodius</p>
+      <p>Raumer, K. von, <a href="#page111">111</a></p>
+      <p>Reclus, E., <a href="#page14">14</a>, <a href="#page96">96</a>, <a href="#page130">130</a></p>
+      <p>Recorde, <a href="#page3">3</a>, <a href="#page58">58</a></p>
+      <p>Reinaud, <a href="#page67">67</a>, <a href="#page74">74</a>, <a href="#page80">80</a></p>
+      <p>Reveillaud, <a href="#page36">36</a></p>
+      <p>Richer, <a href="#page110">110</a>, <a href="#page112">112</a>, <a href="#page115">115</a></p>
+      <p>Riese, A., <a href="#page119">119</a></p>
+      <p>Robertson, <a href="#page81">81</a></p>
+      <p>Robertus Cestrensis, <a href="#page97">97</a>, <a href="#page126">126</a></p>
+      <p>Rodet, <a href="#page5">5</a>, <a href="#page44">44</a></p>
+      <p>Roediger, J., <a href="#page68">68</a></p>
+      <p>Rollandus, <a href="#page144">144</a></p>
+      <p>Romagnosi, <a href="#page81">81</a></p>
+      <p>Rosen, F., <a href="#page5">5</a></p>
+      <p>Rotula, <a href="#page60">60</a></p>
+      <p>Rudolff, <a href="#page85">85</a></p>
+      <p>Rudolph, <a href="#page62">62</a>, <a href="#page67">67</a></p>
+      <p>Ruffi, <a href="#page150">150</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Sachau, <a href="#page6">6</a></p>
+      <p>Sacrobosco, <a href="#page3">3</a>, <a href="#page58">58</a>, <a href="#page133">133</a></p>
+      <p>Sacy, S. de, <a href="#page66">66</a>, <a href="#page70">70</a></p>
+      <p>Sa&#x201B;d&#x12B;, <a href="#page102">102</a></p>
+      <p>&#x15A;aka inscriptions, <a href="#page20">20</a></p>
+      <p><span class="special" title="Samu'il ibn Yahya">Sam&#x16B;'&#x12B;l ibn Ya&#x1E25;y&#x101;</span>, <a href="#page93">93</a></p>
+      <p>&#x15A;&#x101;rad&#x101; characters, <a href="#page55">55</a></p>
+      <p>Savonne, <a href="#page60">60</a></p>
+      <p>Scaliger, J. C., <a href="#page73">73</a></p>
+      <p>Scheubel, <a href="#page62">62</a></p>
+      <p>Schlegel, <a href="#page12">12</a></p>
+      <p>Schmidt, <a href="#page133">133</a></p>
+      <p>Schonerus, <a href="#page87">87</a>, <a href="#page119">119</a></p>
+      <p>Schroeder, L. von, <a href="#page13">13</a></p>
+      <p>Scylax, <a href="#page75">75</a></p>
+      <p>Sedillot, <a href="#page8">8</a>, <a href="#page34">34</a></p>
+      <p>Senart, <a href="#page20">20</a>, <a href="#page24">24</a>, <a href="#page25">25</a></p>
+      <p>Sened ibn &#x201B;Al&#x12B;, <a href="#page10">10</a>, <a href="#page98">98</a></p>
+      <p>Sfortunati, <a href="#page62">62</a>, <a href="#page150">150</a></p>
+      <p>Shelley, W., <a href="#page126">126</a></p>
+      <p>Siamese numerals, <a href="#page36">36</a></p>
+      <p>Siddh&#x101;nta, <a href="#page8">8</a>, <a href="#page18">18</a></p>
+      <p><span class="special" title="Sifr">&#x1E62;ifr</span>, <a href="#page57">57</a></p>
+      <p>Sigsboto, <a href="#page55">55</a></p>
+      <p>Sih&#x101;b al-D&#x12B;n, <a href="#page67">67</a></p>
+      <p>Silberberg, <a href="#page60">60</a></p>
+      <p>Simon, <a href="#page13">13</a></p>
+      <p><span class="special" title="Sinan ibn al-Fath">Sin&#x101;n ibn al-Fat&#x1E25;</span>, <a href="#page93">93</a></p>
+      <p>Sindbad, <a href="#page100">100</a></p>
+      <p>Sindhind, <a href="#page97">97</a></p>
+      <p>Sipos, <a href="#page60">60</a></p>
+      <p>Sirr, H. C., <a href="#page75">75</a></p>
+      <p>Skeel, C. A., <a href="#page74">74</a></p>
+      <p>Smith, D. E., <a href="#page11">11</a>, <a href="#page17">17</a>, <a href="#page53">53</a>, <a href="#page86">86</a>, <a href="#page141">141</a>, <a href="#page143">143</a></p>
+      <p>Smith, V. A., <a href="#page20">20</a>, <a href="#page35">35</a>, <a href="#page46">46</a>, <a href="#page47">47</a></p>
+      <p>Smith, Wm., <a href="#page75">75</a></p>
+      <p><span class="special" title="Smrti">Sm&#x1E5B;ti</span>, <a href="#page17">17</a></p>
+      <p>Spain, <a href="#page64">64</a>, <a href="#page65">65</a>, <a href="#page100">100</a></p>
+      <p>Spitta-Bey, <a href="#page5">5</a></p>
+      <p>Sprenger, <a href="#page94">94</a></p>
+      <p>&#x15A;rautas&#x16B;tra, <a href="#page39">39</a></p>
+      <p>Steffens, F., <a href="#page116">116</a></p>
+      <p>Steinschneider, <a href="#page5">5</a>, <a href="#page57">57</a>, <a href="#page65">65</a>, <a href="#page66">66</a>, <a href="#page98">98</a>, <a href="#page126">126</a></p>
+      <p>Stifel, <a href="#page62">62</a></p>
+<!-- Page 160 --><span class="pagenum"><a name="page160"></a>[160]</span>
+      <p>Subandhus, <a href="#page44">44</a></p>
+      <p>Suetonius, <a href="#page80">80</a></p>
+      <p>Suleim&#x101;n, <a href="#page100">100</a></p>
+      <p>&#x15A;&#x16B;nya, <a href="#page43">43</a>, <a href="#page53">53</a>, <a href="#page57">57</a></p>
+      <p>Suter, <a href="#page5">5</a>, <a href="#page9">9</a>, <a href="#page68">68</a>, <a href="#page69">69</a>, <a href="#page93">93</a>, <a href="#page116">116</a>, <a href="#page131">131</a></p>
+      <p>S&#x16B;tras, <a href="#page13">13</a></p>
+      <p>Sykes, P. M., <a href="#page75">75</a></p>
+      <p>Sylvester II, <i>see</i> Gerbert</p>
+      <p>Symonds, J. A., <a href="#page129">129</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Tannery, P., <a href="#page62">62</a>, <a href="#page84">84</a>, <a href="#page85">85</a></p>
+      <p>Tartaglia, <a href="#page4">4</a>, <a href="#page61">61</a></p>
+      <p>Taylor, I., <a href="#page19">19</a>, <a href="#page30">30</a></p>
+      <p>Teca, <a href="#page55">55</a>, <a href="#page61">61</a></p>
+      <p>Tennent, J. E., <a href="#page75">75</a></p>
+      <p>Texada, <a href="#page60">60</a></p>
+      <p>Theca, <a href="#page58">58</a>, <a href="#page61">61</a></p>
+      <p>Theophanes, <a href="#page64">64</a></p>
+      <p>Thibaut, G., <a href="#page12">12</a>, <a href="#page13">13</a>, <a href="#page16">16</a>, <a href="#page44">44</a>, <a href="#page47">47</a></p>
+      <p>Tibetan numerals, <a href="#page36">36</a></p>
+      <p>Timotheus, <a href="#page103">103</a></p>
+      <p>Tonstall, C., <a href="#page3">3</a>, <a href="#page61">61</a></p>
+      <p>Trenchant, <a href="#page60">60</a></p>
+      <p>Treutlein, <a href="#page5">5</a>, <a href="#page63">63</a>, <a href="#page123">123</a></p>
+      <p>Trevisa, <a href="#page136">136</a></p>
+      <p>Treviso arithmetic, <a href="#page145">145</a></p>
+      <p>Trivium and quadrivium, <a href="#page73">73</a></p>
+      <p>Tsin, <a href="#page56">56</a></p>
+      <p>Tunis, <a href="#page65">65</a></p>
+      <p>Turchill, <a href="#page88">88</a>, <a href="#page118">118</a>, <a href="#page123">123</a></p>
+      <p>Turnour, G., <a href="#page75">75</a></p>
+      <p>Tziphra, <a href="#page57">57</a>, <a href="#page62">62</a></p>
+      <p><span title="tziphra" class="grk">&tau;&zeta;&#x1F77;&phi;&rho;&alpha;</span>, <a href="#page55">55</a>, <a href="#page57">57</a>, <a href="#page62">62</a></p>
+      <p>Tzwivel, <a href="#page61">61</a>, <a href="#page118">118</a>, <a href="#page145">145</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Ujjain, <a href="#page32">32</a></p>
+      <p>Unger, <a href="#page133">133</a></p>
+      <p>Upanishads, <a href="#page12">12</a></p>
+      <p>Usk, <a href="#page121">121</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Valla, G., <a href="#page61">61</a></p>
+      <p>Van der Schuere, <a href="#page62">62</a></p>
+      <p>Var&#x101;ha-Mihira, <a href="#page39">39</a>, <a href="#page44">44</a>, <a href="#page78">78</a></p>
+      <p>V&#x101;savadatt&#x101;, <a href="#page44">44</a></p>
+      <p>Vaux, Carra de, <a href="#page9">9</a>, <a href="#page74">74</a></p>
+      <p>Vaux, W. S. W., <a href="#page91">91</a></p>
+      <p><span class="special" title="Vedangas">Ved&#x101;&#x1E45;gas</span>, <a href="#page17">17</a></p>
+      <p>Vedas, <a href="#page12">12</a>, <a href="#page15">15</a>, <a href="#page17">17</a></p>
+      <p>Vergil, <a href="#page80">80</a></p>
+      <p>Vincent, A. J. H., <a href="#page57">57</a></p>
+      <p>Vogt, <a href="#page13">13</a></p>
+      <p>Voizot, P., <a href="#page36">36</a></p>
+      <p>Vossius, <a href="#page4">4</a>, <a href="#page76">76</a>, <a href="#page81">81</a>, <a href="#page84">84</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Wallis, <a href="#page3">3</a>, <a href="#page62">62</a>, <a href="#page84">84</a>, <a href="#page116">116</a></p>
+      <p>Wappler, E., <a href="#page54">54</a>, <a href="#page126">126</a></p>
+      <p>Wäschke, H., <a href="#page2">2</a>, <a href="#page93">93</a></p>
+      <p>Wattenbach, <a href="#page143">143</a></p>
+      <p>Weber, A., <a href="#page31">31</a></p>
+      <p>Weidler, I. F., <a href="#page34">34</a>, <a href="#page66">66</a></p>
+      <p>Weidler, I. F. and G. I., <a href="#page63">63</a>, <a href="#page66">66</a></p>
+      <p>Weissenborn, <a href="#page85">85</a>, <a href="#page110">110</a></p>
+      <p>Wertheim, G., <a href="#page57">57</a>, <a href="#page61">61</a></p>
+      <p>Whitney, W. D., <a href="#page13">13</a></p>
+      <p>Wilford, F., <a href="#page75">75</a></p>
+      <p>Wilkens, <a href="#page62">62</a></p>
+      <p>Wilkinson, J. G., <a href="#page70">70</a></p>
+      <p>Willichius, <a href="#page3">3</a></p>
+      <p>Woepcke, <a href="#page3">3</a>, <a href="#page6">6</a>, <a href="#page42">42</a>, <a href="#page63">63</a>, <a href="#page64">64</a>, <a href="#page65">65</a>, <a href="#page67">67</a>, <a href="#page69">69</a>, <a href="#page70">70</a>, <a href="#page94">94</a>, <a href="#page113">113</a>, <a href="#page138">138</a></p>
+      <p>Wolack, G., <a href="#page54">54</a></p>
+      <p>Woodruff, C. E., <a href="#page32">32</a></p>
+      <p>Word and letter numerals, <a href="#page38">38</a>, <a href="#page44">44</a></p>
+      <p>Wüstenfeld, <a href="#page74">74</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Yule, H., <a href="#page107">107</a></p>
+    </div>
+
+    <div class="stanza">
+      <p>Zephirum, <a href="#page57">57</a>, <a href="#page58">58</a></p>
+      <p>Zephyr, <a href="#page59">59</a></p>
+      <p>Zepiro, <a href="#page58">58</a></p>
+      <p>Zero, <a href="#page26">26</a>, <a href="#page38">38</a>, <a href="#page40">40</a>, <a href="#page43">43</a>, <a href="#page45">45</a>, <a href="#page49">49</a>, <a href="#page51">51</a>-<a href="#page62">62</a>, <a href="#page67">67</a></p>
+      <p>Zeuero, <a href="#page58">58</a></p>
+    </div>
+  </div>
+<hr class="full" >
+
+<h3>ANNOUNCEMENTS</h3>
+
+<hr class="full" >
+
+<h3>WENTWORTH'S</h3>
+
+<h2>COLLEGE ALGEBRA</h2>
+
+<p class="cenhead">REVISED EDITION</p>
+
+<p class="cenhead">12mo. Half morocco. 530 pages. List price, $1.50; mailing price, $1.65</p>
+
+<hr class="short" >
+
+  <p>This book is a thorough revision of the author's "College Algebra."
+  Some chapters of the old edition have been wholly rewritten, and the
+  other chapters have been rewritten in part and greatly improved. The
+  order of topics has been changed to a certain extent; the plan is to have
+  each chapter as complete in itself as possible, so that the teacher may
+  vary the order of succession at his discretion.</p>
+
+  <p>As the name implies, the work is intended for colleges and scientific
+  schools. The first part is simply a review of the principles of algebra
+  preceding Quadratic Equations, with just enough examples to illustrate
+  and enforce these principles. By this brief treatment of the first
+  chapters sufficient space is allowed, without making the book cumbersome,
+  for a full discussion of Quadratic Equations, The Binomial Theorem,
+  Choice Chance, Series, Determinants, and the General Properties of
+  Equations.</p>
+
+  <p>Every effort has been made to present in the clearest light each
+  subject discussed, and to give in matter and methods the best training in
+  algebraic analysis at present attainable.</p>
+
+<p class="cenhead">ADDITIONAL PUBLICATIONS</p>
+
+<p class="cenhead">By G. A. WENTWORTH</p>
+
+<table class="nobctr">
+<tr><td>&nbsp;</td><td align="right">List<br />price</td><td align="right"> &nbsp; Mailing<br />price</td></tr>
+<tr><td>Plane and Solid Geometry (Revised Edition) &nbsp; </td><td align="right">$1.25</td><td align="right">$1.40</td></tr>
+<tr><td> &nbsp; Plane Geometry (Revised Edition)</td><td align="right">.75</td><td align="right">.85</td></tr>
+<tr><td> &nbsp; Solid Geometry (Revised Edition)</td><td align="right">.75</td><td align="right">.85</td></tr>
+<tr><td>Analytic Geometry</td><td align="right">1.25</td><td align="right">1.35</td></tr>
+<tr><td>Trigonometries (Second Revised Editions)</td></tr>
+</table>
+
+<p class="cenhead"><i>A list of the various editions of Wentworth's Trigonometries will be sent on request.</i></p>
+
+<hr class="full" >
+
+<h2>GINN AND COMPANY <span class="sc">Publishers</span></h2>
+
+<hr class="full" >
+
+<h2>ALGEBRA FOR BEGINNERS</h2>
+
+<p class="cenhead">By DAVID EUGENE SMITH,</p>
+
+<p class="cenhead">Professor of Mathematics in Teachers College, Columbia University</p>
+
+<hr class="short" >
+
+<p class="cenhead">12mo, cloth, 154 pages, 50 cents</p>
+
+<hr class="short" >
+
+  <p>This work is intended to serve as an introduction to the study of
+  algebra, and is adapted to the needs of the seventh or eighth school
+  year. It is arranged in harmony with the leading courses of study that
+  include algebra in the curriculum of the grades.</p>
+
+  <p>The relation of algebra to arithmetic is emphasized, the subject is
+  treated topically, and each important point is touched at least twice.
+  The book begins by showing the uses of algebra, employing such practical
+  applications as are within the pupil's range of knowledge. When an
+  interest has thus been awakened in the subject, the fundamental
+  operations are presented with the simple explanations necessary to make
+  the student independent of dogmatic rules. Throughout the book abundant
+  oral and written drill exercises are provided. The work includes linear
+  equations with two unknown quantities, and easy quadratics.</p>
+
+  <p>The leading features may be summarized as follows: (1) an arrangement
+  in harmony with existing courses of study; (2) a presentation designed to
+  awaken the interest of the pupils; (3) a topical arrangement for each
+  half year, every important topic being repeated; (4) simplicity of
+  explanations; (5) development of the relation of algebra to arithmetic
+  both in theory and in applications; (6) emphasis laid on the importance
+  of oral as well as written algebra.</p>
+
+<hr class="full" >
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+
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+<tr><td>Hall: Aspects of Child Life and Education</td><td align="right">1.50</td></tr>
+<tr><td>Hodge: Nature Study and Life</td><td align="right">1.50</td></tr>
+<tr><td>Johnson: Education by Plays and Games</td><td align="right">.90</td></tr>
+<tr><td>Johnson: What to Do at Recess</td><td align="right">.25</td></tr>
+<tr><td>Kern: Among Country Schools</td><td align="right">1.25</td></tr>
+<tr><td>Mace: Method in History</td><td align="right">1.00</td></tr>
+<tr><td>MacVicar: The Principles of Education</td><td align="right">.60</td></tr>
+<tr><td>Moral Training in the Public Schools</td><td align="right">1.25</td></tr>
+<tr><td>Prince: Courses of Studies and Methods of Teaching</td><td align="right">.75</td></tr>
+<tr><td>Scott: Social Education</td><td align="right">1.25</td></tr>
+<tr><td>Tompkins: Philosophy of School Management</td><td align="right">.75</td></tr>
+<tr><td>Tompkins: Philosophy of Teaching</td><td align="right">.75</td></tr>
+<tr><td>Wiltse: Place of the Story in Early Education, and Other Essays. A Manual for Teachers</td><td align="right">.50</td></tr>
+</table>
+
+<h3>FOR CLASS RECORDS</h3>
+
+<table class="nobctr">
+<tr><td>Comings: Complete Record&mdash;Attendance and Scholarship</td></tr>
+<tr><td> &nbsp; Graded-School Edition</td><td align="right">.30</td></tr>
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+<tr><td>Ginn and Company: Teacher's Class Books</td></tr>
+<tr><td> &nbsp; No. I</td><td align="right">.30</td></tr>
+<tr><td> &nbsp; No. II</td><td align="right">.40</td></tr>
+<tr><td> &nbsp; Twenty Weeks' Class Book</td><td align="right">.30</td></tr>
+</table>
+
+<hr class="full" >
+
+<h2>GINN AND COMPANY Publishers</h2>
+
+<hr class="full" >
+
+<h2>TEXTBOOKS ON MATHEMATICS</h2>
+
+<p class="cenhead">FOR HIGHER SCHOOLS AND COLLEGES</p>
+
+<table class="nobctr">
+<tr><td>&nbsp;</td><td align="right">List price</td></tr>
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+<tr><td>Beman and Smith: Academic Algebra</td><td align="right">1.12</td></tr>
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+<tr><td>Beman and Smith: New Plane and Solid Geometry</td><td align="right">1.25</td></tr>
+<tr><td>Breckenridge, Mersereau, and Moore: Shop Problems in Mathematics</td><td align="right">1.00</td></tr>
+<tr><td>Byerly: Differential Calculus</td><td align="right">2.00</td></tr>
+<tr><td>Byerly: Integral Calculus (Second Edition. Revised and Enlarged)</td><td align="right">2.00</td></tr>
+<tr><td>Eisenhart: Differential Geometry of Curves and Surfaces</td><td align="right">4.50</td></tr>
+<tr><td>Faunce: Descriptive Geometry</td><td align="right">1.25</td></tr>
+<tr><td>Fine: College Algebra</td><td align="right">1.50</td></tr>
+<tr><td>Granville: Plane and Spherical Trigonometry and Tables</td><td align="right">1.25</td></tr>
+<tr><td>Granville: Plane and Spherical Trigonometry Without Tables</td><td align="right">1.00</td></tr>
+<tr><td>Granville: Plane Trigonometry and Tables</td><td align="right">1.00</td></tr>
+<tr><td>Granville: Logarithmic Tables</td><td align="right">.50</td></tr>
+<tr><td>Granville and Smith: Elements of Differential and Integral Calculus</td><td align="right">2.50</td></tr>
+<tr><td>Hardy: Analytic Geometry</td><td align="right">1.50</td></tr>
+<tr><td>Hardy: Elements of the Calculus</td><td align="right">1.50</td></tr>
+<tr><td>Hawkes: Advanced Algebra</td><td align="right">1.40</td></tr>
+<tr><td>Hawkes, Luby, and Touton: First Course in Algebra</td><td align="right">1.00</td></tr>
+<tr><td>Hedrick: Goursat's Mathematical Analysis. Vol. I</td><td align="right">4.00</td></tr>
+<tr><td>Hedrick and Kellogg: Applications of the Calculus to Mechanics</td><td align="right">1.25</td></tr>
+<tr><td>Hooper and Wells: Electrical Problems</td><td align="right">1.25</td></tr>
+<tr><td>Lester: Integrals of Mechanics</td><td align="right">.80</td></tr>
+<tr><td>Peirce, B. O.: Newtonian Potential Function (Third, Revised, and Enlarged Edition)</td><td align="right">2.50</td></tr>
+<tr><td>Peirce, J. M.: Elements of Logarithms</td><td align="right">.50</td></tr>
+<tr><td>Peirce, J. M.: Mathematical Tables</td><td align="right">.40</td></tr>
+<tr><td>Pierpont: Theory of Functions of Real Variables. Vol. I</td><td align="right">4.50</td></tr>
+<tr><td>Shepard: Problems in the Strength of Materials</td><td align="right">1.25</td></tr>
+<tr><td>Slocum and Hancock: Textbook on the Strength of Materials</td><td align="right">3.00</td></tr>
+<tr><td>Smith and Gale: Elements of Analytic Geometry</td><td align="right">2.00</td></tr>
+<tr><td>Smith and Gale: Introduction to Analytic Geometry</td><td align="right">1.25</td></tr>
+<tr><td>Smith and Longley: Theoretical Mechanics</td><td align="right">2.50</td></tr>
+<tr><td>Taylor: Elements of the Calculus (Revised and Enlarged Edition)</td><td align="right">2.00</td></tr>
+<tr><td>Taylor: Plane and Spherical Trigonometry</td><td align="right">1.00</td></tr>
+<tr><td>Taylor: Plane Trigonometry</td><td align="right">.75</td></tr>
+<tr><td>Taylor and Puryear: Elements of Plane and Spherical Trigonometry</td><td align="right">1.25</td></tr>
+<tr><td>Wentworth: Advanced Arithmetic</td><td align="right">1.00</td></tr>
+<tr><td>Wentworth: Analytic Geometry</td><td align="right">1.25</td></tr>
+<tr><td>Wentworth: College Algebra (Revised Edition)</td><td align="right">1.50</td></tr>
+<tr><td>Wentworth: Elementary Algebra</td><td align="right">1.12</td></tr>
+<tr><td>Wentworth: Higher Algebra</td><td align="right">1.40</td></tr>
+<tr><td>Wentworth: New School Algebra</td><td align="right">1.12</td></tr>
+<tr><td>Wentworth: Plane and Solid Geometry (Revised Edition)</td><td align="right">1.25</td></tr>
+<tr><td>Wentworth: Plane Geometry (Revised Edition); Solid Geometry (Revised Edition), each</td><td align="right">.75</td></tr>
+<tr><td>Wentworth: Trigonometries (Second Revised Editions)</td></tr>
+<tr><td align="center">(For list, see Descriptive Catalogue)</td></tr>
+<tr><td>Woods and Bailey: A Course In Mathematics</td></tr>
+<tr><td> &nbsp; Volume I</td><td align="right">2.25</td></tr>
+<tr><td> &nbsp; Volume II</td><td align="right">2.25</td></tr>
+</table>
+
+<hr class="full" >
+
+<h2>GINN AND COMPANY <span class="sc">Publishers</span></h2>
+
+<hr class="full" >
+
+<h3>Notes</h3>
+
+<div class="note">
+  <p><a name="Nt_1" href="#NtA_1">[1]</a> "<i>Discipulus.</i> Quis primus
+  invenit numerum apud Hebræos et Ægyptios? <i>Magister.</i> Abraham primus
+  invenit numerum apud Hebræos, deinde Moses; et Abraham tradidit istam
+  scientiam numeri ad Ægyptios, et docuit eos: deinde Josephus." [Bede,
+  <i>De computo dialogus</i> (doubtfully assigned to him), <i>Opera
+  omnia</i>, Paris, 1862, Vol. I, p. 650.]</p>
+
+  <p>"Alii referunt ad Ph&oelig;nices inventores arithmeticæ, propter
+  eandem commerciorum caussam: Alii ad Indos: Ioannes de Sacrobosco, cujus
+  sepulchrum est Lutetiæ in comitio Maturinensi, refert ad Arabes." [Ramus,
+  <i>Arithmeticæ libri dvo</i>, Basel, 1569, p. 112.]</p>
+
+  <p>Similar notes are given by Peletarius in his commentary on the
+  arithmetic of Gemma Frisius (1563 ed., fol. 77), and in his own work
+  (1570 Lyons ed., p. 14): "La valeur des Figures commence au coste dextre
+  tirant vers le coste senestre: au rebours de notre maniere d'escrire par
+  ce que la premiere prattique est venue des Chaldees: ou des Pheniciens,
+  qui ont été les premiers traffiquers de marchandise."</p>
+
+  <p><a name="Nt_2" href="#NtA_2">[2]</a> Maximus Planudes (c. 1330) states
+  that "the nine symbols come from the Indians." [Wäschke's German
+  translation, Halle, 1878, p. 3.] Willichius speaks of the "Zyphræ
+  Indicæ," in his <i>Arithmeticæ libri tres</i> (Strasburg, 1540, p. 93),
+  and Cataneo of "le noue figure de gli Indi," in his <i>Le pratiche delle
+  dve prime mathematiche</i> (Venice, 1546, fol. 1). Woepcke is not
+  correct, therefore, in saying ("Mémoire sur la propagation des chiffres
+  indiens," hereafter referred to as <i>Propagation</i> [<i>Journal
+  Asiatique</i>, Vol. I (6), 1863, p. 34]) that Wallis (<i>A Treatise on
+  Algebra, both historical and practical</i>, London, 1685, p. 13, and
+  <i>De algebra tractatus</i>, Latin edition in his <i>Opera omnia</i>,
+  1693, Vol. II, p. 10) was one of the first to give the Hindu origin.</p>
+
+  <p><a name="Nt_3" href="#NtA_3">[3]</a> From the 1558 edition of <i>The
+  Grovnd of Artes</i>, fol. C, 5. Similarly Bishop Tonstall writes: "Qui a
+  Chaldeis primum in finitimos, deinde in omnes pene gentes fluxit....
+  Numerandi artem a Chaldeis esse profectam: qui dum scribunt, a dextra
+  incipiunt, et in leuam progrediuntur." [<i>De arte supputandi</i>,
+  London, 1522, fol. B, 3.] Gemma Frisius, the great continental rival of
+  Recorde, had the same idea: "Primùm autem appellamus dexterum locum, eo
+  quòd haec ars vel à Chaldæis, vel ab Hebræis ortum habere credatur, qui
+  etiam eo ordine scribunt"; but this refers more evidently to the Arabic
+  numerals. [<i>Arithmeticæ practicæ methodvs facilis</i>, Antwerp, 1540,
+  fol. 4 of the 1563 ed.] Sacrobosco (c. 1225) mentions the same thing.
+  Even the modern Jewish writers claim that one of their scholars,
+  M&#x101;sh&#x101;ll&#x101;h (c. 800), introduced them to the Mohammedan
+  world. [C. Levias, <i>The Jewish Encyclopedia</i>, New York, 1905, Vol.
+  IX, p. 348.]</p>
+
+  <p><a name="Nt_4" href="#NtA_4">[4]</a> "... &amp; que esto fu trouato di
+  fare da gli Arabi con diece figure." [<i>La prima parte del general
+  trattato di nvmeri, et misvre</i>, Venice, 1556, fol. 9 of the 1592
+  edition.]</p>
+
+  <p><a name="Nt_5" href="#NtA_5">[5]</a> "Vom welchen Arabischen auch disz
+  Kunst entsprungen ist." [<i>Ain nerv geordnet Rechenbiechlin</i>,
+  Augsburg, 1514, fol. 13 of the 1531 edition. The printer used the letters
+  <i>rv</i> for <i>w</i> in "new" in the first edition, as he had no
+  <i>w</i> of the proper font.]</p>
+
+  <p><a name="Nt_6" href="#NtA_6">[6]</a> Among them Glareanus:
+  "Characteres simplices sunt nouem significatiui, ab Indis usque, siue
+  Chaldæis asciti .1.2.3.4.5.6.7.8.9. Est item unus .0 circulus, qui nihil
+  significat." [<i>De VI. Arithmeticae practicae speciebvs</i>, Paris,
+  1539, fol. 9 of the 1543 edition.]</p>
+
+  <p><a name="Nt_7" href="#NtA_7">[7]</a> "Barbarische oder gemeine
+  Ziffern." [Anonymous, <i>Das Einmahl Eins cum notis variorum</i>,
+  Dresden, 1703, p. 3.] So Vossius (<i>De universae matheseos natura et
+  constitutione liber</i>, Amsterdam, 1650, p. 34) calls them "Barbaras
+  numeri notas." The word at that time was possibly synonymous with
+  Arabic.</p>
+
+  <p><a name="Nt_8" href="#NtA_8">[8]</a> His full name was
+  &#x201B;Ab&#x16B; &#x201B;Abdall&#x101;h <span class="special"
+  title="Mohammed">Mo&#x1E25;ammed</span> ibn M&#x16B;s&#x101;
+  al-Khow&#x101;razm&#x12B;. He was born in Khow&#x101;rezm, "the
+  lowlands," the country about the present Khiva and bordering on the Oxus,
+  and lived at Bagdad under the caliph al-M&#x101;m&#x16B;n. He died
+  probably between 220 and 230 of the Mohammedan era, that is, between 835
+  and 845 <span class="scac">A.D.</span>, although some put the date as
+  early as 812. The best account of this great scholar may be found in an
+  article by C. Nallino, "<span class="special"
+  title="Al-Huwarizmi">Al-&#x1E2A;uw&#x101;rizm&#x12B;</span>" in the
+  <i>Atti della R. Accad. dei Lincei</i>, Rome, 1896. See also
+  <i>Verhandlungen des 5. Congresses der Orientalisten</i>, Berlin, 1882,
+  Vol. II, p. 19; W. Spitta-Bey in the <i>Zeitschrift der deutschen
+  Morgenländ. Gesellschaft</i>, Vol. XXXIII, p. 224; Steinschneider in the
+  <i>Zeitschrift der deutschen Morgenländ. Gesellschaft</i>, Vol. L, p.
+  214; Treutlein in the <i>Abhandlungen zur Geschichte der Mathematik</i>,
+  Vol. I, p. 5; Suter, "Die Mathematiker und Astronomen der Araber und ihre
+  Werke," <i>Abhandlungen zur Geschichte der Mathematik</i>, Vol. X,
+  Leipzig, 1900, p. 10, and "Nachträge," in Vol. XIV, p. 158; Cantor,
+  <i>Geschichte der Mathematik</i>, Vol. I, 3d ed., pp. 712-733 etc.; F.
+  Woepcke in <i>Propagation</i>, p. 489. So recently has he become known
+  that Heilbronner, writing in 1742, merely mentions him as "Ben-Musa,
+  inter Arabes celebris Geometra, scripsit de figuris planis &amp;
+  sphericis." [<i>Historia matheseos universæ</i>, Leipzig, 1742, p.
+  438.]</p>
+
+  <p>In this work most of the Arabic names will be transliterated
+  substantially as laid down by Suter in his work <i>Die Mathematiker</i>
+  etc., except where this violates English pronunciation. The scheme of
+  pronunciation of oriental names is set forth in the preface.</p>
+
+  <p><a name="Nt_9" href="#NtA_9">[9]</a> Our word <i>algebra</i> is from
+  the title of one of his works, Al-jabr wa'l-muq&#x101;balah, Completion
+  and Comparison. The work was translated into English by F. Rosen, London,
+  1831, and treated in <i>L'Algèbre d'al-Kh&#x101;rizmi et les méthodes
+  indienne et grecque</i>, Léon Rodet, Paris, 1878, extract from the
+  <i>Journal Asiatique</i>. For the derivation of the word <i>algebra</i>,
+  see Cossali, <i>Scritti Inediti</i>, pp. 381-383, Rome, 1857; Leonardo's
+  <i>Liber Abbaci</i> (1202), p. 410, Rome, 1857; both published by B.
+  Boncompagni. "Almuchabala" also was used as a name for algebra.</p>
+
+  <p><a name="Nt_10" href="#NtA_10">[10]</a> This learned scholar, teacher
+  of O'Creat who wrote the <i>Helceph</i> ("<i>Prologus N. Ocreati in
+  Helceph ad Adelardum Batensem magistrum suum</i>"), studied in Toledo,
+  learned Arabic, traveled as far east as Egypt, and brought from the
+  Levant numerous manuscripts for study and translation. See Henry in the
+  <i>Abhandlungen zur Geschichte der Mathematik</i>, Vol. III, p. 131;
+  Woepcke in <i>Propagation</i>, p. 518.</p>
+
+  <p><a name="Nt_11" href="#NtA_11">[11]</a> The title is <i>Algoritmi de
+  numero Indorum</i>. That he did not make this translation is asserted by
+  Eneström in the <i>Bibliotheca Mathematica</i>, Vol. I (3), p. 520.</p>
+
+  <p><a name="Nt_12" href="#NtA_12">[12]</a> Thus he speaks "de numero
+  indorum per .IX. literas," and proceeds: "Dixit algoritmi: Cum uidissem
+  yndos constituisse .IX. literas in uniuerso numero suo, propter
+  dispositionem suam quam posuerunt, uolui patefacere de opera quod fit per
+  eas aliquid quod esset leuius discentibus, si deus uoluerit."
+  [Boncompagni, <i>Trattati d'Aritmetica</i>,  Rome, 1857.] Discussed by F.
+  Woepcke, <i>Sur l'introduction de l'arithmétique indienne en
+  Occident</i>, Rome, 1859.</p>
+
+  <p><a name="Nt_13" href="#NtA_13">[13]</a> Thus in a commentary by
+  &#x201B;Al&#x12B; ibn Ab&#x12B; Bekr ibn al-Jam&#x101;l <span
+  class="special" title="al-Ansari">al-An&#x1E63;&#x101;r&#x12B;</span>
+  al-Mekk&#x12B; on a treatise on &#x121;ob&#x101;r arithmetic (explained
+  later) called <i>Al-murshidah</i>, found by Woepcke in Paris
+  (<i>Propagation</i>, p. 66), there is mentioned the fact that there are
+  "nine Indian figures" and "a second kind of Indian figures ... although
+  these are the figures of the &#x121;ob&#x101;r writing." So in a
+  commentary by <span class="special" title="Hosein ibn Mohammed al-Mahalli"
+  >&#x1E24;osein ibn Mo&#x1E25;ammed al-Ma&#x1E25;all&#x12B;</span> (died
+  in 1756) on the <i><span class="special" title="Mokhtasar fi`ilm el-hisab"
+  >Mokhta&#x1E63;ar f&#x12B;&#x201B;ilm el-&#x1E25;is&#x101;b</span></i>
+  (Extract from Arithmetic) by &#x201B;Abdalq&#x101;dir ibn
+  &#x201B;Al&#x12B; al-Sakh&#x101;w&#x12B; (died c. 1000) it is related
+  that "the preface treats of the forms of the figures of Hindu signs, such
+  as were established by the Hindu nation." [Woepcke, <i>Propagation</i>,
+  p. 63.]</p>
+
+  <p><a name="Nt_14" href="#NtA_14">[14]</a> See also Woepcke,
+  <i>Propagation</i>, p. 505. The origin is discussed at much length by G.
+  R. Kaye, "Notes on Indian Mathematics.&mdash;Arithmetical Notation,"
+  <i>Journ. and Proc. of the Asiatic Soc. of Bengal</i>, Vol. III, 1907, p.
+  489.</p>
+
+  <p><a name="Nt_15" href="#NtA_15">[15]</a> <i>Alberuni's India</i>,
+  Arabic version, London, 1887; English translation, ibid., 1888.</p>
+
+  <p><a name="Nt_16" href="#NtA_16">[16]</a> <i>Chronology of Ancient
+  Nations</i>, London, 1879. Arabic and English versions, by C. E.
+  Sachau.</p>
+
+  <p><a name="Nt_17" href="#NtA_17">[17]</a> <i>India</i>, Vol. I, chap.
+  xvi.</p>
+
+  <p><a name="Nt_18" href="#NtA_18">[18]</a> The Hindu name for the symbols
+  of the decimal place system.</p>
+
+  <p><a name="Nt_19" href="#NtA_19">[19]</a> Sachau's English edition of
+  the <i>Chronology</i>, p. 64.</p>
+
+  <p><a name="Nt_20" href="#NtA_20">[20]</a> <i>Littérature arabe</i>, Cl.
+  Huart, Paris, 1902.</p>
+
+  <p><a name="Nt_21" href="#NtA_21">[21]</a> Huart, <i>History of Arabic
+  Literature</i>, English ed., New York, 1903, p. 182 seq.</p>
+
+  <p><a name="Nt_22" href="#NtA_22">[22]</a>
+  Al-Mas&#x201B;&#x16B;d&#x12B;'s <i>Meadows of Gold</i>, translated in
+  part by Aloys Sprenger, London, 1841; <i>Les prairies d'or</i>, trad. par
+  C. Barbier de Meynard et Pavet de Courteille, Vols. I to IX, Paris,
+  1861-1877.</p>
+
+  <p><a name="Nt_23" href="#NtA_23">[23]</a> <i>Les prairies d'or</i>, Vol.
+  VIII, p. 289 seq.</p>
+
+  <p><a name="Nt_24" href="#NtA_24">[24]</a> <i>Essays</i>, Vol. II, p.
+  428.</p>
+
+  <p><a name="Nt_25" href="#NtA_25">[25]</a> Loc. cit., p. 504.</p>
+
+  <p><a name="Nt_26" href="#NtA_26">[26]</a> <i>Matériaux pour servir à
+  l'histoire comparée des sciences mathématiques chez les Grecs et les
+  Orientaux</i>, 2 vols., Paris, 1845-1849, pp. 438-439.</p>
+
+  <p><a name="Nt_27" href="#NtA_27">[27]</a> He made an exception, however,
+  in favor of the numerals, loc. cit., Vol. II, p. 503.</p>
+
+  <p><a name="Nt_28" href="#NtA_28">[28]</a> <i>Bibliotheca Arabico-Hispana
+  Escurialensis</i>, Madrid, 1760-1770, pp. 426-427.</p>
+
+  <p><a name="Nt_29" href="#NtA_29">[29]</a> The author, <span
+  class="special" title="Ibn al-Qifti">Ibn al-Qif&#x1E6D;&#x12B;</span>,
+  flourished <span class="scac">A.D.</span> 1198 [Colebrooke, loc. cit.,
+  note Vol. II, p. 510].</p>
+
+  <p><a name="Nt_30" href="#NtA_30">[30]</a> "Liber Artis Logisticae à
+  Mohamado Ben Musa <i>Alkhuarezmita</i> exornatus, qui ceteros omnes
+  brevitate methodi ac facilitate praestat, Indorum que in praeclarissimis
+  inventis ingenium &amp; acumen ostendit." [Casiri, loc. cit., p.
+  427.]</p>
+
+  <p><a name="Nt_31" href="#NtA_31">[31]</a> Maçoudi, <i>Le livre de
+  l'avertissement et de la révision</i>. Translation by B. Carra de Vaux,
+  Paris, 1896.</p>
+
+  <p><a name="Nt_32" href="#NtA_32">[32]</a> Verifying the hypothesis of
+  Woepcke, <i>Propagation</i>, that the Sindhind included a treatment of
+  arithmetic.</p>
+
+  <p><a name="Nt_33" href="#NtA_33">[33]</a> <span class="special"
+  title="Ahmed ibn `Abdallah">A&#x1E25;med ibn
+  &#x201B;Abdall&#x101;h</span>, Suter, <i>Die Mathematiker</i>, etc., p.
+  12.</p>
+
+  <p><a name="Nt_34" href="#NtA_34">[34]</a> <i>India</i>, Vol. II, p.
+  15.</p>
+
+  <p><a name="Nt_35" href="#NtA_35">[35]</a> See H. Suter, "Das
+  Mathematiker-Verzeichniss im Fihrist," <i>Abhandlungen zur Geschichte der
+  Mathematik</i>, Vol. VI, Leipzig, 1892. For further references to early
+  Arabic writers the reader is referred to H. Suter, <i>Die Mathematiker
+  und Astronomen der Araber und ihre Werke</i>. Also "Nachträge und
+  Berichtigungen" to the same (<i>Abhandlungen</i>,  Vol. XIV, 1902, pp.
+  155-186).</p>
+
+  <p><a name="Nt_36" href="#NtA_36">[36]</a> Suter, loc. cit., note 165,
+  pp. 62-63.</p>
+
+  <p><a name="Nt_37" href="#NtA_37">[37]</a> "Send Ben Ali,... tùm
+  arithmetica scripta maximè celebrata, quae publici juris fecit." [Loc.
+  cit., p. 440.]</p>
+
+  <p><a name="Nt_38" href="#NtA_38">[38]</a> <i>Scritti di Leonardo
+  Pisano</i>, Vol. I, <i>Liber Abbaci</i> (1857); Vol. II, <i>Scritti</i>
+  (1862); published by Baldassarre Boncompagni, Rome. Also <i>Tre Scritti
+  Inediti</i>, and <i>Intorno ad Opere di Leonardo Pisano</i>, Rome,
+  1854.</p>
+
+  <p><a name="Nt_39" href="#NtA_39">[39]</a> "Ubi ex mirabili magisterio in
+  arte per novem figuras indorum introductus" etc. In another place, as a
+  heading to a separate division, he writes, "De cognitione novem figurarum
+  yndorum" etc. "Novem figure indorum he sunt 9 8 7 6 5 4 3 2 1."</p>
+
+  <p><a name="Nt_40" href="#NtA_40">[40]</a> See <i>An Ancient English
+  Algorism</i>, by David Eugene Smith, in <i>Festschrift Moritz Cantor</i>,
+  Leipzig, 1909. See also Victor Mortet, "Le plus ancien traité francais
+  d'algorisme," <i>Bibliotheca Mathematica</i>, Vol. IX (3), pp. 55-64.</p>
+
+  <p><a name="Nt_41" href="#NtA_41">[41]</a> These are the two opening
+  lines of the <i>Carmen de Algorismo</i> that the anonymous author is
+  explaining. They should read as follows:</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Haec algorismus ars praesens dicitur, in qua</p>
+      <p>Talibus Indorum fruimur bis quinque figuris.</p>
+    </div>
+  </div>
+  <p>What follows is the translation.</p>
+
+  <p><a name="Nt_42" href="#NtA_42">[42]</a> Thibaut, <i>Astronomie,
+  Astrologie und Mathematik</i>, Strassburg, 1899.</p>
+
+  <p><a name="Nt_43" href="#NtA_43">[43]</a> Gustave Schlegel,
+  <i>Uranographie chinoise ou preuves directes que l'astronomie primitive
+  est originaire de la Chine, et qu'elle a été empruntée par les anciens
+  peuples occidentaux à la sphère chinoise; ouvrage accompagné d'un atlas
+  céleste chinois et grec</i>, The Hague and Leyden, 1875.</p>
+
+  <p><a name="Nt_44" href="#NtA_44">[44]</a> E. W. Hopkins, <i>The
+  Religions of India</i>, Boston, 1898, p. 7.</p>
+
+  <p><a name="Nt_45" href="#NtA_45">[45]</a> R. C. Dutt, <i>History of
+  India</i>, London, 1906.</p>
+
+  <p><a name="Nt_46" href="#NtA_46">[46]</a> W. D. Whitney, <i>Sanskrit
+  Grammar</i>, 3d ed., Leipzig, 1896.</p>
+
+  <p><a name="Nt_47" href="#NtA_47">[47]</a> "Das
+  &#x100;pastamba-&#x15A;ulba-S&#x16B;tra," <i>Zeitschrift der deutschen
+  Morgenländischen Gesellschaft</i>, Vol. LV, p. 543, and Vol. LVI, p.
+  327.</p>
+
+  <p><a name="Nt_48" href="#NtA_48">[48]</a> <i>Geschichte der Math.</i>,
+  Vol. I, 2d ed., p. 595.</p>
+
+  <p><a name="Nt_49" href="#NtA_49">[49]</a> L. von Schroeder,
+  <i>Pythagoras und die Inder</i>, Leipzig, 1884; H. Vogt, "Haben die alten
+  Inder den Pythagoreischen Lehrsatz und das Irrationale gekannt?"
+  <i>Bibliotheca Mathematica</i>, Vol. VII (3), pp. 6-20; A. Bürk, loc.
+  cit.; Max Simon, <i>Geschichte der Mathematik im Altertum</i>, Berlin,
+  1909, pp. 137-165; three S&#x16B;tras are translated in part by Thibaut,
+  <i>Journal of the Asiatic Society of Bengal</i>, 1875, and one appeared
+  in <i>The Pandit</i>, 1875; Beppo Levi, "Osservazioni e congetture sopra
+  la geometria degli indiani," <i>Bibliotheca Mathematica</i>, Vol. IX (3),
+  1908, pp. 97-105.</p>
+
+  <p><a name="Nt_50" href="#NtA_50">[50]</a> Loc. cit.; also <i>Indiens
+  Literatur und Cultur</i>, Leipzig, 1887.</p>
+
+  <p><a name="Nt_51" href="#NtA_51">[51]</a> It is generally agreed that
+  the name of the river Sindhu, corrupted by western peoples to Hindhu,
+  Indos, Indus, is the root of Hindustan and of India. Reclus, <i>Asia</i>,
+  English ed., Vol. III, p. 14.</p>
+
+  <p><a name="Nt_52" href="#NtA_52">[52]</a> See the comments of Oppert,
+  <i>On the Original Inhabitants of <span class="special"
+  title="Bharatavarsa">Bharatavar&#x1E63;a</span> or India</i>, London,
+  1893, p. 1.</p>
+
+  <p><a name="Nt_53" href="#NtA_53">[53]</a> A. Hillebrandt,
+  <i>Alt-Indien</i>, Breslau, 1899, p. 111. Fragmentary records relate that
+  Kh&#x101;ravela, king of <span class="special"
+  title="Kalinga">Kali&#x1E45;ga</span>, learned as a boy
+  <i>lekh&#x101;</i> (writing), <i><span class="special"
+  title="ganana">ga&#x1E47;an&#x101;</span></i> (reckoning), and
+  <i>r&#x16B;pa</i> (arithmetic applied to monetary affairs and
+  mensuration), probably in the 5th century <span class="scac">B.C.</span>
+  [Bühler, <i>Indische Palaeographie</i>, Strassburg, 1896, p. 5.]</p>
+
+  <p><a name="Nt_54" href="#NtA_54">[54]</a> R. C. Dutt, <i>A History of
+  Civilization in Ancient India</i>, London, 1893, Vol. I, p. 174.</p>
+
+  <p><a name="Nt_55" href="#NtA_55">[55]</a> The Buddha. The date of his
+  birth is uncertain. Sir Edwin Arnold put it c. 620 <span
+  class="scac">B.C.</span></p>
+
+  <p><a name="Nt_56" href="#NtA_56">[56]</a> I.e. 100·10<sup>7</sup>.</p>
+
+  <p><a name="Nt_57" href="#NtA_57">[57]</a> There is some uncertainty
+  about this limit.</p>
+
+  <p><a name="Nt_58" href="#NtA_58">[58]</a> This problem deserves more
+  study than has yet been given it. A beginning may be made with Comte
+  Goblet d'Alviella, <i>Ce que l'Inde doit à la Grèce</i>, Paris, 1897, and
+  H. G. Keene's review, "The Greeks in India," in the <i>Calcutta
+  Review</i>, Vol. CXIV, 1902, p. 1. See also F. Woepeke,
+  <i>Propagation</i>, p. 253; G. R. Kaye, loc. cit., p. 475 seq., and "The
+  Source of Hindu Mathematics," <i>Journal of the Royal Asiatic
+  Society</i>, July, 1910, pp. 749-760; G. Thibaut, <i>Astronomie,
+  Astrologie und Mathematik</i>, pp. 43-50 and 76-79. It will be discussed
+  more fully in Chapter VI.</p>
+
+  <p><a name="Nt_59" href="#NtA_59">[59]</a> I.e. to 100,000. The lakh is
+  still the common large unit in India, like the myriad in ancient Greece
+  and the million in the West.</p>
+
+  <p><a name="Nt_60" href="#NtA_60">[60]</a> This again suggests the
+  <i>Psammites</i>, or <i>De harenae numero</i> as it is called in the 1544
+  edition of the <i>Opera</i> of Archimedes, a work in which the great
+  Syracusan proposes to show to the king "by geometric proofs which you can
+  follow, that the numbers which have been named by us ... are sufficient
+  to exceed not only the number of a sand-heap as large as the whole earth,
+  but one as large as the universe." For a list of early editions of this
+  work see D. E. Smith, <i>Rara Arithmetica</i>, Boston, 1909, p. 227.</p>
+
+  <p><a name="Nt_61" href="#NtA_61">[61]</a> I.e. the Wise.</p>
+
+  <p><a name="Nt_62" href="#NtA_62">[62]</a> Sir Monier Monier-Williams,
+  <i>Indian Wisdom</i>, 4th ed., London, 1893, pp. 144, 177. See also J. C.
+  Marshman, <i>Abridgment of the History of India</i>, London, 1893, p.
+  2.</p>
+
+  <p><a name="Nt_63" href="#NtA_63">[63]</a> For a list and for some
+  description of these works see R. C. Dutt, <i>A History of Civilization
+  in Ancient India</i>, Vol. II, p. 121.</p>
+
+  <p><a name="Nt_64" href="#NtA_64">[64]</a> Professor Ramkrishna Gopal
+  Bhandarkar fixes the date as the fifth century <span
+  class="scac">B.C.</span> ["Consideration of the Date of the
+  Mah&#x101;bh&#x101;rata," in the <i>Journal of the Bombay Branch of the
+  R. A. Soc.</i>, Bombay, 1873, Vol. X, p. 2.].</p>
+
+  <p><a name="Nt_65" href="#NtA_65">[65]</a> Marshman, loc. cit., p. 2.</p>
+
+  <p><a name="Nt_66" href="#NtA_66">[66]</a> A. C. Burnell, <i>South Indian
+  Palæography</i>, 2d ed., London, 1878, p. 1, seq.</p>
+
+  <p><a name="Nt_67" href="#NtA_67">[67]</a> This extensive subject of
+  palpable arithmetic, essentially the history of the abacus, deserves to
+  be treated in a work by itself.</p>
+
+  <p><a name="Nt_68" href="#NtA_68">[68]</a> The following are the leading
+  sources of information upon this subject: G. Bühler, <i>Indische
+  Palaeographie</i>, particularly chap. vi; A. C. Burnell, <i>South Indian
+  Palæography</i>, 2d ed., London, 1878, where tables of the various Indian
+  numerals are given in Plate XXIII; E. C. Bayley, "On the Genealogy of
+  Modern Numerals," <i>Journal of the Royal Asiatic Society</i>, Vol. XIV,
+  part 3, and Vol. XV, part 1, and reprint, London, 1882; I. Taylor, in
+  <i>The Academy</i>, January 28, 1882, with a repetition of his argument
+  in his work <i>The Alphabet</i>, London, 1883, Vol. II, p. 265, based on
+  Bayley; G. R. Kaye, loc. cit., in some respects one of the most critical
+  articles thus far published; J. C. Fleet, <i>Corpus inscriptionum
+  Indicarum</i>, London, 1888, Vol. III, with facsimiles of many Indian
+  inscriptions, and <i>Indian Epigraphy</i>, Oxford, 1907, reprinted from
+  the <i>Imperial Gazetteer of India</i>, Vol. II, pp. 1-88, 1907; G.
+  Thibaut, loc. cit., <i>Astronomie</i> etc.; R. Caldwell, <i>Comparative
+  Grammar of the Dravidian Languages</i>, London, 1856, p. 262 seq.; and
+  <i>Epigraphia Indica</i> (official publication of the government of
+  India), Vols. I-IX. Another work of Bühler's, <i>On the Origin of the
+  Indian Br&#x101;hma Alphabet</i>, is also of value.</p>
+
+  <p><a name="Nt_69" href="#NtA_69">[69]</a> The earliest work on the
+  subject was by James Prinsep, "On the Inscriptions of Piyadasi or
+  A&#x15B;oka," etc., <i>Journal of the Asiatic Society of Bengal</i>,
+  1838, following a preliminary suggestion in the same journal in 1837. See
+  also "A&#x15B;oka Notes," by V. A. Smith, <i>The Indian Antiquary</i>,
+  Vol. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, pp. 151-159, June, 1909;
+  <i>The Early History of India</i>, 2d ed., Oxford, 1908, p. 154; J. F.
+  Fleet, "The Last Words of A&#x15B;oka," <i>Journal of the Royal Asiatic
+  Society</i>, October, 1909, pp. 981-1016; E. Senart, <i>Les inscriptions
+  de Piyadasi</i>, 2 vols., Paris, 1887.</p>
+
+  <p><a name="Nt_70" href="#NtA_70">[70]</a> For a discussion of the minor
+  details of this system, see Bühler, loc. cit., p. 73.</p>
+
+  <p><a name="Nt_71" href="#NtA_71">[71]</a> Julius Euting, <i>Nabatäische
+  Inschriften aus Arabien</i>, Berlin, 1885, pp. 96-97, with a table of
+  numerals.</p>
+
+  <p><a name="Nt_72" href="#NtA_72">[72]</a> For the five principal
+  theories see Bühler, loc. cit., p. 10.</p>
+
+  <p><a name="Nt_73" href="#NtA_73">[73]</a> Bayley, loc. cit., reprint p.
+  3.</p>
+
+  <p><a name="Nt_74" href="#NtA_74">[74]</a> Bühler, loc. cit.;
+  <i>Epigraphia Indica</i>, Vol. III, p. 134; <i>Indian Antiquary</i>, Vol.
+  VI, p. 155 seq., and Vol. X, p. 107.</p>
+
+  <p><a name="Nt_75" href="#NtA_75">[75]</a> Pandit Bhagav&#x101;nl&#x101;l
+  Indr&#x101;j&#x12B;, "On Ancient N&#x101;g&#x101;ri Numeration; from an
+  Inscription at N&#x101;negh&#x101;t," <i>Journal of the Bombay Branch of
+  the Royal Asiatic Society</i>, 1876, Vol. XII, p. 404.</p>
+
+  <p><a name="Nt_76" href="#NtA_76">[76]</a> Ib., p. 405. He gives also a
+  plate and an interpretation of each numeral.</p>
+
+  <p><a name="Nt_77" href="#NtA_77">[77]</a> These may be compared with
+  Bühler's drawings, loc. cit.; with Bayley, loc. cit., p. 337 and plates;
+  and with Bayley's article in the <i>Encyclopædia Britannica</i>, 9th ed.,
+  art. "Numerals."</p>
+
+  <p><a name="Nt_78" href="#NtA_78">[78]</a> E. Senart, "The Inscriptions
+  in the Caves at Nasik," <i>Epigraphia Indica</i>, Vol. VIII, pp. 59-96;
+  "The Inscriptions in the Cave at Karle," <i>Epigraphia Indica</i>, Vol.
+  VII, pp. 47-74; Bühler, <i>Palaeographie</i>, Tafel IX.</p>
+
+  <p><a name="Nt_79" href="#NtA_79">[79]</a> See Fleet, loc. cit. See also
+  T. Benfey, <i>Sanskrit Grammar</i>, London, 1863, p. 217; M. R. Kále,
+  <i>Higher Sanskrit Grammar</i>, 2d ed., Bombay, 1898, p. 110, and other
+  authorities as cited.</p>
+
+  <p><a name="Nt_80" href="#NtA_80">[80]</a> <span class="special"
+  title="Kharosthi">Kharo&#x1E63;&#x1E6D;h&#x12B;</span> numerals,
+  A&#x15B;oka inscriptions, c. 250 <span class="scac">B.C.</span> Senart,
+  <i>Notes d'épigraphie indienne</i>. Given by Bühler, loc. cit., Tafel
+  I.</p>
+
+  <p><a name="Nt_81" href="#NtA_81">[81]</a> Same, &#x15A;aka inscriptions,
+  probably of the first century <span class="scac">B.C.</span> Senart, loc.
+  cit.; Bühler, loc. cit.</p>
+
+  <p><a name="Nt_82" href="#NtA_82">[82]</a> Br&#x101;hm&#x12B; numerals,
+  A&#x15B;oka inscriptions, c. 250 <span class="scac">B.C.</span> <i>Indian
+  Antiquary</i>, Vol. VI, p. 155 seq.</p>
+
+  <p><a name="Nt_83" href="#NtA_83">[83]</a> Same, N&#x101;n&#x101;
+  Gh&#x101;t inscriptions, c. 150 <span class="scac">B.C.</span>
+  Bhagav&#x101;nl&#x101;l Indr&#x101;j&#x12B;, <i>On Ancient
+  N&#x101;gar&#x12B; Numeration</i>, loc. cit. Copied from a squeeze of the
+  original.</p>
+
+  <p><a name="Nt_84" href="#NtA_84">[84]</a> Same, Nasik inscription, c.
+  100 <span class="scac">B.C.</span> Burgess, <i>Archeological Survey
+  Report, Western India</i>; Senart, <i>Epigraphia Indica</i>, Vol. VII,
+  pp. 47-79, and Vol. VIII, pp. 59-96.</p>
+
+  <p><a name="Nt_85" href="#NtA_85">[85]</a> <span class="special"
+  title="Ksatrapa">K&#x1E63;atrapa</span> coins, c. 200 <span
+  class="scac">A.D.</span> <i>Journal of the Royal Asiatic Society</i>,
+  1890, p. 639.</p>
+
+  <p><a name="Nt_86" href="#NtA_86">[86]</a> <span class="special"
+  title="Kusana">Ku&#x1E63;ana</span> inscriptions, c. 150 <span
+  class="scac">A.D.</span> <i>Epigraphia Indica</i>, Vol. I, p. 381, and
+  Vol. II, p. 201.</p>
+
+  <p><a name="Nt_87" href="#NtA_87">[87]</a> Gupta Inscriptions, c. 300
+  <span class="scac">A.D.</span> to 450 <span class="scac">A.D.</span>
+  Fleet, loc. cit., Vol. III.</p>
+
+  <p><a name="Nt_88" href="#NtA_88">[88]</a> Valhab&#x12B;, c. 600 <span
+  class="scac">A.D.</span> <i>Corpus</i>, Vol. III.</p>
+
+  <p><a name="Nt_89" href="#NtA_89">[89]</a> Bendall's Table of Numerals,
+  in <i>Cat. Sansk. Budd. MSS.</i>, British Museum.</p>
+
+  <p><a name="Nt_90" href="#NtA_90">[90]</a> <i>Indian Antiquary</i>, Vol.
+  XIII, 120; <i>Epigraphia Indica</i>, Vol. III, 127 ff.</p>
+
+  <p><a name="Nt_91" href="#NtA_91">[91]</a> Fleet, loc. cit.</p>
+
+  <p><a name="Nt_92" href="#NtA_92">[92]</a> Bayley, loc. cit., p. 335.</p>
+
+  <p><a name="Nt_93" href="#NtA_93">[93]</a> From a copper plate of 493
+  <span class="scac">A.D.</span>, found at
+  K&#x101;r&#x12B;tal&#x101;&#x12B;, Central India. [Fleet, loc. cit.,
+  Plate XVI.] It should be stated, however, that many of these copper
+  plates, being deeds of property, have forged dates so as to give the
+  appearance of antiquity of title. On the other hand, as Colebrooke long
+  ago pointed out, a successful forgery has to imitate the writing of the
+  period in question, so that it becomes evidence well worth considering,
+  as shown in Chapter III.</p>
+
+  <p><a name="Nt_94" href="#NtA_94">[94]</a> From a copper plate of 510
+  <span class="scac">A.D.</span>, found at Majhgaw&#x101;in, Central India.
+  [Fleet, loc. cit., Plate XIV.]</p>
+
+  <p><a name="Nt_95" href="#NtA_95">[95]</a> From an inscription of 588
+  <span class="scac">A.D.</span>, found at B&#x14D;dh-Gay&#x101;, Bengal
+  Presidency. [Fleet, loc. cit., Plate XXIV.]</p>
+
+  <p><a name="Nt_96" href="#NtA_96">[96]</a> From a copper plate of 571
+  <span class="scac">A.D.</span>, found at M&#x101;liy&#x101;, Bombay
+  Presidency. [Fleet, loc. cit., Plate XXIV.]</p>
+
+  <p><a name="Nt_97" href="#NtA_97">[97]</a> From a <span class="special"
+  title="Bijayagadh">Bijayaga&#x1E0D;h</span> pillar inscription of 372
+  <span class="scac">A.D.</span> [Fleet, loc. cit., Plate XXXVI, C.]</p>
+
+  <p><a name="Nt_98" href="#NtA_98">[98]</a> From a copper plate of 434
+  <span class="scac">A.D.</span> [<i>Indian Antiquary</i>, Vol. I, p.
+  60.]</p>
+
+  <p><a name="Nt_99" href="#NtA_99">[99]</a> Gadhwa inscription, c. 417
+  <span class="scac">A.D.</span> [Fleet, loc. cit., Plate IV, D.]</p>
+
+  <p><a name="Nt_100" href="#NtA_100">[100]</a>
+  K&#x101;r&#x12B;tal&#x101;&#x12B; plate of 493 <span
+  class="scac">A.D.</span>, referred to above.</p>
+
+  <p><a name="Nt_101" href="#NtA_101">[101]</a> It seems evident that the
+  Chinese four, curiously enough called "eight in the mouth," is only a
+  cursive <a href="images/034b.png"><img src="images/034b.png"
+  class="middle" style="height:1.5ex" alt="4 vertical strokes" /></a>.</p>
+
+  <p><a name="Nt_102" href="#NtA_102">[102]</a> Chalfont, F. H., <i>Memoirs
+  of the Carnegie Museum</i>, Vol. IV, no. 1; J. Hager, <i>An Explanation
+  of the Elementary Characters of the Chinese</i>, London, 1801.</p>
+
+  <p><a name="Nt_103" href="#NtA_103">[103]</a> H. V. Hilprecht,
+  <i>Mathematical, Metrological and Chronological Tablets from the Temple
+  Library at Nippur</i>, Vol. XX, part I, of Series A, Cuneiform Texts
+  Published by the Babylonian Expedition of the University of Pennsylvania,
+  1906; A. Eisenlohr, <i>Ein altbabylonischer Felderplan</i>, Leipzig,
+  1906; Maspero, <i>Dawn of Civilization</i>, p. 773.</p>
+
+  <p><a name="Nt_104" href="#NtA_104">[104]</a> Sir H. H. Howard, "On the
+  Earliest Inscriptions from Chaldea," <i>Proceedings of the Society of
+  Biblical Archæology</i>, XXI, p. 301, London, 1899.</p>
+
+  <p><a name="Nt_105" href="#NtA_105">[105]</a> For a bibliography of the
+  principal hypotheses of this nature see Bühler, loc. cit., p. 77. Bühler
+  (p. 78) feels that of all these hypotheses that which connects the
+  Br&#x101;hm&#x12B; with the Egyptian numerals is the most plausible,
+  although he does not adduce any convincing proof. Th. Henri Martin, "Les
+  signes numéraux et l'arithmétique chez les peuples de l'antiquité et du
+  moyen âge" (being an examination of Cantor's <i>Mathematische Beiträge
+  zum Culturleben der Völker</i>), <i>Annali di matematica pura ed
+  applicata</i>, Vol. V, Rome, 1864, pp. 8, 70. Also, same author,
+  "Recherches nouvelles sur l'origine de notre système de numération
+  écrite," <i>Revue Archéologique</i>, 1857, pp. 36, 55. See also the
+  tables given later in this work.</p>
+
+  <p><a name="Nt_106" href="#NtA_106">[106]</a> <i>Journal of the Royal
+  Asiatic Society, Bombay Branch</i>, Vol. XXIII.</p>
+
+  <p><a name="Nt_107" href="#NtA_107">[107]</a> Loc. cit., reprint, Part I,
+  pp. 12, 17. Bayley's deductions are generally regarded as
+  unwarranted.</p>
+
+  <p><a name="Nt_108" href="#NtA_108">[108]</a> <i>The Alphabet</i>;
+  London, 1883, Vol. II, pp. 265, 266, and <i>The Academy</i> of Jan. 28,
+  1882.</p>
+
+  <p><a name="Nt_109" href="#NtA_109">[109]</a> Taylor, <i>The
+  Alphabet</i>, loc. cit., table on p. 266.</p>
+
+  <p><a name="Nt_110" href="#NtA_110">[110]</a> Bühler, <i>On the Origin of
+  the Indian Br&#x101;hma Alphabet</i>, Strassburg, 1898, footnote, pp. 52,
+  53.</p>
+
+  <p><a name="Nt_111" href="#NtA_111">[111]</a> Albrecht Weber, <i>History
+  of Indian Literature</i>, English ed., Boston, 1878, p. 256: "The Indian
+  figures from 1-9 are abbreviated forms of the initial letters of the
+  numerals themselves...: the zero, too, has arisen out of the first letter
+  of the word <i><span class="special"
+  title="sunya">&#x1E63;unya</span></i> (empty) (it occurs even in
+  Piñgala). It is the decimal place value of these figures which gives them
+  significance." C. Henry, "Sur l'origine de quelques notations
+  mathématiques," <i>Revue Archéologique</i>, June and July, 1879, attempts
+  to derive the Boethian forms from the initials of Latin words. See also
+  J. Prinsep, "Examination of the Inscriptions from Girnar in Gujerat, and
+  Dhauli in Cuttach," <i>Journal of the Asiatic Society of Bengal</i>,
+  1838, especially Plate XX, p. 348; this was the first work on the
+  subject.</p>
+
+  <p><a name="Nt_112" href="#NtA_112">[112]</a> Bühler,
+  <i>Palaeographie</i>, p. 75, gives the list, with the list of letters (p.
+  76) corresponding to the number symbols.</p>
+
+  <p><a name="Nt_113" href="#NtA_113">[113]</a> For a general discussion of
+  the connection between the numerals and the different kinds of alphabets,
+  see the articles by U. Ceretti, "Sulla origine delle cifre numerali
+  moderne," <i>Rivista di fisica, matematica e scienze naturali</i>, Pisa
+  and Pavia, 1909, anno X, numbers 114, 118, 119, and 120, and continuation
+  in 1910.</p>
+
+  <p><a name="Nt_114" href="#NtA_114">[114]</a> This is one of Bühler's
+  hypotheses. See Bayley, loc. cit., reprint p. 4; a good bibliography of
+  original sources is given in this work, p. 38.</p>
+
+  <p><a name="Nt_115" href="#NtA_115">[115]</a> Loc. cit., reprint, part I,
+  pp. 12, 17. See also Burnell, loc. cit., p. 64, and tables in plate
+  XXIII.</p>
+
+  <p><a name="Nt_116" href="#NtA_116">[116]</a> This was asserted by G.
+  Hager (<i>Memoria sulle cifre arabiche</i>, Milan, 1813, also published
+  in <i>Fundgruben des Orients</i>, Vienna, 1811, and in <i>Bibliothèque
+  Britannique</i>, Geneva, 1812). See also the recent article by Major
+  Charles E. Woodruff, "The Evolution of Modern Numerals from Tally Marks,"
+  <i>American Mathematical Monthly</i>, August-September, 1909. Biernatzki,
+  "Die Arithmetik der Chinesen," <i>Crelle's Journal für die reine und
+  angewandte Mathematik</i>, Vol. LII, 1857, pp. 59-96, also asserts the
+  priority of the Chinese claim for a place system and the zero, but upon
+  the flimsiest authority. Ch. de Paravey, <i>Essai sur l'origine unique et
+  hiéroglyphique des chiffres et des lettres de tous les peuples</i>,
+  Paris, 1826; G. Kleinwächter, "The Origin of the Arabic Numerals,"
+  <i>China Review</i>, Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp.
+  28-30; Biot, "Note sur la connaissance que les Chinois ont eue de la
+  valeur de position des chiffres," <i>Journal Asiatique</i>, 1839, pp.
+  497-502. A. Terrien de Lacouperie, "The Old Numerals, the Counting-Rods
+  and the Swan-Pan in China," <i>Numismatic Chronicle</i>, Vol. III (3),
+  pp. 297-340, and Crowder B. Moseley, "Numeral Characters: Theory of
+  Origin and Development," <i>American Antiquarian</i>, Vol. XXII, pp.
+  279-284, both propose to derive our numerals from Chinese characters, in
+  much the same way as is done by Major Woodruff, in the article above
+  cited.</p>
+
+  <p><a name="Nt_117" href="#NtA_117">[117]</a> The Greeks, probably
+  following the Semitic custom, used nine letters of the alphabet for the
+  numerals from 1 to 9, then nine others for 10 to 90, and further letters
+  to represent 100 to 900. As the ordinary Greek alphabet was insufficient,
+  containing only twenty-four letters, an alphabet of twenty-seven letters
+  was used.</p>
+
+  <p><a name="Nt_118" href="#NtA_118">[118]</a> <i>Institutiones
+  mathematicae</i>, 2 vols., Strassburg, 1593-1596, a somewhat rare work
+  from which the following quotation is taken:</p>
+
+  <p>"<i>Quis est harum Cyphrarum autor?</i></p>
+
+  <p>"A quibus hae usitatae syphrarum notae sint inventae: hactenus
+  incertum fuit: meo tamen iudicio, quod exiguum esse fateor: a graecis
+  librarijs (quorum olim magna fuit copia) literae Graecorum quibus veteres
+  Graeci tamquam numerorum notis sunt usi: fuerunt corruptae. vt ex his
+  licet videre.</p>
+
+  <p>"Graecorum Literae corruptae.</p>
+
+  <div class="figright" style="width:20%;">
+      <a href="images/039a.png"><img style="width:100%" src="images/039a.png"
+      alt="Graecorum Literae corruptae." title="Graecorum Literae corruptae." /></a>
+  </div>
+  <p><i>"Sed qua ratione graecorum literae ita fuerunt corruptae?</i></p>
+
+  <p>"Finxerunt has corruptas Graecorum literarum notas: vel abiectione vt
+  in nota binarij numeri, vel additione vt in ternarij, vel inuersione vt
+  in septenarij, numeri nota, nostrae notae, quibus hodie utimur: ab his
+  sola differunt elegantia, vt apparet."</p>
+
+  <p>See also Bayer, <i>Historia regni Graecorum Bactriani</i>, St.
+  Petersburg, 1788, pp. 129-130, quoted by Martin, <i>Recherches
+  nouvelles</i>, etc., loc. cit.</p>
+
+  <p><a name="Nt_119" href="#NtA_119">[119]</a> P. D. Huet, <i>Demonstratio
+  evangelica</i>, Paris, 1769, note to p. 139 on p. 647: "Ab Arabibus vel
+  ab Indis inventas esse, non vulgus eruditorum modo, sed doctissimi quique
+  ad hanc diem arbitrati sunt. Ego vero falsum id esse, merosque esse
+  Graecorum characteres aio; à librariis Graecae linguae ignaris
+  interpolatos, et diuturna scribendi consuetudine corruptos. Nam primum 1
+  apex fuit, seu virgula, nota <span title="monados" class="grk"
+  >&mu;&omicron;&nu;&#x1F71;&delta;&omicron;&sigmaf;</span>. 2, est ipsum
+  <span class="grk">&beta;</span> extremis suis truncatum. <span
+  class="grk">&gamma;</span>, si in sinistram partem inclinaveris &amp;
+  cauda mutilaveris &amp; sinistrum cornu sinistrorsum flexeris, fiet 3.
+  Res ipsa loquitur 4 ipsissimum esse <span class="grk">&Delta;</span>,
+  cujus crus sinistrum erigitur <span title="kata katheton" class="grk"
+  >&kappa;&alpha;&tau;&#x1F70;
+  &kappa;&#x1F71;&theta;&epsilon;&tau;&omicron;&nu;</span>, &amp; infra
+  basim descendit; basis vero ipsa ultra crus producta eminet. Vides quam 5
+  simile sit <span title="tôi" class="grk">&tau;&#x1FF7;</span> <a
+  href="images/040a.png"><img src="images/040a.png" class="middle"
+  style="height:2ex" alt="epsilon" /></a>; infimo tantum semicirculo, qui
+  sinistrorsum patebat, dextrorsum converso. <span title="episêmon bau" class="grk"
+  >&#x1F10;&pi;&#x1F77;&sigma;&eta;&mu;&omicron;&nu;
+  &beta;&alpha;&#x1FE6;</span> quod ita notabatur <a
+  href="images/040b.png"><img src="images/040b.png" class="middle"
+  style="height:2ex" alt="digamma" /></a>, rotundato ventre, pede detracto,
+  peperit <span title="to" class="grk">&tau;&#x1F78;</span> 6. Ex <span
+  class="grk">&Zeta;</span> basi sua mutilato, ortum est <span title="to" class="grk"
+  >&tau;&#x1F78;</span> 7. Si <span class="grk">&Eta;</span> inflexis
+  introrsum apicibus in rotundiorem &amp; commodiorem formam mutaveris,
+  exurget <span title="to" class="grk">&tau;&#x1F78;</span> 8. At 9
+  ipsissimum est <a href="images/040c.png"><img src="images/040c.png"
+  class="middle" style="height:2ex" alt="alt theta" /></a>."</p>
+
+  <p>I. Weidler, <i>Spicilegium observationum ad historiam notarum
+  numeralium</i>, Wittenberg, 1755, derives them from the Hebrew letters;
+  Dom Augustin Calmet, "Recherches sur l'origine des chiffres
+  d'arithmétique," <i>Mémoires pour l'histoire des sciences et des beaux
+  arts</i>, Trévoux, 1707 (pp. 1620-1635, with two plates), derives the
+  current symbols from the Romans, stating that they are relics of the
+  ancient "Notae Tironianae." These "notes" were part of a system of
+  shorthand invented, or at least perfected, by Tiro, a slave who was freed
+  by Cicero. L. A. Sedillot, "Sur l'origine de nos chiffres," <i>Atti dell'
+  Accademia pontificia dei nuovi Lincei</i>, Vol. XVIII, 1864-1865, pp.
+  316-322, derives the Arabic forms from the Roman numerals.</p>
+
+  <p><a name="Nt_120" href="#NtA_120">[120]</a> Athanasius Kircher,
+  <i>Arithmologia sive De abditis Numerorum, mysterijs qua origo,
+  antiquitas &amp; fabrica Numerorum exponitur</i>, Rome, 1665.</p>
+
+  <p><a name="Nt_121" href="#NtA_121">[121]</a> See Suter, <i>Die
+  Mathematiker und Astronomen der Araber</i>, p. 100.</p>
+
+  <p><a name="Nt_122" href="#NtA_122">[122]</a> "Et hi numeri sunt numeri
+  Indiani, a Brachmanis Indiae Sapientibus ex figura circuli secti
+  inuenti."</p>
+
+  <p><a name="Nt_123" href="#NtA_123">[123]</a> V. A. Smith, <i>The Early
+  History of India</i>, Oxford, 2d ed., 1908, p. 333.</p>
+
+  <p><a name="Nt_124" href="#NtA_124">[124]</a> C. J. Ball, "An Inscribed
+  Limestone Tablet from Sippara," <i>Proceedings of the Society of Biblical
+  Archæology</i>, Vol. XX, p. 25 (London, 1898). Terrien de Lacouperie
+  states that the Chinese used the circle for 10 before the beginning of
+  the Christian era. [<i>Catalogue of Chinese Coins</i>, London, 1892, p.
+  xl.]</p>
+
+  <p><a name="Nt_125" href="#NtA_125">[125]</a> For a purely fanciful
+  derivation from the corresponding number of strokes, see W. W. R. Ball,
+  <i>A Short Account of the History of Mathematics</i>, 1st ed., London,
+  1888, p. 147; similarly J. B. Reveillaud, <i>Essai sur les chiffres
+  arabes</i>, Paris, 1883; P. Voizot, "Les chiffres arabes et leur
+  origine," <i>La Nature</i>, 1899, p. 222; G. Dumesnil, "De la forme des
+  chiffres usuels," <i>Annales de l'université de Grenoble</i>, 1907, Vol.
+  XIX, pp. 657-674, also a note in <i>Revue Archéologique</i>, 1890, Vol.
+  XVI (3), pp. 342-348; one of the earliest references to a possible
+  derivation from points is in a work by Bettino entitled <i>Apiaria
+  universae philosophiae mathematicae in quibus paradoxa et noua
+  machinamenta ad usus eximios traducta, et facillimis demonstrationibus
+  confirmata</i>, Bologna, 1545, Vol. II, Apiarium XI, p. 5.</p>
+
+  <p><a name="Nt_126" href="#NtA_126">[126]</a> <i>Alphabetum Barmanum</i>,
+  Romae, <span class="scac">MDCCLXXVI</span>, p. 50. The 1 is evidently
+  Sanskrit, and the 4, 7, and possibly 9 are from India.</p>
+
+  <p><a name="Nt_127" href="#NtA_127">[127]</a> <i>Alphabetum
+  Grandonico-Malabaricum</i>, Romae, <span class="scac">MDCCLXXII</span>,
+  p. 90. The zero is not used, but the symbols for 10, 100, and so on, are
+  joined to the units to make the higher numbers.</p>
+
+  <p><a name="Nt_128" href="#NtA_128">[128]</a> <i>Alphabetum
+  Tangutanum</i>, Romae, <span class="scac">MDCCLXXIII</span>, p. 107. In a
+  Tibetan MS. in the library of Professor Smith, probably of the eighteenth
+  century, substantially these forms are given.</p>
+
+  <p><a name="Nt_129" href="#NtA_129">[129]</a> Bayley, loc. cit., plate
+  II. Similar forms to these here shown, and numerous other forms found in
+  India, as well as those of other oriental countries, are given by A. P.
+  Pihan, <i>Exposé des signes de numération usités chez les peuples
+  orientaux anciens et modernes</i>, Paris, 1860.</p>
+
+  <p><a name="Nt_130" href="#NtA_130">[130]</a> Bühler, loc. cit., p. 80;
+  J. F. Fleet, <i>Corpus inscriptionum Indicarum</i>, Vol. III, Calcutta,
+  1888. Lists of such words are given also by Al-B&#x12B;r&#x16B;n&#x12B;
+  in his work <i>India</i>; by Burnell, loc. cit.; by E. Jacquet, "Mode
+  d'expression symbolique des nombres employé par les Indiens, les
+  Tibétains et les Javanais," <i>Journal Asiatique</i>, Vol. XVI, Paris,
+  1835.</p>
+
+  <p><a name="Nt_131" href="#NtA_131">[131]</a> This date is given by
+  Fleet, loc. cit., Vol. III, p. 73, as the earliest epigraphical instance
+  of this usage in India proper.</p>
+
+  <p><a name="Nt_132" href="#NtA_132">[132]</a> Weber, <i>Indische
+  Studien</i>, Vol. VIII, p. 166 seq.</p>
+
+  <p><a name="Nt_133" href="#NtA_133">[133]</a> <i>Journal of the Royal
+  Asiatic Society</i>, Vol. I (<span class="scac">N.S.</span>), p. 407.</p>
+
+  <p><a name="Nt_134" href="#NtA_134">[134]</a> VIII, 20, 21.</p>
+
+  <p><a name="Nt_135" href="#NtA_135">[135]</a> Th. H. Martin, <i>Les
+  signes numéraux</i> ..., Rome, 1864; Lassen, <i>Indische
+  Alterthumskunde</i>, Vol. II, 2d ed., Leipzig and London, 1874, p.
+  1153.</p>
+
+  <p><a name="Nt_136" href="#NtA_136">[136]</a> But see Burnell, loc. cit.,
+  and Thibaut, <i>Astronomie, Astrologie und Mathematik</i>, p. 71.</p>
+
+  <p><a name="Nt_137" href="#NtA_137">[137]</a> A. Barth, "Inscriptions
+  Sanscrites du Cambodge," in the <i>Notices et extraits des Mss. de la
+  Bibliothèque nationale</i>, Vol. XXVII, Part I, pp. 1-180, 1885; see also
+  numerous articles in <i>Journal Asiatique</i>, by Aymonier.</p>
+
+  <p><a name="Nt_138" href="#NtA_138">[138]</a> Bühler, loc. cit., p.
+  82.</p>
+
+  <p><a name="Nt_139" href="#NtA_139">[139]</a> Loc. cit., p. 79.</p>
+
+  <p><a name="Nt_140" href="#NtA_140">[140]</a> Bühler, loc. cit., p. 83.
+  The Hindu astrologers still use an alphabetical system of numerals.
+  [Burnell, loc. cit., p. 79.]</p>
+
+  <p><a name="Nt_141" href="#NtA_141">[141]</a> Well could Ramus say,
+  "Quicunq; autem fuerit inventor decem notarum laudem magnam meruit."</p>
+
+  <p><a name="Nt_142" href="#NtA_142">[142]</a> Al-B&#x12B;r&#x16B;n&#x12B;
+  gives lists.</p>
+
+  <p><a name="Nt_143" href="#NtA_143">[143]</a> <i>Propagation</i>, loc.
+  cit., p. 443.</p>
+
+  <p><a name="Nt_144" href="#NtA_144">[144]</a> See the quotation from
+  <i>The Light of Asia</i> in Chapter II, p. 16.</p>
+
+  <p><a name="Nt_145" href="#NtA_145">[145]</a> The nine ciphers were
+  called <i><span class="special" title="anka">a&#x1E45;ka</span></i>.</p>
+
+  <p><a name="Nt_146" href="#NtA_146">[146]</a> "Zur Geschichte des
+  indischen Ziffernsystems," <i>Zeitschrift für die Kunde des
+  Morgenlandes</i>, Vol. IV, 1842, pp. 74-83.</p>
+
+  <p><a name="Nt_147" href="#NtA_147">[147]</a> It is found in the <span
+  class="special" title="Bakhsali">Bakh&#x1E63;&#x101;l&#x12B;</span> MS.
+  of an elementary arithmetic which Hoernle placed, at first, about the
+  beginning of our era, but the date is much in question. G. Thibaut, loc.
+  cit., places it between 700 and 900 <span class="scac">A.D.</span>;
+  Cantor places the body of the work about the third or fourth century
+  <span class="scac">A.D.</span>, <i>Geschichte der Mathematik</i>, Vol. I
+  (3), p. 598.</p>
+
+  <p><a name="Nt_148" href="#NtA_148">[148]</a> For the opposite side of
+  the case see G. R. Kaye, "Notes on Indian Mathematics, No. 2.&mdash;<span
+  class="special" title="Aryabhata">&#x100;ryabha&#x1E6D;a</span>,"
+  <i>Journ. and Proc. of the Asiatic Soc. of Bengal</i>, Vol. IV, 1908, pp.
+  111-141.</p>
+
+  <p><a name="Nt_149" href="#NtA_149">[149]</a> He used one of the
+  alphabetic systems explained above. This ran up to 10<sup>18</sup> and
+  was not difficult, beginning as follows:</p>
+
+  <div class="figcenter" style="width:30%;">
+      <a href="images/049a.png"><img style="width:100%" src="images/049a.png"
+      alt="The system of Aryabhata" title="The system of Aryabhata" /></a>
+  </div>
+  <p>the same letter (<i>ka</i>) appearing in the successive consonant
+  forms, <i>ka</i>, <i>kha</i>, <i>ga</i>, <i>gha</i>, etc. See C. I.
+  Gerhardt, <i>Über die Entstehung und Ausbreitung des dekadischen
+  Zahlensystems</i>, Programm, p. 17, Salzwedel, 1853, and <i>Études
+  historiques sur l'arithmétique de position</i>, Programm, p. 24, Berlin,
+  1856; E. Jacquet, <i>Mode d'expression symbolique des nombres</i>, loc.
+  cit., p. 97; L. Rodet, "Sur la véritable signification de la notation
+  numérique inventée par &#x100;ryabhata," <i>Journal Asiatique</i>, Vol.
+  XVI (7), pp. 440-485. On the two <span class="special"
+  title="Aryabhatas">&#x100;ryabha&#x1E6D;as</span> see Kaye, <i>Bibl.
+  Math.</i>, Vol. X (3), p. 289.</p>
+
+  <p><a name="Nt_150" href="#NtA_150">[150]</a> Using <i>kha</i>, a synonym
+  of <i>&#x15B;&#x16B;nya</i>. [Bayley, loc. cit., p. 22, and L. Rodet,
+  <i>Journal Asiatique</i>, Vol. XVI (7), p. 443.]</p>
+
+  <p><a name="Nt_151" href="#NtA_151">[151]</a> Var&#x101;ha-Mihira,
+  <i>Pañcasiddh&#x101;ntik&#x101;</i>, translated by G. Thibaut and M. S.
+  Dvived&#x12B;, Benares, 1889; see Bühler, loc. cit., p. 78; Bayley, loc.
+  cit., p. 23.</p>
+
+  <p><a name="Nt_152" href="#NtA_152">[152]</a> <i><span class="special"
+  title="Brhat Samhita">B&#x1E5B;hat Sa&#x1E43;hit&#x101;</span></i>,
+  translated by Kern, <i>Journal of the Royal Asiatic Society</i>,
+  1870-1875.</p>
+
+  <p><a name="Nt_153" href="#NtA_153">[153]</a> It is stated by Bühler in a
+  personal letter to Bayley (loc. cit., p. 65) that there are hundreds of
+  instances of this usage in the <i><span class="special" title="Brhat Samhita"
+  >B&#x1E5B;hat Sa&#x1E43;hit&#x101;</span></i>. The system was also used
+  in the <i>Pañcasiddh&#x101;ntik&#x101;</i> as early as 505 <span
+  class="scac">A.D.</span> [Bühler, <i>Palaeographie</i>, p. 80, and Fleet,
+  <i>Journal of the Royal Asiatic Society</i>, 1910, p. 819.]</p>
+
+  <p><a name="Nt_154" href="#NtA_154">[154]</a> Cantor, <i>Geschichte der
+  Mathematik</i>, Vol. I (3), p. 608.</p>
+
+  <p><a name="Nt_155" href="#NtA_155">[155]</a> Bühler, loc. cit., p.
+  78.</p>
+
+  <p><a name="Nt_156" href="#NtA_156">[156]</a> Bayley, p. 38.</p>
+
+  <p><a name="Nt_157" href="#NtA_157">[157]</a> Noviomagus, in his <i>De
+  numeris libri duo</i>, Paris, 1539, confesses his ignorance as to the
+  origin of the zero, but says: "D. Henricus Grauius, vir Graecè &amp;
+  Hebraicè eximè doctus, Hebraicam originem ostendit," adding that Valla
+  "Indis Orientalibus gentibus inventionem tribuit."</p>
+
+  <p><a name="Nt_158" href="#NtA_158">[158]</a> See <i>Essays</i>, Vol. II,
+  pp. 287 and 288.</p>
+
+  <p><a name="Nt_159" href="#NtA_159">[159]</a> Vol. XXX, p. 205 seqq.</p>
+
+  <p><a name="Nt_160" href="#NtA_160">[160]</a> Loc. cit., p. 284 seqq.</p>
+
+  <p><a name="Nt_161" href="#NtA_161">[161]</a> Colebrooke, loc. cit., p.
+  288.</p>
+
+  <p><a name="Nt_162" href="#NtA_162">[162]</a> Loc. cit., p. 78.</p>
+
+  <p><a name="Nt_163" href="#NtA_163">[163]</a> Hereafter, unless expressly
+  stated to the contrary, we shall use the word "numerals" to mean numerals
+  with place value.</p>
+
+  <p><a name="Nt_164" href="#NtA_164">[164]</a> "The Gurjaras of
+  R&#x101;jput&#x101;na and Kanauj," in <i>Journal of the Royal Asiatic
+  Society</i>, January and April, 1909.</p>
+
+  <p><a name="Nt_165" href="#NtA_165">[165]</a> Vol. IX, 1908, p. 248.</p>
+
+  <p><a name="Nt_166" href="#NtA_166">[166]</a> <i>Epigraphia Indica</i>,
+  Vol. IX, pp. 193 and 198.</p>
+
+  <p><a name="Nt_167" href="#NtA_167">[167]</a> <i>Epigraphia Indica</i>,
+  Vol. IX, p. 1.</p>
+
+  <p><a name="Nt_168" href="#NtA_168">[168]</a> Loc. cit., p. 71.</p>
+
+  <p><a name="Nt_169" href="#NtA_169">[169]</a> Thibaut, p. 71.</p>
+
+  <p><a name="Nt_170" href="#NtA_170">[170]</a> "Est autem in aliquibus
+  figurarum istaram apud multos diuersitas. Quidam enim septimam hanc
+  figuram representant," etc. [Boncompagni, <i>Trattati</i>, p. 28.]
+  Eneström has shown that very likely this work is incorrectly attributed
+  to Johannes Hispalensis. [<i>Bibliotheca Mathematica</i>, Vol. IX (3), p.
+  2.]</p>
+
+  <p><a name="Nt_171" href="#NtA_171">[171]</a> <i>Indische
+  Palaeographie</i>, Tafel IX.</p>
+
+  <p><a name="Nt_172" href="#NtA_172">[172]</a> Edited by Bloomfield and
+  Garbe, Baltimore, 1901, containing photographic reproductions of the
+  manuscript.</p>
+
+  <p><a name="Nt_173" href="#NtA_173">[173]</a> <span class="special"
+  title="Bakhsali">Bakh&#x1E63;&#x101;l&#x12B;</span> MS. See page 43;
+  Hoernle, R., <i>The Indian Antiquary</i>, Vol. XVII, pp. 33-48, 1 plate;
+  Hoernle, <i>Verhandlungen des VII. Internationalen
+  Orientalisten-Congresses, Arische Section</i>, Vienna, 1888, "On the
+  Baksh&#x101;l&#x12B; Manuscript," pp. 127-147, 3 plates; Bühler, loc.
+  cit.</p>
+
+  <p><a name="Nt_174" href="#NtA_174">[174]</a> 3, 4, 6, from H. H. Dhruva,
+  "Three Land-Grants from Sankheda," <i>Epigraphia Indica</i>, Vol. II, pp.
+  19-24 with plates; date 595 <span class="scac">A.D.</span> 7, 1, 5, from
+  Bhandarkar, "Daulatabad Plates," <i>Epigraphia Indica</i>, Vol. IX, part
+  V; date c. 798 <span class="scac">A.D.</span></p>
+
+  <p><a name="Nt_175" href="#NtA_175">[175]</a> 8, 7, 2, from "Buckhala
+  Inscription of Nagabhatta," Bhandarkar, <i>Epigraphia Indica</i>, Vol.
+  IX, part V; date 815 <span class="scac">A.D.</span> 5 from "The Morbi
+  Copper-Plate," Bhandarkar, <i>The Indian Antiquary</i>, Vol. II, pp.
+  257-258, with plate; date 804 <span class="scac">A.D.</span> See Bühler,
+  loc. cit.</p>
+
+  <p><a name="Nt_176" href="#NtA_176">[176]</a> 8 from the above Morbi
+  Copper-Plate. 4, 5, 7, 9, and 0, from "Asni Inscription of Mahipala,"
+  <i>The Indian Antiquary</i>, Vol. XVI, pp. 174-175; inscription is on red
+  sandstone, date 917 <span class="scac">A.D.</span> See Bühler.</p>
+
+  <p><a name="Nt_177" href="#NtA_177">[177]</a> 8, 9, 4, from "Rashtrakuta
+  Grant of Amoghavarsha," J. F. Fleet, <i>The Indian Antiquary</i>, Vol.
+  XII, pp. 263-272; copper-plate grant of date c. 972 <span
+  class="scac">A.D.</span> See Bühler. 7, 3, 5, from "Torkhede Copper-Plate
+  Grant of the Time of Govindaraja of Gujerat," Fleet, <i>Epigraphia
+  Indica</i>, Vol. III, pp. 53-58. See Bühler.</p>
+
+  <p><a name="Nt_178" href="#NtA_178">[178]</a> From "A Copper-Plate Grant
+  of King Tritochanapâla Chanlukya of <span class="special"
+  title="Latadesa">L&#x101;&#x1E6D;ade&#x15B;a</span>," H.H. Dhruva,
+  <i>Indian Antiquary</i>, Vol. XII, pp. 196-205; date 1050 <span
+  class="scac">A.D.</span> See Bühler.</p>
+
+  <p><a name="Nt_179" href="#NtA_179">[179]</a> Burnell, A. C., <i>South
+  Indian Palæography</i>, plate XXIII, Telugu-Canarese numerals of the
+  eleventh century. See Bühler.</p>
+
+  <p><a name="Nt_180" href="#NtA_180">[180]</a> From a manuscript of the
+  second half of the thirteenth century, reproduced in "Della vita e delle
+  opere di Leonardo Pisano," Baldassare Boncompagni, Rome, 1852, in <i>Atti
+  dell' Accademia Pontificia dei nuovi Lincei</i>, anno V.</p>
+
+  <p><a name="Nt_181" href="#NtA_181">[181]</a> From a fourteenth-century
+  manuscript, as reproduced in <i>Della vita</i> etc., Boncompagni, loc.
+  cit.</p>
+
+  <p><a name="Nt_182" href="#NtA_182">[182]</a> From a Tibetan MS. in the
+  library of D. E. Smith.</p>
+
+  <p><a name="Nt_183" href="#NtA_183">[183]</a> From a Tibetan block-book
+  in the library of D. E. Smith.</p>
+
+  <p><a name="Nt_184" href="#NtA_184">[184]</a> &#x15A;&#x101;rad&#x101;
+  numerals from <i>The Kashmirian Atharva-Veda, reproduced by
+  chromophotography from the manuscript in the University Library at
+  Tübingen</i>, Bloomfield and Garbe, Baltimore, 1901. Somewhat similar
+  forms are given under "Numération Cachemirienne," by Pihan, <i>Exposé</i>
+  etc., p. 84.</p>
+
+  <p><a name="Nt_185" href="#NtA_185">[185]</a> Franz X. Kugler, <i>Die
+  Babylonische Mondrechnung</i>, Freiburg i. Br., 1900, in the numerous
+  plates at the end of the book; practically all of these contain the
+  symbol to which reference is made. Cantor, <i>Geschichte</i>, Vol. I, p.
+  31.</p>
+
+  <p><a name="Nt_186" href="#NtA_186">[186]</a> F. X. Kugler, <i>Sternkunde
+  und Sterndienst in Babel</i>, I. Buch, from the beginnings to the time of
+  Christ, Münster i. Westfalen, 1907. It also has numerous tables
+  containing the above zero.</p>
+
+  <p><a name="Nt_187" href="#NtA_187">[187]</a> From a letter to D. E.
+  Smith, from G. F. Hill of the British Museum. See also his monograph "On
+  the Early Use of Arabic Numerals in Europe," in <i>Archæologia</i>, Vol.
+  LXII (1910), p. 137.</p>
+
+  <p><a name="Nt_188" href="#NtA_188">[188]</a> R. Hoernle, "The
+  Baksh&#x101;l&#x12B; Manuscript," <i>Indian Antiquary</i>, Vol. XVII, pp.
+  33-48 and 275-279, 1888; Thibaut, <i>Astronomie, Astrologie und
+  Mathematik</i>, p. 75; Hoernle, <i>Verhandlungen</i>, loc. cit., p.
+  132.</p>
+
+  <p><a name="Nt_189" href="#NtA_189">[189]</a> Bayley, loc. cit., Vol. XV,
+  p. 29. Also Bendall, "On a System of Numerals used in South India,"
+  <i>Journal of the Royal Asiatic Society</i>, 1896, pp. 789-792.</p>
+
+  <p><a name="Nt_190" href="#NtA_190">[190]</a> V. A. Smith, <i>The Early
+  History of India</i>, 2d ed., Oxford, 1908, p. 14.</p>
+
+  <p><a name="Nt_191" href="#NtA_191">[191]</a> Colebrooke, <i>Algebra,
+  with Arithmetic and Mensuration, from the Sanskrit of Brahmegupta and
+  Bháscara</i>, London, 1817, pp. 339-340.</p>
+
+  <p><a name="Nt_192" href="#NtA_192">[192]</a> Ibid., p. 138.</p>
+
+  <p><a name="Nt_193" href="#NtA_193">[193]</a> D. E. Smith, in the
+  <i>Bibliotheca Mathematica</i>, Vol. IX (3), pp. 106-110.</p>
+
+  <p><a name="Nt_194" href="#NtA_194">[194]</a> As when we use three dots
+  (...).</p>
+
+  <p><a name="Nt_195" href="#NtA_195">[195]</a> "The Hindus call the nought
+  explicitly <i>&#x15B;&#x16B;nyabindu</i> 'the dot marking a blank,' and
+  about 500 <span class="scac">A.D.</span> they marked it by a simple dot,
+  which latter is commonly used in inscriptions and MSS. in order to mark a
+  blank, and which was later converted into a small circle." [Bühler, <i>On
+  the Origin of the Indian Alphabet</i>, p. 53, note.]</p>
+
+  <p><a name="Nt_196" href="#NtA_196">[196]</a> Fazzari, <i>Dell' origine
+  delle parole zero e cifra</i>, Naples, 1903.</p>
+
+  <p><a name="Nt_197" href="#NtA_197">[197]</a> E. Wappler, "Zur Geschichte
+  der Mathematik im 15. Jahrhundert," in the <i>Zeitschrift für Mathematik
+  und Physik</i>, Vol. XLV, <i>Hist.-lit. Abt.</i>, p. 47. The manuscript
+  is No. C. 80, in the Dresden library.</p>
+
+  <p><a name="Nt_198" href="#NtA_198">[198]</a> J. G. Prändel, <i>Algebra
+  nebst ihrer literarischen Geschichte</i>, p. 572, Munich, 1795.</p>
+
+  <p><a name="Nt_199" href="#NtA_199">[199]</a> See the table, p. 23. Does
+  the fact that the early European arithmetics, following the Arab custom,
+  always put the 0 after the 9, suggest that the 0 was derived from the old
+  Hindu symbol for 10?</p>
+
+  <p><a name="Nt_200" href="#NtA_200">[200]</a> Bayley, loc. cit., p. 48.
+  From this fact Delambre (<i>Histoire de l'astronomie ancienne</i>)
+  inferred that Ptolemy knew the zero, a theory accepted by Chasles,
+  <i>Aperçu historique sur l'origine et le développement des méthodes en
+  géométrie</i>, 1875 ed., p. 476; Nesselmann, however, showed (<i>Algebra
+  der Griechen</i>, 1842, p. 138), that Ptolemy merely used <span title="o" class="grk"
+  >&omicron;</span> for <span title="ouden" class="grk"
+  >&omicron;&#x1F50;&delta;&#x1F72;&nu;</span>, with no notion of zero. See
+  also G. Fazzari, "Dell' origine delle parole zero e cifra,"
+  <i>Ateneo</i>, Anno I, No. 11, reprinted at Naples in 1903, where the use
+  of the point and the small cross for zero is also mentioned. Th. H.
+  Martin, <i>Les signes numéraux</i> etc., reprint p. 30, and J. Brandis,
+  <i>Das Münz-, Mass- und Gewichtswesen in Vorderasien bis auf Alexander
+  den Grossen</i>, Berlin, 1866, p. 10, also discuss this usage of <span
+  title="o" class="grk">&omicron;</span>, without the notion of place
+  value, by the Greeks.</p>
+
+  <p><a name="Nt_201" href="#NtA_201">[201]</a> <i>Al-Batt&#x101;n&#x12B;
+  sive Albatenii opus astronomicum</i>. Ad fidem codicis escurialensis
+  arabice editum, latine versum, adnotationibus instructum a Carolo
+  Alphonso Nallino, 1899-1907. Publicazioni del R. Osservatorio di Brera in
+  Milano, No. XL.</p>
+
+  <p><a name="Nt_202" href="#NtA_202">[202]</a> Loc. cit., Vol. II, p.
+  271.</p>
+
+  <p><a name="Nt_203" href="#NtA_203">[203]</a> C. Henry, "Prologus N.
+  Ocreati in Helceph ad Adelardum Batensem magistrum suum," <i>Abhandlungen
+  zur Geschichte der Mathematik</i>, Vol. III, 1880.</p>
+
+  <p><a name="Nt_204" href="#NtA_204">[204]</a> Max. Curtze, "Ueber eine
+  Algorismus-Schrift des XII. Jahrhunderts," <i>Abhandlungen zur Geschichte
+  der Mathematik</i>, Vol. VIII, 1898, pp. 1-27; Alfred Nagl, "Ueber eine
+  Algorismus-Schrift des XII. Jahrhunderts und über die Verbreitung der
+  indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande,"
+  <i>Zeitschrift für Mathematik und Physik, Hist.-lit. Abth.</i>, Vol.
+  XXXIV, pp. 129-146 and 161-170, with one plate.</p>
+
+  <p><a name="Nt_205" href="#NtA_205">[205]</a> "Byzantinische Analekten,"
+  <i>Abhandlungen zur Geschichte der Mathematik</i>, Vol. IX, pp.
+  161-189.</p>
+
+  <p><a name="Nt_206" href="#NtA_206">[206]</a> <a
+  href="images/061d.png"><img src="images/061d.png" class="middle"
+  style="height:2ex" alt="symbol" /></a> or <a href="images/061e.png"><img
+  src="images/061e.png" class="middle" style="height:2ex" alt="symbol"
+  /></a> for 0. <a href="images/061d.png"><img src="images/061d.png"
+  class="middle" style="height:2ex" alt="symbol" /></a> also used for 5. <a
+  href="images/061f.png"><img src="images/061f.png" class="middle"
+  style="height:2ex" alt="symbols" /></a> for 13. [Heiberg, loc. cit.]</p>
+
+  <p><a name="Nt_207" href="#NtA_207">[207]</a> Gerhardt, <i>Études
+  historiques sur l'arithmétique de position</i>, Berlin, 1856, p. 12; J.
+  Bowring, <i>The Decimal System in Numbers, Coins, &amp; Accounts</i>,
+  London, 1854, p. 33.</p>
+
+  <p><a name="Nt_208" href="#NtA_208">[208]</a> Karabacek, <i>Wiener
+  Zeitschrift für die Kunde des Morgenlandes</i>, Vol. XI, p. 13; <i>Führer
+  durch die Papyrus-Ausstellung Erzherzog Rainer</i>, Vienna, 1894, p.
+  216.</p>
+
+  <p><a name="Nt_209" href="#NtA_209">[209]</a> In the library of G. A.
+  Plimpton, Esq.</p>
+
+  <p><a name="Nt_210" href="#NtA_210">[210]</a> Cantor, <i>Geschichte</i>,
+  Vol. I (3), p. 674; Y. Mikami, "A Remark on the Chinese Mathematics in
+  Cantor's Geschichte der Mathematik," <i>Archiv der Mathematik und
+  Physik</i>, Vol. XV (3), pp. 68-70.</p>
+
+  <p><a name="Nt_211" href="#NtA_211">[211]</a> Of course the earlier
+  historians made innumerable guesses as to the origin of the word
+  <i>cipher</i>. E.g. Matthew Hostus, <i>De numeratione emendata</i>,
+  Antwerp, 1582, p. 10, says: "Siphra vox Hebræam originem sapit refértque:
+  &amp; ut docti arbitrantur, à verbo saphar, quod Ordine numerauit
+  significat. Unde Sephar numerus est: hinc Siphra (vulgo corruptius). Etsi
+  verò gens Iudaica his notis, quæ hodie Siphræ vocantur, usa non fuit:
+  mansit tamen rei appellatio apud multas gentes." Dasypodius,
+  <i>Institutiones mathematicae</i>, Vol. I, 1593, gives a large part of
+  this quotation word for word, without any mention of the source.
+  Hermannus Hugo, <i>De prima scribendi origine</i>, Trajecti ad Rhenum,
+  1738, pp. 304-305, and note, p. 305; Karl Krumbacher, "Woher stammt das
+  Wort Ziffer (Chiffre)?", <i>Études de philologie néo-grecque</i>, Paris,
+  1892.</p>
+
+  <p><a name="Nt_212" href="#NtA_212">[212]</a> Bühler, loc. cit., p. 78
+  and p. 86.</p>
+
+  <p><a name="Nt_213" href="#NtA_213">[213]</a> Fazzari, loc. cit., p. 4.
+  So Elia Misrachi (1455-1526) in his posthumous <i>Book of Number</i>,
+  Constantinople, 1534, explains <i>sifra</i> as being Arabic. See also
+  Steinschneider, <i>Bibliotheca Mathematica</i>, 1893, p. 69, and G.
+  Wertheim, <i>Die Arithmetik des Elia Misrachi</i>, Programm, Frankfurt,
+  1893.</p>
+
+  <p><a name="Nt_214" href="#NtA_214">[214]</a> "Cum his novem figuris, et
+  cum hoc signo 0, quod arabice zephirum appellatur, scribitur quilibet
+  numerus."</p>
+
+  <p><a name="Nt_215" href="#NtA_215">[215]</a> <span title="tziphra" class="grk"
+  >&tau;&zeta;&#x1F77;&phi;&rho;&alpha;</span>, a form also used by
+  Neophytos (date unknown, probably c. 1330). It is curious that Finaeus
+  (1555 ed., f. 2) used the form <i>tziphra</i> throughout. A. J. H.
+  Vincent ["Sur l'origine de nos chiffres," <i>Notices et Extraits des
+  MSS.</i>, Paris, 1847, pp. 143-150] says: "Ce cercle fut nommé par les
+  uns, <i>sipos, rota, galgal</i> ...; par les autres <i>tsiphra</i> (de
+  <span lang="he" class="heb" title="TSPR" ><bdo
+  dir="rtl">&#x5E6;&#x5E4;&#x5E8;</bdo></span>, <i>couronne</i> ou
+  <i>diadème</i>) ou <i>ciphra</i> (de <span lang="he" class="heb"
+  title="SPR" ><bdo dir="rtl">&#x5E1;&#x5E4;&#x5E8;</bdo></span>,
+  <i>numération</i>)." Ch. de Paravey, <i>Essai sur l'origine unique et
+  hiéroglyphique des chiffres et des lettres de tous les peuples</i>,
+  Paris, 1826, p. 165, a rather fanciful work, gives "vase, vase arrondi et
+  fermé par un couvercle, qui est le symbole de la 10<sup>e</sup> Heure, <a
+  href="images/063a.png"><img src="images/063a.png" class="middle"
+  style="height:2ex" alt="symbol" /></a>," among the Chinese; also
+  "Tsiphron Zéron, ou tout à fait vide en arabe, <span title="tziphra" class="grk"
+  >&tau;&zeta;&#x1F77;&phi;&rho;&alpha;</span> en grec ... d'où chiffre
+  (qui dérive plutôt, suivant nous, de l'Hébreu <i>Sepher</i>,
+  compter.")</p>
+
+  <p><a name="Nt_216" href="#NtA_216">[216]</a> "Compilatus a Magistro
+  Jacobo de Florentia apud montem pesalanum," and described by G. Lami in
+  his <i>Catalogus codicum manuscriptorum qui in bibliotheca Riccardiana
+  Florentiæ adservantur</i>. See Fazzari, loc. cit., p. 5.</p>
+
+  <p><a name="Nt_217" href="#NtA_217">[217]</a> "Et doveto sapere chel
+  zeuero per se solo non significa nulla ma è potentia di fare significare,
+  ... Et decina o centinaia o migliaia non si puote scrivere senza questo
+  segno 0. la quale si chiama zeuero." [Fazzari, loc. cit., p. 5.]</p>
+
+  <p><a name="Nt_218" href="#NtA_218">[218]</a> Ibid., p. 6.</p>
+
+  <p><a name="Nt_219" href="#NtA_219">[219]</a> Avicenna (980-1036),
+  translation by Gasbarri et François, "più il punto (gli Arabi adoperavano
+  il punto in vece dello zero il cui segno 0 in arabo si chiama
+  <i>zepiro</i> donde il vocabolo zero), che per sè stesso non esprime
+  nessun numero." This quotation is taken from D. C. Martines, <i>Origine e
+  progressi dell' aritmetica</i>, Messina, 1865.</p>
+
+  <p><a name="Nt_220" href="#NtA_220">[220]</a> Leo Jordan, "Materialien
+  zur Geschichte der arabischen Zahlzeichen in Frankreich," <i>Archiv für
+  Kulturgeschichte</i>, Berlin, 1905, pp. 155-195, gives the following two
+  schemes of derivation, (1) "zefiro, zeviro, zeiro, zero," (2) "zefiro,
+  zefro, zevro, zero."</p>
+
+  <p><a name="Nt_221" href="#NtA_221">[221]</a> Köbel (1518 ed., f. A_4)
+  speaks of the numerals in general as "die der gemain man Zyfer nendt."
+  Recorde (<i>Grounde of Artes</i>, 1558 ed., f. B_6) says that the zero is
+  "called priuatly a Cyphar, though all the other sometimes be likewise
+  named."</p>
+
+  <p><a name="Nt_222" href="#NtA_222">[222]</a> "Decimo X 0 theca, circul<a
+  href="images/064a.png"><img src="images/064a.png" class="middle"
+  style="height:2ex" alt="us" /></a> cifra sive figura nihili
+  appelat&prime;." [<i>Enchiridion Algorismi</i>, Cologne, 1501.] Later,
+  "quoniam de integris tam in cifris quam in proiectilibus,"&mdash;the word
+  <i>proiectilibus</i> referring to markers "thrown" and used on an abacus,
+  whence the French <i>jetons</i> and the English expression "to
+  <i>cast</i> an account."</p>
+
+  <p><a name="Nt_223" href="#NtA_223">[223]</a> "Decima vero o dicitur
+  teca, circulus, vel cyfra vel figura nichili." [Maximilian Curtze,
+  <i>Petri Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco
+  commentarius, una cum Algorismo ipso</i>, Copenhagen, 1897, p. 2.] Curtze
+  cites five manuscripts (fourteenth and fifteenth centuries) of Dacia's
+  commentary in the libraries at Erfurt, Leipzig, and Salzburg, in addition
+  to those given by Eneström, <i>Öfversigt af Kongl. Vetenskaps-Akademiens
+  Förhandlingar</i>, 1885, pp. 15-27, 65-70; 1886, pp. 57-60.</p>
+
+  <p><a name="Nt_224" href="#NtA_224">[224]</a> Curtze, loc. cit., p. <span
+  class="scac">VI</span>.</p>
+
+  <p><a name="Nt_225" href="#NtA_225">[225]</a> <i>Rara Mathematica</i>,
+  London, 1841, chap, i, "Joannis de Sacro-Bosco Tractatus de Arte
+  Numerandi."</p>
+
+  <p><a name="Nt_226" href="#NtA_226">[226]</a> Smith, <i>Rara
+  Arithmetica</i>, Boston, 1909.</p>
+
+  <p><a name="Nt_227" href="#NtA_227">[227]</a> In the 1484 edition, Borghi
+  uses the form "çefiro: ouero nulla:" while in the 1488 edition he uses
+  "zefiro: ouero nulla," and in the 1540 edition, f. 3, appears "Chiamata
+  zero, ouero nulla." Woepcke asserted that it first appeared in Calandri
+  (1491) in this sentence: "Sono dieci le figure con le quali ciascuno
+  numero si può significare: delle quali n'è una che si chiama zero: et per
+  se sola nulla significa." (f. 4). [See <i>Propagation</i>, p. 522.]</p>
+
+  <p><a name="Nt_228" href="#NtA_228">[228]</a> Boncompagni
+  <i>Bulletino</i>, Vol. XVI, pp. 673-685.</p>
+
+  <p><a name="Nt_229" href="#NtA_229">[229]</a> Leo Jordan, loc. cit. In
+  the <i>Catalogue of MSS., Bibl. de l'Arsenal</i>, Vol. III, pp. 154-156,
+  this work is No. 2904 (184 S.A.F.), Bibl. Nat., and is also called
+  <i>Petit traicté de algorisme</i>.</p>
+
+  <p><a name="Nt_230" href="#NtA_230">[230]</a> Texada (1546) says that
+  there are "nueue letros yvn zero o cifra" (f. 3).</p>
+
+  <p><a name="Nt_231" href="#NtA_231">[231]</a> Savonne (1563, 1751 ed., f.
+  1): "Vne ansi formee (o) qui s'appelle nulle, &amp; entre marchans zero,"
+  showing the influence of Italian names on French mercantile customs.
+  Trenchant (Lyons, 1566, 1578 ed., p. 12) also says: "La derniere qui
+  s'apele nulle, ou zero;" but Champenois, his contemporary, writing in
+  Paris in 1577 (although the work was not published until 1578), uses
+  "cipher," the Italian influence showing itself less in this center of
+  university culture than in the commercial atmosphere of Lyons.</p>
+
+  <p><a name="Nt_232" href="#NtA_232">[232]</a> Thus Radulph of Laon (c.
+  1100): "Inscribitur in ultimo ordine et figura <a
+  href="images/066a.png"><img src="images/066a.png" class="middle"
+  style="height:3ex" alt="symbol" /></a> sipos nomine, quae, licet numerum
+  nullum signitet, tantum ad alia quaedam utilis, ut insequentibus
+  declarabitur." ["Der Arithmetische Tractat des Radulph von Laon,"
+  <i>Abhandlungen zur Geschichte der Mathematik</i>, Vol. V, p. 97, from a
+  manuscript of the thirteenth century.] Chasles (<i>Comptes rendus</i>, t.
+  16, 1843, pp. 1393, 1408) calls attention to the fact that Radulph did
+  not know how to use the zero, and he doubts if the sipos was really
+  identical with it. Radulph says: "... figuram, cui sipos nomen est <a
+  href="images/066b.png"><img src="images/066b.png" class="middle"
+  style="height:3ex" alt="symbol" /></a> in motum rotulae formatam nullius
+  numeri significatione inscribi solere praediximus," and thereafter uses
+  <i>rotula</i>. He uses the sipos simply as a kind of marker on the
+  abacus.</p>
+
+  <p><a name="Nt_233" href="#NtA_233">[233]</a> Rabbi ben Ezra (1092-1168)
+  used both <span lang="he" class="heb" title="GLGL" ><bdo
+  dir="rtl">&#x5D2;&#x5DC;&#x5D2;&#x5DC;</bdo></span>, <i>galgal</i> (the
+  Hebrew for <i>wheel</i>), and <span lang="he" class="heb" title="SPR'"
+  ><bdo dir="rtl">&#x5E1;&#x5E4;&#x5E8;&#x5D0;</bdo></span>, <i>sifra</i>.
+  See M. Steinschneider, "Die Mathematik bei den Juden," in <i>Bibliotheca
+  Mathematica</i>, 1893, p. 69, and Silberberg, <i>Das Buch der Zahl des R.
+  Abraham ibn Esra</i>, Frankfurt a. M., 1895, p. 96, note 23; in this work
+  the Hebrew letters are used for numerals with place value, having the
+  zero.</p>
+
+  <p><a name="Nt_234" href="#NtA_234">[234]</a> E.g., in the
+  twelfth-century <i>Liber aligorismi</i> (see Boncompagni's
+  <i>Trattati</i>, II, p. 28). So Ramus (<i>Libri II</i>, 1569 ed., p. 1)
+  says: "Circulus quæ nota est ultima: nil per se significat." (See also
+  the Schonerus ed. of Ramus, 1586, p. 1.)</p>
+
+  <p><a name="Nt_235" href="#NtA_235">[235]</a> "Und wirt das ringlein o.
+  die Ziffer genant die nichts bedeut." [Köbel's <i>Rechenbuch</i>, 1549
+  ed., f. 10, and other editions.]</p>
+
+  <p><a name="Nt_236" href="#NtA_236">[236]</a> I.e. "circular figure," our
+  word <i>notation</i> having come from the medieval <i>nota</i>. Thus
+  Tzwivel (1507, f. 2) says: "Nota autem circularis .o. per se sumpta nihil
+  vsus habet. alijs tamen adiuncta earum significantiam et auget et ordinem
+  permutat quantum quo ponit ordinem. vt adiuncta note binarij hoc modo 20
+  facit eam significare bis decem etc." Also (ibid., f. 4), "figura
+  circularis," "circularis nota." Clichtoveus (1503 ed., f. <span
+  class="scac">XXXVII</span>) calls it "nota aut circularis o," "circularis
+  nota," and "figura circularis." Tonstall (1522, f. B_3) says of it:
+  "Decimo uero nota ad formam <a href="images/067a.png"><img
+  src="images/067a.png" class="middle" style="height:2ex" alt="symbol"
+  /></a> litteræ circulari figura est: quam alij circulum, uulgus cyphram
+  uocat," and later (f. C_4) speaks of the "circulos." Grammateus, in his
+  <i>Algorismus de integris</i> (Erfurt, 1523, f. A_2), speaking of the
+  nine significant figures, remarks: "His autem superadditur decima figura
+  circularis ut 0 existens que ratione sua nihil significat." Noviomagus
+  (<i>De Numeris libri II</i>, Paris, 1539, chap. xvi, "De notis numerorum,
+  quas zyphras vocant") calls it "circularis nota, quam ex his solam, alij
+  sipheram, Georgius Valla zyphram."</p>
+
+  <p><a name="Nt_237" href="#NtA_237">[237]</a> Huswirt, as above. Ramus
+  (<i>Scholae mathematicae</i>, 1569 ed., p. 112) discusses the name
+  interestingly, saying: "Circulum appellamus cum multis, quam alii thecam,
+  alii figuram nihili, alii figuram privationis, seu figuram nullam vocant,
+  alii ciphram, cùm tamen hodie omnes hæ notæ vulgò ciphræ nominentur,
+  &amp; his notis numerare idem sit quod ciphrare." Tartaglia (1592 ed., f.
+  9) says: "si chiama da alcuni tecca, da alcuni circolo, da altri cifra,
+  da altri zero, &amp; da alcuni altri nulla."</p>
+
+  <p><a name="Nt_238" href="#NtA_238">[238]</a> "Quare autem aliis
+  nominibus vocetur, non dicit auctor, quia omnia alia nomina habent
+  rationem suae lineationis sive figurationis. Quia rotunda est, dicitur
+  haec figura teca ad similitudinem tecae. Teca enim est ferrum figurae
+  rotundae, quod ignitum solet in quibusdam regionibus imprimi fronti vel
+  maxillae furis seu latronum." [Loc. cit., p. 26.] But in Greek
+  <i>theca</i> (<a href="images/067b.png"><img src="images/067b.png"
+  class="middle" style="height:2ex" alt="THEKE" /></a>, <span title="thêkê" class="grk"
+  >&theta;&#x1F75;&kappa;&eta;</span>) is a place to put something, a
+  receptacle. If a vacant column, e.g. in the abacus, was so called, the
+  initial might have given the early forms <a href="images/067c.png"><img
+  src="images/067c.png" class="middle" style="height:2ex" alt="symbol"
+  /></a> and <a href="images/067d.png"><img src="images/067d.png"
+  class="middle" style="height:2ex" alt="symbol" /></a> for the zero.</p>
+
+  <p><a name="Nt_239" href="#NtA_239">[239]</a> Buteo, <i>Logistica</i>,
+  Lyons, 1559. See also Wertheim in the <i>Bibliotheca Mathematica</i>,
+  1901, p. 214.</p>
+
+  <p><a name="Nt_240" href="#NtA_240">[240]</a> "0 est appellee chiffre ou
+  nulle ou figure de nulle valeur." [La Roche, <i>L'arithmétique</i>,
+  Lyons, 1520.]</p>
+
+  <p><a name="Nt_241" href="#NtA_241">[241]</a> "Decima autem figura nihil
+  uocata," "figura nihili (quam etiam cifram uocant)." [Stifel,
+  <i>Arithmetica integra</i>, 1544, f. 1.]</p>
+
+  <p><a name="Nt_242" href="#NtA_242">[242]</a> "Zifra, &amp; Nulla uel
+  figura Nihili." [Scheubel, 1545, p. 1 of ch. 1.] <i>Nulla</i> is also
+  used by Italian writers. Thus Sfortunati (1545 ed., f. 4) says: "et la
+  decima nulla &amp; e chiamata questa decima zero;" Cataldi (1602, p. 1):
+  "La prima, che è o, si chiama nulla, ouero zero, ouero niente." It also
+  found its way into the Dutch arithmetics, e.g. Raets (1576, 1580 ed., f.
+  A_3): "Nullo dat ist niet;" Van der Schuere (1600, 1624 ed., f. 7);
+  Wilkens (1669 ed., p. 1). In Germany Johann Albert (Wittenberg, 1534) and
+  Rudolff (1526) both adopted the Italian <i>nulla</i> and popularized it.
+  (See also Kuckuck, <i>Die Rechenkunst im sechzehnten Jahrhundert</i>,
+  Berlin, 1874, p. 7; Günther, <i>Geschichte</i>, p. 316.)</p>
+
+  <p><a name="Nt_243" href="#NtA_243">[243]</a> "La dixième s'appelle
+  chifre vulgairement: les vns l'appellant zero: nous la pourrons appeller
+  vn Rien." [Peletier, 1607 ed., p. 14.]</p>
+
+  <p><a name="Nt_244" href="#NtA_244">[244]</a> It appears in the Polish
+  arithmetic of K<span class="over">l</span>os (1538) as <i>cyfra</i>. "The
+  Ciphra 0 augmenteth places, but of himselfe signifieth not," Digges,
+  1579, p. 1. Hodder (10th ed., 1672, p. 2) uses only this word (cypher or
+  cipher), and the same is true of the first native American arithmetic,
+  written by Isaac Greenwood (1729, p. 1). Petrus de Dacia derives
+  <i>cyfra</i> from circumference. "Vocatur etiam cyfra, quasi circumfacta
+  vel circumferenda, quod idem est, quod circulus non habito respectu ad
+  centrum." [Loc. cit., p. 26.]</p>
+
+  <p><a name="Nt_245" href="#NtA_245">[245]</a> <i>Opera mathematica</i>,
+  1695, Oxford, Vol. I, chap. ix, <i>Mathesis universalis</i>, "De figuris
+  numeralibus," pp. 46-49; Vol. II, <i>Algebra</i>, p. 10.</p>
+
+  <p><a name="Nt_246" href="#NtA_246">[246]</a> Martin, <i>Origine de notre
+  système de numération écrite</i>, note 149, p. 36 of reprint, spells
+  <span title="tsiphra" class="grk"
+  >&tau;&sigma;&#x1F77;&phi;&rho;&alpha;</span> from Maximus Planudes,
+  citing Wallis as an authority. This is an error, for Wallis gives the
+  correct form as above.</p>
+
+  <p>Alexander von Humboldt, "Über die bei verschiedenen Völkern üblichen
+  Systeme von Zahlzeichen und über den Ursprung des Stellenwerthes in den
+  indischen Zahlen," Crelle's <i>Journal für reine und angewandte
+  Mathematik</i>, Vol. IV, 1829, called attention to the work <span
+  title="arithmoi Indikoi" class="grk"
+  >&#x1F00;&rho;&iota;&theta;&mu;&omicron;&#x1F76;
+  &#x1F38;&nu;&delta;&iota;&kappa;&omicron;&#x1F77;</span> of the monk
+  Neophytos, supposed to be of the fourteenth century. In this work the
+  forms <span title="tzuphra" class="grk"
+  >&tau;&zeta;&#x1F7B;&phi;&rho;&alpha;</span> and <span title="tzumphra" class="grk"
+  >&tau;&zeta;&#x1F7B;&mu;&phi;&rho;&alpha;</span> appear. See also Boeckh,
+  <i>De abaco Graecorum</i>, Berlin, 1841, and Tannery, "Le Scholie du
+  moine Néophytos," <i>Revue Archéologique</i>, 1885, pp. 99-102. Jordan,
+  loc. cit., gives from twelfth and thirteenth century manuscripts the
+  forms <i>cifra</i>, <i>ciffre</i>, <i>chifras</i>, and <i>cifrus</i>. Du
+  Cange, <i>Glossarium mediae et infimae Latinitatis</i>, Paris, 1842,
+  gives also <i>chilerae</i>. Dasypodius, <i>Institutiones
+  Mathematicae</i>, Strassburg, 1593-1596, adds the forms <i>zyphra</i> and
+  <i>syphra</i>. Boissière, <i>L'art d'arythmetique contenant toute
+  dimention, tres-singulier et commode, tant pour l'art militaire que
+  autres calculations</i>, Paris, 1554: "Puis y en a vn autre dict zero
+  lequel ne designe nulle quantité par soy, ains seulement les loges
+  vuides."</p>
+
+  <p><a name="Nt_247" href="#NtA_247">[247]</a> <i>Propagation</i>, pp. 27,
+  234, 442. Treutlein, "Das Rechnen im 16. Jahrhundert," <i>Abhandlungen
+  zur Geschichte der Mathematik</i>, Vol. I, p. 5, favors the same view. It
+  is combated by many writers, e.g. A. C. Burnell, loc. cit., p. 59. Long
+  before Woepcke, I. F. and G. I. Weidler, <i>De characteribus numerorum
+  vulgaribus et eorum aetatibus</i>, Wittenberg, 1727, asserted the
+  possibility of their introduction into Greece by Pythagoras or one of his
+  followers: "Potuerunt autem ex oriente, uel ex phoenicia, ad graecos
+  traduci, uel Pythagorae, uel eius discipulorum auxilio, cum aliquis eo,
+  proficiendi in literis causa, iter faceret, et hoc quoque inuentum
+  addisceret."</p>
+
+  <p><a name="Nt_248" href="#NtA_248">[248]</a> E.g., they adopted the
+  Greek numerals in use in Damascus and Syria, and the Coptic in Egypt.
+  Theophanes (758-818 <span class="scac">A.D.</span>),
+  <i>Chronographia</i>, Scriptores Historiae Byzantinae, Vol. XXXIX,
+  Bonnae, 1839, p. 575, relates that in 699 <span class="scac">A.D.</span>
+  the caliph Wal&#x12B;d forbade the use of the Greek language in the
+  bookkeeping of the treasury of the caliphate, but permitted the use of
+  the Greek alphabetic numerals, since the Arabs had no convenient number
+  notation: <span title="kai ekôluse graphesthai Hellênisti tous dêmosious tôn logothesiôn kôdikas, all' Arabiois auta parasêmainesthai, chôris tôn psêphôn, epeidê adunaton têi ekeinôn glôssêi monada ê duada ê triada ê oktô hêmisu ê tria graphesthai; dio kai heôs sêmeron eisin sun autois notarioi Christianoi." class="grk"
+  >&kappa;&alpha;&#x1F76;
+  &#x1F10;&kappa;&#x1F7D;&lambda;&upsilon;&sigma;&epsilon;
+  &gamma;&rho;&#x1F71;&phi;&epsilon;&sigma;&theta;&alpha;&iota;
+  &#x1F19;&lambda;&lambda;&eta;&nu;&iota;&sigma;&tau;&#x1F76;
+  &tau;&omicron;&#x1F7A;&sigmaf;
+  &delta;&eta;&mu;&omicron;&sigma;&#x1F77;&omicron;&upsilon;&sigmaf;
+  &tau;&#x1FF6;&nu;
+  &lambda;&omicron;&gamma;&omicron;&theta;&epsilon;&sigma;&#x1F77;&omega;&nu;
+  &kappa;&#x1F7D;&delta;&iota;&kappa;&alpha;&sigmaf;,
+  &#x1F00;&lambda;&lambda;'
+  &#x1F08;&rho;&alpha;&beta;&#x1F77;&omicron;&iota;&sigmaf;
+  &alpha;&#x1F50;&tau;&#x1F70;
+  &pi;&alpha;&rho;&alpha;&sigma;&eta;&mu;&alpha;&#x1F77;&nu;&epsilon;&sigma;&theta;&alpha;&iota;,
+  &chi;&omega;&rho;&#x1F76;&sigmaf; &tau;&#x1FF6;&nu;
+  &psi;&#x1F75;&phi;&omega;&nu;, &#x1F10;&pi;&epsilon;&iota;&delta;&#x1F74;
+  &#x1F00;&delta;&#x1F7B;&nu;&alpha;&tau;&omicron;&nu; &tau;&#x1FC7;
+  &#x1F10;&kappa;&epsilon;&#x1F77;&nu;&omega;&nu;
+  &gamma;&lambda;&#x1F7D;&sigma;&sigma;&#x1FC3;
+  &mu;&omicron;&nu;&#x1F71;&delta;&alpha; &#x1F22;
+  &delta;&upsilon;&#x1F71;&delta;&alpha; &#x1F22;
+  &tau;&rho;&iota;&#x1F71;&delta;&alpha; &#x1F22;
+  &#x1F40;&kappa;&tau;&#x1F7C; &#x1F25;&mu;&iota;&sigma;&upsilon; &#x1F22;
+  &tau;&rho;&#x1F77;&alpha;
+  &gamma;&rho;&#x1F71;&phi;&epsilon;&sigma;&theta;&alpha;&iota;&#x387;
+  &delta;&iota;&#x1F78; &kappa;&alpha;&#x1F76; &#x1F15;&omega;&sigmaf;
+  &sigma;&#x1F75;&mu;&epsilon;&rho;&#x1F79;&nu;
+  &epsilon;&#x1F30;&sigma;&iota;&nu; &sigma;&#x1F7A;&nu;
+  &alpha;&#x1F50;&tau;&omicron;&#x1FD6;&sigmaf;
+  &nu;&omicron;&tau;&#x1F71;&rho;&iota;&omicron;&iota;
+  &Chi;&rho;&iota;&sigma;&tau;&iota;&alpha;&nu;&omicron;&#x1F77;.</span>
+  The importance of this contemporaneous document was pointed out by
+  Martin, loc. cit. Karabacek, "Die Involutio im arabischen Schriftwesen,"
+  Vol. CXXXV of <i>Sitzungsberichte d. phil.-hist. Classe d. k. Akad. d.
+  Wiss.</i>, Vienna, 1896, p. 25, gives an Arabic date of 868 <span
+  class="scac">A.D.</span> in Greek letters.</p>
+
+  <p><a name="Nt_249" href="#NtA_249">[249]</a> <i>The Origin and History
+  of Our Numerals</i> (in Russian), Kiev, 1908; <i>The Independence of
+  European Arithmetic</i> (in Russian), Kiev.</p>
+
+  <p><a name="Nt_250" href="#NtA_250">[250]</a> Woepcke, loc. cit., pp.
+  462, 262.</p>
+
+  <p><a name="Nt_251" href="#NtA_251">[251]</a> Woepcke, loc. cit., p. 240.
+  <i><span class="special"
+  title="Hisab-al-Gobar">&#x1E24;is&#x101;b-al-&#x120;ob&#x101;r</span></i>,
+  by an anonymous author, probably Ab&#x16B; Sahl Dunash ibn Tamim, is
+  given by Steinschneider, "Die Mathematik bei den Juden," <i>Bibliotheca
+  Mathematica</i>, 1896, p. 26.</p>
+
+  <p><a name="Nt_252" href="#NtA_252">[252]</a> Steinschneider in the
+  <i>Abhandlungen</i>, Vol. III, p. 110.</p>
+
+  <p><a name="Nt_253" href="#NtA_253">[253]</a> See his <i>Grammaire
+  arabe</i>, Vol. I, Paris, 1810, plate VIII; Gerhardt, <i>Études</i>, pp.
+  9-11, and <i>Entstehung</i> etc., p. 8; I. F. Weidler, <i>Spicilegium
+  observationum ad historiam notarum numeralium pertinentium</i>,
+  Wittenberg, 1755, speaks of the "figura cifrarum Saracenicarum" as being
+  different from that of the "characterum Boethianorum," which are similar
+  to the "vulgar" or common numerals; see also Humboldt, loc. cit.</p>
+
+  <p><a name="Nt_254" href="#NtA_254">[254]</a> Gerhardt mentions it in his
+  <i>Entstehung</i> etc., p. 8; Woepcke, <i>Propagation</i>, states that
+  these numerals were used not for calculation, but very much as we use
+  Roman numerals. These superposed dots are found with both forms of
+  numerals (<i>Propagation</i>, pp. 244-246).</p>
+
+  <p><a name="Nt_255" href="#NtA_255">[255]</a> Gerhardt (<i>Études</i>, p.
+  9) from a manuscript in the Bibliothèque Nationale. The numeral forms are
+  <a href="images/072c.png"><img src="images/072c.png" class="middle"
+  style="height:2ex" alt="symbols" /></a>, 20 being indicated by <a
+  href="images/072d.png"><img src="images/072d.png" class="middle"
+  style="height:2.2ex" alt="symbol with dot" /></a> and 200 by <a
+  href="images/072e.png"><img src="images/072e.png" class="middle"
+  style="height:2.2ex" alt="symbol with 2 dots" /></a>. This scheme of zero
+  dots was also adopted by the Byzantine Greeks, for a manuscript of
+  Planudes in the Bibliothèque Nationale has numbers like <a
+  href="images/072f.png"><img src="images/072f.png" class="middle"
+  style="height:2ex" alt="pi alpha with 4 dots" /></a> for 8,100,000,000.
+  See Gerhardt, <i>Études</i>, p. 19. Pihan, <i>Exposé</i> etc., p. 208,
+  gives two forms, Asiatic and Maghrebian, of "Ghob&#x101;r" numerals.</p>
+
+  <p><a name="Nt_256" href="#NtA_256">[256]</a> See Chap. IV.</p>
+
+  <p><a name="Nt_257" href="#NtA_257">[257]</a> Possibly as early as the
+  third century <span class="scac">A.D.</span>, but probably of the eighth
+  or ninth. See Cantor, I (3), p. 598.</p>
+
+  <p><a name="Nt_258" href="#NtA_258">[258]</a> Ascribed by the Arabic
+  writer to India.</p>
+
+  <p><a name="Nt_259" href="#NtA_259">[259]</a> See Woepcke's description
+  of a manuscript in the Chasles library, "Recherches sur l'histoire des
+  sciences mathématiques chez les orientaux," <i>Journal Asiatique</i>, IV
+  (5), 1859, p. 358, note.</p>
+
+  <p><a name="Nt_260" href="#NtA_260">[260]</a> P. 56.</p>
+
+  <p><a name="Nt_261" href="#NtA_261">[261]</a> Reinaud, <i>Mémoire sur
+  l'Inde</i>, p. 399. In the fourteenth century one Sih&#x101;b
+  al-D&#x12B;n wrote a work on which, a scholiast to the Bodleian
+  manuscript remarks: "The science is called Algobar because the inventor
+  had the habit of writing the figures on a tablet covered with sand."
+  [Gerhardt, <i>Études, </i>p. 11, note.]</p>
+
+  <p><a name="Nt_262" href="#NtA_262">[262]</a> Gerhardt, <i>Entstehung
+  </i>etc., p. 20.</p>
+
+  <p><a name="Nt_263" href="#NtA_263">[263]</a> H. Suter, "Das Rechenbuch
+  des <span class="special" title="Abu Zakarija el-Hassar">Ab&#x16B;
+  Zakar&#x12B;j&#x101; el-&#x1E24;a&#x1E63;&#x1E63;&#x101;r</span>,"
+  <i>Bibliotheca Mathematica</i>, Vol. II (3), p. 15.</p>
+
+  <p><a name="Nt_264" href="#NtA_264">[264]</a> A. Devoulx, "Les chiffres
+  arabes," <i>Revue Africaine</i>, Vol. XVI, pp. 455-458.</p>
+
+  <p><a name="Nt_265" href="#NtA_265">[265]</a> <i>Kit&#x101;b
+  al-Fihrist</i>, G. Flügel, Leipzig, Vol. I, 1871, and Vol. II, 1872. This
+  work was published after Professor Flügel's death by J. Roediger and A.
+  Mueller. The first volume contains the Arabic text and the second volume
+  contains critical notes upon it.</p>
+
+  <p><a name="Nt_266" href="#NtA_266">[266]</a> Like those of line 5 in the
+  illustration on page <a href="#page69">69</a>.</p>
+
+  <p><a name="Nt_267" href="#NtA_267">[267]</a> Woepcke, <i>Recherches sur
+  l'histoire des sciences mathématiques chez les orientaux</i>, loc. cit.;
+  <i>Propagation, </i>p. 57.</p>
+
+  <p><a name="Nt_268" href="#NtA_268">[268]</a> <span class="special"
+  title="Al-Hassar's">Al-&#x1E24;a&#x1E63;&#x1E63;&#x101;r's</span> forms,
+  Suter, <i>Bibliotheca Mathematica</i>, Vol. II (3), p. 15.</p>
+
+  <p><a name="Nt_269" href="#NtA_269">[269]</a> Woepcke, <i>Sur une donnée
+  historique</i>, etc., loc. cit. The name <i>&#x121;ob&#x101;r</i> is not
+  used in the text. The manuscript from which these are taken is the oldest
+  (970 <span class="scac">A.D.</span>) Arabic document known to contain all
+  of the numerals.</p>
+
+  <p><a name="Nt_270" href="#NtA_270">[270]</a> Silvestre de Sacy, loc.
+  cit. He gives the ordinary modern Arabic forms, calling them
+  <i>Indien</i>.</p>
+
+  <p><a name="Nt_271" href="#NtA_271">[271]</a> Woepcke, "Introduction au
+  calcul Gob&#x101;r&#x12B; et Haw&#x101;&#x12B;," <i>Atti dell' accademia
+  pontificia dei nuovi Lincei</i>, Vol. XIX. The adjective applied to the
+  forms in 5 is <i>gob&#x101;r&#x12B;</i> and to those in 6
+  <i>indienne</i>. This is the direct opposite of Woepcke's use of these
+  adjectives in the <i>Recherches sur l'histoire</i> cited above, in which
+  the ordinary Arabic forms (like those in row 5) are called
+  <i>indiens</i>.</p>
+
+  <p>These forms are usually written from right to left.</p>
+
+  <p><a name="Nt_272" href="#NtA_272">[272]</a> J. G. Wilkinson, <i>The
+  Manners and Customs of the Ancient Egyptians</i>, revised by S. Birch,
+  London, 1878, Vol. II, p. 493, plate XVI.</p>
+
+  <p><a name="Nt_273" href="#NtA_273">[273]</a> There is an extensive
+  literature on this "Boethius-Frage." The reader who cares to go fully
+  into it should consult the various volumes of the <i>Jahrbuch über die
+  Fortschritte der Mathematik</i>.</p>
+
+  <p><a name="Nt_274" href="#NtA_274">[274]</a> This title was first
+  applied to Roman emperors in posthumous coins of Julius Cæsar.
+  Subsequently the emperors assumed it during their own lifetimes, thus
+  deifying themselves. See F. Gnecchi, <i>Monete romane</i>, 2d ed., Milan,
+  1900, p. 299.</p>
+
+  <p><a name="Nt_275" href="#NtA_275">[275]</a> This is the common spelling
+  of the name, although the more correct Latin form is Boëtius. See
+  Harper's <i>Dict. of Class. Lit. and Antiq.</i>, New York, 1897, Vol. I,
+  p. 213. There is much uncertainty as to his life. A good summary of the
+  evidence is given in the last two editions of the <i>Encyclopædia
+  Britannica</i>.</p>
+
+  <p><a name="Nt_276" href="#NtA_276">[276]</a> His father, Flavius Manlius
+  Boethius, was consul in 487.</p>
+
+  <p><a name="Nt_277" href="#NtA_277">[277]</a> There is, however, no good
+  historic evidence of this sojourn in Athens.</p>
+
+  <p><a name="Nt_278" href="#NtA_278">[278]</a> His arithmetic is dedicated
+  to Symmachus: "Domino suo patricio Symmacho Boetius." [Friedlein ed., p.
+  3.]</p>
+
+  <p><a name="Nt_279" href="#NtA_279">[279]</a> It was while here that he
+  wrote <i>De consolatione philosophiae</i>.</p>
+
+  <p><a name="Nt_280" href="#NtA_280">[280]</a> It is sometimes given as
+  525.</p>
+
+  <p><a name="Nt_281" href="#NtA_281">[281]</a> There was a medieval
+  tradition that he was executed because of a work on the Trinity.</p>
+
+  <p><a name="Nt_282" href="#NtA_282">[282]</a> Hence the <i>Divus</i> in
+  his name.</p>
+
+  <p><a name="Nt_283" href="#NtA_283">[283]</a> Thus Dante, speaking of his
+  burial place in the monastery of St. Pietro in Ciel d'Oro, at Pavia,
+  says:</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p class="i4">"The saintly soul, that shows</p>
+      <p>The world's deceitfulness, to all who hear him,</p>
+      <p>Is, with the sight of all the good that is,</p>
+      <p>Blest there. The limbs, whence it was driven, lie</p>
+      <p>Down in Cieldauro; and from martyrdom</p>
+      <p>And exile came it here."&mdash;<i>Paradiso</i>, Canto X.</p>
+    </div>
+  </div>
+  <p><a name="Nt_284" href="#NtA_284">[284]</a> Not, however, in the
+  mercantile schools. The arithmetic of Boethius would have been about the
+  last book to be thought of in such institutions. While referred to by
+  Bæda (672-735) and Hrabanus Maurus (c. 776-856), it was only after
+  Gerbert's time that the <i>Boëtii de institutione arithmetica libri
+  duo</i> was really a common work.</p>
+
+  <p><a name="Nt_285" href="#NtA_285">[285]</a> Also spelled
+  Cassiodorius.</p>
+
+  <p><a name="Nt_286" href="#NtA_286">[286]</a> As a matter of fact,
+  Boethius could not have translated any work by Pythagoras on music,
+  because there was no such work, but he did make the theories of the
+  Pythagoreans known. Neither did he translate Nicomachus, although he
+  embodied many of the ideas of the Greek writer in his own arithmetic.
+  Gibbon follows Cassiodorus in these statements in his <i>Decline and Fall
+  of the Roman Empire</i>, chap. xxxix. Martin pointed out with
+  positiveness the similarity of the first book of Boethius to the first
+  five books of Nicomachus. [<i>Les signes numéraux</i> etc., reprint, p.
+  4.]</p>
+
+  <p><a name="Nt_287" href="#NtA_287">[287]</a> The general idea goes back
+  to Pythagoras, however.</p>
+
+  <p><a name="Nt_288" href="#NtA_288">[288]</a> J. C. Scaliger in his
+  <i>Poëtice</i> also said of him: "Boethii Severini ingenium, eruditio,
+  ars, sapientia facile provocat omnes auctores, sive illi Graeci sint,
+  sive Latini" [Heilbronner, <i>Hist. math. univ.</i>, p. 387]. Libri,
+  speaking of the time of Boethius, remarks: "Nous voyons du temps de
+  Théodoric, les lettres reprendre une nouvelle vie en Italie, les écoles
+  florissantes et les savans honorés. Et certes les ouvrages de Boëce, de
+  Cassiodore, de Symmaque, surpassent de beaucoup toutes les productions du
+  siècle précédent." [<i>Histoire des mathématiques</i>, Vol. I, p.
+  78.]</p>
+
+  <p><a name="Nt_289" href="#NtA_289">[289]</a> Carra de Vaux,
+  <i>Avicenne</i>, Paris, 1900; Woepcke, <i>Sur l'introduction</i>, etc.;
+  Gerhardt, <i>Entstehung</i> etc., p. 20. Avicenna is a corruption from
+  Ibn S&#x12B;n&#x101;, as pointed out by Wüstenfeld, <i>Geschichte der
+  arabischen Aerzte und Naturforscher</i>, Göttingen, 1840. His full name
+  is <span class="special" title="Abu `Ali al-Hosein ibn Sina">Ab&#x16B;
+  &#x201B;Al&#x12B; al-&#x1E24;osein ibn S&#x12B;n&#x101;</span>. For notes
+  on Avicenna's arithmetic, see Woepcke, <i>Propagation</i>, p. 502.</p>
+
+  <p><a name="Nt_290" href="#NtA_290">[290]</a> On the early travel between
+  the East and the West the following works may be consulted: A.
+  Hillebrandt, <i>Alt-Indien</i>, containing "Chinesische Reisende in
+  Indien," Breslau, 1899, p. 179; C. A. Skeel, <i>Travel in the First
+  Century after Christ</i>, Cambridge, 1901, p. 142; M. Reinaud, "Relations
+  politiques et commerciales de l'empire romain avec l'Asie orientale," in
+  the <i>Journal Asiatique</i>, Mars-Avril, 1863, Vol. I (6), p. 93;
+  Beazley, <i>Dawn of Modern Geography, a History of Exploration and
+  Geographical Science from the Conversion of the Roman Empire to <span
+  class="scac">A.D.</span> 1420</i>, London, 1897-1906, 3 vols.; Heyd,
+  <i>Geschichte des Levanthandels im Mittelalter</i>, Stuttgart, 1897; J.
+  Keane, <i>The Evolution of Geography</i>, London, 1899, p. 38; A.
+  Cunningham, <i>Corpus inscriptionum Indicarum</i>, Calcutta, 1877, Vol.
+  I; A. Neander, <i>General History of the Christian Religion and
+  Church</i>, 5th American ed., Boston, 1855, Vol. III, p. 89; R. C. Dutt,
+  <i>A History of Civilization in Ancient India</i>, Vol. II, Bk. V, chap,
+  ii; E. C. Bayley, loc. cit., p. 28 et seq.; A. C. Burnell, loc. cit., p.
+  3; J. E. Tennent, <i>Ceylon</i>, London, 1859, Vol. I, p. 159; Geo.
+  Turnour, <i>Epitome of the History of Ceylon</i>, London, n.d., preface;
+  "Philalethes," <i>History of Ceylon</i>, London, 1816, chap, i; H. C.
+  Sirr, <i>Ceylon and the Cingalese</i>, London, 1850, Vol. I, chap. ix. On
+  the Hindu knowledge of the Nile see F. Wilford, <i>Asiatick
+  Researches</i>, Vol. III, p. 295, Calcutta, 1792.</p>
+
+  <p><a name="Nt_291" href="#NtA_291">[291]</a> G. Oppert, <i>On the
+  Ancient Commerce of India</i>, Madras, 1879, p. 8.</p>
+
+  <p><a name="Nt_292" href="#NtA_292">[292]</a> Gerhardt, <i>Études</i>
+  etc., pp. 8, 11.</p>
+
+  <p><a name="Nt_293" href="#NtA_293">[293]</a> See Smith's <i>Dictionary
+  of Greek and Roman Biography and Mythology</i>.</p>
+
+  <p><a name="Nt_294" href="#NtA_294">[294]</a> P. M. Sykes, <i>Ten
+  Thousand Miles in Persia, or Eight Years in Irán</i>, London, 1902, p.
+  167. Sykes was the first European to follow the course of Alexander's
+  army across eastern Persia.</p>
+
+  <p><a name="Nt_295" href="#NtA_295">[295]</a> Bühler, <i>Indian
+  Br&#x101;hma Alphabet</i>, note, p. 27; <i>Palaeographie</i>, p. 2;
+  <i>Herodoti Halicarnassei historia</i>, Amsterdam, 1763, Bk. IV, p. 300;
+  Isaac Vossius, <i>Periplus Scylacis Caryandensis</i>, 1639. It is
+  doubtful whether the work attributed to Scylax was written by him, but in
+  any case the work dates back to the fourth century <span
+  class="scac">B.C.</span> See Smith's <i>Dictionary of Greek and Roman
+  Biography</i>.</p>
+
+  <p><a name="Nt_296" href="#NtA_296">[296]</a> Herodotus, Bk. III.</p>
+
+  <p><a name="Nt_297" href="#NtA_297">[297]</a> Rameses II(?), the
+  <i>Sesoosis</i> of Diodorus Siculus.</p>
+
+  <p><a name="Nt_298" href="#NtA_298">[298]</a> <i>Indian Antiquary</i>,
+  Vol. I, p. 229; F. B. Jevons, <i>Manual of Greek Antiquities</i>, London,
+  1895, p. 386. On the relations, political and commercial, between India
+  and Egypt c. 72 <span class="scac">B.C.</span>, under Ptolemy Auletes,
+  see the <i>Journal Asiatique</i>, 1863, p. 297.</p>
+
+  <p><a name="Nt_299" href="#NtA_299">[299]</a> Sikandar, as the name still
+  remains in northern India.</p>
+
+  <p><a name="Nt_300" href="#NtA_300">[300]</a> <i>Harper's Classical
+  Dict.</i>, New York, 1897, Vol. I, p. 724; F. B. Jevons, loc. cit., p.
+  389; J. C. Marshman, <i>Abridgment of the History of India</i>, chaps. i
+  and ii.</p>
+
+  <p><a name="Nt_301" href="#NtA_301">[301]</a> Oppert, loc. cit., p. 11.
+  It was at or near this place that the first great Indian mathematician,
+  <span class="special" title="Aryabhata">&#x100;ryabha&#x1E6D;a</span>,
+  was born in 476 <span class="scac">A.D.</span></p>
+
+  <p><a name="Nt_302" href="#NtA_302">[302]</a> Bühler,
+  <i>Palaeographie</i>, p. 2, speaks of Greek coins of a period anterior to
+  Alexander, found in northern India. More complete information may be
+  found in <i>Indian Coins</i>, by E. J. Rapson, Strassburg, 1898, pp.
+  3-7.</p>
+
+  <p><a name="Nt_303" href="#NtA_303">[303]</a> Oppert, loc. cit., p. 14;
+  and to him is due other similar information.</p>
+
+  <p><a name="Nt_304" href="#NtA_304">[304]</a> J. Beloch, <i>Griechische
+  Geschichte</i>, Vol. III, Strassburg, 1904, pp. 30-31.</p>
+
+  <p><a name="Nt_305" href="#NtA_305">[305]</a> E.g., the denarius, the
+  words for hour and minute (<span title="hôra, lepton" class="grk"
+  >&#x1F65;&rho;&alpha;, &lambda;&epsilon;&pi;&tau;&#x1F79;&nu;</span>),
+  and possibly the signs of the zodiac. [R. Caldwell, <i>Comparative
+  Grammar of the Dravidian Languages</i>, London, 1856, p. 438.] On the
+  probable Chinese origin of the zodiac see Schlegel, loc. cit.</p>
+
+  <p><a name="Nt_306" href="#NtA_306">[306]</a> Marie, Vol. II, p. 73; R.
+  Caldwell, loc. cit.</p>
+
+  <p><a name="Nt_307" href="#NtA_307">[307]</a> A. Cunningham, loc. cit.,
+  p. 50.</p>
+
+  <p><a name="Nt_308" href="#NtA_308">[308]</a> C. A. J. Skeel,
+  <i>Travel</i>, loc. cit., p. 14.</p>
+
+  <p><a name="Nt_309" href="#NtA_309">[309]</a> <i>Inchiver</i>, from
+  <i>inchi</i>, "the green root." [<i>Indian Antiquary</i>, Vol. I, p.
+  352.]</p>
+
+  <p><a name="Nt_310" href="#NtA_310">[310]</a> In China dating only from
+  the second century <span class="scac">A.D.</span>, however.</p>
+
+  <p><a name="Nt_311" href="#NtA_311">[311]</a> The Italian
+  <i>morra</i>.</p>
+
+  <p><a name="Nt_312" href="#NtA_312">[312]</a> J. Bowring, <i>The Decimal
+  System</i>, London, 1854, p. 2.</p>
+
+  <p><a name="Nt_313" href="#NtA_313">[313]</a> H. A. Giles, lecture at
+  Columbia University, March 12, 1902, on "China and Ancient Greece."</p>
+
+  <p><a name="Nt_314" href="#NtA_314">[314]</a> Giles, loc. cit.</p>
+
+  <p><a name="Nt_315" href="#NtA_315">[315]</a> E.g., the names for grape,
+  radish (<i>la-po</i>, <span title="rhaphê" class="grk"
+  >&#x1FE5;&#x1F71;&phi;&eta;</span>), water-lily (<i>si-kua</i>,  "west
+  gourds"; <span title="sikua" class="grk"
+  >&sigma;&iota;&kappa;&#x1F7B;&alpha;</span>, "gourds"), are much alike.
+  [Giles, loc. cit.]</p>
+
+  <p><a name="Nt_316" href="#NtA_316">[316]</a> <i>Epistles</i>, I, 1,
+  45-46. On the Roman trade routes, see Beazley, loc. cit., Vol. I, p.
+  179.</p>
+
+  <p><a name="Nt_317" href="#NtA_317">[317]</a> <i>Am. Journ. of
+  Archeol.</i>, Vol. IV, p. 366.</p>
+
+  <p><a name="Nt_318" href="#NtA_318">[318]</a> M. Perrot gives this
+  conjectural restoration of his words: "Ad me ex India regum legationes
+  saepe missi sunt numquam antea visae apud quemquam principem Romanorum."
+  [M. Reinaud, "Relations politiques et commerciales de l'empire romain
+  avec l'Asie orientale," <i>Journ. Asiat.</i>, Vol. I (6), p. 93.]</p>
+
+  <p><a name="Nt_319" href="#NtA_319">[319]</a> Reinaud, loc. cit., p. 189.
+  Florus, II, 34 (IV, 12), refers to it: "Seres etiam habitantesque sub
+  ipso sole Indi, cum gemmis et margaritis elephantes quoque inter munera
+  trahentes nihil magis quam longinquitatem viae imputabant." Horace shows
+  his geographical knowledge by saying: "Not those who drink of the deep
+  Danube shall now break the Julian edicts; not the Getae, not the Seres,
+  nor the perfidious Persians, nor those born on the river Tanaïs."
+  [<i>Odes</i>, Bk. IV, Ode 15, 21-24.]</p>
+
+  <p><a name="Nt_320" href="#NtA_320">[320]</a> "Qua virtutis
+  moderationisque fama Indos etiam ac Scythas auditu modo cognitos pellexit
+  ad amicitiam suam populique Romani ultro per legatos petendam." [Reinaud,
+  loc. cit., p. 180.]</p>
+
+  <p><a name="Nt_321" href="#NtA_321">[321]</a> Reinaud, loc. cit., p.
+  180.</p>
+
+  <p><a name="Nt_322" href="#NtA_322">[322]</a> <i>Georgics</i>, II,
+  170-172. So Propertius (<i>Elegies</i>, III, 4):</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Arma deus Caesar dites meditatur ad Indos</p>
+      <p class="i2">Et freta gemmiferi findere classe maris.</p>
+    </div>
+  </div>
+  <p>"The divine Cæsar meditated carrying arms against opulent India, and
+  with his ships to cut the gem-bearing seas."</p>
+
+  <p><a name="Nt_323" href="#NtA_323">[323]</a> Heyd, loc. cit., Vol. I, p.
+  4.</p>
+
+  <p><a name="Nt_324" href="#NtA_324">[324]</a> Reinaud, loc. cit., p.
+  393.</p>
+
+  <p><a name="Nt_325" href="#NtA_325">[325]</a> The title page of Calandri
+  (1491), for example, represents Pythagoras with these numerals before
+  him. [Smith, <i>Rara Arithmetica</i>, p. 46.] Isaacus Vossius,
+  <i>Observationes ad Pomponium Melam de situ orbis</i>, 1658, maintained
+  that the Arabs derived these numerals from the west. A learned
+  dissertation to this effect, but deriving them from the Romans instead of
+  the Greeks, was written by Ginanni in 1753 (<i>Dissertatio mathematica
+  critica de numeralium notarum minuscularum origine</i>, Venice, 1753).
+  See also Mannert, <i>De numerorum quos arabicos vocant vera origine
+  Pythagorica</i>, Nürnberg, 1801. Even as late as 1827 Romagnosi (in his
+  supplement to <i>Ricerche storiche sull' India</i> etc., by Robertson,
+  Vol. II, p. 580, 1827) asserted that Pythagoras originated them. [R.
+  Bombelli, <i>L'antica numerazione italica</i>, Rome, 1876, p. 59.] Gow
+  (<i>Hist. of Greek Math.</i>, p. 98) thinks that Iamblichus must have
+  known a similar system in order to have worked out certain of his
+  theorems, but this is an unwarranted deduction from the passage
+  given.</p>
+
+  <p><a name="Nt_326" href="#NtA_326">[326]</a> A. Hillebrandt,
+  <i>Alt-Indien</i>, p. 179.</p>
+
+  <p><a name="Nt_327" href="#NtA_327">[327]</a> J. C. Marshman, loc. cit.,
+  chaps. i and ii.</p>
+
+  <p><a name="Nt_328" href="#NtA_328">[328]</a> He reigned 631-579 <span
+  class="scac">A.D.</span>; called Nu&#x15B;&#x12B;rw&#x101;n, <i>the holy
+  one</i>.</p>
+
+  <p><a name="Nt_329" href="#NtA_329">[329]</a> J. Keane, <i>The Evolution
+  of Geography</i>, London, 1899, p. 38.</p>
+
+  <p><a name="Nt_330" href="#NtA_330">[330]</a> The Arabs who lived in and
+  about Mecca.</p>
+
+  <p><a name="Nt_331" href="#NtA_331">[331]</a> S. Guyard, in <i>Encyc.
+  Brit.</i>, 9th ed., Vol. XVI, p. 597.</p>
+
+  <p><a name="Nt_332" href="#NtA_332">[332]</a> Oppert, loc. cit., p.
+  29.</p>
+
+  <p><a name="Nt_333" href="#NtA_333">[333]</a> "At non credendum est id in
+  Autographis contigisse, aut vetustioribus Codd. MSS." [Wallis, <i>Opera
+  omnia</i>, Vol. II, p. 11.]</p>
+
+  <p><a name="Nt_334" href="#NtA_334">[334]</a> In <i>Observationes ad
+  Pomponium Melam de situ orbis</i>. The question was next taken up in a
+  large way by Weidler, loc. cit., <i>De characteribus</i> etc., 1727, and
+  in <i>Spicilegium</i> etc., 1755.</p>
+
+  <p><a name="Nt_335" href="#NtA_335">[335]</a> The best edition of these
+  works is that of G. Friedlein, <i>Anicii Manlii Torquati Severini Boetii
+  de institutione arithmetica libri duo, de institutione musica libri
+  quinque. Accedit geometria quae fertur Boetii</i>.... Leipzig.... <span
+  class="scac">MDCCCLXVII</span>.</p>
+
+  <p><a name="Nt_336" href="#NtA_336">[336]</a> See also P. Tannery, "Notes
+  sur la pseudo-géometrie de Boèce," in <i>Bibliotheca Mathematica</i>,
+  Vol. I (3), p. 39. This is not the geometry in two books in which are
+  mentioned the numerals. There is a manuscript of this pseudo-geometry of
+  the ninth century, but the earliest one of the other work is of the
+  eleventh century (Tannery), unless the Vatican codex is of the tenth
+  century as Friedlein (p. 372) asserts.</p>
+
+  <p><a name="Nt_337" href="#NtA_337">[337]</a> Friedlein feels that it is
+  partly spurious, but he says: "Eorum librorum, quos Boetius de geometria
+  scripsisse dicitur, investigare veram inscriptionem nihil aliud esset
+  nisi operam et tempus perdere." [Preface, p. v.] N. Bubnov in the Russian
+  <i>Journal of the Ministry of Public Instruction</i>, 1907, in an article
+  of which a synopsis is given in the <i>Jahrbuch über die Fortschritte der
+  Mathematik</i> for 1907, asserts that the geometry was written in the
+  eleventh century.</p>
+
+  <p><a name="Nt_338" href="#NtA_338">[338]</a> The most noteworthy of
+  these was for a long time Cantor (<i>Geschichte</i>, Vol. I., 3d ed., pp.
+  587-588), who in his earlier days even believed that Pythagoras had known
+  them. Cantor says (<i>Die römischen Agrimensoren</i>, Leipzig, 1875, p.
+  130): "Uns also, wir wiederholen es, ist die Geometrie des Boetius echt,
+  dieselbe Schrift, welche er nach Euklid bearbeitete, von welcher ein
+  Codex bereits in Jahre 821 im  Kloster Reichenau vorhanden war, von
+  welcher ein anderes Exemplar im Jahre 982 zu Mantua in die Hände
+  Gerbert's gelangte, von welcher mannigfache Handschriften noch heute
+  vorhanden sind." But against this opinion of the antiquity of MSS.
+  containing these numerals is the important statement of P. Tannery,
+  perhaps the most critical of modern historians of mathematics, that none
+  exists earlier than the eleventh century. See also J. L. Heiberg in
+  <i>Philologus, Zeitschrift f. d. klass. Altertum</i>, Vol. XLIII, p.
+  508.</p>
+
+  <p>Of Cantor's predecessors, Th. H. Martin was one of the most prominent,
+  his argument for authenticity appearing in the <i>Revue Archéologique</i>
+  for 1856-1857, and in his treatise <i>Les signes numéraux</i> etc. See
+  also M. Chasles, "De la connaissance qu'ont eu les anciens d'une
+  numération décimale écrite qui fait usage de neuf chiffres prenant les
+  valeurs de position," <i>Comptes rendus</i>, Vol. VI, pp. 678-680; "Sur
+  l'origine de notre système de numération," <i>Comptes rendus</i>, Vol.
+  VIII, pp. 72-81; and note "Sur le passage du premier livre de la
+  géométrie de Boèce, relatif à un nouveau système de numération," in his
+  work <i>Aperçu historique sur l'origine et le devéloppement des méthodes
+  en géométrie</i>, of which the first edition appeared in 1837.</p>
+
+  <p><a name="Nt_339" href="#NtA_339">[339]</a> J. L. Heiberg places the
+  book in the eleventh century on philological grounds, <i>Philologus</i>,
+  loc. cit.; Woepcke, in <i>Propagation</i>, p. 44; Blume, Lachmann, and
+  Rudorff, <i>Die Schriften der römischen Feldmesser</i>, Berlin, 1848;
+  Boeckh, <i>De abaco graecorum</i>, Berlin, 1841; Friedlein, in his
+  Leipzig edition of 1867; Weissenborn, <i>Abhandlungen</i>, Vol. II, p.
+  185, his <i>Gerbert</i>, pp. 1, 247, and his <i>Geschichte der Einführung
+  der jetzigen Ziffern in Europa durch Gerbert</i>, Berlin, 1892, p. 11;
+  Bayley, loc. cit., p. 59; Gerhardt, <i>Études</i>, p. 17, <i>Entstehung
+  und Ausbreitung</i>, p. 14; Nagl, <i>Gerbert</i>, p. 57; Bubnov, loc.
+  cit. See also the discussion by Chasles, Halliwell, and Libri, in the
+  <i>Comptes rendus</i>, 1839, Vol. IX, p. 447, and in Vols. VIII, XVI,
+  XVII of the same journal.</p>
+
+  <p><a name="Nt_340" href="#NtA_340">[340]</a> J. Marquardt, <i>La vie
+  privée des Romains</i>, Vol. II (French trans.), p. 505, Paris, 1893.</p>
+
+  <p><a name="Nt_341" href="#NtA_341">[341]</a> In a Plimpton manuscript of
+  the arithmetic of Boethius of the thirteenth century, for example, the
+  Roman numerals are all replaced by the Arabic, and the same is true in
+  the first printed edition of the book. (See Smith's <i>Rara
+  Arithmetica</i>, pp. 434, 25-27.) D. E. Smith also copied from a
+  manuscript of the arithmetic in the Laurentian library at Florence, of
+  1370, the following forms, <a href="images/092a.png"><img
+  src="images/092a.png" class="middle" style="height:2.5ex" alt="Forged
+  numerals" /></a> which, of course, are interpolations. An interesting
+  example of a forgery in ecclesiastical matters is in the charter said to
+  have been given by St. Patrick, granting indulgences to the benefactors
+  of Glastonbury, dated "In nomine domini nostri Jhesu Christi Ego
+  Patricius humilis servunculus Dei anno incarnationis ejusdem ccccxxx."
+  Now if the Benedictines are right in saying that Dionysius Exiguus, a
+  Scythian monk, first arranged the Christian chronology c. 532 <span
+  class="scac">A.D.</span>, this can hardly be other than spurious. See
+  Arbuthnot, loc. cit., p. 38.</p>
+
+  <p><a name="Nt_342" href="#NtA_342">[342]</a> Halliwell, in his <i>Rara
+  Mathematica, </i>p. 107, states that the disputed passage is not in a
+  manuscript belonging to Mr. Ames, nor in one at Trinity College. See also
+  Woepcke, in <i>Propagation</i>, pp. 37 and 42. It was the evident
+  corruption of the texts in such editions of Boethius as those of Venice,
+  1499, Basel, 1546 and 1570, that led Woepcke to publish his work <i>Sur
+  l'introduction de l'arithmétique indienne en Occident</i>.</p>
+
+  <p><a name="Nt_343" href="#NtA_343">[343]</a> They are found in none of
+  the very ancient manuscripts, as, for example, in the ninth-century (?)
+  codex in the Laurentian library which one of the authors has examined. It
+  should be said, however, that the disputed passage was written after the
+  arithmetic, for it contains a reference to that work. See the Friedlein
+  ed., p. 397.</p>
+
+  <p><a name="Nt_344" href="#NtA_344">[344]</a> Smith, <i>Rara
+  Arithmetica</i>, p. 66.</p>
+
+  <p><a name="Nt_345" href="#NtA_345">[345]</a> J. L. Heiberg,
+  <i>Philologus</i>, Vol. XLIII, p. 507.</p>
+
+  <p><a name="Nt_346" href="#NtA_346">[346]</a> "Nosse autem huius artis
+  dispicientem, quid sint digiti, quid articuli, quid compositi, quid
+  incompositi numeri." [Friedlein ed., p. 395.]</p>
+
+  <p><a name="Nt_347" href="#NtA_347">[347]</a> <i>De ratione abaci.</i> In
+  this he describes "quandam formulam, quam ob honorem sui praeceptoris
+  mensam Pythagoream nominabant ... a posterioribus appellabatur abacus."
+  This, as pictured in the text, is the common Gerbert abacus. In the
+  edition in Migne's <i>Patrologia Latina</i>, Vol. LXIII, an ordinary
+  multiplication table (sometimes called Pythagorean abacus) is given in
+  the illustration.</p>
+
+  <p><a name="Nt_348" href="#NtA_348">[348]</a> "Habebant enim diverse
+  formatos apices vel caracteres." See the reference to Gerbert on p.
+  117.</p>
+
+  <p><a name="Nt_349" href="#NtA_349">[349]</a> C. Henry, "Sur l'origine de
+  quelques notations mathématiques," <i>Revue Archéologique</i>, 1879,
+  derives these from the initial letters used as abbreviations for the
+  names of the numerals, a theory that finds few supporters.</p>
+
+  <p><a name="Nt_350" href="#NtA_350">[350]</a> E.g., it appears in
+  Schonerus, <i>Algorithmus Demonstratus</i>, Nürnberg, 1534, f. A4. In
+  England it appeared in the earliest English arithmetical manuscript
+  known, <i>The Crafte of Nombrynge</i>: "¶ fforthermore ye most
+  vndirstonde that in this craft ben vsid teen figurys, as here bene writen
+  for ensampul, <a href="images/093a.png"><img src="images/093a.png"
+  class="middle" style="height:2ex" alt="Numerals" /></a> ... in the quych
+  we vse teen figurys of Inde. Questio. ¶ why ten fyguris of Inde? Solucio.
+  for as I have sayd afore thei were fonde fyrst in Inde of a kynge of that
+  Cuntre, that was called Algor." See Smith, <i>An Early English
+  Algorism</i>, loc. cit.</p>
+
+  <p><a name="Nt_351" href="#NtA_351">[351]</a> Friedlein ed., p. 397.</p>
+
+  <p><a name="Nt_352" href="#NtA_352">[352]</a> Carlsruhe codex of
+  Gerlando.</p>
+
+  <p><a name="Nt_353" href="#NtA_353">[353]</a> Munich codex of
+  Gerlando.</p>
+
+  <p><a name="Nt_354" href="#NtA_354">[354]</a> Carlsruhe codex of
+  Bernelinus.</p>
+
+  <p><a name="Nt_355" href="#NtA_355">[355]</a> Munich codex of
+  Bernelinus.</p>
+
+  <p><a name="Nt_356" href="#NtA_356">[356]</a> Turchill, c. 1200.</p>
+
+  <p><a name="Nt_357" href="#NtA_357">[357]</a> Anon. MS., thirteenth
+  century, Alexandrian Library, Rome.</p>
+
+  <p><a name="Nt_358" href="#NtA_358">[358]</a> Twelfth-century Boethius,
+  Friedlein, p. 396.</p>
+
+  <p><a name="Nt_359" href="#NtA_359">[359]</a> Vatican codex, tenth
+  century, Boethius.</p>
+
+  <p><a name="Nt_360" href="#NtA_360">[360]</a> a, h, i, are from the
+  Friedlein ed.; the original in the manuscript from which a is taken
+  contains a zero symbol, as do all of the six plates given by Friedlein.
+  b-e from the Boncompagni <i>Bulletino</i>, Vol. X, p. 596; f ibid., Vol.
+  XV, p. 186; g <i>Memorie della classe di sci., Reale Acc. dei Lincei</i>,
+  An. CCLXXIV (1876-1877), April, 1877. A twelfth-century arithmetician,
+  possibly John of Luna (Hispalensis, of Seville, c. 1150), speaks of the
+  great diversity of these forms even in his day, saying: "Est autem in
+  aliquibus figuram istarum apud multos diuersitas. Quidam enim septimam
+  hanc figuram representant <a href="images/094j.png"><img
+  src="images/094j.png" class="middle" style="height:2.5ex" alt="Symbol"
+  /></a> alii autem sic <a href="images/094k.png"><img
+  src="images/094k.png" class="middle" style="height:2.5ex" alt="Symbol"
+  /></a>, uel sic <a href="images/094l.png"><img src="images/094l.png"
+  class="middle" style="height:2.5ex" alt="Symbol" /></a>. Quidam vero
+  quartam sic <a href="images/094m.png"><img src="images/094m.png"
+  class="middle" style="height:2.5ex" alt="Symbol" /></a>." [Boncompagni,
+  <i>Trattati</i>, Vol. II, p. 28.]</p>
+
+  <p><a name="Nt_361" href="#NtA_361">[361]</a> Loc. cit., p. 59.</p>
+
+  <p><a name="Nt_362" href="#NtA_362">[362]</a> Ibid., p. 101.</p>
+
+  <p><a name="Nt_363" href="#NtA_363">[363]</a> Loc. cit., p. 396.</p>
+
+  <p><a name="Nt_364" href="#NtA_364">[364]</a> Khosr&#x16B; I, who began
+  to reign in 531 <span class="scac">A.D.</span> See W. S. W Vaux,
+  <i>Persia, </i>London, 1875, p. 169; Th. Nöldeke, <i>Aufsätze zur
+  persichen Geschichte</i>, Leipzig, 1887, p. 113, and his article in the
+  ninth edition of the <i>Encyclopædia Britannica</i>.</p>
+
+  <p><a name="Nt_365" href="#NtA_365">[365]</a> Colebrooke, <i>Essays</i>,
+  Vol. II, p. 504, on the authority of Ibn al-Adam&#x12B;, astronomer, in a
+  work published by his continuator Al-Q&#x101;sim in 920 <span
+  class="scac">A.D.</span>; Al-B&#x12B;r&#x16B;n&#x12B;, <i>India, </i>Vol.
+  II, p. 15.</p>
+
+  <p><a name="Nt_366" href="#NtA_366">[366]</a> H. Suter, <i>Die
+  Mathematiker</i> etc., pp. 4-5, states that Al-Faz&#x101;r&#x12B; died
+  between 796 and 806.</p>
+
+  <p><a name="Nt_367" href="#NtA_367">[367]</a> Suter, loc. cit., p.
+  63.</p>
+
+  <p><a name="Nt_368" href="#NtA_368">[368]</a> Suter, loc. cit., p.
+  74.</p>
+
+  <p><a name="Nt_369" href="#NtA_369">[369]</a> Suter, <i>Das
+  Mathematiker-Verzeichniss im Fihrist</i>. The references to Suter, unless
+  otherwise stated, are to his later work <i>Die Mathematiker und
+  Astronomen der Araber</i> etc.</p>
+
+  <p><a name="Nt_370" href="#NtA_370">[370]</a> Suter, <i>Fihrist</i>, p.
+  37, no date.</p>
+
+  <p><a name="Nt_371" href="#NtA_371">[371]</a> Suter, <i>Fihrist</i>, p.
+  38, no date.</p>
+
+  <p><a name="Nt_372" href="#NtA_372">[372]</a> Possibly late tenth, since
+  he refers to one arithmetical work which is entitled <i>Book of the
+  Cyphers</i> in his <i>Chronology</i>, English ed., p. 132. Suter, <i>Die
+  Mathematiker</i> etc., pp. 98-100, does not mention this work; see the
+  <i>Nachträge und Berichtigungen</i>, pp. 170-172.</p>
+
+  <p><a name="Nt_373" href="#NtA_373">[373]</a> Suter, pp. 96-97.</p>
+
+  <p><a name="Nt_374" href="#NtA_374">[374]</a> Suter, p. 111.</p>
+
+  <p><a name="Nt_375" href="#NtA_375">[375]</a> Suter, p. 124. As the name
+  shows, he came from the West.</p>
+
+  <p><a name="Nt_376" href="#NtA_376">[376]</a> Suter, p. 138.</p>
+
+  <p><a name="Nt_377" href="#NtA_377">[377]</a> Hankel, <i>Zur Geschichte
+  der Mathematik</i>, p. 256, refers to him as writing on the Hindu art of
+  reckoning; Suter, p. 162.</p>
+
+  <p><a name="Nt_378" href="#NtA_378">[378]</a> <span title="Psêphophoria kat' Indous" class="grk"
+  >&Psi;&eta;&phi;&omicron;&phi;&omicron;&rho;&#x1F77;&alpha;
+  &kappa;&alpha;&tau;' &#x1F38;&nu;&delta;&omicron;&#x1F7B;&sigmaf;</span>,
+  Greek ed., C. I. Gerhardt, Halle, 1865; and German translation, <i>Das
+  Rechenbuch des Maximus Planudes</i>, H. Wäschke, Halle, 1878.</p>
+
+  <p><a name="Nt_379" href="#NtA_379">[379]</a> "Sur une donnée historique
+  relative à l'emploi des chiffres indiens par les Arabes," Tortolini's
+  <i>Annali di scienze mat. e fis.</i>, 1855.</p>
+
+  <p><a name="Nt_380" href="#NtA_380">[380]</a> Suter, p. 80.</p>
+
+  <p><a name="Nt_381" href="#NtA_381">[381]</a> Suter, p. 68.</p>
+
+  <p><a name="Nt_382" href="#NtA_382">[382]</a> Sprenger also calls
+  attention to this fact, in the <i>Zeitschrift d. deutschen morgenländ.
+  Gesellschaft</i>, Vol. XLV, p. 367.</p>
+
+  <p><a name="Nt_383" href="#NtA_383">[383]</a> Libri, <i>Histoire des
+  mathématiques</i>, Vol. I, p. 147.</p>
+
+  <p><a name="Nt_384" href="#NtA_384">[384]</a> "Dictant la paix à
+  l'empereur de Constantinople, l'Arabe victorieux demandait des manuscrits
+  et des savans." [Libri, loc. cit., p. 108.]</p>
+
+  <p><a name="Nt_385" href="#NtA_385">[385]</a> Persian <i>bagadata</i>,
+  "God-given."</p>
+
+  <p><a name="Nt_386" href="#NtA_386">[386]</a> One of the Abbassides, the
+  (at least pretended) descendants of &#x201B;Al-Abb&#x101;s, uncle and
+  adviser of <span class="special"
+  title="Mohammed">Mo&#x1E25;ammed</span>.</p>
+
+  <p><a name="Nt_387" href="#NtA_387">[387]</a> E. Reclus, <i>Asia</i>,
+  American ed., N. Y., 1891, Vol. IV, p. 227.</p>
+
+  <p><a name="Nt_388" href="#NtA_388">[388]</a> <i>Historical Sketches</i>,
+  Vol. III, chap. iii.</p>
+
+  <p><a name="Nt_389" href="#NtA_389">[389]</a> On its prominence at that
+  period see Villicus, p. 70.</p>
+
+  <p><a name="Nt_390" href="#NtA_390">[390]</a> See pp. 4-5.</p>
+
+  <p><a name="Nt_391" href="#NtA_391">[391]</a> Smith, D. E., in the
+  <i>Cantor Festschrift</i>, 1909, note pp. 10-11. See also F. Woepcke,
+  <i>Propagation</i>.</p>
+
+  <p><a name="Nt_392" href="#NtA_392">[392]</a> Eneström, in <i>Bibliotheca
+  Mathematica</i>, Vol. I (3), p. 499; Cantor, <i>Geschichte</i>, Vol. I
+  (3), p. 671.</p>
+
+  <p><a name="Nt_393" href="#NtA_393">[393]</a> Cited in Chapter I. It
+  begins: "Dixit algoritmi: laudes deo rectori nostro atque defensori
+  dicamus dignas." It is devoted entirely to the fundamental operations and
+  contains no applications.</p>
+
+  <p><a name="Nt_394" href="#NtA_394">[394]</a> M. Steinschneider, "Die
+  Mathematik bei den Juden," <i>Bibliotheca Mathematica</i>, Vol. VIII (2),
+  p. 99. See also the reference to this writer in Chapter I.</p>
+
+  <p><a name="Nt_395" href="#NtA_395">[395]</a> Part of this work has been
+  translated from a Leyden MS. by F. Woepcke, <i>Propagation</i>, and more
+  recently by H. Suter, <i>Bibliotheca Mathematica</i>, Vol. VII (3), pp.
+  113-119.</p>
+
+  <p><a name="Nt_396" href="#NtA_396">[396]</a> A. Neander, <i>General
+  History of the Christian Religion and Church</i>, 5th American ed.,
+  Boston, 1855, Vol. III, p. 335.</p>
+
+  <p><a name="Nt_397" href="#NtA_397">[397]</a> Beazley, loc. cit., Vol. I,
+  p. 49.</p>
+
+  <p><a name="Nt_398" href="#NtA_398">[398]</a> Beazley, loc. cit., Vol. I,
+  pp. 50, 460.</p>
+
+  <p><a name="Nt_399" href="#NtA_399">[399]</a> See pp. <a
+  href="#page7">7</a>-<a href="#page8">8</a>.</p>
+
+  <p><a name="Nt_400" href="#NtA_400">[400]</a> The name also appears as
+  <span class="special" title="Mohammed">Mo&#x1E25;ammed</span>
+  Ab&#x16B;'l-Q&#x101;sim, and Ibn Hauqal. Beazley, loc. cit., Vol. I, p.
+  45.</p>
+
+  <p><a name="Nt_401" href="#NtA_401">[401]</a> <i>Kit&#x101;b
+  al-mas&#x101;lik wa'l-mam&#x101;lik.</i></p>
+
+  <p><a name="Nt_402" href="#NtA_402">[402]</a> Reinaud, <i>Mém. sur
+  l'Inde</i>; in Gerhardt, <i>Études</i>, p. 18.</p>
+
+  <p><a name="Nt_403" href="#NtA_403">[403]</a> Born at Shiraz in 1193. He
+  himself had traveled from India to Europe.</p>
+
+  <p><a name="Nt_404" href="#NtA_404">[404]</a> <i>Gulistan</i> (<i>Rose
+  Garden</i>), Gateway the third, XXII. Sir Edwin Arnold's translation, N.
+  Y., 1899, p. 177.</p>
+
+  <p><a name="Nt_405" href="#NtA_405">[405]</a> Cunningham, loc. cit., p.
+  81.</p>
+
+  <p><a name="Nt_406" href="#NtA_406">[406]</a> Putnam, <i>Books</i>, Vol.
+  I, p. 227:</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p class="hg3">"Non semel externas peregrino tramite terras</p>
+      <p>Jam peragravit ovans, sophiae deductus amore,</p>
+      <p>Si quid forte novi librorum seu studiorum</p>
+      <p>Quod secum ferret, terris reperiret in illis.</p>
+      <p>Hic quoque Romuleum venit devotus ad urbem."</p>
+    </div>
+  </div>
+  <p>("More than once he has traveled joyfully through remote regions and
+  by strange roads, led on by his zeal for knowledge and seeking to
+  discover in foreign lands novelties in books or in studies which he could
+  take back with him. And this zealous student journeyed to the city of
+  Romulus.")</p>
+
+  <p><a name="Nt_407" href="#NtA_407">[407]</a> A. Neander, <i>General
+  History of the Christian Religion and Church</i>, 5th American ed.,
+  Boston, 1855, Vol. III, p. 89, note 4; Libri, <i>Histoire</i>, Vol. I, p.
+  143.</p>
+
+  <p><a name="Nt_408" href="#NtA_408">[408]</a> Cunningham, loc. cit., p.
+  81.</p>
+
+  <p><a name="Nt_409" href="#NtA_409">[409]</a> Heyd, loc. cit., Vol. I, p.
+  4.</p>
+
+  <p><a name="Nt_410" href="#NtA_410">[410]</a> Ibid., p. 5.</p>
+
+  <p><a name="Nt_411" href="#NtA_411">[411]</a> Ibid., p. 21.</p>
+
+  <p><a name="Nt_412" href="#NtA_412">[412]</a> Ibid., p. 23.</p>
+
+  <p><a name="Nt_413" href="#NtA_413">[413]</a> Libri, <i>Histoire</i>,
+  Vol. I, p. 167.</p>
+
+  <p><a name="Nt_414" href="#NtA_414">[414]</a> Picavet, <i>Gerbert, un
+  pape philosophe, d'après l'histoire et d'après la légende</i>, Paris,
+  1897, p. 19.</p>
+
+  <p><a name="Nt_415" href="#NtA_415">[415]</a> Beazley, loc. cit., Vol. I,
+  chap, i, and p. 54 seq.</p>
+
+  <p><a name="Nt_416" href="#NtA_416">[416]</a> Ibid., p. 57.</p>
+
+  <p><a name="Nt_417" href="#NtA_417">[417]</a> Libri, <i>Histoire</i>,
+  Vol. I, p. 110, n., citing authorities, and p. 152.</p>
+
+  <p><a name="Nt_418" href="#NtA_418">[418]</a> Possibly the old tradition,
+  "Prima dedit nautis usum magnetis Amalphis," is true so far as it means
+  the modern form of compass card. See Beazley, loc. cit., Vol. II, p.
+  398.</p>
+
+  <p><a name="Nt_419" href="#NtA_419">[419]</a> R. C. Dutt, loc. cit., Vol.
+  II, p. 312.</p>
+
+  <p><a name="Nt_420" href="#NtA_420">[420]</a> E. J. Payne, in <i>The
+  Cambridge Modern History</i>, London, 1902, Vol. I, chap. i.</p>
+
+  <p><a name="Nt_421" href="#NtA_421">[421]</a> Geo. Phillips, "The
+  Identity of Marco Polo's Zaitun with Changchau, in T'oung pao,"
+  <i>Archives pour servir à l'étude de l'histoire de l'Asie orientale</i>,
+  Leyden, 1890, Vol. I, p. 218. W. Heyd, <i>Geschichte des Levanthandels im
+  Mittelalter</i>, Vol. II, p. 216.</p>
+
+  <p>The Palazzo dei Poli, where Marco was born and died, still stands in
+  the Corte del Milione, in Venice. The best description of the Polo
+  travels, and of other travels of the later Middle Ages, is found in C. R.
+  Beazley's <i>Dawn of Modern Geography</i>, Vol. III, chap, ii, and Part
+  II.</p>
+
+  <p><a name="Nt_422" href="#NtA_422">[422]</a> Heyd, loc. cit., Vol. II,
+  p. 220; H. Yule, in <i>Encyclopædia Britannica</i>, 9th (10th) or 11th
+  ed., article "China." The handbook cited is Pegolotti's <i>Libro di
+  divisamenti di paesi</i>, chapters i-ii, where it is implied that $60,000
+  would be a likely amount for a merchant going to China to invest in his
+  trip.</p>
+
+  <p><a name="Nt_423" href="#NtA_423">[423]</a> Cunningham, loc. cit., p.
+  194.</p>
+
+  <p><a name="Nt_424" href="#NtA_424">[424]</a> I.e. a commission
+  house.</p>
+
+  <p><a name="Nt_425" href="#NtA_425">[425]</a> Cunningham, loc. cit., p.
+  186.</p>
+
+  <p><a name="Nt_426" href="#NtA_426">[426]</a> J. R. Green, <i>Short
+  History of the English People</i>, New York, 1890, p. 66.</p>
+
+  <p><a name="Nt_427" href="#NtA_427">[427]</a> W. Besant, <i>London</i>,
+  New York, 1892, p. 43.</p>
+
+  <p><a name="Nt_428" href="#NtA_428">[428]</a> <i>Baldakin</i>,
+  <i>baldekin</i>, <i>baldachino</i>.</p>
+
+  <p><a name="Nt_429" href="#NtA_429">[429]</a> Italian
+  <i>Baldacco</i>.</p>
+
+  <p><a name="Nt_430" href="#NtA_430">[430]</a> J. K. Mumford, <i>Oriental
+  Rugs</i>, New York, 1901, p. 18.</p>
+
+  <p><a name="Nt_431" href="#NtA_431">[431]</a> Or Girbert, the Latin forms
+  <i>Gerbertus</i> and <i>Girbertus</i> appearing indifferently in the
+  documents of his time.</p>
+
+  <p><a name="Nt_432" href="#NtA_432">[432]</a> See, for example, J. C.
+  Heilbronner, <i>Historia matheseos universæ</i>, p. 740.</p>
+
+  <p><a name="Nt_433" href="#NtA_433">[433]</a> "Obscuro loco natum," as an
+  old chronicle of Aurillac has it.</p>
+
+  <p><a name="Nt_434" href="#NtA_434">[434]</a> N. Bubnov, <i>Gerberti
+  postea Silvestri II papae opera mathematica</i>, Berlin, 1899, is the
+  most complete and reliable source of information; Picavet, loc. cit.,
+  <i>Gerbert</i> etc.; Olleris, <i>&OElig;uvres de Gerbert</i>, Paris,
+  1867; Havet, <i>Lettres de Gerbert</i>, Paris, 1889 ; H. Weissenborn,
+  <i>Gerbert; Beiträge zur Kenntnis der Mathematik des Mittelalters</i>,
+  Berlin, 1888, and <i>Zur Geschichte der Einführung der jetzigen Ziffern
+  in Europa durch Gerbert</i>, Berlin, 1892; Büdinger, <i>Ueber Gerberts
+  wissenschaftliche und politische Stellung</i>, Cassel, 1851; Richer,
+  "Historiarum liber III," in Bubnov, loc. cit., pp. 376-381; Nagl,
+  <i>Gerbert und die Rechenkunst des 10. Jahrhunderts</i>, Vienna,
+  1888.</p>
+
+  <p><a name="Nt_435" href="#NtA_435">[435]</a> Richer tells of the visit
+  to Aurillac by Borel, a Spanish nobleman, just as Gerbert was entering
+  into young manhood. He relates how affectionately the abbot received him,
+  asking if there were men in Spain well versed in the arts. Upon Borel's
+  reply in the affirmative, the abbot asked that one of his young men might
+  accompany him upon his return, that he might carry on his studies
+  there.</p>
+
+  <p><a name="Nt_436" href="#NtA_436">[436]</a> Vicus Ausona. Hatto also
+  appears as Atton and Hatton.</p>
+
+  <p><a name="Nt_437" href="#NtA_437">[437]</a> This is all that we know of
+  his sojourn in Spain, and this comes from his pupil Richer. The stories
+  told by Adhemar of Chabanois, an apparently ignorant and certainly
+  untrustworthy contemporary, of his going to Cordova, are unsupported.
+  (See e.g. Picavet, p. 34.) Nevertheless this testimony is still accepted:
+  K. von Raumer, for example (<i>Geschichte der Pädagogik</i>, 6th ed.,
+  1890, Vol. I, p. 6), says "Mathematik studierte man im Mittelalter bei
+  den Arabern in Spanien. Zu ihnen gieng Gerbert, nachmaliger Pabst
+  Sylvester II."</p>
+
+  <p><a name="Nt_438" href="#NtA_438">[438]</a> Thus in a letter to
+  Aldaberon he says: "Quos post repperimus speretis, id est VIII volumina
+  Boeti de astrologia, praeclarissima quoque figurarum geometriæ, aliaque
+  non minus admiranda" (Epist. 8). Also in a letter to Rainard (Epist.
+  130), he says: "Ex tuis sumptibus fac ut michi scribantur M. Manlius
+  (Manilius in one MS.) de astrologia."</p>
+
+  <p><a name="Nt_439" href="#NtA_439">[439]</a> Picavet, loc. cit., p.
+  31.</p>
+
+  <p><a name="Nt_440" href="#NtA_440">[440]</a> Picavet, loc. cit., p.
+  36.</p>
+
+  <p><a name="Nt_441" href="#NtA_441">[441]</a> Havet, loc. cit., p.
+  vii.</p>
+
+  <p><a name="Nt_442" href="#NtA_442">[442]</a> Picavet, loc. cit., p.
+  37.</p>
+
+  <p><a name="Nt_443" href="#NtA_443">[443]</a> "Con sinistre arti
+  conseguri la dignita del Pontificato.... Lasciato poi l' abito, e 'l
+  monasterio, e datosi tutto in potere del diavolo." [Quoted in Bombelli,
+  <i>L'antica numerazione Italica</i>, Rome, 1876, p. 41 n.]</p>
+
+  <p><a name="Nt_444" href="#NtA_444">[444]</a> He writes from Rheims in
+  984 to one Lupitus, in Barcelona, saying: "Itaque librum de astrologia
+  translatum a te michi petenti dirige," presumably referring to some
+  Arabic treatise. [Epist. no. 24 of the Havet collection, p. 19.]</p>
+
+  <p><a name="Nt_445" href="#NtA_445">[445]</a> See Bubnov, loc. cit., p.
+  x.</p>
+
+  <p><a name="Nt_446" href="#NtA_446">[446]</a> Olleris, loc. cit., p. 361,
+  l. 15, for Bernelinus; and Bubnov, loc. cit., p. 381, l. 4, for
+  Richer.</p>
+
+  <p><a name="Nt_447" href="#NtA_447">[447]</a> Woepcke found this in a
+  Paris MS. of Radulph of Laon, c. 1100. [<i>Propagation</i>, p. 246.] "Et
+  prima quidem trium spaciorum superductio unitatis caractere inscribitur,
+  qui chaldeo nomine dicitur igin." See also Alfred Nagl, "Der
+  arithmetische Tractat des Radulph von Laon" (<i>Abhandlungen zur
+  Geschichte der Mathematik</i>, Vol. V, pp. 85-133), p. 97.</p>
+
+  <p><a name="Nt_448" href="#NtA_448">[448]</a> Weissenborn, loc. cit., p.
+  239. When Olleris (<i>&OElig;uvres de Gerbert</i>, Paris, 1867, p. cci)
+  says, "C'est à lui et non point aux Arabes, que l'Europe doit son système
+  et ses signes de numération," he exaggerates, since the evidence is all
+  against his knowing the place value. Friedlein emphasizes this in the
+  <i>Zeitschrift für Mathematik und Physik</i>, Vol. XII (1867),
+  <i>Literaturzeitung</i>, p. 70: "Für das <i>System</i> unserer Numeration
+  ist die <i>Null</i> das wesentlichste Merkmal, und diese kannte Gerbert
+  nicht. Er selbst schrieb alle Zahlen mit den römischen Zahlzeichen und
+  man kann ihm also nicht verdanken, was er selbst nicht kannte."</p>
+
+  <p><a name="Nt_449" href="#NtA_449">[449]</a> E.g., Chasles, Büdinger,
+  Gerhardt, and Richer. So Martin (<i>Recherches nouvelles</i> etc.)
+  believes that Gerbert received them from Boethius or his followers. See
+  Woepcke, <i>Propagation</i>, p. 41.</p>
+
+  <p><a name="Nt_450" href="#NtA_450">[450]</a> Büdinger, loc. cit., p. 10.
+  Nevertheless, in Gerbert's time one <span class="special"
+  title="Al-Mansur">Al-Man&#x1E63;&#x16B;r</span>, governing Spain under
+  the name of Hish&#x101;m (976-1002), called from the Orient
+  Al-Be&#x121;&#x101;n&#x12B; to teach his son, so that scholars were
+  recognized. [Picavet, p. 36.]</p>
+
+  <p><a name="Nt_451" href="#NtA_451">[451]</a> Weissenborn, loc. cit., p.
+  235.</p>
+
+  <p><a name="Nt_452" href="#NtA_452">[452]</a> Ibid., p. 234.</p>
+
+  <p><a name="Nt_453" href="#NtA_453">[453]</a> These letters, of the
+  period 983-997, were edited by Havet, loc. cit., and, less completely, by
+  Olleris, loc. cit. Those touching mathematical topics were edited by
+  Bubnov, loc. cit., pp. 98-106.</p>
+
+  <p><a name="Nt_454" href="#NtA_454">[454]</a> He published it in the
+  <i>Monumenta Germaniae historica</i>, "Scriptores," Vol. III, and at
+  least three other editions have since appeared, viz. those by Guadet in
+  1845, by Poinsignon in 1855, and by Waitz in 1877.</p>
+
+  <p><a name="Nt_455" href="#NtA_455">[455]</a> Domino ac beatissimo Patri
+  Gerberto, Remorum archiepiscopo, Richerus Monchus, Gallorum congressibus
+  in volumine regerendis, imperii tui, pater sanctissime Gerberte,
+  auctoritas seminarium dedit.</p>
+
+  <p><a name="Nt_456" href="#NtA_456">[456]</a> In epistle 17 (Havet
+  collection) he speaks of the "De multiplicatione et divisione numerorum
+  libellum a Joseph Ispano editum abbas Warnerius" (a person otherwise
+  unknown). In epistle 25 he says: "De multiplicatione et divisione
+  numerorum, Joseph Sapiens sententias quasdam edidit."</p>
+
+  <p><a name="Nt_457" href="#NtA_457">[457]</a> H. Suter, "Zur Frage über
+  den Josephus Sapiens," <i>Bibliotheca Mathematica</i>, Vol. VIII (2), p.
+  84; Weissenborn, <i>Einführung</i>, p. 14; also his <i>Gerbert</i>; M.
+  Steinschneider, in <i>Bibliotheca Mathematica</i>, 1893, p. 68. Wallis
+  (<i>Algebra</i>, 1685, chap. 14) went over the list of Spanish Josephs
+  very carefully, but could find nothing save that "Josephus Hispanus seu
+  Josephus sapiens videtur aut Maurus fuisse aut alius quis in
+  Hispania."</p>
+
+  <p><a name="Nt_458" href="#NtA_458">[458]</a> P. Ewald, <i>Mittheilungen,
+  Neues Archiv d. Gesellschaft für ältere deutsche Geschichtskunde</i>,
+  Vol. VIII, 1883, pp. 354-364. One of the manuscripts is of 976 <span
+  class="scac">A.D.</span> and the other of 992 <span
+  class="scac">A.D.</span> See also Franz Steffens, <i>Lateinische
+  Paläographie</i>, Freiburg (Schweiz), 1903, pp. xxxix-xl. The forms are
+  reproduced in the plate on page 140.</p>
+
+  <p><a name="Nt_459" href="#NtA_459">[459]</a> It is entitled
+  <i>Constantino suo Gerbertus scolasticus</i>, because it was addressed to
+  Constantine, a monk of the Abbey of Fleury. The text of the letter to
+  Constantine, preceding the treatise on the Abacus, is given in the
+  <i>Comptes rendus</i>, Vol. XVI (1843), p. 295. This book seems to have
+  been written c. 980 <span class="scac">A.D.</span> [Bubnov, loc. cit., p.
+  6.]</p>
+
+  <p><a name="Nt_460" href="#NtA_460">[460]</a> "Histoire de
+  l'Arithmétique," <i>Comptes rendus</i>, Vol. XVI (1843), pp. 156,
+  281.</p>
+
+  <p><a name="Nt_461" href="#NtA_461">[461]</a> Loc. cit., <i>Gerberti
+  Opera</i> etc.</p>
+
+  <p><a name="Nt_462" href="#NtA_462">[462]</a> Friedlein thought it
+  spurious. See <i>Zeitschrift für Mathematik und Physik</i>, Vol. XII
+  (1867), Hist.-lit. suppl., p. 74. It was discovered in the library of the
+  Benedictine monastry of St. Peter, at Salzburg, and was published by
+  Peter Bernhard Pez in 1721. Doubt was first cast upon it in the Olleris
+  edition (<i>&OElig;uvres de Gerbert</i>). See Weissenborn,
+  <i>Gerbert</i>, pp. 2, 6, 168, and Picavet, p. 81. Hock, Cantor, and Th.
+  Martin place the composition of the work at c. 996 when Gerbert was in
+  Germany, while Olleris and Picavet refer it to the period when he was at
+  Rheims.</p>
+
+  <p><a name="Nt_463" href="#NtA_463">[463]</a> Picavet, loc. cit., p.
+  182.</p>
+
+  <p><a name="Nt_464" href="#NtA_464">[464]</a> Who wrote after Gerbert
+  became pope, for he uses, in his preface, the words, "a domino pape
+  Gerberto." He was quite certainly not later than the eleventh century; we
+  do not have exact information about the time in which he lived.</p>
+
+  <p><a name="Nt_465" href="#NtA_465">[465]</a> Picavet, loc. cit., p. 182.
+  Weissenborn, <i>Gerbert</i>, p. 227. In Olleris, <i>Liber Abaci</i> (of
+  Bernelinus), p. 361.</p>
+
+  <p><a name="Nt_466" href="#NtA_466">[466]</a> Richer, in Bubnov, loc.
+  cit., p. 381.</p>
+
+  <p><a name="Nt_467" href="#NtA_467">[467]</a> Weissenborn,
+  <i>Gerbert</i>, p. 241.</p>
+
+  <p><a name="Nt_468" href="#NtA_468">[468]</a> Writers on numismatics are
+  quite uncertain as to their use. See F. Gnecchi, <i>Monete Romane</i>, 2d
+  ed., Milan, 1900, cap. XXXVII. For pictures of old Greek tesserae of
+  Sarmatia, see S. Ambrosoli, <i>Monete Greche</i>, Milan, 1899, p.
+  202.</p>
+
+  <p><a name="Nt_469" href="#NtA_469">[469]</a> Thus Tzwivel's arithmetic
+  of 1507, fol. 2, v., speaks of the ten figures as "characteres sive
+  numerorum apices a diuo Seuerino Boetio."</p>
+
+  <p><a name="Nt_470" href="#NtA_470">[470]</a> Weissenborn uses
+  <i>sipos</i> for 0. It is not given by Bernelinus, and appears in Radulph
+  of Laon, in the twelfth century. See Günther's <i>Geschichte</i>, p. 98,
+  n.; Weissenborn, p. 11; Pihan, <i>Exposé</i> etc., pp. xvi-xxii.</p>
+
+  <p>In Friedlein's <i>Boetius</i>, p. 396, the plate shows that all of the
+  six important manuscripts from which the illustrations are taken contain
+  the symbol, while four out of five which give the words use the word
+  <i>sipos</i> for 0. The names appear in a twelfth-century anonymous
+  manuscript in the Vatican, in a passage beginning</p>
+
+  <div class="poem">
+    <div class="stanza">
+      <p>Ordine primigeno sibi nomen possidet igin.</p>
+      <p>Andras ecce locum mox uendicat ipse secundum</p>
+      <p>Ormis post numeros incompositus sibi primus.</p>
+    </div>
+  </div>
+  <p>[Boncompagni <i>Buttetino</i>, XV, p. 132.] Turchill (twelfth century)
+  gives the names Igin, andras, hormis, arbas, quimas, caletis, zenis,
+  temenias, celentis, saying: "Has autem figuras, ut donnus [dominus]
+  Gvillelmus Rx testatur, a pytagoricis habemus, nomina uero ab arabibus."
+  (Who the William R. was is not known. Boncompagni <i>Bulletino</i> XV, p.
+  136.) Radulph of Laon (d. 1131) asserted that they were Chaldean
+  (<i>Propagation</i>, p. 48 n.). A discussion of the whole question is
+  also given in E. C. Bayley, loc. cit. Huet, writing in 1679, asserted
+  that they were of Semitic origin, as did Nesselmann in spite of his
+  despair over ormis, calctis, and celentis; see Woepcke,
+  <i>Propagation</i>, p. 48. The names were used as late as the fifteenth
+  century, without the zero, but with the superscript dot for 10's, two
+  dots for 100's, etc., as among the early Arabs. Gerhardt mentions having
+  seen a fourteenth or fifteenth century manuscript in the Bibliotheca
+  Amploniana with the names "Ingnin, andras, armis, arbas, quinas, calctis,
+  zencis, zemenias, zcelentis," and the statement "Si unum punctum super
+  ingnin ponitur, X significat.... Si duo puncta super ... figuras
+  superponunter, fiet decuplim illius quod cum uno puncto significabatur,"
+  in <i>Monatsberichte der K. P. Akad. d. Wiss.</i>, Berlin, 1867, p.
+  40.</p>
+
+  <p><a name="Nt_471" href="#NtA_471">[471]</a> <i>A chart of ten numerals
+  in 200 tongues</i>, by Rev. R. Patrick, London, 1812.</p>
+
+  <p><a name="Nt_472" href="#NtA_472">[472]</a> "Numeratio figuralis est
+  cuiusuis numeri per notas, et figuras numerates descriptio."
+  [Clichtoveus, edition of c. 1507, fol. C ii, v.] "Aristoteles enim uoces
+  rerum <span title="sumbola" class="grk"
+  >&sigma;&#x1F7B;&mu;&beta;&omicron;&lambda;&alpha;</span> uocat: id
+  translatum, sonat notas." [Noviomagus, <i>De Numeris Libri II</i>, cap.
+  vi.] "Alphabetum decem notarum." [Schonerus, notes to Ramus, 1586, p. 3
+  seq.] Richer says: "novem numero notas omnem numerum significantes."
+  [Bubnov, loc. cit., p. 381.]</p>
+
+  <p><a name="Nt_473" href="#NtA_473">[473]</a> "Il y a dix Characteres,
+  autrement Figures, Notes, ou Elements." [Peletier, edition of 1607, p.
+  13.] "Numerorum notas alij figuras, alij signa, alij characteres uocant."
+  [Glareanus, 1545 edition, f. 9, r.] "Per figuras (quas zyphras uocant)
+  assignationem, quales sunt hæ notulæ, 1. 2. 3. 4...." [Noviomagus, <i>De
+  Numeris Libri II</i>, cap. vi.] Gemma Frisius also uses <i>elementa</i>
+  and Cardan uses <i>literae</i>. In the first arithmetic by an American
+  (Greenwood, 1729) the author speaks of "a few Arabian <i>Charecters</i>
+  or Numeral Figures, called <i>Digits</i>" (p. 1), and as late as 1790, in
+  the third edition of J. J. Blassière's arithmetic (1st ed. 1769), the
+  name <i>characters</i> is still in use, both for "de Latynsche en de
+  Arabische" (p. 4), as is also the term "Cyfferletters" (p. 6, n.).
+  <i>Ziffer</i>, the modern German form of cipher, was commonly used to
+  designate any of the nine figures, as by Boeschenstein and Riese,
+  although others, like Köbel, used it only for the zero. So <i>zifre</i>
+  appears in the arithmetic by Borgo, 1550 ed. In a Munich codex of the
+  twelfth century, attributed to Gerland, they are called <i>characters</i>
+  only: "Usque ad VIIII. enim porrigitur omnis numerus et qui supercrescit
+  eisdem designator Karacteribus." [Boncompagni <i>Bulletino</i>, Vol. X.
+  p. 607.]</p>
+
+  <p><a name="Nt_474" href="#NtA_474">[474]</a> The title of his work is
+  <i>Prologus N. Ocreati in Helceph</i> (Arabic <i>al-qeif</i>,
+  investigation or memoir) <i>ad Adelardum Batensem magistrum suum</i>. The
+  work was made known by C. Henry, in the <i>Zeitschrift für Mathematik und
+  Physik</i>, Vol. XXV, p. 129, and in the <i>Abhandlungen zur Geschichte
+  der Mathematik</i>, Vol. III; Weissenborn, <i>Gerbert</i>, p. 188.</p>
+
+  <p><a name="Nt_475" href="#NtA_475">[475]</a> The zero is indicated by a
+  vacant column.</p>
+
+  <p><a name="Nt_476" href="#NtA_476">[476]</a> Leo Jordan, loc. cit., p.
+  170. "Chifre en augorisme" is the expression used, while a century later
+  "giffre en argorisme" and "cyffres d'augorisme" are similarly used.</p>
+
+  <p><a name="Nt_477" href="#NtA_477">[477]</a> <i>The Works of Geoffrey
+  Chaucer</i>, edited by W. W. Skeat, Vol. IV, Oxford, 1894, p. 92.</p>
+
+  <p><a name="Nt_478" href="#NtA_478">[478]</a> Loc. cit., Vol. III, pp.
+  179 and 180.</p>
+
+  <p><a name="Nt_479" href="#NtA_479">[479]</a> In Book II, chap, vii, of
+  <i>The Testament of Love</i>, printed with Chaucer's Works, loc. cit.,
+  Vol. VII, London, 1897.</p>
+
+  <p><a name="Nt_480" href="#NtA_480">[480]</a> <i>Liber Abacci</i>,
+  published in Olleris, <i>&OElig;uvres de Gerbert</i>, pp. 357-400.</p>
+
+  <p><a name="Nt_481" href="#NtA_481">[481]</a> G. R. Kaye, "The Use of the
+  Abacus in Ancient India," <i>Journal and Proceedings of the Asiatic
+  Society of Bengal</i>, 1908, pp. 293-297.</p>
+
+  <p><a name="Nt_482" href="#NtA_482">[482]</a> <i>Liber Abbaci</i>, by
+  Leonardo Pisano, loc. cit., p. 1.</p>
+
+  <p><a name="Nt_483" href="#NtA_483">[483]</a> Friedlein, "Die
+  Entwickelung des Rechnens mit Columnen," <i>Zeitschrift für Mathematik
+  und Physik</i>, Vol. X, p. 247.</p>
+
+  <p><a name="Nt_484" href="#NtA_484">[484]</a> The divisor 6 or 16 being
+  increased by the difference 4, to 10 or 20 respectively.</p>
+
+  <p><a name="Nt_485" href="#NtA_485">[485]</a> E.g. Cantor, Vol. I, p.
+  882.</p>
+
+  <p><a name="Nt_486" href="#NtA_486">[486]</a> Friedlein, loc. cit.;
+  Friedlein, "Gerbert's Regeln der Division" and "Das Rechnen mit Columnen
+  vor dem 10. Jahrhundert," <i>Zeitschrift für Mathematik und Physik</i>,
+  Vol. IX; Bubnov, loc. cit., pp. 197-245; M. Chasles, "Histoire de
+  l'arithmétique. Recherches des traces du système de l'abacus, après que
+  cette méthode a pris le nom d'Algorisme.&mdash;Preuves qu'à toutes les
+  époques, jusq'au <span class="scac">XVI</span><sup>e</sup> siècle, on a
+  su que l'arithmétique vulgaire avait pour origine cette méthode
+  ancienne," <i>Comptes rendus</i>, Vol. XVII, pp. 143-154, also "Règles de
+  l'abacus," <i>Comptes rendus</i>, Vol. XVI, pp. 218-246, and "Analyse et
+  explication du traité de Gerbert," <i>Comptes rendus</i>, Vol. XVI, pp.
+  281-299.</p>
+
+  <p><a name="Nt_487" href="#NtA_487">[487]</a> Bubnov, loc. cit., pp.
+  203-204, "Abbonis abacus."</p>
+
+  <p><a name="Nt_488" href="#NtA_488">[488]</a> "Regulae de numerorum abaci
+  rationibus," in Bubnov, loc. cit., pp. 205-225.</p>
+
+  <p><a name="Nt_489" href="#NtA_489">[489]</a> P. Treutlein, "Intorno ad
+  alcuni scritti inediti relativi al calcolo dell' abaco," <i>Bulletino di
+  bibliografia e di storia delle scienze matematiche e fisiche</i>, Vol. X,
+  pp. 589-647.</p>
+
+  <p><a name="Nt_490" href="#NtA_490">[490]</a> "Intorno ad uno scritto
+  inedito di Adelhardo di Bath intitolato 'Regulae Abaci,'" B. Boncompagni,
+  in his <i>Bulletino</i>, Vol. XIV, pp. 1-134.</p>
+
+  <p><a name="Nt_491" href="#NtA_491">[491]</a> Treutlein, loc. cit.;
+  Boncompagni, "Intorno al Tractatus de Abaco di Gerlando,"
+  <i>Bulletino</i>, Vol. X, pp. 648-656.</p>
+
+  <p><a name="Nt_492" href="#NtA_492">[492]</a> E. Narducci, "Intorno a due
+  trattati inediti d'abaco contenuti in due codici Vaticani del secolo
+  XII," Boncompagni <i>Bulletino</i>, Vol. XV, pp. 111-162.</p>
+
+  <p><a name="Nt_493" href="#NtA_493">[493]</a> See Molinier, <i>Les
+  sources de l'histoire de France</i>, Vol. II, Paris, 1902, pp. 2, 3.</p>
+
+  <p><a name="Nt_494" href="#NtA_494">[494]</a> Cantor, <i>Geschichte</i>,
+  Vol. I, p. 762. A. Nagl in the <i>Abhandlungen zur Geschichte der
+  Mathematik</i>, Vol. V, p. 85.</p>
+
+  <p><a name="Nt_495" href="#NtA_495">[495]</a> 1030-1117.</p>
+
+  <p><a name="Nt_496" href="#NtA_496">[496]</a> <i>Abhandlungen zur
+  Geschichte der Mathematik</i>, Vol. V, pp. 85-133. The work begins
+  "Incipit Liber Radulfi laudunensis de abaco."</p>
+
+  <p><a name="Nt_497" href="#NtA_497">[497]</a> <i>Materialien zur
+  Geschichte der arabischen Zahlzeichen in Frankreich</i>, loc. cit.</p>
+
+  <p><a name="Nt_498" href="#NtA_498">[498]</a> Who died in 1202.</p>
+
+  <p><a name="Nt_499" href="#NtA_499">[499]</a> Cantor, <i>Geschichte</i>,
+  Vol. I (3), pp. 800-803; Boncompagni, <i>Trattati</i>, Part II. M.
+  Steinschneider ("Die Mathematik bei den Juden," <i>Bibliotheca
+  Mathematica</i>, Vol. X (2), p. 79) ingeniously derives another name by
+  which he is called (Abendeuth) from Ibn Da&#x16B;d (Son of David). See
+  also <i>Abhandlungen</i>, Vol. III, p. 110.</p>
+
+  <p><a name="Nt_500" href="#NtA_500">[500]</a> John is said to have died
+  in 1157.</p>
+
+  <p><a name="Nt_501" href="#NtA_501">[501]</a> For it says, "Incipit
+  prologus in libro alghoarismi de practica arismetrice. Qui editus est a
+  magistro Johanne yspalensi." It is published in full in the second part
+  of Boncompagni's <i>Trattati d'aritmetica</i>.</p>
+
+  <p><a name="Nt_502" href="#NtA_502">[502]</a> Possibly, indeed, the
+  meaning of "libro alghoarismi" is not "to Al-Khow&#x101;razm&#x12B;'s
+  book," but "to a book of algorism." John of Luna says of it: "Hoc idem
+  est illud etiam quod ... alcorismus dicere videtur." [<i>Trattati</i>, p.
+  68.]</p>
+
+  <p><a name="Nt_503" href="#NtA_503">[503]</a> For a résumé, see Cantor,
+  Vol. I (3), pp. 800-803. As to the author, see Eneström in the
+  <i>Bibliotheca Mathematica</i>, Vol. VI (3), p. 114, and Vol. IX (3), p.
+  2.</p>
+
+  <p><a name="Nt_504" href="#NtA_504">[504]</a> Born at Cremona (although
+  some have asserted at Carmona, in Andalusia) in 1114; died at Toledo in
+  1187. Cantor, loc. cit.; Boncompagni, <i>Atti d. R. Accad. d. n.
+  Lincei</i>, 1851.</p>
+
+  <p><a name="Nt_505" href="#NtA_505">[505]</a> See <i>Abhandlungen zur
+  Geschichte der Mathematik</i>, Vol. XIV, p. 149; <i>Bibliotheca
+  Mathematica</i>, Vol. IV (3), p. 206. Boncompagni had a
+  fourteenth-century manuscript of his work, <i>Gerardi Cremonensis artis
+  metrice practice</i>. See also T. L. Heath, <i>The Thirteen Books of
+  Euclid's Elements</i>, 3 vols., Cambridge, 1908, Vol. I, pp. 92-94 ; A.
+  A. Björnbo, "Gerhard von Cremonas Übersetzung von Alkwarizmis Algebra und
+  von Euklids Elementen," <i>Bibliotheca Mathematica</i>, Vol. VI (3), pp.
+  239-248.</p>
+
+  <p><a name="Nt_506" href="#NtA_506">[506]</a> Wallis, <i>Algebra</i>,
+  1685, p. 12 seq.</p>
+
+  <p><a name="Nt_507" href="#NtA_507">[507]</a> Cantor, <i>Geschichte</i>,
+  Vol. I (3), p. 906; A. A. Björnbo, "Al-Chw&#x101;rizm&#x12B;'s
+  trigonometriske Tavler," <i>Festskrift til H. G. Zeuthen</i>, Copenhagen,
+  1909, pp. 1-17.</p>
+
+  <p><a name="Nt_508" href="#NtA_508">[508]</a> Heath, loc. cit., pp.
+  93-96.</p>
+
+  <p><a name="Nt_509" href="#NtA_509">[509]</a> M. Steinschneider,
+  <i>Zeitschrift der deutschen morgenländischen Gesellschaft</i>, Vol. XXV,
+  1871, p. 104, and <i>Zeitschrift für Mathematik und Physik</i>, Vol. XVI,
+  1871, pp. 392-393; M. Curtze, <i>Centralblatt für Bibliothekswesen</i>,
+  1899, p. 289; E. Wappler, <i>Zur Geschichte der deutschen Algebra im 15.
+  Jahrhundert</i>, Programm, Zwickau, 1887; L. C. Karpinski, "Robert of
+  Chester's Translation of the Algebra of Al-Khow&#x101;razm&#x12B;,"
+  <i>Bibliotheca Mathematica</i>, Vol. XI (3), p. 125. He is also known as
+  Robertus Retinensis, or Robert of Reading.</p>
+
+  <p><a name="Nt_510" href="#NtA_510">[510]</a> Nagl, A., "Ueber eine
+  Algorismus-Schrift des XII. Jahrhunderts und über die Verbreitung der
+  indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande,"
+  in the <i>Zeitschrift für Mathematik und Physik, Hist.-lit. Abth.</i>,
+  Vol. XXXIV, p. 129. Curtze, <i>Abhandlungen zur Geschichte der
+  Mathematik</i>, Vol. VIII, pp. 1-27.</p>
+
+  <p><a name="Nt_511" href="#NtA_511">[511]</a> See line <i>a</i> in the
+  plate on p. <a href="#page143">143</a>.</p>
+
+  <p><a name="Nt_512" href="#NtA_512">[512]</a> <i>Sefer ha-Mispar, Das
+  Buch der Zahl, ein hebräisch-arithmetisches Werk des R. Abraham ibn
+  Esra</i>, Moritz Silberberg, Frankfurt a. M., 1895.</p>
+
+  <p><a name="Nt_513" href="#NtA_513">[513]</a> Browning's "Rabbi ben
+  Ezra."</p>
+
+  <p><a name="Nt_514" href="#NtA_514">[514]</a> "Darum haben auch die
+  Weisen Indiens all ihre Zahlen durch neun bezeichnet und Formen für die 9
+  Ziffern gebildet." [<i>Sefer ha-Mispar</i>, loc. cit., p. 2.]</p>
+
+  <p><a name="Nt_515" href="#NtA_515">[515]</a> F. Bonaini, "Memoria unica
+  sincrona di Leonardo Fibonacci," Pisa, 1858, republished in 1867, and
+  appearing in the <i>Giornale Arcadico</i>, Vol. CXCVII (N.S. LII);
+  Gaetano Milanesi, <i>Documento inedito e sconosciuto a Lionardo
+  Fibonacci</i>, Roma, 1867; Guglielmini, <i>Elogio di Lionardo Pisano</i>,
+  Bologna, 1812, p. 35; Libri, <i>Histoire des sciences mathématiques</i>,
+  Vol. II, p. 25; D. Martines, <i>Origine e progressi dell' aritmetica</i>,
+  Messina, 1865, p. 47; Lucas, in Boncompagni <i>Bulletino</i>, Vol. X, pp.
+  129, 239; Besagne, ibid., Vol. IX, p. 583; Boncompagni, three works as
+  cited in Chap. I; G. Eneström, "Ueber zwei angebliche mathematische
+  Schulen im christlichen Mittelalter," <i>Bibliotheca Mathematica</i>,
+  Vol. VIII (3), pp. 252-262; Boncompagni, "Della vita e delle opere di
+  Leonardo Pisano," loc. cit.</p>
+
+  <p><a name="Nt_516" href="#NtA_516">[516]</a> The date is purely
+  conjectural. See the <i>Bibliotheca Mathematica</i>, Vol. IV (3), p.
+  215.</p>
+
+  <p><a name="Nt_517" href="#NtA_517">[517]</a> An old chronicle relates
+  that in 1063 Pisa fought a great battle with the Saracens at Palermo,
+  capturing six ships, one being "full of wondrous treasure," and this was
+  devoted to building the cathedral.</p>
+
+  <p><a name="Nt_518" href="#NtA_518">[518]</a> Heyd, loc. cit., Vol. I, p.
+  149.</p>
+
+  <p><a name="Nt_519" href="#NtA_519">[519]</a> Ibid., p. 211.</p>
+
+  <p><a name="Nt_520" href="#NtA_520">[520]</a> J. A. Symonds,
+  <i>Renaissance in Italy. The Age of Despots.</i> New York, 1883, p.
+  62.</p>
+
+  <p><a name="Nt_521" href="#NtA_521">[521]</a> Symonds, loc. cit., p.
+  79.</p>
+
+  <p><a name="Nt_522" href="#NtA_522">[522]</a> J. A. Froude, <i>The
+  Science of History</i>, London, 1864. "Un brevet d'apothicaire n'empêcha
+  pas Dante d'être le plus grand poète de l'Italie, et ce fut un petit
+  marchand de Pise qui donna l'algèbre aux Chrétiens." [Libri,
+  <i>Histoire</i>, Vol. I, p. xvi.]</p>
+
+  <p><a name="Nt_523" href="#NtA_523">[523]</a> A document of 1226, found
+  and published in 1858, reads: "Leonardo bigollo quondam Guilielmi."</p>
+
+  <p><a name="Nt_524" href="#NtA_524">[524]</a> "Bonaccingo germano
+  suo."</p>
+
+  <p><a name="Nt_525" href="#NtA_525">[525]</a> E.g. Libri, Guglielmini,
+  Tiraboschi.</p>
+
+  <p><a name="Nt_526" href="#NtA_526">[526]</a> Latin,
+  <i>Bonaccius</i>.</p>
+
+  <p><a name="Nt_527" href="#NtA_527">[527]</a> Boncompagni and
+  Milanesi.</p>
+
+  <p><a name="Nt_528" href="#NtA_528">[528]</a> Reprint, p. 5.</p>
+
+  <p><a name="Nt_529" href="#NtA_529">[529]</a> Whence the French name for
+  candle.</p>
+
+  <p><a name="Nt_530" href="#NtA_530">[530]</a> Now part of Algiers.</p>
+
+  <p><a name="Nt_531" href="#NtA_531">[531]</a> E. Reclus, <i>Africa</i>,
+  New York, 1893, Vol. II, p. 253.</p>
+
+  <p><a name="Nt_532" href="#NtA_532">[532]</a> "Sed hoc totum et
+  algorismum atque arcus pictagore quasi errorem computavi respectu modi
+  indorum." Woepcke, <i>Propagation</i> etc., regards this as referring to
+  two different systems, but the expression may very well mean algorism as
+  performed upon the Pythagorean arcs (or table).</p>
+
+  <p><a name="Nt_533" href="#NtA_533">[533]</a> "Book of the Abacus," this
+  term then being used, and long afterwards in Italy, to mean merely the
+  arithmetic of computation.</p>
+
+  <p><a name="Nt_534" href="#NtA_534">[534]</a> "Incipit liber Abaci a
+  Leonardo filio Bonacci compositus anno 1202 et correctus ab eodem anno
+  1228." Three MSS. of the thirteenth century are known, viz. at Milan, at
+  Siena, and in the Vatican library. The work was first printed by
+  Boncompagni in 1857.</p>
+
+  <p><a name="Nt_535" href="#NtA_535">[535]</a> I.e. in relation to the
+  quadrivium. "Non legant in festivis diebus, nisi Philosophos et
+  rhetoricas et quadrivalia et barbarismum et ethicam, si placet." Suter,
+  <i>Die Mathematik auf den Universitäten des Mittelalters</i>, Zürich,
+  1887, p. 56. Roger Bacon gives a still more gloomy view of Oxford in his
+  time in his <i>Opus minus</i>, in the <i>Rerum Britannicarum medii aevi
+  scriptores</i>, London, 1859, Vol. I, p. 327. For a picture of Cambridge
+  at this time consult F. W. Newman, <i>The English Universities,
+  translated from the German of V. A. Huber</i>, London, 1843, Vol. I, p.
+  61; W. W. R. Ball, <i>History of Mathematics at Cambridge</i>, 1889; S.
+  Günther, <i>Geschichte des mathematischen Unterrichts im deutschen
+  Mittelalter bis zum Jahre 1525</i>, Berlin, 1887, being Vol. III of
+  <i>Monumenta Germaniae paedagogica</i>.</p>
+
+  <p><a name="Nt_536" href="#NtA_536">[536]</a> On the commercial activity
+  of the period, it is known that bills of exchange passed between Messina
+  and Constantinople in 1161, and that a bank was founded at Venice in
+  1170, the Bank of San Marco being established in the following year. The
+  activity of Pisa was very manifest at this time. Heyd, loc. cit., Vol.
+  II, p. 5; V. Casagrandi, <i>Storia e cronologia</i>, 3d ed., Milan, 1901,
+  p. 56.</p>
+
+  <p><a name="Nt_537" href="#NtA_537">[537]</a> J. A. Symonds, loc. cit.,
+  Vol. II, p. 127.</p>
+
+  <p><a name="Nt_538" href="#NtA_538">[538]</a> I. Taylor, <i>The
+  Alphabet</i>, London, 1883, Vol. II, p. 263.</p>
+
+  <p><a name="Nt_539" href="#NtA_539">[539]</a> Cited by Unger's History,
+  p. 15. The Arabic numerals appear in a Regensburg chronicle of 1167 and
+  in Silesia in 1340. See Schmidt's <i>Encyclopädie der Erziehung</i>, Vol.
+  VI, p. 726; A. Kuckuk, "Die Rechenkunst im sechzehnten Jahrhundert,"
+  <i>Festschrift zur dritten Säcularfeier des Berlinischen Gymnasiums zum
+  grauen Kloster</i>, Berlin, 1874, p. 4.</p>
+
+  <p><a name="Nt_540" href="#NtA_540">[540]</a> The text is given in
+  Halliwell, <i>Rara Mathematica</i>, London, 1839.</p>
+
+  <p><a name="Nt_541" href="#NtA_541">[541]</a> Seven are given in
+  Ashmole's <i>Catalogue of Manuscripts in the Oxford Library</i>,
+  1845.</p>
+
+  <p><a name="Nt_542" href="#NtA_542">[542]</a> Maximilian Curtze, <i>Petri
+  Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco
+  commentarius, una cum Algorismo ipso</i>, Copenhagen, 1897; L. C.
+  Karpinski, "Jordanus Nemorarius and John of Halifax," <i>American
+  Mathematical Monthly</i>, Vol. XVII, pp. 108-113.</p>
+
+  <p><a name="Nt_543" href="#NtA_543">[543]</a> J. Aschbach, <i>Geschichte
+  der Wiener Universität im ersten Jahrhunderte ihres Bestehens</i>, Wien,
+  1865, p. 93.</p>
+
+  <p><a name="Nt_544" href="#NtA_544">[544]</a> Curtze, loc. cit., gives
+  the text.</p>
+
+  <p><a name="Nt_545" href="#NtA_545">[545]</a> Curtze, loc. cit., found
+  some forty-five copies of the <i>Algorismus</i> in three libraries of
+  Munich, Venice, and Erfurt (Amploniana). Examination of two manuscripts
+  from the Plimpton collection and the Columbia library shows such marked
+  divergence from each other and from the text published by Curtze that the
+  conclusion seems legitimate that these were students' lecture notes. The
+  shorthand character of the writing further confirms this view, as it
+  shows that they were written largely for the personal use of the
+  writers.</p>
+
+  <p><a name="Nt_546" href="#NtA_546">[546]</a> "Quidam philosophus edidit
+  nomine Algus, unde et Algorismus nuncupatur." [Curtze, loc. cit., p.
+  1.]</p>
+
+  <p><a name="Nt_547" href="#NtA_547">[547]</a> "Sinistrorsum autera
+  scribimus in hac arte more arabico sive iudaico, huius scientiae
+  inventorum." [Curtze, loc. cit., p. 7.] The Plimpton manuscript omits the
+  words "sive iudaico."</p>
+
+  <p><a name="Nt_548" href="#NtA_548">[548]</a> "Non enim omnis numerus per
+  quascumque figuras Indorum repraesentatur, sed tantum determinatus per
+  determinatam, ut 4 non per 5,..." [Curtze, loc. cit., p. 25.]</p>
+
+  <p><a name="Nt_549" href="#NtA_549">[549]</a> C. Henry, "Sur les deux
+  plus anciens traités français d'algorisme et de géométrie," Boncompagni
+  <i>Bulletino</i>, Vol. XV, p. 49; Victor Mortet, "Le plus ancien traité
+  français d'algorisme," loc. cit.</p>
+
+  <p><a name="Nt_550" href="#NtA_550">[550]</a> <i>L'État des sciences en
+  France, depute la mort du Roy Robert, arrivée en 1031, jusqu'à celle de
+  Philippe le Bel, arrivée en 1314</i>, Paris, 1741.</p>
+
+  <p><a name="Nt_551" href="#NtA_551">[551]</a> <i>Discours sur l'état des
+  lettres en France au XIII<sup>e</sup> siecle</i>, Paris, 1824.</p>
+
+  <p><a name="Nt_552" href="#NtA_552">[552]</a> <i>Aperçu historique</i>,
+  Paris, 1876 ed., p. 464.</p>
+
+  <p><a name="Nt_553" href="#NtA_553">[553]</a> Ranulf Higden, a native of
+  the west of England, entered St. Werburgh's monastery at Chester in 1299.
+  He was a Benedictine monk and chronicler, and died in 1364. His
+  <i>Polychronicon</i>, a history in seven books, was printed by Caxton in
+  1480.</p>
+
+  <p><a name="Nt_554" href="#NtA_554">[554]</a> Trevisa's translation,
+  Higden having written in Latin.</p>
+
+  <p><a name="Nt_555" href="#NtA_555">[555]</a> An illustration of this
+  feeling is seen in the writings of Prosdocimo de' Beldomandi (b. c.
+  1370-1380, d. 1428): "Inveni in quam pluribus libris algorismi nuncupatis
+  mores circa numeros operandi satis varios atque diversos, qui licet boni
+  existerent atque veri erant, tamen fastidiosi, tum propter ipsarum
+  regularum multitudinem, tum propter earum deleationes, tum etiam propter
+  ipsarum operationum probationes, utrum si bone fuerint vel ne. Erant et
+  etiam isti modi interim fastidiosi, quod si in aliquo calculo astroloico
+  error contigisset, calculatorem operationem suam a capite incipere
+  oportebat, dato quod error suus adhuc satis propinquus existeret; et hoc
+  propter figuras in sua operatione deletas. Indigebat etiam calculator
+  semper aliquo lapide vel sibi conformi, super quo scribere atque
+  faciliter delere posset figuras cum quibus operabatur in calculo suo. Et
+  quia haec omnia satis fastidiosa atque laboriosa mihi visa sunt, disposui
+  libellum edere in quo omnia ista abicerentur: qui etiam algorismus sive
+  liber de numeris denominari poterit. Scias tamen quod in hoc libello
+  ponere non intendo nisi ea quae ad calculum necessaria sunt, alia quae in
+  aliis libris practice arismetrice tanguntur, ad calculum non necessaria,
+  propter brevitatem dimitendo." [Quoted by A. Nagl, <i>Zeitschrift für
+  Mathematik und Physik, Hist.-lit. Abth.</i>, Vol. XXXIV, p. 143; Smith,
+  <i>Rara Arithmetica</i>, p. 14, in facsimile.]</p>
+
+  <p><a name="Nt_556" href="#NtA_556">[556]</a> P. Ewald, loc. cit.; Franz
+  Steffens, <i>Lateinische Paläographie</i>, pp. xxxix-xl. We are indebted
+  to Professor J. M. Burnam for a photograph of this rare manuscript.</p>
+
+  <p><a name="Nt_557" href="#NtA_557">[557]</a> See the plate of forms on
+  p. <a href="#page88">88</a>.</p>
+
+  <p><a name="Nt_558" href="#NtA_558">[558]</a> Karabacek, loc. cit., p.
+  56; Karpinski, "Hindu Numerals in the Fihrist," <i>Bibliotheca
+  Mathematica</i>, Vol. XI (3), p. 121.</p>
+
+  <p><a name="Nt_559" href="#NtA_559">[559]</a> Woepcke, "Sur une donnée
+  historique," etc., loc. cit., and "Essai d'une restitution de travaux
+  perdus d'Apollonius sur les quantités irrationnelles, d'après des
+  indications tirées d'un manuscrit arabe," <i>Tome XIV des Mémoires
+  présentés par divers savants à l'Académie des sciences</i>, Paris, 1856,
+  note, pp. 6-14.</p>
+
+  <p><a name="Nt_560" href="#NtA_560">[560]</a> <i>Archeological Report of
+  the Egypt Exploration Fund for 1908-1909</i>, London, 1910, p. 18.</p>
+
+  <p><a name="Nt_561" href="#NtA_561">[561]</a> There was a set of
+  astronomical tables in Boncompagni's library bearing this date: "Nota
+  quod anno d<span class="over">n</span>i <span class="over">n</span>ri ihû
+  x<span class="over">p</span>i. 1264. perfecto." See Narducci's
+  <i>Catalogo</i>, p. 130.</p>
+
+  <p><a name="Nt_562" href="#NtA_562">[562]</a> "On the Early use of Arabic
+  Numerals in Europe," read before the Society of Antiquaries April 14,
+  1910, and published in <i>Archæologia</i> in the same year.</p>
+
+  <p><a name="Nt_563" href="#NtA_563">[563]</a> Ibid., p. 8, n. The date is
+  part of an Arabic inscription.</p>
+
+  <p><a name="Nt_564" href="#NtA_564">[564]</a> O. Codrington, <i>A Manual
+  of Musalman Numismatics</i>, London, 1904.</p>
+
+  <p><a name="Nt_565" href="#NtA_565">[565]</a> See Arbuthnot, <i>The
+  Mysteries of Chronology</i>, London, 1900, pp. 75, 78, 98; F. Pichler,
+  <i>Repertorium der steierischen Münzkunde</i>, Grätz, 1875, where the
+  claim is made of an Austrian coin of 1458; <i>Bibliotheca
+  Mathematica</i>, Vol. X (2), p. 120, and Vol. XII (2), p. 120. There is a
+  Brabant piece of 1478 in the collection of D. E. Smith.</p>
+
+  <p><a name="Nt_566" href="#NtA_566">[566]</a> A specimen is in the
+  British Museum. [Arbuthnot, p. 79.]</p>
+
+  <p><a name="Nt_567" href="#NtA_567">[567]</a> Ibid., p. 79.</p>
+
+  <p><a name="Nt_568" href="#NtA_568">[568]</a> <i>Liber de Remediis
+  utriusque fortunae Coloniae.</i></p>
+
+  <p><a name="Nt_569" href="#NtA_569">[569]</a> Fr. Walthern et Hans
+  Hurning, Nördlingen.</p>
+
+  <p><a name="Nt_570" href="#NtA_570">[570]</a> <i>Ars Memorandi</i>, one
+  of the oldest European block-books.</p>
+
+  <p><a name="Nt_571" href="#NtA_571">[571]</a> Eusebius Caesariensis,
+  <i>De praeparatione evangelica</i>, Venice, Jenson, 1470. The above
+  statement holds for copies in the Astor Library and in the Harvard
+  University Library.</p>
+
+  <p><a name="Nt_572" href="#NtA_572">[572]</a> Francisco de Retza,
+  <i>Comestorium vitiorum</i>, Nürnberg, 1470. The copy referred to is in
+  the Astor Library.</p>
+
+  <p><a name="Nt_573" href="#NtA_573">[573]</a> See Mauch, "Ueber den
+  Gebrauch arabischer Ziffern und die Veränderungen derselben," <i>Anzeiger
+  für Kunde der deutschen Vorzeit</i>, 1861, columns 46, 81, 116, 151, 189,
+  229, and 268; Calmet, <i>Recherches sur l'origine des chiffres
+  d'arithmétique</i>, plate, loc. cit.</p>
+
+  <p><a name="Nt_574" href="#NtA_574">[574]</a> Günther, <i>Geschichte</i>,
+  p. 175, n.; Mauch, loc. cit.</p>
+
+  <p><a name="Nt_575" href="#NtA_575">[575]</a> These are given by W. R.
+  Lethaby, from drawings by J. T. Irvine, in the <i>Proceedings of the
+  Society of Antiquaries</i>, 1906, p. 200.</p>
+
+  <p><a name="Nt_576" href="#NtA_576">[576]</a> There are some
+  ill-tabulated forms to be found in J. Bowring, <i>The Decimal System</i>,
+  London, 1854, pp. 23, 25, and in L. A. Chassant, <i>Dictionnaire des
+  abréviations latines et françaises ... du moyen âge</i>, Paris, <span
+  class="scac">MDCCCLXVI</span>, p. 113. The best sources we have at
+  present, aside from the Hill monograph, are P. Treutlein, <i>Geschichte
+  unserer Zahlzeichen</i>, Karlsruhe, 1875; Cantor's <i>Geschichte</i>,
+  Vol. I, table; M. Prou, <i>Manuel de paléographie latine et
+  française</i>, 2d ed., Paris, 1892, p. 164; A. Cappelli, <i>Dizionario di
+  abbreviature latine ed italiane</i>, Milan, 1899. An interesting early
+  source is found in the rare Caxton work of 1480, <i>The Myrrour of the
+  World</i>. In Chap. X is a cut with the various numerals, the chapter
+  beginning "The fourth scyence is called arsmetrique." Two of the fifteen
+  extant copies of this work are at present in the library of Mr. J. P.
+  Morgan, in New York.</p>
+
+  <p><a name="Nt_577" href="#NtA_577">[577]</a> From the twelfth-century
+  manuscript on arithmetic, Curtze, loc. cit., <i>Abhandlungen</i>, and
+  Nagl, loc. cit. The forms are copied from Plate VII in <i>Zeitschrift für
+  Mathematik und Physik</i>, Vol. XXXIV.</p>
+
+  <p><a name="Nt_578" href="#NtA_578">[578]</a> From the Regensburg
+  chronicle. Plate containing some of these numerals in <i>Monumenta
+  Germaniae historica</i>, "Scriptores" Vol. XVII, plate to p. 184;
+  Wattenbach, <i>Anleitung zur lateinischen Palaeographie</i>, Leipzig,
+  1886, p. 102; Boehmer, <i>Fontes rerum Germanicarum</i>, Vol. III,
+  Stuttgart, 1852, p. lxv.</p>
+
+  <p><a name="Nt_579" href="#NtA_579">[579]</a> French Algorismus of 1275;
+  from an unpublished photograph of the original, in the possession of D.
+  E. Smith. See also p. 135.</p>
+
+  <p><a name="Nt_580" href="#NtA_580">[580]</a> From a manuscript of
+  Boethius c. 1294, in Mr. Plimpton's library. Smith, <i>Rara
+  Arithmetica</i>, Plate I.</p>
+
+  <p><a name="Nt_581" href="#NtA_581">[581]</a> Numerals in a 1303
+  manuscript in Sigmaringen, copied from Wattenbach, loc. cit., p. 102.</p>
+
+  <p><a name="Nt_582" href="#NtA_582">[582]</a> From a manuscript, Add.
+  Manuscript 27,589, British Museum, 1360 <span class="scac">A.D.</span>
+  The work is a computus in which the date 1360 appears, assigned in the
+  British Museum catalogue to the thirteenth century.</p>
+
+  <p><a name="Nt_583" href="#NtA_583">[583]</a> From the copy of
+  Sacrabosco's <i>Algorismus</i> in Mr. Plimpton's library. Date c. 1442.
+  See Smith, <i>Rara Arithmetica</i>, p. 450.</p>
+
+  <p><a name="Nt_584" href="#NtA_584">[584]</a> See <i>Rara
+  Arithmetica</i>, pp. 446-447.</p>
+
+  <p><a name="Nt_585" href="#NtA_585">[585]</a> Ibid., pp. 469-470.</p>
+
+  <p><a name="Nt_586" href="#NtA_586">[586]</a> Ibid., pp. 477-478.</p>
+
+  <p><a name="Nt_587" href="#NtA_587">[587]</a> The i is used for "one" in
+  the Treviso arithmetic (1478), Clichtoveus (c. 1507 ed., where both i and
+  j are so used), Chiarini (1481), Sacrobosco (1488 ed.), and Tzwivel (1507
+  ed., where jj and jz are used for 11 and 12). This was not universal,
+  however, for the <i>Algorithmus linealis</i> of c. 1488 has a special
+  type for 1. In a student's notebook of lectures taken at the University
+  of Würzburg in 1660, in Mr. Plimpton's library, the ones are all in the
+  form of i.</p>
+
+  <p><a name="Nt_588" href="#NtA_588">[588]</a> Thus the date <a
+  href="images/151c.png"><img src="images/151c.png" class="middle"
+  style="height:2ex" alt="Numerals 1580" /></a>, for 1580, appears in a MS.
+  in the Laurentian library at Florence. The second and the following five
+  characters are taken from Cappelli's <i>Dizionario</i>, p. 380, and are
+  from manuscripts of the twelfth, thirteenth, fourteenth, sixteenth,
+  seventeenth, and eighteenth centuries, respectively.</p>
+
+  <p><a name="Nt_589" href="#NtA_589">[589]</a> E.g. Chiarini's work of
+  1481; Clichtoveus (c. 1507).</p>
+
+  <p><a name="Nt_590" href="#NtA_590">[590]</a> The first is from an
+  algorismus of the thirteenth century, in the Hannover Library. [See
+  Gerhardt, "Ueber die Entstehung und Ausbreitung des dekadischen
+  Zahlensystems," loc. cit., p. 28.] The second character is from a French
+  algorismus, c. 1275. [Boncompagni <i>Bulletino</i>, Vol. XV, p. 51.] The
+  third and the following sixteen characters are given by Cappelli, loc.
+  cit., and are from manuscripts of the twelfth (1), thirteenth (2),
+  fourteenth (7), fifteenth (3), sixteenth (1), seventeenth (2), and
+  eighteenth (1) centuries, respectively.</p>
+
+  <p><a name="Nt_591" href="#NtA_591">[591]</a> Thus Chiarini (1481) has <a
+  href="images/152j.png"><img src="images/152j.png" class="middle"
+  style="height:1.5ex" alt="Symbol" /></a> for 23.</p>
+
+  <p><a name="Nt_592" href="#NtA_592">[592]</a> The first of these is from
+  a French algorismus, c. 1275. The second and the following eight
+  characters are given by Cappelli, loc. cit., and are from manuscripts of
+  the twelfth (2), thirteenth, fourteenth, fifteenth (3), seventeenth, and
+  eighteenth centuries, respectively.</p>
+
+  <p><a name="Nt_593" href="#NtA_593">[593]</a> See Nagl, loc. cit.</p>
+
+  <p><a name="Nt_594" href="#NtA_594">[594]</a> Hannover algorismus,
+  thirteenth century.</p>
+
+  <p><a name="Nt_595" href="#NtA_595">[595]</a> See the Dagomari
+  manuscript, in <i>Rara Arithmetica</i>, pp. 435, 437-440.</p>
+
+  <p><a name="Nt_596" href="#NtA_596">[596]</a> But in the woodcuts of the
+  <i>Margarita Philosophica</i> (1503) the old forms are used, although the
+  new ones appear in the text. In Caxton's <i>Myrrour of the World</i>
+  (1480) the old form is used.</p>
+
+  <p><a name="Nt_597" href="#NtA_597">[597]</a> Cappelli, loc. cit. They
+  are partly from manuscripts of the tenth, twelfth, thirteenth (3),
+  fourteenth (7), fifteenth (6), and eighteenth centuries, respectively.
+  Those in the third line are from Chassant's <i>Dictionnaire</i>, p. 113,
+  without mention of dates.</p>
+
+  <p><a name="Nt_598" href="#NtA_598">[598]</a> The first is from the
+  Hannover algorismus, thirteenth century. The second is taken from the
+  Rollandus manuscript, 1424. The others in the first two lines are from
+  Cappelli, twelfth (3), fourteenth (6), fifteenth (13) centuries,
+  respectively. The third line is from Chassant, loc. cit., p. 113, no
+  mention of dates.</p>
+
+  <p><a name="Nt_599" href="#NtA_599">[599]</a> The first of these forms is
+  from the Hannover algorismus, thirteenth century. The following are from
+  Cappelli, fourteenth (3), fifteenth, sixteenth (2), and eighteenth
+  centuries, respectively.</p>
+
+  <p><a name="Nt_600" href="#NtA_600">[600]</a> The first of these is taken
+  from the Hannover algorismus, thirteenth century. The following forms are
+  from Cappelli, twelfth, thirteenth, fourteenth (5), fifteenth (2),
+  seventeenth, and eighteenth centuries, respectively.</p>
+
+  <p><a name="Nt_601" href="#NtA_601">[601]</a> All of these are given by
+  Cappelli, thirteenth, fourteenth, fifteenth (2), and sixteenth centuries,
+  respectively.</p>
+
+  <p><a name="Nt_602" href="#NtA_602">[602]</a> Smith, <i>Rara
+  Arithmetica</i>, p. 489. This is also seen in several of the Plimpton
+  manuscripts, as in one written at Ancona in 1684. See also Cappelli, loc.
+  cit.</p>
+
+  <p><a name="Nt_603" href="#NtA_603">[603]</a> French algorismus, c. 1275,
+  for the first of these forms. Cappelli, thirteenth, fourteenth, fifteenth
+  (3), and seventeenth centuries, respectively. The last three are taken
+  from <i>Byzantinische Analekten</i>, J. L. Heiberg, being forms of the
+  fifteenth century, but not at all common. <a href="images/154e.png"><img
+  src="images/154e.png" class="middle" style="height:1.8ex" alt="Symbol:
+  Qoppa" /></a> was the old Greek symbol for 90.</p>
+
+  <p><a name="Nt_604" href="#NtA_604">[604]</a> For the first of these the
+  reader is referred to the forms ascribed to Boethius, in the illustration
+  on p. <a href="#page88">88</a>; for the second, to Radulph of Laon, see
+  p. <a href="#Nt_232">60</a>. The third is used occasionally in the
+  Rollandus (1424) manuscript, in Mr. Plimpton's library. The remaining
+  three are from Cappelli, fourteenth (2) and seventeenth centuries.</p>
+
+  <p><a name="Nt_605" href="#NtA_605">[605]</a> Smith, <i>An Early English
+  Algorism</i>.</p>
+
+  <p><a name="Nt_606" href="#NtA_606">[606]</a> Kuckuck, p. 5.</p>
+
+  <p><a name="Nt_607" href="#NtA_607">[607]</a> A. Cappelli, loc. cit., p.
+  372.</p>
+
+  <p><a name="Nt_608" href="#NtA_608">[608]</a> Smith, <i>Rara
+  Arithmetica</i>, p. 443.</p>
+
+  <p><a name="Nt_609" href="#NtA_609">[609]</a> Curtze, <i>Petri Philomeni
+  de Dacia</i> etc., p. <span class="scac">IX</span>.</p>
+
+  <p><a name="Nt_610" href="#NtA_610">[610]</a> Cappelli, loc. cit., p.
+  376.</p>
+
+  <p><a name="Nt_611" href="#NtA_611">[611]</a> Curtze, loc. cit., pp.
+  <span class="scac">VIII-IX</span>, note.</p>
+
+  <p><a name="Nt_612" href="#NtA_612">[612]</a> Edition of 1544-1545, f.
+  52.</p>
+
+  <p><a name="Nt_613" href="#NtA_613">[613]</a> <i>De numeris libri II</i>,
+  1544 ed., cap. <span class="scac">XV</span>. Heilbronner, loc. cit., p.
+  736, also gives them, and compares this with other systems.</p>
+
+  <p><a name="Nt_614" href="#NtA_614">[614]</a> Noviomagus says of them:
+  "De quibusdam Astrologicis, sive Chaldaicis numerorum notis.... Sunt
+  &amp; aliæ quædam notæ, quibus Chaldaei &amp; Astrologii quemlibet
+  numerum artificiose &amp; arguté describunt, scitu periucundae, quas
+  nobis communicauit Rodolphus Paludanus Nouiomagus."</p>
+
+</div>
+
+
+
+
+
+
+
+<pre>
+
+
+
+
+
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+</body>
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@@ -0,0 +1,7037 @@
+The Project Gutenberg EBook of The Hindu-Arabic Numerals, by
+David Eugene Smith and Louis Charles Karpinski
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: The Hindu-Arabic Numerals
+
+Author: David Eugene Smith
+        Louis Charles Karpinski
+
+Release Date: September 14, 2007 [EBook #22599]
+
+Language: English
+
+Character set encoding: ASCII
+
+*** START OF THIS PROJECT GUTENBERG EBOOK THE HINDU-ARABIC NUMERALS ***
+
+
+
+
+Produced by David Newman, Chuck Greif, Keith Edkins and
+the Online Distributed Proofreading Team at
+https://www.pgdp.net (This file was produced from images
+from the Cornell University Library: Historical Mathematics
+Monographs collection.)
+
+
+
+
+
+Transcriber's Note:
+
+The following codes are used for characters that are not present in the
+character set used for this version of the book.
+
+  [=a] a with macron (etc.)
+  [.g] g with dot above (etc.)
+  ['s] s with acute accent
+  [d.] d with dot below (etc.)
+  [d=] d with line below
+  [H)] H with breve below
+
+
+
+
+
+THE
+
+HINDU-ARABIC NUMERALS
+
+BY
+DAVID EUGENE SMITH
+AND
+LOUIS CHARLES KARPINSKI
+
+BOSTON AND LONDON
+GINN AND COMPANY, PUBLISHERS
+1911
+
+COPYRIGHT, 1911, BY DAVID EUGENE SMITH
+AND LOUIS CHARLES KARPINSKI
+ALL RIGHTS RESERVED
+811.7
+
+THE ATHENAEUM PRESS
+GINN AND COMPANY . PROPRIETORS
+BOSTON . U.S.A.
+
+       *       *       *       *       *
+
+
+{iii}
+
+PREFACE
+
+So familiar are we with the numerals that bear the misleading name of
+Arabic, and so extensive is their use in Europe and the Americas, that it
+is difficult for us to realize that their general acceptance in the
+transactions of commerce is a matter of only the last four centuries, and
+that they are unknown to a very large part of the human race to-day. It
+seems strange that such a labor-saving device should have struggled for
+nearly a thousand years after its system of place value was perfected
+before it replaced such crude notations as the one that the Roman conqueror
+made substantially universal in Europe. Such, however, is the case, and
+there is probably no one who has not at least some slight passing interest
+in the story of this struggle. To the mathematician and the student of
+civilization the interest is generally a deep one; to the teacher of the
+elements of knowledge the interest may be less marked, but nevertheless it
+is real; and even the business man who makes daily use of the curious
+symbols by which we express the numbers of commerce, cannot fail to have
+some appreciation for the story of the rise and progress of these tools of
+his trade.
+
+This story has often been told in part, but it is a long time since any
+effort has been made to bring together the fragmentary narrations and to
+set forth the general problem of the origin and development of these {iv}
+numerals. In this little work we have attempted to state the history of
+these forms in small compass, to place before the student materials for the
+investigation of the problems involved, and to express as clearly as
+possible the results of the labors of scholars who have studied the subject
+in different parts of the world. We have had no theory to exploit, for the
+history of mathematics has seen too much of this tendency already, but as
+far as possible we have weighed the testimony and have set forth what seem
+to be the reasonable conclusions from the evidence at hand.
+
+To facilitate the work of students an index has been prepared which we hope
+may be serviceable. In this the names of authors appear only when some use
+has been made of their opinions or when their works are first mentioned in
+full in a footnote.
+
+If this work shall show more clearly the value of our number system, and
+shall make the study of mathematics seem more real to the teacher and
+student, and shall offer material for interesting some pupil more fully in
+his work with numbers, the authors will feel that the considerable labor
+involved in its preparation has not been in vain.
+
+We desire to acknowledge our especial indebtedness to Professor Alexander
+Ziwet for reading all the proof, as well as for the digest of a Russian
+work, to Professor Clarence L. Meader for Sanskrit transliterations, and to
+Mr. Steven T. Byington for Arabic transliterations and the scheme of
+pronunciation of Oriental names, and also our indebtedness to other
+scholars in Oriental learning for information.
+
+DAVID EUGENE SMITH
+
+LOUIS CHARLES KARPINSKI
+
+       *       *       *       *       *
+
+
+{v}
+
+CONTENTS
+
+  CHAPTER
+
+        PRONUNCIATION OF ORIENTAL NAMES                              vi
+
+  I.    EARLY IDEAS OF THEIR ORIGIN                                   1
+
+  II.   EARLY HINDU FORMS WITH NO PLACE VALUE                        12
+
+  III.  LATER HINDU FORMS, WITH A PLACE VALUE                        38
+
+  IV.   THE SYMBOL ZERO                                              51
+
+  V.    THE QUESTION OF THE INTRODUCTION OF THE
+        NUMERALS INTO EUROPE BY BOETHIUS                             63
+
+  VI.   THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS              91
+
+  VII.  THE DEFINITE INTRODUCTION OF THE NUMERALS INTO EUROPE        99
+
+  VIII. THE SPREAD OF THE NUMERALS IN EUROPE                        128
+
+  INDEX                                                             153
+
+       *       *       *       *       *
+
+
+{vi}
+
+PRONUNCIATION OF ORIENTAL NAMES
+
+(S) = in Sanskrit names and words; (A) = in Arabic names and words.
+
+B, D, F, G, H, J, L, M, N, P, SH (A), T, TH (A), V, W, X, Z, as in English.
+
+A, (S) like _u_ in _but_: thus _pandit_, pronounced _pundit_. (A) like _a_
+in _ask_ or in _man_. [=A], as in _father_.
+
+C, (S) like _ch_ in _church_ (Italian _c_ in _cento_).
+
+[D.], [N.], [S.], [T.], (S) _d_, _n_, _sh_, _t_, made with the tip of the
+tongue turned up and back into the dome of the palate. [D.], [S.], [T.],
+[Z.], (A) _d_, _s_, _t_, _z_, made with the tongue spread so that the
+sounds are produced largely against the side teeth. Europeans commonly
+pronounce [D.], [N.], [S.], [T.], [Z.], both (S) and (A), as simple _d_,
+_n_, _sh_ (S) or _s_ (A), _t_, _z_. [D=] (A), like _th_ in _this_.
+
+E, (S) as in _they_. (A) as in _bed_.
+
+[.G], (A) a voiced consonant formed below the vocal cords; its sound is
+compared by some to a _g_, by others to a guttural _r_; in Arabic words
+adopted into English it is represented by _gh_ (e.g. _ghoul_), less often
+_r_ (e.g. _razzia_).
+
+H preceded by _b_, _c_, _t_, _[t.]_, etc. does not form a single sound with
+these letters, but is a more or less distinct _h_ sound following them; cf.
+the sounds in _abhor, boathook_, etc., or, more accurately for (S), the
+"bhoys" etc. of Irish brogue. H (A) retains its consonant sound at the end
+of a word. [H.], (A) an unvoiced consonant formed below the vocal cords;
+its sound is sometimes compared to German hard _ch_, and may be represented
+by an _h_ as strong as possible. In Arabic words adopted into English it is
+represented by _h_, e.g. in _sahib_, _hakeem_. [H.] (S) is final consonant
+_h_, like final _h_ (A).
+
+I, as in _pin_. [=I], as in _pique_.
+
+K, as in _kick_.
+
+KH, (A) the hard _ch_ of Scotch _loch_, German _ach_, especially of German
+as pronounced by the Swiss.
+
+[.M], [.N], (S) like French final _m_ or _n_, nasalizing the preceding
+vowel.
+
+[N.], see [D.]. N, like _ng_ in _singing_.
+
+O, (S) as in _so_. (A) as in _obey_.
+
+Q, (A) like _k_ (or _c_) in _cook_; further back in the mouth than in
+_kick_.
+
+R, (S) English _r_, smooth and untrilled. (A) stronger. [R.], (S) r used as
+vowel, as in _apron_ when pronounced _aprn_ and not _apern_; modern Hindus
+say _ri_, hence our _amrita_, _Krishna_, for _a-m[r.]ta, K[r.][s.][n.]a_.
+
+S, as in _same_. [S.], see [D.]. ['S], (S) English _sh_ (German _sch_).
+
+[T.], see [D.].
+
+U, as in _put_. [=U], as in _rule_.
+
+Y, as in _you_.
+
+[Z.], see [D.].
+
+`, (A) a sound kindred to the spiritus lenis (that is, to our ears, the
+mere distinct separation of a vowel from the preceding sound, as at the
+beginning of a word in German) and to _[h.]_. The ` is a very distinct
+sound in Arabic, but is more nearly represented by the spiritus lenis than
+by any sound that we can produce without much special training. That is, it
+should be treated as silent, but the sounds that precede and follow it
+should not run together. In Arabic words adopted into English it is treated
+as silent, e.g. in _Arab_, _amber_, _Caaba_ (_`Arab_, _`anbar_, _ka`abah_).
+
+(A) A final long vowel is shortened before _al_ (_'l_) or _ibn_ (whose _i_
+is then silent).
+
+Accent: (S) as if Latin; in determining the place of the accent _[.m]_ and
+_[.n]_ count as consonants, but _h_ after another consonant does not. (A),
+on the last syllable that contains a long vowel or a vowel followed by two
+consonants, except that a final long vowel is not ordinarily accented; if
+there is no long vowel nor two consecutive consonants, the accent falls on
+the first syllable. The words _al_ and _ibn_ are never accented.
+
+       *       *       *       *       *
+
+
+{1}
+
+THE HINDU-ARABIC NUMERALS
+
+CHAPTER I
+
+EARLY IDEAS OF THEIR ORIGIN
+
+It has long been recognized that the common numerals used in daily life are
+of comparatively recent origin. The number of systems of notation employed
+before the Christian era was about the same as the number of written
+languages, and in some cases a single language had several systems. The
+Egyptians, for example, had three systems of writing, with a numerical
+notation for each; the Greeks had two well-defined sets of numerals, and
+the Roman symbols for number changed more or less from century to century.
+Even to-day the number of methods of expressing numerical concepts is much
+greater than one would believe before making a study of the subject, for
+the idea that our common numerals are universal is far from being correct.
+It will be well, then, to think of the numerals that we still commonly call
+Arabic, as only one of many systems in use just before the Christian era.
+As it then existed the system was no better than many others, it was of
+late origin, it contained no zero, it was cumbersome and little used, {2}
+and it had no particular promise. Not until centuries later did the system
+have any standing in the world of business and science; and had the place
+value which now characterizes it, and which requires a zero, been worked
+out in Greece, we might have been using Greek numerals to-day instead of
+the ones with which we are familiar.
+
+Of the first number forms that the world used this is not the place to
+speak. Many of them are interesting, but none had much scientific value. In
+Europe the invention of notation was generally assigned to the eastern
+shores of the Mediterranean until the critical period of about a century
+ago,--sometimes to the Hebrews, sometimes to the Egyptians, but more often
+to the early trading Phoenicians.[1]
+
+The idea that our common numerals are Arabic in origin is not an old one.
+The mediaeval and Renaissance writers generally recognized them as Indian,
+and many of them expressly stated that they were of Hindu origin.[2] {3}
+Others argued that they were probably invented by the Chaldeans or the Jews
+because they increased in value from right to left, an argument that would
+apply quite as well to the Roman and Greek systems, or to any other. It
+was, indeed, to the general idea of notation that many of these writers
+referred, as is evident from the words of England's earliest arithmetical
+textbook-maker, Robert Recorde (c. 1542): "In that thinge all men do agree,
+that the Chaldays, whiche fyrste inuented thys arte, did set these figures
+as thei set all their letters. for they wryte backwarde as you tearme it,
+and so doo they reade. And that may appeare in all Hebrewe, Chaldaye and
+Arabike bookes ... where as the Greekes, Latines, and all nations of
+Europe, do wryte and reade from the lefte hand towarde the ryghte."[3]
+Others, and {4} among them such influential writers as Tartaglia[4] in
+Italy and Koebel[5] in Germany, asserted the Arabic origin of the numerals,
+while still others left the matter undecided[6] or simply dismissed them as
+"barbaric."[7] Of course the Arabs themselves never laid claim to the
+invention, always recognizing their indebtedness to the Hindus both for the
+numeral forms and for the distinguishing feature of place value. Foremost
+among these writers was the great master of the golden age of Bagdad, one
+of the first of the Arab writers to collect the mathematical classics of
+both the East and the West, preserving them and finally passing them on to
+awakening Europe. This man was Mo[h.]ammed the Son of Moses, from
+Khow[=a]rezm, or, more after the manner of the Arab, Mo[h.]ammed ibn
+M[=u]s[=a] al-Khow[=a]razm[=i],[8] a man of great {5} learning and one to
+whom the world is much indebted for its present knowledge of algebra[9] and
+of arithmetic. Of him there will often be occasion to speak; and in the
+arithmetic which he wrote, and of which Adelhard of Bath[10] (c. 1130) may
+have made the translation or paraphrase,[11] he stated distinctly that the
+numerals were due to the Hindus.[12] This is as plainly asserted by later
+Arab {6} writers, even to the present day.[13] Indeed the phrase _`ilm
+hind[=i]_, "Indian science," is used by them for arithmetic, as also the
+adjective _hind[=i]_ alone.[14]
+
+Probably the most striking testimony from Arabic sources is that given by
+the Arabic traveler and scholar Mohammed ibn A[h.]med, Ab[=u]
+'l-R[=i][h.][=a]n al-B[=i]r[=u]n[=i] (973-1048), who spent many years in
+Hindustan. He wrote a large work on India,[15] one on ancient
+chronology,[16] the "Book of the Ciphers," unfortunately lost, which
+treated doubtless of the Hindu art of calculating, and was the author of
+numerous other works. Al-B[=i]r[=u]n[=i] was a man of unusual attainments,
+being versed in Arabic, Persian, Sanskrit, Hebrew, and Syriac, as well as
+in astronomy, chronology, and mathematics. In his work on India he gives
+detailed information concerning the language and {7} customs of the people
+of that country, and states explicitly[17] that the Hindus of his time did
+not use the letters of their alphabet for numerical notation, as the Arabs
+did. He also states that the numeral signs called _a[.n]ka_[18] had
+different shapes in various parts of India, as was the case with the
+letters. In his _Chronology of Ancient Nations_ he gives the sum of a
+geometric progression and shows how, in order to avoid any possibility of
+error, the number may be expressed in three different systems: with Indian
+symbols, in sexagesimal notation, and by an alphabet system which will be
+touched upon later. He also speaks[19] of "179, 876, 755, expressed in
+Indian ciphers," thus again attributing these forms to Hindu sources.
+
+Preceding Al-B[=i]r[=u]n[=i] there was another Arabic writer of the tenth
+century, Mo[t.]ahhar ibn [T.][=a]hir,[20] author of the _Book of the
+Creation and of History_, who gave as a curiosity, in Indian (N[=a]gar[=i])
+symbols, a large number asserted by the people of India to represent the
+duration of the world. Huart feels positive that in Mo[t.]ahhar's time the
+present Arabic symbols had not yet come into use, and that the Indian
+symbols, although known to scholars, were not current. Unless this were the
+case, neither the author nor his readers would have found anything
+extraordinary in the appearance of the number which he cites.
+
+Mention should also be made of a widely-traveled student, Al-Mas`[=u]d[=i]
+(885?-956), whose journeys carried him from Bagdad to Persia, India,
+Ceylon, and even {8} across the China sea, and at other times to
+Madagascar, Syria, and Palestine.[21] He seems to have neglected no
+accessible sources of information, examining also the history of the
+Persians, the Hindus, and the Romans. Touching the period of the Caliphs
+his work entitled _Meadows of Gold_ furnishes a most entertaining fund of
+information. He states[22] that the wise men of India, assembled by the
+king, composed the _Sindhind_. Further on[23] he states, upon the authority
+of the historian Mo[h.]ammed ibn `Al[=i] `Abd[=i], that by order of
+Al-Man[s.][=u]r many works of science and astrology were translated into
+Arabic, notably the _Sindhind_ (_Siddh[=a]nta_). Concerning the meaning and
+spelling of this name there is considerable diversity of opinion.
+Colebrooke[24] first pointed out the connection between _Siddh[=a]nta_ and
+_Sindhind_. He ascribes to the word the meaning "the revolving ages."[25]
+Similar designations are collected by Sedillot,[26] who inclined to the
+Greek origin of the sciences commonly attributed to the Hindus.[27]
+Casiri,[28] citing the _T[=a]r[=i]kh al-[h.]okam[=a]_ or _Chronicles of the
+Learned_,[29] refers to the work {9} as the _Sindum-Indum_ with the meaning
+"perpetuum aeternumque." The reference[30] in this ancient Arabic work to
+Al-Khow[=a]razm[=i] is worthy of note.
+
+This _Sindhind_ is the book, says Mas`[=u]d[=i],[31] which gives all that
+the Hindus know of the spheres, the stars, arithmetic,[32] and the other
+branches of science. He mentions also Al-Khow[=a]razm[=i] and [H.]abash[33]
+as translators of the tables of the _Sindhind_. Al-B[=i]r[=u]n[=i][34]
+refers to two other translations from a work furnished by a Hindu who came
+to Bagdad as a member of the political mission which Sindh sent to the
+caliph Al-Man[s.][=u]r, in the year of the Hejira 154 (A.D. 771).
+
+The oldest work, in any sense complete, on the history of Arabic literature
+and history is the _Kit[=a]b al-Fihrist_, written in the year 987 A.D., by
+Ibn Ab[=i] Ya`q[=u]b al-Nad[=i]m. It is of fundamental importance for the
+history of Arabic culture. Of the ten chief divisions of the work, the
+seventh demands attention in this discussion for the reason that its second
+subdivision treats of mathematicians and astronomers.[35]
+
+{10}
+
+The first of the Arabic writers mentioned is Al-Kind[=i] (800-870 A.D.),
+who wrote five books on arithmetic and four books on the use of the Indian
+method of reckoning. Sened ibn `Al[=i], the Jew, who was converted to Islam
+under the caliph Al-M[=a]m[=u]n, is also given as the author of a work on
+the Hindu method of reckoning. Nevertheless, there is a possibility[36]
+that some of the works ascribed to Sened ibn `Al[=i] are really works of
+Al-Khow[=a]razm[=i], whose name immediately precedes his. However, it is to
+be noted in this connection that Casiri[37] also mentions the same writer
+as the author of a most celebrated work on arithmetic.
+
+To Al-[S.][=u]f[=i], who died in 986 A.D., is also credited a large work on
+the same subject, and similar treatises by other writers are mentioned. We
+are therefore forced to the conclusion that the Arabs from the early ninth
+century on fully recognized the Hindu origin of the new numerals.
+
+Leonard of Pisa, of whom we shall speak at length in the chapter on the
+Introduction of the Numerals into Europe, wrote his _Liber Abbaci_[38] in
+1202. In this work he refers frequently to the nine Indian figures,[39]
+thus showing again the general consensus of opinion in the Middle Ages that
+the numerals were of Hindu origin.
+
+Some interest also attaches to the oldest documents on arithmetic in our
+own language. One of the earliest {11} treatises on algorism is a
+commentary[40] on a set of verses called the _Carmen de Algorismo_, written
+by Alexander de Villa Dei (Alexandra de Ville-Dieu), a Minorite monk of
+about 1240 A.D. The text of the first few lines is as follows:
+
+ "Hec algorism' ars p'sens dicit' in qua
+  Talib; indor[um] fruim bis quinq; figuris.[41]
+
+"This boke is called the boke of algorim or augrym after lewder use. And
+this boke tretys of the Craft of Nombryng, the quych crafte is called also
+Algorym. Ther was a kyng of Inde the quich heyth Algor & he made this
+craft.... Algorisms, in the quych we use teen figurys of Inde."
+
+       *       *       *       *       *
+
+
+{12}
+
+CHAPTER II
+
+EARLY HINDU FORMS WITH NO PLACE VALUE
+
+While it is generally conceded that the scientific development of astronomy
+among the Hindus towards the beginning of the Christian era rested upon
+Greek[42] or Chinese[43] sources, yet their ancient literature testifies to
+a high state of civilization, and to a considerable advance in sciences, in
+philosophy, and along literary lines, long before the golden age of Greece.
+From the earliest times even up to the present day the Hindu has been wont
+to put his thought into rhythmic form. The first of this poetry--it well
+deserves this name, being also worthy from a metaphysical point of
+view[44]--consists of the Vedas, hymns of praise and poems of worship,
+collected during the Vedic period which dates from approximately 2000 B.C.
+to 1400 B.C.[45] Following this work, or possibly contemporary with it, is
+the Brahmanic literature, which is partly ritualistic (the
+Br[=a]hma[n.]as), and partly philosophical (the Upanishads). Our especial
+interest is {13} in the S[=u]tras, versified abridgments of the ritual and
+of ceremonial rules, which contain considerable geometric material used in
+connection with altar construction, and also numerous examples of rational
+numbers the sum of whose squares is also a square, i.e. "Pythagorean
+numbers," although this was long before Pythagoras lived. Whitney[46]
+places the whole of the Veda literature, including the Vedas, the
+Br[=a]hma[n.]as, and the S[=u]tras, between 1500 B.C. and 800 B.C., thus
+agreeing with Buerk[47] who holds that the knowledge of the Pythagorean
+theorem revealed in the S[=u]tras goes back to the eighth century B.C.
+
+The importance of the S[=u]tras as showing an independent origin of Hindu
+geometry, contrary to the opinion long held by Cantor[48] of a Greek
+origin, has been repeatedly emphasized in recent literature,[49] especially
+since the appearance of the important work of Von Schroeder.[50] Further
+fundamental mathematical notions such as the conception of irrationals and
+the use of gnomons, as well as the philosophical doctrine of the
+transmigration of souls,--all of these having long been attributed to the
+Greeks,--are shown in these works to be native to India. Although this
+discussion does not bear directly upon the {14} origin of our numerals, yet
+it is highly pertinent as showing the aptitude of the Hindu for
+mathematical and mental work, a fact further attested by the independent
+development of the drama and of epic and lyric poetry.
+
+It should be stated definitely at the outset, however, that we are not at
+all sure that the most ancient forms of the numerals commonly known as
+Arabic had their origin in India. As will presently be seen, their forms
+may have been suggested by those used in Egypt, or in Eastern Persia, or in
+China, or on the plains of Mesopotamia. We are quite in the dark as to
+these early steps; but as to their development in India, the approximate
+period of the rise of their essential feature of place value, their
+introduction into the Arab civilization, and their spread to the West, we
+have more or less definite information. When, therefore, we consider the
+rise of the numerals in the land of the Sindhu,[51] it must be understood
+that it is only the large movement that is meant, and that there must
+further be considered the numerous possible sources outside of India itself
+and long anterior to the first prominent appearance of the number symbols.
+
+No one attempts to examine any detail in the history of ancient India
+without being struck with the great dearth of reliable material.[52] So
+little sympathy have the people with any save those of their own caste that
+a general literature is wholly lacking, and it is only in the observations
+of strangers that any all-round view of scientific progress is to be found.
+There is evidence that primary schools {15} existed in earliest times, and
+of the seventy-two recognized sciences writing and arithmetic were the most
+prized.[53] In the Vedic period, say from 2000 to 1400 B.C., there was the
+same attention to astronomy that was found in the earlier civilizations of
+Babylon, China, and Egypt, a fact attested by the Vedas themselves.[54]
+Such advance in science presupposes a fair knowledge of calculation, but of
+the manner of calculating we are quite ignorant and probably always shall
+be. One of the Buddhist sacred books, the _Lalitavistara_, relates that
+when the B[=o]dhisattva[55] was of age to marry, the father of Gopa, his
+intended bride, demanded an examination of the five hundred suitors, the
+subjects including arithmetic, writing, the lute, and archery. Having
+vanquished his rivals in all else, he is matched against Arjuna the great
+arithmetician and is asked to express numbers greater than 100 kotis.[56]
+In reply he gave a scheme of number names as high as 10^{53}, adding that
+he could proceed as far as 10^{421},[57] all of which suggests the system
+of Archimedes and the unsettled question of the indebtedness of the West to
+the East in the realm of ancient mathematics.[58] Sir Edwin Arnold, {16} in
+_The Light of Asia_, does not mention this part of the contest, but he
+speaks of Buddha's training at the hands of the learned Vi[s.]vamitra:
+
+  "And Viswamitra said, 'It is enough,
+  Let us to numbers. After me repeat
+  Your numeration till we reach the lakh,[59]
+  One, two, three, four, to ten, and then by tens
+  To hundreds, thousands.' After him the child
+  Named digits, decads, centuries, nor paused,
+  The round lakh reached, but softly murmured on,
+  Then comes the k[=o]ti, nahut, ninnahut,
+  Khamba, viskhamba, abab, attata,
+  To kumuds, gundhikas, and utpalas,
+  By pundar[=i]kas into padumas,
+  Which last is how you count the utmost grains
+  Of Hastagiri ground to finest dust;[60]
+  But beyond that a numeration is,
+  The K[=a]tha, used to count the stars of night,
+  The K[=o]ti-K[=a]tha, for the ocean drops;
+  Ingga, the calculus of circulars;
+  Sarvanikchepa, by the which you deal
+  With all the sands of Gunga, till we come
+  To Antah-Kalpas, where the unit is
+  The sands of the ten crore Gungas.  If one seeks
+  More comprehensive scale, th' arithmic mounts
+  By the Asankya, which is the tale
+  Of all the drops that in ten thousand years
+  Would fall on all the worlds by daily rain;
+  Thence unto Maha Kalpas, by the which
+  The gods compute their future and their past.'"
+
+{17}
+
+Thereupon Vi[s.]vamitra [=A]c[=a]rya[61] expresses his approval of the
+task, and asks to hear the "measure of the line" as far as y[=o]jana, the
+longest measure bearing name. This given, Buddha adds:
+
+  ... "'And master! if it please,
+  I shall recite how many sun-motes lie
+  From end to end within a y[=o]jana.'
+  Thereat, with instant skill, the little prince
+  Pronounced the total of the atoms true.
+  But Viswamitra heard it on his face
+  Prostrate before the boy; 'For thou,' he cried,
+  'Art Teacher of thy teachers--thou, not I,
+  Art G[=u]r[=u].'"
+
+It is needless to say that this is far from being history. And yet it puts
+in charming rhythm only what the ancient _Lalitavistara_ relates of the
+number-series of the Buddha's time. While it extends beyond all reason,
+nevertheless it reveals a condition that would have been impossible unless
+arithmetic had attained a considerable degree of advancement.
+
+To this pre-Christian period belong also the _Ved[=a][.n]gas_, or "limbs
+for supporting the Veda," part of that great branch of Hindu literature
+known as _Sm[r.]iti_ (recollection), that which was to be handed down by
+tradition. Of these the sixth is known as _Jyoti[s.]a_ (astronomy), a short
+treatise of only thirty-six verses, written not earlier than 300 B.C., and
+affording us some knowledge of the extent of number work in that
+period.[62] The Hindus {18} also speak of eighteen ancient Siddh[=a]ntas or
+astronomical works, which, though mostly lost, confirm this evidence.[63]
+
+As to authentic histories, however, there exist in India none relating to
+the period before the Mohammedan era (622 A.D.). About all that we know of
+the earlier civilization is what we glean from the two great epics, the
+Mah[=a]bh[=a]rata[64] and the R[=a]m[=a]yana, from coins, and from a few
+inscriptions.[65]
+
+It is with this unsatisfactory material, then, that we have to deal in
+searching for the early history of the Hindu-Arabic numerals, and the fact
+that many unsolved problems exist and will continue to exist is no longer
+strange when we consider the conditions. It is rather surprising that so
+much has been discovered within a century, than that we are so uncertain as
+to origins and dates and the early spread of the system. The probability
+being that writing was not introduced into India before the close of the
+fourth century B.C., and literature existing only in spoken form prior to
+that period,[66] the number work was doubtless that of all primitive
+peoples, palpable, merely a matter of placing sticks or cowries or pebbles
+on the ground, of marking a sand-covered board, or of cutting notches or
+tying cords as is still done in parts of Southern India to-day.[67]
+
+{19}
+
+The early Hindu numerals[68] may be classified into three great groups, (1)
+the Kharo[s.][t.]h[=i], (2) the Br[=a]hm[=i], and (3) the word and letter
+forms; and these will be considered in order.
+
+The Kharo[s.][t.]h[=i] numerals are found in inscriptions formerly known as
+Bactrian, Indo-Bactrian, and Aryan, and appearing in ancient Gandh[=a]ra,
+now eastern Afghanistan and northern Punjab. The alphabet of the language
+is found in inscriptions dating from the fourth century B.C. to the third
+century A.D., and from the fact that the words are written from right to
+left it is assumed to be of Semitic origin. No numerals, however, have been
+found in the earliest of these inscriptions, number-names probably having
+been written out in words as was the custom with many ancient peoples. Not
+until the time of the powerful King A['s]oka, in the third century B.C., do
+numerals appear in any inscriptions thus far discovered; and then only in
+the primitive form of marks, quite as they would be found in Egypt, Greece,
+Rome, or in {20} various other parts of the world. These A['s]oka[69]
+inscriptions, some thirty in all, are found in widely separated parts of
+India, often on columns, and are in the various vernaculars that were
+familiar to the people. Two are in the Kharo[s.][t.]h[=i] characters, and
+the rest in some form of Br[=a]hm[=i]. In the Kharo[s.][t.]h[=i]
+inscriptions only four numerals have been found, and these are merely
+vertical marks for one, two, four, and five, thus:
+
+  |     ||     |||     ||||
+
+In the so-called ['S]aka inscriptions, possibly of the first century B.C.,
+more numerals are found, and in more highly developed form, the
+right-to-left system appearing, together with evidences of three different
+scales of counting,--four, ten, and twenty. The numerals of this period are
+as follows:
+
+[Illustration]
+
+There are several noteworthy points to be observed in studying this system.
+In the first place, it is probably not as early as that shown in the
+N[=a]n[=a] Gh[=a]t forms hereafter given, although the inscriptions
+themselves at N[=a]n[=a] Gh[=a]t are later than those of the A['s]oka
+period. The {21} four is to this system what the X was to the Roman,
+probably a canceling of three marks as a workman does to-day for five, or a
+laying of one stick across three others. The ten has never been
+satisfactorily explained. It is similar to the A of the Kharo[s.][t.]h[=i]
+alphabet, but we have no knowledge as to why it was chosen. The twenty is
+evidently a ligature of two tens, and this in turn suggested a kind of
+radix, so that ninety was probably written in a way reminding one of the
+quatre-vingt-dix of the French. The hundred is unexplained, although it
+resembles the letter _ta_ or _tra_ of the Br[=a]hm[=i] alphabet with 1
+before (to the right of) it. The two hundred is only a variant of the
+symbol for hundred, with two vertical marks.[70]
+
+This system has many points of similarity with the Nabatean numerals[71] in
+use in the first centuries of the Christian era. The cross is here used for
+four, and the Kharo[s.][t.]h[=i] form is employed for twenty. In addition
+to this there is a trace of an analogous use of a scale of twenty. While
+the symbol for 100 is quite different, the method of forming the other
+hundreds is the same. The correspondence seems to be too marked to be
+wholly accidental.
+
+It is not in the Kharo[s.][t.]h[=i] numerals, therefore, that we can hope
+to find the origin of those used by us, and we turn to the second of the
+Indian types, the Br[=a]hm[=i] characters. The alphabet attributed to
+Brahm[=a] is the oldest of the several known in India, and was used from
+the earliest historic times. There are various theories of its origin, {22}
+none of which has as yet any wide acceptance,[72] although the problem
+offers hope of solution in due time. The numerals are not as old as the
+alphabet, or at least they have not as yet been found in inscriptions
+earlier than those in which the edicts of A['s]oka appear, some of these
+having been incised in Br[=a]hm[=i] as well as Kharo[s.][t.]h[=i]. As
+already stated, the older writers probably wrote the numbers in words, as
+seems to have been the case in the earliest Pali writings of Ceylon.[73]
+
+The following numerals are, as far as known, the only ones to appear in the
+A['s]oka edicts:[74]
+
+[Illustration]
+
+These fragments from the third century B.C., crude and unsatisfactory as
+they are, are the undoubted early forms from which our present system
+developed. They next appear in the second century B.C. in some inscriptions
+in the cave on the top of the N[=a]n[=a] Gh[=a]t hill, about seventy-five
+miles from Poona in central India. These inscriptions may be memorials of
+the early Andhra dynasty of southern India, but their chief interest lies
+in the numerals which they contain.
+
+The cave was made as a resting-place for travelers ascending the hill,
+which lies on the road from Kaly[=a]na to Junar. It seems to have been cut
+out by a descendant {23} of King ['S][=a]tav[=a]hana,[75] for inside the
+wall opposite the entrance are representations of the members of his
+family, much defaced, but with the names still legible. It would seem that
+the excavation was made by order of a king named Vedisiri, and "the
+inscription contains a list of gifts made on the occasion of the
+performance of several _yagnas_ or religious sacrifices," and numerals are
+to be seen in no less than thirty places.[76]
+
+There is considerable dispute as to what numerals are really found in these
+inscriptions, owing to the difficulty of deciphering them; but the
+following, which have been copied from a rubbing, are probably number
+forms:[77]
+
+[Illustration]
+
+The inscription itself, so important as containing the earliest
+considerable Hindu numeral system connected with our own, is of sufficient
+interest to warrant reproducing part of it in facsimile, as is done on page
+24.
+
+{24}
+
+[Illustration]
+
+The next very noteworthy evidence of the numerals, and this quite complete
+as will be seen, is found in certain other cave inscriptions dating back to
+the first or second century A.D. In these, the Nasik[78] cave inscriptions,
+the forms are as follows:
+
+[Illustration]
+
+From this time on, until the decimal system finally adopted the first nine
+characters and replaced the rest of the Br[=a]hm[=i] notation by adding the
+zero, the progress of these forms is well marked. It is therefore well to
+present synoptically the best-known specimens that have come down to us,
+and this is done in the table on page 25.[79]
+
+{25}
+
+TABLE SHOWING THE PROGRESS OF NUMBER FORMS IN INDIA
+
+  NUMERALS 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 1000
+  A['s]oka[80]                [Illustration]
+  ['S]aka[81]                 [Illustration]
+  A['s]oka[82]                [Illustration]
+  N[=a]gar[=i][83]            [Illustration]
+  Nasik[84]                   [Illustration]
+  K[s.]atrapa[85]             [Illustration]
+  Ku[s.]ana [86]              [Illustration]
+  Gupta[87]                   [Illustration]
+  Valhab[=i][88]              [Illustration]
+  Nepal [89]                  [Illustration]
+  Kali[.n]ga[90]              [Illustration]
+  V[=a]k[=a][t.]aka[91]       [Illustration]
+
+[Most of these numerals are given by Buehler, loc. cit., Tafel IX.]
+
+{26} With respect to these numerals it should first be noted that no zero
+appears in the table, and as a matter of fact none existed in any of the
+cases cited. It was therefore impossible to have any place value, and the
+numbers like twenty, thirty, and other multiples of ten, one hundred, and
+so on, required separate symbols except where they were written out in
+words. The ancient Hindus had no less than twenty of these symbols,[92] a
+number that was afterward greatly increased. The following are examples of
+their method of indicating certain numbers between one hundred and one
+thousand:
+
+  [93] [Numerals] for 174
+  [94] [Numerals] for 191
+  [95] [Numerals] for 269
+  [96] [Numerals] for 252
+  [97] [Numerals] for 400
+  [98] [Numerals] for 356
+
+{27}
+
+To these may be added the following numerals below one hundred, similar to
+those in the table:
+
+  [Numerals][99]    for 90
+  [Numerals][100]   for 70
+
+We have thus far spoken of the Kharo[s.][t.]h[=i] and Br[=a]hm[=i]
+numerals, and it remains to mention the third type, the word and letter
+forms. These are, however, so closely connected with the perfecting of the
+system by the invention of the zero that they are more appropriately
+considered in the next chapter, particularly as they have little relation
+to the problem of the origin of the forms known as the Arabic.
+
+Having now examined types of the early forms it is appropriate to turn our
+attention to the question of their origin. As to the first three there is
+no question. The [1 vertical stroke] or [1 horizontal stroke] is simply one
+stroke, or one stick laid down by the computer. The [2 vertical strokes] or
+[2 horizontal strokes] represents two strokes or two sticks, and so for the
+[3 vertical strokes] and [3 horizontal strokes]. From some primitive [2
+vertical strokes] came the two of Egypt, of Rome, of early Greece, and of
+various other civilizations. It appears in the three Egyptian numeral
+systems in the following forms:
+
+  Hieroglyphic [2 vertical strokes]
+  Hieratic     [Hieratic 2]
+  Demotic      [Demotic 2]
+
+The last of these is merely a cursive form as in the Arabic [Arabic 2],
+which becomes our 2 if tipped through a right angle. From some primitive [2
+horizontal strokes] came the Chinese {28} symbol, which is practically
+identical with the symbols found commonly in India from 150 B.C. to 700
+A.D. In the cursive form it becomes [2 horizontal strokes joined], and this
+was frequently used for two in Germany until the 18th century. It finally
+went into the modern form 2, and the [3 horizontal strokes] in the same way
+became our 3.
+
+There is, however, considerable ground for interesting speculation with
+respect to these first three numerals. The earliest Hindu forms were
+perpendicular. In the N[=a]n[=a] Gh[=a]t inscriptions they are vertical.
+But long before either the A['s]oka or the N[=a]n[=a] Gh[=a]t inscriptions
+the Chinese were using the horizontal forms for the first three numerals,
+but a vertical arrangement for four.[101] Now where did China get these
+forms? Surely not from India, for she had them, as her monuments and
+literature[102] show, long before the Hindus knew them. The tradition is
+that China brought her civilization around the north of Tibet, from
+Mongolia, the primitive habitat being Mesopotamia, or possibly the oases of
+Turkestan. Now what numerals did Mesopotamia use? The Babylonian system,
+simple in its general principles but very complicated in many of its
+details, is now well known.[103] In particular, one, two, and three were
+represented by vertical arrow-heads. Why, then, did the Chinese write {29}
+theirs horizontally? The problem now takes a new interest when we find that
+these Babylonian forms were not the primitive ones of this region, but that
+the early Sumerian forms were horizontal.[104]
+
+What interpretation shall be given to these facts? Shall we say that it was
+mere accident that one people wrote "one" vertically and that another wrote
+it horizontally? This may be the case; but it may also be the case that the
+tribal migrations that ended in the Mongol invasion of China started from
+the Euphrates while yet the Sumerian civilization was prominent, or from
+some common source in Turkestan, and that they carried to the East the
+primitive numerals of their ancient home, the first three, these being all
+that the people as a whole knew or needed. It is equally possible that
+these three horizontal forms represent primitive stick-laying, the most
+natural position of a stick placed in front of a calculator being the
+horizontal one. When, however, the cuneiform writing developed more fully,
+the vertical form may have been proved the easier to make, so that by the
+time the migrations to the West began these were in use, and from them came
+the upright forms of Egypt, Greece, Rome, and other Mediterranean lands,
+and those of A['s]oka's time in India. After A['s]oka, and perhaps among
+the merchants of earlier centuries, the horizontal forms may have come down
+into India from China, thus giving those of the N[=a]n[=a] Gh[=a]t cave and
+of later inscriptions. This is in the realm of speculation, but it is not
+improbable that further epigraphical studies may confirm the hypothesis.
+
+{30}
+
+As to the numerals above three there have been very many conjectures. The
+figure one of the Demotic looks like the one of the Sanskrit, the two
+(reversed) like that of the Arabic, the four has some resemblance to that
+in the Nasik caves, the five (reversed) to that on the K[s.]atrapa coins,
+the nine to that of the Ku[s.]ana inscriptions, and other points of
+similarity have been imagined. Some have traced resemblance between the
+Hieratic five and seven and those of the Indian inscriptions. There have
+not, therefore, been wanting those who asserted an Egyptian origin for
+these numerals.[105] There has already been mentioned the fact that the
+Kharo[s.][t.]h[=i] numerals were formerly known as Bactrian, Indo-Bactrian,
+and Aryan. Cunningham[106] was the first to suggest that these numerals
+were derived from the alphabet of the Bactrian civilization of Eastern
+Persia, perhaps a thousand years before our era, and in this he was
+supported by the scholarly work of Sir E. Clive Bayley,[107] who in turn
+was followed by Canon Taylor.[108] The resemblance has not proved
+convincing, however, and Bayley's drawings {31} have been criticized as
+being affected by his theory. The following is part of the hypothesis:[109]
+
+  _Numeral_      _Hindu_       _Bactrian_       _Sanskrit_
+     4           [Symbol]      [Symbol] = ch    chatur, Lat. quattuor
+     5           [Symbol]      [Symbol] = p     pancha, Gk. [Greek:p/ente]
+     6           [Symbol]      [Symbol] = s     [s.]a[s.]
+     7           [Symbol]      [Symbol] = [s.]  sapta
+     ( the s and [s.] are interchanged as occasionally in N. W. India)
+
+Buehler[110] rejects this hypothesis, stating that in four cases (four, six,
+seven, and ten) the facts are absolutely against it.
+
+While the relation to ancient Bactrian forms has been generally doubted, it
+is agreed that most of the numerals resemble Br[=a]hm[=i] letters, and we
+would naturally expect them to be initials.[111] But, knowing the ancient
+pronunciation of most of the number names,[112] we find this not to be the
+case. We next fall back upon the hypothesis {32} that they represent the
+order of letters[113] in the ancient alphabet. From what we know of this
+order, however, there seems also no basis for this assumption. We have,
+therefore, to confess that we are not certain that the numerals were
+alphabetic at all, and if they were alphabetic we have no evidence at
+present as to the basis of selection. The later forms may possibly have
+been alphabetical expressions of certain syllables called _ak[s.]aras_,
+which possessed in Sanskrit fixed numerical values,[114] but this is
+equally uncertain with the rest. Bayley also thought[115] that some of the
+forms were Phoenician, as notably the use of a circle for twenty, but the
+resemblance is in general too remote to be convincing.
+
+There is also some slight possibility that Chinese influence is to be seen
+in certain of the early forms of Hindu numerals.[116]
+
+{33}
+
+More absurd is the hypothesis of a Greek origin, supposedly supported by
+derivation of the current symbols from the first nine letters of the Greek
+alphabet.[117] This difficult feat is accomplished by twisting some of the
+letters, cutting off, adding on, and effecting other changes to make the
+letters fit the theory. This peculiar theory was first set up by
+Dasypodius[118] (Conrad Rauhfuss), and was later elaborated by Huet.[119]
+
+{34}
+
+A bizarre derivation based upon early Arabic (c. 1040 A.D.) sources is
+given by Kircher in his work[120] on number mysticism. He quotes from
+Abenragel,[121] giving the Arabic and a Latin translation[122] and stating
+that the ordinary Arabic forms are derived from sectors of a circle,
+[circle].
+
+Out of all these conflicting theories, and from all the resemblances seen
+or imagined between the numerals of the West and those of the East, what
+conclusions are we prepared to draw as the evidence now stands? Probably
+none that is satisfactory. Indeed, upon the evidence at {35} hand we might
+properly feel that everything points to the numerals as being substantially
+indigenous to India. And why should this not be the case? If the king
+Srong-tsan-Gampo (639 A.D.), the founder of Lh[=a]sa,[123] could have set
+about to devise a new alphabet for Tibet, and if the Siamese, and the
+Singhalese, and the Burmese, and other peoples in the East, could have
+created alphabets of their own, why should not the numerals also have been
+fashioned by some temple school, or some king, or some merchant guild? By
+way of illustration, there are shown in the table on page 36 certain
+systems of the East, and while a few resemblances are evident, it is also
+evident that the creators of each system endeavored to find original forms
+that should not be found in other systems. This, then, would seem to be a
+fair interpretation of the evidence. A human mind cannot readily create
+simple forms that are absolutely new; what it fashions will naturally
+resemble what other minds have fashioned, or what it has known through
+hearsay or through sight. A circle is one of the world's common stock of
+figures, and that it should mean twenty in Phoenicia and in India is hardly
+more surprising than that it signified ten at one time in Babylon.[124] It
+is therefore quite probable that an extraneous origin cannot be found for
+the very sufficient reason that none exists.
+
+Of absolute nonsense about the origin of the symbols which we use much has
+been written. Conjectures, {36} however, without any historical evidence
+for support, have no place in a serious discussion of the gradual evolution
+of the present numeral forms.[125]
+
+  TABLE OF CERTAIN EASTERN SYSTEMS
+  Siam            [Illustration: numerals]
+  Burma[126]      [Illustration: numerals]
+  Malabar[127]    [Illustration: numerals]
+  Tibet[128]      [Illustration: numerals]
+  Ceylon[129]     [Illustration: numerals]
+  Malayalam[129]  [Illustration: numerals]
+
+{37}
+
+We may summarize this chapter by saying that no one knows what suggested
+certain of the early numeral forms used in India. The origin of some is
+evident, but the origin of others will probably never be known. There is no
+reason why they should not have been invented by some priest or teacher or
+guild, by the order of some king, or as part of the mysticism of some
+temple. Whatever the origin, they were no better than scores of other
+ancient systems and no better than the present Chinese system when written
+without the zero, and there would never have been any chance of their
+triumphal progress westward had it not been for this relatively late
+symbol. There could hardly be demanded a stronger proof of the Hindu origin
+of the character for zero than this, and to it further reference will be
+made in Chapter IV.
+
+       *       *       *       *       *
+
+
+{38}
+
+CHAPTER III
+
+LATER HINDU FORMS, WITH A PLACE VALUE
+
+Before speaking of the perfected Hindu numerals with the zero and the place
+value, it is necessary to consider the third system mentioned on page
+19,--the word and letter forms. The use of words with place value began at
+least as early as the 6th century of the Christian era. In many of the
+manuals of astronomy and mathematics, and often in other works in
+mentioning dates, numbers are represented by the names of certain objects
+or ideas. For example, zero is represented by "the void" (_['s][=u]nya_),
+or "heaven-space" (_ambara [=a]k[=a]['s]a_); one by "stick" (_rupa_),
+"moon" (_indu ['s]a['s]in_), "earth" (_bh[=u]_), "beginning" (_[=a]di_),
+"Brahma," or, in general, by anything markedly unique; two by "the twins"
+(_yama_), "hands" (_kara_), "eyes" (_nayana_), etc.; four by "oceans," five
+by "senses" (_vi[s.]aya_) or "arrows" (the five arrows of K[=a]mad[=e]va);
+six by "seasons" or "flavors"; seven by "mountain" (_aga_), and so on.[130]
+These names, accommodating themselves to the verse in which scientific
+works were written, had the additional advantage of not admitting, as did
+the figures, easy alteration, since any change would tend to disturb the
+meter.
+
+{39}
+
+As an example of this system, the date "['S]aka Sa[m.]vat, 867" (A.D. 945
+or 946), is given by "_giri-ra[s.]a-vasu_," meaning "the mountains"
+(seven), "the flavors" (six), and the gods "_Vasu_" of which there were
+eight. In reading the date these are read from right to left.[131] The
+period of invention of this system is uncertain. The first trace seems to
+be in the _['S]rautas[=u]tra_ of K[=a]ty[=a]yana and
+L[=a][t.]y[=a]yana.[132] It was certainly known to Var[=a]ha-Mihira (d.
+587),[133] for he used it in the _B[r.]hat-Sa[m.]hit[=a]._[134] It has also
+been asserted[135] that [=A]ryabha[t.]a (c. 500 A.D.) was familiar with
+this system, but there is nothing to prove the statement.[136] The earliest
+epigraphical examples of the system are found in the Bayang (Cambodia)
+inscriptions of 604 and 624 A.D.[137]
+
+Mention should also be made, in this connection, of a curious system of
+alphabetic numerals that sprang up in southern India. In this we have the
+numerals represented by the letters as given in the following table:
+
+  1        2       3       4       5        6       7      8     9      0
+  k        kh      g       gh      [.n]     c       ch     j     jh     n
+  [t.]     [t.]h   [d.]    [d.]h   [n.]     t       th     d     th     n
+  p        ph      b       bh      m
+  y        r       l       v       ['s]     [s.]    s      h      l
+
+{40}
+
+By this plan a numeral might be represented by any one of several letters,
+as shown in the preceding table, and thus it could the more easily be
+formed into a word for mnemonic purposes. For example, the word
+
+    2     3      1     5    6       5       1
+  _kha_ _gont_ _yan_ _me_ _[s.]a_ _m[=a]_ _pa_
+
+has the value 1,565,132, reading from right to left.[138] This, the oldest
+specimen (1184 A.D.) known of this notation, is given in a commentary on
+the Rigveda, representing the number of days that had elapsed from the
+beginning of the Kaliyuga. Burnell[139] states that this system is even yet
+in use for remembering rules to calculate horoscopes, and for astronomical
+tables.
+
+A second system of this kind is still used in the pagination of manuscripts
+in Ceylon, Siam, and Burma, having also had its rise in southern India. In
+this the thirty-four consonants when followed by _a_ (as _ka_ ... _la_)
+designate the numbers 1-34; by _[=a]_ (as _k[=a]_ ... _l[=a]_), those from
+35 to 68; by _i_ (_ki_ ... _li_), those from 69 to 102, inclusive; and so
+on.[140]
+
+As already stated, however, the Hindu system as thus far described was no
+improvement upon many others of the ancients, such as those used by the
+Greeks and the Hebrews. Having no zero, it was impracticable to designate
+the tens, hundreds, and other units of higher order by the same symbols
+used for the units from one to nine. In other words, there was no
+possibility of place value without some further improvement. So the
+N[=a]n[=a] Gh[=a]t {41} symbols required the writing of "thousand seven
+twenty-four" about like T 7, tw, 4 in modern symbols, instead of 7024, in
+which the seven of the thousands, the two of the tens (concealed in the
+word twenty, being originally "twain of tens," the _-ty_ signifying ten),
+and the four of the units are given as spoken and the order of the unit
+(tens, hundreds, etc.) is given by the place. To complete the system only
+the zero was needed; but it was probably eight centuries after the
+N[=a]n[=a] Gh[=a]t inscriptions were cut, before this important symbol
+appeared; and not until a considerably later period did it become well
+known. Who it was to whom the invention is due, or where he lived, or even
+in what century, will probably always remain a mystery.[141] It is possible
+that one of the forms of ancient abacus suggested to some Hindu astronomer
+or mathematician the use of a symbol to stand for the vacant line when the
+counters were removed. It is well established that in different parts of
+India the names of the higher powers took different forms, even the order
+being interchanged.[142] Nevertheless, as the significance of the name of
+the unit was given by the order in reading, these variations did not lead
+to error. Indeed the variation itself may have necessitated the
+introduction of a word to signify a vacant place or lacking unit, with the
+ultimate introduction of a zero symbol for this word.
+
+To enable us to appreciate the force of this argument a large number,
+8,443,682,155, may be considered as the Hindus wrote and read it, and then,
+by way of contrast, as the Greeks and Arabs would have read it.
+
+{42}
+
+_Modern American reading_, 8 billion, 443 million, 682 thousand, 155.
+
+_Hindu_, 8 padmas, 4 vyarbudas, 4 k[=o][t.]is, 3 prayutas, 6 lak[s.]as, 8
+ayutas, 2 sahasra, 1 ['s]ata, 5 da['s]an, 5.
+
+_Arabic and early German_, eight thousand thousand thousand and four
+hundred thousand thousand and forty-three thousand thousand, and six
+hundred thousand and eighty-two thousand and one hundred fifty-five (or
+five and fifty).
+
+_Greek_, eighty-four myriads of myriads and four thousand three hundred
+sixty-eight myriads and two thousand and one hundred fifty-five.
+
+As Woepcke[143] pointed out, the reading of numbers of this kind shows that
+the notation adopted by the Hindus tended to bring out the place idea. No
+other language than the Sanskrit has made such consistent application, in
+numeration, of the decimal system of numbers. The introduction of myriads
+as in the Greek, and thousands as in Arabic and in modern numeration, is
+really a step away from a decimal scheme. So in the numbers below one
+hundred, in English, eleven and twelve are out of harmony with the rest of
+the -teens, while the naming of all the numbers between ten and twenty is
+not analogous to the naming of the numbers above twenty. To conform to our
+written system we should have ten-one, ten-two, ten-three, and so on, as we
+have twenty-one, twenty-two, and the like. The Sanskrit is consistent, the
+units, however, preceding the tens and hundreds. Nor did any other ancient
+people carry the numeration as far as did the Hindus.[144]
+
+{43}
+
+When the _a[.n]kapalli_,[145] the decimal-place system of writing numbers,
+was perfected, the tenth symbol was called the _['s][=u]nyabindu_,
+generally shortened to _['s][=u]nya_ (the void). Brockhaus[146] has well
+said that if there was any invention for which the Hindus, by all their
+philosophy and religion, were well fitted, it was the invention of a symbol
+for zero. This making of nothingness the crux of a tremendous achievement
+was a step in complete harmony with the genius of the Hindu.
+
+It is generally thought that this _['s][=u]nya_ as a symbol was not used
+before about 500 A.D., although some writers have placed it earlier.[147]
+Since [=A]ryabha[t.]a gives our common method of extracting roots, it would
+seem that he may have known a decimal notation,[148] although he did not
+use the characters from which our numerals are derived.[149] Moreover, he
+frequently speaks of the {44} void.[150] If he refers to a symbol this
+would put the zero as far back as 500 A.D., but of course he may have
+referred merely to the concept of nothingness.
+
+A little later, but also in the sixth century, Var[=a]ha-Mihira[151] wrote
+a work entitled _B[r.]hat Sa[m.]hit[=a]_[152] in which he frequently uses
+_['s][=u]nya_ in speaking of numerals, so that it has been thought that he
+was referring to a definite symbol. This, of course, would add to the
+probability that [=A]ryabha[t.]a was doing the same.
+
+It should also be mentioned as a matter of interest, and somewhat related
+to the question at issue, that Var[=a]ha-Mihira used the word-system with
+place value[153] as explained above.
+
+The first kind of alphabetic numerals and also the word-system (in both of
+which the place value is used) are plays upon, or variations of, position
+arithmetic, which would be most likely to occur in the country of its
+origin.[154]
+
+At the opening of the next century (c. 620 A.D.) B[=a][n.]a[155] wrote of
+Subandhus's _V[=a]savadatt[=a]_ as a celebrated work, {45} and mentioned
+that the stars dotting the sky are here compared with zeros, these being
+points as in the modern Arabic system. On the other hand, a strong argument
+against any Hindu knowledge of the symbol zero at this time is the fact
+that about 700 A.D. the Arabs overran the province of Sind and thus had an
+opportunity of knowing the common methods used there for writing numbers.
+And yet, when they received the complete system in 776 they looked upon it
+as something new.[156] Such evidence is not conclusive, but it tends to
+show that the complete system was probably not in common use in India at
+the beginning of the eighth century. On the other hand, we must bear in
+mind the fact that a traveler in Germany in the year 1700 would probably
+have heard or seen nothing of decimal fractions, although these were
+perfected a century before that date. The elite of the mathematicians may
+have known the zero even in [=A]ryabha[t.]a's time, while the merchants and
+the common people may not have grasped the significance of the novelty
+until a long time after. On the whole, the evidence seems to point to the
+west coast of India as the region where the complete system was first
+seen.[157] As mentioned above, traces of the numeral words with place
+value, which do not, however, absolutely require a decimal place-system of
+symbols, are found very early in Cambodia, as well as in India.
+
+Concerning the earliest epigraphical instances of the use of the nine
+symbols, plus the zero, with place value, there {46} is some question.
+Colebrooke[158] in 1807 warned against the possibility of forgery in many
+of the ancient copper-plate land grants. On this account Fleet, in the
+_Indian Antiquary_,[159] discusses at length this phase of the work of the
+epigraphists in India, holding that many of these forgeries were made about
+the end of the eleventh century. Colebrooke[160] takes a more rational view
+of these forgeries than does Kaye, who seems to hold that they tend to
+invalidate the whole Indian hypothesis. "But even where that may be
+suspected, the historical uses of a monument fabricated so much nearer to
+the times to which it assumes to belong, will not be entirely superseded.
+The necessity of rendering the forged grant credible would compel a
+fabricator to adhere to history, and conform to established notions: and
+the tradition, which prevailed in his time, and by which he must be guided,
+would probably be so much nearer to the truth, as it was less remote from
+the period which it concerned."[161] Buehler[162] gives the copper-plate
+Gurjara inscription of Cedi-sa[m.]vat 346 (595 A.D.) as the oldest
+epigraphical use of the numerals[163] "in which the symbols correspond to
+the alphabet numerals of the period and the place." Vincent A. Smith[164]
+quotes a stone inscription of 815 A.D., dated Sa[m.]vat 872. So F. Kielhorn
+in the _Epigraphia Indica_[165] gives a Pathari pillar inscription of
+Parabala, dated Vikrama-sa[m.]vat 917, which corresponds to 861 A.D., {47}
+and refers also to another copper-plate inscription dated Vikrama-sa[m.]vat
+813 (756 A.D.). The inscription quoted by V. A. Smith above is that given
+by D. R. Bhandarkar,[166] and another is given by the same writer as of
+date Saka-sa[m.]vat 715 (798 A.D.), being incised on a pilaster.
+Kielhorn[167] also gives two copper-plate inscriptions of the time of
+Mahendrapala of Kanauj, Valhab[=i]-sa[m.]vat 574 (893 A.D.) and
+Vikrama-sa[m.]vat 956 (899 A.D.). That there should be any inscriptions of
+date as early even as 750 A.D., would tend to show that the system was at
+least a century older. As will be shown in the further development, it was
+more than two centuries after the introduction of the numerals into Europe
+that they appeared there upon coins and inscriptions. While Thibaut[168]
+does not consider it necessary to quote any specific instances of the use
+of the numerals, he states that traces are found from 590 A.D. on. "That
+the system now in use by all civilized nations is of Hindu origin cannot be
+doubted; no other nation has any claim upon its discovery, especially since
+the references to the origin of the system which are found in the nations
+of western Asia point unanimously towards India."[169]
+
+The testimony and opinions of men like Buehler, Kielhorn, V. A. Smith,
+Bhandarkar, and Thibaut are entitled to the most serious consideration. As
+authorities on ancient Indian epigraphy no others rank higher. Their work
+is accepted by Indian scholars the world over, and their united judgment as
+to the rise of the system with a place value--that it took place in India
+as early as the {48} sixth century A.D.--must stand unless new evidence of
+great weight can be submitted to the contrary.
+
+Many early writers remarked upon the diversity of Indian numeral forms.
+Al-B[=i]r[=u]n[=i] was probably the first; noteworthy is also Johannes
+Hispalensis,[170] who gives the variant forms for seven and four. We insert
+on p. 49 a table of numerals used with place value. While the chief
+authority for this is Buehler,[171] several specimens are given which are
+not found in his work and which are of unusual interest.
+
+The ['S][=a]rad[=a] forms given in the table use the circle as a symbol for
+1 and the dot for zero. They are taken from the paging and text of _The
+Kashmirian Atharva-Veda_[172], of which the manuscript used is certainly
+four hundred years old. Similar forms are found in a manuscript belonging
+to the University of Tuebingen. Two other series presented are from Tibetan
+books in the library of one of the authors.
+
+For purposes of comparison the modern Sanskrit and Arabic numeral forms are
+added.
+
+  Sanskrit, [Illustration]
+  Arabic, [Illustration]
+
+{49}
+
+NUMERALS USED WITH PLACE VALUE
+
+      1  2  3  4  5  6  7  8  9  0
+  a[173] [Illustration]
+  b[174] [Illustration]
+  c[175] [Illustration]
+  d[176] [Illustration]
+  e[177] [Illustration]
+  f[178] [Illustration]
+  g[179] [Illustration]
+  h[180] [Illustration]
+  i[180] [Illustration]
+  j[181] [Illustration]
+  k[181] [Illustration]
+  l[182] [Illustration]
+  m[183] [Illustration]
+  n[184] [Illustration]
+
+       *       *       *       *       *
+
+
+{51}
+
+CHAPTER IV
+
+THE SYMBOL ZERO
+
+What has been said of the improved Hindu system with a place value does not
+touch directly the origin of a symbol for zero, although it assumes that
+such a symbol exists. The importance of such a sign, the fact that it is a
+prerequisite to a place-value system, and the further fact that without it
+the Hindu-Arabic numerals would never have dominated the computation system
+of the western world, make it proper to devote a chapter to its origin and
+history.
+
+It was some centuries after the primitive Br[=a]hm[=i] and
+Kharo[s.][t.]h[=i] numerals had made their appearance in India that the
+zero first appeared there, although such a character was used by the
+Babylonians[185] in the centuries immediately preceding the Christian era.
+The symbol is [Babylonian zero symbol] or [Babylonian zero symbol], and
+apparently it was not used in calculation. Nor does it always occur when
+units of any order are lacking; thus 180 is written [Babylonian numerals
+180] with the meaning three sixties and no units, since 181 immediately
+following is [Babylonian numerals 181], three sixties and one unit.[186]
+The main {52} use of this Babylonian symbol seems to have been in the
+fractions, 60ths, 3600ths, etc., and somewhat similar to the Greek use of
+[Greek: o], for [Greek: ouden], with the meaning _vacant_.
+
+"The earliest undoubted occurrence of a zero in India is an inscription at
+Gwalior, dated Samvat 933 (876 A.D.). Where 50 garlands are mentioned (line
+20), 50 is written [Gwalior numerals 50]. 270 (line 4) is written [Gwalior
+numerals 270]."[187] The Bakh[s.][=a]l[=i] Manuscript[188] probably
+antedates this, using the point or dot as a zero symbol. Bayley mentions a
+grant of Jaika Rashtrakuta of Bharuj, found at Okamandel, of date 738 A.D.,
+which contains a zero, and also a coin with indistinct Gupta date 707 (897
+A.D.), but the reliability of Bayley's work is questioned. As has been
+noted, the appearance of the numerals in inscriptions and on coins would be
+of much later occurrence than the origin and written exposition of the
+system. From the period mentioned the spread was rapid over all of India,
+save the southern part, where the Tamil and Malayalam people retain the old
+system even to the present day.[189]
+
+Aside from its appearance in early inscriptions, there is still another
+indication of the Hindu origin of the symbol in the special treatment of
+the concept zero in the early works on arithmetic. Brahmagupta, who lived
+in Ujjain, the center of Indian astronomy,[190] in the early part {53} of
+the seventh century, gives in his arithmetic[191] a distinct treatment of
+the properties of zero. He does not discuss a symbol, but he shows by his
+treatment that in some way zero had acquired a special significance not
+found in the Greek or other ancient arithmetics. A still more scientific
+treatment is given by Bh[=a]skara,[192] although in one place he permits
+himself an unallowed liberty in dividing by zero. The most recently
+discovered work of ancient Indian mathematical lore, the
+Ganita-S[=a]ra-Sa[.n]graha[193] of Mah[=a]v[=i]r[=a]c[=a]rya (c. 830 A.D.),
+while it does not use the numerals with place value, has a similar
+discussion of the calculation with zero.
+
+What suggested the form for the zero is, of course, purely a matter of
+conjecture. The dot, which the Hindus used to fill up lacunae in their
+manuscripts, much as we indicate a break in a sentence,[194] would have
+been a more natural symbol; and this is the one which the Hindus first
+used[195] and which most Arabs use to-day. There was also used for this
+purpose a cross, like our X, and this is occasionally found as a zero
+symbol.[196] In the Bakh[s.][=a]l[=i] manuscript above mentioned, the word
+_['s][=u]nya_, with the dot as its symbol, is used to denote the unknown
+quantity, as well as to denote zero. An analogous use of the {54} zero, for
+the unknown quantity in a proportion, appears in a Latin manuscript of some
+lectures by Gottfried Wolack in the University of Erfurt in 1467 and
+1468.[197] The usage was noted even as early as the eighteenth
+century.[198]
+
+The small circle was possibly suggested by the spurred circle which was
+used for ten.[199] It has also been thought that the omicron used by
+Ptolemy in his _Almagest_, to mark accidental blanks in the sexagesimal
+system which he employed, may have influenced the Indian writers.[200] This
+symbol was used quite generally in Europe and Asia, and the Arabic
+astronomer Al-Batt[=a]n[=i][201] (died 929 A.D.) used a similar symbol in
+connection with the alphabetic system of numerals. The occasional use by
+Al-Batt[=a]n[=i] of the Arabic negative, _l[=a]_, to indicate the absence
+of minutes {55} (or seconds), is noted by Nallino.[202] Noteworthy is also
+the use of the [Circle] for unity in the ['S][=a]rad[=a] characters of the
+Kashmirian Atharva-Veda, the writing being at least 400 years old.
+Bh[=a]skara (c. 1150) used a small circle above a number to indicate
+subtraction, and in the Tartar writing a redundant word is removed by
+drawing an oval around it. It would be interesting to know whether our
+score mark [score mark], read "four in the hole," could trace its pedigree
+to the same sources. O'Creat[203] (c. 1130), in a letter to his teacher,
+Adelhard of Bath, uses [Greek: t] for zero, being an abbreviation for the
+word _teca_ which we shall see was one of the names used for zero, although
+it could quite as well be from [Greek: tziphra]. More rarely O'Creat uses
+[circle with bar], applying the name _cyfra_ to both forms. Frater
+Sigsboto[204] (c. 1150) uses the same symbol. Other peculiar forms are
+noted by Heiberg[205] as being in use among the Byzantine Greeks in the
+fifteenth century. It is evident from the text that some of these writers
+did not understand the import of the new system.[206]
+
+Although the dot was used at first in India, as noted above, the small
+circle later replaced it and continues in use to this day. The Arabs,
+however, did not adopt the {56} circle, since it bore some resemblance to
+the letter which expressed the number five in the alphabet system.[207] The
+earliest Arabic zero known is the dot, used in a manuscript of 873
+A.D.[208] Sometimes both the dot and the circle are used in the same work,
+having the same meaning, which is the case in an Arabic MS., an abridged
+arithmetic of Jamshid,[209] 982 A.H. (1575 A.D.). As given in this work the
+numerals are [symbols]. The form for 5 varies, in some works becoming
+[symbol] or [symbol]; [symbol] is found in Egypt and [symbol] appears in
+some fonts of type. To-day the Arabs use the 0 only when, under European
+influence, they adopt the ordinary system. Among the Chinese the first
+definite trace of zero is in the work of Tsin[210] of 1247 A.D. The form is
+the circular one of the Hindus, and undoubtedly was brought to China by
+some traveler.
+
+The name of this all-important symbol also demands some attention,
+especially as we are even yet quite undecided as to what to call it. We
+speak of it to-day as _zero, naught_, and even _cipher_; the telephone
+operator often calls it _O_, and the illiterate or careless person calls it
+_aught_. In view of all this uncertainty we may well inquire what it has
+been called in the past.[211]
+
+{57}
+
+As already stated, the Hindus called it _['s][=u]nya_, "void."[212] This
+passed over into the Arabic as _a[s.]-[s.]ifr_ or _[s.]ifr_.[213] When
+Leonard of Pisa (1202) wrote upon the Hindu numerals he spoke of this
+character as _zephirum_.[214] Maximus Planudes (1330), writing under both
+the Greek and the Arabic influence, called it _tziphra_.[215] In a treatise
+on arithmetic written in the Italian language by Jacob of Florence[216]
+{58} (1307) it is called _zeuero_,[217] while in an arithmetic of Giovanni
+di Danti of Arezzo (1370) the word appears as _ceuero_.[218] Another form
+is _zepiro_,[219] which was also a step from _zephirum_ to zero.[220]
+
+Of course the English _cipher_, French _chiffre_, is derived from the same
+Arabic word, _a[s.]-[s.]ifr_, but in several languages it has come to mean
+the numeral figures in general. A trace of this appears in our word
+_ciphering_, meaning figuring or computing.[221] Johann Huswirt[222] uses
+the word with both meanings; he gives for the tenth character the four
+names _theca, circulus, cifra_, and _figura nihili_. In this statement
+Huswirt probably follows, as did many writers of that period, the
+_Algorismus_ of Johannes de Sacrobosco (c. 1250 A.D.), who was also known
+as John of Halifax or John of Holywood. The commentary of {59} Petrus de
+Dacia[223] (c. 1291 A.D.) on the _Algorismus vulgaris_ of Sacrobosco was
+also widely used. The widespread use of this Englishman's work on
+arithmetic in the universities of that time is attested by the large
+number[224] of MSS. from the thirteenth to the seventeenth century still
+extant, twenty in Munich, twelve in Vienna, thirteen in Erfurt, several in
+England given by Halliwell,[225] ten listed in Coxe's _Catalogue of the
+Oxford College Library_, one in the Plimpton collection,[226] one in the
+Columbia University Library, and, of course, many others.
+
+From _a[s.]-[s.]ifr _has come _zephyr, cipher,_ and finally the abridged
+form _zero_. The earliest printed work in which is found this final form
+appears to be Calandri's arithmetic of 1491,[227] while in manuscript it
+appears at least as early as the middle of the fourteenth century.[228] It
+also appears in a work, _Le Kadran des marchans_, by Jehan {60}
+Certain,[229] written in 1485. This word soon became fairly well known in
+Spain[230] and France.[231] The medieval writers also spoke of it as the
+_sipos_,[232] and occasionally as the _wheel_,[233] _circulus_[234] (in
+German _das Ringlein_[235]), _circular {61} note_,[236] _theca_,[237] long
+supposed to be from its resemblance to the Greek theta, but explained by
+Petrus de Dacia as being derived from the name of the iron[238] used to
+brand thieves and robbers with a circular mark placed on the forehead or on
+the cheek. It was also called _omicron_[239] (the Greek _o_), being
+sometimes written o or [Greek: ph] to distinguish it from the letter _o_.
+It also went by the name _null_[240] (in the Latin books {62} _nihil_[241]
+or _nulla_,[242] and in the French _rien_[243]), and very commonly by the
+name _cipher_.[244] Wallis[245] gives one of the earliest extended
+discussions of the various forms of the word, giving certain other
+variations worthy of note, as _ziphra_, _zifera_, _siphra_, _ciphra_,
+_tsiphra_, _tziphra,_ and the Greek [Greek: tziphra].[246]
+
+       *       *       *       *       *
+
+
+{63}
+
+CHAPTER V
+
+THE QUESTION OF THE INTRODUCTION OF THE NUMERALS INTO EUROPE BY BOETHIUS
+
+Just as we were quite uncertain as to the origin of the numeral forms, so
+too are we uncertain as to the time and place of their introduction into
+Europe. There are two general theories as to this introduction. The first
+is that they were carried by the Moors to Spain in the eighth or ninth
+century, and thence were transmitted to Christian Europe, a theory which
+will be considered later. The second, advanced by Woepcke,[247] is that
+they were not brought to Spain by the Moors, but that they were already in
+Spain when the Arabs arrived there, having reached the West through the
+Neo-Pythagoreans. There are two facts to support this second theory: (1)
+the forms of these numerals are characteristic, differing materially from
+those which were brought by Leonardo of Pisa from Northern Africa early in
+the thirteenth century (before 1202 A.D.); (2) they are essentially those
+which {64} tradition has so persistently assigned to Boethius (c. 500
+A.D.), and which he would naturally have received, if at all, from these
+same Neo-Pythagoreans or from the sources from which they derived them.
+Furthermore, Woepcke points out that the Arabs on entering Spain (711 A.D.)
+would naturally have followed their custom of adopting for the computation
+of taxes the numerical systems of the countries they conquered,[248] so
+that the numerals brought from Spain to Italy, not having undergone the
+same modifications as those of the Eastern Arab empire, would have
+differed, as they certainly did, from those that came through Bagdad. The
+theory is that the Hindu system, without the zero, early reached Alexandria
+(say 450 A.D.), and that the Neo-Pythagorean love for the mysterious and
+especially for the Oriental led to its use as something bizarre and
+cabalistic; that it was then passed along the Mediterranean, reaching
+Boethius in Athens or in Rome, and to the schools of Spain, being
+discovered in Africa and Spain by the Arabs even before they themselves
+knew the improved system with the place value.
+
+{65}
+
+A recent theory set forth by Bubnov[249] also deserves mention, chiefly
+because of the seriousness of purpose shown by this well-known writer.
+Bubnov holds that the forms first found in Europe are derived from ancient
+symbols used on the abacus, but that the zero is of Hindu origin. This
+theory does not seem tenable, however, in the light of the evidence already
+set forth.
+
+Two questions are presented by Woepcke's theory: (1) What was the nature of
+these Spanish numerals, and how were they made known to Italy? (2) Did
+Boethius know them?
+
+The Spanish forms of the numerals were called the _[h.]ur[=u]f
+al-[.g]ob[=a]r_, the [.g]ob[=a]r or dust numerals, as distinguished from
+the _[h.]ur[=u]f al-jumal_ or alphabetic numerals. Probably the latter,
+under the influence of the Syrians or Jews,[250] were also used by the
+Arabs. The significance of the term [.g]ob[=a]r is doubtless that these
+numerals were written on the dust abacus, this plan being distinct from the
+counter method of representing numbers. It is also worthy of note that
+Al-B[=i]r[=u]n[=i] states that the Hindus often performed numerical
+computations in the sand. The term is found as early as c. 950, in the
+verses of an anonymous writer of Kairw[=a]n, in Tunis, in which the author
+speaks of one of his works on [.g]ob[=a]r calculation;[251] and, much
+later, the Arab writer Ab[=u] Bekr Mo[h.]ammed ibn `Abdall[=a]h, surnamed
+al-[H.]a[s.][s.][=a]r {66} (the arithmetician), wrote a work of which the
+second chapter was "On the dust figures."[252]
+
+The [.g]ob[=a]r numerals themselves were first made known to modern
+scholars by Silvestre de Sacy, who discovered them in an Arabic manuscript
+from the library of the ancient abbey of St.-Germain-des-Pres.[253] The
+system has nine characters, but no zero. A dot above a character indicates
+tens, two dots hundreds, and so on, [5 with dot] meaning 50, and [5 with 3
+dots] meaning 5000. It has been suggested that possibly these dots,
+sprinkled like dust above the numerals, gave rise to the word
+_[.g]ob[=a]r_,[254] but this is not at all probable. This system of dots is
+found in Persia at a much later date with numerals quite like the modern
+Arabic;[255] but that it was used at all is significant, for it is hardly
+likely that the western system would go back to Persia, when the perfected
+Hindu one was near at hand.
+
+At first sight there would seem to be some reason for believing that this
+feature of the [.g]ob[=a]r system was of {67} Arabic origin, and that the
+present zero of these people,[256] the dot, was derived from it. It was
+entirely natural that the Semitic people generally should have adopted such
+a scheme, since their diacritical marks would suggest it, not to speak of
+the possible influence of the Greek accents in the Hellenic number system.
+When we consider, however, that the dot is found for zero in the
+Bakh[s.][=a]l[=i] manuscript,[257] and that it was used in subscript form
+in the _Kit[=a]b al-Fihrist_[258] in the tenth century, and as late as the
+sixteenth century,[259] although in this case probably under Arabic
+influence, we are forced to believe that this form may also have been of
+Hindu origin.
+
+The fact seems to be that, as already stated,[260] the Arabs did not
+immediately adopt the Hindu zero, because it resembled their 5; they used
+the superscript dot as serving their purposes fairly well; they may,
+indeed, have carried this to the west and have added it to the [.g]ob[=a]r
+forms already there, just as they transmitted it to the Persians.
+Furthermore, the Arab and Hebrew scholars of Northern Africa in the tenth
+century knew these numerals as Indian forms, for a commentary on the
+_S[=e]fer Ye[s.][=i]r[=a]h_ by Ab[=u] Sahl ibn Tamim (probably composed at
+Kairw[=a]n, c. 950) speaks of "the Indian arithmetic known under the name
+of _[.g]ob[=a]r_ or dust calculation."[261] All this suggests that the
+Arabs may very {68} likely have known the [.g]ob[=a]r forms before the
+numerals reached them again in 773.[262] The term "[.g]ob[=a]r numerals"
+was also used without any reference to the peculiar use of dots.[263] In
+this connection it is worthy of mention that the Algerians employed two
+different forms of numerals in manuscripts even of the fourteenth
+century,[264] and that the Moroccans of to-day employ the European forms
+instead of the present Arabic.
+
+The Indian use of subscript dots to indicate the tens, hundreds, thousands,
+etc., is established by a passage in the _Kit[=a]b al-Fihrist_[265] (987
+A.D.) in which the writer discusses the written language of the people of
+India. Notwithstanding the importance of this reference for the early
+history of the numerals, it has not been mentioned by previous writers on
+this subject. The numeral forms given are those which have usually been
+called Indian,[266] in opposition to [.g]ob[=a]r. In this document the dots
+are placed below the characters, instead of being superposed as described
+above. The significance was the same.
+
+In form these [.g]ob[=a]r numerals resemble our own much more closely than
+the Arab numerals do. They varied more or less, but were substantially as
+follows:
+
+{69}
+
+  1[267][Illustration]
+  2[268][Illustration]
+  3[269][Illustration]
+  4[270][Illustration]
+  5[271][Illustration]
+  6[271][Illustration]
+
+The question of the possible influence of the Egyptian demotic and hieratic
+ordinal forms has been so often suggested that it seems well to introduce
+them at this point, for comparison with the [.g]ob[=a]r forms. They would
+as appropriately be used in connection with the Hindu forms, and the
+evidence of a relation of the first three with all these systems is
+apparent. The only further resemblance is in the Demotic 4 and in the 9, so
+that the statement that the Hindu forms in general came from {70} this
+source has no foundation. The first four Egyptian cardinal numerals[272]
+resemble more the modern Arabic.
+
+[Illustration: DEMOTIC AND HIERATIC ORDINALS]
+
+This theory of the very early introduction of the numerals into Europe
+fails in several points. In the first place the early Western forms are not
+known; in the second place some early Eastern forms are like the
+[.g]ob[=a]r, as is seen in the third line on p. 69, where the forms are
+from a manuscript written at Shiraz about 970 A.D., and in which some
+western Arabic forms, e.g. [symbol] for 2, are also used. Probably most
+significant of all is the fact that the [.g]ob[=a]r numerals as given by
+Sacy are all, with the exception of the symbol for eight, either single
+Arabic letters or combinations of letters. So much for the Woepcke theory
+and the meaning of the [.g]ob[=a]r numerals. We now have to consider the
+question as to whether Boethius knew these [.g]ob[=a]r forms, or forms akin
+to them.
+
+This large question[273] suggests several minor ones: (1) Who was Boethius?
+(2) Could he have known these numerals? (3) Is there any positive or strong
+circumstantial evidence that he did know them? (4) What are the
+probabilities in the case?
+
+{71}
+
+First, who was Boethius,--Divus[274] Boethius as he was called in the
+Middle Ages? Anicius Manlius Severinus Boethius[275] was born at Rome c.
+475. He was a member of the distinguished family of the Anicii,[276] which
+had for some time before his birth been Christian. Early left an orphan,
+the tradition is that he was taken to Athens at about the age of ten, and
+that he remained there eighteen years.[277] He married Rusticiana, daughter
+of the senator Symmachus, and this union of two such powerful families
+allowed him to move in the highest circles.[278] Standing strictly for the
+right, and against all iniquity at court, he became the object of hatred on
+the part of all the unscrupulous element near the throne, and his bold
+defense of the ex-consul Albinus, unjustly accused of treason, led to his
+imprisonment at Pavia[279] and his execution in 524.[280] Not many
+generations after his death, the period being one in which historical
+criticism was at its lowest ebb, the church found it profitable to look
+upon his execution as a martyrdom.[281] He was {72} accordingly looked upon
+as a saint,[282] his bones were enshrined,[283] and as a natural
+consequence his books were among the classics in the church schools for a
+thousand years.[284] It is pathetic, however, to think of the medieval
+student trying to extract mental nourishment from a work so abstract, so
+meaningless, so unnecessarily complicated, as the arithmetic of Boethius.
+
+He was looked upon by his contemporaries and immediate successors as a
+master, for Cassiodorus[285] (c. 490-c. 585 A.D.) says to him: "Through
+your translations the music of Pythagoras and the astronomy of Ptolemy are
+read by those of Italy, and the arithmetic of Nicomachus and the geometry
+of Euclid are known to those of the West."[286] Founder of the medieval
+scholasticism, {73} distinguishing the trivium and quadrivium,[287] writing
+the only classics of his time, Gibbon well called him "the last of the
+Romans whom Cato or Tully could have acknowledged for their
+countryman."[288]
+
+The second question relating to Boethius is this: Could he possibly have
+known the Hindu numerals? In view of the relations that will be shown to
+have existed between the East and the West, there can only be an
+affirmative answer to this question. The numerals had existed, without the
+zero, for several centuries; they had been well known in India; there had
+been a continued interchange of thought between the East and West; and
+warriors, ambassadors, scholars, and the restless trader, all had gone back
+and forth, by land or more frequently by sea, between the Mediterranean
+lands and the centers of Indian commerce and culture. Boethius could very
+well have learned one or more forms of Hindu numerals from some traveler or
+merchant.
+
+To justify this statement it is necessary to speak more fully of these
+relations between the Far East and Europe. It is true that we have no
+records of the interchange of learning, in any large way, between eastern
+Asia and central Europe in the century preceding the time of Boethius. But
+it is one of the mistakes of scholars to believe that they are the sole
+transmitters of knowledge. {74} As a matter of fact there is abundant
+reason for believing that Hindu numerals would naturally have been known to
+the Arabs, and even along every trade route to the remote west, long before
+the zero entered to make their place-value possible, and that the
+characters, the methods of calculating, the improvements that took place
+from time to time, the zero when it appeared, and the customs as to solving
+business problems, would all have been made known from generation to
+generation along these same trade routes from the Orient to the Occident.
+It must always be kept in mind that it was to the tradesman and the
+wandering scholar that the spread of such learning was due, rather than to
+the school man. Indeed, Avicenna[289] (980-1037 A.D.) in a short biography
+of himself relates that when his people were living at Bokh[=a]ra his
+father sent him to the house of a grocer to learn the Hindu art of
+reckoning, in which this grocer (oil dealer, possibly) was expert. Leonardo
+of Pisa, too, had a similar training.
+
+The whole question of this spread of mercantile knowledge along the trade
+routes is so connected with the [.g]ob[=a]r numerals, the Boethius
+question, Gerbert, Leonardo of Pisa, and other names and events, that a
+digression for its consideration now becomes necessary.[290]
+
+{75}
+
+Even in very remote times, before the Hindu numerals were sculptured in the
+cave of N[=a]n[=a] Gh[=a]t, there were trade relations between Arabia and
+India. Indeed, long before the Aryans went to India the great Turanian race
+had spread its civilization from the Mediterranean to the Indus.[291] At a
+much later period the Arabs were the intermediaries between Egypt and Syria
+on the west, and the farther Orient.[292] In the sixth century B.C.,
+Hecataeus,[293] the father of geography, was acquainted not only with the
+Mediterranean lands but with the countries as far as the Indus,[294] and in
+Biblical times there were regular triennial voyages to India. Indeed, the
+story of Joseph bears witness to the caravan trade from India, across
+Arabia, and on to the banks of the Nile. About the same time as Hecataeus,
+Scylax, a Persian admiral under Darius, from Caryanda on the coast of Asia
+Minor, traveled to {76} northwest India and wrote upon his ventures.[295]
+He induced the nations along the Indus to acknowledge the Persian
+supremacy, and such number systems as there were in these lands would
+naturally have been known to a man of his attainments.
+
+A century after Scylax, Herodotus showed considerable knowledge of India,
+speaking of its cotton and its gold,[296] telling how Sesostris[297] fitted
+out ships to sail to that country, and mentioning the routes to the east.
+These routes were generally by the Red Sea, and had been followed by the
+Phoenicians and the Sabaeans, and later were taken by the Greeks and
+Romans.[298]
+
+In the fourth century B.C. the West and East came into very close
+relations. As early as 330, Pytheas of Massilia (Marseilles) had explored
+as far north as the northern end of the British Isles and the coasts of the
+German Sea, while Macedon, in close touch with southern France, was also
+sending her armies under Alexander[299] through Afghanistan as far east as
+the Punjab.[300] Pliny tells us that Alexander the Great employed surveyors
+to measure {77} the roads of India; and one of the great highways is
+described by Megasthenes, who in 295 B.C., as the ambassador of Seleucus,
+resided at P[=a]tal[=i]pu[t.]ra, the present Patna.[301]
+
+The Hindus also learned the art of coining from the Greeks, or possibly
+from the Chinese, and the stores of Greco-Hindu coins still found in
+northern India are a constant source of historical information.[302] The
+R[=a]m[=a]yana speaks of merchants traveling in great caravans and
+embarking by sea for foreign lands.[303] Ceylon traded with Malacca and
+Siam, and Java was colonized by Hindu traders, so that mercantile knowledge
+was being spread about the Indies during all the formative period of the
+numerals.
+
+Moreover the results of the early Greek invasion were embodied by
+Dicaearchus of Messana (about 320 B.C.) in a map that long remained a
+standard. Furthermore, Alexander did not allow his influence on the East to
+cease. He divided India into three satrapies,[304] placing Greek governors
+over two of them and leaving a Hindu ruler in charge of the third, and in
+Bactriana, a part of Ariana or ancient Persia, he left governors; and in
+these the western civilization was long in evidence. Some of the Greek and
+Roman metrical and astronomical terms {78} found their way, doubtless at
+this time, into the Sanskrit language.[305] Even as late as from the second
+to the fifth centuries A.D., Indian coins showed the Hellenic influence.
+The Hindu astronomical terminology reveals the same relationship to western
+thought, for Var[=a]ha-Mihira (6th century A.D.), a contemporary of
+[=A]ryabha[t.]a, entitled a work of his the _B[r.]hat-Sa[m.]hit[=a]_, a
+literal translation of [Greek: megale suntaxis] of Ptolemy;[306] and in
+various ways is this interchange of ideas apparent.[307] It could not have
+been at all unusual for the ancient Greeks to go to India, for Strabo lays
+down the route, saying that all who make the journey start from Ephesus and
+traverse Phrygia and Cappadocia before taking the direct road.[308] The
+products of the East were always finding their way to the West, the Greeks
+getting their ginger[309] from Malabar, as the Phoenicians had long before
+brought gold from Malacca.
+
+Greece must also have had early relations with China, for there is a
+notable similarity between the Greek and Chinese life, as is shown in their
+houses, their domestic customs, their marriage ceremonies, the public
+story-tellers, the puppet shows which Herodotus says were introduced from
+Egypt, the street jugglers, the games of dice,[310] the game of
+finger-guessing,[311] the water clock, the {79} music system, the use of
+the myriad,[312] the calendars, and in many other ways.[313] In passing
+through the suburbs of Peking to-day, on the way to the Great Bell temple,
+one is constantly reminded of the semi-Greek architecture of Pompeii, so
+closely does modern China touch the old classical civilization of the
+Mediterranean. The Chinese historians tell us that about 200 B.C. their
+arms were successful in the far west, and that in 180 B.C. an ambassador
+went to Bactria, then a Greek city, and reported that Chinese products were
+on sale in the markets there.[314] There is also a noteworthy resemblance
+between certain Greek and Chinese words,[315] showing that in remote times
+there must have been more or less interchange of thought.
+
+The Romans also exchanged products with the East. Horace says, "A busy
+trader, you hasten to the farthest Indies, flying from poverty over sea,
+over crags, over fires."[316] The products of the Orient, spices and jewels
+from India, frankincense from Persia, and silks from China, being more in
+demand than the exports from the Mediterranean lands, the balance of trade
+was against the West, and thus Roman coin found its way eastward. In 1898,
+for example, a number of Roman coins dating from 114 B.C. to Hadrian's time
+were found at Pakl[=i], a part of the Haz[=a]ra district, sixteen miles
+north of Abbott[=a]b[=a]d,[317] and numerous similar discoveries have been
+made from time to time.
+
+{80}
+
+Augustus speaks of envoys received by him from India, a thing never before
+known,[318] and it is not improbable that he also received an embassy from
+China.[319] Suetonius (first century A.D.) speaks in his history of these
+relations,[320] as do several of his contemporaries,[321] and Vergil[322]
+tells of Augustus doing battle in Persia. In Pliny's time the trade of the
+Roman Empire with Asia amounted to a million and a quarter dollars a year,
+a sum far greater relatively then than now,[323] while by the time of
+Constantine Europe was in direct communication with the Far East.[324]
+
+In view of these relations it is not beyond the range of possibility that
+proof may sometime come to light to show that the Greeks and Romans knew
+something of the {81} number system of India, as several writers have
+maintained.[325]
+
+Returning to the East, there are many evidences of the spread of knowledge
+in and about India itself. In the third century B.C. Buddhism began to be a
+connecting medium of thought. It had already permeated the Himalaya
+territory, had reached eastern Turkestan, and had probably gone thence to
+China. Some centuries later (in 62 A.D.) the Chinese emperor sent an
+ambassador to India, and in 67 A.D. a Buddhist monk was invited to
+China.[326] Then, too, in India itself A['s]oka, whose name has already
+been mentioned in this work, extended the boundaries of his domains even
+into Afghanistan, so that it was entirely possible for the numerals of the
+Punjab to have worked their way north even at that early date.[327]
+
+Furthermore, the influence of Persia must not be forgotten in considering
+this transmission of knowledge. In the fifth century the Persian medical
+school at Jondi-Sapur admitted both the Hindu and the Greek doctrines, and
+Firdus[=i] tells us that during the brilliant reign of {82} Khosr[=u]
+I,[328] the golden age of Pahlav[=i] literature, the Hindu game of chess
+was introduced into Persia, at a time when wars with the Greeks were
+bringing prestige to the Sassanid dynasty.
+
+Again, not far from the time of Boethius, in the sixth century, the
+Egyptian monk Cosmas, in his earlier years as a trader, made journeys to
+Abyssinia and even to India and Ceylon, receiving the name _Indicopleustes_
+(the Indian traveler). His map (547 A.D.) shows some knowledge of the earth
+from the Atlantic to India. Such a man would, with hardly a doubt, have
+observed every numeral system used by the people with whom he
+sojourned,[329] and whether or not he recorded his studies in permanent
+form he would have transmitted such scraps of knowledge by word of mouth.
+
+As to the Arabs, it is a mistake to feel that their activities began with
+Mohammed. Commerce had always been held in honor by them, and the
+Qoreish[330] had annually for many generations sent caravans bearing the
+spices and textiles of Yemen to the shores of the Mediterranean. In the
+fifth century they traded by sea with India and even with China, and
+[H.]ira was an emporium for the wares of the East,[331] so that any numeral
+system of any part of the trading world could hardly have remained
+isolated.
+
+Long before the warlike activity of the Arabs, Alexandria had become the
+great market-place of the world. From this center caravans traversed Arabia
+to Hadramaut, where they met ships from India. Others went north to
+Damascus, while still others made their way {83} along the southern shores
+of the Mediterranean. Ships sailed from the isthmus of Suez to all the
+commercial ports of Southern Europe and up into the Black Sea. Hindus were
+found among the merchants[332] who frequented the bazaars of Alexandria,
+and Brahmins were reported even in Byzantium.
+
+Such is a very brief resume of the evidence showing that the numerals of
+the Punjab and of other parts of India as well, and indeed those of China
+and farther Persia, of Ceylon and the Malay peninsula, might well have been
+known to the merchants of Alexandria, and even to those of any other
+seaport of the Mediterranean, in the time of Boethius. The Br[=a]hm[=i]
+numerals would not have attracted the attention of scholars, for they had
+no zero so far as we know, and therefore they were no better and no worse
+than those of dozens of other systems. If Boethius was attracted to them it
+was probably exactly as any one is naturally attracted to the bizarre or
+the mystic, and he would have mentioned them in his works only
+incidentally, as indeed they are mentioned in the manuscripts in which they
+occur.
+
+In answer therefore to the second question, Could Boethius have known the
+Hindu numerals? the reply must be, without the slightest doubt, that he
+could easily have known them, and that it would have been strange if a man
+of his inquiring mind did not pick up many curious bits of information of
+this kind even though he never thought of making use of them.
+
+Let us now consider the third question, Is there any positive or strong
+circumstantial evidence that Boethius did know these numerals? The question
+is not new, {84} nor is it much nearer being answered than it was over two
+centuries ago when Wallis (1693) expressed his doubts about it[333] soon
+after Vossius (1658) had called attention to the matter.[334] Stated
+briefly, there are three works on mathematics attributed to Boethius:[335]
+(1) the arithmetic, (2) a work on music, and (3) the geometry.[336]
+
+The genuineness of the arithmetic and the treatise on music is generally
+recognized, but the geometry, which contains the Hindu numerals with the
+zero, is under suspicion.[337] There are plenty of supporters of the idea
+that Boethius knew the numerals and included them in this book,[338] and on
+the other hand there are as many who {85} feel that the geometry, or at
+least the part mentioning the numerals, is spurious.[339] The argument of
+those who deny the authenticity of the particular passage in question may
+briefly be stated thus:
+
+1. The falsification of texts has always been the subject of complaint. It
+was so with the Romans,[340] it was common in the Middle Ages,[341] and it
+is much more prevalent {86} to-day than we commonly think. We have but to
+see how every hymn-book compiler feels himself authorized to change at will
+the classics of our language, and how unknown editors have mutilated
+Shakespeare, to see how much more easy it was for medieval scribes to
+insert or eliminate paragraphs without any protest from critics.[342]
+
+2. If Boethius had known these numerals he would have mentioned them in his
+arithmetic, but he does not do so.[343]
+
+3. If he had known them, and had mentioned them in any of his works, his
+contemporaries, disciples, and successors would have known and mentioned
+them. But neither Capella (c. 475)[344] nor any of the numerous medieval
+writers who knew the works of Boethius makes any reference to the
+system.[345]
+
+{87}
+
+4. The passage in question has all the appearance of an interpolation by
+some scribe. Boethius is speaking of angles, in his work on geometry, when
+the text suddenly changes to a discussion of classes of numbers.[346] This
+is followed by a chapter in explanation of the abacus,[347] in which are
+described those numeral forms which are called _apices_ or
+_caracteres_.[348] The forms[349] of these characters vary in different
+manuscripts, but in general are about as shown on page 88. They are
+commonly written with the 9 at the left, decreasing to the unit at the
+right, numerous writers stating that this was because they were derived
+from Semitic sources in which the direction of writing is the opposite of
+our own. This practice continued until the sixteenth century.[350] The
+writer then leaves the subject entirely, using the Roman numerals for the
+rest of his discussion, a proceeding so foreign to the method of Boethius
+as to be inexplicable on the hypothesis of authenticity. Why should such a
+scholarly writer have given them with no mention of their origin or use?
+Either he would have mentioned some historical interest attaching to them,
+or he would have used them in some discussion; he certainly would not have
+left the passage as it is.
+
+{88}
+
+FORMS OF THE NUMERALS, LARGELY FROM WORKS ON THE ABACUS[351]
+
+  a[352] [Illustration]
+  b[353] [Illustration]
+  c[354] [Illustration]
+  d[355] [Illustration]
+  e[356] [Illustration]
+  f[357] [Illustration]
+  g[358] [Illustration]
+  h[359] [Illustration]
+  i[360] [Illustration]
+
+{89}
+
+Sir E. Clive Bayley has added[361] a further reason for believing them
+spurious, namely that the 4 is not of the N[=a]n[=a] Gh[=a]t type, but of
+the Kabul form which the Arabs did not receive until 776;[362] so that it
+is not likely, even if the characters were known in Europe in the time of
+Boethius, that this particular form was recognized. It is worthy of
+mention, also, that in the six abacus forms from the chief manuscripts as
+given by Friedlein,[363] each contains some form of zero, which symbol
+probably originated in India about this time or later. It could hardly have
+reached Europe so soon.
+
+As to the fourth question, Did Boethius probably know the numerals? It
+seems to be a fair conclusion, according to our present evidence, that (1)
+Boethius might very easily have known these numerals without the zero, but,
+(2) there is no reliable evidence that he did know them. And just as
+Boethius might have come in contact with them, so any other inquiring mind
+might have done so either in his time or at any time before they definitely
+appeared in the tenth century. These centuries, five in number, represented
+the darkest of the Dark Ages, and even if these numerals were occasionally
+met and studied, no trace of them would be likely to show itself in the
+{90} literature of the period, unless by chance it should get into the
+writings of some man like Alcuin. As a matter of fact, it was not until the
+ninth or tenth century that there is any tangible evidence of their
+presence in Christendom. They were probably known to merchants here and
+there, but in their incomplete state they were not of sufficient importance
+to attract any considerable attention.
+
+As a result of this brief survey of the evidence several conclusions seem
+reasonable: (1) commerce, and travel for travel's sake, never died out
+between the East and the West; (2) merchants had every opportunity of
+knowing, and would have been unreasonably stupid if they had not known, the
+elementary number systems of the peoples with whom they were trading, but
+they would not have put this knowledge in permanent written form; (3)
+wandering scholars would have known many and strange things about the
+peoples they met, but they too were not, as a class, writers; (4) there is
+every reason a priori for believing that the [.g]ob[=a]r numerals would
+have been known to merchants, and probably to some of the wandering
+scholars, long before the Arabs conquered northern Africa; (5) the wonder
+is not that the Hindu-Arabic numerals were known about 1000 A.D., and that
+they were the subject of an elaborate work in 1202 by Fibonacci, but rather
+that more extended manuscript evidence of their appearance before that time
+has not been found. That they were more or less known early in the Middle
+Ages, certainly to many merchants of Christian Europe, and probably to
+several scholars, but without the zero, is hardly to be doubted. The lack
+of documentary evidence is not at all strange, in view of all of the
+circumstances.
+
+       *       *       *       *       *
+
+
+{91}
+
+CHAPTER VI
+
+THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS
+
+If the numerals had their origin in India, as seems most probable, when did
+the Arabs come to know of them? It is customary to say that it was due to
+the influence of Mohammedanism that learning spread through Persia and
+Arabia; and so it was, in part. But learning was already respected in these
+countries long before Mohammed appeared, and commerce flourished all
+through this region. In Persia, for example, the reign of Khosr[=u]
+Nu['s][=i]rw[=a]n,[364] the great contemporary of Justinian the law-maker,
+was characterized not only by an improvement in social and economic
+conditions, but by the cultivation of letters. Khosr[=u] fostered learning,
+inviting to his court scholars from Greece, and encouraging the
+introduction of culture from the West as well as from the East. At this
+time Aristotle and Plato were translated, and portions of the
+_Hito-pad[=e]['s]a_, or Fables of Pilpay, were rendered from the Sanskrit
+into Persian. All this means that some three centuries before the great
+intellectual ascendancy of Bagdad a similar fostering of learning was
+taking place in Persia, and under pre-Mohammedan influences.
+
+{92}
+
+The first definite trace that we have of the introduction of the Hindu
+system into Arabia dates from 773 A.D.,[365] when an Indian astronomer
+visited the court of the caliph, bringing with him astronomical tables
+which at the caliph's command were translated into Arabic by
+Al-Faz[=a]r[=i].[366] Al-Khow[=a]razm[=i] and [H.]abash (A[h.]med ibn
+`Abdall[=a]h, died c. 870) based their well-known tables upon the work of
+Al-F[=a]zar[=i]. It may be asserted as highly probable that the numerals
+came at the same time as the tables. They were certainly known a few
+decades later, and before 825 A.D., about which time the original of the
+_Algoritmi de numero Indorum_ was written, as that work makes no pretense
+of being the first work to treat of the Hindu numerals.
+
+The three writers mentioned cover the period from the end of the eighth to
+the end of the ninth century. While the historians Al-Ma['s]`[=u]d[=i] and
+Al-B[=i]r[=u]n[=i] follow quite closely upon the men mentioned, it is well
+to note again the Arab writers on Hindu arithmetic, contemporary with
+Al-Khow[=a]razm[=i], who were mentioned in chapter I, viz. Al-Kind[=i],
+Sened ibn `Al[=i], and Al-[S.][=u]f[=i].
+
+For over five hundred years Arabic writers and others continued to apply to
+works on arithmetic the name "Indian." In the tenth century such writers
+are `Abdall[=a]h ibn al-[H.]asan, Ab[=u] 'l-Q[=a]sim[367] (died 987 A.D.)
+of Antioch, and Mo[h.]ammed ibn `Abdall[=a]h, Ab[=u] Na[s.]r[368] (c. 982),
+of Kalw[=a]d[=a] near Bagdad. Others of the same period or {93} earlier
+(since they are mentioned in the _Fihrist_,[369] 987 A.D.), who explicitly
+use the word "Hindu" or "Indian," are Sin[=a]n ibn al-Fat[h.][370] of
+[H.]arr[=a]n, and Ahmed ibn `Omar, al-Kar[=a]b[=i]s[=i].[371] In the
+eleventh century come Al-B[=i]r[=u]n[=i][372] (973-1048) and `Ali ibn
+A[h.]med, Ab[=u] 'l-[H.]asan, Al-Nasaw[=i][373] (c. 1030). The following
+century brings similar works by Ish[=a]q ibn Y[=u]suf al-[S.]ardaf[=i][374]
+and Sam[=u]'[=i]l ibn Ya[h.]y[=a] ibn `Abb[=a]s al-Ma[.g]reb[=i]
+al-Andalus[=i][375] (c. 1174), and in the thirteenth century are
+`Abdallat[=i]f ibn Y[=u]suf ibn Mo[h.]ammed, Muwaffaq al-D[=i]n Ab[=u]
+Mo[h.]ammed al-Ba[.g]d[=a]d[=i][376] (c. 1231), and Ibn al-Bann[=a].[377]
+
+The Greek monk Maximus Planudes, writing in the first half of the
+fourteenth century, followed the Arabic usage in calling his work _Indian
+Arithmetic_.[378] There were numerous other Arabic writers upon arithmetic,
+as that subject occupied one of the high places among the sciences, but
+most of them did not feel it necessary to refer to the origin of the
+symbols, the knowledge of which might well have been taken for granted.
+
+{94}
+
+One document, cited by Woepcke,[379] is of special interest since it shows
+at an early period, 970 A.D., the use of the ordinary Arabic forms
+alongside the [.g]ob[=a]r. The title of the work is _Interesting and
+Beautiful Problems on Numbers_ copied by A[h.]med ibn Mo[h.]ammed ibn
+`Abdaljal[=i]l, Ab[=u] Sa`[=i]d, al-Sijz[=i],[380] (951-1024) from a work
+by a priest and physician, Na[z.][=i]f ibn Yumn,[381] al-Qass (died c.
+990). Suter does not mention this work of Na[z.][=i]f.
+
+The second reason for not ascribing too much credit to the purely Arab
+influence is that the Arab by himself never showed any intellectual
+strength. What took place after Mo[h.]ammed had lighted the fire in the
+hearts of his people was just what always takes place when different types
+of strong races blend,--a great renaissance in divers lines. It was seen in
+the blending of such types at Miletus in the time of Thales, at Rome in the
+days of the early invaders, at Alexandria when the Greek set firm foot on
+Egyptian soil, and we see it now when all the nations mingle their vitality
+in the New World. So when the Arab culture joined with the Persian, a new
+civilization rose and flourished.[382] The Arab influence came not from its
+purity, but from its intermingling with an influence more cultured if less
+virile.
+
+As a result of this interactivity among peoples of diverse interests and
+powers, Mohammedanism was to the world from the eighth to the thirteenth
+century what Rome and Athens and the Italo-Hellenic influence generally had
+{95} been to the ancient civilization. "If they did not possess the spirit
+of invention which distinguished the Greeks and the Hindus, if they did not
+show the perseverance in their observations that characterized the Chinese
+astronomers, they at least possessed the virility of a new and victorious
+people, with a desire to understand what others had accomplished, and a
+taste which led them with equal ardor to the study of algebra and of
+poetry, of philosophy and of language."[383]
+
+It was in 622 A.D. that Mo[h.]ammed fled from Mecca, and within a century
+from that time the crescent had replaced the cross in Christian Asia, in
+Northern Africa, and in a goodly portion of Spain. The Arab empire was an
+ellipse of learning with its foci at Bagdad and Cordova, and its rulers not
+infrequently took pride in demanding intellectual rather than commercial
+treasure as the result of conquest.[384]
+
+It was under these influences, either pre-Mohammedan or later, that the
+Hindu numerals found their way to the North. If they were known before
+Mo[h.]ammed's time, the proof of this fact is now lost. This much, however,
+is known, that in the eighth century they were taken to Bagdad. It was
+early in that century that the Mohammedans obtained their first foothold in
+northern India, thus foreshadowing an epoch of supremacy that endured with
+varied fortunes until after the golden age of Akbar the Great (1542-1605)
+and Shah Jehan. They also conquered Khorassan and Afghanistan, so that the
+learning and the commercial customs of India at once found easy {96} access
+to the newly-established schools and the bazaars of Mesopotamia and western
+Asia. The particular paths of conquest and of commerce were either by way
+of the Khyber Pass and through Kabul, Herat and Khorassan, or by sea
+through the strait of Ormuz to Basra (Busra) at the head of the Persian
+Gulf, and thence to Bagdad. As a matter of fact, one form of Arabic
+numerals, the one now in use by the Arabs, is attributed to the influence
+of Kabul, while the other, which eventually became our numerals, may very
+likely have reached Arabia by the other route. It is in Bagdad,[385] D[=a]r
+al-Sal[=a]m--"the Abode of Peace," that our special interest in the
+introduction of the numerals centers. Built upon the ruins of an ancient
+town by Al-Man[s.][=u]r[386] in the second half of the eighth century, it
+lies in one of those regions where the converging routes of trade give rise
+to large cities.[387] Quite as well of Bagdad as of Athens might Cardinal
+Newman have said:[388]
+
+"What it lost in conveniences of approach, it gained in its neighborhood to
+the traditions of the mysterious East, and in the loveliness of the region
+in which it lay. Hither, then, as to a sort of ideal land, where all
+archetypes of the great and the fair were found in substantial being, and
+all departments of truth explored, and all diversities of intellectual
+power exhibited, where taste and philosophy were majestically enthroned as
+in a royal court, where there was no sovereignty but that of mind, and no
+nobility but that of genius, where professors were {97} rulers, and princes
+did homage, thither flocked continually from the very corners of the _orbis
+terrarum_ the many-tongued generation, just rising, or just risen into
+manhood, in order to gain wisdom." For here it was that Al-Man[s.][=u]r and
+Al-M[=a]m[=u]n and H[=a]r[=u]n al-Rash[=i]d (Aaron the Just) made for a
+time the world's center of intellectual activity in general and in the
+domain of mathematics in particular.[389] It was just after the _Sindhind_
+was brought to Bagdad that Mo[h.]ammed ibn M[=u]s[=a] al-Khow[=a]razm[=i],
+whose name has already been mentioned,[390] was called to that city. He was
+the most celebrated mathematician of his time, either in the East or West,
+writing treatises on arithmetic, the sundial, the astrolabe, chronology,
+geometry, and algebra, and giving through the Latin transliteration of his
+name, _algoritmi_, the name of algorism to the early arithmetics using the
+new Hindu numerals.[391] Appreciating at once the value of the position
+system so recently brought from India, he wrote an arithmetic based upon
+these numerals, and this was translated into Latin in the time of Adelhard
+of Bath (c. 1180), although possibly by his contemporary countryman Robert
+Cestrensis.[392] This translation was found in Cambridge and was published
+by Boncompagni in 1857.[393]
+
+Contemporary with Al-Khow[=a]razm[=i], and working also under
+Al-M[=a]m[=u]n, was a Jewish astronomer, Ab[=u] 'l-[T.]eiyib, {98} Sened
+ibn `Al[=i], who is said to have adopted the Mohammedan religion at the
+caliph's request. He also wrote a work on Hindu arithmetic,[394] so that
+the subject must have been attracting considerable attention at that time.
+Indeed, the struggle to have the Hindu numerals replace the Arabic did not
+cease for a long time thereafter. `Al[=i] ibn A[h.]med al-Nasaw[=i], in his
+arithmetic of c. 1025, tells us that the symbolism of number was still
+unsettled in his day, although most people preferred the strictly Arabic
+forms.[395]
+
+We thus have the numerals in Arabia, in two forms: one the form now used
+there, and the other the one used by Al-Khow[=a]razm[=i]. The question then
+remains, how did this second form find its way into Europe? and this
+question will be considered in the next chapter.
+
+       *       *       *       *       *
+
+
+{99}
+
+CHAPTER VII
+
+THE DEFINITE INTRODUCTION OF THE NUMERALS INTO EUROPE
+
+It being doubtful whether Boethius ever knew the Hindu numeral forms,
+certainly without the zero in any case, it becomes necessary now to
+consider the question of their definite introduction into Europe. From what
+has been said of the trade relations between the East and the West, and of
+the probability that it was the trader rather than the scholar who carried
+these numerals from their original habitat to various commercial centers,
+it is evident that we shall never know when they first made their
+inconspicuous entrance into Europe. Curious customs from the East and from
+the tropics,--concerning games, social peculiarities, oddities of dress,
+and the like,--are continually being related by sailors and traders in
+their resorts in New York, London, Hamburg, and Rotterdam to-day, customs
+that no scholar has yet described in print and that may not become known
+for many years, if ever. And if this be so now, how much more would it have
+been true a thousand years before the invention of printing, when learning
+was at its lowest ebb. It was at this period of low esteem of culture that
+the Hindu numerals undoubtedly made their first appearance in Europe.
+
+There were many opportunities for such knowledge to reach Spain and Italy.
+In the first place the Moors went into Spain as helpers of a claimant of
+the throne, and {100} remained as conquerors. The power of the Goths, who
+had held Spain for three centuries, was shattered at the battle of Jerez de
+la Frontera in 711, and almost immediately the Moors became masters of
+Spain and so remained for five hundred years, and masters of Granada for a
+much longer period. Until 850 the Christians were absolutely free as to
+religion and as to holding political office, so that priests and monks were
+not infrequently skilled both in Latin and Arabic, acting as official
+translators, and naturally reporting directly or indirectly to Rome. There
+was indeed at this time a complaint that Christian youths cultivated too
+assiduously a love for the literature of the Saracen, and married too
+frequently the daughters of the infidel.[396] It is true that this happy
+state of affairs was not permanent, but while it lasted the learning and
+the customs of the East must have become more or less the property of
+Christian Spain. At this time the [.g]ob[=a]r numerals were probably in
+that country, and these may well have made their way into Europe from the
+schools of Cordova, Granada, and Toledo.
+
+Furthermore, there was abundant opportunity for the numerals of the East to
+reach Europe through the journeys of travelers and ambassadors. It was from
+the records of Suleim[=a]n the Merchant, a well-known Arab trader of the
+ninth century, that part of the story of Sindb[=a]d the Sailor was
+taken.[397] Such a merchant would have been particularly likely to know the
+numerals of the people whom he met, and he is a type of man that may well
+have taken such symbols to European markets. A little later, {101} Ab[=u]
+'l-[H.]asan `Al[=i] al-Mas`[=u]d[=i] (d. 956) of Bagdad traveled to the
+China Sea on the east, at least as far south as Zanzibar, and to the
+Atlantic on the west,[398] and he speaks of the nine figures with which the
+Hindus reckoned.[399]
+
+There was also a Bagdad merchant, one Ab[=u] 'l-Q[=a]sim `Obeidall[=a]h ibn
+A[h.]med, better known by his Persian name Ibn Khord[=a][d.]beh,[400] who
+wrote about 850 A.D. a work entitled _Book of Roads and Provinces_[401] in
+which the following graphic account appears:[402] "The Jewish merchants
+speak Persian, Roman (Greek and Latin), Arabic, French, Spanish, and
+Slavic. They travel from the West to the East, and from the East to the
+West, sometimes by land, sometimes by sea. They take ship from France on
+the Western Sea, and they voyage to Farama (near the ruins of the ancient
+Pelusium); there they transfer their goods to caravans and go by land to
+Colzom (on the Red Sea). They there reembark on the Oriental (Red) Sea and
+go to Hejaz and to Jiddah, and thence to the Sind, India, and China.
+Returning, they bring back the products of the oriental lands.... These
+journeys are also made by land. The merchants, leaving France and Spain,
+cross to Tangier and thence pass through the African provinces and Egypt.
+They then go to Ramleh, visit Damascus, Kufa, Bagdad, and Basra, penetrate
+into Ahwaz, Fars, Kerman, Sind, and thus reach India and China." Such
+travelers, about 900 A.D., must necessarily have spread abroad a knowledge
+of all number {102} systems used in recording prices or in the computations
+of the market. There is an interesting witness to this movement, a
+cruciform brooch now in the British Museum. It is English, certainly as
+early as the eleventh century, but it is inlaid with a piece of paste on
+which is the Mohammedan inscription, in Kufic characters, "There is no God
+but God." How did such an inscription find its way, perhaps in the time of
+Alcuin of York, to England? And if these Kufic characters reached there,
+then why not the numeral forms as well?
+
+Even in literature of the better class there appears now and then some
+stray proof of the important fact that the great trade routes to the far
+East were never closed for long, and that the customs and marks of trade
+endured from generation to generation. The _Gulist[=a]n_ of the Persian
+poet Sa`d[=i][403] contains such a passage:
+
+"I met a merchant who owned one hundred and forty camels, and fifty slaves
+and porters.... He answered to me: 'I want to carry sulphur of Persia to
+China, which in that country, as I hear, bears a high price; and thence to
+take Chinese ware to Roum; and from Roum to load up with brocades for Hind;
+and so to trade Indian steel (_pulab_) to Halib. From Halib I will convey
+its glass to Yeman, and carry the painted cloths of Yeman back to
+Persia.'"[404] On the other hand, these men were not of the learned class,
+nor would they preserve in treatises any knowledge that they might have,
+although this knowledge would occasionally reach the ears of the learned as
+bits of curious information.
+
+{103}
+
+There were also ambassadors passing back and forth from time to time,
+between the East and the West, and in particular during the period when
+these numerals probably began to enter Europe. Thus Charlemagne (c. 800)
+sent emissaries to Bagdad just at the time of the opening of the
+mathematical activity there.[405] And with such ambassadors must have gone
+the adventurous scholar, inspired, as Alcuin says of Archbishop Albert of
+York (766-780),[406] to seek the learning of other lands. Furthermore, the
+Nestorian communities, established in Eastern Asia and in India at this
+time, were favored both by the Persians and by their Mohammedan conquerors.
+The Nestorian Patriarch of Syria, Timotheus (778-820), sent missionaries
+both to India and to China, and a bishop was appointed for the latter
+field. Ibn Wahab, who traveled to China in the ninth century, found images
+of Christ and the apostles in the Emperor's court.[407] Such a learned body
+of men, knowing intimately the countries in which they labored, could
+hardly have failed to make strange customs known as they returned to their
+home stations. Then, too, in Alfred's time (849-901) emissaries went {104}
+from England as far as India,[408] and generally in the Middle Ages
+groceries came to Europe from Asia as now they come from the colonies and
+from America. Syria, Asia Minor, and Cyprus furnished sugar and wool, and
+India yielded her perfumes and spices, while rich tapestries for the courts
+and the wealthy burghers came from Persia and from China.[409] Even in the
+time of Justinian (c. 550) there seems to have been a silk trade with
+China, which country in turn carried on commerce with Ceylon,[410] and
+reached out to Turkestan where other merchants transmitted the Eastern
+products westward. In the seventh century there was a well-defined commerce
+between Persia and India, as well as between Persia and
+Constantinople.[411] The Byzantine _commerciarii_ were stationed at the
+outposts not merely as customs officers but as government purchasing
+agents.[412]
+
+Occasionally there went along these routes of trade men of real learning,
+and such would surely have carried the knowledge of many customs back and
+forth. Thus at a period when the numerals are known to have been partly
+understood in Italy, at the opening of the eleventh century, one
+Constantine, an African, traveled from Italy through a great part of Africa
+and Asia, even on to India, for the purpose of learning the sciences of the
+Orient. He spent thirty-nine years in travel, having been hospitably
+received in Babylon, and upon his return he was welcomed with great honor
+at Salerno.[413]
+
+A very interesting illustration of this intercourse also appears in the
+tenth century, when the son of Otto I {105} (936-973) married a princess
+from Constantinople. This monarch was in touch with the Moors of Spain and
+invited to his court numerous scholars from abroad,[414] and his
+intercourse with the East as well as the West must have brought together
+much of the learning of each.
+
+Another powerful means for the circulation of mysticism and philosophy, and
+more or less of culture, took its start just before the conversion of
+Constantine (c. 312), in the form of Christian pilgrim travel. This was a
+feature peculiar to the zealots of early Christianity, found in only a
+slight degree among their Jewish predecessors in the annual pilgrimage to
+Jerusalem, and almost wholly wanting in other pre-Christian peoples. Chief
+among these early pilgrims were the two Placentians, John and Antonine the
+Elder (c. 303), who, in their wanderings to Jerusalem, seem to have started
+a movement which culminated centuries later in the crusades.[415] In 333 a
+Bordeaux pilgrim compiled the first Christian guide-book, the _Itinerary
+from Bordeaux to Jerusalem_,[416] and from this time on the holy pilgrimage
+never entirely ceased.
+
+Still another certain route for the entrance of the numerals into Christian
+Europe was through the pillaging and trading carried on by the Arabs on the
+northern shores of the Mediterranean. As early as 652 A.D., in the
+thirtieth year of the Hejira, the Mohammedans descended upon the shores of
+Sicily and took much spoil. Hardly had the wretched Constans given place to
+the {106} young Constantine IV when they again attacked the island and
+plundered ancient Syracuse. Again in 827, under Asad, they ravaged the
+coasts. Although at this time they failed to conquer Syracuse, they soon
+held a good part of the island, and a little later they successfully
+besieged the city. Before Syracuse fell, however, they had plundered the
+shores of Italy, even to the walls of Rome itself; and had not Leo IV, in
+849, repaired the neglected fortifications, the effects of the Moslem raid
+of that year might have been very far-reaching. Ibn Khord[=a][d.]beh, who
+left Bagdad in the latter part of the ninth century, gives a picture of the
+great commercial activity at that time in the Saracen city of Palermo. In
+this same century they had established themselves in Piedmont, and in 906
+they pillaged Turin.[417] On the Sorrento peninsula the traveler who climbs
+the hill to the beautiful Ravello sees still several traces of the Arab
+architecture, reminding him of the fact that about 900 A.D. Amalfi was a
+commercial center of the Moors.[418] Not only at this time, but even a
+century earlier, the artists of northern India sold their wares at such
+centers, and in the courts both of H[=a]r[=u]n al-Rash[=i]d and of
+Charlemagne.[419] Thus the Arabs dominated the Mediterranean Sea long
+before Venice
+
+              "held the gorgeous East in fee
+  And was the safeguard of the West,"
+
+and long before Genoa had become her powerful rival.[420]
+
+{107}
+
+Only a little later than this the brothers Nicolo and Maffeo Polo entered
+upon their famous wanderings.[421] Leaving Constantinople in 1260, they
+went by the Sea of Azov to Bokhara, and thence to the court of Kublai Khan,
+penetrating China, and returning by way of Acre in 1269 with a commission
+which required them to go back to China two years later. This time they
+took with them Nicolo's son Marco, the historian of the journey, and went
+across the plateau of Pamir; they spent about twenty years in China, and
+came back by sea from China to Persia.
+
+The ventures of the Poli were not long unique, however: the thirteenth
+century had not closed before Roman missionaries and the merchant Petrus de
+Lucolongo had penetrated China. Before 1350 the company of missionaries was
+large, converts were numerous, churches and Franciscan convents had been
+organized in the East, travelers were appealing for the truth of their
+accounts to the "many" persons in Venice who had been in China,
+Tsuan-chau-fu had a European merchant community, and Italian trade and
+travel to China was a thing that occupied two chapters of a commercial
+handbook.[422]
+
+{108}
+
+It is therefore reasonable to conclude that in the Middle Ages, as in the
+time of Boethius, it was a simple matter for any inquiring scholar to
+become acquainted with such numerals of the Orient as merchants may have
+used for warehouse or price marks. And the fact that Gerbert seems to have
+known only the forms of the simplest of these, not comprehending their full
+significance, seems to prove that he picked them up in just this way.
+
+Even if Gerbert did not bring his knowledge of the Oriental numerals from
+Spain, he may easily have obtained them from the marks on merchant's goods,
+had he been so inclined. Such knowledge was probably obtainable in various
+parts of Italy, though as parts of mere mercantile knowledge the forms
+might soon have been lost, it needing the pen of the scholar to preserve
+them. Trade at this time was not stagnant. During the eleventh and twelfth
+centuries the Slavs, for example, had very great commercial interests,
+their trade reaching to Kiev and Novgorod, and thence to the East.
+Constantinople was a great clearing-house of commerce with the Orient,[423]
+and the Byzantine merchants must have been entirely familiar with the
+various numerals of the Eastern peoples. In the eleventh century the
+Italian town of Amalfi established a factory[424] in Constantinople, and
+had trade relations with Antioch and Egypt. Venice, as early as the ninth
+century, had a valuable trade with Syria and Cairo.[425] Fifty years after
+Gerbert died, in the time of Cnut, the Dane and the Norwegian pushed their
+commerce far beyond the northern seas, both by caravans through Russia to
+the Orient, and by their venturesome barks which {109} sailed through the
+Strait of Gibraltar into the Mediterranean.[426] Only a little later,
+probably before 1200 A.D., a clerk in the service of Thomas a Becket,
+present at the latter's death, wrote a life of the martyr, to which
+(fortunately for our purposes) he prefixed a brief eulogy of the city of
+London.[427] This clerk, William Fitz Stephen by name, thus speaks of the
+British capital:
+
+  Aurum mittit Arabs: species et thura Sabaeus:
+  Arma Sythes: oleum palmarum divite sylva
+  Pingue solum Babylon: Nilus lapides pretiosos:
+  Norwegi, Russi, varium grisum, sabdinas:
+  Seres, purpureas vestes: Galli, sua vina.
+
+Although, as a matter of fact, the Arabs had no gold to send, and the
+Scythians no arms, and Egypt no precious stones save only the turquoise,
+the Chinese (_Seres_) may have sent their purple vestments, and the north
+her sables and other furs, and France her wines. At any rate the verses
+show very clearly an extensive foreign trade.
+
+Then there were the Crusades, which in these times brought the East in
+touch with the West. The spirit of the Orient showed itself in the songs of
+the troubadours, and the _baudekin_,[428] the canopy of Bagdad,[429] became
+common in the churches of Italy. In Sicily and in Venice the textile
+industries of the East found place, and made their way even to the
+Scandinavian peninsula.[430]
+
+We therefore have this state of affairs: There was abundant intercourse
+between the East and West for {110} some centuries before the Hindu
+numerals appear in any manuscripts in Christian Europe. The numerals must
+of necessity have been known to many traders in a country like Italy at
+least as early as the ninth century, and probably even earlier, but there
+was no reason for preserving them in treatises. Therefore when a man like
+Gerbert made them known to the scholarly circles, he was merely describing
+what had been familiar in a small way to many people in a different walk of
+life.
+
+Since Gerbert[431] was for a long time thought to have been the one to
+introduce the numerals into Italy,[432] a brief sketch of this unique
+character is proper. Born of humble parents,[433] this remarkable man
+became the counselor and companion of kings, and finally wore the papal
+tiara as Sylvester II, from 999 until his death in 1003.[434] He was early
+brought under the influence of the monks at Aurillac, and particularly of
+Raimund, who had been a pupil of Odo of Cluny, and there in due time he
+himself took holy orders. He visited Spain in about 967 in company with
+Count Borel,[435] remaining there three years, {111} and studying under
+Bishop Hatto of Vich,[436] a city in the province of Barcelona,[437] then
+entirely under Christian rule. Indeed, all of Gerbert's testimony is as to
+the influence of the Christian civilization upon his education. Thus he
+speaks often of his study of Boethius,[438] so that if the latter knew the
+numerals Gerbert would have learned them from him.[439] If Gerbert had
+studied in any Moorish schools he would, under the decree of the emir
+Hish[=a]m (787-822), have been obliged to know Arabic, which would have
+taken most of his three years in Spain, and of which study we have not the
+slightest hint in any of his letters.[440] On the other hand, Barcelona was
+the only Christian province in immediate touch with the Moorish
+civilization at that time.[441] Furthermore we know that earlier in the
+same century King Alonzo of Asturias (d. 910) confided the education of his
+son Ordono to the Arab scholars of the court of the {112} w[=a]l[=i] of
+Saragossa,[442] so that there was more or less of friendly relation between
+Christian and Moor.
+
+After his three years in Spain, Gerbert went to Italy, about 970, where he
+met Pope John XIII, being by him presented to the emperor Otto I. Two years
+later (972), at the emperor's request, he went to Rheims, where he studied
+philosophy, assisting to make of that place an educational center; and in
+983 he became abbot at Bobbio. The next year he returned to Rheims, and
+became archbishop of that diocese in 991. For political reasons he returned
+to Italy in 996, became archbishop of Ravenna in 998, and the following
+year was elected to the papal chair. Far ahead of his age in wisdom, he
+suffered as many such scholars have even in times not so remote by being
+accused of heresy and witchcraft. As late as 1522, in a biography published
+at Venice, it is related that by black art he attained the papacy, after
+having given his soul to the devil.[443] Gerbert was, however, interested
+in astrology,[444] although this was merely the astronomy of that time and
+was such a science as any learned man would wish to know, even as to-day we
+wish to be reasonably familiar with physics and chemistry.
+
+That Gerbert and his pupils knew the [.g]ob[=a]r numerals is a fact no
+longer open to controversy.[445] Bernelinus and Richer[446] call them by
+the well-known name of {113} "caracteres," a word used by Radulph of Laon
+in the same sense a century later.[447] It is probable that Gerbert was the
+first to describe these [.g]ob[=a]r numerals in any scientific way in
+Christian Europe, but without the zero. If he knew the latter he certainly
+did not understand its use.[448]
+
+The question still to be settled is as to where he found these numerals.
+That he did not bring them from Spain is the opinion of a number of careful
+investigators.[449] This is thought to be the more probable because most of
+the men who made Spain famous for learning lived after Gerbert was there.
+Such were Ibn S[=i]n[=a] (Avicenna) who lived at the beginning, and Gerber
+of Seville who flourished in the middle, of the eleventh century, and
+Ab[=u] Roshd (Averroes) who lived at the end of the twelfth.[450] Others
+hold that his proximity to {114} the Arabs for three years makes it
+probable that he assimilated some of their learning, in spite of the fact
+that the lines between Christian and Moor at that time were sharply
+drawn.[451] Writers fail, however, to recognize that a commercial numeral
+system would have been more likely to be made known by merchants than by
+scholars. The itinerant peddler knew no forbidden pale in Spain, any more
+than he has known one in other lands. If the [.g]ob[=a]r numerals were used
+for marking wares or keeping simple accounts, it was he who would have
+known them, and who would have been the one rather than any Arab scholar to
+bring them to the inquiring mind of the young French monk. The facts that
+Gerbert knew them only imperfectly, that he used them solely for
+calculations, and that the forms are evidently like the Spanish
+[.g]ob[=a]r, make it all the more probable that it was through the small
+tradesman of the Moors that this versatile scholar derived his knowledge.
+Moreover the part of the geometry bearing his name, and that seems
+unquestionably his, shows the Arab influence, proving that he at least came
+into contact with the transplanted Oriental learning, even though
+imperfectly.[452] There was also the persistent Jewish merchant trading
+with both peoples then as now, always alive to the acquiring of useful
+knowledge, and it would be very natural for a man like Gerbert to welcome
+learning from such a source.
+
+On the other hand, the two leading sources of information as to the life of
+Gerbert reveal practically nothing to show that he came within the Moorish
+sphere of influence during his sojourn in Spain. These sources {115} are
+his letters and the history written by Richer. Gerbert was a master of the
+epistolary art, and his exalted position led to the preservation of his
+letters to a degree that would not have been vouchsafed even by their
+classic excellence.[453] Richer was a monk at St. Remi de Rheims, and was
+doubtless a pupil of Gerbert. The latter, when archbishop of Rheims, asked
+Richer to write a history of his times, and this was done. The work lay in
+manuscript, entirely forgotten until Pertz discovered it at Bamberg in
+1833.[454] The work is dedicated to Gerbert as archbishop of Rheims,[455]
+and would assuredly have testified to such efforts as he may have made to
+secure the learning of the Moors.
+
+Now it is a fact that neither the letters nor this history makes any
+statement as to Gerbert's contact with the Saracens. The letters do not
+speak of the Moors, of the Arab numerals, nor of Cordova. Spain is not
+referred to by that name, and only one Spanish scholar is mentioned. In one
+of his letters he speaks of Joseph Ispanus,[456] or Joseph Sapiens, but who
+this Joseph the Wise of Spain may have been we do not know. Possibly {116}
+it was he who contributed the morsel of knowledge so imperfectly
+assimilated by the young French monk.[457] Within a few years after
+Gerbert's visit two young Spanish monks of lesser fame, and doubtless with
+not that keen interest in mathematical matters which Gerbert had, regarded
+the apparently slight knowledge which they had of the Hindu numeral forms
+as worthy of somewhat permanent record[458] in manuscripts which they were
+transcribing. The fact that such knowledge had penetrated to their modest
+cloisters in northern Spain--the one Albelda or Albaida--indicates that it
+was rather widely diffused.
+
+Gerbert's treatise _Libellus de numerorum divisione_[459] is characterized
+by Chasles as "one of the most obscure documents in the history of
+science."[460] The most complete information in regard to this and the
+other mathematical works of Gerbert is given by Bubnov,[461] who considers
+this work to be genuine.[462]
+
+{117}
+
+So little did Gerbert appreciate these numerals that in his works known as
+the _Regula de abaco computi_ and the _Libellus_ he makes no use of them at
+all, employing only the Roman forms.[463] Nevertheless Bernelinus[464]
+refers to the nine [.g]ob[=a]r characters.[465] These Gerbert had marked on
+a thousand _jetons_ or counters,[466] using the latter on an abacus which
+he had a sign-maker prepare for him.[467] Instead of putting eight counters
+in say the tens' column, Gerbert would put a single counter marked 8, and
+so for the other places, leaving the column empty where we would place a
+zero, but where he, lacking the zero, had no counter to place. These
+counters he possibly called _caracteres_, a name which adhered also to the
+figures themselves. It is an interesting speculation to consider whether
+these _apices_, as they are called in the Boethius interpolations, were in
+any way suggested by those Roman jetons generally known in numismatics as
+_tesserae_, and bearing the figures I-XVI, the sixteen referring to the
+number of _assi_ in a _sestertius_.[468] The {118} name _apices_ adhered to
+the Hindu-Arabic numerals until the sixteenth century.[469]
+
+To the figures on the _apices_ were given the names Igin, andras, ormis,
+arbas, quimas, calctis or caltis, zenis, temenias, celentis, sipos,[470]
+the origin and meaning of which still remain a mystery. The Semitic origin
+of several of the words seems probable. _Wahud_, _thaneine_, {119}
+_thalata_, _arba_, _kumsa_, _setta_, _sebba_, _timinia_, _taseud_ are given
+by the Rev. R. Patrick[471] as the names, in an Arabic dialect used in
+Morocco, for the numerals from one to nine. Of these the words for four,
+five, and eight are strikingly like those given above.
+
+The name _apices_ was not, however, a common one in later times. _Notae_
+was more often used, and it finally gave the name to notation.[472] Still
+more common were the names _figures_, _ciphers_, _signs_, _elements_, and
+_characters_.[473]
+
+So little effect did the teachings of Gerbert have in making known the new
+numerals, that O'Creat, who lived a century later, a friend and pupil of
+Adelhard {120} of Bath, used the zero with the Roman characters, in
+contrast to Gerbert's use of the [.g]ob[=a]r forms without the zero.[474]
+O'Creat uses three forms for zero, o, [=o], and [Greek: t], as in Maximus
+Planudes. With this use of the zero goes, naturally, a place value, for he
+writes III III for 33, ICCOO and I. II. [tau]. [tau] for 1200,
+I. O. VIII. IX for 1089, and I. IIII. IIII. [tau][tau][tau][tau] for the
+square of 1200.
+
+The period from the time of Gerbert until after the appearance of
+Leonardo's monumental work may be called the period of the abacists. Even
+for many years after the appearance early in the twelfth century of the
+books explaining the Hindu art of reckoning, there was strife between the
+abacists, the advocates of the abacus, and the algorists, those who favored
+the new numerals. The words _cifra_ and _algorismus cifra_ were used with a
+somewhat derisive significance, indicative of absolute uselessness, as
+indeed the zero is useless on an abacus in which the value of any unit is
+given by the column which it occupies.[475] So Gautier de Coincy
+(1177-1236) in a work on the miracles of Mary says:
+
+  A horned beast, a sheep,
+  An algorismus-cipher,
+  Is a priest, who on such a feast day
+  Does not celebrate the holy Mother.[476]
+
+So the abacus held the field for a long time, even against the new algorism
+employing the new numerals. {121} Geoffrey Chaucer[477] describes in _The
+Miller's Tale_ the clerk with
+
+ "His Almageste and bokes grete and smale,
+  His astrelabie, longinge for his art,
+  His augrim-stones layen faire apart
+  On shelves couched at his beddes heed."
+
+So, too, in Chaucer's explanation of the astrolabe,[478] written for his
+son Lewis, the number of degrees is expressed on the instrument in
+Hindu-Arabic numerals: "Over the whiche degrees ther ben noumbres of
+augrim, that devyden thilke same degrees fro fyve to fyve," and "... the
+nombres ... ben writen in augrim," meaning in the way of the algorism.
+Thomas Usk about 1387 writes:[479] "a sypher in augrim have no might in
+signification of it-selve, yet he yeveth power in signification to other."
+So slow and so painful is the assimilation of new ideas.
+
+Bernelinus[480] states that the abacus is a well-polished board (or table),
+which is covered with blue sand and used by geometers in drawing
+geometrical figures. We have previously mentioned the fact that the Hindus
+also performed mathematical computations in the sand, although there is no
+evidence to show that they had any column abacus.[481] For the purposes of
+computation, Bernelinus continues, the board is divided into thirty
+vertical columns, three of which are reserved for fractions. Beginning with
+the units columns, each set of {122} three columns (_lineae_ is the word
+which Bernelinus uses) is grouped together by a semicircular arc placed
+above them, while a smaller arc is placed over the units column and another
+joins the tens and hundreds columns. Thus arose the designation _arcus
+pictagore_[482] or sometimes simply _arcus_.[483] The operations of
+addition, subtraction, and multiplication upon this form of the abacus
+required little explanation, although they were rather extensively treated,
+especially the multiplication of different orders of numbers. But the
+operation of division was effected with some difficulty. For the
+explanation of the method of division by the use of the complementary
+difference,[484] long the stumbling-block in the way of the medieval
+arithmetician, the reader is referred to works on the history of
+mathematics[485] and to works relating particularly to the abacus.[486]
+
+Among the writers on the subject may be mentioned Abbo[487] of Fleury (c.
+970), Heriger[488] of Lobbes or Laubach {123} (c. 950-1007), and Hermannus
+Contractus[489] (1013-1054), all of whom employed only the Roman numerals.
+Similarly Adelhard of Bath (c. 1130), in his work _Regulae Abaci_,[490]
+gives no reference to the new numerals, although it is certain that he knew
+them. Other writers on the abacus who used some form of Hindu numerals were
+Gerland[491] (first half of twelfth century) and Turchill[492] (c. 1200).
+For the forms used at this period the reader is referred to the plate on
+page 88.
+
+After Gerbert's death, little by little the scholars of Europe came to know
+the new figures, chiefly through the introduction of Arab learning. The
+Dark Ages had passed, although arithmetic did not find another advocate as
+prominent as Gerbert for two centuries. Speaking of this great revival,
+Raoul Glaber[493] (985-c. 1046), a monk of the great Benedictine abbey of
+Cluny, of the eleventh century, says: "It was as though the world had
+arisen and tossed aside the worn-out garments of ancient time, and wished
+to apparel itself in a white robe of churches." And with this activity in
+religion came a corresponding interest in other lines. Algorisms began to
+appear, and knowledge from the outside world found {124} interested
+listeners. Another Raoul, or Radulph, to whom we have referred as Radulph
+of Laon,[494] a teacher in the cloister school of his city, and the brother
+of Anselm of Laon[495] the celebrated theologian, wrote a treatise on
+music, extant but unpublished, and an arithmetic which Nagl first published
+in 1890.[496] The latter work, preserved to us in a parchment manuscript of
+seventy-seven leaves, contains a curious mixture of Roman and [.g]ob[=a]r
+numerals, the former for expressing large results, the latter for practical
+calculation. These [.g]ob[=a]r "caracteres" include the sipos (zero),
+[Symbol], of which, however, Radulph did not know the full significance;
+showing that at the opening of the twelfth century the system was still
+uncertain in its status in the church schools of central France.
+
+At the same time the words _algorismus_ and _cifra_ were coming into
+general use even in non-mathematical literature. Jordan [497] cites
+numerous instances of such use from the works of Alanus ab Insulis[498]
+(Alain de Lille), Gautier de Coincy (1177-1236), and others.
+
+Another contributor to arithmetic during this interesting period was a
+prominent Spanish Jew called variously John of Luna, John of Seville,
+Johannes Hispalensis, Johannes Toletanus, and Johannes Hispanensis de
+Luna.[499] {125} His date is rather closely fixed by the fact that he
+dedicated a work to Raimund who was archbishop of Toledo between 1130 and
+1150.[500] His interests were chiefly in the translation of Arabic works,
+especially such as bore upon the Aristotelian philosophy. From the
+standpoint of arithmetic, however, the chief interest centers about a
+manuscript entitled _Joannis Hispalensis liber Algorismi de Practica
+Arismetrice_ which Boncompagni found in what is now the _Bibliotheque
+nationale_ at Paris. Although this distinctly lays claim to being
+Al-Khow[=a]razm[=i]'s work,[501] the evidence is altogether against the
+statement,[502] but the book is quite as valuable, since it represents the
+knowledge of the time in which it was written. It relates to the operations
+with integers and sexagesimal fractions, including roots, and contains no
+applications.[503]
+
+Contemporary with John of Luna, and also living in Toledo, was Gherard of
+Cremona,[504] who has sometimes been identified, but erroneously, with
+Gernardus,[505] the {126} author of a work on algorism. He was a physician,
+an astronomer, and a mathematician, translating from the Arabic both in
+Italy and in Spain. In arithmetic he was influential in spreading the ideas
+of algorism.
+
+Four Englishmen--Adelhard of Bath (c. 1130), Robert of Chester (Robertus
+Cestrensis, c. 1143), William Shelley, and Daniel Morley (1180)--are
+known[506] to have journeyed to Spain in the twelfth century for the
+purpose of studying mathematics and Arabic. Adelhard of Bath made
+translations from Arabic into Latin of Al-Khow[=a]razm[=i]'s astronomical
+tables[507] and of Euclid's Elements,[508] while Robert of Chester is known
+as the translator of Al-Khow[=a]razm[=i]'s algebra.[509] There is no reason
+to doubt that all of these men, and others, were familiar with the numerals
+which the Arabs were using.
+
+The earliest trace we have of computation with Hindu numerals in Germany is
+in an Algorismus of 1143, now in the Hofbibliothek in Vienna.[510] It is
+bound in with a {127} _Computus_ by the same author and bearing the date
+given. It contains chapters "De additione," "De diminutione," "De
+mediatione," "De divisione," and part of a chapter on multiplication. The
+numerals are in the usual medieval forms except the 2 which, as will be
+seen from the illustration,[511] is somewhat different, and the 3, which
+takes the peculiar shape [Symbol], a form characteristic of the twelfth
+century.
+
+It was about the same time that the _Sefer ha-Mispar_,[512] the Book of
+Number, appeared in the Hebrew language. The author, Rabbi Abraham ibn Meir
+ibn Ezra,[513] was born in Toledo (c. 1092). In 1139 he went to Egypt,
+Palestine, and the Orient, spending also some years in Italy. Later he
+lived in southern France and in England. He died in 1167. The probability
+is that he acquired his knowledge of the Hindu arithmetic[514] in his
+native town of Toledo, but it is also likely that the knowledge of other
+systems which he acquired on travels increased his appreciation of this
+one. We have mentioned the fact that he used the first letters of the
+Hebrew alphabet, [Hebrew: A B G D H W Z CH T`], for the numerals 9 8 7 6 5
+4 3 2 1, and a circle for the zero. The quotation in the note given below
+shows that he knew of the Hindu origin; but in his manuscript, although he
+set down the Hindu forms, he used the above nine Hebrew letters with place
+value for all computations.
+
+       *       *       *       *       *
+
+
+{128}
+
+CHAPTER VIII
+
+THE SPREAD OF THE NUMERALS IN EUROPE
+
+Of all the medieval writers, probably the one most influential in
+introducing the new numerals to the scholars of Europe was Leonardo
+Fibonacci, of Pisa.[515] This remarkable man, the most noteworthy
+mathematical genius of the Middle Ages, was born at Pisa about 1175.[516]
+
+The traveler of to-day may cross the Via Fibonacci on his way to the Campo
+Santo, and there he may see at the end of the long corridor, across the
+quadrangle, the statue of Leonardo in scholars garb. Few towns have honored
+a mathematician more, and few mathematicians have so distinctly honored
+their birthplace. Leonardo was born in the golden age of this city, the
+period of its commercial, religious, and intellectual prosperity.[517]
+{129} Situated practically at the mouth of the Arno, Pisa formed with Genoa
+and Venice the trio of the greatest commercial centers of Italy at the
+opening of the thirteenth century. Even before Venice had captured the
+Levantine trade, Pisa had close relations with the East. An old Latin
+chronicle relates that in 1005 "Pisa was captured by the Saracens," that in
+the following year "the Pisans overthrew the Saracens at Reggio," and that
+in 1012 "the Saracens came to Pisa and destroyed it." The city soon
+recovered, however, sending no fewer than a hundred and twenty ships to
+Syria in 1099,[518] founding a merchant colony in Constantinople a few
+years later,[519] and meanwhile carrying on an interurban warfare in Italy
+that seemed to stimulate it to great activity.[520] A writer of 1114 tells
+us that at that time there were many heathen people--Turks, Libyans,
+Parthians, and Chaldeans--to be found in Pisa. It was in the midst of such
+wars, in a cosmopolitan and commercial town, in a center where literary
+work was not appreciated,[521] that the genius of Leonardo appears as one
+of the surprises of history, warning us again that "we should draw no
+horoscope; that we should expect little, for what we expect will not come
+to pass."[522]
+
+Leonardo's father was one William,[523] and he had a brother named
+Bonaccingus,[524] but nothing further is {130} known of his family. As to
+Fibonacci, most writers[525] have assumed that his father's name was
+Bonaccio,[526] whence _filius Bonaccii_, or Fibonacci. Others[527] believe
+that the name, even in the Latin form of _filius Bonaccii_ as used in
+Leonardo's work, was simply a general one, like our Johnson or Bronson
+(Brown's son); and the only contemporary evidence that we have bears out
+this view. As to the name Bigollo, used by Leonardo, some have thought it a
+self-assumed one meaning blockhead, a term that had been applied to him by
+the commercial world or possibly by the university circle, and taken by him
+that he might prove what a blockhead could do. Milanesi,[528] however, has
+shown that the word Bigollo (or Pigollo) was used in Tuscany to mean a
+traveler, and was naturally assumed by one who had studied, as Leonardo
+had, in foreign lands.
+
+Leonardo's father was a commercial agent at Bugia, the modern Bougie,[529]
+the ancient Saldae on the coast of Barbary,[530] a royal capital under the
+Vandals and again, a century before Leonardo, under the Beni Hammad. It had
+one of the best harbors on the coast, sheltered as it is by Mt. Lalla
+Guraia,[531] and at the close of the twelfth century it was a center of
+African commerce. It was here that Leonardo was taken as a child, and here
+he went to school to a Moorish master. When he reached the years of young
+manhood he started on a tour of the Mediterranean Sea, and visited Egypt,
+Syria, Greece, Sicily, and Provence, meeting with scholars as well as with
+{131} merchants, and imbibing a knowledge of the various systems of numbers
+in use in the centers of trade. All these systems, however, he says he
+counted almost as errors compared with that of the Hindus.[532] Returning
+to Pisa, he wrote his _Liber Abaci_[533] in 1202, rewriting it in
+1228.[534] In this work the numerals are explained and are used in the
+usual computations of business. Such a treatise was not destined to be
+popular, however, because it was too advanced for the mercantile class, and
+too novel for the conservative university circles. Indeed, at this time
+mathematics had only slight place in the newly established universities, as
+witness the oldest known statute of the Sorbonne at Paris, dated 1215,
+where the subject is referred to only in an incidental way.[535] The period
+was one of great commercial activity, and on this very {132} account such a
+book would attract even less attention than usual.[536]
+
+It would now be thought that the western world would at once adopt the new
+numerals which Leonardo had made known, and which were so much superior to
+anything that had been in use in Christian Europe. The antagonism of the
+universities would avail but little, it would seem, against such an
+improvement. It must be remembered, however, that there was great
+difficulty in spreading knowledge at this time, some two hundred and fifty
+years before printing was invented. "Popes and princes and even great
+religious institutions possessed far fewer books than many farmers of the
+present age. The library belonging to the Cathedral Church of San Martino
+at Lucca in the ninth century contained only nineteen volumes of
+abridgments from ecclesiastical commentaries."[537] Indeed, it was not
+until the early part of the fifteenth century that Palla degli Strozzi took
+steps to carry out the project that had been in the mind of Petrarch, the
+founding of a public library. It was largely by word of mouth, therefore,
+that this early knowledge had to be transmitted. Fortunately the presence
+of foreign students in Italy at this time made this transmission feasible.
+(If human nature was the same then as now, it is not impossible that the
+very opposition of the faculties to the works of Leonardo led the students
+to investigate {133} them the more zealously.) At Vicenza in 1209, for
+example, there were Bohemians, Poles, Frenchmen, Burgundians, Germans, and
+Spaniards, not to speak of representatives of divers towns of Italy; and
+what was true there was also true of other intellectual centers. The
+knowledge could not fail to spread, therefore, and as a matter of fact we
+find numerous bits of evidence that this was the case. Although the bankers
+of Florence were forbidden to use these numerals in 1299, and the statutes
+of the university of Padua required stationers to keep the price lists of
+books "non per cifras, sed per literas claros,"[538] the numerals really
+made much headway from about 1275 on.
+
+It was, however, rather exceptional for the common people of Germany to use
+the Arabic numerals before the sixteenth century, a good witness to this
+fact being the popular almanacs. Calendars of 1457-1496[539] have generally
+the Roman numerals, while Koebel's calendar of 1518 gives the Arabic forms
+as subordinate to the Roman. In the register of the Kreuzschule at Dresden
+the Roman forms were used even until 1539.
+
+While not minimizing the importance of the scientific work of Leonardo of
+Pisa, we may note that the more popular treatises by Alexander de Villa Dei
+(c. 1240 A.D.) and John of Halifax (Sacrobosco, c. 1250 A.D.) were much
+more widely used, and doubtless contributed more to the spread of the
+numerals among the common people.
+
+{134}
+
+The _Carmen de Algorismo_[540] of Alexander de Villa Dei was written in
+verse, as indeed were many other textbooks of that time. That it was widely
+used is evidenced by the large number of manuscripts[541] extant in
+European libraries. Sacrobosco's _Algorismus_,[542] in which some lines
+from the Carmen are quoted, enjoyed a wide popularity as a textbook for
+university instruction.[543] The work was evidently written with this end
+in view, as numerous commentaries by university lecturers are found.
+Probably the most widely used of these was that of Petrus de Dacia[544]
+written in 1291. These works throw an interesting light upon the method of
+instruction in mathematics in use in the universities from the thirteenth
+even to the sixteenth century. Evidently the text was first read and copied
+by students.[545] Following this came line by line an exposition of the
+text, such as is given in Petrus de Dacia's commentary.
+
+Sacrobosco's work is of interest also because it was probably due to the
+extended use of this work that the {135} term _Arabic numerals_ became
+common. In two places there is mention of the inventors of this system. In
+the introduction it is stated that this science of reckoning was due to a
+philosopher named Algus, whence the name _algorismus_,[546] and in the
+section on numeration reference is made to the Arabs as the inventors of
+this science.[547] While some of the commentators, Petrus de Dacia[548]
+among them, knew of the Hindu origin, most of them undoubtedly took the
+text as it stood; and so the Arabs were credited with the invention of the
+system.
+
+The first definite trace that we have of an algorism in the French language
+is found in a manuscript written about 1275.[549] This interesting leaf,
+for the part on algorism consists of a single folio, was noticed by the
+Abbe Leboeuf as early as 1741,[550] and by Daunou in 1824.[551] It then
+seems to have been lost in the multitude of Paris manuscripts; for although
+Chasles[552] relates his vain search for it, it was not rediscovered until
+1882. In that year M. Ch. Henry found it, and to his care we owe our
+knowledge of the interesting manuscript. The work is anonymous and is
+devoted almost entirely to geometry, only {136} two pages (one folio)
+relating to arithmetic. In these the forms of the numerals are given, and a
+very brief statement as to the operations, it being evident that the writer
+himself had only the slightest understanding of the subject.
+
+Once the new system was known in France, even thus superficially, it would
+be passed across the Channel to England. Higden,[553] writing soon after
+the opening of the fourteenth century, speaks of the French influence at
+that time and for some generations preceding:[554] "For two hundred years
+children in scole, agenst the usage and manir of all other nations beeth
+compelled for to leave hire own language, and for to construe hir lessons
+and hire thynges in Frensche.... Gentilmen children beeth taught to speke
+Frensche from the tyme that they bith rokked in hir cradell; and
+uplondissche men will likne himself to gentylmen, and fondeth with greet
+besynesse for to speke Frensche."
+
+The question is often asked, why did not these new numerals attract more
+immediate attention? Why did they have to wait until the sixteenth century
+to be generally used in business and in the schools? In reply it may be
+said that in their elementary work the schools always wait upon the demands
+of trade. That work which pretends to touch the life of the people must
+come reasonably near doing so. Now the computations of business until about
+1500 did not demand the new figures, for two reasons: First, cheap paper
+was not known. Paper-making of any kind was not introduced into Europe
+until {137} the twelfth century, and cheap paper is a product of the
+nineteenth. Pencils, too, of the modern type, date only from the sixteenth
+century. In the second place, modern methods of operating, particularly of
+multiplying and dividing (operations of relatively greater importance when
+all measures were in compound numbers requiring reductions at every step),
+were not yet invented. The old plan required the erasing of figures after
+they had served their purpose, an operation very simple with counters,
+since they could be removed. The new plan did not as easily permit this.
+Hence we find the new numerals very tardily admitted to the counting-house,
+and not welcomed with any enthusiasm by teachers.[555]
+
+Aside from their use in the early treatises on the new art of reckoning,
+the numerals appeared from time to time in the dating of manuscripts and
+upon monuments. The oldest definitely dated European document known {138}
+to contain the numerals is a Latin manuscript,[556] the Codex Vigilanus,
+written in the Albelda Cloister not far from Logrono in Spain, in 976 A.D.
+The nine characters (of [.g]ob[=a]r type), without the zero, are given as
+an addition to the first chapters of the third book of the _Origines_ by
+Isidorus of Seville, in which the Roman numerals are under discussion.
+Another Spanish copy of the same work, of 992 A.D., contains the numerals
+in the corresponding section. The writer ascribes an Indian origin to them
+in the following words: "Item de figuris arithmetic[e,]. Scire debemus in
+Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et
+geometria et ceteris liberalibus disciplinis concedere. Et hoc manifestum
+est in nobem figuris, quibus designant unumquemque gradum cuiuslibet
+gradus. Quarum hec sunt forma." The nine [.g]ob[=a]r characters follow.
+Some of the abacus forms[557] previously given are doubtless also of the
+tenth century. The earliest Arabic documents containing the numerals are
+two manuscripts of 874 and 888 A.D.[558] They appear about a century later
+in a work[559] written at Shiraz in 970 A.D. There is also an early trace
+of their use on a pillar recently discovered in a church apparently
+destroyed as early as the tenth century, not far from the Jeremias
+Monastery, in Egypt. {139} A graffito in Arabic on this pillar has the date
+349 A.H., which corresponds to 961 A.D.[560] For the dating of Latin
+documents the Arabic forms were used as early as the thirteenth
+century.[561]
+
+On the early use of these numerals in Europe the only scientific study
+worthy the name is that made by Mr. G. F. Hill of the British Museum.[562]
+From his investigations it appears that the earliest occurrence of a date
+in these numerals on a coin is found in the reign of Roger of Sicily in
+1138.[563] Until recently it was thought that the earliest such date was
+1217 A.D. for an Arabic piece and 1388 for a Turkish one.[564] Most of the
+seals and medals containing dates that were at one time thought to be very
+early have been shown by Mr. Hill to be of relatively late workmanship.
+There are, however, in European manuscripts, numerous instances of the use
+of these numerals before the twelfth century. Besides the example in the
+Codex Vigilanus, another of the tenth century has been found in the St.
+Gall MS. now in the University Library at Zuerich, the forms differing
+materially from those in the Spanish codex.
+
+The third specimen in point of time in Mr. Hill's list is from a Vatican
+MS. of 1077. The fourth and fifth specimens are from the Erlangen MS. of
+Boethius, of the same {140} (eleventh) century, and the sixth and seventh
+are also from an eleventh-century MS. of Boethius at Chartres. These and
+other early forms are given by Mr. Hill in this table, which is reproduced
+with his kind permission.
+
+EARLIEST MANUSCRIPT FORMS
+
+[Illustration]
+
+This is one of more than fifty tables given in Mr. Hill's valuable paper,
+and to this monograph students {141} are referred for details as to the
+development of number-forms in Europe from the tenth to the sixteenth
+century. It is of interest to add that he has found that among the earliest
+dates of European coins or medals in these numerals, after the Sicilian one
+already mentioned, are the following: Austria, 1484; Germany, 1489
+(Cologne); Switzerland, 1424 (St. Gall); Netherlands, 1474; France, 1485;
+Italy, 1390.[565]
+
+The earliest English coin dated in these numerals was struck in 1551,[566]
+although there is a Scotch piece of 1539.[567] In numbering pages of a
+printed book these numerals were first used in a work of Petrarch's
+published at Cologne in 1471.[568] The date is given in the following form
+in the _Biblia Pauperum_,[569] a block-book of 1470,
+
+[Illustration]
+
+while in another block-book which possibly goes back to c. 1430[570] the
+numerals appear in several illustrations, with forms as follows:
+
+[Illustration]
+
+Many printed works anterior to 1471 have pages or chapters numbered by
+hand, but many of these numerals are {142} of date much later than the
+printing of the work. Other works were probably numbered directly after
+printing. Thus the chapters 2, 3, 4, 5, 6 in a book of 1470[571] are
+numbered as follows: Capitulem [Symbol 2]m.,... [Symbol 3]m.,... 4m.,...
+v,... vi, and followed by Roman numerals. This appears in the body of the
+text, in spaces left by the printer to be filled in by hand. Another
+book[572] of 1470 has pages numbered by hand with a mixture of Roman and
+Hindu numerals, thus,
+
+  [Illustration] for 125      [Illustration] for 150
+  [Illustration] for 147      [Illustration] for 202
+
+As to monumental inscriptions,[573] there was once thought to be a
+gravestone at Katharein, near Troppau, with the date 1007, and one at
+Biebrich of 1299. There is no doubt, however, of one at Pforzheim of 1371
+and one at Ulm of 1388.[574] Certain numerals on Wells Cathedral have been
+assigned to the thirteenth century, but they are undoubtedly considerably
+later.[575]
+
+The table on page 143 will serve to supplement that from Mr. Hill's
+work.[576]
+
+{143}
+
+EARLY MANUSCRIPT FORMS
+
+  [577] [Illustration] Twelfth century A.D.
+  [578] [Illustration] 1197 A.D.
+  [579] [Illustration] 1275 A.D.
+  [580] [Illustration] c. 1294 A.D.
+  [581] [Illustration] c. 1303 A.D.
+  [582] [Illustration] c. 1360 A.D.
+  [583] [Illustration] c. 1442 A.D.
+
+{144}
+
+[Illustration]
+
+For the sake of further comparison, three illustrations from works in Mr.
+Plimpton's library, reproduced from the _Rara Arithmetica_, may be
+considered. The first is from a Latin manuscript on arithmetic,[584] of
+which the original was written at Paris in 1424 by Rollandus, a Portuguese
+physician, who prepared the work at the command of John of Lancaster, Duke
+of Bedford, at one time Protector of England and Regent of France, to whom
+the work is dedicated. The figures show the successive powers of 2. The
+second illustration is from Luca da Firenze's _Inprencipio darte
+dabacho_,[585] c. 1475, and the third is from an anonymous manuscript[586]
+of about 1500.
+
+[Illustration]
+
+As to the forms of the numerals, fashion played a leading part until
+printing was invented. This tended to fix these forms, although in writing
+there is still a great variation, as witness the French 5 and the German 7
+and 9. Even in printing there is not complete uniformity, {145} and it is
+often difficult for a foreigner to distinguish between the 3 and 5 of the
+French types.
+
+[Illustration]
+
+As to the particular numerals, the following are some of the forms to be
+found in the later manuscripts and in the early printed books.
+
+1. In the early printed books "one" was often i, perhaps to save types,
+just as some modern typewriters use the same character for l and 1.[587] In
+the manuscripts the "one" appears in such forms as[588]
+
+[Illustration]
+
+2. "Two" often appears as z in the early printed books, 12 appearing as
+iz.[589] In the medieval manuscripts the following forms are common:[590]
+
+[Illustration]
+
+{146}
+
+It is evident, from the early traces, that it is merely a cursive form for
+the primitive [2 horizontal strokes], just as 3 comes from [3 horizontal
+strokes], as in the N[=a]n[=a] Gh[=a]t inscriptions.
+
+3. "Three" usually had a special type in the first printed books, although
+occasionally it appears as [Symbol].[591] In the medieval manuscripts it
+varied rather less than most of the others. The following are common
+forms:[592]
+
+[Illustration]
+
+4. "Four" has changed greatly; and one of the first tests as to the age of
+a manuscript on arithmetic, and the place where it was written, is the
+examination of this numeral. Until the time of printing the most common
+form was [Symbol], although the Florentine manuscript of Leonard of Pisa's
+work has the form [Symbol];[593] but the manuscripts show that the
+Florentine arithmeticians and astronomers rather early began to straighten
+the first of these forms up to forms like [Symbol][594] and [Symbol][594]
+or [Symbol],[595] more closely resembling our own. The first printed books
+generally used our present form[596] with the closed top [Symbol], the open
+top used in writing ( [Symbol]) being {147} purely modern. The following
+are other forms of the four, from various manuscripts:[597]
+
+[Illustration]
+
+5. "Five" also varied greatly before the time of printing. The following
+are some of the forms:[598]
+
+[Illustration]
+
+6. "Six" has changed rather less than most of the others. The chief
+variation has been in the slope of the top, as will be seen in the
+following:[599]
+
+[Illustration]
+
+7. "Seven," like "four," has assumed its present erect form only since the
+fifteenth century. In medieval times it appeared as follows:[600]
+
+[Illustration]
+
+{148}
+
+8. "Eight," like "six," has changed but little. In medieval times there are
+a few variants of interest as follows:[601]
+
+[Illustration]
+
+In the sixteenth century, however, there was manifested a tendency to write
+it [Symbol].[602]
+
+9. "Nine" has not varied as much as most of the others. Among the medieval
+forms are the following:[603]
+
+[Illustration]
+
+0. The shape of the zero also had a varied history. The following are
+common medieval forms:[604]
+
+[Illustration]
+
+The explanation of the place value was a serious matter to most of the
+early writers. If they had been using an abacus constructed like the
+Russian chotue, and had placed this before all learners of the positional
+system, there would have been little trouble. But the medieval {149}
+line-reckoning, where the lines stood for powers of 10 and the spaces for
+half of such powers, did not lend itself to this comparison. Accordingly we
+find such labored explanations as the following, from _The Crafte of
+Nombrynge_:
+
+"Euery of these figuris bitokens hym selfe & no more, yf he stonde in the
+first place of the rewele....
+
+"If it stonde in the secunde place of the rewle, he betokens ten tymes hym
+selfe, as this figure 2 here 20 tokens ten tyme hym selfe, that is twenty,
+for he hym selfe betokens tweyne, & ten tymes twene is twenty. And for he
+stondis on the lyft side & in the secunde place, he betokens ten tyme hym
+selfe. And so go forth....
+
+"Nil cifra significat sed dat signare sequenti. Expone this verse. A cifre
+tokens no[gh]t, bot he makes the figure to betoken that comes after hym
+more than he shuld & he were away, as thus 10. here the figure of one
+tokens ten, & yf the cifre were away & no figure byfore hym he schuld token
+bot one, for than he schuld stonde in the first place...."[605]
+
+It would seem that a system that was thus used for dating documents, coins,
+and monuments, would have been generally adopted much earlier than it was,
+particularly in those countries north of Italy where it did not come into
+general use until the sixteenth century. This, however, has been the fate
+of many inventions, as witness our neglect of logarithms and of contracted
+processes to-day.
+
+As to Germany, the fifteenth century saw the rise of the new symbolism; the
+sixteenth century saw it slowly {150} gain the mastery; the seventeenth
+century saw it finally conquer the system that for two thousand years had
+dominated the arithmetic of business. Not a little of the success of the
+new plan was due to Luther's demand that all learning should go into the
+vernacular.[606]
+
+During the transition period from the Roman to the Arabic numerals, various
+anomalous forms found place. For example, we have in the fourteenth century
+c[alpha] for 104;[607] 1000. 300. 80 et 4 for 1384;[608] and in a
+manuscript of the fifteenth century 12901 for 1291.[609] In the same
+century m. cccc. 8II appears for 1482,[610] while M^oCCCC^o50 (1450) and
+MCCCCXL6 (1446) are used by Theodoricus Ruffi about the same time.[611] To
+the next century belongs the form 1vojj for 1502. Even in Sfortunati's
+_Nuovo lume_[612] the use of ordinals is quite confused, the propositions
+on a single page being numbered "tertia," "4," and "V."
+
+Although not connected with the Arabic numerals in any direct way, the
+medieval astrological numerals may here be mentioned. These are given by
+several early writers, but notably by Noviomagus (1539),[613] as
+follows[614]:
+
+[Illustration]
+
+{151}
+
+Thus we find the numerals gradually replacing the Roman forms all over
+Europe, from the time of Leonardo of Pisa until the seventeenth century.
+But in the Far East to-day they are quite unknown in many countries, and
+they still have their way to make. In many parts of India, among the common
+people of Japan and China, in Siam and generally about the Malay Peninsula,
+in Tibet, and among the East India islands, the natives still adhere to
+their own numeral forms. Only as Western civilization is making its way
+into the commercial life of the East do the numerals as used by us find
+place, save as the Sanskrit forms appear in parts of India. It is therefore
+with surprise that the student of mathematics comes to realize how modern
+are these forms so common in the West, how limited is their use even at the
+present time, and how slow the world has been and is in adopting such a
+simple device as the Hindu-Arabic numerals.
+
+       *       *       *       *       *
+
+
+{153}
+
+INDEX
+
+_Transcriber's note: many of the entries refer to footnotes linked from the
+page numbers given._
+
+  Abbo of Fleury, 122
+  `Abdall[=a]h ibn al-[H.]asan, 92
+  `Abdallat[=i]f ibn Y[=u]suf, 93
+  `Abdalq[=a]dir ibn `Al[=i] al-Sakh[=a]w[=i], 6
+  Abenragel, 34
+  Abraham ibn Meir ibn Ezra, _see_ Rabbi ben Ezra
+  Ab[=u] `Al[=i] al-[H.]osein ibn S[=i]n[=a], 74
+  Ab[=u] 'l-[H.]asan, 93, 100
+  Ab[=u] 'l-Q[=a]sim, 92
+  Ab[=u] 'l-[T.]eiyib, 97
+  Ab[=u] Na[s.]r, 92
+  Ab[=u] Roshd, 113
+  Abu Sahl Dunash ibn Tamim, 65, 67
+  Adelhard of Bath, 5, 55, 97, 119, 123, 126
+  Adhemar of Chabanois, 111
+  A[h.]med al-Nasaw[=i], 98
+  A[h.]med ibn `Abdall[=a]h, 9, 92
+  A[h.]med ibn Mo[h.]ammed, 94
+  A[h.]med ibn `Omar, 93
+  Ak[s.]aras, 32
+  Alanus ab Insulis, 124
+  Al-Ba[.g]d[=a]d[=i], 93
+  Al-Batt[=a]n[=i], 54
+  Albelda (Albaida) MS., 116
+  Albert, J., 62
+  Albert of York, 103
+  Al-B[=i]r[=u]n[=i], 6, 41, 49, 65, 92, 93
+  Alcuin, 103
+  Alexander the Great, 76
+  Alexander de Villa Dei, 11, 133
+  Alexandria, 64, 82
+  Al-Faz[=a]r[=i], 92
+  Alfred, 103
+  Algebra, etymology, 5
+  Algerian numerals, 68
+  Algorism, 97
+  Algorismus, 124, 126, 135
+  Algorismus cifra, 120
+  Al-[H.]a[s.][s.][=a]r, 65
+  `Al[=i] ibn Ab[=i] Bekr, 6
+  `Al[=i] ibn A[h.]med, 93, 98
+  Al-Kar[=a]b[=i]s[=i], 93
+  Al-Khow[=a]razm[=i], 4, 9, 10, 92, 97, 98, 125, 126
+  Al-Kind[=i], 10, 92
+  Almagest, 54
+  Al-Ma[.g]reb[=i], 93
+  Al-Ma[h.]all[=i], 6
+  Al-M[=a]m[=u]n, 10, 97
+  Al-Man[s.][=u]r, 96, 97
+  Al-Mas`[=u]d[=i], 7, 92
+  Al-Nad[=i]m, 9
+  Al-Nasaw[=i], 93, 98
+  Alphabetic numerals, 39, 40, 43
+  Al-Q[=a]sim, 92
+  Al-Qass, 94
+  Al-Sakh[=a]w[=i], 6
+  Al-[S.]ardaf[=i], 93
+  Al-Sijz[=i], 94
+  Al-S[=u]f[=i], 10, 92
+  Ambrosoli, 118
+  A[.n]kapalli, 43
+  Apices, 87, 117, 118
+  Arabs, 91-98
+  Arbuthnot, 141
+  {154}
+  Archimedes, 15, 16
+  Arcus Pictagore, 122
+  Arjuna, 15
+  Arnold, E., 15, 102
+  Ars memorandi, 141
+  [=A]ryabha[t.]a, 39, 43, 44
+  Aryan numerals, 19
+  Aschbach, 134
+  Ashmole, 134
+  A['s]oka, 19, 20, 22, 81
+  A[s.]-[s.]ifr, 57, 58
+  Astrological numerals, 150
+  Atharva-Veda, 48, 49, 55
+  Augustus, 80
+  Averroes, 113
+  Avicenna, 58, 74, 113
+
+  Babylonian numerals, 28
+  Babylonian zero, 51
+  Bacon, R., 131
+  Bactrian numerals, 19, 30
+  Baeda, 2, 72
+  Bagdad, 4, 96
+  Bakh[s.][=a]l[=i] manuscript, 43, 49, 52, 53
+  Ball, C. J., 35
+  Ball, W. W. R., 36, 131
+  B[=a][n.]a, 44
+  Barth, A., 39
+  Bayang inscriptions, 39
+  Bayer, 33
+  Bayley, E. C., 19, 23, 30, 32, 52, 89
+  Beazley, 75
+  Bede, _see_ Baeda
+  Beldomandi, 137
+  Beloch, J., 77
+  Bendall, 25, 52
+  Benfey, T., 26
+  Bernelinus, 88, 112, 117, 121
+  Besagne, 128
+  Besant, W., 109
+  Bettino, 36
+  Bhandarkar, 18, 47, 49
+  Bh[=a]skara, 53, 55
+  Biernatzki, 32
+  Biot, 32
+  Bjoernbo, A. A., 125, 126
+  Blassiere, 119
+  Bloomfield, 48
+  Blume, 85
+  Boeckh, 62
+  Boehmer, 143
+  Boeschenstein, 119
+  Boethius, 63, 70-73, 83-90
+  Boissiere, 63
+  Bombelli, 81
+  Bonaini, 128
+  Boncompagni, 5, 6, 10, 48, 49, 123, 125
+  Borghi, 59
+  Borgo, 119
+  Bougie, 130
+  Bowring, J., 56
+  Brahmagupta, 52
+  Br[=a]hma[n.]as, 12, 13
+  Br[=a]hm[=i], 19, 20, 31, 83
+  Brandis, J., 54
+  B[r.]hat-Sa[m.]hita, 39, 44, 78
+  Brockhaus, 43
+  Bubnov, 65, 84, 110, 116
+  Buddha, education of, 15, 16
+  Buedinger, 110
+  Bugia, 130
+  Buehler, G., 15, 19, 22, 31, 44, 49
+  Burgess, 25
+  Buerk, 13
+  Burmese numerals, 36
+  Burnell, A. C., 18, 40
+  Buteo, 61
+
+  Calandri, 59, 81
+  Caldwell, R., 19
+  Calendars, 133
+  Calmet, 34
+  Cantor, M., 5, 13, 30, 43, 84
+  {155}
+  Capella, 86
+  Cappelli, 143
+  Caracteres, 87, 113, 117, 119
+  Cardan, 119
+  Carmen de Algorismo, 11, 134
+  Casagrandi, 132
+  Casiri, 8, 10
+  Cassiodorus, 72
+  Cataldi, 62
+  Cataneo, 3
+  Caxton, 143, 146
+  Ceretti, 32
+  Ceylon numerals, 36
+  Chalfont, F. H., 28
+  Champenois, 60
+  Characters, _see_ Caracteres
+  Charlemagne, 103
+  Chasles, 54, 60, 85, 116, 122, 135
+  Chassant, L. A., 142
+  Chaucer, 121
+  Chiarini, 145, 146
+  Chiffre, 58
+  Chinese numerals, 28, 56
+  Chinese zero, 56
+  Cifra, 120, 124
+  Cipher, 58
+  Circulus, 58, 60
+  Clichtoveus, 61, 119, 145
+  Codex Vigilanus, 138
+  Codrington, O., 139
+  Coins dated, 141
+  Colebrooke, 8, 26, 46, 53
+  Constantine, 104, 105
+  Cosmas, 82
+  Cossali, 5
+  Counters, 117
+  Courteille, 8
+  Coxe, 59
+  Crafte of Nombrynge, 11, 87, 149
+  Crusades, 109
+  Cunningham, A., 30, 75
+  Curtze, 55, 59, 126, 134
+  Cyfra, 55
+
+  Dagomari, 146
+  D'Alviella, 15
+  Dante, 72
+  Dasypodius, 33, 67, 63
+  Daunou, 135
+  Delambre, 54
+  Devan[=a]gar[=i], 7
+  Devoulx, A., 68
+  Dhruva, 49
+  Dicaearchus of Messana, 77
+  Digits, 119
+  Diodorus Siculus, 76
+  Du Cange, 62
+  Dumesnil, 36
+  Dutt, R. C., 12, 15, 18, 75
+  Dvived[=i], 44
+
+  East and West, relations, 73-81, 100-109
+  Egyptian numerals, 27
+  Eisenlohr, 28
+  Elia Misrachi, 57
+  Enchiridion Algorismi, 58
+  Enestroem, 5, 48, 59, 97, 125, 128
+  Europe, numerals in, 63, 99, 128, 136
+  Eusebius Caesariensis, 142
+  Euting, 21
+  Ewald, P., 116
+
+  Fazzari, 53, 54
+  Fibonacci, _see_ Leonardo of Pisa
+  Figura nihili, 58
+  Figures, 119. _See_ numerals.
+  Fihrist, 67, 68, 93
+  Finaeus, 57
+  Firdus[=i], 81
+  Fitz Stephen, W., 109
+  Fleet, J. C., 19, 20, 49
+  {156}
+  Florus, 80
+  Fluegel, G., 68
+  Francisco de Retza, 142
+  Francois, 58
+  Friedlein, G., 84, 113, 116, 122
+  Froude, J. A., 129
+
+  Gandh[=a]ra, 19
+  Garbe, 48
+  Gasbarri, 58
+  Gautier de Coincy, 120, 124
+  Gemma Frisius, 2, 3, 119
+  Gerber, 113
+  Gerbert, 108, 110-120, 122
+  Gerhardt, C. I., 43, 56, 93, 118
+  Gerland, 88, 123
+  Gherard of Cremona, 125
+  Gibbon, 72
+  Giles, H. A., 79
+  Ginanni, 81
+  Giovanni di Danti, 58
+  Glareanus, 4, 119
+  Gnecchi, 71, 117
+  [.G]ob[=a]r numerals, 65, 100, 112, 124, 138
+  Gow, J., 81
+  Grammateus, 61
+  Greek origin, 33
+  Green, J. R., 109
+  Greenwood, I., 62, 119
+  Guglielmini, 128
+  Gulist[=a]n, 102
+  Guenther, S., 131
+  Guyard, S., 82
+
+  [H.]abash, 9, 92
+  Hager, J. (G.), 28, 32
+  Halliwell, 59, 85
+  Hankel, 93
+  H[=a]r[=u]n al-Rash[=i]d, 97, 106
+  Havet, 110
+  Heath, T. L., 125
+  Hebrew numerals, 127
+  Hecataeus, 75
+  Heiberg, J. L., 55, 85, 148
+  Heilbronner, 5
+  Henry, C., 5, 31, 55, 87, 120, 135
+  Heriger, 122
+  Hermannus Contractus, 123
+  Herodotus, 76, 78
+  Heyd, 75
+  Higden, 136
+  Hill, G. F., 52, 139, 142
+  Hillebrandt, A., 15, 74
+  Hilprecht, H. V., 28
+  Hindu forms, early, 12
+  Hindu number names, 42
+  Hodder, 62
+  Hoernle, 43, 49
+  Holywood, _see_ Sacrobosco
+  Hopkins, E. W., 12
+  Horace, 79, 80
+  [H.]osein ibn Mo[h.]ammed al-Ma[h.]all[=i], 6
+  Hostus, M., 56
+  Howard, H. H., 29
+  Hrabanus Maurus, 72
+  Huart, 7
+  Huet, 33
+  Hugo, H., 57
+  Humboldt, A. von, 62
+  Huswirt, 58
+
+  Iamblichus, 81
+  Ibn Ab[=i] Ya`q[=u]b, 9
+  Ibn al-Adam[=i], 92
+  Ibn al-Bann[=a], 93
+  Ibn Khord[=a][d.]beh, 101, 106
+  Ibn Wahab, 103
+  India, history of, 14
+    writing in, 18
+  Indicopleustes, 83
+  Indo-Bactrian numerals, 19
+  {157}
+  Indr[=a]j[=i], 23
+  Is[h.][=a]q ibn Y[=u]suf al-[S.]ardaf[=i], 93
+
+  Jacob of Florence, 57
+  Jacquet, E., 38
+  Jamshid, 56
+  Jehan Certain, 59
+  Jetons, 58, 117
+  Jevons, F. B., 76
+  Johannes Hispalensis, 48, 88, 124
+  John of Halifax, _see_ Sacrobosco
+  John of Luna, _see_ Johannes Hispalensis
+  Jordan, L., 58, 124
+  Joseph Ispanus (Joseph Sapiens), 115
+  Justinian, 104
+
+  Kale, M. R., 26
+  Karabacek, 56
+  Karpinski, L. C., 126, 134, 138
+  K[=a]ty[=a]yana, 39
+  Kaye, C. R., 6, 16, 43, 46, 121
+  Keane, J., 75, 82
+  Keene, H. G., 15
+  Kern, 44
+  Kharo[s.][t.]h[=i], 19, 20
+  Khosr[=u], 82, 91
+  Kielhorn, F., 46, 47
+  Kircher, A., 34
+  Kit[=a]b al-Fihrist, _see_ Fihrist
+  Kleinwaechter, 32
+  K[=l]os, 62
+  Koebel, 4, 58, 60, 119, 123
+  Krumbacher, K., 57
+  Kuckuck, 62, 133
+  Kugler, F. X., 51
+
+  Lachmann, 85
+  Lacouperie, 33, 35
+  Lalitavistara, 15, 17
+  Lami, G., 57
+  La Roche, 61
+  Lassen, 39
+  L[=a][t.]y[=a]yana, 39
+  Leboeuf, 135
+  Leonardo of Pisa, 5, 10, 57, 64, 74, 120, 128-133
+  Lethaby, W. R., 142
+  Levi, B., 13
+  Levias, 3
+  Libri, 73, 85, 95
+  Light of Asia, 16
+  Luca da Firenze, 144
+  Lucas, 128
+
+  Mah[=a]bh[=a]rata, 18
+  Mah[=a]v[=i]r[=a]c[=a]rya, 53
+  Malabar numerals, 36
+  Malayalam numerals, 36
+  Mannert, 81
+  Margarita Philosophica, 146
+  Marie, 78
+  Marquardt, J., 85
+  Marshman, J. C., 17
+  Martin, T. H., 30, 62, 85, 113
+  Martines, D. C., 58
+  M[=a]sh[=a]ll[=a]h, 3
+  Maspero, 28
+  Mauch, 142
+  Maximus Planudes, 2, 57, 66, 93, 120
+  Megasthenes, 77
+  Merchants, 114
+  Meynard, 8
+  Migne, 87
+  Mikami, Y., 56
+  Milanesi, 128
+  Mo[h.]ammed ibn `Abdall[=a]h, 92
+  Mo[h.]ammed ibn A[h.]med, 6
+  Mo[h.]ammed ibn `Al[=i] `Abd[=i], 8
+  Mo[h.]ammed ibn M[=u]s[=a], _see_ Al-Khow[=a]razm[=i]
+  Molinier, 123
+  Monier-Williams, 17
+  {158}
+  Morley, D., 126
+  Moroccan numerals, 68, 119
+  Mortet, V., 11
+  Moseley, C. B., 33
+  Mo[t.]ahhar ibn [T.][=a]hir, 7
+  Mueller, A., 68
+  Mumford, J. K., 109
+  Muwaffaq al-D[=i]n, 93
+
+  Nabatean forms, 21
+  Nallino, 4, 54, 55
+  Nagl, A., 55, 110, 113, 126
+  N[=a]n[=a] Gh[=a]t inscriptions, 20, 22, 23, 40
+  Narducci, 123
+  Nasik cave inscriptions, 24
+  Na[z.][=i]f ibn Yumn, 94
+  Neander, A., 75
+  Neophytos, 57, 62
+  Neo-Pythagoreans, 64
+  Nesselmann, 58
+  Newman, Cardinal, 96
+  Newman, F. W., 131
+  Noeldeke, Th., 91
+  Notation, 61
+  Note, 61, 119
+  Noviomagus, 45, 61, 119, 150
+  Null, 61
+  Numerals,
+    Algerian, 68
+    astrological, 150
+    Br[=a]hm[=i], 19-22, 83
+    early ideas of origin, 1
+    Hindu, 26
+    Hindu, classified, 19, 38
+    Kharo[s.][t.]h[=i], 19-22
+    Moroccan, 68
+    Nabatean, 21
+    origin, 27, 30, 31, 37
+    supposed Arabic origin, 2
+    supposed Babylonian origin, 28
+    supposed Chaldean and Jewish origin, 3
+    supposed Chinese origin, 28, 32
+    supposed Egyptian origin, 27, 30, 69, 70
+    supposed Greek origin, 33
+    supposed Phoenician origin, 32
+    tables of, 22-27, 36, 48, 49, 69, 88, 140, 143, 145-148
+
+  O'Creat, 5, 55, 119, 120
+  Olleris, 110, 113
+  Oppert, G., 14, 75
+
+  Pali, 22
+  Pancasiddh[=a]ntik[=a], 44
+  Paravey, 32, 57
+  P[=a]tal[=i]pu[t.]ra, 77
+  Patna, 77
+  Patrick, R., 119
+  Payne, E. J., 106
+  Pegolotti, 107
+  Peletier, 2, 62
+  Perrot, 80
+  Persia, 66, 91, 107
+  Pertz, 115
+  Petrus de Dacia, 59, 61, 62
+  Pez, P. B., 117
+  "Philalethes," 75
+  Phillips, G., 107
+  Picavet, 105
+  Pichler, F., 141
+  Pihan, A. P., 36
+  Pisa, 128
+  Place value, 26, 42, 46, 48
+  Planudes, _see_ Maximus Planudes
+  Plimpton, G. A., 56, 59, 85, 143, 144, 145, 148
+  Pliny, 76
+  Polo, N. and M., 107
+  {159}
+  Praendel, J. G., 54
+  Prinsep, J., 20, 31
+  Propertius, 80
+  Prosdocimo de' Beldomandi, 137
+  Prou, 143
+  Ptolemy, 54, 78
+  Putnam, 103
+  Pythagoras, 63
+  Pythagorean numbers, 13
+  Pytheas of Massilia, 76
+
+  Rabbi ben Ezra, 60, 127
+  Radulph of Laon, 60, 113, 118, 124
+  Raets, 62
+  Rainer, _see_ Gemma Frisius
+  R[=a]m[=a]yana, 18
+  Ramus, 2, 41, 60, 61
+  Raoul Glaber, 123
+  Rapson, 77
+  Rauhfuss, _see_ Dasypodius
+  Raumer, K. von, 111
+  Reclus, E., 14, 96, 130
+  Recorde, 3, 58
+  Reinaud, 67, 74, 80
+  Reveillaud, 36
+  Richer, 110, 112, 115
+  Riese, A., 119
+  Robertson, 81
+  Robertus Cestrensis, 97, 126
+  Rodet, 5, 44
+  Roediger, J., 68
+  Rollandus, 144
+  Romagnosi, 81
+  Rosen, F., 5
+  Rotula, 60
+  Rudolff, 85
+  Rudolph, 62, 67
+  Ruffi, 150
+
+  Sachau, 6
+  Sacrobosco, 3, 58, 133
+  Sacy, S. de, 66, 70
+  Sa`d[=i], 102
+  ['S]aka inscriptions, 20
+  Sam[=u]'[=i]l ibn Ya[h.]y[=a], 93
+  ['S][=a]rad[=a] characters, 55
+  Savonne, 60
+  Scaliger, J. C., 73
+  Scheubel, 62
+  Schlegel, 12
+  Schmidt, 133
+  Schonerus, 87, 119
+  Schroeder, L. von, 13
+  Scylax, 75
+  Sedillot, 8, 34
+  Senart, 20, 24, 25
+  Sened ibn `Al[=i], 10, 98
+  Sfortunati, 62, 150
+  Shelley, W., 126
+  Siamese numerals, 36
+  Siddh[=a]nta, 8, 18
+  [S.]ifr, 57
+  Sigsboto, 55
+  Sih[=a]b al-D[=i]n, 67
+  Silberberg, 60
+  Simon, 13
+  Sin[=a]n ibn al-Fat[h.], 93
+  Sindbad, 100
+  Sindhind, 97
+  Sipos, 60
+  Sirr, H. C., 75
+  Skeel, C. A., 74
+  Smith, D. E., 11, 17, 53, 86, 141, 143
+  Smith, V. A., 20, 35, 46, 47
+  Smith, Wm., 75
+  Sm[r.]ti, 17
+  Spain, 64, 65, 100
+  Spitta-Bey, 5
+  Sprenger, 94
+  ['S]rautas[=u]tra, 39
+  Steffens, F., 116
+  Steinschneider, 5, 57, 65, 66, 98, 126
+  Stifel, 62
+  {160}
+  Subandhus, 44
+  Suetonius, 80
+  Suleim[=a]n, 100
+  ['S][=u]nya, 43, 53, 57
+  Suter, 5, 9, 68, 69, 93, 116, 131
+  S[=u]tras, 13
+  Sykes, P. M., 75
+  Sylvester II, _see_ Gerbert
+  Symonds, J. A., 129
+
+  Tannery, P., 62, 84, 85
+  Tartaglia, 4, 61
+  Taylor, I., 19, 30
+  Teca, 55, 61
+  Tennent, J. E., 75
+  Texada, 60
+  Theca, 58, 61
+  Theophanes, 64
+  Thibaut, G., 12, 13, 16, 44, 47
+  Tibetan numerals, 36
+  Timotheus, 103
+  Tonstall, C., 3, 61
+  Trenchant, 60
+  Treutlein, 5, 63, 123
+  Trevisa, 136
+  Treviso arithmetic, 145
+  Trivium and quadrivium, 73
+  Tsin, 56
+  Tunis, 65
+  Turchill, 88, 118, 123
+  Turnour, G., 75
+  Tziphra, 57, 62
+  [Greek: tziphra], 55, 57, 62
+  Tzwivel, 61, 118, 145
+
+  Ujjain, 32
+  Unger, 133
+  Upanishads, 12
+  Usk, 121
+
+  Valla, G., 61
+  Van der Schuere, 62
+  Var[=a]ha-Mihira, 39, 44, 78
+  V[=a]savadatt[=a], 44
+  Vaux, Carra de, 9, 74
+  Vaux, W. S. W., 91
+  Ved[=a][.n]gas, 17
+  Vedas, 12, 15, 17
+  Vergil, 80
+  Vincent, A. J. H., 57
+  Vogt, 13
+  Voizot, P., 36
+  Vossius, 4, 76, 81, 84
+
+  Wallis, 3, 62, 84, 116
+  Wappler, E., 54, 126
+  Waeschke, H., 2, 93
+  Wattenbach, 143
+  Weber, A., 31
+  Weidler, I. F., 34, 66
+  Weidler, I. F. and G. I., 63, 66
+  Weissenborn, 85, 110
+  Wertheim, G., 57, 61
+  Whitney, W. D., 13
+  Wilford, F., 75
+  Wilkens, 62
+  Wilkinson, J. G., 70
+  Willichius, 3
+  Woepcke, 3, 6, 42, 63, 64, 65, 67, 69, 70, 94, 113, 138
+  Wolack, G., 54
+  Woodruff, C. E., 32
+  Word and letter numerals, 38, 44
+  Wuestenfeld, 74
+
+  Yule, H., 107
+
+  Zephirum, 57, 58
+  Zephyr, 59
+  Zepiro, 58
+  Zero, 26, 38, 40, 43, 45, 49, 51-62, 67
+  Zeuero, 58
+
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+Notes
+
+al-Mekk[=i] on a treatise on [.g]ob[=a]r arithmetic (explained later)
+called _Al-murshidah_, found by Woepcke in Paris (_Propagation_, p. 66),
+there is mentioned the fact that there are "nine Indian figures" and "a
+second kind of Indian figures ... although these are the figures of the
+[.g]ob[=a]r writing." So in a commentary by [H.]osein ibn Mo[h.]ammed
+al-Ma[h.]all[=i] (died in 1756) on the _Mokhta[s.]ar f[=i]`ilm
+el-[h.]is[=a]b_ (Extract from Arithmetic) by `Abdalq[=a]dir ibn `Al[=i]
+al-Sakh[=a]w[=i] (died c. 1000) it is related that "the preface treats of
+the forms of the figures of Hindu signs, such as were established by the
+Hindu nation." [Woepcke, _Propagation_, p. 63.]]
+
+which, of course, are interpolations. An interesting example of a forgery
+in ecclesiastical matters is in the charter said to have been given by St.
+Patrick, granting indulgences to the benefactors of Glastonbury, dated "In
+nomine domini nostri Jhesu Christi Ego Patricius humilis servunculus Dei
+anno incarnationis ejusdem ccccxxx." Now if the Benedictines are right in
+saying that Dionysius Exiguus, a Scythian monk, first arranged the
+Christian chronology c. 532 A.D., this can hardly be other than spurious.
+See Arbuthnot, loc. cit., p. 38.
+
+[1] "_Discipulus._ Quis primus invenit numerum apud Hebraeos et AEgyptios?
+_Magister._ Abraham primus invenit numerum apud Hebraeos, deinde Moses; et
+Abraham tradidit istam scientiam numeri ad AEgyptios, et docuit eos: deinde
+Josephus." [Bede, _De computo dialogus_ (doubtfully assigned to him),
+_Opera omnia_, Paris, 1862, Vol. I, p. 650.]
+
+"Alii referunt ad Phoenices inventores arithmeticae, propter eandem
+commerciorum caussam: Alii ad Indos: Ioannes de Sacrobosco, cujus
+sepulchrum est Lutetiae in comitio Maturinensi, refert ad Arabes." [Ramus,
+_Arithmeticae libri dvo_, Basel, 1569, p. 112.]
+
+Similar notes are given by Peletarius in his commentary on the arithmetic
+of Gemma Frisius (1563 ed., fol. 77), and in his own work (1570 Lyons ed.,
+p. 14): "La valeur des Figures commence au coste dextre tirant vers le
+coste senestre: au rebours de notre maniere d'escrire par ce que la
+premiere prattique est venue des Chaldees: ou des Pheniciens, qui ont ete
+les premiers traffiquers de marchandise."
+
+[2] Maximus Planudes (c. 1330) states that "the nine symbols come from the
+Indians." [Waeschke's German translation, Halle, 1878, p. 3.] Willichius
+speaks of the "Zyphrae Indicae," in his _Arithmeticae libri tres_ (Strasburg,
+1540, p. 93), and Cataneo of "le noue figure de gli Indi," in his _Le
+pratiche delle dve prime mathematiche_ (Venice, 1546, fol. 1). Woepcke is
+not correct, therefore, in saying ("Memoire sur la propagation des chiffres
+indiens," hereafter referred to as _Propagation_ [_Journal Asiatique_, Vol.
+I (6), 1863, p. 34]) that Wallis (_A Treatise on Algebra, both historical
+and practical_, London, 1685, p. 13, and _De algebra tractatus_, Latin
+edition in his _Opera omnia_, 1693, Vol. II, p. 10) was one of the first to
+give the Hindu origin.
+
+[3] From the 1558 edition of _The Grovnd of Artes_, fol. C, 5. Similarly
+Bishop Tonstall writes: "Qui a Chaldeis primum in finitimos, deinde in
+omnes pene gentes fluxit.... Numerandi artem a Chaldeis esse profectam: qui
+dum scribunt, a dextra incipiunt, et in leuam progrediuntur." [_De arte
+supputandi_, London, 1522, fol. B, 3.] Gemma Frisius, the great continental
+rival of Recorde, had the same idea: "Primum autem appellamus dexterum
+locum, eo quod haec ars vel a Chaldaeis, vel ab Hebraeis ortum habere
+credatur, qui etiam eo ordine scribunt"; but this refers more evidently to
+the Arabic numerals. [_Arithmeticae practicae methodvs facilis_, Antwerp,
+1540, fol. 4 of the 1563 ed.] Sacrobosco (c. 1225) mentions the same thing.
+Even the modern Jewish writers claim that one of their scholars,
+M[=a]sh[=a]ll[=a]h (c. 800), introduced them to the Mohammedan world. [C.
+Levias, _The Jewish Encyclopedia_, New York, 1905, Vol. IX, p. 348.]
+
+[4] "... & que esto fu trouato di fare da gli Arabi con diece figure." [_La
+prima parte del general trattato di nvmeri, et misvre_, Venice, 1556, fol.
+9 of the 1592 edition.]
+
+[5] "Vom welchen Arabischen auch disz Kunst entsprungen ist." [_Ain nerv
+geordnet Rechenbiechlin_, Augsburg, 1514, fol. 13 of the 1531 edition. The
+printer used the letters _rv_ for _w_ in "new" in the first edition, as he
+had no _w_ of the proper font.]
+
+[6] Among them Glareanus: "Characteres simplices sunt nouem significatiui,
+ab Indis usque, siue Chaldaeis asciti .1.2.3.4.5.6.7.8.9. Est item unus .0
+circulus, qui nihil significat." [_De VI. Arithmeticae practicae
+speciebvs_, Paris, 1539, fol. 9 of the 1543 edition.]
+
+[7] "Barbarische oder gemeine Ziffern." [Anonymous, _Das Einmahl Eins cum
+notis variorum_, Dresden, 1703, p. 3.] So Vossius (_De universae matheseos
+natura et constitutione liber_, Amsterdam, 1650, p. 34) calls them
+"Barbaras numeri notas." The word at that time was possibly synonymous with
+Arabic.
+
+[8] His full name was `Ab[=u] `Abdall[=a]h Mo[h.]ammed ibn M[=u]s[=a]
+al-Khow[=a]razm[=i]. He was born in Khow[=a]rezm, "the lowlands," the
+country about the present Khiva and bordering on the Oxus, and lived at
+Bagdad under the caliph al-M[=a]m[=u]n. He died probably between 220 and
+230 of the Mohammedan era, that is, between 835 and 845 A.D., although some
+put the date as early as 812. The best account of this great scholar may be
+found in an article by C. Nallino, "Al-[H)]uw[=a]rizm[=i]" in the _Atti
+della R. Accad. dei Lincei_, Rome, 1896. See also _Verhandlungen des 5.
+Congresses der Orientalisten_, Berlin, 1882, Vol. II, p. 19; W. Spitta-Bey
+in the _Zeitschrift der deutschen Morgenlaend. Gesellschaft_, Vol. XXXIII,
+p. 224; Steinschneider in the _Zeitschrift der deutschen Morgenlaend.
+Gesellschaft_, Vol. L, p. 214; Treutlein in the _Abhandlungen zur
+Geschichte der Mathematik_, Vol. I, p. 5; Suter, "Die Mathematiker und
+Astronomen der Araber und ihre Werke," _Abhandlungen zur Geschichte der
+Mathematik_, Vol. X, Leipzig, 1900, p. 10, and "Nachtraege," in Vol. XIV, p.
+158; Cantor, _Geschichte der Mathematik_, Vol. I, 3d ed., pp. 712-733 etc.;
+F. Woepcke in _Propagation_, p. 489. So recently has he become known that
+Heilbronner, writing in 1742, merely mentions him as "Ben-Musa, inter
+Arabes celebris Geometra, scripsit de figuris planis & sphericis."
+[_Historia matheseos universae_, Leipzig, 1742, p. 438.]
+
+In this work most of the Arabic names will be transliterated substantially
+as laid down by Suter in his work _Die Mathematiker_ etc., except where
+this violates English pronunciation. The scheme of pronunciation of
+oriental names is set forth in the preface.
+
+[9] Our word _algebra_ is from the title of one of his works, Al-jabr
+wa'l-muq[=a]balah, Completion and Comparison. The work was translated into
+English by F. Rosen, London, 1831, and treated in _L'Algebre
+d'al-Kh[=a]rizmi et les methodes indienne et grecque_, Leon Rodet, Paris,
+1878, extract from the _Journal Asiatique_. For the derivation of the word
+_algebra_, see Cossali, _Scritti Inediti_, pp. 381-383, Rome, 1857;
+Leonardo's _Liber Abbaci_ (1202), p. 410, Rome, 1857; both published by B.
+Boncompagni. "Almuchabala" also was used as a name for algebra.
+
+[10] This learned scholar, teacher of O'Creat who wrote the _Helceph_
+("_Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum_"),
+studied in Toledo, learned Arabic, traveled as far east as Egypt, and
+brought from the Levant numerous manuscripts for study and translation. See
+Henry in the _Abhandlungen zur Geschichte der Mathematik_, Vol. III, p.
+131; Woepcke in _Propagation_, p. 518.
+
+[11] The title is _Algoritmi de numero Indorum_. That he did not make this
+translation is asserted by Enestroem in the _Bibliotheca Mathematica_, Vol.
+I (3), p. 520.
+
+[12] Thus he speaks "de numero indorum per .IX. literas," and proceeds:
+"Dixit algoritmi: Cum uidissem yndos constituisse .IX. literas in uniuerso
+numero suo, propter dispositionem suam quam posuerunt, uolui patefacere de
+opera quod fit per eas aliquid quod esset leuius discentibus, si deus
+uoluerit." [Boncompagni, _Trattati d'Aritmetica_,  Rome, 1857.] Discussed
+by F. Woepcke, _Sur l'introduction de l'arithmetique indienne en Occident_,
+Rome, 1859.
+
+[13] Thus in a commentary by `Al[=i] ibn Ab[=i] Bekr ibn al-Jam[=a]l
+al-An[s.][=a]r[=i
+
+[14] See also Woepcke, _Propagation_, p. 505. The origin is discussed at
+much length by G. R. Kaye, "Notes on Indian Mathematics.--Arithmetical
+Notation," _Journ. and Proc. of the Asiatic Soc. of Bengal_, Vol. III,
+1907, p. 489.
+
+[15] _Alberuni's India_, Arabic version, London, 1887; English translation,
+ibid., 1888.
+
+[16] _Chronology of Ancient Nations_, London, 1879. Arabic and English
+versions, by C. E. Sachau.
+
+[17] _India_, Vol. I, chap. xvi.
+
+[18] The Hindu name for the symbols of the decimal place system.
+
+[19] Sachau's English edition of the _Chronology_, p. 64.
+
+[20] _Litterature arabe_, Cl. Huart, Paris, 1902.
+
+[21] Huart, _History of Arabic Literature_, English ed., New York, 1903, p.
+182 seq.
+
+[22] Al-Mas`[=u]d[=i]'s _Meadows of Gold_, translated in part by Aloys
+Sprenger, London, 1841; _Les prairies d'or_, trad. par C. Barbier de
+Meynard et Pavet de Courteille, Vols. I to IX, Paris, 1861-1877.
+
+[23] _Les prairies d'or_, Vol. VIII, p. 289 seq.
+
+[24] _Essays_, Vol. II, p. 428.
+
+[25] Loc. cit., p. 504.
+
+[26] _Materiaux pour servir a l'histoire comparee des sciences
+mathematiques chez les Grecs et les Orientaux_, 2 vols., Paris, 1845-1849,
+pp. 438-439.
+
+[27] He made an exception, however, in favor of the numerals, loc. cit.,
+Vol. II, p. 503.
+
+[28] _Bibliotheca Arabico-Hispana Escurialensis_, Madrid, 1760-1770, pp.
+426-427.
+
+[29] The author, Ibn al-Qif[t.][=i], flourished A.D. 1198 [Colebrooke, loc.
+cit., note Vol. II, p. 510].
+
+[30] "Liber Artis Logisticae a Mohamado Ben Musa _Alkhuarezmita_ exornatus,
+qui ceteros omnes brevitate methodi ac facilitate praestat, Indorum que in
+praeclarissimis inventis ingenium & acumen ostendit." [Casiri, loc. cit.,
+p. 427.]
+
+[31] Macoudi, _Le livre de l'avertissement et de la revision_. Translation
+by B. Carra de Vaux, Paris, 1896.
+
+[32] Verifying the hypothesis of Woepcke, _Propagation_, that the Sindhind
+included a treatment of arithmetic.
+
+[33] A[h.]med ibn `Abdall[=a]h, Suter, _Die Mathematiker_, etc., p. 12.
+
+[34] _India_, Vol. II, p. 15.
+
+[35] See H. Suter, "Das Mathematiker-Verzeichniss im Fihrist,"
+_Abhandlungen zur Geschichte der Mathematik_, Vol. VI, Leipzig, 1892. For
+further references to early Arabic writers the reader is referred to H.
+Suter, _Die Mathematiker und Astronomen der Araber und ihre Werke_. Also
+"Nachtraege und Berichtigungen" to the same (_Abhandlungen_,  Vol. XIV,
+1902, pp. 155-186).
+
+[36] Suter, loc. cit., note 165, pp. 62-63.
+
+[37] "Send Ben Ali,... tum arithmetica scripta maxime celebrata, quae
+publici juris fecit." [Loc. cit., p. 440.]
+
+[38] _Scritti di Leonardo Pisano_, Vol. I, _Liber Abbaci_ (1857); Vol. II,
+_Scritti_ (1862); published by Baldassarre Boncompagni, Rome. Also _Tre
+Scritti Inediti_, and _Intorno ad Opere di Leonardo Pisano_, Rome, 1854.
+
+[39] "Ubi ex mirabili magisterio in arte per novem figuras indorum
+introductus" etc. In another place, as a heading to a separate division, he
+writes, "De cognitione novem figurarum yndorum" etc. "Novem figure indorum
+he sunt 9 8 7 6 5 4 3 2 1."
+
+[40] See _An Ancient English Algorism_, by David Eugene Smith, in
+_Festschrift Moritz Cantor_, Leipzig, 1909. See also Victor Mortet, "Le
+plus ancien traite francais d'algorisme," _Bibliotheca Mathematica_, Vol.
+IX (3), pp. 55-64.
+
+[41] These are the two opening lines of the _Carmen de Algorismo_ that the
+anonymous author is explaining. They should read as follows:
+
+  Haec algorismus ars praesens dicitur, in qua
+  Talibus Indorum fruimur bis quinque figuris.
+
+What follows is the translation.
+
+[42] Thibaut, _Astronomie, Astrologie und Mathematik_, Strassburg, 1899.
+
+[43] Gustave Schlegel, _Uranographie chinoise ou preuves directes que
+l'astronomie primitive est originaire de la Chine, et qu'elle a ete
+empruntee par les anciens peuples occidentaux a la sphere chinoise; ouvrage
+accompagne d'un atlas celeste chinois et grec_, The Hague and Leyden, 1875.
+
+[44] E. W. Hopkins, _The Religions of India_, Boston, 1898, p. 7.
+
+[45] R. C. Dutt, _History of India_, London, 1906.
+
+[46] W. D. Whitney, _Sanskrit Grammar_, 3d ed., Leipzig, 1896.
+
+[47] "Das [=A]pastamba-['S]ulba-S[=u]tra," _Zeitschrift der deutschen
+Morgenlaendischen Gesellschaft_, Vol. LV, p. 543, and Vol. LVI, p. 327.
+
+[48] _Geschichte der Math._, Vol. I, 2d ed., p. 595.
+
+[49] L. von Schroeder, _Pythagoras und die Inder_, Leipzig, 1884; H. Vogt,
+"Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale
+gekannt?" _Bibliotheca Mathematica_, Vol. VII (3), pp. 6-20; A. Buerk, loc.
+cit.; Max Simon, _Geschichte der Mathematik im Altertum_, Berlin, 1909, pp.
+137-165; three S[=u]tras are translated in part by Thibaut, _Journal of the
+Asiatic Society of Bengal_, 1875, and one appeared in _The Pandit_, 1875;
+Beppo Levi, "Osservazioni e congetture sopra la geometria degli indiani,"
+_Bibliotheca Mathematica_, Vol. IX (3), 1908, pp. 97-105.
+
+[50] Loc. cit.; also _Indiens Literatur und Cultur_, Leipzig, 1887.
+
+[51] It is generally agreed that the name of the river Sindhu, corrupted by
+western peoples to Hindhu, Indos, Indus, is the root of Hindustan and of
+India. Reclus, _Asia_, English ed., Vol. III, p. 14.
+
+[52] See the comments of Oppert, _On the Original Inhabitants of
+Bharatavar[s.]a or India_, London, 1893, p. 1.
+
+[53] A. Hillebrandt, _Alt-Indien_, Breslau, 1899, p. 111. Fragmentary
+records relate that Kh[=a]ravela, king of Kali[.n]ga, learned as a boy
+_lekh[=a]_ (writing), _ga[n.]an[=a]_ (reckoning), and _r[=u]pa_ (arithmetic
+applied to monetary affairs and mensuration), probably in the 5th century
+B.C. [Buehler, _Indische Palaeographie_, Strassburg, 1896, p. 5.]
+
+[54] R. C. Dutt, _A History of Civilization in Ancient India_, London,
+1893, Vol. I, p. 174.
+
+[55] The Buddha. The date of his birth is uncertain. Sir Edwin Arnold put
+it c. 620 B.C.
+
+[56] I.e. 100.10^7.
+
+[57] There is some uncertainty about this limit.
+
+[58] This problem deserves more study than has yet been given it. A
+beginning may be made with Comte Goblet d'Alviella, _Ce que l'Inde doit a
+la Grece_, Paris, 1897, and H. G. Keene's review, "The Greeks in India," in
+the _Calcutta Review_, Vol. CXIV, 1902, p. 1. See also F. Woepeke,
+_Propagation_, p. 253; G. R. Kaye, loc. cit., p. 475 seq., and "The Source
+of Hindu Mathematics," _Journal of the Royal Asiatic Society_, July, 1910,
+pp. 749-760; G. Thibaut, _Astronomie, Astrologie und Mathematik_, pp. 43-50
+and 76-79. It will be discussed more fully in Chapter VI.
+
+[59] I.e. to 100,000. The lakh is still the common large unit in India,
+like the myriad in ancient Greece and the million in the West.
+
+[60] This again suggests the _Psammites_, or _De harenae numero_ as it is
+called in the 1544 edition of the _Opera_ of Archimedes, a work in which
+the great Syracusan proposes to show to the king "by geometric proofs which
+you can follow, that the numbers which have been named by us ... are
+sufficient to exceed not only the number of a sand-heap as large as the
+whole earth, but one as large as the universe." For a list of early
+editions of this work see D. E. Smith, _Rara Arithmetica_, Boston, 1909, p.
+227.
+
+[61] I.e. the Wise.
+
+[62] Sir Monier Monier-Williams, _Indian Wisdom_, 4th ed., London, 1893,
+pp. 144, 177. See also J. C. Marshman, _Abridgment of the History of
+India_, London, 1893, p. 2.
+
+[63] For a list and for some description of these works see R. C. Dutt, _A
+History of Civilization in Ancient India_, Vol. II, p. 121.
+
+[64] Professor Ramkrishna Gopal Bhandarkar fixes the date as the fifth
+century B.C. ["Consideration of the Date of the Mah[=a]bh[=a]rata," in the
+_Journal of the Bombay Branch of the R. A. Soc._, Bombay, 1873, Vol. X, p.
+2.].
+
+[65] Marshman, loc. cit., p. 2.
+
+[66] A. C. Burnell, _South Indian Palaeography_, 2d ed., London, 1878, p. 1,
+seq.
+
+[67] This extensive subject of palpable arithmetic, essentially the history
+of the abacus, deserves to be treated in a work by itself.
+
+[68] The following are the leading sources of information upon this
+subject: G. Buehler, _Indische Palaeographie_, particularly chap. vi; A. C.
+Burnell, _South Indian Palaeography_, 2d ed., London, 1878, where tables of
+the various Indian numerals are given in Plate XXIII; E. C. Bayley, "On the
+Genealogy of Modern Numerals," _Journal of the Royal Asiatic Society_, Vol.
+XIV, part 3, and Vol. XV, part 1, and reprint, London, 1882; I. Taylor, in
+_The Academy_, January 28, 1882, with a repetition of his argument in his
+work _The Alphabet_, London, 1883, Vol. II, p. 265, based on Bayley; G. R.
+Kaye, loc. cit., in some respects one of the most critical articles thus
+far published; J. C. Fleet, _Corpus inscriptionum Indicarum_, London, 1888,
+Vol. III, with facsimiles of many Indian inscriptions, and _Indian
+Epigraphy_, Oxford, 1907, reprinted from the _Imperial Gazetteer of India_,
+Vol. II, pp. 1-88, 1907; G. Thibaut, loc. cit., _Astronomie_ etc.; R.
+Caldwell, _Comparative Grammar of the Dravidian Languages_, London, 1856,
+p. 262 seq.; and _Epigraphia Indica_ (official publication of the
+government of India), Vols. I-IX. Another work of Buehler's, _On the Origin
+of the Indian Br[=a]hma Alphabet_, is also of value.
+
+[69] The earliest work on the subject was by James Prinsep, "On the
+Inscriptions of Piyadasi or A['s]oka," etc., _Journal of the Asiatic
+Society of Bengal_, 1838, following a preliminary suggestion in the same
+journal in 1837. See also "A['s]oka Notes," by V. A. Smith, _The Indian
+Antiquary_, Vol. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, pp. 151-159, June,
+1909; _The Early History of India_, 2d ed., Oxford, 1908, p. 154; J. F.
+Fleet, "The Last Words of A['s]oka," _Journal of the Royal Asiatic
+Society_, October, 1909, pp. 981-1016; E. Senart, _Les inscriptions de
+Piyadasi_, 2 vols., Paris, 1887.
+
+[70] For a discussion of the minor details of this system, see Buehler, loc.
+cit., p. 73.
+
+[71] Julius Euting, _Nabataeische Inschriften aus Arabien_, Berlin, 1885,
+pp. 96-97, with a table of numerals.
+
+[72] For the five principal theories see Buehler, loc. cit., p. 10.
+
+[73] Bayley, loc. cit., reprint p. 3.
+
+[74] Buehler, loc. cit.; _Epigraphia Indica_, Vol. III, p. 134; _Indian
+Antiquary_, Vol. VI, p. 155 seq., and Vol. X, p. 107.
+
+[75] Pandit Bhagav[=a]nl[=a]l Indr[=a]j[=i], "On Ancient N[=a]g[=a]ri
+Numeration; from an Inscription at N[=a]negh[=a]t," _Journal of the Bombay
+Branch of the Royal Asiatic Society_, 1876, Vol. XII, p. 404.
+
+[76] Ib., p. 405. He gives also a plate and an interpretation of each
+numeral.
+
+[77] These may be compared with Buehler's drawings, loc. cit.; with Bayley,
+loc. cit., p. 337 and plates; and with Bayley's article in the
+_Encyclopaedia Britannica_, 9th ed., art. "Numerals."
+
+[78] E. Senart, "The Inscriptions in the Caves at Nasik," _Epigraphia
+Indica_, Vol. VIII, pp. 59-96; "The Inscriptions in the Cave at Karle,"
+_Epigraphia Indica_, Vol. VII, pp. 47-74; Buehler, _Palaeographie_, Tafel
+IX.
+
+[79] See Fleet, loc. cit. See also T. Benfey, _Sanskrit Grammar_, London,
+1863, p. 217; M. R. Kale, _Higher Sanskrit Grammar_, 2d ed., Bombay, 1898,
+p. 110, and other authorities as cited.
+
+[80] Kharo[s.][t.]h[=i] numerals, A['s]oka inscriptions, c. 250 B.C.
+Senart, _Notes d'epigraphie indienne_. Given by Buehler, loc. cit., Tafel I.
+
+[81] Same, ['S]aka inscriptions, probably of the first century B.C. Senart,
+loc. cit.; Buehler, loc. cit.
+
+[82] Br[=a]hm[=i] numerals, A['s]oka inscriptions, c. 250 B.C. _Indian
+Antiquary_, Vol. VI, p. 155 seq.
+
+[83] Same, N[=a]n[=a] Gh[=a]t inscriptions, c. 150 B.C. Bhagav[=a]nl[=a]l
+Indr[=a]j[=i], _On Ancient N[=a]gar[=i] Numeration_, loc. cit. Copied from
+a squeeze of the original.
+
+[84] Same, Nasik inscription, c. 100 B.C. Burgess, _Archeological Survey
+Report, Western India_; Senart, _Epigraphia Indica_, Vol. VII, pp. 47-79,
+and Vol. VIII, pp. 59-96.
+
+[85] K[s.]atrapa coins, c. 200 A.D. _Journal of the Royal Asiatic Society_,
+1890, p. 639.
+
+[86] Ku[s.]ana inscriptions, c. 150 A.D. _Epigraphia Indica_, Vol. I, p.
+381, and Vol. II, p. 201.
+
+[87] Gupta Inscriptions, c. 300 A.D. to 450 A.D. Fleet, loc. cit., Vol.
+III.
+
+[88] Valhab[=i], c. 600 A.D. _Corpus_, Vol. III.
+
+[89] Bendall's Table of Numerals, in _Cat. Sansk. Budd. MSS._, British
+Museum.
+
+[90] _Indian Antiquary_, Vol. XIII, 120; _Epigraphia Indica_, Vol. III, 127
+ff.
+
+[91] Fleet, loc. cit.
+
+[92] Bayley, loc. cit., p. 335.
+
+[93] From a copper plate of 493 A.D., found at K[=a]r[=i]tal[=a][=i],
+Central India. [Fleet, loc. cit., Plate XVI.] It should be stated, however,
+that many of these copper plates, being deeds of property, have forged
+dates so as to give the appearance of antiquity of title. On the other
+hand, as Colebrooke long ago pointed out, a successful forgery has to
+imitate the writing of the period in question, so that it becomes evidence
+well worth considering, as shown in Chapter III.
+
+[94] From a copper plate of 510 A.D., found at Majhgaw[=a]in, Central
+India. [Fleet, loc. cit., Plate XIV.]
+
+[95] From an inscription of 588 A.D., found at B[=o]dh-Gay[=a], Bengal
+Presidency. [Fleet, loc. cit., Plate XXIV.]
+
+[96] From a copper plate of 571 A.D., found at M[=a]liy[=a], Bombay
+Presidency. [Fleet, loc. cit., Plate XXIV.]
+
+[97] From a Bijayaga[d.]h pillar inscription of 372 A.D. [Fleet, loc. cit.,
+Plate XXXVI, C.]
+
+[98] From a copper plate of 434 A.D. [_Indian Antiquary_, Vol. I, p. 60.]
+
+[99] Gadhwa inscription, c. 417 A.D. [Fleet, loc. cit., Plate IV, D.]
+
+[100] K[=a]r[=i]tal[=a][=i] plate of 493 A.D., referred to above.
+
+[101] It seems evident that the Chinese four, curiously enough called
+"eight in the mouth," is only a cursive [4 vertical strokes].
+
+[102] Chalfont, F. H., _Memoirs of the Carnegie Museum_, Vol. IV, no. 1; J.
+Hager, _An Explanation of the Elementary Characters of the Chinese_,
+London, 1801.
+
+[103] H. V. Hilprecht, _Mathematical, Metrological and Chronological
+Tablets from the Temple Library at Nippur_, Vol. XX, part I, of Series A,
+Cuneiform Texts Published by the Babylonian Expedition of the University of
+Pennsylvania, 1906; A. Eisenlohr, _Ein altbabylonischer Felderplan_,
+Leipzig, 1906; Maspero, _Dawn of Civilization_, p. 773.
+
+[104] Sir H. H. Howard, "On the Earliest Inscriptions from Chaldea,"
+_Proceedings of the Society of Biblical Archaeology_, XXI, p. 301, London,
+1899.
+
+[105] For a bibliography of the principal hypotheses of this nature see
+Buehler, loc. cit., p. 77. Buehler (p. 78) feels that of all these hypotheses
+that which connects the Br[=a]hm[=i] with the Egyptian numerals is the most
+plausible, although he does not adduce any convincing proof. Th. Henri
+Martin, "Les signes numeraux et l'arithmetique chez les peuples de
+l'antiquite et du moyen age" (being an examination of Cantor's
+_Mathematische Beitraege zum Culturleben der Voelker_), _Annali di matematica
+pura ed applicata_, Vol. V, Rome, 1864, pp. 8, 70. Also, same author,
+"Recherches nouvelles sur l'origine de notre systeme de numeration ecrite,"
+_Revue Archeologique_, 1857, pp. 36, 55. See also the tables given later in
+this work.
+
+[106] _Journal of the Royal Asiatic Society, Bombay Branch_, Vol. XXIII.
+
+[107] Loc. cit., reprint, Part I, pp. 12, 17. Bayley's deductions are
+generally regarded as unwarranted.
+
+[108] _The Alphabet_; London, 1883, Vol. II, pp. 265, 266, and _The
+Academy_ of Jan. 28, 1882.
+
+[109] Taylor, _The Alphabet_, loc. cit., table on p. 266.
+
+[110] Buehler, _On the Origin of the Indian Br[=a]hma Alphabet_, Strassburg,
+1898, footnote, pp. 52, 53.
+
+[111] Albrecht Weber, _History of Indian Literature_, English ed., Boston,
+1878, p. 256: "The Indian figures from 1-9 are abbreviated forms of the
+initial letters of the numerals themselves...: the zero, too, has arisen
+out of the first letter of the word _[s.]unya_ (empty) (it occurs even in
+Pingala). It is the decimal place value of these figures which gives them
+significance." C. Henry, "Sur l'origine de quelques notations
+mathematiques," _Revue Archeologique_, June and July, 1879, attempts to
+derive the Boethian forms from the initials of Latin words. See also J.
+Prinsep, "Examination of the Inscriptions from Girnar in Gujerat, and
+Dhauli in Cuttach," _Journal of the Asiatic Society of Bengal_, 1838,
+especially Plate XX, p. 348; this was the first work on the subject.
+
+[112] Buehler, _Palaeographie_, p. 75, gives the list, with the list of
+letters (p. 76) corresponding to the number symbols.
+
+[113] For a general discussion of the connection between the numerals and
+the different kinds of alphabets, see the articles by U. Ceretti, "Sulla
+origine delle cifre numerali moderne," _Rivista di fisica, matematica e
+scienze naturali_, Pisa and Pavia, 1909, anno X, numbers 114, 118, 119, and
+120, and continuation in 1910.
+
+[114] This is one of Buehler's hypotheses. See Bayley, loc. cit., reprint p.
+4; a good bibliography of original sources is given in this work, p. 38.
+
+[115] Loc. cit., reprint, part I, pp. 12, 17. See also Burnell, loc. cit.,
+p. 64, and tables in plate XXIII.
+
+[116] This was asserted by G. Hager (_Memoria sulle cifre arabiche_, Milan,
+1813, also published in _Fundgruben des Orients_, Vienna, 1811, and in
+_Bibliotheque Britannique_, Geneva, 1812). See also the recent article by
+Major Charles E. Woodruff, "The Evolution of Modern Numerals from Tally
+Marks," _American Mathematical Monthly_, August-September, 1909.
+Biernatzki, "Die Arithmetik der Chinesen," _Crelle's Journal fuer die reine
+und angewandte Mathematik_, Vol. LII, 1857, pp. 59-96, also asserts the
+priority of the Chinese claim for a place system and the zero, but upon the
+flimsiest authority. Ch. de Paravey, _Essai sur l'origine unique et
+hieroglyphique des chiffres et des lettres de tous les peuples_, Paris,
+1826; G. Kleinwaechter, "The Origin of the Arabic Numerals," _China Review_,
+Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp. 28-30; Biot, "Note sur la
+connaissance que les Chinois ont eue de la valeur de position des
+chiffres," _Journal Asiatique_, 1839, pp. 497-502. A. Terrien de
+Lacouperie, "The Old Numerals, the Counting-Rods and the Swan-Pan in
+China," _Numismatic Chronicle_, Vol. III (3), pp. 297-340, and Crowder B.
+Moseley, "Numeral Characters: Theory of Origin and Development," _American
+Antiquarian_, Vol. XXII, pp. 279-284, both propose to derive our numerals
+from Chinese characters, in much the same way as is done by Major Woodruff,
+in the article above cited.
+
+[117] The Greeks, probably following the Semitic custom, used nine letters
+of the alphabet for the numerals from 1 to 9, then nine others for 10 to
+90, and further letters to represent 100 to 900. As the ordinary Greek
+alphabet was insufficient, containing only twenty-four letters, an alphabet
+of twenty-seven letters was used.
+
+[118] _Institutiones mathematicae_, 2 vols., Strassburg, 1593-1596, a
+somewhat rare work from which the following quotation is taken:
+
+"_Quis est harum Cyphrarum autor?_
+
+"A quibus hae usitatae syphrarum notae sint inventae: hactenus incertum
+fuit: meo tamen iudicio, quod exiguum esse fateor: a graecis librarijs
+(quorum olim magna fuit copia) literae Graecorum quibus veteres Graeci
+tamquam numerorum notis sunt usi: fuerunt corruptae. vt ex his licet
+videre.
+
+"Graecorum Literae corruptae.
+
+[Illustration]
+
+_"Sed qua ratione graecorum literae ita fuerunt corruptae?_
+
+"Finxerunt has corruptas Graecorum literarum notas: vel abiectione vt in
+nota binarij numeri, vel additione vt in ternarij, vel inuersione vt in
+septenarij, numeri nota, nostrae notae, quibus hodie utimur: ab his sola
+differunt elegantia, vt apparet."
+
+See also Bayer, _Historia regni Graecorum Bactriani_, St. Petersburg, 1788,
+pp. 129-130, quoted by Martin, _Recherches nouvelles_, etc., loc. cit.
+
+[119] P. D. Huet, _Demonstratio evangelica_, Paris, 1769, note to p. 139 on
+p. 647: "Ab Arabibus vel ab Indis inventas esse, non vulgus eruditorum
+modo, sed doctissimi quique ad hanc diem arbitrati sunt. Ego vero falsum id
+esse, merosque esse Graecorum characteres aio; a librariis Graecae linguae
+ignaris interpolatos, et diuturna scribendi consuetudine corruptos. Nam
+primum 1 apex fuit, seu virgula, nota [Greek: monados]. 2, est ipsum [beta]
+extremis suis truncatum. [gamma], si in sinistram partem inclinaveris &
+cauda mutilaveris & sinistrum cornu sinistrorsum flexeris, fiet 3. Res ipsa
+loquitur 4 ipsissimum esse [Delta], cujus crus sinistrum erigitur [Greek:
+kata katheton], & infra basim descendit; basis vero ipsa ultra crus
+producta eminet. Vides quam 5 simile sit [Greek: toi] [epsilon]; infimo
+tantum semicirculo, qui sinistrorsum patebat, dextrorsum converso. [Greek:
+episemon bau] quod ita notabatur [digamma], rotundato ventre, pede
+detracto, peperit [Greek: to] 6. Ex [Zeta] basi sua mutilato, ortum est
+[Greek: to] 7. Si [Eta] inflexis introrsum apicibus in rotundiorem &
+commodiorem formam mutaveris, exurget [Greek: to] 8. At 9 ipsissimum est
+[alt theta]."
+
+I. Weidler, _Spicilegium observationum ad historiam notarum numeralium_,
+Wittenberg, 1755, derives them from the Hebrew letters; Dom Augustin
+Calmet, "Recherches sur l'origine des chiffres d'arithmetique," _Memoires
+pour l'histoire des sciences et des beaux arts_, Trevoux, 1707 (pp.
+1620-1635, with two plates), derives the current symbols from the Romans,
+stating that they are relics of the ancient "Notae Tironianae." These
+"notes" were part of a system of shorthand invented, or at least perfected,
+by Tiro, a slave who was freed by Cicero. L. A. Sedillot, "Sur l'origine de
+nos chiffres," _Atti dell' Accademia pontificia dei nuovi Lincei_, Vol.
+XVIII, 1864-1865, pp. 316-322, derives the Arabic forms from the Roman
+numerals.
+
+[120] Athanasius Kircher, _Arithmologia sive De abditis Numerorum,
+mysterijs qua origo, antiquitas & fabrica Numerorum exponitur_, Rome, 1665.
+
+[121] See Suter, _Die Mathematiker und Astronomen der Araber_, p. 100.
+
+[122] "Et hi numeri sunt numeri Indiani, a Brachmanis Indiae Sapientibus ex
+figura circuli secti inuenti."
+
+[123] V. A. Smith, _The Early History of India_, Oxford, 2d ed., 1908, p.
+333.
+
+[124] C. J. Ball, "An Inscribed Limestone Tablet from Sippara,"
+_Proceedings of the Society of Biblical Archaeology_, Vol. XX, p. 25
+(London, 1898). Terrien de Lacouperie states that the Chinese used the
+circle for 10 before the beginning of the Christian era. [_Catalogue of
+Chinese Coins_, London, 1892, p. xl.]
+
+[125] For a purely fanciful derivation from the corresponding number of
+strokes, see W. W. R. Ball, _A Short Account of the History of
+Mathematics_, 1st ed., London, 1888, p. 147; similarly J. B. Reveillaud,
+_Essai sur les chiffres arabes_, Paris, 1883; P. Voizot, "Les chiffres
+arabes et leur origine," _La Nature_, 1899, p. 222; G. Dumesnil, "De la
+forme des chiffres usuels," _Annales de l'universite de Grenoble_, 1907,
+Vol. XIX, pp. 657-674, also a note in _Revue Archeologique_, 1890, Vol. XVI
+(3), pp. 342-348; one of the earliest references to a possible derivation
+from points is in a work by Bettino entitled _Apiaria universae
+philosophiae mathematicae in quibus paradoxa et noua machinamenta ad usus
+eximios traducta, et facillimis demonstrationibus confirmata_, Bologna,
+1545, Vol. II, Apiarium XI, p. 5.
+
+[126] _Alphabetum Barmanum_, Romae, MDCCLXXVI, p. 50. The 1 is evidently
+Sanskrit, and the 4, 7, and possibly 9 are from India.
+
+[127] _Alphabetum Grandonico-Malabaricum_, Romae, MDCCLXXII, p. 90. The
+zero is not used, but the symbols for 10, 100, and so on, are joined to the
+units to make the higher numbers.
+
+[128] _Alphabetum Tangutanum_, Romae, MDCCLXXIII, p. 107. In a Tibetan MS.
+in the library of Professor Smith, probably of the eighteenth century,
+substantially these forms are given.
+
+[129] Bayley, loc. cit., plate II. Similar forms to these here shown, and
+numerous other forms found in India, as well as those of other oriental
+countries, are given by A. P. Pihan, _Expose des signes de numeration
+usites chez les peuples orientaux anciens et modernes_, Paris, 1860.
+
+[130] Buehler, loc. cit., p. 80; J. F. Fleet, _Corpus inscriptionum
+Indicarum_, Vol. III, Calcutta, 1888. Lists of such words are given also by
+Al-B[=i]r[=u]n[=i] in his work _India_; by Burnell, loc. cit.; by E.
+Jacquet, "Mode d'expression symbolique des nombres employe par les Indiens,
+les Tibetains et les Javanais," _Journal Asiatique_, Vol. XVI, Paris, 1835.
+
+[131] This date is given by Fleet, loc. cit., Vol. III, p. 73, as the
+earliest epigraphical instance of this usage in India proper.
+
+[132] Weber, _Indische Studien_, Vol. VIII, p. 166 seq.
+
+[133] _Journal of the Royal Asiatic Society_, Vol. I (N.S.), p. 407.
+
+[134] VIII, 20, 21.
+
+[135] Th. H. Martin, _Les signes numeraux_ ..., Rome, 1864; Lassen,
+_Indische Alterthumskunde_, Vol. II, 2d ed., Leipzig and London, 1874, p.
+1153.
+
+[136] But see Burnell, loc. cit., and Thibaut, _Astronomie, Astrologie und
+Mathematik_, p. 71.
+
+[137] A. Barth, "Inscriptions Sanscrites du Cambodge," in the _Notices et
+extraits des Mss. de la Bibliotheque nationale_, Vol. XXVII, Part I, pp.
+1-180, 1885; see also numerous articles in _Journal Asiatique_, by
+Aymonier.
+
+[138] Buehler, loc. cit., p. 82.
+
+[139] Loc. cit., p. 79.
+
+[140] Buehler, loc. cit., p. 83. The Hindu astrologers still use an
+alphabetical system of numerals. [Burnell, loc. cit., p. 79.]
+
+[141] Well could Ramus say, "Quicunq; autem fuerit inventor decem notarum
+laudem magnam meruit."
+
+[142] Al-B[=i]r[=u]n[=i] gives lists.
+
+[143] _Propagation_, loc. cit., p. 443.
+
+[144] See the quotation from _The Light of Asia_ in Chapter II, p. 16.
+
+[145] The nine ciphers were called _a[.n]ka_.
+
+[146] "Zur Geschichte des indischen Ziffernsystems," _Zeitschrift fuer die
+Kunde des Morgenlandes_, Vol. IV, 1842, pp. 74-83.
+
+[147] It is found in the Bakh[s.][=a]l[=i] MS. of an elementary arithmetic
+which Hoernle placed, at first, about the beginning of our era, but the
+date is much in question. G. Thibaut, loc. cit., places it between 700 and
+900 A.D.; Cantor places the body of the work about the third or fourth
+century A.D., _Geschichte der Mathematik_, Vol. I (3), p. 598.
+
+[148] For the opposite side of the case see G. R. Kaye, "Notes on Indian
+Mathematics, No. 2.--[=A]ryabha[t.]a," _Journ. and Proc. of the Asiatic
+Soc. of Bengal_, Vol. IV, 1908, pp. 111-141.
+
+[149] He used one of the alphabetic systems explained above. This ran up to
+10^{18} and was not difficult, beginning as follows:
+
+[Illustration]
+
+the same letter (_ka_) appearing in the successive consonant forms, _ka_,
+_kha_, _ga_, _gha_, etc. See C. I. Gerhardt, _Ueber die Entstehung und
+Ausbreitung des dekadischen Zahlensystems_, Programm, p. 17, Salzwedel,
+1853, and _Etudes historiques sur l'arithmetique de position_, Programm, p.
+24, Berlin, 1856; E. Jacquet, _Mode d'expression symbolique des nombres_,
+loc. cit., p. 97; L. Rodet, "Sur la veritable signification de la notation
+numerique inventee par [=A]ryabhata," _Journal Asiatique_, Vol. XVI (7),
+pp. 440-485. On the two [=A]ryabha[t.]as see Kaye, _Bibl. Math._, Vol. X
+(3), p. 289.
+
+[150] Using _kha_, a synonym of _['s][=u]nya_. [Bayley, loc. cit., p. 22,
+and L. Rodet, _Journal Asiatique_, Vol. XVI (7), p. 443.]
+
+[151] Var[=a]ha-Mihira, _Pancasiddh[=a]ntik[=a]_, translated by G. Thibaut
+and M. S. Dvived[=i], Benares, 1889; see Buehler, loc. cit., p. 78; Bayley,
+loc. cit., p. 23.
+
+[152] _B[r.]hat Sa[m.]hit[=a]_, translated by Kern, _Journal of the Royal
+Asiatic Society_, 1870-1875.
+
+[153] It is stated by Buehler in a personal letter to Bayley (loc. cit., p.
+65) that there are hundreds of instances of this usage in the _B[r.]hat
+Sa[m.]hit[=a]_. The system was also used in the _Pancasiddh[=a]ntik[=a]_ as
+early as 505 A.D. [Buehler, _Palaeographie_, p. 80, and Fleet, _Journal of
+the Royal Asiatic Society_, 1910, p. 819.]
+
+[154] Cantor, _Geschichte der Mathematik_, Vol. I (3), p. 608.
+
+[155] Buehler, loc. cit., p. 78.
+
+[156] Bayley, p. 38.
+
+[157] Noviomagus, in his _De numeris libri duo_, Paris, 1539, confesses his
+ignorance as to the origin of the zero, but says: "D. Henricus Grauius, vir
+Graece & Hebraice exime doctus, Hebraicam originem ostendit," adding that
+Valla "Indis Orientalibus gentibus inventionem tribuit."
+
+[158] See _Essays_, Vol. II, pp. 287 and 288.
+
+[159] Vol. XXX, p. 205 seqq.
+
+[160] Loc. cit., p. 284 seqq.
+
+[161] Colebrooke, loc. cit., p. 288.
+
+[162] Loc. cit., p. 78.
+
+[163] Hereafter, unless expressly stated to the contrary, we shall use the
+word "numerals" to mean numerals with place value.
+
+[164] "The Gurjaras of R[=a]jput[=a]na and Kanauj," in _Journal of the
+Royal Asiatic Society_, January and April, 1909.
+
+[165] Vol. IX, 1908, p. 248.
+
+[166] _Epigraphia Indica_, Vol. IX, pp. 193 and 198.
+
+[167] _Epigraphia Indica_, Vol. IX, p. 1.
+
+[168] Loc. cit., p. 71.
+
+[169] Thibaut, p. 71.
+
+[170] "Est autem in aliquibus figurarum istaram apud multos diuersitas.
+Quidam enim septimam hanc figuram representant," etc. [Boncompagni,
+_Trattati_, p. 28.] Enestroem has shown that very likely this work is
+incorrectly attributed to Johannes Hispalensis. [_Bibliotheca Mathematica_,
+Vol. IX (3), p. 2.]
+
+[171] _Indische Palaeographie_, Tafel IX.
+
+[172] Edited by Bloomfield and Garbe, Baltimore, 1901, containing
+photographic reproductions of the manuscript.
+
+[173] Bakh[s.][=a]l[=i] MS. See page 43; Hoernle, R., _The Indian
+Antiquary_, Vol. XVII, pp. 33-48, 1 plate; Hoernle, _Verhandlungen des VII.
+Internationalen Orientalisten-Congresses, Arische Section_, Vienna, 1888,
+"On the Baksh[=a]l[=i] Manuscript," pp. 127-147, 3 plates; Buehler, loc.
+cit.
+
+[174] 3, 4, 6, from H. H. Dhruva, "Three Land-Grants from Sankheda,"
+_Epigraphia Indica_, Vol. II, pp. 19-24 with plates; date 595 A.D. 7, 1, 5,
+from Bhandarkar, "Daulatabad Plates," _Epigraphia Indica_, Vol. IX, part V;
+date c. 798 A.D.
+
+[175] 8, 7, 2, from "Buckhala Inscription of Nagabhatta," Bhandarkar,
+_Epigraphia Indica_, Vol. IX, part V; date 815 A.D. 5 from "The Morbi
+Copper-Plate," Bhandarkar, _The Indian Antiquary_, Vol. II, pp. 257-258,
+with plate; date 804 A.D. See Buehler, loc. cit.
+
+[176] 8 from the above Morbi Copper-Plate. 4, 5, 7, 9, and 0, from "Asni
+Inscription of Mahipala," _The Indian Antiquary_, Vol. XVI, pp. 174-175;
+inscription is on red sandstone, date 917 A.D. See Buehler.
+
+[177] 8, 9, 4, from "Rashtrakuta Grant of Amoghavarsha," J. F. Fleet, _The
+Indian Antiquary_, Vol. XII, pp. 263-272; copper-plate grant of date c. 972
+A.D. See Buehler. 7, 3, 5, from "Torkhede Copper-Plate Grant of the Time of
+Govindaraja of Gujerat," Fleet, _Epigraphia Indica_, Vol. III, pp. 53-58.
+See Buehler.
+
+[178] From "A Copper-Plate Grant of King Tritochanapala Chanlukya of
+L[=a][t.]ade['s]a," H.H. Dhruva, _Indian Antiquary_, Vol. XII, pp. 196-205;
+date 1050 A.D. See Buehler.
+
+[179] Burnell, A. C., _South Indian Palaeography_, plate XXIII,
+Telugu-Canarese numerals of the eleventh century. See Buehler.
+
+[180] From a manuscript of the second half of the thirteenth century,
+reproduced in "Della vita e delle opere di Leonardo Pisano," Baldassare
+Boncompagni, Rome, 1852, in _Atti dell' Accademia Pontificia dei nuovi
+Lincei_, anno V.
+
+[181] From a fourteenth-century manuscript, as reproduced in _Della vita_
+etc., Boncompagni, loc. cit.
+
+[182] From a Tibetan MS. in the library of D. E. Smith.
+
+[183] From a Tibetan block-book in the library of D. E. Smith.
+
+[184] ['S][=a]rad[=a] numerals from _The Kashmirian Atharva-Veda,
+reproduced by chromophotography from the manuscript in the University
+Library at Tuebingen_, Bloomfield and Garbe, Baltimore, 1901. Somewhat
+similar forms are given under "Numeration Cachemirienne," by Pihan,
+_Expose_ etc., p. 84.
+
+[185] Franz X. Kugler, _Die Babylonische Mondrechnung_, Freiburg i. Br.,
+1900, in the numerous plates at the end of the book; practically all of
+these contain the symbol to which reference is made. Cantor, _Geschichte_,
+Vol. I, p. 31.
+
+[186] F. X. Kugler, _Sternkunde und Sterndienst in Babel_, I. Buch, from
+the beginnings to the time of Christ, Muenster i. Westfalen, 1907. It also
+has numerous tables containing the above zero.
+
+[187] From a letter to D. E. Smith, from G. F. Hill of the British Museum.
+See also his monograph "On the Early Use of Arabic Numerals in Europe," in
+_Archaeologia_, Vol. LXII (1910), p. 137.
+
+[188] R. Hoernle, "The Baksh[=a]l[=i] Manuscript," _Indian Antiquary_, Vol.
+XVII, pp. 33-48 and 275-279, 1888; Thibaut, _Astronomie, Astrologie und
+Mathematik_, p. 75; Hoernle, _Verhandlungen_, loc. cit., p. 132.
+
+[189] Bayley, loc. cit., Vol. XV, p. 29. Also Bendall, "On a System of
+Numerals used in South India," _Journal of the Royal Asiatic Society_,
+1896, pp. 789-792.
+
+[190] V. A. Smith, _The Early History of India_, 2d ed., Oxford, 1908, p.
+14.
+
+[191] Colebrooke, _Algebra, with Arithmetic and Mensuration, from the
+Sanskrit of Brahmegupta and Bhascara_, London, 1817, pp. 339-340.
+
+[192] Ibid., p. 138.
+
+[193] D. E. Smith, in the _Bibliotheca Mathematica_, Vol. IX (3), pp.
+106-110.
+
+[194] As when we use three dots (...).
+
+[195] "The Hindus call the nought explicitly _['s][=u]nyabindu_ 'the dot
+marking a blank,' and about 500 A.D. they marked it by a simple dot, which
+latter is commonly used in inscriptions and MSS. in order to mark a blank,
+and which was later converted into a small circle." [Buehler, _On the Origin
+of the Indian Alphabet_, p. 53, note.]
+
+[196] Fazzari, _Dell' origine delle parole zero e cifra_, Naples, 1903.
+
+[197] E. Wappler, "Zur Geschichte der Mathematik im 15. Jahrhundert," in
+the _Zeitschrift fuer Mathematik und Physik_, Vol. XLV, _Hist.-lit. Abt._,
+p. 47. The manuscript is No. C. 80, in the Dresden library.
+
+[198] J. G. Praendel, _Algebra nebst ihrer literarischen Geschichte_, p.
+572, Munich, 1795.
+
+[199] See the table, p. 23. Does the fact that the early European
+arithmetics, following the Arab custom, always put the 0 after the 9,
+suggest that the 0 was derived from the old Hindu symbol for 10?
+
+[200] Bayley, loc. cit., p. 48. From this fact Delambre (_Histoire de
+l'astronomie ancienne_) inferred that Ptolemy knew the zero, a theory
+accepted by Chasles, _Apercu historique sur l'origine et le developpement
+des methodes en geometrie_, 1875 ed., p. 476; Nesselmann, however, showed
+(_Algebra der Griechen_, 1842, p. 138), that Ptolemy merely used [Greek: o]
+for [Greek: ouden], with no notion of zero. See also G. Fazzari, "Dell'
+origine delle parole zero e cifra," _Ateneo_, Anno I, No. 11, reprinted at
+Naples in 1903, where the use of the point and the small cross for zero is
+also mentioned. Th. H. Martin, _Les signes numeraux_ etc., reprint p. 30,
+and J. Brandis, _Das Muenz-, Mass- und Gewichtswesen in Vorderasien bis auf
+Alexander den Grossen_, Berlin, 1866, p. 10, also discuss this usage of
+[Greek: o], without the notion of place value, by the Greeks.
+
+[201] _Al-Batt[=a]n[=i] sive Albatenii opus astronomicum_. Ad fidem codicis
+escurialensis arabice editum, latine versum, adnotationibus instructum a
+Carolo Alphonso Nallino, 1899-1907. Publicazioni del R. Osservatorio di
+Brera in Milano, No. XL.
+
+[202] Loc. cit., Vol. II, p. 271.
+
+[203] C. Henry, "Prologus N. Ocreati in Helceph ad Adelardum Batensem
+magistrum suum," _Abhandlungen zur Geschichte der Mathematik_, Vol. III,
+1880.
+
+[204] Max. Curtze, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts,"
+_Abhandlungen zur Geschichte der Mathematik_, Vol. VIII, 1898, pp. 1-27;
+Alfred Nagl, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und ueber
+die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im
+christl. Abendlande," _Zeitschrift fuer Mathematik und Physik, Hist.-lit.
+Abth._, Vol. XXXIV, pp. 129-146 and 161-170, with one plate.
+
+[205] "Byzantinische Analekten," _Abhandlungen zur Geschichte der
+Mathematik_, Vol. IX, pp. 161-189.
+
+[206] [symbol] or [symbol] for 0. [symbol] also used for 5. [symbols] for
+13. [Heiberg, loc. cit.]
+
+[207] Gerhardt, _Etudes historiques sur l'arithmetique de position_,
+Berlin, 1856, p. 12; J. Bowring, _The Decimal System in Numbers, Coins, &
+Accounts_, London, 1854, p. 33.
+
+[208] Karabacek, _Wiener Zeitschrift fuer die Kunde des Morgenlandes_, Vol.
+XI, p. 13; _Fuehrer durch die Papyrus-Ausstellung Erzherzog Rainer_, Vienna,
+1894, p. 216.
+
+[209] In the library of G. A. Plimpton, Esq.
+
+[210] Cantor, _Geschichte_, Vol. I (3), p. 674; Y. Mikami, "A Remark on the
+Chinese Mathematics in Cantor's Geschichte der Mathematik," _Archiv der
+Mathematik und Physik_, Vol. XV (3), pp. 68-70.
+
+[211] Of course the earlier historians made innumerable guesses as to the
+origin of the word _cipher_. E.g. Matthew Hostus, _De numeratione
+emendata_, Antwerp, 1582, p. 10, says: "Siphra vox Hebraeam originem sapit
+refertque: & ut docti arbitrantur, a verbo saphar, quod Ordine numerauit
+significat. Unde Sephar numerus est: hinc Siphra (vulgo corruptius). Etsi
+vero gens Iudaica his notis, quae hodie Siphrae vocantur, usa non fuit:
+mansit tamen rei appellatio apud multas gentes." Dasypodius, _Institutiones
+mathematicae_, Vol. I, 1593, gives a large part of this quotation word for
+word, without any mention of the source. Hermannus Hugo, _De prima
+scribendi origine_, Trajecti ad Rhenum, 1738, pp. 304-305, and note, p.
+305; Karl Krumbacher, "Woher stammt das Wort Ziffer (Chiffre)?", _Etudes de
+philologie neo-grecque_, Paris, 1892.
+
+[212] Buehler, loc. cit., p. 78 and p. 86.
+
+[213] Fazzari, loc. cit., p. 4. So Elia Misrachi (1455-1526) in his
+posthumous _Book of Number_, Constantinople, 1534, explains _sifra_ as
+being Arabic. See also Steinschneider, _Bibliotheca Mathematica_, 1893, p.
+69, and G. Wertheim, _Die Arithmetik des Elia Misrachi_, Programm,
+Frankfurt, 1893.
+
+[214] "Cum his novem figuris, et cum hoc signo 0, quod arabice zephirum
+appellatur, scribitur quilibet numerus."
+
+[215] [Greek: tziphra], a form also used by Neophytos (date unknown,
+probably c. 1330). It is curious that Finaeus (1555 ed., f. 2) used the
+form _tziphra_ throughout. A. J. H. Vincent ["Sur l'origine de nos
+chiffres," _Notices et Extraits des MSS._, Paris, 1847, pp. 143-150] says:
+"Ce cercle fut nomme par les uns, _sipos, rota, galgal_ ...; par les autres
+_tsiphra_ (de [Hebrew: TSPR], _couronne_ ou _diademe_) ou _ciphra_ (de
+[Hebrew: SPR], _numeration_)." Ch. de Paravey, _Essai sur l'origine unique
+et hieroglyphique des chiffres et des lettres de tous les peuples_, Paris,
+1826, p. 165, a rather fanciful work, gives "vase, vase arrondi et ferme
+par un couvercle, qui est le symbole de la 10^e Heure, [symbol]," among the
+Chinese; also "Tsiphron Zeron, ou tout a fait vide en arabe, [Greek:
+tziphra] en grec ... d'ou chiffre (qui derive plutot, suivant nous, de
+l'Hebreu _Sepher_, compter.")
+
+[216] "Compilatus a Magistro Jacobo de Florentia apud montem pesalanum,"
+and described by G. Lami in his _Catalogus codicum manuscriptorum qui in
+bibliotheca Riccardiana Florentiae adservantur_. See Fazzari, loc. cit., p.
+5.
+
+[217] "Et doveto sapere chel zeuero per se solo non significa nulla ma e
+potentia di fare significare, ... Et decina o centinaia o migliaia non si
+puote scrivere senza questo segno 0. la quale si chiama zeuero." [Fazzari,
+loc. cit., p. 5.]
+
+[218] Ibid., p. 6.
+
+[219] Avicenna (980-1036), translation by Gasbarri et Francois, "piu il
+punto (gli Arabi adoperavano il punto in vece dello zero il cui segno 0 in
+arabo si chiama _zepiro_ donde il vocabolo zero), che per se stesso non
+esprime nessun numero." This quotation is taken from D. C. Martines,
+_Origine e progressi dell' aritmetica_, Messina, 1865.
+
+[220] Leo Jordan, "Materialien zur Geschichte der arabischen Zahlzeichen in
+Frankreich," _Archiv fuer Kulturgeschichte_, Berlin, 1905, pp. 155-195,
+gives the following two schemes of derivation, (1) "zefiro, zeviro, zeiro,
+zero," (2) "zefiro, zefro, zevro, zero."
+
+[221] Koebel (1518 ed., f. A_4) speaks of the numerals in general as "die
+der gemain man Zyfer nendt." Recorde (_Grounde of Artes_, 1558 ed., f. B_6)
+says that the zero is "called priuatly a Cyphar, though all the other
+sometimes be likewise named."
+
+[222] "Decimo X 0 theca, circul[us] cifra sive figura nihili appelat'."
+[_Enchiridion Algorismi_, Cologne, 1501.] Later, "quoniam de integris tam
+in cifris quam in proiectilibus,"--the word _proiectilibus_ referring to
+markers "thrown" and used on an abacus, whence the French _jetons_ and the
+English expression "to _cast_ an account."
+
+[223] "Decima vero o dicitur teca, circulus, vel cyfra vel figura nichili."
+[Maximilian Curtze, _Petri Philomeni de Dacia in Algorismum Vulgarem
+Johannis de Sacrobosco commentarius, una cum Algorismo ipso_, Copenhagen,
+1897, p. 2.] Curtze cites five manuscripts (fourteenth and fifteenth
+centuries) of Dacia's commentary in the libraries at Erfurt, Leipzig, and
+Salzburg, in addition to those given by Enestroem, _Oefversigt af Kongl.
+Vetenskaps-Akademiens Foerhandlingar_, 1885, pp. 15-27, 65-70; 1886, pp.
+57-60.
+
+[224] Curtze, loc. cit., p. VI.
+
+[225] _Rara Mathematica_, London, 1841, chap, i, "Joannis de Sacro-Bosco
+Tractatus de Arte Numerandi."
+
+[226] Smith, _Rara Arithmetica_, Boston, 1909.
+
+[227] In the 1484 edition, Borghi uses the form "cefiro: ouero nulla:"
+while in the 1488 edition he uses "zefiro: ouero nulla," and in the 1540
+edition, f. 3, appears "Chiamata zero, ouero nulla." Woepcke asserted that
+it first appeared in Calandri (1491) in this sentence: "Sono dieci le
+figure con le quali ciascuno numero si puo significare: delle quali n'e una
+che si chiama zero: et per se sola nulla significa." (f. 4). [See
+_Propagation_, p. 522.]
+
+[228] Boncompagni _Bulletino_, Vol. XVI, pp. 673-685.
+
+[229] Leo Jordan, loc. cit. In the _Catalogue of MSS., Bibl. de l'Arsenal_,
+Vol. III, pp. 154-156, this work is No. 2904 (184 S.A.F.), Bibl. Nat., and
+is also called _Petit traicte de algorisme_.
+
+[230] Texada (1546) says that there are "nueue letros yvn zero o cifra" (f.
+3).
+
+[231] Savonne (1563, 1751 ed., f. 1): "Vne ansi formee (o) qui s'appelle
+nulle, & entre marchans zero," showing the influence of Italian names on
+French mercantile customs. Trenchant (Lyons, 1566, 1578 ed., p. 12) also
+says: "La derniere qui s'apele nulle, ou zero;" but Champenois, his
+contemporary, writing in Paris in 1577 (although the work was not published
+until 1578), uses "cipher," the Italian influence showing itself less in
+this center of university culture than in the commercial atmosphere of
+Lyons.
+
+[232] Thus Radulph of Laon (c. 1100): "Inscribitur in ultimo ordine et
+figura [symbol] sipos nomine, quae, licet numerum nullum signitet, tantum
+ad alia quaedam utilis, ut insequentibus declarabitur." ["Der Arithmetische
+Tractat des Radulph von Laon," _Abhandlungen zur Geschichte der
+Mathematik_, Vol. V, p. 97, from a manuscript of the thirteenth century.]
+Chasles (_Comptes rendus_, t. 16, 1843, pp. 1393, 1408) calls attention to
+the fact that Radulph did not know how to use the zero, and he doubts if
+the sipos was really identical with it. Radulph says: "... figuram, cui
+sipos nomen est [symbol] in motum rotulae formatam nullius numeri
+significatione inscribi solere praediximus," and thereafter uses _rotula_.
+He uses the sipos simply as a kind of marker on the abacus.
+
+[233] Rabbi ben Ezra (1092-1168) used both [Hebrew: GLGL], _galgal_ (the
+Hebrew for _wheel_), and [Hebrew: SPR'], _sifra_. See M. Steinschneider,
+"Die Mathematik bei den Juden," in _Bibliotheca Mathematica_, 1893, p. 69,
+and Silberberg, _Das Buch der Zahl des R. Abraham ibn Esra_, Frankfurt a.
+M., 1895, p. 96, note 23; in this work the Hebrew letters are used for
+numerals with place value, having the zero.
+
+[234] E.g., in the twelfth-century _Liber aligorismi_ (see Boncompagni's
+_Trattati_, II, p. 28). So Ramus (_Libri II_, 1569 ed., p. 1) says:
+"Circulus quae nota est ultima: nil per se significat." (See also the
+Schonerus ed. of Ramus, 1586, p. 1.)
+
+[235] "Und wirt das ringlein o. die Ziffer genant die nichts bedeut."
+[Koebel's _Rechenbuch_, 1549 ed., f. 10, and other editions.]
+
+[236] I.e. "circular figure," our word _notation_ having come from the
+medieval _nota_. Thus Tzwivel (1507, f. 2) says: "Nota autem circularis .o.
+per se sumpta nihil vsus habet. alijs tamen adiuncta earum significantiam
+et auget et ordinem permutat quantum quo ponit ordinem. vt adiuncta note
+binarij hoc modo 20 facit eam significare bis decem etc." Also (ibid., f.
+4), "figura circularis," "circularis nota." Clichtoveus (1503 ed., f.
+XXXVII) calls it "nota aut circularis o," "circularis nota," and "figura
+circularis." Tonstall (1522, f. B_3) says of it: "Decimo uero nota ad
+formam [symbol] litterae circulari figura est: quam alij circulum, uulgus
+cyphram uocat," and later (f. C_4) speaks of the "circulos." Grammateus, in
+his _Algorismus de integris_ (Erfurt, 1523, f. A_2), speaking of the nine
+significant figures, remarks: "His autem superadditur decima figura
+circularis ut 0 existens que ratione sua nihil significat." Noviomagus (_De
+Numeris libri II_, Paris, 1539, chap. xvi, "De notis numerorum, quas
+zyphras vocant") calls it "circularis nota, quam ex his solam, alij
+sipheram, Georgius Valla zyphram."
+
+[237] Huswirt, as above. Ramus (_Scholae mathematicae_, 1569 ed., p. 112)
+discusses the name interestingly, saying: "Circulum appellamus cum multis,
+quam alii thecam, alii figuram nihili, alii figuram privationis, seu
+figuram nullam vocant, alii ciphram, cum tamen hodie omnes hae notae vulgo
+ciphrae nominentur, & his notis numerare idem sit quod ciphrare." Tartaglia
+(1592 ed., f. 9) says: "si chiama da alcuni tecca, da alcuni circolo, da
+altri cifra, da altri zero, & da alcuni altri nulla."
+
+[238] "Quare autem aliis nominibus vocetur, non dicit auctor, quia omnia
+alia nomina habent rationem suae lineationis sive figurationis. Quia
+rotunda est, dicitur haec figura teca ad similitudinem tecae. Teca enim est
+ferrum figurae rotundae, quod ignitum solet in quibusdam regionibus imprimi
+fronti vel maxillae furis seu latronum." [Loc. cit., p. 26.] But in Greek
+_theca_ ([THEKE], [Greek: theke]) is a place to put something, a
+receptacle. If a vacant column, e.g. in the abacus, was so called, the
+initial might have given the early forms [symbol] and [symbol] for the
+zero.
+
+[239] Buteo, _Logistica_, Lyons, 1559. See also Wertheim in the
+_Bibliotheca Mathematica_, 1901, p. 214.
+
+[240] "0 est appellee chiffre ou nulle ou figure de nulle valeur." [La
+Roche, _L'arithmetique_, Lyons, 1520.]
+
+[241] "Decima autem figura nihil uocata," "figura nihili (quam etiam cifram
+uocant)." [Stifel, _Arithmetica integra_, 1544, f. 1.]
+
+[242] "Zifra, & Nulla uel figura Nihili." [Scheubel, 1545, p. 1 of ch. 1.]
+_Nulla_ is also used by Italian writers. Thus Sfortunati (1545 ed., f. 4)
+says: "et la decima nulla & e chiamata questa decima zero;" Cataldi (1602,
+p. 1): "La prima, che e o, si chiama nulla, ouero zero, ouero niente." It
+also found its way into the Dutch arithmetics, e.g. Raets (1576, 1580 ed.,
+f. A_3): "Nullo dat ist niet;" Van der Schuere (1600, 1624 ed., f. 7);
+Wilkens (1669 ed., p. 1). In Germany Johann Albert (Wittenberg, 1534) and
+Rudolff (1526) both adopted the Italian _nulla_ and popularized it. (See
+also Kuckuck, _Die Rechenkunst im sechzehnten Jahrhundert_, Berlin, 1874,
+p. 7; Guenther, _Geschichte_, p. 316.)
+
+[243] "La dixieme s'appelle chifre vulgairement: les vns l'appellant zero:
+nous la pourrons appeller vn Rien." [Peletier, 1607 ed., p. 14.]
+
+[244] It appears in the Polish arithmetic of K[=l]os (1538) as _cyfra_.
+"The Ciphra 0 augmenteth places, but of himselfe signifieth not," Digges,
+1579, p. 1. Hodder (10th ed., 1672, p. 2) uses only this word (cypher or
+cipher), and the same is true of the first native American arithmetic,
+written by Isaac Greenwood (1729, p. 1). Petrus de Dacia derives _cyfra_
+from circumference. "Vocatur etiam cyfra, quasi circumfacta vel
+circumferenda, quod idem est, quod circulus non habito respectu ad
+centrum." [Loc. cit., p. 26.]
+
+[245] _Opera mathematica_, 1695, Oxford, Vol. I, chap. ix, _Mathesis
+universalis_, "De figuris numeralibus," pp. 46-49; Vol. II, _Algebra_, p.
+10.
+
+[246] Martin, _Origine de notre systeme de numeration ecrite_, note 149, p.
+36 of reprint, spells [Greek: tsiphra] from Maximus Planudes, citing Wallis
+as an authority. This is an error, for Wallis gives the correct form as
+above.
+
+Alexander von Humboldt, "Ueber die bei verschiedenen Voelkern ueblichen
+Systeme von Zahlzeichen und ueber den Ursprung des Stellenwerthes in den
+indischen Zahlen," Crelle's _Journal fuer reine und angewandte Mathematik_,
+Vol. IV, 1829, called attention to the work [Greek: arithmoi Indikoi] of
+the monk Neophytos, supposed to be of the fourteenth century. In this work
+the forms [Greek: tzuphra] and [Greek: tzumphra] appear. See also Boeckh,
+_De abaco Graecorum_, Berlin, 1841, and Tannery, "Le Scholie du moine
+Neophytos," _Revue Archeologique_, 1885, pp. 99-102. Jordan, loc. cit.,
+gives from twelfth and thirteenth century manuscripts the forms _cifra_,
+_ciffre_, _chifras_, and _cifrus_. Du Cange, _Glossarium mediae et infimae
+Latinitatis_, Paris, 1842, gives also _chilerae_. Dasypodius,
+_Institutiones Mathematicae_, Strassburg, 1593-1596, adds the forms
+_zyphra_ and _syphra_. Boissiere, _L'art d'arythmetique contenant toute
+dimention, tres-singulier et commode, tant pour l'art militaire que autres
+calculations_, Paris, 1554: "Puis y en a vn autre dict zero lequel ne
+designe nulle quantite par soy, ains seulement les loges vuides."
+
+[247] _Propagation_, pp. 27, 234, 442. Treutlein, "Das Rechnen im 16.
+Jahrhundert," _Abhandlungen zur Geschichte der Mathematik_, Vol. I, p. 5,
+favors the same view. It is combated by many writers, e.g. A. C. Burnell,
+loc. cit., p. 59. Long before Woepcke, I. F. and G. I. Weidler, _De
+characteribus numerorum vulgaribus et eorum aetatibus_, Wittenberg, 1727,
+asserted the possibility of their introduction into Greece by Pythagoras or
+one of his followers: "Potuerunt autem ex oriente, uel ex phoenicia, ad
+graecos traduci, uel Pythagorae, uel eius discipulorum auxilio, cum aliquis
+eo, proficiendi in literis causa, iter faceret, et hoc quoque inuentum
+addisceret."
+
+[248] E.g., they adopted the Greek numerals in use in Damascus and Syria,
+and the Coptic in Egypt. Theophanes (758-818 A.D.), _Chronographia_,
+Scriptores Historiae Byzantinae, Vol. XXXIX, Bonnae, 1839, p. 575, relates
+that in 699 A.D. the caliph Wal[=i]d forbade the use of the Greek language
+in the bookkeeping of the treasury of the caliphate, but permitted the use
+of the Greek alphabetic numerals, since the Arabs had no convenient number
+notation: [Greek: kai ekoluse graphesthai Hellenisti tous demosious ton
+logothesion kodikas, all' Arabiois auta parasemainesthai, choris ton
+psephon, epeide adunaton tei ekeinon glossei monada e duada e triada e okto
+hemisu e tria graphesthai; dio kai heos semeron eisin sun autois notarioi
+Christianoi.] The importance of this contemporaneous document was pointed
+out by Martin, loc. cit. Karabacek, "Die Involutio im arabischen
+Schriftwesen," Vol. CXXXV of _Sitzungsberichte d. phil.-hist. Classe d. k.
+Akad. d. Wiss._, Vienna, 1896, p. 25, gives an Arabic date of 868 A.D. in
+Greek letters.
+
+[249] _The Origin and History of Our Numerals_ (in Russian), Kiev, 1908;
+_The Independence of European Arithmetic_ (in Russian), Kiev.
+
+[250] Woepcke, loc. cit., pp. 462, 262.
+
+[251] Woepcke, loc. cit., p. 240. _[H.]is[=a]b-al-[.G]ob[=a]r_, by an
+anonymous author, probably Ab[=u] Sahl Dunash ibn Tamim, is given by
+Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca Mathematica_,
+1896, p. 26.
+
+[252] Steinschneider in the _Abhandlungen_, Vol. III, p. 110.
+
+[253] See his _Grammaire arabe_, Vol. I, Paris, 1810, plate VIII; Gerhardt,
+_Etudes_, pp. 9-11, and _Entstehung_ etc., p. 8; I. F. Weidler,
+_Spicilegium observationum ad historiam notarum numeralium pertinentium_,
+Wittenberg, 1755, speaks of the "figura cifrarum Saracenicarum" as being
+different from that of the "characterum Boethianorum," which are similar to
+the "vulgar" or common numerals; see also Humboldt, loc. cit.
+
+[254] Gerhardt mentions it in his _Entstehung_ etc., p. 8; Woepcke,
+_Propagation_, states that these numerals were used not for calculation,
+but very much as we use Roman numerals. These superposed dots are found
+with both forms of numerals (_Propagation_, pp. 244-246).
+
+[255] Gerhardt (_Etudes_, p. 9) from a manuscript in the Bibliotheque
+Nationale. The numeral forms are [symbols], 20 being indicated by [symbol
+with dot] and 200 by [symbol with 2 dots]. This scheme of zero dots was
+also adopted by the Byzantine Greeks, for a manuscript of Planudes in the
+Bibliotheque Nationale has numbers like [pi alpha with 4 dots] for
+8,100,000,000. See Gerhardt, _Etudes_, p. 19. Pihan, _Expose_ etc., p. 208,
+gives two forms, Asiatic and Maghrebian, of "Ghob[=a]r" numerals.
+
+[256] See Chap. IV.
+
+[257] Possibly as early as the third century A.D., but probably of the
+eighth or ninth. See Cantor, I (3), p. 598.
+
+[258] Ascribed by the Arabic writer to India.
+
+[259] See Woepcke's description of a manuscript in the Chasles library,
+"Recherches sur l'histoire des sciences mathematiques chez les orientaux,"
+_Journal Asiatique_, IV (5), 1859, p. 358, note.
+
+[260] P. 56.
+
+[261] Reinaud, _Memoire sur l'Inde_, p. 399. In the fourteenth century one
+Sih[=a]b al-D[=i]n wrote a work on which, a scholiast to the Bodleian
+manuscript remarks: "The science is called Algobar because the inventor had
+the habit of writing the figures on a tablet covered with sand." [Gerhardt,
+_Etudes, _p. 11, note.]
+
+[262] Gerhardt, _Entstehung _etc., p. 20.
+
+[263] H. Suter, "Das Rechenbuch des Ab[=u] Zakar[=i]j[=a]
+el-[H.]a[s.][s.][=a]r," _Bibliotheca Mathematica_, Vol. II (3), p. 15.
+
+[264] A. Devoulx, "Les chiffres arabes," _Revue Africaine_, Vol. XVI, pp.
+455-458.
+
+[265] _Kit[=a]b al-Fihrist_, G. Fluegel, Leipzig, Vol. I, 1871, and Vol. II,
+1872. This work was published after Professor Fluegel's death by J. Roediger
+and A. Mueller. The first volume contains the Arabic text and the second
+volume contains critical notes upon it.
+
+[266] Like those of line 5 in the illustration on page 69.
+
+[267] Woepcke, _Recherches sur l'histoire des sciences mathematiques chez
+les orientaux_, loc. cit.; _Propagation, _p. 57.
+
+[268] Al-[H.]a[s.][s.][=a]r's forms, Suter, _Bibliotheca Mathematica_, Vol.
+II (3), p. 15.
+
+[269] Woepcke, _Sur une donnee historique_, etc., loc. cit. The name
+_[.g]ob[=a]r_ is not used in the text. The manuscript from which these are
+taken is the oldest (970 A.D.) Arabic document known to contain all of the
+numerals.
+
+[270] Silvestre de Sacy, loc. cit. He gives the ordinary modern Arabic
+forms, calling them _Indien_.
+
+[271] Woepcke, "Introduction au calcul Gob[=a]r[=i] et Haw[=a][=i]," _Atti
+dell' accademia pontificia dei nuovi Lincei_, Vol. XIX. The adjective
+applied to the forms in 5 is _gob[=a]r[=i]_ and to those in 6 _indienne_.
+This is the direct opposite of Woepcke's use of these adjectives in the
+_Recherches sur l'histoire_ cited above, in which the ordinary Arabic forms
+(like those in row 5) are called _indiens_.
+
+These forms are usually written from right to left.
+
+[272] J. G. Wilkinson, _The Manners and Customs of the Ancient Egyptians_,
+revised by S. Birch, London, 1878, Vol. II, p. 493, plate XVI.
+
+[273] There is an extensive literature on this "Boethius-Frage." The reader
+who cares to go fully into it should consult the various volumes of the
+_Jahrbuch ueber die Fortschritte der Mathematik_.
+
+[274] This title was first applied to Roman emperors in posthumous coins of
+Julius Caesar. Subsequently the emperors assumed it during their own
+lifetimes, thus deifying themselves. See F. Gnecchi, _Monete romane_, 2d
+ed., Milan, 1900, p. 299.
+
+[275] This is the common spelling of the name, although the more correct
+Latin form is Boetius. See Harper's _Dict. of Class. Lit. and Antiq._, New
+York, 1897, Vol. I, p. 213. There is much uncertainty as to his life. A
+good summary of the evidence is given in the last two editions of the
+_Encyclopaedia Britannica_.
+
+[276] His father, Flavius Manlius Boethius, was consul in 487.
+
+[277] There is, however, no good historic evidence of this sojourn in
+Athens.
+
+[278] His arithmetic is dedicated to Symmachus: "Domino suo patricio
+Symmacho Boetius." [Friedlein ed., p. 3.]
+
+[279] It was while here that he wrote _De consolatione philosophiae_.
+
+[280] It is sometimes given as 525.
+
+[281] There was a medieval tradition that he was executed because of a work
+on the Trinity.
+
+[282] Hence the _Divus_ in his name.
+
+[283] Thus Dante, speaking of his burial place in the monastery of St.
+Pietro in Ciel d'Oro, at Pavia, says:
+
+      "The saintly soul, that shows
+  The world's deceitfulness, to all who hear him,
+  Is, with the sight of all the good that is,
+  Blest there. The limbs, whence it was driven, lie
+  Down in Cieldauro; and from martyrdom
+  And exile came it here."--_Paradiso_, Canto X.
+
+[284] Not, however, in the mercantile schools. The arithmetic of Boethius
+would have been about the last book to be thought of in such institutions.
+While referred to by Baeda (672-735) and Hrabanus Maurus (c. 776-856), it
+was only after Gerbert's time that the _Boetii de institutione arithmetica
+libri duo_ was really a common work.
+
+[285] Also spelled Cassiodorius.
+
+[286] As a matter of fact, Boethius could not have translated any work by
+Pythagoras on music, because there was no such work, but he did make the
+theories of the Pythagoreans known. Neither did he translate Nicomachus,
+although he embodied many of the ideas of the Greek writer in his own
+arithmetic. Gibbon follows Cassiodorus in these statements in his _Decline
+and Fall of the Roman Empire_, chap. xxxix. Martin pointed out with
+positiveness the similarity of the first book of Boethius to the first five
+books of Nicomachus. [_Les signes numeraux_ etc., reprint, p. 4.]
+
+[287] The general idea goes back to Pythagoras, however.
+
+[288] J. C. Scaliger in his _Poetice_ also said of him: "Boethii Severini
+ingenium, eruditio, ars, sapientia facile provocat omnes auctores, sive
+illi Graeci sint, sive Latini" [Heilbronner, _Hist. math. univ._, p. 387].
+Libri, speaking of the time of Boethius, remarks: "Nous voyons du temps de
+Theodoric, les lettres reprendre une nouvelle vie en Italie, les ecoles
+florissantes et les savans honores. Et certes les ouvrages de Boece, de
+Cassiodore, de Symmaque, surpassent de beaucoup toutes les productions du
+siecle precedent." [_Histoire des mathematiques_, Vol. I, p. 78.]
+
+[289] Carra de Vaux, _Avicenne_, Paris, 1900; Woepcke, _Sur
+l'introduction_, etc.; Gerhardt, _Entstehung_ etc., p. 20. Avicenna is a
+corruption from Ibn S[=i]n[=a], as pointed out by Wuestenfeld, _Geschichte
+der arabischen Aerzte und Naturforscher_, Goettingen, 1840. His full name is
+Ab[=u] `Al[=i] al-[H.]osein ibn S[=i]n[=a]. For notes on Avicenna's
+arithmetic, see Woepcke, _Propagation_, p. 502.
+
+[290] On the early travel between the East and the West the following works
+may be consulted: A. Hillebrandt, _Alt-Indien_, containing "Chinesische
+Reisende in Indien," Breslau, 1899, p. 179; C. A. Skeel, _Travel in the
+First Century after Christ_, Cambridge, 1901, p. 142; M. Reinaud,
+"Relations politiques et commerciales de l'empire romain avec l'Asie
+orientale," in the _Journal Asiatique_, Mars-Avril, 1863, Vol. I (6), p.
+93; Beazley, _Dawn of Modern Geography, a History of Exploration and
+Geographical Science from the Conversion of the Roman Empire to A.D. 1420_,
+London, 1897-1906, 3 vols.; Heyd, _Geschichte des Levanthandels im
+Mittelalter_, Stuttgart, 1897; J. Keane, _The Evolution of Geography_,
+London, 1899, p. 38; A. Cunningham, _Corpus inscriptionum Indicarum_,
+Calcutta, 1877, Vol. I; A. Neander, _General History of the Christian
+Religion and Church_, 5th American ed., Boston, 1855, Vol. III, p. 89; R.
+C. Dutt, _A History of Civilization in Ancient India_, Vol. II, Bk. V,
+chap, ii; E. C. Bayley, loc. cit., p. 28 et seq.; A. C. Burnell, loc. cit.,
+p. 3; J. E. Tennent, _Ceylon_, London, 1859, Vol. I, p. 159; Geo. Turnour,
+_Epitome of the History of Ceylon_, London, n.d., preface; "Philalethes,"
+_History of Ceylon_, London, 1816, chap, i; H. C. Sirr, _Ceylon and the
+Cingalese_, London, 1850, Vol. I, chap. ix. On the Hindu knowledge of the
+Nile see F. Wilford, _Asiatick Researches_, Vol. III, p. 295, Calcutta,
+1792.
+
+[291] G. Oppert, _On the Ancient Commerce of India_, Madras, 1879, p. 8.
+
+[292] Gerhardt, _Etudes_ etc., pp. 8, 11.
+
+[293] See Smith's _Dictionary of Greek and Roman Biography and Mythology_.
+
+[294] P. M. Sykes, _Ten Thousand Miles in Persia, or Eight Years in Iran_,
+London, 1902, p. 167. Sykes was the first European to follow the course of
+Alexander's army across eastern Persia.
+
+[295] Buehler, _Indian Br[=a]hma Alphabet_, note, p. 27; _Palaeographie_, p.
+2; _Herodoti Halicarnassei historia_, Amsterdam, 1763, Bk. IV, p. 300;
+Isaac Vossius, _Periplus Scylacis Caryandensis_, 1639. It is doubtful
+whether the work attributed to Scylax was written by him, but in any case
+the work dates back to the fourth century B.C. See Smith's _Dictionary of
+Greek and Roman Biography_.
+
+[296] Herodotus, Bk. III.
+
+[297] Rameses II(?), the _Sesoosis_ of Diodorus Siculus.
+
+[298] _Indian Antiquary_, Vol. I, p. 229; F. B. Jevons, _Manual of Greek
+Antiquities_, London, 1895, p. 386. On the relations, political and
+commercial, between India and Egypt c. 72 B.C., under Ptolemy Auletes, see
+the _Journal Asiatique_, 1863, p. 297.
+
+[299] Sikandar, as the name still remains in northern India.
+
+[300] _Harper's Classical Dict._, New York, 1897, Vol. I, p. 724; F. B.
+Jevons, loc. cit., p. 389; J. C. Marshman, _Abridgment of the History of
+India_, chaps. i and ii.
+
+[301] Oppert, loc. cit., p. 11. It was at or near this place that the first
+great Indian mathematician, [=A]ryabha[t.]a, was born in 476 A.D.
+
+[302] Buehler, _Palaeographie_, p. 2, speaks of Greek coins of a period
+anterior to Alexander, found in northern India. More complete information
+may be found in _Indian Coins_, by E. J. Rapson, Strassburg, 1898, pp. 3-7.
+
+[303] Oppert, loc. cit., p. 14; and to him is due other similar
+information.
+
+[304] J. Beloch, _Griechische Geschichte_, Vol. III, Strassburg, 1904, pp.
+30-31.
+
+[305] E.g., the denarius, the words for hour and minute ([Greek: hora,
+lepton]), and possibly the signs of the zodiac. [R. Caldwell, _Comparative
+Grammar of the Dravidian Languages_, London, 1856, p. 438.] On the probable
+Chinese origin of the zodiac see Schlegel, loc. cit.
+
+[306] Marie, Vol. II, p. 73; R. Caldwell, loc. cit.
+
+[307] A. Cunningham, loc. cit., p. 50.
+
+[308] C. A. J. Skeel, _Travel_, loc. cit., p. 14.
+
+[309] _Inchiver_, from _inchi_, "the green root." [_Indian Antiquary_, Vol.
+I, p. 352.]
+
+[310] In China dating only from the second century A.D., however.
+
+[311] The Italian _morra_.
+
+[312] J. Bowring, _The Decimal System_, London, 1854, p. 2.
+
+[313] H. A. Giles, lecture at Columbia University, March 12, 1902, on
+"China and Ancient Greece."
+
+[314] Giles, loc. cit.
+
+[315] E.g., the names for grape, radish (_la-po_, [Greek: rhaphe]),
+water-lily (_si-kua_,  "west gourds"; [Greek: sikua], "gourds"), are much
+alike. [Giles, loc. cit.]
+
+[316] _Epistles_, I, 1, 45-46. On the Roman trade routes, see Beazley, loc.
+cit., Vol. I, p. 179.
+
+[317] _Am. Journ. of Archeol._, Vol. IV, p. 366.
+
+[318] M. Perrot gives this conjectural restoration of his words: "Ad me ex
+India regum legationes saepe missi sunt numquam antea visae apud quemquam
+principem Romanorum." [M. Reinaud, "Relations politiques et commerciales de
+l'empire romain avec l'Asie orientale," _Journ. Asiat._, Vol. I (6), p.
+93.]
+
+[319] Reinaud, loc. cit., p. 189. Florus, II, 34 (IV, 12), refers to it:
+"Seres etiam habitantesque sub ipso sole Indi, cum gemmis et margaritis
+elephantes quoque inter munera trahentes nihil magis quam longinquitatem
+viae imputabant." Horace shows his geographical knowledge by saying: "Not
+those who drink of the deep Danube shall now break the Julian edicts; not
+the Getae, not the Seres, nor the perfidious Persians, nor those born on
+the river Tanais." [_Odes_, Bk. IV, Ode 15, 21-24.]
+
+[320] "Qua virtutis moderationisque fama Indos etiam ac Scythas auditu modo
+cognitos pellexit ad amicitiam suam populique Romani ultro per legatos
+petendam." [Reinaud, loc. cit., p. 180.]
+
+[321] Reinaud, loc. cit., p. 180.
+
+[322] _Georgics_, II, 170-172. So Propertius (_Elegies_, III, 4):
+
+  Arma deus Caesar dites meditatur ad Indos
+    Et freta gemmiferi findere classe maris.
+
+"The divine Caesar meditated carrying arms against opulent India, and with
+his ships to cut the gem-bearing seas."
+
+[323] Heyd, loc. cit., Vol. I, p. 4.
+
+[324] Reinaud, loc. cit., p. 393.
+
+[325] The title page of Calandri (1491), for example, represents Pythagoras
+with these numerals before him. [Smith, _Rara Arithmetica_, p. 46.] Isaacus
+Vossius, _Observationes ad Pomponium Melam de situ orbis_, 1658, maintained
+that the Arabs derived these numerals from the west. A learned dissertation
+to this effect, but deriving them from the Romans instead of the Greeks,
+was written by Ginanni in 1753 (_Dissertatio mathematica critica de
+numeralium notarum minuscularum origine_, Venice, 1753). See also Mannert,
+_De numerorum quos arabicos vocant vera origine Pythagorica_, Nuernberg,
+1801. Even as late as 1827 Romagnosi (in his supplement to _Ricerche
+storiche sull' India_ etc., by Robertson, Vol. II, p. 580, 1827) asserted
+that Pythagoras originated them. [R. Bombelli, _L'antica numerazione
+italica_, Rome, 1876, p. 59.] Gow (_Hist. of Greek Math._, p. 98) thinks
+that Iamblichus must have known a similar system in order to have worked
+out certain of his theorems, but this is an unwarranted deduction from the
+passage given.
+
+[326] A. Hillebrandt, _Alt-Indien_, p. 179.
+
+[327] J. C. Marshman, loc. cit., chaps. i and ii.
+
+[328] He reigned 631-579 A.D.; called Nu['s][=i]rw[=a]n, _the holy one_.
+
+[329] J. Keane, _The Evolution of Geography_, London, 1899, p. 38.
+
+[330] The Arabs who lived in and about Mecca.
+
+[331] S. Guyard, in _Encyc. Brit._, 9th ed., Vol. XVI, p. 597.
+
+[332] Oppert, loc. cit., p. 29.
+
+[333] "At non credendum est id in Autographis contigisse, aut vetustioribus
+Codd. MSS." [Wallis, _Opera omnia_, Vol. II, p. 11.]
+
+[334] In _Observationes ad Pomponium Melam de situ orbis_. The question was
+next taken up in a large way by Weidler, loc. cit., _De characteribus_
+etc., 1727, and in _Spicilegium_ etc., 1755.
+
+[335] The best edition of these works is that of G. Friedlein, _Anicii
+Manlii Torquati Severini Boetii de institutione arithmetica libri duo, de
+institutione musica libri quinque. Accedit geometria quae fertur
+Boetii_.... Leipzig.... MDCCCLXVII.
+
+[336] See also P. Tannery, "Notes sur la pseudo-geometrie de Boece," in
+_Bibliotheca Mathematica_, Vol. I (3), p. 39. This is not the geometry in
+two books in which are mentioned the numerals. There is a manuscript of
+this pseudo-geometry of the ninth century, but the earliest one of the
+other work is of the eleventh century (Tannery), unless the Vatican codex
+is of the tenth century as Friedlein (p. 372) asserts.
+
+[337] Friedlein feels that it is partly spurious, but he says: "Eorum
+librorum, quos Boetius de geometria scripsisse dicitur, investigare veram
+inscriptionem nihil aliud esset nisi operam et tempus perdere." [Preface,
+p. v.] N. Bubnov in the Russian _Journal of the Ministry of Public
+Instruction_, 1907, in an article of which a synopsis is given in the
+_Jahrbuch ueber die Fortschritte der Mathematik_ for 1907, asserts that the
+geometry was written in the eleventh century.
+
+[338] The most noteworthy of these was for a long time Cantor
+(_Geschichte_, Vol. I., 3d ed., pp. 587-588), who in his earlier days even
+believed that Pythagoras had known them. Cantor says (_Die roemischen
+Agrimensoren_, Leipzig, 1875, p. 130): "Uns also, wir wiederholen es, ist
+die Geometrie des Boetius echt, dieselbe Schrift, welche er nach Euklid
+bearbeitete, von welcher ein Codex bereits in Jahre 821 im  Kloster
+Reichenau vorhanden war, von welcher ein anderes Exemplar im Jahre 982 zu
+Mantua in die Haende Gerbert's gelangte, von welcher mannigfache
+Handschriften noch heute vorhanden sind." But against this opinion of the
+antiquity of MSS. containing these numerals is the important statement of
+P. Tannery, perhaps the most critical of modern historians of mathematics,
+that none exists earlier than the eleventh century. See also J. L. Heiberg
+in _Philologus, Zeitschrift f. d. klass. Altertum_, Vol. XLIII, p. 508.
+
+Of Cantor's predecessors, Th. H. Martin was one of the most prominent, his
+argument for authenticity appearing in the _Revue Archeologique_  for
+1856-1857, and in his treatise _Les signes numeraux_ etc. See also M.
+Chasles, "De la connaissance qu'ont eu les anciens d'une numeration
+decimale ecrite qui fait usage de neuf chiffres prenant les valeurs de
+position," _Comptes rendus_, Vol. VI, pp. 678-680; "Sur l'origine de notre
+systeme de numeration," _Comptes rendus_, Vol. VIII, pp. 72-81; and note
+"Sur le passage du premier livre de la geometrie de Boece, relatif a un
+nouveau systeme de numeration," in his work _Apercu historique sur
+l'origine et le developpement des methodes en geometrie_, of which the
+first edition appeared in 1837.
+
+[339] J. L. Heiberg places the book in the eleventh century on philological
+grounds, _Philologus_, loc. cit.; Woepcke, in _Propagation_, p. 44; Blume,
+Lachmann, and Rudorff, _Die Schriften der roemischen Feldmesser_, Berlin,
+1848; Boeckh, _De abaco graecorum_, Berlin, 1841; Friedlein, in his Leipzig
+edition of 1867; Weissenborn, _Abhandlungen_, Vol. II, p. 185, his
+_Gerbert_, pp. 1, 247, and his _Geschichte der Einfuehrung der jetzigen
+Ziffern in Europa durch Gerbert_, Berlin, 1892, p. 11; Bayley, loc. cit.,
+p. 59; Gerhardt, _Etudes_, p. 17, _Entstehung und Ausbreitung_, p. 14;
+Nagl, _Gerbert_, p. 57; Bubnov, loc. cit. See also the discussion by
+Chasles, Halliwell, and Libri, in the _Comptes rendus_, 1839, Vol. IX, p.
+447, and in Vols. VIII, XVI, XVII of the same journal.
+
+[340] J. Marquardt, _La vie privee des Romains_, Vol. II (French trans.),
+p. 505, Paris, 1893.
+
+[341] In a Plimpton manuscript of the arithmetic of Boethius of the
+thirteenth century, for example, the Roman numerals are all replaced by the
+Arabic, and the same is true in the first printed edition of the book. (See
+Smith's _Rara Arithmetica_, pp. 434, 25-27.) D. E. Smith also copied from a
+manuscript of the arithmetic in the Laurentian library at Florence, of
+1370, the following forms, [Forged numerals
+
+[342] Halliwell, in his _Rara Mathematica, _p. 107, states that the
+disputed passage is not in a manuscript belonging to Mr. Ames, nor in one
+at Trinity College. See also Woepcke, in _Propagation_, pp. 37 and 42. It
+was the evident corruption of the texts in such editions of Boethius as
+those of Venice, 1499, Basel, 1546 and 1570, that led Woepcke to publish
+his work _Sur l'introduction de l'arithmetique indienne en Occident_.
+
+[343] They are found in none of the very ancient manuscripts, as, for
+example, in the ninth-century (?) codex in the Laurentian library which one
+of the authors has examined. It should be said, however, that the disputed
+passage was written after the arithmetic, for it contains a reference to
+that work. See the Friedlein ed., p. 397.
+
+[344] Smith, _Rara Arithmetica_, p. 66.
+
+[345] J. L. Heiberg, _Philologus_, Vol. XLIII, p. 507.
+
+[346] "Nosse autem huius artis dispicientem, quid sint digiti, quid
+articuli, quid compositi, quid incompositi numeri." [Friedlein ed., p.
+395.]
+
+[347] _De ratione abaci._ In this he describes "quandam formulam, quam ob
+honorem sui praeceptoris mensam Pythagoream nominabant ... a posterioribus
+appellabatur abacus." This, as pictured in the text, is the common Gerbert
+abacus. In the edition in Migne's _Patrologia Latina_, Vol. LXIII, an
+ordinary multiplication table (sometimes called Pythagorean abacus) is
+given in the illustration.
+
+[348] "Habebant enim diverse formatos apices vel caracteres." See the
+reference to Gerbert on p. 117.
+
+[349] C. Henry, "Sur l'origine de quelques notations mathematiques," _Revue
+Archeologique_, 1879, derives these from the initial letters used as
+abbreviations for the names of the numerals, a theory that finds few
+supporters.
+
+[350] E.g., it appears in Schonerus, _Algorithmus Demonstratus_, Nuernberg,
+1534, f. A4. In England it appeared in the earliest English arithmetical
+manuscript known, _The Crafte of Nombrynge_: "¶ fforthermore ye most
+vndirstonde that in this craft ben vsid teen figurys, as here bene writen
+for ensampul, [Numerals] ... in the quych we vse teen figurys of Inde.
+Questio. ¶ why ten fyguris of Inde? Solucio. for as I have sayd afore thei
+were fonde fyrst in Inde of a kynge of that Cuntre, that was called Algor."
+See Smith, _An Early English Algorism_, loc. cit.
+
+[351] Friedlein ed., p. 397.
+
+[352] Carlsruhe codex of Gerlando.
+
+[353] Munich codex of Gerlando.
+
+[354] Carlsruhe codex of Bernelinus.
+
+[355] Munich codex of Bernelinus.
+
+[356] Turchill, c. 1200.
+
+[357] Anon. MS., thirteenth century, Alexandrian Library, Rome.
+
+[358] Twelfth-century Boethius, Friedlein, p. 396.
+
+[359] Vatican codex, tenth century, Boethius.
+
+[360] a, h, i, are from the Friedlein ed.; the original in the manuscript
+from which a is taken contains a zero symbol, as do all of the six plates
+given by Friedlein. b-e from the Boncompagni _Bulletino_, Vol. X, p. 596; f
+ibid., Vol. XV, p. 186; g _Memorie della classe di sci., Reale Acc. dei
+Lincei_, An. CCLXXIV (1876-1877), April, 1877. A twelfth-century
+arithmetician, possibly John of Luna (Hispalensis, of Seville, c. 1150),
+speaks of the great diversity of these forms even in his day, saying: "Est
+autem in aliquibus figuram istarum apud multos diuersitas. Quidam enim
+septimam hanc figuram representant [Symbol] alii autem sic [Symbol], uel
+sic [Symbol]. Quidam vero quartam sic [Symbol]." [Boncompagni, _Trattati_,
+Vol. II, p. 28.]
+
+[361] Loc. cit., p. 59.
+
+[362] Ibid., p. 101.
+
+[363] Loc. cit., p. 396.
+
+[364] Khosr[=u] I, who began to reign in 531 A.D. See W. S. W Vaux,
+_Persia, _London, 1875, p. 169; Th. Noeldeke, _Aufsaetze zur persichen
+Geschichte_, Leipzig, 1887, p. 113, and his article in the ninth edition of
+the _Encyclopaedia Britannica_.
+
+[365] Colebrooke, _Essays_, Vol. II, p. 504, on the authority of Ibn
+al-Adam[=i], astronomer, in a work published by his continuator Al-Q[=a]sim
+in 920 A.D.; Al-B[=i]r[=u]n[=i], _India, _Vol. II, p. 15.
+
+[366] H. Suter, _Die Mathematiker_ etc., pp. 4-5, states that
+Al-Faz[=a]r[=i] died between 796 and 806.
+
+[367] Suter, loc. cit., p. 63.
+
+[368] Suter, loc. cit., p. 74.
+
+[369] Suter, _Das Mathematiker-Verzeichniss im Fihrist_. The references to
+Suter, unless otherwise stated, are to his later work _Die Mathematiker und
+Astronomen der Araber_ etc.
+
+[370] Suter, _Fihrist_, p. 37, no date.
+
+[371] Suter, _Fihrist_, p. 38, no date.
+
+[372] Possibly late tenth, since he refers to one arithmetical work which
+is entitled _Book of the Cyphers_ in his _Chronology_, English ed., p. 132.
+Suter, _Die Mathematiker_ etc., pp. 98-100, does not mention this work; see
+the _Nachtraege und Berichtigungen_, pp. 170-172.
+
+[373] Suter, pp. 96-97.
+
+[374] Suter, p. 111.
+
+[375] Suter, p. 124. As the name shows, he came from the West.
+
+[376] Suter, p. 138.
+
+[377] Hankel, _Zur Geschichte der Mathematik_, p. 256, refers to him as
+writing on the Hindu art of reckoning; Suter, p. 162.
+
+[378] [Greek: Psephophoria kat' Indous], Greek ed., C. I. Gerhardt, Halle,
+1865; and German translation, _Das Rechenbuch des Maximus Planudes_, H.
+Waeschke, Halle, 1878.
+
+[379] "Sur une donnee historique relative a l'emploi des chiffres indiens
+par les Arabes," Tortolini's _Annali di scienze mat. e fis._, 1855.
+
+[380] Suter, p. 80.
+
+[381] Suter, p. 68.
+
+[382] Sprenger also calls attention to this fact, in the _Zeitschrift d.
+deutschen morgenlaend. Gesellschaft_, Vol. XLV, p. 367.
+
+[383] Libri, _Histoire des mathematiques_, Vol. I, p. 147.
+
+[384] "Dictant la paix a l'empereur de Constantinople, l'Arabe victorieux
+demandait des manuscrits et des savans." [Libri, loc. cit., p. 108.]
+
+[385] Persian _bagadata_, "God-given."
+
+[386] One of the Abbassides, the (at least pretended) descendants of
+`Al-Abb[=a]s, uncle and adviser of Mo[h.]ammed.
+
+[387] E. Reclus, _Asia_, American ed., N. Y., 1891, Vol. IV, p. 227.
+
+[388] _Historical Sketches_, Vol. III, chap. iii.
+
+[389] On its prominence at that period see Villicus, p. 70.
+
+[390] See pp. 4-5.
+
+[391] Smith, D. E., in the _Cantor Festschrift_, 1909, note pp. 10-11. See
+also F. Woepcke, _Propagation_.
+
+[392] Enestroem, in _Bibliotheca Mathematica_, Vol. I (3), p. 499; Cantor,
+_Geschichte_, Vol. I (3), p. 671.
+
+[393] Cited in Chapter I. It begins: "Dixit algoritmi: laudes deo rectori
+nostro atque defensori dicamus dignas." It is devoted entirely to the
+fundamental operations and contains no applications.
+
+[394] M. Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca
+Mathematica_, Vol. VIII (2), p. 99. See also the reference to this writer
+in Chapter I.
+
+[395] Part of this work has been translated from a Leyden MS. by F.
+Woepcke, _Propagation_, and more recently by H. Suter, _Bibliotheca
+Mathematica_, Vol. VII (3), pp. 113-119.
+
+[396] A. Neander, _General History of the Christian Religion and Church_,
+5th American ed., Boston, 1855, Vol. III, p. 335.
+
+[397] Beazley, loc. cit., Vol. I, p. 49.
+
+[398] Beazley, loc. cit., Vol. I, pp. 50, 460.
+
+[399] See pp. 7-8.
+
+[400] The name also appears as Mo[h.]ammed Ab[=u]'l-Q[=a]sim, and Ibn
+Hauqal. Beazley, loc. cit., Vol. I, p. 45.
+
+[401] _Kit[=a]b al-mas[=a]lik wa'l-mam[=a]lik._
+
+[402] Reinaud, _Mem. sur l'Inde_; in Gerhardt, _Etudes_, p. 18.
+
+[403] Born at Shiraz in 1193. He himself had traveled from India to Europe.
+
+[404] _Gulistan_ (_Rose Garden_), Gateway the third, XXII. Sir Edwin
+Arnold's translation, N. Y., 1899, p. 177.
+
+[405] Cunningham, loc. cit., p. 81.
+
+[406] Putnam, _Books_, Vol. I, p. 227:
+
+ "Non semel externas peregrino tramite terras
+  Jam peragravit ovans, sophiae deductus amore,
+  Si quid forte novi librorum seu studiorum
+  Quod secum ferret, terris reperiret in illis.
+  Hic quoque Romuleum venit devotus ad urbem."
+
+("More than once he has traveled joyfully through remote regions and by
+strange roads, led on by his zeal for knowledge and seeking to discover in
+foreign lands novelties in books or in studies which he could take back
+with him. And this zealous student journeyed to the city of Romulus.")
+
+[407] A. Neander, _General History of the Christian Religion and Church_,
+5th American ed., Boston, 1855, Vol. III, p. 89, note 4; Libri, _Histoire_,
+Vol. I, p. 143.
+
+[408] Cunningham, loc. cit., p. 81.
+
+[409] Heyd, loc. cit., Vol. I, p. 4.
+
+[410] Ibid., p. 5.
+
+[411] Ibid., p. 21.
+
+[412] Ibid., p. 23.
+
+[413] Libri, _Histoire_, Vol. I, p. 167.
+
+[414] Picavet, _Gerbert, un pape philosophe, d'apres l'histoire et d'apres
+la legende_, Paris, 1897, p. 19.
+
+[415] Beazley, loc. cit., Vol. I, chap, i, and p. 54 seq.
+
+[416] Ibid., p. 57.
+
+[417] Libri, _Histoire_, Vol. I, p. 110, n., citing authorities, and p.
+152.
+
+[418] Possibly the old tradition, "Prima dedit nautis usum magnetis
+Amalphis," is true so far as it means the modern form of compass card. See
+Beazley, loc. cit., Vol. II, p. 398.
+
+[419] R. C. Dutt, loc. cit., Vol. II, p. 312.
+
+[420] E. J. Payne, in _The Cambridge Modern History_, London, 1902, Vol. I,
+chap. i.
+
+[421] Geo. Phillips, "The Identity of Marco Polo's Zaitun with Changchau,
+in T'oung pao," _Archives pour servir a l'etude de l'histoire de l'Asie
+orientale_, Leyden, 1890, Vol. I, p. 218. W. Heyd, _Geschichte des
+Levanthandels im Mittelalter_, Vol. II, p. 216.
+
+The Palazzo dei Poli, where Marco was born and died, still stands in the
+Corte del Milione, in Venice. The best description of the Polo travels, and
+of other travels of the later Middle Ages, is found in C. R. Beazley's
+_Dawn of Modern Geography_, Vol. III, chap, ii, and Part II.
+
+[422] Heyd, loc. cit., Vol. II, p. 220; H. Yule, in _Encyclopaedia
+Britannica_, 9th (10th) or 11th ed., article "China." The handbook cited is
+Pegolotti's _Libro di divisamenti di paesi_, chapters i-ii, where it is
+implied that $60,000 would be a likely amount for a merchant going to China
+to invest in his trip.
+
+[423] Cunningham, loc. cit., p. 194.
+
+[424] I.e. a commission house.
+
+[425] Cunningham, loc. cit., p. 186.
+
+[426] J. R. Green, _Short History of the English People_, New York, 1890,
+p. 66.
+
+[427] W. Besant, _London_, New York, 1892, p. 43.
+
+[428] _Baldakin_, _baldekin_, _baldachino_.
+
+[429] Italian _Baldacco_.
+
+[430] J. K. Mumford, _Oriental Rugs_, New York, 1901, p. 18.
+
+[431] Or Girbert, the Latin forms _Gerbertus_ and _Girbertus_ appearing
+indifferently in the documents of his time.
+
+[432] See, for example, J. C. Heilbronner, _Historia matheseos universae_,
+p. 740.
+
+[433] "Obscuro loco natum," as an old chronicle of Aurillac has it.
+
+[434] N. Bubnov, _Gerberti postea Silvestri II papae opera mathematica_,
+Berlin, 1899, is the most complete and reliable source of information;
+Picavet, loc. cit., _Gerbert_ etc.; Olleris, _Oeuvres de Gerbert_, Paris,
+1867; Havet, _Lettres de Gerbert_, Paris, 1889 ; H. Weissenborn, _Gerbert;
+Beitraege zur Kenntnis der Mathematik des Mittelalters_, Berlin, 1888, and
+_Zur Geschichte der Einfuehrung der jetzigen Ziffern in Europa durch
+Gerbert_, Berlin, 1892; Buedinger, _Ueber Gerberts wissenschaftliche und
+politische Stellung_, Cassel, 1851; Richer, "Historiarum liber III," in
+Bubnov, loc. cit., pp. 376-381; Nagl, _Gerbert und die Rechenkunst des 10.
+Jahrhunderts_, Vienna, 1888.
+
+[435] Richer tells of the visit to Aurillac by Borel, a Spanish nobleman,
+just as Gerbert was entering into young manhood. He relates how
+affectionately the abbot received him, asking if there were men in Spain
+well versed in the arts. Upon Borel's reply in the affirmative, the abbot
+asked that one of his young men might accompany him upon his return, that
+he might carry on his studies there.
+
+[436] Vicus Ausona. Hatto also appears as Atton and Hatton.
+
+[437] This is all that we know of his sojourn in Spain, and this comes from
+his pupil Richer. The stories told by Adhemar of Chabanois, an apparently
+ignorant and certainly untrustworthy contemporary, of his going to Cordova,
+are unsupported. (See e.g. Picavet, p. 34.) Nevertheless this testimony is
+still accepted: K. von Raumer, for example (_Geschichte der Paedagogik_, 6th
+ed., 1890, Vol. I, p. 6), says "Mathematik studierte man im Mittelalter bei
+den Arabern in Spanien. Zu ihnen gieng Gerbert, nachmaliger Pabst Sylvester
+II."
+
+[438] Thus in a letter to Aldaberon he says: "Quos post repperimus
+speretis, id est VIII volumina Boeti de astrologia, praeclarissima quoque
+figurarum geometriae, aliaque non minus admiranda" (Epist. 8). Also in a
+letter to Rainard (Epist. 130), he says: "Ex tuis sumptibus fac ut michi
+scribantur M. Manlius (Manilius in one MS.) de astrologia."
+
+[439] Picavet, loc. cit., p. 31.
+
+[440] Picavet, loc. cit., p. 36.
+
+[441] Havet, loc. cit., p. vii.
+
+[442] Picavet, loc. cit., p. 37.
+
+[443] "Con sinistre arti conseguri la dignita del Pontificato.... Lasciato
+poi l' abito, e 'l monasterio, e datosi tutto in potere del diavolo."
+[Quoted in Bombelli, _L'antica numerazione Italica_, Rome, 1876, p. 41 n.]
+
+[444] He writes from Rheims in 984 to one Lupitus, in Barcelona, saying:
+"Itaque librum de astrologia translatum a te michi petenti dirige,"
+presumably referring to some Arabic treatise. [Epist. no. 24 of the Havet
+collection, p. 19.]
+
+[445] See Bubnov, loc. cit., p. x.
+
+[446] Olleris, loc. cit., p. 361, l. 15, for Bernelinus; and Bubnov, loc.
+cit., p. 381, l. 4, for Richer.
+
+[447] Woepcke found this in a Paris MS. of Radulph of Laon, c. 1100.
+[_Propagation_, p. 246.] "Et prima quidem trium spaciorum superductio
+unitatis caractere inscribitur, qui chaldeo nomine dicitur igin." See also
+Alfred Nagl, "Der arithmetische Tractat des Radulph von Laon"
+(_Abhandlungen zur Geschichte der Mathematik_, Vol. V, pp. 85-133), p. 97.
+
+[448] Weissenborn, loc. cit., p. 239. When Olleris (_Oeuvres de Gerbert_,
+Paris, 1867, p. cci) says, "C'est a lui et non point aux Arabes, que
+l'Europe doit son systeme et ses signes de numeration," he exaggerates,
+since the evidence is all against his knowing the place value. Friedlein
+emphasizes this in the _Zeitschrift fuer Mathematik und Physik_, Vol. XII
+(1867), _Literaturzeitung_, p. 70: "Fuer das _System_ unserer Numeration ist
+die _Null_ das wesentlichste Merkmal, und diese kannte Gerbert nicht. Er
+selbst schrieb alle Zahlen mit den roemischen Zahlzeichen und man kann ihm
+also nicht verdanken, was er selbst nicht kannte."
+
+[449] E.g., Chasles, Buedinger, Gerhardt, and Richer. So Martin (_Recherches
+nouvelles_ etc.) believes that Gerbert received them from Boethius or his
+followers. See Woepcke, _Propagation_, p. 41.
+
+[450] Buedinger, loc. cit., p. 10. Nevertheless, in Gerbert's time one
+Al-Man[s.][=u]r, governing Spain under the name of Hish[=a]m (976-1002),
+called from the Orient Al-Be[.g][=a]n[=i] to teach his son, so that
+scholars were recognized. [Picavet, p. 36.]
+
+[451] Weissenborn, loc. cit., p. 235.
+
+[452] Ibid., p. 234.
+
+[453] These letters, of the period 983-997, were edited by Havet, loc.
+cit., and, less completely, by Olleris, loc. cit. Those touching
+mathematical topics were edited by Bubnov, loc. cit., pp. 98-106.
+
+[454] He published it in the _Monumenta Germaniae historica_, "Scriptores,"
+Vol. III, and at least three other editions have since appeared, viz. those
+by Guadet in 1845, by Poinsignon in 1855, and by Waitz in 1877.
+
+[455] Domino ac beatissimo Patri Gerberto, Remorum archiepiscopo, Richerus
+Monchus, Gallorum congressibus in volumine regerendis, imperii tui, pater
+sanctissime Gerberte, auctoritas seminarium dedit.
+
+[456] In epistle 17 (Havet collection) he speaks of the "De multiplicatione
+et divisione numerorum libellum a Joseph Ispano editum abbas Warnerius" (a
+person otherwise unknown). In epistle 25 he says: "De multiplicatione et
+divisione numerorum, Joseph Sapiens sententias quasdam edidit."
+
+[457] H. Suter, "Zur Frage ueber den Josephus Sapiens," _Bibliotheca
+Mathematica_, Vol. VIII (2), p. 84; Weissenborn, _Einfuehrung_, p. 14; also
+his _Gerbert_; M. Steinschneider, in _Bibliotheca Mathematica_, 1893, p.
+68. Wallis (_Algebra_, 1685, chap. 14) went over the list of Spanish
+Josephs very carefully, but could find nothing save that "Josephus Hispanus
+seu Josephus sapiens videtur aut Maurus fuisse aut alius quis in Hispania."
+
+[458] P. Ewald, _Mittheilungen, Neues Archiv d. Gesellschaft fuer aeltere
+deutsche Geschichtskunde_, Vol. VIII, 1883, pp. 354-364. One of the
+manuscripts is of 976 A.D. and the other of 992 A.D. See also Franz
+Steffens, _Lateinische Palaeographie_, Freiburg (Schweiz), 1903, pp.
+xxxix-xl. The forms are reproduced in the plate on page 140.
+
+[459] It is entitled _Constantino suo Gerbertus scolasticus_, because it
+was addressed to Constantine, a monk of the Abbey of Fleury. The text of
+the letter to Constantine, preceding the treatise on the Abacus, is given
+in the _Comptes rendus_, Vol. XVI (1843), p. 295. This book seems to have
+been written c. 980 A.D. [Bubnov, loc. cit., p. 6.]
+
+[460] "Histoire de l'Arithmetique," _Comptes rendus_, Vol. XVI (1843), pp.
+156, 281.
+
+[461] Loc. cit., _Gerberti Opera_ etc.
+
+[462] Friedlein thought it spurious. See _Zeitschrift fuer Mathematik und
+Physik_, Vol. XII (1867), Hist.-lit. suppl., p. 74. It was discovered in
+the library of the Benedictine monastry of St. Peter, at Salzburg, and was
+published by Peter Bernhard Pez in 1721. Doubt was first cast upon it in
+the Olleris edition (_Oeuvres de Gerbert_). See Weissenborn, _Gerbert_, pp.
+2, 6, 168, and Picavet, p. 81. Hock, Cantor, and Th. Martin place the
+composition of the work at c. 996 when Gerbert was in Germany, while
+Olleris and Picavet refer it to the period when he was at Rheims.
+
+[463] Picavet, loc. cit., p. 182.
+
+[464] Who wrote after Gerbert became pope, for he uses, in his preface, the
+words, "a domino pape Gerberto." He was quite certainly not later than the
+eleventh century; we do not have exact information about the time in which
+he lived.
+
+[465] Picavet, loc. cit., p. 182. Weissenborn, _Gerbert_, p. 227. In
+Olleris, _Liber Abaci_ (of Bernelinus), p. 361.
+
+[466] Richer, in Bubnov, loc. cit., p. 381.
+
+[467] Weissenborn, _Gerbert_, p. 241.
+
+[468] Writers on numismatics are quite uncertain as to their use. See F.
+Gnecchi, _Monete Romane_, 2d ed., Milan, 1900, cap. XXXVII. For pictures of
+old Greek tesserae of Sarmatia, see S. Ambrosoli, _Monete Greche_, Milan,
+1899, p. 202.
+
+[469] Thus Tzwivel's arithmetic of 1507, fol. 2, v., speaks of the ten
+figures as "characteres sive numerorum apices a diuo Seuerino Boetio."
+
+[470] Weissenborn uses _sipos_ for 0. It is not given by Bernelinus, and
+appears in Radulph of Laon, in the twelfth century. See Guenther's
+_Geschichte_, p. 98, n.; Weissenborn, p. 11; Pihan, _Expose_ etc., pp.
+xvi-xxii.
+
+In Friedlein's _Boetius_, p. 396, the plate shows that all of the six
+important manuscripts from which the illustrations are taken contain the
+symbol, while four out of five which give the words use the word _sipos_
+for 0. The names appear in a twelfth-century anonymous manuscript in the
+Vatican, in a passage beginning
+
+  Ordine primigeno sibi nomen possidet igin.
+  Andras ecce locum mox uendicat ipse secundum
+  Ormis post numeros incompositus sibi primus.
+
+[Boncompagni _Buttetino_, XV, p. 132.] Turchill (twelfth century) gives the
+names Igin, andras, hormis, arbas, quimas, caletis, zenis, temenias,
+celentis, saying: "Has autem figuras, ut donnus [dominus] Gvillelmus Rx
+testatur, a pytagoricis habemus, nomina uero ab arabibus." (Who the William
+R. was is not known. Boncompagni _Bulletino_ XV, p. 136.) Radulph of Laon
+(d. 1131) asserted that they were Chaldean (_Propagation_, p. 48 n.). A
+discussion of the whole question is also given in E. C. Bayley, loc. cit.
+Huet, writing in 1679, asserted that they were of Semitic origin, as did
+Nesselmann in spite of his despair over ormis, calctis, and celentis; see
+Woepcke, _Propagation_, p. 48. The names were used as late as the fifteenth
+century, without the zero, but with the superscript dot for 10's, two dots
+for 100's, etc., as among the early Arabs. Gerhardt mentions having seen a
+fourteenth or fifteenth century manuscript in the Bibliotheca Amploniana
+with the names "Ingnin, andras, armis, arbas, quinas, calctis, zencis,
+zemenias, zcelentis," and the statement "Si unum punctum super ingnin
+ponitur, X significat.... Si duo puncta super ... figuras superponunter,
+fiet decuplim illius quod cum uno puncto significabatur," in
+_Monatsberichte der K. P. Akad. d. Wiss._, Berlin, 1867, p. 40.
+
+[471] _A chart of ten numerals in 200 tongues_, by Rev. R. Patrick, London,
+1812.
+
+[472] "Numeratio figuralis est cuiusuis numeri per notas, et figuras
+numerates descriptio." [Clichtoveus, edition of c. 1507, fol. C ii, v.]
+"Aristoteles enim uoces rerum [Greek: sumbola] uocat: id translatum, sonat
+notas." [Noviomagus, _De Numeris Libri II_, cap. vi.] "Alphabetum decem
+notarum." [Schonerus, notes to Ramus, 1586, p. 3 seq.] Richer says: "novem
+numero notas omnem numerum significantes." [Bubnov, loc. cit., p. 381.]
+
+[473] "Il y a dix Characteres, autrement Figures, Notes, ou Elements."
+[Peletier, edition of 1607, p. 13.] "Numerorum notas alij figuras, alij
+signa, alij characteres uocant." [Glareanus, 1545 edition, f. 9, r.] "Per
+figuras (quas zyphras uocant) assignationem, quales sunt hae notulae, 1. 2.
+3. 4...." [Noviomagus, _De Numeris Libri II_, cap. vi.] Gemma Frisius also
+uses _elementa_ and Cardan uses _literae_. In the first arithmetic by an
+American (Greenwood, 1729) the author speaks of "a few Arabian _Charecters_
+or Numeral Figures, called _Digits_" (p. 1), and as late as 1790, in the
+third edition of J. J. Blassiere's arithmetic (1st ed. 1769), the name
+_characters_ is still in use, both for "de Latynsche en de Arabische" (p.
+4), as is also the term "Cyfferletters" (p. 6, n.). _Ziffer_, the modern
+German form of cipher, was commonly used to designate any of the nine
+figures, as by Boeschenstein and Riese, although others, like Koebel, used
+it only for the zero. So _zifre_ appears in the arithmetic by Borgo, 1550
+ed. In a Munich codex of the twelfth century, attributed to Gerland, they
+are called _characters_ only: "Usque ad VIIII. enim porrigitur omnis
+numerus et qui supercrescit eisdem designator Karacteribus." [Boncompagni
+_Bulletino_, Vol. X. p. 607.]
+
+[474] The title of his work is _Prologus N. Ocreati in Helceph_ (Arabic
+_al-qeif_, investigation or memoir) _ad Adelardum Batensem magistrum suum_.
+The work was made known by C. Henry, in the _Zeitschrift fuer Mathematik und
+Physik_, Vol. XXV, p. 129, and in the _Abhandlungen zur Geschichte der
+Mathematik_, Vol. III; Weissenborn, _Gerbert_, p. 188.
+
+[475] The zero is indicated by a vacant column.
+
+[476] Leo Jordan, loc. cit., p. 170. "Chifre en augorisme" is the
+expression used, while a century later "giffre en argorisme" and "cyffres
+d'augorisme" are similarly used.
+
+[477] _The Works of Geoffrey Chaucer_, edited by W. W. Skeat, Vol. IV,
+Oxford, 1894, p. 92.
+
+[478] Loc. cit., Vol. III, pp. 179 and 180.
+
+[479] In Book II, chap, vii, of _The Testament of Love_, printed with
+Chaucer's Works, loc. cit., Vol. VII, London, 1897.
+
+[480] _Liber Abacci_, published in Olleris, _Oeuvres de Gerbert_, pp.
+357-400.
+
+[481] G. R. Kaye, "The Use of the Abacus in Ancient India," _Journal and
+Proceedings of the Asiatic Society of Bengal_, 1908, pp. 293-297.
+
+[482] _Liber Abbaci_, by Leonardo Pisano, loc. cit., p. 1.
+
+[483] Friedlein, "Die Entwickelung des Rechnens mit Columnen," _Zeitschrift
+fuer Mathematik und Physik_, Vol. X, p. 247.
+
+[484] The divisor 6 or 16 being increased by the difference 4, to 10 or 20
+respectively.
+
+[485] E.g. Cantor, Vol. I, p. 882.
+
+[486] Friedlein, loc. cit.; Friedlein, "Gerbert's Regeln der Division" and
+"Das Rechnen mit Columnen vor dem 10. Jahrhundert," _Zeitschrift fuer
+Mathematik und Physik_, Vol. IX; Bubnov, loc. cit., pp. 197-245; M.
+Chasles, "Histoire de l'arithmetique. Recherches des traces du systeme de
+l'abacus, apres que cette methode a pris le nom d'Algorisme.--Preuves qu'a
+toutes les epoques, jusq'au XVI^e siecle, on a su que l'arithmetique
+vulgaire avait pour origine cette methode ancienne," _Comptes rendus_, Vol.
+XVII, pp. 143-154, also "Regles de l'abacus," _Comptes rendus_, Vol. XVI,
+pp. 218-246, and "Analyse et explication du traite de Gerbert," _Comptes
+rendus_, Vol. XVI, pp. 281-299.
+
+[487] Bubnov, loc. cit., pp. 203-204, "Abbonis abacus."
+
+[488] "Regulae de numerorum abaci rationibus," in Bubnov, loc. cit., pp.
+205-225.
+
+[489] P. Treutlein, "Intorno ad alcuni scritti inediti relativi al calcolo
+dell' abaco," _Bulletino di bibliografia e di storia delle scienze
+matematiche e fisiche_, Vol. X, pp. 589-647.
+
+[490] "Intorno ad uno scritto inedito di Adelhardo di Bath intitolato
+'Regulae Abaci,'" B. Boncompagni, in his _Bulletino_, Vol. XIV, pp. 1-134.
+
+[491] Treutlein, loc. cit.; Boncompagni, "Intorno al Tractatus de Abaco di
+Gerlando," _Bulletino_, Vol. X, pp. 648-656.
+
+[492] E. Narducci, "Intorno a due trattati inediti d'abaco contenuti in due
+codici Vaticani del secolo XII," Boncompagni _Bulletino_, Vol. XV, pp.
+111-162.
+
+[493] See Molinier, _Les sources de l'histoire de France_, Vol. II, Paris,
+1902, pp. 2, 3.
+
+[494] Cantor, _Geschichte_, Vol. I, p. 762. A. Nagl in the _Abhandlungen
+zur Geschichte der Mathematik_, Vol. V, p. 85.
+
+[495] 1030-1117.
+
+[496] _Abhandlungen zur Geschichte der Mathematik_, Vol. V, pp. 85-133. The
+work begins "Incipit Liber Radulfi laudunensis de abaco."
+
+[497] _Materialien zur Geschichte der arabischen Zahlzeichen in
+Frankreich_, loc. cit.
+
+[498] Who died in 1202.
+
+[499] Cantor, _Geschichte_, Vol. I (3), pp. 800-803; Boncompagni,
+_Trattati_, Part II. M. Steinschneider ("Die Mathematik bei den Juden,"
+_Bibliotheca Mathematica_, Vol. X (2), p. 79) ingeniously derives another
+name by which he is called (Abendeuth) from Ibn Da[=u]d (Son of David). See
+also _Abhandlungen_, Vol. III, p. 110.
+
+[500] John is said to have died in 1157.
+
+[501] For it says, "Incipit prologus in libro alghoarismi de practica
+arismetrice. Qui editus est a magistro Johanne yspalensi." It is published
+in full in the second part of Boncompagni's _Trattati d'aritmetica_.
+
+[502] Possibly, indeed, the meaning of "libro alghoarismi" is not "to
+Al-Khow[=a]razm[=i]'s book," but "to a book of algorism." John of Luna says
+of it: "Hoc idem est illud etiam quod ... alcorismus dicere videtur."
+[_Trattati_, p. 68.]
+
+[503] For a resume, see Cantor, Vol. I (3), pp. 800-803. As to the author,
+see Enestroem in the _Bibliotheca Mathematica_, Vol. VI (3), p. 114, and
+Vol. IX (3), p. 2.
+
+[504] Born at Cremona (although some have asserted at Carmona, in
+Andalusia) in 1114; died at Toledo in 1187. Cantor, loc. cit.; Boncompagni,
+_Atti d. R. Accad. d. n. Lincei_, 1851.
+
+[505] See _Abhandlungen zur Geschichte der Mathematik_, Vol. XIV, p. 149;
+_Bibliotheca Mathematica_, Vol. IV (3), p. 206. Boncompagni had a
+fourteenth-century manuscript of his work, _Gerardi Cremonensis artis
+metrice practice_. See also T. L. Heath, _The Thirteen Books of Euclid's
+Elements_, 3 vols., Cambridge, 1908, Vol. I, pp. 92-94 ; A. A. Bjoernbo,
+"Gerhard von Cremonas Uebersetzung von Alkwarizmis Algebra und von Euklids
+Elementen," _Bibliotheca Mathematica_, Vol. VI (3), pp. 239-248.
+
+[506] Wallis, _Algebra_, 1685, p. 12 seq.
+
+[507] Cantor, _Geschichte_, Vol. I (3), p. 906; A. A. Bjoernbo,
+"Al-Chw[=a]rizm[=i]'s trigonometriske Tavler," _Festskrift til H. G.
+Zeuthen_, Copenhagen, 1909, pp. 1-17.
+
+[508] Heath, loc. cit., pp. 93-96.
+
+[509] M. Steinschneider, _Zeitschrift der deutschen morgenlaendischen
+Gesellschaft_, Vol. XXV, 1871, p. 104, and _Zeitschrift fuer Mathematik und
+Physik_, Vol. XVI, 1871, pp. 392-393; M. Curtze, _Centralblatt fuer
+Bibliothekswesen_, 1899, p. 289; E. Wappler, _Zur Geschichte der deutschen
+Algebra im 15. Jahrhundert_, Programm, Zwickau, 1887; L. C. Karpinski,
+"Robert of Chester's Translation of the Algebra of Al-Khow[=a]razm[=i],"
+_Bibliotheca Mathematica_, Vol. XI (3), p. 125. He is also known as
+Robertus Retinensis, or Robert of Reading.
+
+[510] Nagl, A., "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und
+ueber die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im
+christl. Abendlande," in the _Zeitschrift fuer Mathematik und Physik,
+Hist.-lit. Abth._, Vol. XXXIV, p. 129. Curtze, _Abhandlungen zur Geschichte
+der Mathematik_, Vol. VIII, pp. 1-27.
+
+[511] See line _a_ in the plate on p. 143.
+
+[512] _Sefer ha-Mispar, Das Buch der Zahl, ein hebraeisch-arithmetisches
+Werk des R. Abraham ibn Esra_, Moritz Silberberg, Frankfurt a. M., 1895.
+
+[513] Browning's "Rabbi ben Ezra."
+
+[514] "Darum haben auch die Weisen Indiens all ihre Zahlen durch neun
+bezeichnet und Formen fuer die 9 Ziffern gebildet." [_Sefer ha-Mispar_, loc.
+cit., p. 2.]
+
+[515] F. Bonaini, "Memoria unica sincrona di Leonardo Fibonacci," Pisa,
+1858, republished in 1867, and appearing in the _Giornale Arcadico_, Vol.
+CXCVII (N.S. LII); Gaetano Milanesi, _Documento inedito e sconosciuto a
+Lionardo Fibonacci_, Roma, 1867; Guglielmini, _Elogio di Lionardo Pisano_,
+Bologna, 1812, p. 35; Libri, _Histoire des sciences mathematiques_, Vol.
+II, p. 25; D. Martines, _Origine e progressi dell' aritmetica_, Messina,
+1865, p. 47; Lucas, in Boncompagni _Bulletino_, Vol. X, pp. 129, 239;
+Besagne, ibid., Vol. IX, p. 583; Boncompagni, three works as cited in Chap.
+I; G. Enestroem, "Ueber zwei angebliche mathematische Schulen im
+christlichen Mittelalter," _Bibliotheca Mathematica_, Vol. VIII (3), pp.
+252-262; Boncompagni, "Della vita e delle opere di Leonardo Pisano," loc.
+cit.
+
+[516] The date is purely conjectural. See the _Bibliotheca Mathematica_,
+Vol. IV (3), p. 215.
+
+[517] An old chronicle relates that in 1063 Pisa fought a great battle with
+the Saracens at Palermo, capturing six ships, one being "full of wondrous
+treasure," and this was devoted to building the cathedral.
+
+[518] Heyd, loc. cit., Vol. I, p. 149.
+
+[519] Ibid., p. 211.
+
+[520] J. A. Symonds, _Renaissance in Italy. The Age of Despots._ New York,
+1883, p. 62.
+
+[521] Symonds, loc. cit., p. 79.
+
+[522] J. A. Froude, _The Science of History_, London, 1864. "Un brevet
+d'apothicaire n'empecha pas Dante d'etre le plus grand poete de l'Italie,
+et ce fut un petit marchand de Pise qui donna l'algebre aux Chretiens."
+[Libri, _Histoire_, Vol. I, p. xvi.]
+
+[523] A document of 1226, found and published in 1858, reads: "Leonardo
+bigollo quondam Guilielmi."
+
+[524] "Bonaccingo germano suo."
+
+[525] E.g. Libri, Guglielmini, Tiraboschi.
+
+[526] Latin, _Bonaccius_.
+
+[527] Boncompagni and Milanesi.
+
+[528] Reprint, p. 5.
+
+[529] Whence the French name for candle.
+
+[530] Now part of Algiers.
+
+[531] E. Reclus, _Africa_, New York, 1893, Vol. II, p. 253.
+
+[532] "Sed hoc totum et algorismum atque arcus pictagore quasi errorem
+computavi respectu modi indorum." Woepcke, _Propagation_ etc., regards this
+as referring to two different systems, but the expression may very well
+mean algorism as performed upon the Pythagorean arcs (or table).
+
+[533] "Book of the Abacus," this term then being used, and long afterwards
+in Italy, to mean merely the arithmetic of computation.
+
+[534] "Incipit liber Abaci a Leonardo filio Bonacci compositus anno 1202 et
+correctus ab eodem anno 1228." Three MSS. of the thirteenth century are
+known, viz. at Milan, at Siena, and in the Vatican library. The work was
+first printed by Boncompagni in 1857.
+
+[535] I.e. in relation to the quadrivium. "Non legant in festivis diebus,
+nisi Philosophos et rhetoricas et quadrivalia et barbarismum et ethicam, si
+placet." Suter, _Die Mathematik auf den Universitaeten des Mittelalters_,
+Zuerich, 1887, p. 56. Roger Bacon gives a still more gloomy view of Oxford
+in his time in his _Opus minus_, in the _Rerum Britannicarum medii aevi
+scriptores_, London, 1859, Vol. I, p. 327. For a picture of Cambridge at
+this time consult F. W. Newman, _The English Universities, translated from
+the German of V. A. Huber_, London, 1843, Vol. I, p. 61; W. W. R. Ball,
+_History of Mathematics at Cambridge_, 1889; S. Guenther, _Geschichte des
+mathematischen Unterrichts im deutschen Mittelalter bis zum Jahre 1525_,
+Berlin, 1887, being Vol. III of _Monumenta Germaniae paedagogica_.
+
+[536] On the commercial activity of the period, it is known that bills of
+exchange passed between Messina and Constantinople in 1161, and that a bank
+was founded at Venice in 1170, the Bank of San Marco being established in
+the following year. The activity of Pisa was very manifest at this time.
+Heyd, loc. cit., Vol. II, p. 5; V. Casagrandi, _Storia e cronologia_, 3d
+ed., Milan, 1901, p. 56.
+
+[537] J. A. Symonds, loc. cit., Vol. II, p. 127.
+
+[538] I. Taylor, _The Alphabet_, London, 1883, Vol. II, p. 263.
+
+[539] Cited by Unger's History, p. 15. The Arabic numerals appear in a
+Regensburg chronicle of 1167 and in Silesia in 1340. See Schmidt's
+_Encyclopaedie der Erziehung_, Vol. VI, p. 726; A. Kuckuk, "Die Rechenkunst
+im sechzehnten Jahrhundert," _Festschrift zur dritten Saecularfeier des
+Berlinischen Gymnasiums zum grauen Kloster_, Berlin, 1874, p. 4.
+
+[540] The text is given in Halliwell, _Rara Mathematica_, London, 1839.
+
+[541] Seven are given in Ashmole's _Catalogue of Manuscripts in the Oxford
+Library_, 1845.
+
+[542] Maximilian Curtze, _Petri Philomeni de Dacia in Algorismum Vulgarem
+Johannis de Sacrobosco commentarius, una cum Algorismo ipso_, Copenhagen,
+1897; L. C. Karpinski, "Jordanus Nemorarius and John of Halifax," _American
+Mathematical Monthly_, Vol. XVII, pp. 108-113.
+
+[543] J. Aschbach, _Geschichte der Wiener Universitaet im ersten
+Jahrhunderte ihres Bestehens_, Wien, 1865, p. 93.
+
+[544] Curtze, loc. cit., gives the text.
+
+[545] Curtze, loc. cit., found some forty-five copies of the _Algorismus_
+in three libraries of Munich, Venice, and Erfurt (Amploniana). Examination
+of two manuscripts from the Plimpton collection and the Columbia library
+shows such marked divergence from each other and from the text published by
+Curtze that the conclusion seems legitimate that these were students'
+lecture notes. The shorthand character of the writing further confirms this
+view, as it shows that they were written largely for the personal use of
+the writers.
+
+[546] "Quidam philosophus edidit nomine Algus, unde et Algorismus
+nuncupatur." [Curtze, loc. cit., p. 1.]
+
+[547] "Sinistrorsum autera scribimus in hac arte more arabico sive iudaico,
+huius scientiae inventorum." [Curtze, loc. cit., p. 7.] The Plimpton
+manuscript omits the words "sive iudaico."
+
+[548] "Non enim omnis numerus per quascumque figuras Indorum
+repraesentatur, sed tantum determinatus per determinatam, ut 4 non per
+5,..." [Curtze, loc. cit., p. 25.]
+
+[549] C. Henry, "Sur les deux plus anciens traites francais d'algorisme et
+de geometrie," Boncompagni _Bulletino_, Vol. XV, p. 49; Victor Mortet, "Le
+plus ancien traite francais d'algorisme," loc. cit.
+
+[550] _L'Etat des sciences en France, depute la mort du Roy Robert, arrivee
+en 1031, jusqu'a celle de Philippe le Bel, arrivee en 1314_, Paris, 1741.
+
+[551] _Discours sur l'etat des lettres en France au XIII^e siecle_, Paris,
+1824.
+
+[552] _Apercu historique_, Paris, 1876 ed., p. 464.
+
+[553] Ranulf Higden, a native of the west of England, entered St.
+Werburgh's monastery at Chester in 1299. He was a Benedictine monk and
+chronicler, and died in 1364. His _Polychronicon_, a history in seven
+books, was printed by Caxton in 1480.
+
+[554] Trevisa's translation, Higden having written in Latin.
+
+[555] An illustration of this feeling is seen in the writings of Prosdocimo
+de' Beldomandi (b. c. 1370-1380, d. 1428): "Inveni in quam pluribus libris
+algorismi nuncupatis mores circa numeros operandi satis varios atque
+diversos, qui licet boni existerent atque veri erant, tamen fastidiosi, tum
+propter ipsarum regularum multitudinem, tum propter earum deleationes, tum
+etiam propter ipsarum operationum probationes, utrum si bone fuerint vel
+ne. Erant et etiam isti modi interim fastidiosi, quod si in aliquo calculo
+astroloico error contigisset, calculatorem operationem suam a capite
+incipere oportebat, dato quod error suus adhuc satis propinquus existeret;
+et hoc propter figuras in sua operatione deletas. Indigebat etiam
+calculator semper aliquo lapide vel sibi conformi, super quo scribere atque
+faciliter delere posset figuras cum quibus operabatur in calculo suo. Et
+quia haec omnia satis fastidiosa atque laboriosa mihi visa sunt, disposui
+libellum edere in quo omnia ista abicerentur: qui etiam algorismus sive
+liber de numeris denominari poterit. Scias tamen quod in hoc libello ponere
+non intendo nisi ea quae ad calculum necessaria sunt, alia quae in aliis
+libris practice arismetrice tanguntur, ad calculum non necessaria, propter
+brevitatem dimitendo." [Quoted by A. Nagl, _Zeitschrift fuer Mathematik und
+Physik, Hist.-lit. Abth._, Vol. XXXIV, p. 143; Smith, _Rara Arithmetica_,
+p. 14, in facsimile.]
+
+[556] P. Ewald, loc. cit.; Franz Steffens, _Lateinische Palaeographie_, pp.
+xxxix-xl. We are indebted to Professor J. M. Burnam for a photograph of
+this rare manuscript.
+
+[557] See the plate of forms on p. 88.
+
+[558] Karabacek, loc. cit., p. 56; Karpinski, "Hindu Numerals in the
+Fihrist," _Bibliotheca Mathematica_, Vol. XI (3), p. 121.
+
+[559] Woepcke, "Sur une donnee historique," etc., loc. cit., and "Essai
+d'une restitution de travaux perdus d'Apollonius sur les quantites
+irrationnelles, d'apres des indications tirees d'un manuscrit arabe," _Tome
+XIV des Memoires presentes par divers savants a l'Academie des sciences_,
+Paris, 1856, note, pp. 6-14.
+
+[560] _Archeological Report of the Egypt Exploration Fund for 1908-1909_,
+London, 1910, p. 18.
+
+[561] There was a set of astronomical tables in Boncompagni's library
+bearing this date: "Nota quod anno d[=n]i [=n]ri ihu x[=p]i. 1264.
+perfecto." See Narducci's _Catalogo_, p. 130.
+
+[562] "On the Early use of Arabic Numerals in Europe," read before the
+Society of Antiquaries April 14, 1910, and published in _Archaeologia_ in
+the same year.
+
+[563] Ibid., p. 8, n. The date is part of an Arabic inscription.
+
+[564] O. Codrington, _A Manual of Musalman Numismatics_, London, 1904.
+
+[565] See Arbuthnot, _The Mysteries of Chronology_, London, 1900, pp. 75,
+78, 98; F. Pichler, _Repertorium der steierischen Muenzkunde_, Graetz, 1875,
+where the claim is made of an Austrian coin of 1458; _Bibliotheca
+Mathematica_, Vol. X (2), p. 120, and Vol. XII (2), p. 120. There is a
+Brabant piece of 1478 in the collection of D. E. Smith.
+
+[566] A specimen is in the British Museum. [Arbuthnot, p. 79.]
+
+[567] Ibid., p. 79.
+
+[568] _Liber de Remediis utriusque fortunae Coloniae._
+
+[569] Fr. Walthern et Hans Hurning, Noerdlingen.
+
+[570] _Ars Memorandi_, one of the oldest European block-books.
+
+[571] Eusebius Caesariensis, _De praeparatione evangelica_, Venice, Jenson,
+1470. The above statement holds for copies in the Astor Library and in the
+Harvard University Library.
+
+[572] Francisco de Retza, _Comestorium vitiorum_, Nuernberg, 1470. The copy
+referred to is in the Astor Library.
+
+[573] See Mauch, "Ueber den Gebrauch arabischer Ziffern und die
+Veraenderungen derselben," _Anzeiger fuer Kunde der deutschen Vorzeit_, 1861,
+columns 46, 81, 116, 151, 189, 229, and 268; Calmet, _Recherches sur
+l'origine des chiffres d'arithmetique_, plate, loc. cit.
+
+[574] Guenther, _Geschichte_, p. 175, n.; Mauch, loc. cit.
+
+[575] These are given by W. R. Lethaby, from drawings by J. T. Irvine, in
+the _Proceedings of the Society of Antiquaries_, 1906, p. 200.
+
+[576] There are some ill-tabulated forms to be found in J. Bowring, _The
+Decimal System_, London, 1854, pp. 23, 25, and in L. A. Chassant,
+_Dictionnaire des abreviations latines et francaises ... du moyen age_,
+Paris, MDCCCLXVI, p. 113. The best sources we have at present, aside from
+the Hill monograph, are P. Treutlein, _Geschichte unserer Zahlzeichen_,
+Karlsruhe, 1875; Cantor's _Geschichte_, Vol. I, table; M. Prou, _Manuel de
+paleographie latine et francaise_, 2d ed., Paris, 1892, p. 164; A.
+Cappelli, _Dizionario di abbreviature latine ed italiane_, Milan, 1899. An
+interesting early source is found in the rare Caxton work of 1480, _The
+Myrrour of the World_. In Chap. X is a cut with the various numerals, the
+chapter beginning "The fourth scyence is called arsmetrique." Two of the
+fifteen extant copies of this work are at present in the library of Mr. J.
+P. Morgan, in New York.
+
+[577] From the twelfth-century manuscript on arithmetic, Curtze, loc. cit.,
+_Abhandlungen_, and Nagl, loc. cit. The forms are copied from Plate VII in
+_Zeitschrift fuer Mathematik und Physik_, Vol. XXXIV.
+
+[578] From the Regensburg chronicle. Plate containing some of these
+numerals in _Monumenta Germaniae historica_, "Scriptores" Vol. XVII, plate
+to p. 184; Wattenbach, _Anleitung zur lateinischen Palaeographie_, Leipzig,
+1886, p. 102; Boehmer, _Fontes rerum Germanicarum_, Vol. III, Stuttgart,
+1852, p. lxv.
+
+[579] French Algorismus of 1275; from an unpublished photograph of the
+original, in the possession of D. E. Smith. See also p. 135.
+
+[580] From a manuscript of Boethius c. 1294, in Mr. Plimpton's library.
+Smith, _Rara Arithmetica_, Plate I.
+
+[581] Numerals in a 1303 manuscript in Sigmaringen, copied from Wattenbach,
+loc. cit., p. 102.
+
+[582] From a manuscript, Add. Manuscript 27,589, British Museum, 1360 A.D.
+The work is a computus in which the date 1360 appears, assigned in the
+British Museum catalogue to the thirteenth century.
+
+[583] From the copy of Sacrabosco's _Algorismus_ in Mr. Plimpton's library.
+Date c. 1442. See Smith, _Rara Arithmetica_, p. 450.
+
+[584] See _Rara Arithmetica_, pp. 446-447.
+
+[585] Ibid., pp. 469-470.
+
+[586] Ibid., pp. 477-478.
+
+[587] The i is used for "one" in the Treviso arithmetic (1478), Clichtoveus
+(c. 1507 ed., where both i and j are so used), Chiarini (1481), Sacrobosco
+(1488 ed.), and Tzwivel (1507 ed., where jj and jz are used for 11 and 12).
+This was not universal, however, for the _Algorithmus linealis_ of c. 1488
+has a special type for 1. In a student's notebook of lectures taken at the
+University of Wuerzburg in 1660, in Mr. Plimpton's library, the ones are all
+in the form of i.
+
+[588] Thus the date [Numerals 1580], for 1580, appears in a MS. in the
+Laurentian library at Florence. The second and the following five
+characters are taken from Cappelli's _Dizionario_, p. 380, and are from
+manuscripts of the twelfth, thirteenth, fourteenth, sixteenth, seventeenth,
+and eighteenth centuries, respectively.
+
+[589] E.g. Chiarini's work of 1481; Clichtoveus (c. 1507).
+
+[590] The first is from an algorismus of the thirteenth century, in the
+Hannover Library. [See Gerhardt, "Ueber die Entstehung und Ausbreitung des
+dekadischen Zahlensystems," loc. cit., p. 28.] The second character is from
+a French algorismus, c. 1275. [Boncompagni _Bulletino_, Vol. XV, p. 51.]
+The third and the following sixteen characters are given by Cappelli, loc.
+cit., and are from manuscripts of the twelfth (1), thirteenth (2),
+fourteenth (7), fifteenth (3), sixteenth (1), seventeenth (2), and
+eighteenth (1) centuries, respectively.
+
+[591] Thus Chiarini (1481) has [Symbol] for 23.
+
+[592] The first of these is from a French algorismus, c. 1275. The second
+and the following eight characters are given by Cappelli, loc. cit., and
+are from manuscripts of the twelfth (2), thirteenth, fourteenth, fifteenth
+(3), seventeenth, and eighteenth centuries, respectively.
+
+[593] See Nagl, loc. cit.
+
+[594] Hannover algorismus, thirteenth century.
+
+[595] See the Dagomari manuscript, in _Rara Arithmetica_, pp. 435, 437-440.
+
+[596] But in the woodcuts of the _Margarita Philosophica_ (1503) the old
+forms are used, although the new ones appear in the text. In Caxton's
+_Myrrour of the World_ (1480) the old form is used.
+
+[597] Cappelli, loc. cit. They are partly from manuscripts of the tenth,
+twelfth, thirteenth (3), fourteenth (7), fifteenth (6), and eighteenth
+centuries, respectively. Those in the third line are from Chassant's
+_Dictionnaire_, p. 113, without mention of dates.
+
+[598] The first is from the Hannover algorismus, thirteenth century. The
+second is taken from the Rollandus manuscript, 1424. The others in the
+first two lines are from Cappelli, twelfth (3), fourteenth (6), fifteenth
+(13) centuries, respectively. The third line is from Chassant, loc. cit.,
+p. 113, no mention of dates.
+
+[599] The first of these forms is from the Hannover algorismus, thirteenth
+century. The following are from Cappelli, fourteenth (3), fifteenth,
+sixteenth (2), and eighteenth centuries, respectively.
+
+[600] The first of these is taken from the Hannover algorismus, thirteenth
+century. The following forms are from Cappelli, twelfth, thirteenth,
+fourteenth (5), fifteenth (2), seventeenth, and eighteenth centuries,
+respectively.
+
+[601] All of these are given by Cappelli, thirteenth, fourteenth, fifteenth
+(2), and sixteenth centuries, respectively.
+
+[602] Smith, _Rara Arithmetica_, p. 489. This is also seen in several of
+the Plimpton manuscripts, as in one written at Ancona in 1684. See also
+Cappelli, loc. cit.
+
+[603] French algorismus, c. 1275, for the first of these forms. Cappelli,
+thirteenth, fourteenth, fifteenth (3), and seventeenth centuries,
+respectively. The last three are taken from _Byzantinische Analekten_, J.
+L. Heiberg, being forms of the fifteenth century, but not at all common.
+[Symbol: Qoppa] was the old Greek symbol for 90.
+
+[604] For the first of these the reader is referred to the forms ascribed
+to Boethius, in the illustration on p. 88; for the second, to Radulph of
+Laon, see p. 60. The third is used occasionally in the Rollandus (1424)
+manuscript, in Mr. Plimpton's library. The remaining three are from
+Cappelli, fourteenth (2) and seventeenth centuries.
+
+[605] Smith, _An Early English Algorism_.
+
+[606] Kuckuck, p. 5.
+
+[607] A. Cappelli, loc. cit., p. 372.
+
+[608] Smith, _Rara Arithmetica_, p. 443.
+
+[609] Curtze, _Petri Philomeni de Dacia_ etc., p. IX.
+
+[610] Cappelli, loc. cit., p. 376.
+
+[611] Curtze, loc. cit., pp. VIII-IX, note.
+
+[612] Edition of 1544-1545, f. 52.
+
+[613] _De numeris libri II_, 1544 ed., cap. XV. Heilbronner, loc. cit., p.
+736, also gives them, and compares this with other systems.
+
+[614] Noviomagus says of them: "De quibusdam Astrologicis, sive Chaldaicis
+numerorum notis.... Sunt & aliae quaedam notae, quibus Chaldaei & Astrologii
+quemlibet numerum artificiose & argute describunt, scitu periucundae, quas
+nobis communicauit Rodolphus Paludanus Nouiomagus."
+
+
+
+
+
+
+End of the Project Gutenberg EBook of The Hindu-Arabic Numerals, by
+David Eugene Smith and Louis Charles Karpinski
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