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diff --git a/22599-8.txt b/22599-8.txt new file mode 100644 index 0000000..3093a11 --- /dev/null +++ b/22599-8.txt @@ -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: 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 + + * * * * * + + +ANNOUNCEMENTS + + * * * * * + + +WENTWORTH'S + +COLLEGE ALGEBRA + +REVISED EDITION + +12mo. Half morocco. 530 pages. List price, $1.50; mailing price, $1.65 + + * * * * * + +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. + +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. + +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. + +ADDITIONAL PUBLICATIONS + +By G. A. 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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 + +*** END OF THIS PROJECT GUTENBERG EBOOK THE HINDU-ARABIC NUMERALS *** + +***** This file should be named 22599-8.txt or 22599-8.zip ***** +This and all associated files of various formats will be found in: + https://www.gutenberg.org/2/2/5/9/22599/ + +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.) + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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