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text-indent: 1em;} + .figure p.in, .figcenter p.in, .figright p.in, .figleft p.in + {margin: 0; text-indent: 8em;} + .figcenter p.poem + {margin-left: 1em; text-align: left; text-indent: 0;} + .figcenter {margin: auto;} + .figright {float: right;} + .figleft {float: left;} + img.middle { border: none; vertical-align: middle } + + // --> + </style> + </head> +<body> + + +<pre> + +The Project Gutenberg EBook of The Hindu-Arabic Numerals, by +David Eugene Smith and Louis Charles Karpinski + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: The Hindu-Arabic Numerals + +Author: David Eugene Smith + Louis Charles Karpinski + +Release Date: September 14, 2007 [EBook #22599] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE HINDU-ARABIC NUMERALS *** + + + + +Produced by David Newman, Chuck Greif, Keith Edkins and +the Online Distributed Proofreading Team at +https://www.pgdp.net (This file was produced from images +from the Cornell University Library: Historical Mathematics +Monographs collection.) + + + + + + +</pre> + + +<table border="0" cellpadding="10" style="background-color: #ccccff;"> +<tr> +<td style="width:25%; vertical-align:top"> +Transcriber's note: +</td> +<td> +Sections in Greek or Hebrew will yield a transliteration +when the pointer is moved over them, and words using diacritic characters in the +Latin Extended Additional block, which may not display in some fonts or browsers, will +display an unaccented version. +</td> +</tr> +</table> + +<h3>THE</h3> + +<h2>HINDU-ARABIC NUMERALS</h2> + +<p class="cenhead">BY<br /> +DAVID EUGENE SMITH<br /> +AND<br /> +LOUIS CHARLES KARPINSKI</p> + +<p class="cenhead">BOSTON AND LONDON<br /> +GINN AND COMPANY, PUBLISHERS<br /> +1911</p> + +<p class="cenhead">COPYRIGHT, 1911, BY DAVID EUGENE SMITH<br /> +AND LOUIS CHARLES KARPINSKI<br /> +ALL RIGHTS RESERVED<br /> +811.7</p> + +<p class="cenhead"><b>The Athenæum Press</b><br /> +GINN AND COMPANY · PROPRIETORS<br /> +BOSTON · U.S.A.</p> + +<hr class="full" > + +<p><!-- Page iii --><span class="pagenum"><a name="pageiii"></a>[iii]</span></p> + +<h3>PREFACE</h3> + + <p>So familiar are we with the numerals that bear the misleading name of + Arabic, and so extensive is their use in Europe and the Americas, that it + is difficult for us to realize that their general acceptance in the + transactions of commerce is a matter of only the last four centuries, and + that they are unknown to a very large part of the human race to-day. It + seems strange that such a labor-saving device should have struggled for + nearly a thousand years after its system of place value was perfected + before it replaced such crude notations as the one that the Roman + conqueror made substantially universal in Europe. Such, however, is the + case, and there is probably no one who has not at least some slight + passing interest in the story of this struggle. To the mathematician and + the student of civilization the interest is generally a deep one; to the + teacher of the elements of knowledge the interest may be less marked, but + nevertheless it is real; and even the business man who makes daily use of + the curious symbols by which we express the numbers of commerce, cannot + fail to have some appreciation for the story of the rise and progress of + these tools of his trade.</p> + + <p>This story has often been told in part, but it is a long time since + any effort has been made to bring together the fragmentary narrations and + to set forth the general problem of the origin and development of these + <!-- Page iv --><span class="pagenum"><a + name="pageiv"></a>[iv]</span>numerals. In this little work we have + attempted to state the history of these forms in small compass, to place + before the student materials for the investigation of the problems + involved, and to express as clearly as possible the results of the labors + of scholars who have studied the subject in different parts of the world. + We have had no theory to exploit, for the history of mathematics has seen + too much of this tendency already, but as far as possible we have weighed + the testimony and have set forth what seem to be the reasonable + conclusions from the evidence at hand.</p> + + <p>To facilitate the work of students an index has been prepared which we + hope may be serviceable. In this the names of authors appear only when + some use has been made of their opinions or when their works are first + mentioned in full in a footnote.</p> + + <p>If this work shall show more clearly the value of our number system, + and shall make the study of mathematics seem more real to the teacher and + student, and shall offer material for interesting some pupil more fully + in his work with numbers, the authors will feel that the considerable + labor involved in its preparation has not been in vain.</p> + + <p>We desire to acknowledge our especial indebtedness to Professor + Alexander Ziwet for reading all the proof, as well as for the digest of a + Russian work, to Professor Clarence L. Meader for Sanskrit + transliterations, and to Mr. Steven T. Byington for Arabic + transliterations and the scheme of pronunciation of Oriental names, and + also our indebtedness to other scholars in Oriental learning for + information.</p> + + <p class="author">DAVID EUGENE SMITH + + <p class="author">LOUIS CHARLES KARPINSKI + +<hr class="full" > + +<p><!-- Page v --><span class="pagenum"><a name="pagev"></a>[v]</span></p> + +<h3>CONTENTS</h3> + + <div class="poem"> + <div class="stanza"> + <p>CHAPTER</p> + </div> + + <div class="stanza"> + <p class="i6">PRONUNCIATION OF ORIENTAL NAMES <a href="#pagevi">vi</a></p> + </div> + + <div class="stanza"> + <p>I. EARLY IDEAS OF THEIR ORIGIN <a href="#page1">1</a></p> + </div> + + <div class="stanza"> + <p>II. EARLY HINDU FORMS WITH NO PLACE VALUE <a href="#page12">12</a></p> + </div> + + <div class="stanza"> + <p>III. LATER HINDU FORMS, WITH A PLACE VALUE <a href="#page38">38</a></p> + </div> + + <div class="stanza"> + <p>IV. THE SYMBOL ZERO <a href="#page51">51</a></p> + </div> + + <div class="stanza"> + <p>V. THE QUESTION OF THE INTRODUCTION OF THE</p> + <p class="i6">NUMERALS INTO EUROPE BY BOETHIUS <a href="#page63">63</a></p> + </div> + + <div class="stanza"> + <p>VI. THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS <a href="#page91">91</a></p> + </div> + + <div class="stanza"> + <p>VII. THE DEFINITE INTRODUCTION OF THE NUMERALS INTO EUROPE <a href="#page99">99</a></p> + </div> + + <div class="stanza"> + <p>VIII. THE SPREAD OF THE NUMERALS IN EUROPE <a href="#page128">128</a></p> + </div> + + <div class="stanza"> + <p>INDEX <a href="#page153">153</a></p> + </div> + </div> +<hr class="full" > + +<p><!-- Page vi --><span class="pagenum"><a name="pagevi"></a>[vi]</span></p> + +<h3>PRONUNCIATION OF ORIENTAL NAMES</h3> + + <p>(S) = in Sanskrit names and words; (A) = in Arabic names and + words.</p> + + <p><b>b</b>, <b>d</b>, <b>f</b>, <b>g</b>, <b>h</b>, <b>j</b>, <b>l</b>, + <b>m</b>, <b>n</b>, <b>p</b>, <b>sh</b> (A), <b>t</b>, <b>th</b> (A), + <b>v</b>, <b>w</b>, <b>x</b>, <b>z</b>, as in English.</p> + + <p><b>a</b>, (S) like <i>u</i> in <i>but</i>: thus <i>pandit</i>, + pronounced <i>pundit</i>. (A) like <i>a</i> in <i>ask</i> or in + <i>man</i>. <b>ā</b>, as in <i>father</i>.</p> + + <p><b>c</b>, (S) like <i>ch</i> in <i>church</i> (Italian <i>c</i> in + <i>cento</i>).</p> + + <p><b><span class="special" title="d-dot-below">ḍ</span></b>, + <b><span class="special" title="n-dot-below">ṇ</span></b>, + <b><span class="special" title="s-dot-below">ṣ</span></b>, + <b><span class="special" title="t-dot-below">ṭ</span></b>, (S) + <i>d</i>, <i>n</i>, <i>sh</i>, <i>t</i>, made with the tip of the tongue + turned up and back into the dome of the palate. <b><span class="special" + title="d-dot-below">ḍ</span></b>, <b><span class="special" + title="s-dot-below">ṣ</span></b>, <b><span class="special" + title="t-dot-below">ṭ</span></b>, <b><span class="special" + title="z-dot-below">ẓ</span></b>, (A) <i>d</i>, <i>s</i>, + <i>t</i>, <i>z</i>, made with the tongue spread so that the sounds are + produced largely against the side teeth. Europeans commonly pronounce + <b><span class="special" title="d-dot-below">ḍ</span></b>, + <b><span class="special" title="n-dot-below">ṇ</span></b>, + <b><span class="special" title="s-dot-below">ṣ</span></b>, + <b><span class="special" title="t-dot-below">ṭ</span></b>, + <b><span class="special" title="z-dot-below">ẓ</span></b>, both + (S) and (A), as simple <i>d</i>, <i>n</i>, <i>sh</i> (S) or <i>s</i> (A), + <i>t</i>, <i>z</i>. <b><span class="special" + title="d-line-below">ḏ</span></b> (A), like <i>th</i> in + <i>this</i>.</p> + + <p><b>e</b>, (S) as in <i>they</i>. (A) as in <i>bed</i>.</p> + + <p><b>ġ</b>, (A) a voiced consonant formed below the vocal cords; + its sound is compared by some to a <i>g</i>, by others to a guttural + <i>r</i>; in Arabic words adopted into English it is represented by + <i>gh</i> (e.g. <i>ghoul</i>), less often <i>r</i> (e.g. + <i>razzia</i>).</p> + + <p><b>h</b> preceded by <i>b</i>, <i>c</i>, <i>t</i>, <i><span + class="special" title="t-dot-below">ṭ</span></i>, etc. does not + form a single sound with these letters, but is a more or less distinct + <i>h</i> sound following them; cf. the sounds in <i>abhor, boathook</i>, + etc., or, more accurately for (S), the "bhoys" etc. of Irish brogue. + <b>h</b> (A) retains its consonant sound at the end of a word. <b><span + class="special" title="h-dot-below">ḥ</span></b>, (A) an unvoiced + consonant formed below the vocal cords; its sound is sometimes compared + to German hard <i>ch</i>, and may be represented by an <i>h</i> as strong + as possible. In Arabic words adopted into English it is represented by + <i>h</i>, e.g. in <i>sahib</i>, <i>hakeem</i>. <b><span class="special" + title="h-dot-below">ḥ</span></b> (S) is final consonant <i>h</i>, + like final <i>h</i> (A).</p> + + <p><b>i</b>, as in <i>pin</i>. <b>ī</b>, as in <i>pique</i>.</p> + + <p><b>k</b>, as in <i>kick</i>.</p> + + <p><b>kh</b>, (A) the hard <i>ch</i> of Scotch <i>loch</i>, German + <i>ach</i>, especially of German as pronounced by the Swiss.</p> + + <p><b><span class="special" title="m-dot-above">ṁ</span></b>, + <b><span class="special" title="n-dot-above">ṅ</span></b>, (S) + like French final <i>m</i> or <i>n</i>, nasalizing the preceding + vowel.</p> + + <p><b><span class="special" title="n-dot-below">ṇ</span></b>, see + <b><span class="special" title="d-dot-below">ḍ</span></b>. + <b>ñ</b>, like <i>ng</i> in <i>singing</i>.</p> + + <p><b>o</b>, (S) as in <i>so</i>. (A) as in <i>obey</i>.</p> + + <p><b>q</b>, (A) like <i>k</i> (or <i>c</i>) in <i>cook</i>; further back + in the mouth than in <i>kick</i>.</p> + + <p><b>r</b>, (S) English <i>r</i>, smooth and untrilled. (A) stronger. + <b><span class="special" title="r-dot-below">ṛ</span></b>, (S) r + used as vowel, as in <i>apron</i> when pronounced <i>aprn</i> and not + <i>apern</i>; modern Hindus say <i>ri</i>, hence our <i>amrita</i>, + <i>Krishna</i>, for <i><span class="special" + title="a-mrta">a-mṛta</span>, <span class="special" + title="Krsna">Kṛṣṇa</span></i>.</p> + + <p><b>s</b>, as in <i>same</i>. <b><span class="special" + title="s-dot-below">ṣ</span></b>, see <b><span class="special" + title="d-dot-below">ḍ</span></b>. <b>ś</b>, (S) English + <i>sh</i> (German <i>sch</i>).</p> + + <p><b><span class="special" title="t-dot-below">ṭ</span></b>, see + <b><span class="special" title="d-dot-below">ḍ</span></b>.</p> + + <p><b>u</b>, as in <i>put</i>. <b>ū</b>, as in <i>rule</i>.</p> + + <p><b>y</b>, as in <i>you</i>.</p> + + <p><b><span class="special" title="z-dot-below">ẓ</span></b>, see + <b><span class="special" title="d-dot-below">ḍ</span></b>.</p> + + <p><b>‛</b>, (A) a sound kindred to the spiritus lenis (that is, + to our ears, the mere distinct separation of a vowel from the preceding + sound, as at the beginning of a word in German) and to <i><span + class="special" title="h-dot-below">ḥ</span></i>. 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 <i>Arab</i>, <i>amber</i>, + <i>Caaba</i> (<i>‛Arab</i>, <i>‛anbar</i>, + <i>ka‛abah</i>).</p> + + <p>(A) A final long vowel is shortened before <i>al</i> (<i>'l</i>) or + <i>ibn</i> (whose <i>i</i> is then silent).</p> + + <p>Accent: (S) as if Latin; in determining the place of the accent + <i><span class="special" title="m-dot-above">ṁ</span></i> and + <i><span class="special" title="n-dot-above">ṅ</span></i> count as + consonants, but <i>h</i> after another consonant does not. (A), on the + last syllable that contains a long vowel or a vowel followed by two + consonants, except that a final long vowel is not ordinarily accented; if + there is no long vowel nor two consecutive consonants, the accent falls + on the first syllable. The words <i>al</i> and <i>ibn</i> are never + accented.</p> + +<hr class="full" > + +<p><!-- Page 1 --><span class="pagenum"><a name="page1"></a>[1]</span></p> + +<h2>THE HINDU-ARABIC NUMERALS</h2> + +<h3>CHAPTER I</h3> + +<p class="cenhead">EARLY IDEAS OF THEIR ORIGIN</p> + + <p>It has long been recognized that the common numerals used in daily + life are of comparatively recent origin. The number of systems of + notation employed before the Christian era was about the same as the + number of written languages, and in some cases a single language had + several systems. The Egyptians, for example, had three systems of + writing, with a numerical notation for each; the Greeks had two + well-defined sets of numerals, and the Roman symbols for number changed + more or less from century to century. Even to-day the number of methods + of expressing numerical concepts is much greater than one would believe + before making a study of the subject, for the idea that our common + numerals are universal is far from being correct. It will be well, then, + to think of the numerals that we still commonly call Arabic, as only one + of many systems in use just before the Christian era. As it then existed + the system was no better than many others, it was of late origin, it + contained no zero, it was cumbersome and little used, <!-- Page 2 + --><span class="pagenum"><a name="page2"></a>[2]</span>and it had no + particular promise. Not until centuries later did the system have any + standing in the world of business and science; and had the place value + which now characterizes it, and which requires a zero, been worked out in + Greece, we might have been using Greek numerals to-day instead of the + ones with which we are familiar.</p> + + <p>Of the first number forms that the world used this is not the place to + speak. Many of them are interesting, but none had much scientific value. + In Europe the invention of notation was generally assigned to the eastern + shores of the Mediterranean until the critical period of about a century + ago,—sometimes to the Hebrews, sometimes to the Egyptians, but more + often to the early trading Phœnicians.<a name="NtA_1" + href="#Nt_1"><sup>[1]</sup></a></p> + + <p>The idea that our common numerals are Arabic in origin is not an old + one. The mediæval and Renaissance writers generally recognized them as + Indian, and many of them expressly stated that they were of Hindu + origin.<a name="NtA_2" href="#Nt_2"><sup>[2]</sup></a> <!-- Page 3 + --><span class="pagenum"><a name="page3"></a>[3]</span>Others argued that + they were probably invented by the Chaldeans or the Jews because they + increased in value from right to left, an argument that would apply quite + as well to the Roman and Greek systems, or to any other. It was, indeed, + to the general idea of notation that many of these writers referred, as + is evident from the words of England's earliest arithmetical + textbook-maker, Robert Recorde (c. 1542): "In that thinge all men do + agree, that the Chaldays, whiche fyrste inuented thys arte, did set these + figures as thei set all their letters. for they wryte backwarde as you + tearme it, and so doo they reade. And that may appeare in all Hebrewe, + Chaldaye and Arabike bookes ... where as the Greekes, Latines, and all + nations of Europe, do wryte and reade from the lefte hand towarde the + ryghte."<a name="NtA_3" href="#Nt_3"><sup>[3]</sup></a> Others, and <!-- + Page 4 --><span class="pagenum"><a name="page4"></a>[4]</span>among them + such influential writers as Tartaglia<a name="NtA_4" + href="#Nt_4"><sup>[4]</sup></a> in Italy and Köbel<a name="NtA_5" + href="#Nt_5"><sup>[5]</sup></a> in Germany, asserted the Arabic origin of + the numerals, while still others left the matter undecided<a name="NtA_6" + href="#Nt_6"><sup>[6]</sup></a> or simply dismissed them as "barbaric."<a + name="NtA_7" href="#Nt_7"><sup>[7]</sup></a> Of course the Arabs + themselves never laid claim to the invention, always recognizing their + indebtedness to the Hindus both for the numeral forms and for the + distinguishing feature of place value. Foremost among these writers was + the great master of the golden age of Bagdad, one of the first of the + Arab writers to collect the mathematical classics of both the East and + the West, preserving them and finally passing them on to awakening + Europe. This man was <span class="special" + title="Mohammed">Moḥammed</span> the Son of Moses, from + Khowārezm, or, more after the manner of the Arab, <span + class="special" title="Mohammed">Moḥammed</span> ibn + Mūsā al-Khowārazmī,<a name="NtA_8" + href="#Nt_8"><sup>[8]</sup></a> a man of great <!-- Page 5 --><span + class="pagenum"><a name="page5"></a>[5]</span>learning and one to whom + the world is much indebted for its present knowledge of algebra<a + name="NtA_9" href="#Nt_9"><sup>[9]</sup></a> and of arithmetic. Of him + there will often be occasion to speak; and in the arithmetic which he + wrote, and of which Adelhard of Bath<a name="NtA_10" + href="#Nt_10"><sup>[10]</sup></a> (c. 1130) may have made the translation + or paraphrase,<a name="NtA_11" href="#Nt_11"><sup>[11]</sup></a> he + stated distinctly that the numerals were due to the Hindus.<a + name="NtA_12" href="#Nt_12"><sup>[12]</sup></a> This is as plainly + asserted by later Arab <!-- Page 6 --><span class="pagenum"><a + name="page6"></a>[6]</span>writers, even to the present day.<a + name="NtA_13" href="#Nt_13"><sup>[13]</sup></a> Indeed the phrase + <i>‛ilm hindī</i>, "Indian science," is used by them for + arithmetic, as also the adjective <i>hindī</i> alone.<a + name="NtA_14" href="#Nt_14"><sup>[14]</sup></a></p> + + <p>Probably the most striking testimony from Arabic sources is that given + by the Arabic traveler and scholar <span class="special" title="Mohammed ibn Ahmed" + >Mohammed ibn Aḥmed</span>, <span class="special" title="Abu 'l-Rihan al-Biruni" + >Abū 'l-Rīḥān al-Bīrūnī</span> + (973-1048), who spent many years in Hindustan. He wrote a large work on + India,<a name="NtA_15" href="#Nt_15"><sup>[15]</sup></a> one on ancient + chronology,<a name="NtA_16" href="#Nt_16"><sup>[16]</sup></a> the "Book + of the Ciphers," unfortunately lost, which treated doubtless of the Hindu + art of calculating, and was the author of numerous other works. + Al-Bīrūnī was a man of unusual attainments, being + versed in Arabic, Persian, Sanskrit, Hebrew, and Syriac, as well as in + astronomy, chronology, and mathematics. In his work on India he gives + detailed information concerning the language and <!-- Page 7 --><span + class="pagenum"><a name="page7"></a>[7]</span>customs of the people of + that country, and states explicitly<a name="NtA_17" + href="#Nt_17"><sup>[17]</sup></a> that the Hindus of his time did not use + the letters of their alphabet for numerical notation, as the Arabs did. + He also states that the numeral signs called <i><span class="special" + title="anka">aṅka</span></i><a name="NtA_18" + href="#Nt_18"><sup>[18]</sup></a> had different shapes in various parts + of India, as was the case with the letters. In his <i>Chronology of + Ancient Nations</i> he gives the sum of a geometric progression and shows + how, in order to avoid any possibility of error, the number may be + expressed in three different systems: with Indian symbols, in sexagesimal + notation, and by an alphabet system which will be touched upon later. He + also speaks<a name="NtA_19" href="#Nt_19"><sup>[19]</sup></a> of "179, + 876, 755, expressed in Indian ciphers," thus again attributing these + forms to Hindu sources.</p> + + <p>Preceding Al-Bīrūnī there was another Arabic writer + of the tenth century, <span class="special" title="Motahhar ibn Tahir" + >Moṭahhar ibn Ṭāhir</span>,<a name="NtA_20" + href="#Nt_20"><sup>[20]</sup></a> author of the <i>Book of the Creation + and of History</i>, who gave as a curiosity, in Indian + (Nāgarī) symbols, a large number asserted by the people of + India to represent the duration of the world. Huart feels positive that + in <span class="special" title="Motahhar's">Moṭahhar's</span> time + the present Arabic symbols had not yet come into use, and that the Indian + symbols, although known to scholars, were not current. Unless this were + the case, neither the author nor his readers would have found anything + extraordinary in the appearance of the number which he cites.</p> + + <p>Mention should also be made of a widely-traveled student, + Al-Mas‛ūdī (885?-956), whose journeys carried him from + Bagdad to Persia, India, Ceylon, and even <!-- Page 8 --><span + class="pagenum"><a name="page8"></a>[8]</span>across the China sea, and + at other times to Madagascar, Syria, and Palestine.<a name="NtA_21" + href="#Nt_21"><sup>[21]</sup></a> He seems to have neglected no + accessible sources of information, examining also the history of the + Persians, the Hindus, and the Romans. Touching the period of the Caliphs + his work entitled <i>Meadows of Gold</i> furnishes a most entertaining + fund of information. He states<a name="NtA_22" + href="#Nt_22"><sup>[22]</sup></a> that the wise men of India, assembled + by the king, composed the <i>Sindhind</i>. Further on<a name="NtA_23" + href="#Nt_23"><sup>[23]</sup></a> he states, upon the authority of the + historian <span class="special" title="Mohammed">Moḥammed</span> + ibn ‛Alī ‛Abdī, that by order of <span + class="special" title="Al-Mansur">Al-Manṣūr</span> many + works of science and astrology were translated into Arabic, notably the + <i>Sindhind</i> (<i>Siddhānta</i>). Concerning the meaning and + spelling of this name there is considerable diversity of opinion. + Colebrooke<a name="NtA_24" href="#Nt_24"><sup>[24]</sup></a> first + pointed out the connection between <i>Siddhānta</i> and + <i>Sindhind</i>. He ascribes to the word the meaning "the revolving + ages."<a name="NtA_25" href="#Nt_25"><sup>[25]</sup></a> Similar + designations are collected by Sédillot,<a name="NtA_26" + href="#Nt_26"><sup>[26]</sup></a> who inclined to the Greek origin of the + sciences commonly attributed to the Hindus.<a name="NtA_27" + href="#Nt_27"><sup>[27]</sup></a> Casiri,<a name="NtA_28" + href="#Nt_28"><sup>[28]</sup></a> citing the <i><span class="special" + title="Tarikh al-hokama">Tārīkh + al-ḥokamā</span></i> or <i>Chronicles of the Learned</i>,<a + name="NtA_29" href="#Nt_29"><sup>[29]</sup></a> refers to the work <!-- + Page 9 --><span class="pagenum"><a name="page9"></a>[9]</span>as the + <i>Sindum-Indum</i> with the meaning "perpetuum æternumque." The + reference<a name="NtA_30" href="#Nt_30"><sup>[30]</sup></a> in this + ancient Arabic work to Al-Khowārazmī is worthy of note.</p> + + <p>This <i>Sindhind</i> is the book, says Mas‛ūdī,<a + name="NtA_31" href="#Nt_31"><sup>[31]</sup></a> which gives all that the + Hindus know of the spheres, the stars, arithmetic,<a name="NtA_32" + href="#Nt_32"><sup>[32]</sup></a> and the other branches of science. He + mentions also Al-Khowārazmī and <span class="special" + title="Habash">Ḥabash</span><a name="NtA_33" + href="#Nt_33"><sup>[33]</sup></a> as translators of the tables of the + <i>Sindhind</i>. Al-Bīrūnī<a name="NtA_34" + href="#Nt_34"><sup>[34]</sup></a> refers to two other translations from a + work furnished by a Hindu who came to Bagdad as a member of the political + mission which Sindh sent to the caliph <span class="special" + title="Al-Mansur">Al-Manṣūr</span>, in the year of the + Hejira 154 (<span class="scac">A.D.</span> 771).</p> + + <p>The oldest work, in any sense complete, on the history of Arabic + literature and history is the <i>Kitāb al-Fihrist</i>, written in + the year 987 <span class="scac">A.D.</span>, by Ibn Abī + Ya‛qūb al-Nadīm. It is of fundamental importance for + the history of Arabic culture. Of the ten chief divisions of the work, + the seventh demands attention in this discussion for the reason that its + second subdivision treats of mathematicians and astronomers.<a + name="NtA_35" href="#Nt_35"><sup>[35]</sup></a></p> + +<p><!-- Page 10 --><span class="pagenum"><a name="page10"></a>[10]</span></p> + + <p>The first of the Arabic writers mentioned is Al-Kindī (800-870 + <span class="scac">A.D.</span>), who wrote five books on arithmetic and + four books on the use of the Indian method of reckoning. Sened ibn + ‛Alī, the Jew, who was converted to Islam under the caliph + Al-Māmūn, is also given as the author of a work on the Hindu + method of reckoning. Nevertheless, there is a possibility<a name="NtA_36" + href="#Nt_36"><sup>[36]</sup></a> that some of the works ascribed to + Sened ibn ‛Alī are really works of + Al-Khowārazmī, whose name immediately precedes his. However, + it is to be noted in this connection that Casiri<a name="NtA_37" + href="#Nt_37"><sup>[37]</sup></a> also mentions the same writer as the + author of a most celebrated work on arithmetic.</p> + + <p>To <span class="special" + title="Al-Sufi">Al-Ṣūfī</span>, who died in 986 <span + class="scac">A.D.</span>, is also credited a large work on the same + subject, and similar treatises by other writers are mentioned. We are + therefore forced to the conclusion that the Arabs from the early ninth + century on fully recognized the Hindu origin of the new numerals.</p> + + <p>Leonard of Pisa, of whom we shall speak at length in the chapter on + the Introduction of the Numerals into Europe, wrote his <i>Liber + Abbaci</i><a name="NtA_38" href="#Nt_38"><sup>[38]</sup></a> in 1202. In + this work he refers frequently to the nine Indian figures,<a + name="NtA_39" href="#Nt_39"><sup>[39]</sup></a> thus showing again the + general consensus of opinion in the Middle Ages that the numerals were of + Hindu origin.</p> + + <p>Some interest also attaches to the oldest documents on arithmetic in + our own language. One of the earliest <!-- Page 11 --><span + class="pagenum"><a name="page11"></a>[11]</span>treatises on algorism is + a commentary<a name="NtA_40" href="#Nt_40"><sup>[40]</sup></a> on a set + of verses called the <i>Carmen de Algorismo</i>, written by Alexander de + Villa Dei (Alexandra de Ville-Dieu), a Minorite monk of about 1240 <span + class="scac">A.D.</span> The text of the first few lines is as + follows:</p> + + <div class="poem"> + <div class="stanza"> + <p class="hg3">"Hec algorism' ars p'sens dicit' in qua</p> + <p>Talib; indor<a href="images/017a.png"><img src="images/017a.png" class="middle" style="height:1.2ex" alt="um" /></a> fruim bis quinq; figuris.<a name="NtA_41" href="#Nt_41"><sup>[41]</sup></a></p> + </div> + </div> + <p>"This boke is called the boke of algorim or augrym after lewder use. + And this boke tretys of the Craft of Nombryng, the quych crafte is called + also Algorym. Ther was a kyng of Inde the quich heyth Algor & he made + this craft.... Algorisms, in the quych we use teen figurys of Inde."</p> + +<hr class="full" > + +<p><!-- Page 12 --><span class="pagenum"><a name="page12"></a>[12]</span></p> + +<h3>CHAPTER II</h3> + +<p class="cenhead">EARLY HINDU FORMS WITH NO PLACE VALUE</p> + + <p>While it is generally conceded that the scientific development of + astronomy among the Hindus towards the beginning of the Christian era + rested upon Greek<a name="NtA_42" href="#Nt_42"><sup>[42]</sup></a> or + Chinese<a name="NtA_43" href="#Nt_43"><sup>[43]</sup></a> sources, yet + their ancient literature testifies to a high state of civilization, and + to a considerable advance in sciences, in philosophy, and along literary + lines, long before the golden age of Greece. From the earliest times even + up to the present day the Hindu has been wont to put his thought into + rhythmic form. The first of this poetry—it well deserves this name, + being also worthy from a metaphysical point of view<a name="NtA_44" + href="#Nt_44"><sup>[44]</sup></a>—consists of the Vedas, hymns of + praise and poems of worship, collected during the Vedic period which + dates from approximately 2000 <span class="scac">B.C.</span> to 1400 + <span class="scac">B.C.</span><a name="NtA_45" + href="#Nt_45"><sup>[45]</sup></a> Following this work, or possibly + contemporary with it, is the Brahmanic literature, which is partly + ritualistic (the <span class="special" + title="Brahmanas">Brāhmaṇas</span>), and partly + philosophical (the Upanishads). Our especial interest is <!-- Page 13 + --><span class="pagenum"><a name="page13"></a>[13]</span>in the + Sūtras, versified abridgments of the ritual and of ceremonial + rules, which contain considerable geometric material used in connection + with altar construction, and also numerous examples of rational numbers + the sum of whose squares is also a square, i.e. "Pythagorean numbers," + although this was long before Pythagoras lived. Whitney<a name="NtA_46" + href="#Nt_46"><sup>[46]</sup></a> places the whole of the Veda + literature, including the Vedas, the <span class="special" + title="Brahmanas">Brāhmaṇas</span>, and the Sūtras, + between 1500 <span class="scac">B.C.</span> and 800 <span + class="scac">B.C.</span>, thus agreeing with Bürk<a name="NtA_47" + href="#Nt_47"><sup>[47]</sup></a> who holds that the knowledge of the + Pythagorean theorem revealed in the Sūtras goes back to the eighth + century <span class="scac">B.C.</span></p> + + <p>The importance of the Sūtras as showing an independent origin of + Hindu geometry, contrary to the opinion long held by Cantor<a + name="NtA_48" href="#Nt_48"><sup>[48]</sup></a> of a Greek origin, has + been repeatedly emphasized in recent literature,<a name="NtA_49" + href="#Nt_49"><sup>[49]</sup></a> especially since the appearance of the + important work of Von Schroeder.<a name="NtA_50" + href="#Nt_50"><sup>[50]</sup></a> Further fundamental mathematical + notions such as the conception of irrationals and the use of gnomons, as + well as the philosophical doctrine of the transmigration of + souls,—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 <!-- Page 14 --><span + class="pagenum"><a name="page14"></a>[14]</span>origin of our numerals, + yet it is highly pertinent as showing the aptitude of the Hindu for + mathematical and mental work, a fact further attested by the independent + development of the drama and of epic and lyric poetry.</p> + + <p>It should be stated definitely at the outset, however, that we are not + at all sure that the most ancient forms of the numerals commonly known as + Arabic had their origin in India. As will presently be seen, their forms + may have been suggested by those used in Egypt, or in Eastern Persia, or + in China, or on the plains of Mesopotamia. We are quite in the dark as to + these early steps; but as to their development in India, the approximate + period of the rise of their essential feature of place value, their + introduction into the Arab civilization, and their spread to the West, we + have more or less definite information. When, therefore, we consider the + rise of the numerals in the land of the Sindhu,<a name="NtA_51" + href="#Nt_51"><sup>[51]</sup></a> it must be understood that it is only + the large movement that is meant, and that there must further be + considered the numerous possible sources outside of India itself and long + anterior to the first prominent appearance of the number symbols.</p> + + <p>No one attempts to examine any detail in the history of ancient India + without being struck with the great dearth of reliable material.<a + name="NtA_52" href="#Nt_52"><sup>[52]</sup></a> So little sympathy have + the people with any save those of their own caste that a general + literature is wholly lacking, and it is only in the observations of + strangers that any all-round view of scientific progress is to be found. + There is evidence that primary schools <!-- Page 15 --><span + class="pagenum"><a name="page15"></a>[15]</span>existed in earliest + times, and of the seventy-two recognized sciences writing and arithmetic + were the most prized.<a name="NtA_53" href="#Nt_53"><sup>[53]</sup></a> + In the Vedic period, say from 2000 to 1400 <span + class="scac">B.C.</span>, there was the same attention to astronomy that + was found in the earlier civilizations of Babylon, China, and Egypt, a + fact attested by the Vedas themselves.<a name="NtA_54" + href="#Nt_54"><sup>[54]</sup></a> Such advance in science presupposes a + fair knowledge of calculation, but of the manner of calculating we are + quite ignorant and probably always shall be. One of the Buddhist sacred + books, the <i>Lalitavistara</i>, relates that when the + Bōdhisattva<a name="NtA_55" href="#Nt_55"><sup>[55]</sup></a> was + of age to marry, the father of Gopa, his intended bride, demanded an + examination of the five hundred suitors, the subjects including + arithmetic, writing, the lute, and archery. Having vanquished his rivals + in all else, he is matched against Arjuna the great arithmetician and is + asked to express numbers greater than 100 kotis.<a name="NtA_56" + href="#Nt_56"><sup>[56]</sup></a> In reply he gave a scheme of number + names as high as 10<sup>53</sup>, adding that he could proceed as far as + 10<sup>421</sup>,<a name="NtA_57" href="#Nt_57"><sup>[57]</sup></a> all + of which suggests the system of Archimedes and the unsettled question of + the indebtedness of the West to the East in the realm of ancient + mathematics.<a name="NtA_58" href="#Nt_58"><sup>[58]</sup></a> Sir Edwin + Arnold, <!-- Page 16 --><span class="pagenum"><a + name="page16"></a>[16]</span>in <i>The Light of Asia</i>, does not + mention this part of the contest, but he speaks of Buddha's training at + the hands of the learned <span class="special" + title="Visvamitra">Viṣvamitra</span>:</p> + + <div class="poem"> + <div class="stanza"> + <p>"And Viswamitra said, 'It is enough,</p> + <p>Let us to numbers. After me repeat</p> + <p>Your numeration till we reach the lakh,<a name="NtA_59" href="#Nt_59"><sup>[59]</sup></a></p> + <p>One, two, three, four, to ten, and then by tens</p> + <p>To hundreds, thousands.' After him the child</p> + <p>Named digits, decads, centuries, nor paused,</p> + <p>The round lakh reached, but softly murmured on,</p> + <p>Then comes the kōti, nahut, ninnahut,</p> + <p>Khamba, viskhamba, abab, attata,</p> + <p>To kumuds, gundhikas, and utpalas,</p> + <p>By pundarīkas into padumas,</p> + <p>Which last is how you count the utmost grains</p> + <p>Of Hastagiri ground to finest dust;<a name="NtA_60" href="#Nt_60"><sup>[60]</sup></a></p> + <p>But beyond that a numeration is,</p> + <p>The Kātha, used to count the stars of night,</p> + <p>The Kōti-Kātha, for the ocean drops;</p> + <p>Ingga, the calculus of circulars;</p> + <p>Sarvanikchepa, by the which you deal</p> + <p>With all the sands of Gunga, till we come</p> + <p>To Antah-Kalpas, where the unit is</p> + <p>The sands of the ten crore Gungas. If one seeks</p> + <p>More comprehensive scale, th' arithmic mounts</p> + <p>By the Asankya, which is the tale</p> + <p>Of all the drops that in ten thousand years</p> + <p>Would fall on all the worlds by daily rain;</p> + <p>Thence unto Maha Kalpas, by the which</p> + <p>The gods compute their future and their past.'"</p> + </div> + </div> +<p><!-- Page 17 --><span class="pagenum"><a name="page17"></a>[17]</span></p> + + <p>Thereupon <span class="special" title="Visvamitra Acarya" + >Viṣvamitra Ācārya</span><a name="NtA_61" + href="#Nt_61"><sup>[61]</sup></a> expresses his approval of the task, and + asks to hear the "measure of the line" as far as yōjana, the + longest measure bearing name. This given, Buddha adds:</p> + + <div class="poem"> + <div class="stanza"> + <p>... "'And master! if it please,</p> + <p>I shall recite how many sun-motes lie</p> + <p>From end to end within a yōjana.'</p> + <p>Thereat, with instant skill, the little prince</p> + <p>Pronounced the total of the atoms true.</p> + <p>But Viswamitra heard it on his face</p> + <p>Prostrate before the boy; 'For thou,' he cried,</p> + <p>'Art Teacher of thy teachers—thou, not I,</p> + <p>Art Gūrū.'"</p> + </div> + </div> + <p>It is needless to say that this is far from being history. And yet it + puts in charming rhythm only what the ancient <i>Lalitavistara</i> + relates of the number-series of the Buddha's time. While it extends + beyond all reason, nevertheless it reveals a condition that would have + been impossible unless arithmetic had attained a considerable degree of + advancement.</p> + + <p>To this pre-Christian period belong also the <i><span class="special" + title="Vedangas">Vedāṅgas</span></i>, or "limbs for + supporting the Veda," part of that great branch of Hindu literature known + as <i><span class="special" title="Smriti">Smṛiti</span></i> + (recollection), that which was to be handed down by tradition. Of these + the sixth is known as <i><span class="special" + title="Jyotisa">Jyotiṣa</span></i> (astronomy), a short treatise + of only thirty-six verses, written not earlier than 300 <span + class="scac">B.C.</span>, and affording us some knowledge of the extent + of number work in that period.<a name="NtA_62" + href="#Nt_62"><sup>[62]</sup></a> The Hindus <!-- Page 18 --><span + class="pagenum"><a name="page18"></a>[18]</span>also speak of eighteen + ancient Siddhāntas or astronomical works, which, though mostly + lost, confirm this evidence.<a name="NtA_63" + href="#Nt_63"><sup>[63]</sup></a></p> + + <p>As to authentic histories, however, there exist in India none relating + to the period before the Mohammedan era (622 <span + class="scac">A.D.</span>). About all that we know of the earlier + civilization is what we glean from the two great epics, the + Mahābhārata<a name="NtA_64" href="#Nt_64"><sup>[64]</sup></a> + and the Rāmāyana, from coins, and from a few inscriptions.<a + name="NtA_65" href="#Nt_65"><sup>[65]</sup></a></p> + + <p>It is with this unsatisfactory material, then, that we have to deal in + searching for the early history of the Hindu-Arabic numerals, and the + fact that many unsolved problems exist and will continue to exist is no + longer strange when we consider the conditions. It is rather surprising + that so much has been discovered within a century, than that we are so + uncertain as to origins and dates and the early spread of the system. The + probability being that writing was not introduced into India before the + close of the fourth century <span class="scac">B.C.</span>, and + literature existing only in spoken form prior to that period,<a + name="NtA_66" href="#Nt_66"><sup>[66]</sup></a> the number work was + doubtless that of all primitive peoples, palpable, merely a matter of + placing sticks or cowries or pebbles on the ground, of marking a + sand-covered board, or of cutting notches or tying cords as is still done + in parts of Southern India to-day.<a name="NtA_67" + href="#Nt_67"><sup>[67]</sup></a></p> + +<p><!-- Page 19 --><span class="pagenum"><a name="page19"></a>[19]</span></p> + + <p>The early Hindu numerals<a name="NtA_68" + href="#Nt_68"><sup>[68]</sup></a> may be classified into three great + groups, (1) the <span class="special" + title="Kharosthi">Kharoṣṭhī</span>, (2) the + Brāhmī, and (3) the word and letter forms; and these will be + considered in order.</p> + + <p>The <span class="special" + title="Kharosthi">Kharoṣṭhī</span> numerals are found + in inscriptions formerly known as Bactrian, Indo-Bactrian, and Aryan, and + appearing in ancient Gandhāra, now eastern Afghanistan and northern + Punjab. The alphabet of the language is found in inscriptions dating from + the fourth century <span class="scac">B.C.</span> to the third century + <span class="scac">A.D.</span>, and from the fact that the words are + written from right to left it is assumed to be of Semitic origin. No + numerals, however, have been found in the earliest of these inscriptions, + number-names probably having been written out in words as was the custom + with many ancient peoples. Not until the time of the powerful King + Aśoka, in the third century <span class="scac">B.C.</span>, do + numerals appear in any inscriptions thus far discovered; and then only in + the primitive form of marks, quite as they would be found in Egypt, + Greece, Rome, or in <!-- Page 20 --><span class="pagenum"><a + name="page20"></a>[20]</span>various other parts of the world. These + Aśoka<a name="NtA_69" href="#Nt_69"><sup>[69]</sup></a> + inscriptions, some thirty in all, are found in widely separated parts of + India, often on columns, and are in the various vernaculars that were + familiar to the people. Two are in the <span class="special" + title="Kharosthi">Kharoṣṭhī</span> characters, and + the rest in some form of Brāhmī. In the <span class="special" + title="Kharosthi">Kharoṣṭhī</span> inscriptions only + four numerals have been found, and these are merely vertical marks for + one, two, four, and five, thus:</p> + + <div class="figcenter" style="width:25%;"> + <a href="images/026a.png"><img style="width:100%" src="images/026a.png" + alt="Numerals in Kharosthi inscriptions." title="Numerals in Kharosthi inscriptions." /></a> + </div> + <p>In the so-called Śaka inscriptions, possibly of the first + century <span class="scac">B.C.</span>, more numerals are found, and in + more highly developed form, the right-to-left system appearing, together + with evidences of three different scales of counting,—four, ten, + and twenty. The numerals of this period are as follows:</p> + + <div class="figcenter" style="width:50%;"> + <a href="images/026b.png"><img style="width:100%" src="images/026b.png" + alt="Numerals in Saka inscriptions." title="Numerals in Saka inscriptions." /></a> + </div> + <p>There are several noteworthy points to be observed in studying this + system. In the first place, it is probably not as early as that shown in + the Nānā Ghāt forms hereafter given, although the + inscriptions themselves at Nānā Ghāt are later than + those of the Aśoka period. The <!-- Page 21 --><span + class="pagenum"><a name="page21"></a>[21]</span>four is to this system + what the X was to the Roman, probably a canceling of three marks as a + workman does to-day for five, or a laying of one stick across three + others. The ten has never been satisfactorily explained. It is similar to + the A of the <span class="special" + title="Kharosthi">Kharoṣṭhī</span> alphabet, but we + have no knowledge as to why it was chosen. The twenty is evidently a + ligature of two tens, and this in turn suggested a kind of radix, so that + ninety was probably written in a way reminding one of the + quatre-vingt-dix of the French. The hundred is unexplained, although it + resembles the letter <i>ta</i> or <i>tra</i> of the Brāhmī + alphabet with 1 before (to the right of) it. The two hundred is only a + variant of the symbol for hundred, with two vertical marks.<a + name="NtA_70" href="#Nt_70"><sup>[70]</sup></a></p> + + <p>This system has many points of similarity with the Nabatean numerals<a + name="NtA_71" href="#Nt_71"><sup>[71]</sup></a> in use in the first + centuries of the Christian era. The cross is here used for four, and the + <span class="special" + title="Kharosthi">Kharoṣṭhī</span> form is employed + for twenty. In addition to this there is a trace of an analogous use of a + scale of twenty. While the symbol for 100 is quite different, the method + of forming the other hundreds is the same. The correspondence seems to be + too marked to be wholly accidental.</p> + + <p>It is not in the <span class="special" + title="Kharosthi">Kharoṣṭhī</span> numerals, + therefore, that we can hope to find the origin of those used by us, and + we turn to the second of the Indian types, the Brāhmī + characters. The alphabet attributed to Brahmā is the oldest of the + several known in India, and was used from the earliest historic times. + There are various theories of its origin, <!-- Page 22 --><span + class="pagenum"><a name="page22"></a>[22]</span>none of which has as yet + any wide acceptance,<a name="NtA_72" href="#Nt_72"><sup>[72]</sup></a> + although the problem offers hope of solution in due time. The numerals + are not as old as the alphabet, or at least they have not as yet been + found in inscriptions earlier than those in which the edicts of + Aśoka appear, some of these having been incised in + Brāhmī as well as <span class="special" + title="Kharosthi">Kharoṣṭhī</span>. As already + stated, the older writers probably wrote the numbers in words, as seems + to have been the case in the earliest Pali writings of Ceylon.<a + name="NtA_73" href="#Nt_73"><sup>[73]</sup></a></p> + + <p>The following numerals are, as far as known, the only ones to appear + in the Aśoka edicts:<a name="NtA_74" + href="#Nt_74"><sup>[74]</sup></a></p> + + <div class="figcenter" style="width:50%;"> + <a href="images/028a.png"><img style="width:100%" src="images/028a.png" + alt="Numerals in Asoka edicts." title="Numerals in Asoka edicts." /></a> + </div> + <p>These fragments from the third century <span class="scac">B.C.</span>, + crude and unsatisfactory as they are, are the undoubted early forms from + which our present system developed. They next appear in the second + century <span class="scac">B.C.</span> in some inscriptions in the cave + on the top of the Nānā Ghāt hill, about seventy-five + miles from Poona in central India. These inscriptions may be memorials of + the early Andhra dynasty of southern India, but their chief interest lies + in the numerals which they contain.</p> + + <p>The cave was made as a resting-place for travelers ascending the hill, + which lies on the road from Kalyāna to Junar. It seems to have been + cut out by a descendant <!-- Page 23 --><span class="pagenum"><a + name="page23"></a>[23]</span>of King Śātavāhana,<a + name="NtA_75" href="#Nt_75"><sup>[75]</sup></a> for inside the wall + opposite the entrance are representations of the members of his family, + much defaced, but with the names still legible. It would seem that the + excavation was made by order of a king named Vedisiri, and "the + inscription contains a list of gifts made on the occasion of the + performance of several <i>yagnas</i> or religious sacrifices," and + numerals are to be seen in no less than thirty places.<a name="NtA_76" + href="#Nt_76"><sup>[76]</sup></a></p> + + <p>There is considerable dispute as to what numerals are really found in + these inscriptions, owing to the difficulty of deciphering them; but the + following, which have been copied from a rubbing, are probably number + forms:<a name="NtA_77" href="#Nt_77"><sup>[77]</sup></a></p> + + <div class="figcenter" style="width:50%;"> + <a href="images/029a.png"><img style="width:100%" src="images/029a.png" + alt="Numerals from Nana Ghat inscriptions." title="Numerals from Nana Ghat inscriptions." /></a> + </div> + <p>The inscription itself, so important as containing the earliest + considerable Hindu numeral system connected with our own, is of + sufficient interest to warrant reproducing part of it in facsimile, as is + done on page 24.</p> + +<p><!-- Page 24 --><span class="pagenum"><a name="page24"></a>[24]</span></p> + + <div class="figcenter" style="width:50%;"> + <a href="images/030a.png"><img style="width:100%" src="images/030a.png" + alt="Nânâghât Inscriptions." title="Nânâghât Inscriptions." /></a> + </div> + <p>The next very noteworthy evidence of the numerals, and this quite + complete as will be seen, is found in certain other cave inscriptions + dating back to the first or second century <span class="scac">A.D.</span> + In these, the Nasik<a name="NtA_78" href="#Nt_78"><sup>[78]</sup></a> + cave inscriptions, the forms are as follows:</p> + + <div class="figcenter" style="width:50%;"> + <a href="images/030b.png"><img style="width:100%" src="images/030b.png" + alt="Numerals from Nasik cave inscriptions." title="Numerals from Nasik cave inscriptions." /></a> + </div> + <p>From this time on, until the decimal system finally adopted the first + nine characters and replaced the rest of the Brāhmī notation + by adding the zero, the progress of these forms is well marked. It is + therefore well to present synoptically the best-known specimens that have + come down to us, and this is done in the table on page 25.<a + name="NtA_79" href="#Nt_79"><sup>[79]</sup></a></p> + +<p><!-- Page 25 --><span class="pagenum"><a name="page25"></a>[25]</span></p> + +<h3><span class="sc">Table showing the Progress of Number Forms in India</span></h3> + +<table class="nobctr"> +<tr><td><span class="sc">Numerals</span></td><td><a href="images/031.png"><img src="images/031.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr> +<tr><td>Aśoka<a name="NtA_80" href="#Nt_80"><sup>[80]</sup></a></td><td><a href="images/031a.png"><img src="images/031a.png" class="middle" style="height:4.5ex" alt="Asoka" /></a></td></tr> +<tr><td>Śaka<a name="NtA_81" href="#Nt_81"><sup>[81]</sup></a></td><td><a href="images/031b.png"><img src="images/031b.png" class="middle" style="height:4.5ex" alt="Saka" /></a></td></tr> +<tr><td>Aśoka<a name="NtA_82" href="#Nt_82"><sup>[82]</sup></a></td><td><a href="images/031c.png"><img src="images/031c.png" class="middle" style="height:4.5ex" alt="Asoka" /></a></td></tr> +<tr><td>Nāgarī<a name="NtA_83" href="#Nt_83"><sup>[83]</sup></a></td><td><a href="images/031d.png"><img src="images/031d.png" class="middle" style="height:4.5ex" alt="Nagari" /></a></td></tr> +<tr><td>Nasik<a name="NtA_84" href="#Nt_84"><sup>[84]</sup></a></td><td><a href="images/031e.png"><img src="images/031e.png" class="middle" style="height:4.5ex" alt="Nasik" /></a></td></tr> +<tr><td><span class="special" title="Ksatrapa">Kṣatrapa</span><a name="NtA_85" href="#Nt_85"><sup>[85]</sup></a></td><td><a href="images/031f.png"><img src="images/031f.png" class="middle" style="height:4.5ex" alt="Ksatrapa" /></a></td></tr> +<tr><td><span class="special" title="Kusana">Kuṣana</span><a name="NtA_86" href="#Nt_86"><sup>[86]</sup></a></td><td><a href="images/031g.png"><img src="images/031g.png" class="middle" style="height:4.5ex" alt="Kusana" /></a></td></tr> +<tr><td>Gupta<a name="NtA_87" href="#Nt_87"><sup>[87]</sup></a></td><td><a href="images/031h.png"><img src="images/031h.png" class="middle" style="height:4.5ex" alt="Gupta" /></a></td></tr> +<tr><td>Valhabī<a name="NtA_88" href="#Nt_88"><sup>[88]</sup></a></td><td><a href="images/031i.png"><img src="images/031i.png" class="middle" style="height:4.5ex" alt="Valhabi" /></a></td></tr> +<tr><td>Nepal<a name="NtA_89" href="#Nt_89"><sup>[89]</sup></a></td><td><a href="images/031j.png"><img src="images/031j.png" class="middle" style="height:4.5ex" alt="Nepal" /></a></td></tr> +<tr><td><span class="special" title="Kalinga">Kaliṅga</span><a name="NtA_90" href="#Nt_90"><sup>[90]</sup></a></td><td><a href="images/031k.png"><img src="images/031k.png" class="middle" style="height:4.5ex" alt="Kalinga" /></a></td></tr> +<tr><td><span class="special" title="Vakataka">Vākāṭaka</span><a name="NtA_91" href="#Nt_91"><sup>[91]</sup></a></td><td><a href="images/031l.png"><img src="images/031l.png" class="middle" style="height:4.5ex" alt="Vakataka" /></a></td></tr> +</table> + + <p>[Most of these numerals are given by Bühler, loc. cit., Tafel IX.]</p> + +<p><!-- Page 26 --><span class="pagenum"><a name="page26"></a>[26]</span></p> + + <p>With respect to these numerals it should first be noted that no zero + appears in the table, and as a matter of fact none existed in any of the + cases cited. It was therefore impossible to have any place value, and the + numbers like twenty, thirty, and other multiples of ten, one hundred, and + so on, required separate symbols except where they were written out in + words. The ancient Hindus had no less than twenty of these symbols,<a + name="NtA_92" href="#Nt_92"><sup>[92]</sup></a> a number that was + afterward greatly increased. The following are examples of their method + of indicating certain numbers between one hundred and one thousand:</p> + + <div class="poem"> + <div class="stanza"> + <p><a name="NtA_93" href="#Nt_93"><sup>[93]</sup></a> <a href="images/032a.png"><img src="images/032a.png" class="middle" style="height:3ex" alt="Numerals" /></a> for 174</p> + <p><a name="NtA_94" href="#Nt_94"><sup>[94]</sup></a> <a href="images/032b.png"><img src="images/032b.png" class="middle" style="height:3ex" alt="Numerals" /></a> for 191</p> + <p><a name="NtA_95" href="#Nt_95"><sup>[95]</sup></a> <a href="images/032c.png"><img src="images/032c.png" class="middle" style="height:3ex" alt="Numerals" /></a> for 269</p> + <p><a name="NtA_96" href="#Nt_96"><sup>[96]</sup></a> <a href="images/032d.png"><img src="images/032d.png" class="middle" style="height:3ex" alt="Numerals" /></a> for 252</p> + <p><a name="NtA_97" href="#Nt_97"><sup>[97]</sup></a> <a href="images/032e.png"><img src="images/032e.png" class="middle" style="height:3ex" alt="Numerals" /></a> for 400</p> + <p><a name="NtA_98" href="#Nt_98"><sup>[98]</sup></a> <a href="images/032f.png"><img src="images/032f.png" class="middle" style="height:3.5ex" alt="Numerals" /></a> for 356</p> + </div> + </div> +<p><!-- Page 27 --><span class="pagenum"><a name="page27"></a>[27]</span></p> + + <p>To these may be added the following numerals below one hundred, + similar to those in the table:</p> + + <div class="poem"> + <div class="stanza"> + <p><a href="images/033a.png"><img src="images/033a.png" class="middle" style="height:3ex" alt="Numerals" /></a><a name="NtA_99" href="#Nt_99"><sup>[99]</sup></a> for 90</p> + <p><a href="images/033b.png"><img src="images/033b.png" class="middle" style="height:3ex" alt="Numerals" /></a><a name="NtA_100" href="#Nt_100"><sup>[100]</sup></a> for 70</p> + </div> + </div> + <p>We have thus far spoken of the <span class="special" + title="Kharosthi">Kharoṣṭhī</span> and + Brāhmī numerals, and it remains to mention the third type, + the word and letter forms. These are, however, so closely connected with + the perfecting of the system by the invention of the zero that they are + more appropriately considered in the next chapter, particularly as they + have little relation to the problem of the origin of the forms known as + the Arabic.</p> + + <p>Having now examined types of the early forms it is appropriate to turn + our attention to the question of their origin. As to the first three + there is no question. The <a href="images/033c.png"><img + src="images/033c.png" class="middle" style="height:1.5ex" alt="1 vertical + stroke" /></a> or <a href="images/033d.png"><img src="images/033d.png" + class="middle" style="height:1.5ex" alt="1 horizontal stroke" /></a> is + simply one stroke, or one stick laid down by the computer. The <a + href="images/033e.png"><img src="images/033e.png" class="middle" + style="height:1.5ex" alt="2 vertical strokes" /></a> or <a + href="images/033f.png"><img src="images/033f.png" class="middle" + style="height:1.5ex" alt="2 horizontal strokes" /></a> represents two + strokes or two sticks, and so for the <a href="images/033g.png"><img + src="images/033g.png" class="middle" style="height:1.5ex" alt="3 vertical + strokes" /></a> and <a href="images/033h.png"><img src="images/033h.png" + class="middle" style="height:1.5ex" alt="3 horizontal strokes" /></a>. + From some primitive <a href="images/033e.png"><img src="images/033e.png" + class="middle" style="height:1.5ex" alt="2 vertical strokes" /></a> came + the two of Egypt, of Rome, of early Greece, and of various other + civilizations. It appears in the three Egyptian numeral systems in the + following forms:</p> + + <div class="poem"> + <div class="stanza"> + <p>Hieroglyphic <a href="images/033e.png"><img src="images/033e.png" class="middle" style="height:1.5ex" alt="2 vertical strokes" /></a></p> + <p>Hieratic <a href="images/033i.png"><img src="images/033i.png" class="middle" style="height:1.7ex" alt="Hieratic 2" /></a></p> + <p>Demotic <a href="images/033j.png"><img src="images/033j.png" class="middle" style="height:1.7ex" alt="Demotic 2" /></a></p> + </div> + </div> + <p>The last of these is merely a cursive form as in the Arabic <a + href="images/033k.png"><img src="images/033k.png" class="middle" + style="height:1.7ex" alt="Arabic 2" /></a>, which becomes our 2 if tipped + through a right angle. From some primitive <a href="images/033f.png"><img + src="images/033f.png" class="middle" style="height:1.5ex" alt="2 + horizontal strokes" /></a> came the Chinese <!-- Page 28 --><span + class="pagenum"><a name="page28"></a>[28]</span>symbol, which is + practically identical with the symbols found commonly in India from 150 + <span class="scac">B.C.</span> to 700 <span class="scac">A.D.</span> In + the cursive form it becomes <a href="images/034a.png"><img + src="images/034a.png" class="middle" style="height:1.5ex" alt="2 + horizontal strokes joined" /></a>, and this was frequently used for two + in Germany until the 18th century. It finally went into the modern form + 2, and the <a href="images/033h.png"><img src="images/033h.png" + class="middle" style="height:1.5ex" alt="3 horizontal strokes" /></a> in + the same way became our 3.</p> + + <p>There is, however, considerable ground for interesting speculation + with respect to these first three numerals. The earliest Hindu forms were + perpendicular. In the Nānā Ghāt inscriptions they are + vertical. But long before either the Aśoka or the Nānā + Ghāt inscriptions the Chinese were using the horizontal forms for + the first three numerals, but a vertical arrangement for four.<a + name="NtA_101" href="#Nt_101"><sup>[101]</sup></a> Now where did China + get these forms? Surely not from India, for she had them, as her + monuments and literature<a name="NtA_102" + href="#Nt_102"><sup>[102]</sup></a> show, long before the Hindus knew + them. The tradition is that China brought her civilization around the + north of Tibet, from Mongolia, the primitive habitat being Mesopotamia, + or possibly the oases of Turkestan. Now what numerals did Mesopotamia + use? The Babylonian system, simple in its general principles but very + complicated in many of its details, is now well known.<a name="NtA_103" + href="#Nt_103"><sup>[103]</sup></a> In particular, one, two, and three + were represented by vertical arrow-heads. Why, then, did the Chinese + write <!-- Page 29 --><span class="pagenum"><a + name="page29"></a>[29]</span>theirs horizontally? The problem now takes a + new interest when we find that these Babylonian forms were not the + primitive ones of this region, but that the early Sumerian forms were + horizontal.<a name="NtA_104" href="#Nt_104"><sup>[104]</sup></a></p> + + <p>What interpretation shall be given to these facts? Shall we say that + it was mere accident that one people wrote "one" vertically and that + another wrote it horizontally? This may be the case; but it may also be + the case that the tribal migrations that ended in the Mongol invasion of + China started from the Euphrates while yet the Sumerian civilization was + prominent, or from some common source in Turkestan, and that they carried + to the East the primitive numerals of their ancient home, the first + three, these being all that the people as a whole knew or needed. It is + equally possible that these three horizontal forms represent primitive + stick-laying, the most natural position of a stick placed in front of a + calculator being the horizontal one. When, however, the cuneiform writing + developed more fully, the vertical form may have been proved the easier + to make, so that by the time the migrations to the West began these were + in use, and from them came the upright forms of Egypt, Greece, Rome, and + other Mediterranean lands, and those of Aśoka's time in India. + After Aśoka, and perhaps among the merchants of earlier centuries, + the horizontal forms may have come down into India from China, thus + giving those of the Nānā Ghāt cave and of later + inscriptions. This is in the realm of speculation, but it is not + improbable that further epigraphical studies may confirm the + hypothesis.</p> + +<p><!-- Page 30 --><span class="pagenum"><a name="page30"></a>[30]</span></p> + + <p>As to the numerals above three there have been very many conjectures. + The figure one of the Demotic looks like the one of the Sanskrit, the two + (reversed) like that of the Arabic, the four has some resemblance to that + in the Nasik caves, the five (reversed) to that on the <span + class="special" title="Ksatrapa">Kṣatrapa</span> coins, the nine + to that of the <span class="special" title="Kusana">Kuṣana</span> + inscriptions, and other points of similarity have been imagined. Some + have traced resemblance between the Hieratic five and seven and those of + the Indian inscriptions. There have not, therefore, been wanting those + who asserted an Egyptian origin for these numerals.<a name="NtA_105" + href="#Nt_105"><sup>[105]</sup></a> There has already been mentioned the + fact that the <span class="special" + title="Kharosthi">Kharoṣṭhī</span> numerals were + formerly known as Bactrian, Indo-Bactrian, and Aryan. Cunningham<a + name="NtA_106" href="#Nt_106"><sup>[106]</sup></a> was the first to + suggest that these numerals were derived from the alphabet of the + Bactrian civilization of Eastern Persia, perhaps a thousand years before + our era, and in this he was supported by the scholarly work of Sir E. + Clive Bayley,<a name="NtA_107" href="#Nt_107"><sup>[107]</sup></a> who in + turn was followed by Canon Taylor.<a name="NtA_108" + href="#Nt_108"><sup>[108]</sup></a> The resemblance has not proved + convincing, however, and Bayley's drawings <!-- Page 31 --><span + class="pagenum"><a name="page31"></a>[31]</span>have been criticized as + being affected by his theory. The following is part of the hypothesis:<a + name="NtA_109" href="#Nt_109"><sup>[109]</sup></a></p> + +<table class="nobctr"> +<tr><td><i>Numeral</i></td><td><i>Hindu</i></td><td><i>Bactrian</i></td><td><i>Sanskrit</i></td></tr> +<tr><td align="center">4</td><td align="center"><a href="images/037a.png"><img src="images/037a.png" class="middle" style="height:3.6ex" alt="Symbol" /></a></td><td align="center"><a href="images/037b.png"><img src="images/037b.png" class="middle" style="height:3.6ex" alt="Symbol" /></a> = ch</td><td>chatur, Lat. quattuor</td></tr> +<tr><td align="center">5</td><td align="center"><a href="images/037c.png"><img src="images/037c.png" class="middle" style="height:3.6ex" alt="Symbol" /></a></td><td align="center"><a href="images/037d.png"><img src="images/037d.png" class="middle" style="height:3.6ex" alt="Symbol" /></a> = p</td><td>pancha, Gk. <span title="pente" class="grk">πέντε</span></td></tr> +<tr><td align="center">6</td><td align="center"><a href="images/037e.png"><img src="images/037e.png" class="middle" style="height:3.6ex" alt="Symbol" /></a></td><td align="center"><a href="images/037f.png"><img src="images/037f.png" class="middle" style="height:3.6ex" alt="Symbol" /></a> = s</td><td><span class="special" title="sas">ṣaṣ</span></td></tr> +<tr><td align="center">7</td><td align="center"><a href="images/037g.png"><img src="images/037g.png" class="middle" style="height:3.6ex" alt="Symbol" /></a></td><td align="center"><a href="images/037h.png"><img src="images/037h.png" class="middle" style="height:3.6ex" alt="Symbol" /></a> = <span class="special" title="s-dot-below">ṣ</span></td><td>sapta</td></tr> +<tr><td align="center" colspan="4">(the s and <span class="special" title="s-dot-below">ṣ</span> are interchanged as occasionally in N. W. India)</td></tr> +</table> + + <p>Bühler<a name="NtA_110" href="#Nt_110"><sup>[110]</sup></a> rejects + this hypothesis, stating that in four cases (four, six, seven, and ten) + the facts are absolutely against it.</p> + + <p>While the relation to ancient Bactrian forms has been generally + doubted, it is agreed that most of the numerals resemble + Brāhmī letters, and we would naturally expect them to be + initials.<a name="NtA_111" href="#Nt_111"><sup>[111]</sup></a> But, + knowing the ancient pronunciation of most of the number names,<a + name="NtA_112" href="#Nt_112"><sup>[112]</sup></a> we find this not to be + the case. We next fall back upon the hypothesis <!-- Page 32 --><span + class="pagenum"><a name="page32"></a>[32]</span>that they represent the + order of letters<a name="NtA_113" href="#Nt_113"><sup>[113]</sup></a> in + the ancient alphabet. From what we know of this order, however, there + seems also no basis for this assumption. We have, therefore, to confess + that we are not certain that the numerals were alphabetic at all, and if + they were alphabetic we have no evidence at present as to the basis of + selection. The later forms may possibly have been alphabetical + expressions of certain syllables called <i><span class="special" + title="aksaras">akṣaras</span></i>, which possessed in Sanskrit + fixed numerical values,<a name="NtA_114" + href="#Nt_114"><sup>[114]</sup></a> but this is equally uncertain with + the rest. Bayley also thought<a name="NtA_115" + href="#Nt_115"><sup>[115]</sup></a> that some of the forms were + Phœnician, as notably the use of a circle for twenty, but the + resemblance is in general too remote to be convincing.</p> + + <p>There is also some slight possibility that Chinese influence is to be + seen in certain of the early forms of Hindu numerals.<a name="NtA_116" + href="#Nt_116"><sup>[116]</sup></a></p> + +<p><!-- Page 33 --><span class="pagenum"><a name="page33"></a>[33]</span></p> + + <p>More absurd is the hypothesis of a Greek origin, supposedly supported + by derivation of the current symbols from the first nine letters of the + Greek alphabet.<a name="NtA_117" href="#Nt_117"><sup>[117]</sup></a> This + difficult feat is accomplished by twisting some of the letters, cutting + off, adding on, and effecting other changes to make the letters fit the + theory. This peculiar theory was first set up by Dasypodius<a + name="NtA_118" href="#Nt_118"><sup>[118]</sup></a> (Conrad Rauhfuss), and + was later elaborated by Huet.<a name="NtA_119" + href="#Nt_119"><sup>[119]</sup></a></p> + +<p><!-- Page 34 --><span class="pagenum"><a name="page34"></a>[34]</span></p> + + <p>A bizarre derivation based upon early Arabic (c. 1040 <span + class="scac">A.D.</span>) sources is given by Kircher in his work<a + name="NtA_120" href="#Nt_120"><sup>[120]</sup></a> on number mysticism. + He quotes from Abenragel,<a name="NtA_121" + href="#Nt_121"><sup>[121]</sup></a> giving the Arabic and a Latin + translation<a name="NtA_122" href="#Nt_122"><sup>[122]</sup></a> and + stating that the ordinary Arabic forms are derived from sectors of a + circle, <a href="images/040d.png"><img src="images/040d.png" + class="middle" style="height:2ex" alt="circle" /></a>.</p> + + <p>Out of all these conflicting theories, and from all the resemblances + seen or imagined between the numerals of the West and those of the East, + what conclusions are we prepared to draw as the evidence now stands? + Probably none that is satisfactory. Indeed, upon the evidence at <!-- + Page 35 --><span class="pagenum"><a name="page35"></a>[35]</span>hand we + might properly feel that everything points to the numerals as being + substantially indigenous to India. And why should this not be the case? + If the king Srong-tsan-Gampo (639 <span class="scac">A.D.</span>), the + founder of Lhāsa,<a name="NtA_123" + href="#Nt_123"><sup>[123]</sup></a> could have set about to devise a new + alphabet for Tibet, and if the Siamese, and the Singhalese, and the + Burmese, and other peoples in the East, could have created alphabets of + their own, why should not the numerals also have been fashioned by some + temple school, or some king, or some merchant guild? By way of + illustration, there are shown in the table on page 36 certain systems of + the East, and while a few resemblances are evident, it is also evident + that the creators of each system endeavored to find original forms that + should not be found in other systems. This, then, would seem to be a fair + interpretation of the evidence. A human mind cannot readily create simple + forms that are absolutely new; what it fashions will naturally resemble + what other minds have fashioned, or what it has known through hearsay or + through sight. A circle is one of the world's common stock of figures, + and that it should mean twenty in Phœnicia and in India is hardly + more surprising than that it signified ten at one time in Babylon.<a + name="NtA_124" href="#Nt_124"><sup>[124]</sup></a> It is therefore quite + probable that an extraneous origin cannot be found for the very + sufficient reason that none exists.</p> + + <p>Of absolute nonsense about the origin of the symbols which we use much + has been written. Conjectures, <!-- Page 36 --><span class="pagenum"><a + name="page36"></a>[36]</span>however, without any historical evidence for + support, have no place in a serious discussion of the gradual evolution + of the present numeral forms.<a name="NtA_125" + href="#Nt_125"><sup>[125]</sup></a></p> + +<h3><span class="sc">Table of Certain Eastern Systems</span></h3> + +<table class="nobctr"> +<tr><td> </td><td><a href="images/042.png"><img src="images/042.png" class="middle" style="height:3.6ex" alt="0 12 3 4 5 6 7 8 9 10" /></a></td></tr> +<tr><td valign="middle">Siam</td><td><a href="images/042a.png"><img src="images/042a.png" class="middle" style="height:5.2ex" alt="Siamese numerals" /></a></td></tr> +<tr><td valign="middle">Burma<a name="NtA_126" href="#Nt_126"><sup>[126]</sup></a></td><td><a href="images/042b.png"><img src="images/042b.png" class="middle" style="height:5.2ex" alt="Burmese numerals" /></a></td></tr> +<tr><td valign="middle">Malabar<a name="NtA_127" href="#Nt_127"><sup>[127]</sup></a></td><td><a href="images/042c.png"><img src="images/042c.png" class="middle" style="height:5.2ex" alt="Malaberese numerals" /></a></td></tr> +<tr><td valign="middle">Tibet<a name="NtA_128" href="#Nt_128"><sup>[128]</sup></a></td><td><a href="images/042d.png"><img src="images/042d.png" class="middle" style="height:5.2ex" alt="Tibetan numerals" /></a></td></tr> +<tr><td valign="middle">Ceylon<a name="NtA_129" href="#Nt_129"><sup>[129]</sup></a></td><td><a href="images/042e.png"><img src="images/042e.png" class="middle" style="height:5.2ex" alt="Celanese numerals" /></a></td></tr> +<tr><td valign="middle">Malayalam<a href="#Nt_129"><sup>[129]</sup></a></td><td><a href="images/042f.png"><img src="images/042f.png" class="middle" style="height:5.2ex" alt="Malayalam numerals" /></a></td></tr> +</table> + +<p><!-- Page 37 --><span class="pagenum"><a name="page37"></a>[37]</span></p> + + <p>We may summarize this chapter by saying that no one knows what + suggested certain of the early numeral forms used in India. The origin of + some is evident, but the origin of others will probably never be known. + There is no reason why they should not have been invented by some priest + or teacher or guild, by the order of some king, or as part of the + mysticism of some temple. Whatever the origin, they were no better than + scores of other ancient systems and no better than the present Chinese + system when written without the zero, and there would never have been any + chance of their triumphal progress westward had it not been for this + relatively late symbol. There could hardly be demanded a stronger proof + of the Hindu origin of the character for zero than this, and to it + further reference will be made in Chapter IV.</p> + +<hr class="full" > + +<p><!-- Page 38 --><span class="pagenum"><a name="page38"></a>[38]</span></p> + +<h3>CHAPTER III</h3> + +<p class="cenhead">LATER HINDU FORMS, WITH A PLACE VALUE</p> + + <p>Before speaking of the perfected Hindu numerals with the zero and the + place value, it is necessary to consider the third system mentioned on + page 19,—the word and letter forms. The use of words with place + value began at least as early as the 6th century of the Christian era. In + many of the manuals of astronomy and mathematics, and often in other + works in mentioning dates, numbers are represented by the names of + certain objects or ideas. For example, zero is represented by "the void" + (<i>śūnya</i>), or "heaven-space" (<i>ambara + ākāśa</i>); one by "stick" (<i>rupa</i>), "moon" + (<i>indu śaśin</i>), "earth" (<i>bhū</i>), "beginning" + (<i>ādi</i>), "Brahma," or, in general, by anything markedly + unique; two by "the twins" (<i>yama</i>), "hands" (<i>kara</i>), "eyes" + (<i>nayana</i>), etc.; four by "oceans," five by "senses" (<i><span + class="special" title="visaya">viṣaya</span></i>) or "arrows" (the + five arrows of Kāmadēva); six by "seasons" or "flavors"; + seven by "mountain" (<i>aga</i>), and so on.<a name="NtA_130" + href="#Nt_130"><sup>[130]</sup></a> These names, accommodating themselves + to the verse in which scientific works were written, had the additional + advantage of not admitting, as did the figures, easy alteration, since + any change would tend to disturb the meter.</p> + +<p><!-- Page 39 --><span class="pagenum"><a name="page39"></a>[39]</span></p> + + <p>As an example of this system, the date "<span class="special" + title="Saka Samvat">Śaka Saṃvat</span>, 867" (<span + class="scac">A.D.</span> 945 or 946), is given by "<i><span + class="special" title="giri-rasa-vasu">giri-raṣa-vasu</span></i>," + meaning "the mountains" (seven), "the flavors" (six), and the gods + "<i>Vasu</i>" of which there were eight. In reading the date these are + read from right to left.<a name="NtA_131" + href="#Nt_131"><sup>[131]</sup></a> The period of invention of this + system is uncertain. The first trace seems to be in the + <i>Śrautasūtra</i> of Kātyāyana and <span + class="special" title="Latyayana">Lāṭyāyana</span>.<a + name="NtA_132" href="#Nt_132"><sup>[132]</sup></a> It was certainly known + to Varāha-Mihira (d. 587),<a name="NtA_133" + href="#Nt_133"><sup>[133]</sup></a> for he used it in the <i><span + class="special" + title="Brhat-Samhita">Bṛhat-Saṃhitā</span>.</i><a + name="NtA_134" href="#Nt_134"><sup>[134]</sup></a> It has also been + asserted<a name="NtA_135" href="#Nt_135"><sup>[135]</sup></a> that <span + class="special" title="Aryabhata">Āryabhaṭa</span> (c. 500 + <span class="scac">A.D.</span>) was familiar with this system, but there + is nothing to prove the statement.<a name="NtA_136" + href="#Nt_136"><sup>[136]</sup></a> The earliest epigraphical examples of + the system are found in the Bayang (Cambodia) inscriptions of 604 and 624 + <span class="scac">A.D.</span><a name="NtA_137" + href="#Nt_137"><sup>[137]</sup></a></p> + + <p>Mention should also be made, in this connection, of a curious system + of alphabetic numerals that sprang up in southern India. In this we have + the numerals represented by the letters as given in the following + table:</p> + +<table class="nobctr"> +<tr><td style="width:6%">1</td><td style="width:6%">2</td><td style="width:6%">3</td><td style="width:6%">4</td><td style="width:6%">5</td><td style="width:6%">6</td><td style="width:6%">7</td><td style="width:6%">8</td><td style="width:6%">9</td><td style="width:6%">0</td></tr> +<tr><td>k</td><td>kh</td><td>g</td><td>gh</td><td> <span class="special" title="n-dot-above">ṅ</span></td><td>c</td><td>ch</td><td>j</td><td>jh</td><td>ñ</td></tr> +<tr><td><span class="special" title="t-dot-below">ṭ</span></td><td><span class="special" title="t-dot-below h">ṭh</span></td><td><span class="special" title="d-dot-below">ḍ</span></td><td><span class="special" title="d-dot-below h">ḍh</span></td><td><span class="special" title="n-dot-below">ṇ</span></td><td>t</td><td>th</td><td>d</td><td>th</td><td>n</td></tr> +<tr><td>p</td><td>ph</td><td>b</td><td>bh</td><td>m</td></tr> +<tr><td>y</td><td>r</td><td>l</td><td>v</td><td>ś</td><td><span class="special" title="s-dot-below">ṣ</span></td><td>s</td><td>h</td><td>l</td></tr> +</table> + +<p><!-- Page 40 --><span class="pagenum"><a name="page40"></a>[40]</span></p> + + <p>By this plan a numeral might be represented by any one of several + letters, as shown in the preceding table, and thus it could the more + easily be formed into a word for mnemonic purposes. For example, the + word</p> + +<table class="nobctr"> +<tr><td>2</td><td>3</td><td>1</td><td>5</td><td>6</td><td>5</td><td>1</td></tr> +<tr><td><i>kha</i></td><td><i>gont</i></td><td><i>yan</i></td><td><i>me</i></td><td><i><span class="special" title="s-dot-under a">ṣa</span></i></td><td><i>mā</i></td><td><i>pa</i></td></tr> +</table> + + <p>has the value 1,565,132, reading from right to left.<a name="NtA_138" + href="#Nt_138"><sup>[138]</sup></a> This, the oldest specimen (1184 <span + class="scac">A.D.</span>) known of this notation, is given in a + commentary on the Rigveda, representing the number of days that had + elapsed from the beginning of the Kaliyuga. Burnell<a name="NtA_139" + href="#Nt_139"><sup>[139]</sup></a> states that this system is even yet + in use for remembering rules to calculate horoscopes, and for + astronomical tables.</p> + + <p>A second system of this kind is still used in the pagination of + manuscripts in Ceylon, Siam, and Burma, having also had its rise in + southern India. In this the thirty-four consonants when followed by + <i>a</i> (as <i>ka</i> ... <i>la</i>) designate the numbers 1-34; by + <i>ā</i> (as <i>kā</i> ... <i>lā</i>), those from 35 to + 68; by <i>i</i> (<i>ki</i> ... <i>li</i>), those from 69 to 102, + inclusive; and so on.<a name="NtA_140" + href="#Nt_140"><sup>[140]</sup></a></p> + + <p>As already stated, however, the Hindu system as thus far described was + no improvement upon many others of the ancients, such as those used by + the Greeks and the Hebrews. Having no zero, it was impracticable to + designate the tens, hundreds, and other units of higher order by the same + symbols used for the units from one to nine. In other words, there was no + possibility of place value without some further improvement. So the + Nānā Ghāt <!-- Page 41 --><span class="pagenum"><a + name="page41"></a>[41]</span>symbols required the writing of "thousand + seven twenty-four" about like T 7, tw, 4 in modern symbols, instead of + 7024, in which the seven of the thousands, the two of the tens (concealed + in the word twenty, being originally "twain of tens," the <i>-ty</i> + signifying ten), and the four of the units are given as spoken and the + order of the unit (tens, hundreds, etc.) is given by the place. To + complete the system only the zero was needed; but it was probably eight + centuries after the Nānā Ghāt inscriptions were cut, + before this important symbol appeared; and not until a considerably later + period did it become well known. Who it was to whom the invention is due, + or where he lived, or even in what century, will probably always remain a + mystery.<a name="NtA_141" href="#Nt_141"><sup>[141]</sup></a> It is + possible that one of the forms of ancient abacus suggested to some Hindu + astronomer or mathematician the use of a symbol to stand for the vacant + line when the counters were removed. It is well established that in + different parts of India the names of the higher powers took different + forms, even the order being interchanged.<a name="NtA_142" + href="#Nt_142"><sup>[142]</sup></a> Nevertheless, as the significance of + the name of the unit was given by the order in reading, these variations + did not lead to error. Indeed the variation itself may have necessitated + the introduction of a word to signify a vacant place or lacking unit, + with the ultimate introduction of a zero symbol for this word.</p> + + <p>To enable us to appreciate the force of this argument a large number, + 8,443,682,155, may be considered as the Hindus wrote and read it, and + then, by way of contrast, as the Greeks and Arabs would have read it.</p> + +<p><!-- Page 42 --><span class="pagenum"><a name="page42"></a>[42]</span></p> + + <p><i>Modern American reading</i>, 8 billion, 443 million, 682 thousand, + 155.</p> + + <p><i>Hindu</i>, 8 padmas, 4 vyarbudas, 4 <span class="special" + title="kotis">kōṭis</span>, 3 prayutas, 6 <span + class="special" title="laksas">lakṣas</span>, 8 ayutas, 2 sahasra, + 1 śata, 5 daśan, 5.</p> + + <p><i>Arabic and early German</i>, eight thousand thousand thousand and + four hundred thousand thousand and forty-three thousand thousand, and six + hundred thousand and eighty-two thousand and one hundred fifty-five (or + five and fifty).</p> + + <p><i>Greek</i>, eighty-four myriads of myriads and four thousand three + hundred sixty-eight myriads and two thousand and one hundred + fifty-five.</p> + + <p>As Woepcke<a name="NtA_143" href="#Nt_143"><sup>[143]</sup></a> + pointed out, the reading of numbers of this kind shows that the notation + adopted by the Hindus tended to bring out the place idea. No other + language than the Sanskrit has made such consistent application, in + numeration, of the decimal system of numbers. The introduction of myriads + as in the Greek, and thousands as in Arabic and in modern numeration, is + really a step away from a decimal scheme. So in the numbers below one + hundred, in English, eleven and twelve are out of harmony with the rest + of the -teens, while the naming of all the numbers between ten and twenty + is not analogous to the naming of the numbers above twenty. To conform to + our written system we should have ten-one, ten-two, ten-three, and so on, + as we have twenty-one, twenty-two, and the like. The Sanskrit is + consistent, the units, however, preceding the tens and hundreds. Nor did + any other ancient people carry the numeration as far as did the Hindus.<a + name="NtA_144" href="#Nt_144"><sup>[144]</sup></a></p> + +<p><!-- Page 43 --><span class="pagenum"><a name="page43"></a>[43]</span></p> + + <p>When the <i><span class="special" + title="ankapalli">aṅkapalli</span></i>,<a name="NtA_145" + href="#Nt_145"><sup>[145]</sup></a> the decimal-place system of writing + numbers, was perfected, the tenth symbol was called the + <i>śūnyabindu</i>, generally shortened to + <i>śūnya</i> (the void). Brockhaus<a name="NtA_146" + href="#Nt_146"><sup>[146]</sup></a> has well said that if there was any + invention for which the Hindus, by all their philosophy and religion, + were well fitted, it was the invention of a symbol for zero. This making + of nothingness the crux of a tremendous achievement was a step in + complete harmony with the genius of the Hindu.</p> + + <p>It is generally thought that this <i>śūnya</i> as a symbol + was not used before about 500 <span class="scac">A.D.</span>, although + some writers have placed it earlier.<a name="NtA_147" + href="#Nt_147"><sup>[147]</sup></a> Since <span class="special" + title="Aryabhata">Āryabhaṭa</span> gives our common method + of extracting roots, it would seem that he may have known a decimal + notation,<a name="NtA_148" href="#Nt_148"><sup>[148]</sup></a> although + he did not use the characters from which our numerals are derived.<a + name="NtA_149" href="#Nt_149"><sup>[149]</sup></a> Moreover, he + frequently speaks of the <!-- Page 44 --><span class="pagenum"><a + name="page44"></a>[44]</span>void.<a name="NtA_150" + href="#Nt_150"><sup>[150]</sup></a> If he refers to a symbol this would + put the zero as far back as 500 <span class="scac">A.D.</span>, but of + course he may have referred merely to the concept of nothingness.</p> + + <p>A little later, but also in the sixth century, Varāha-Mihira<a + name="NtA_151" href="#Nt_151"><sup>[151]</sup></a> wrote a work entitled + <i><span class="special" title="Brhat Samhita">Bṛhat + Saṃhitā</span></i><a name="NtA_152" + href="#Nt_152"><sup>[152]</sup></a> in which he frequently uses + <i>śūnya</i> in speaking of numerals, so that it has been + thought that he was referring to a definite symbol. This, of course, + would add to the probability that <span class="special" + title="Aryabhata">Āryabhaṭa</span> was doing the same.</p> + + <p>It should also be mentioned as a matter of interest, and somewhat + related to the question at issue, that Varāha-Mihira used the + word-system with place value<a name="NtA_153" + href="#Nt_153"><sup>[153]</sup></a> as explained above.</p> + + <p>The first kind of alphabetic numerals and also the word-system (in + both of which the place value is used) are plays upon, or variations of, + position arithmetic, which would be most likely to occur in the country + of its origin.<a name="NtA_154" href="#Nt_154"><sup>[154]</sup></a></p> + + <p>At the opening of the next century (c. 620 <span + class="scac">A.D.</span>) <span class="special" + title="Bana">Bāṇa</span><a name="NtA_155" + href="#Nt_155"><sup>[155]</sup></a> wrote of Subandhus's + <i>Vāsavadattā</i> as a celebrated work, <!-- Page 45 + --><span class="pagenum"><a name="page45"></a>[45]</span>and mentioned + that the stars dotting the sky are here compared with zeros, these being + points as in the modern Arabic system. On the other hand, a strong + argument against any Hindu knowledge of the symbol zero at this time is + the fact that about 700 <span class="scac">A.D.</span> the Arabs overran + the province of Sind and thus had an opportunity of knowing the common + methods used there for writing numbers. And yet, when they received the + complete system in 776 they looked upon it as something new.<a + name="NtA_156" href="#Nt_156"><sup>[156]</sup></a> Such evidence is not + conclusive, but it tends to show that the complete system was probably + not in common use in India at the beginning of the eighth century. On the + other hand, we must bear in mind the fact that a traveler in Germany in + the year 1700 would probably have heard or seen nothing of decimal + fractions, although these were perfected a century before that date. The + élite of the mathematicians may have known the zero even in <span + class="special" title="Aryabhata">Āryabhaṭa</span>'s time, + while the merchants and the common people may not have grasped the + significance of the novelty until a long time after. On the whole, the + evidence seems to point to the west coast of India as the region where + the complete system was first seen.<a name="NtA_157" + href="#Nt_157"><sup>[157]</sup></a> As mentioned above, traces of the + numeral words with place value, which do not, however, absolutely require + a decimal place-system of symbols, are found very early in Cambodia, as + well as in India.</p> + + <p>Concerning the earliest epigraphical instances of the use of the nine + symbols, plus the zero, with place value, there <!-- Page 46 --><span + class="pagenum"><a name="page46"></a>[46]</span>is some question. + Colebrooke<a name="NtA_158" href="#Nt_158"><sup>[158]</sup></a> in 1807 + warned against the possibility of forgery in many of the ancient + copper-plate land grants. On this account Fleet, in the <i>Indian + Antiquary</i>,<a name="NtA_159" href="#Nt_159"><sup>[159]</sup></a> + discusses at length this phase of the work of the epigraphists in India, + holding that many of these forgeries were made about the end of the + eleventh century. Colebrooke<a name="NtA_160" + href="#Nt_160"><sup>[160]</sup></a> takes a more rational view of these + forgeries than does Kaye, who seems to hold that they tend to invalidate + the whole Indian hypothesis. "But even where that may be suspected, the + historical uses of a monument fabricated so much nearer to the times to + which it assumes to belong, will not be entirely superseded. The + necessity of rendering the forged grant credible would compel a + fabricator to adhere to history, and conform to established notions: and + the tradition, which prevailed in his time, and by which he must be + guided, would probably be so much nearer to the truth, as it was less + remote from the period which it concerned."<a name="NtA_161" + href="#Nt_161"><sup>[161]</sup></a> Bühler<a name="NtA_162" + href="#Nt_162"><sup>[162]</sup></a> gives the copper-plate Gurjara + inscription of <span class="special" + title="Cedi-samvat">Cedi-saṃvat</span> 346 (595 <span + class="scac">A.D.</span>) as the oldest epigraphical use of the + numerals<a name="NtA_163" href="#Nt_163"><sup>[163]</sup></a> "in which + the symbols correspond to the alphabet numerals of the period and the + place." Vincent A. Smith<a name="NtA_164" + href="#Nt_164"><sup>[164]</sup></a> quotes a stone inscription of 815 + <span class="scac">A.D.</span>, dated <span class="special" + title="Samvat">Saṃvat</span> 872. So F. Kielhorn in the + <i>Epigraphia Indica</i><a name="NtA_165" + href="#Nt_165"><sup>[165]</sup></a> gives a Pathari pillar inscription of + Parabala, dated <span class="special" + title="Vikrama-samvat">Vikrama-saṃvat</span> 917, which + corresponds to 861 <span class="scac">A.D.</span>, <!-- Page 47 --><span + class="pagenum"><a name="page47"></a>[47]</span>and refers also to + another copper-plate inscription dated <span class="special" + title="Vikrama-samvat">Vikrama-saṃvat</span> 813 (756 <span + class="scac">A.D.</span>). The inscription quoted by V. A. Smith above is + that given by D. R. Bhandarkar,<a name="NtA_166" + href="#Nt_166"><sup>[166]</sup></a> and another is given by the same + writer as of date <span class="special" + title="Saka-samvat">Saka-saṃvat</span> 715 (798 <span + class="scac">A.D.</span>), being incised on a pilaster. Kielhorn<a + name="NtA_167" href="#Nt_167"><sup>[167]</sup></a> also gives two + copper-plate inscriptions of the time of Mahendrapala of Kanauj, <span + class="special" title="Valhabi-samvat">Valhabī-saṃvat</span> + 574 (893 <span class="scac">A.D.</span>) and <span class="special" + title="Vikrama-samvat">Vikrama-saṃvat</span> 956 (899 <span + class="scac">A.D.</span>). That there should be any inscriptions of date + as early even as 750 <span class="scac">A.D.</span>, would tend to show + that the system was at least a century older. As will be shown in the + further development, it was more than two centuries after the + introduction of the numerals into Europe that they appeared there upon + coins and inscriptions. While Thibaut<a name="NtA_168" + href="#Nt_168"><sup>[168]</sup></a> does not consider it necessary to + quote any specific instances of the use of the numerals, he states that + traces are found from 590 <span class="scac">A.D.</span> on. "That the + system now in use by all civilized nations is of Hindu origin cannot be + doubted; no other nation has any claim upon its discovery, especially + since the references to the origin of the system which are found in the + nations of western Asia point unanimously towards India."<a + name="NtA_169" href="#Nt_169"><sup>[169]</sup></a></p> + + <p>The testimony and opinions of men like Bühler, Kielhorn, V. A. Smith, + Bhandarkar, and Thibaut are entitled to the most serious consideration. + As authorities on ancient Indian epigraphy no others rank higher. Their + work is accepted by Indian scholars the world over, and their united + judgment as to the rise of the system with a place value—that it + took place in India as early as the <!-- Page 48 --><span + class="pagenum"><a name="page48"></a>[48]</span>sixth century <span + class="scac">A.D.</span>—must stand unless new evidence of great + weight can be submitted to the contrary.</p> + + <p>Many early writers remarked upon the diversity of Indian numeral + forms. Al-Bīrūnī was probably the first; noteworthy is + also Johannes Hispalensis,<a name="NtA_170" + href="#Nt_170"><sup>[170]</sup></a> who gives the variant forms for seven + and four. We insert on p. 49 a table of numerals used with place value. + While the chief authority for this is Bühler,<a name="NtA_171" + href="#Nt_171"><sup>[171]</sup></a> several specimens are given which are + not found in his work and which are of unusual interest.</p> + + <p>The Śāradā forms given in the table use the circle + as a symbol for 1 and the dot for zero. They are taken from the paging + and text of <i>The Kashmirian Atharva-Veda</i><a name="NtA_172" + href="#Nt_172"><sup>[172]</sup></a>, of which the manuscript used is + certainly four hundred years old. Similar forms are found in a manuscript + belonging to the University of Tübingen. Two other series presented are + from Tibetan books in the library of one of the authors.</p> + + <p>For purposes of comparison the modern Sanskrit and Arabic numeral + forms are added.</p> + +<table class="nobctr"> +<tr><td valign="middle">Sanskrit,</td><td><a href="images/054a.png"><img src="images/054a.png" class="middle" style="height:6ex" alt="Sanskrit" /></a></td></tr> +<tr><td valign="middle">Arabic,</td><td><a href="images/054b.png"><img src="images/054b.png" class="middle" style="height:6ex" alt="Sanskrit" /></a></td></tr> +</table> + +<p><!-- Page 49 --><span class="pagenum"><a name="page49"></a>[49]</span></p> + +<h3><span class="sc">Numerals used with Place Value</span></h3> + +<table class="nobctr"> +<tr><td> </td><td><a href="images/055.png"><img src="images/055.png" class="middle" style="height:5ex" alt="1 2 3 4 5 6 7 8 9 0" /></a></td></tr> +<tr><td valign="middle">a <a name="NtA_173" href="#Nt_173"><sup>[173]</sup></a></td><td><a href="images/055a.png"><img src="images/055a.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">b <a name="NtA_174" href="#Nt_174"><sup>[174]</sup></a></td><td><a href="images/055b.png"><img src="images/055b.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">c <a name="NtA_175" href="#Nt_175"><sup>[175]</sup></a></td><td><a href="images/055c.png"><img src="images/055c.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">d <a name="NtA_176" href="#Nt_176"><sup>[176]</sup></a></td><td><a href="images/055d.png"><img src="images/055d.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">e <a name="NtA_177" href="#Nt_177"><sup>[177]</sup></a></td><td><a href="images/055e.png"><img src="images/055e.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">f <a name="NtA_178" href="#Nt_178"><sup>[178]</sup></a></td><td><a href="images/055f.png"><img src="images/055f.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">g <a name="NtA_179" href="#Nt_179"><sup>[179]</sup></a></td><td><a href="images/055g.png"><img src="images/055g.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">h <a name="NtA_180" href="#Nt_180"><sup>[180]</sup></a></td><td><a href="images/055h.png"><img src="images/055h.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">i <a href="#Nt_180"><sup>[180]</sup></a></td><td><a href="images/055i.png"><img src="images/055i.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">j <a name="NtA_181" href="#Nt_181"><sup>[181]</sup></a></td><td><a href="images/055j.png"><img src="images/055j.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">k <a href="#Nt_181"><sup>[181]</sup></a></td><td><a href="images/055k.png"><img src="images/055k.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">l <a name="NtA_182" href="#Nt_182"><sup>[182]</sup></a></td><td><a href="images/055l.png"><img src="images/055l.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">m <a name="NtA_183" href="#Nt_183"><sup>[183]</sup></a></td><td><a href="images/055m.png"><img src="images/055m.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">n <a name="NtA_184" href="#Nt_184"><sup>[184]</sup></a></td><td><a href="images/055n.png"><img src="images/055n.png" class="middle" style="height:5ex" alt="Numerals" /></a></td></tr> +</table> + +<hr class="full" > + +<p><!-- Page 51 --><span class="pagenum"><a name="page51"></a>[51]</span></p> + +<h3>CHAPTER IV</h3> + +<p class="cenhead">THE SYMBOL ZERO</p> + + <p>What has been said of the improved Hindu system with a place value + does not touch directly the origin of a symbol for zero, although it + assumes that such a symbol exists. The importance of such a sign, the + fact that it is a prerequisite to a place-value system, and the further + fact that without it the Hindu-Arabic numerals would never have dominated + the computation system of the western world, make it proper to devote a + chapter to its origin and history.</p> + + <p>It was some centuries after the primitive Brāhmī and <span + class="special" title="Kharosthi">Kharoṣṭhī</span> + numerals had made their appearance in India that the zero first appeared + there, although such a character was used by the Babylonians<a + name="NtA_185" href="#Nt_185"><sup>[185]</sup></a> in the centuries + immediately preceding the Christian era. The symbol is <a + href="images/057a.png"><img src="images/057a.png" class="middle" + style="height:2ex" alt="Babylonian zero symbol" /></a> or <a + href="images/057b.png"><img src="images/057b.png" class="middle" + style="height:2ex" alt="Babylonian zero symbol" /></a>, and apparently it + was not used in calculation. Nor does it always occur when units of any + order are lacking; thus 180 is written <a href="images/057c.png"><img + src="images/057c.png" class="middle" style="height:2.2ex" alt="Babylonian + numerals 180" /></a> with the meaning three sixties and no units, since + 181 immediately following is <a href="images/057d.png"><img + src="images/057d.png" class="middle" style="height:2.2ex" alt="Babylonian + numerals 181" /></a>, three sixties and one unit.<a name="NtA_186" + href="#Nt_186"><sup>[186]</sup></a> The main <!-- Page 52 --><span + class="pagenum"><a name="page52"></a>[52]</span>use of this Babylonian + symbol seems to have been in the fractions, 60ths, 3600ths, etc., and + somewhat similar to the Greek use of <span title="o" class="grk" + >ο</span>, for <span title="ouden" class="grk" + >οὐδέν</span>, with the meaning + <i>vacant</i>.</p> + + <p>"The earliest undoubted occurrence of a zero in India is an + inscription at Gwalior, dated Samvat 933 (876 <span + class="scac">A.D.</span>). Where 50 garlands are mentioned (line 20), 50 + is written <a href="images/058a.png"><img src="images/058a.png" + class="middle" style="height:2ex" alt="Gwalior numerals 50" /></a>. 270 + (line 4) is written <a href="images/058b.png"><img src="images/058b.png" + class="middle" style="height:2ex" alt="Gwalior numerals 270" /></a>."<a + name="NtA_187" href="#Nt_187"><sup>[187]</sup></a> The <span + class="special" title="Bakhsali">Bakhṣālī</span> + Manuscript<a name="NtA_188" href="#Nt_188"><sup>[188]</sup></a> probably + antedates this, using the point or dot as a zero symbol. Bayley mentions + a grant of Jaika Rashtrakúta of Bharuj, found at Okamandel, of date 738 + <span class="scac">A.D.</span>, which contains a zero, and also a coin + with indistinct Gupta date 707 (897 <span class="scac">A.D.</span>), but + the reliability of Bayley's work is questioned. As has been noted, the + appearance of the numerals in inscriptions and on coins would be of much + later occurrence than the origin and written exposition of the system. + From the period mentioned the spread was rapid over all of India, save + the southern part, where the Tamil and Malayalam people retain the old + system even to the present day.<a name="NtA_189" + href="#Nt_189"><sup>[189]</sup></a></p> + + <p>Aside from its appearance in early inscriptions, there is still + another indication of the Hindu origin of the symbol in the special + treatment of the concept zero in the early works on arithmetic. + Brahmagupta, who lived in Ujjain, the center of Indian astronomy,<a + name="NtA_190" href="#Nt_190"><sup>[190]</sup></a> in the early part <!-- + Page 53 --><span class="pagenum"><a name="page53"></a>[53]</span>of the + seventh century, gives in his arithmetic<a name="NtA_191" + href="#Nt_191"><sup>[191]</sup></a> a distinct treatment of the + properties of zero. He does not discuss a symbol, but he shows by his + treatment that in some way zero had acquired a special significance not + found in the Greek or other ancient arithmetics. A still more scientific + treatment is given by Bhāskara,<a name="NtA_192" + href="#Nt_192"><sup>[192]</sup></a> although in one place he permits + himself an unallowed liberty in dividing by zero. The most recently + discovered work of ancient Indian mathematical lore, the <span + class="special" + title="Ganita-Sara-Sangraha">Ganita-Sāra-Saṅgraha</span><a + name="NtA_193" href="#Nt_193"><sup>[193]</sup></a> of + Mahāvīrācārya (c. 830 <span + class="scac">A.D.</span>), while it does not use the numerals with place + value, has a similar discussion of the calculation with zero.</p> + + <p>What suggested the form for the zero is, of course, purely a matter of + conjecture. The dot, which the Hindus used to fill up lacunæ in their + manuscripts, much as we indicate a break in a sentence,<a name="NtA_194" + href="#Nt_194"><sup>[194]</sup></a> would have been a more natural + symbol; and this is the one which the Hindus first used<a name="NtA_195" + href="#Nt_195"><sup>[195]</sup></a> and which most Arabs use to-day. + There was also used for this purpose a cross, like our X, and this is + occasionally found as a zero symbol.<a name="NtA_196" + href="#Nt_196"><sup>[196]</sup></a> In the <span class="special" + title="Bakhsali">Bakhṣālī</span> manuscript above + mentioned, the word <i>śūnya</i>, with the dot as its symbol, + is used to denote the unknown quantity, as well as to denote zero. An + analogous use of the <!-- Page 54 --><span class="pagenum"><a + name="page54"></a>[54]</span>zero, for the unknown quantity in a + proportion, appears in a Latin manuscript of some lectures by Gottfried + Wolack in the University of Erfurt in 1467 and 1468.<a name="NtA_197" + href="#Nt_197"><sup>[197]</sup></a> The usage was noted even as early as + the eighteenth century.<a name="NtA_198" + href="#Nt_198"><sup>[198]</sup></a></p> + + <p>The small circle was possibly suggested by the spurred circle which + was used for ten.<a name="NtA_199" href="#Nt_199"><sup>[199]</sup></a> It + has also been thought that the omicron used by Ptolemy in his + <i>Almagest</i>, to mark accidental blanks in the sexagesimal system + which he employed, may have influenced the Indian writers.<a + name="NtA_200" href="#Nt_200"><sup>[200]</sup></a> This symbol was used + quite generally in Europe and Asia, and the Arabic astronomer + Al-Battānī<a name="NtA_201" + href="#Nt_201"><sup>[201]</sup></a> (died 929 <span + class="scac">A.D.</span>) used a similar symbol in connection with the + alphabetic system of numerals. The occasional use by + Al-Battānī of the Arabic negative, <i>lā</i>, to + indicate the absence of minutes <!-- Page 55 --><span class="pagenum"><a + name="page55"></a>[55]</span>(or seconds), is noted by Nallino.<a + name="NtA_202" href="#Nt_202"><sup>[202]</sup></a> Noteworthy is also the + use of the <a href="images/061a.png"><img src="images/061a.png" + class="middle" style="height:1.5ex" alt="Circle" /></a> for unity in the + Śāradā characters of the Kashmirian Atharva-Veda, the + writing being at least 400 years old. Bhāskara (c. 1150) used a + small circle above a number to indicate subtraction, and in the Tartar + writing a redundant word is removed by drawing an oval around it. It + would be interesting to know whether our score mark <a + href="images/061b.png"><img src="images/061b.png" class="middle" + style="height:2.2ex" alt="score mark" /></a>, read "four in the hole," + could trace its pedigree to the same sources. O'Creat<a name="NtA_203" + href="#Nt_203"><sup>[203]</sup></a> (c. 1130), in a letter to his + teacher, Adelhard of Bath, uses <span title="t" class="grk">τ</span> + for zero, being an abbreviation for the word <i>teca</i> which we shall + see was one of the names used for zero, although it could quite as well + be from <span title="tziphra" class="grk" + >τζίφρα</span>. More rarely O'Creat uses <a + href="images/061c.png"><img src="images/061c.png" class="middle" + style="height:1.8ex" alt="circle with bar" /></a>, applying the name + <i>cyfra</i> to both forms. Frater Sigsboto<a name="NtA_204" + href="#Nt_204"><sup>[204]</sup></a> (c. 1150) uses the same symbol. Other + peculiar forms are noted by Heiberg<a name="NtA_205" + href="#Nt_205"><sup>[205]</sup></a> as being in use among the Byzantine + Greeks in the fifteenth century. It is evident from the text that some of + these writers did not understand the import of the new system.<a + name="NtA_206" href="#Nt_206"><sup>[206]</sup></a></p> + + <p>Although the dot was used at first in India, as noted above, the small + circle later replaced it and continues in use to this day. The Arabs, + however, did not adopt the <!-- Page 56 --><span class="pagenum"><a + name="page56"></a>[56]</span>circle, since it bore some resemblance to + the letter which expressed the number five in the alphabet system.<a + name="NtA_207" href="#Nt_207"><sup>[207]</sup></a> The earliest Arabic + zero known is the dot, used in a manuscript of 873 <span + class="scac">A.D.</span><a name="NtA_208" + href="#Nt_208"><sup>[208]</sup></a> Sometimes both the dot and the circle + are used in the same work, having the same meaning, which is the case in + an Arabic MS., an abridged arithmetic of Jamshid,<a name="NtA_209" + href="#Nt_209"><sup>[209]</sup></a> 982 A.H. (1575 <span + class="scac">A.D.</span>). As given in this work the numerals are <a + href="images/062a.png"><img src="images/062a.png" class="middle" + style="height:2.8ex" alt="symbols" /></a>. The form for 5 varies, in some + works becoming <a href="images/062b.png"><img src="images/062b.png" + class="middle" style="height:2ex" alt="symbol" /></a> or <a + href="images/062c.png"><img src="images/062c.png" class="middle" + style="height:2ex" alt="symbol" /></a>; <a href="images/062d.png"><img + src="images/062d.png" class="middle" style="height:2ex" alt="symbol" + /></a> is found in Egypt and <a href="images/062e.png"><img + src="images/062e.png" class="middle" style="height:2ex" alt="symbol" + /></a> appears in some fonts of type. To-day the Arabs use the 0 only + when, under European influence, they adopt the ordinary system. Among the + Chinese the first definite trace of zero is in the work of Tsin<a + name="NtA_210" href="#Nt_210"><sup>[210]</sup></a> of 1247 <span + class="scac">A.D.</span> The form is the circular one of the Hindus, and + undoubtedly was brought to China by some traveler.</p> + + <p>The name of this all-important symbol also demands some attention, + especially as we are even yet quite undecided as to what to call it. We + speak of it to-day as <i>zero, naught</i>, and even <i>cipher</i>; the + telephone operator often calls it <i>O</i>, and the illiterate or + careless person calls it <i>aught</i>. In view of all this uncertainty we + may well inquire what it has been called in the past.<a name="NtA_211" + href="#Nt_211"><sup>[211]</sup></a></p> + +<p><!-- Page 57 --><span class="pagenum"><a name="page57"></a>[57]</span></p> + + <p>As already stated, the Hindus called it <i>śūnya</i>, + "void."<a name="NtA_212" href="#Nt_212"><sup>[212]</sup></a> This passed + over into the Arabic as <i><span class="special" + title="as-sifr">aṣ-ṣifr</span></i> or <i><span + class="special" title="sifr">ṣifr</span></i>.<a name="NtA_213" + href="#Nt_213"><sup>[213]</sup></a> When Leonard of Pisa (1202) wrote + upon the Hindu numerals he spoke of this character as <i>zephirum</i>.<a + name="NtA_214" href="#Nt_214"><sup>[214]</sup></a> Maximus Planudes + (1330), writing under both the Greek and the Arabic influence, called it + <i>tziphra</i>.<a name="NtA_215" href="#Nt_215"><sup>[215]</sup></a> In a + treatise on arithmetic written in the Italian language by Jacob of + Florence<a name="NtA_216" href="#Nt_216"><sup>[216]</sup></a> <!-- Page + 58 --><span class="pagenum"><a name="page58"></a>[58]</span>(1307) it is + called <i>zeuero</i>,<a name="NtA_217" + href="#Nt_217"><sup>[217]</sup></a> while in an arithmetic of Giovanni di + Danti of Arezzo (1370) the word appears as <i>çeuero</i>.<a + name="NtA_218" href="#Nt_218"><sup>[218]</sup></a> Another form is + <i>zepiro</i>,<a name="NtA_219" href="#Nt_219"><sup>[219]</sup></a> which + was also a step from <i>zephirum</i> to zero.<a name="NtA_220" + href="#Nt_220"><sup>[220]</sup></a></p> + + <p>Of course the English <i>cipher</i>, French <i>chiffre</i>, is derived + from the same Arabic word, <i><span class="special" + title="as-sifr">aṣ-ṣifr</span></i>, but in several + languages it has come to mean the numeral figures in general. A trace of + this appears in our word <i>ciphering</i>, meaning figuring or + computing.<a name="NtA_221" href="#Nt_221"><sup>[221]</sup></a> Johann + Huswirt<a name="NtA_222" href="#Nt_222"><sup>[222]</sup></a> uses the + word with both meanings; he gives for the tenth character the four names + <i>theca, circulus, cifra</i>, and <i>figura nihili</i>. In this + statement Huswirt probably follows, as did many writers of that period, + the <i>Algorismus</i> of Johannes de Sacrobosco (c. 1250 <span + class="scac">A.D.</span>), who was also known as John of Halifax or John + of Holywood. The commentary of <!-- Page 59 --><span class="pagenum"><a + name="page59"></a>[59]</span>Petrus de Dacia<a name="NtA_223" + href="#Nt_223"><sup>[223]</sup></a> (c. 1291 <span + class="scac">A.D.</span>) on the <i>Algorismus vulgaris</i> of Sacrobosco + was also widely used. The widespread use of this Englishman's work on + arithmetic in the universities of that time is attested by the large + number<a name="NtA_224" href="#Nt_224"><sup>[224]</sup></a> of MSS. from + the thirteenth to the seventeenth century still extant, twenty in Munich, + twelve in Vienna, thirteen in Erfurt, several in England given by + Halliwell,<a name="NtA_225" href="#Nt_225"><sup>[225]</sup></a> ten + listed in Coxe's <i>Catalogue of the Oxford College Library</i>, one in + the Plimpton collection,<a name="NtA_226" + href="#Nt_226"><sup>[226]</sup></a> one in the Columbia University + Library, and, of course, many others.</p> + + <p>From <i><span class="special" + title="as-sifr">aṣ-ṣifr</span> </i>has come <i>zephyr, + cipher,</i> and finally the abridged form <i>zero</i>. The earliest + printed work in which is found this final form appears to be Calandri's + arithmetic of 1491,<a name="NtA_227" href="#Nt_227"><sup>[227]</sup></a> + while in manuscript it appears at least as early as the middle of the + fourteenth century.<a name="NtA_228" href="#Nt_228"><sup>[228]</sup></a> + It also appears in a work, <i>Le Kadran des marchans</i>, by Jehan <!-- + Page 60 --><span class="pagenum"><a + name="page60"></a>[60]</span>Certain,<a name="NtA_229" + href="#Nt_229"><sup>[229]</sup></a> written in 1485. This word soon + became fairly well known in Spain<a name="NtA_230" + href="#Nt_230"><sup>[230]</sup></a> and France.<a name="NtA_231" + href="#Nt_231"><sup>[231]</sup></a> The medieval writers also spoke of it + as the <i>sipos</i>,<a name="NtA_232" href="#Nt_232"><sup>[232]</sup></a> + and occasionally as the <i>wheel</i>,<a name="NtA_233" + href="#Nt_233"><sup>[233]</sup></a> <i>circulus</i><a name="NtA_234" + href="#Nt_234"><sup>[234]</sup></a> (in German <i>das Ringlein</i><a + name="NtA_235" href="#Nt_235"><sup>[235]</sup></a>), <i>circular <!-- + Page 61 --><span class="pagenum"><a + name="page61"></a>[61]</span>note</i>,<a name="NtA_236" + href="#Nt_236"><sup>[236]</sup></a> <i>theca</i>,<a name="NtA_237" + href="#Nt_237"><sup>[237]</sup></a> long supposed to be from its + resemblance to the Greek theta, but explained by Petrus de Dacia as being + derived from the name of the iron<a name="NtA_238" + href="#Nt_238"><sup>[238]</sup></a> used to brand thieves and robbers + with a circular mark placed on the forehead or on the cheek. It was also + called <i>omicron</i><a name="NtA_239" + href="#Nt_239"><sup>[239]</sup></a> (the Greek <i>o</i>), being sometimes + written õ or <span title="ph" class="grk">φ</span> to distinguish it + from the letter <i>o</i>. It also went by the name <i>null</i><a + name="NtA_240" href="#Nt_240"><sup>[240]</sup></a> (in the Latin books + <!-- Page 62 --><span class="pagenum"><a + name="page62"></a>[62]</span><i>nihil</i><a name="NtA_241" + href="#Nt_241"><sup>[241]</sup></a> or <i>nulla</i>,<a name="NtA_242" + href="#Nt_242"><sup>[242]</sup></a> and in the French <i>rien</i><a + name="NtA_243" href="#Nt_243"><sup>[243]</sup></a>), and very commonly by + the name <i>cipher</i>.<a name="NtA_244" + href="#Nt_244"><sup>[244]</sup></a> Wallis<a name="NtA_245" + href="#Nt_245"><sup>[245]</sup></a> gives one of the earliest extended + discussions of the various forms of the word, giving certain other + variations worthy of note, as <i>ziphra</i>, <i>zifera</i>, + <i>siphra</i>, <i>ciphra</i>, <i>tsiphra</i>, <i>tziphra,</i> and the + Greek <span title="tziphra" class="grk" + >τζίφρα</span>.<a name="NtA_246" + href="#Nt_246"><sup>[246]</sup></a></p> + +<hr class="full" > + +<p><!-- Page 63 --><span class="pagenum"><a name="page63"></a>[63]</span></p> + +<h3>CHAPTER V</h3> + +<p class="cenhead">THE QUESTION OF THE INTRODUCTION OF THE +NUMERALS INTO EUROPE BY BOETHIUS</p> + + <p>Just as we were quite uncertain as to the origin of the numeral forms, + so too are we uncertain as to the time and place of their introduction + into Europe. There are two general theories as to this introduction. The + first is that they were carried by the Moors to Spain in the eighth or + ninth century, and thence were transmitted to Christian Europe, a theory + which will be considered later. The second, advanced by Woepcke,<a + name="NtA_247" href="#Nt_247"><sup>[247]</sup></a> is that they were not + brought to Spain by the Moors, but that they were already in Spain when + the Arabs arrived there, having reached the West through the + Neo-Pythagoreans. There are two facts to support this second theory: (1) + the forms of these numerals are characteristic, differing materially from + those which were brought by Leonardo of Pisa from Northern Africa early + in the thirteenth century (before 1202 <span class="scac">A.D.</span>); + (2) they are essentially those which <!-- Page 64 --><span + class="pagenum"><a name="page64"></a>[64]</span>tradition has so + persistently assigned to Boethius (c. 500 <span + class="scac">A.D.</span>), and which he would naturally have received, if + at all, from these same Neo-Pythagoreans or from the sources from which + they derived them. Furthermore, Woepcke points out that the Arabs on + entering Spain (711 <span class="scac">A.D.</span>) would naturally have + followed their custom of adopting for the computation of taxes the + numerical systems of the countries they conquered,<a name="NtA_248" + href="#Nt_248"><sup>[248]</sup></a> so that the numerals brought from + Spain to Italy, not having undergone the same modifications as those of + the Eastern Arab empire, would have differed, as they certainly did, from + those that came through Bagdad. The theory is that the Hindu system, + without the zero, early reached Alexandria (say 450 <span + class="scac">A.D.</span>), and that the Neo-Pythagorean love for the + mysterious and especially for the Oriental led to its use as something + bizarre and cabalistic; that it was then passed along the Mediterranean, + reaching Boethius in Athens or in Rome, and to the schools of Spain, + being discovered in Africa and Spain by the Arabs even before they + themselves knew the improved system with the place value.</p> + +<p><!-- Page 65 --><span class="pagenum"><a name="page65"></a>[65]</span></p> + + <p>A recent theory set forth by Bubnov<a name="NtA_249" + href="#Nt_249"><sup>[249]</sup></a> also deserves mention, chiefly + because of the seriousness of purpose shown by this well-known writer. + Bubnov holds that the forms first found in Europe are derived from + ancient symbols used on the abacus, but that the zero is of Hindu origin. + This theory does not seem tenable, however, in the light of the evidence + already set forth.</p> + + <p>Two questions are presented by Woepcke's theory: (1) What was the + nature of these Spanish numerals, and how were they made known to Italy? + (2) Did Boethius know them?</p> + + <p>The Spanish forms of the numerals were called the <i><span + class="special" title="huruf al-gobar">ḥurūf + al-ġobār</span></i>, the ġobār or dust numerals, + as distinguished from the <i><span class="special" title="huruf al-jumal" + >ḥurūf al-jumal</span></i> or alphabetic numerals. Probably + the latter, under the influence of the Syrians or Jews,<a name="NtA_250" + href="#Nt_250"><sup>[250]</sup></a> were also used by the Arabs. The + significance of the term ġobār 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-Bīrūnī 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 Kairwān, in Tunis, in which + the author speaks of one of his works on ġobār calculation;<a + name="NtA_251" href="#Nt_251"><sup>[251]</sup></a> and, much later, the + Arab writer <span class="special" title="Abu Bekr Mohammed ibn `Abdallah" + >Abū Bekr Moḥammed ibn ‛Abdallāh</span>, + surnamed <span class="special" + title="al-Hassar">al-Ḥaṣṣār</span> <!-- Page + 66 --><span class="pagenum"><a name="page66"></a>[66]</span>(the + arithmetician), wrote a work of which the second chapter was "On the dust + figures."<a name="NtA_252" href="#Nt_252"><sup>[252]</sup></a></p> + + <p>The ġobār numerals themselves were first made known to + modern scholars by Silvestre de Sacy, who discovered them in an Arabic + manuscript from the library of the ancient abbey of + St.-Germain-des-Prés.<a name="NtA_253" + href="#Nt_253"><sup>[253]</sup></a> The system has nine characters, but + no zero. A dot above a character indicates tens, two dots hundreds, and + so on, <a href="images/072a.png"><img src="images/072a.png" + class="middle" style="height:2ex" alt="5 with dot" /></a> meaning 50, and + <a href="images/072b.png"><img src="images/072b.png" class="middle" + style="height:2ex" alt="5 with 3 dots" /></a> meaning 5000. It has been + suggested that possibly these dots, sprinkled like dust above the + numerals, gave rise to the word <i>ġobār</i>,<a + name="NtA_254" href="#Nt_254"><sup>[254]</sup></a> but this is not at all + probable. This system of dots is found in Persia at a much later date + with numerals quite like the modern Arabic;<a name="NtA_255" + href="#Nt_255"><sup>[255]</sup></a> but that it was used at all is + significant, for it is hardly likely that the western system would go + back to Persia, when the perfected Hindu one was near at hand.</p> + + <p>At first sight there would seem to be some reason for believing that + this feature of the ġobār system was of <!-- Page 67 --><span + class="pagenum"><a name="page67"></a>[67]</span>Arabic origin, and that + the present zero of these people,<a name="NtA_256" + href="#Nt_256"><sup>[256]</sup></a> the dot, was derived from it. It was + entirely natural that the Semitic people generally should have adopted + such a scheme, since their diacritical marks would suggest it, not to + speak of the possible influence of the Greek accents in the Hellenic + number system. When we consider, however, that the dot is found for zero + in the <span class="special" + title="Bakhsali">Bakhṣālī</span> manuscript,<a + name="NtA_257" href="#Nt_257"><sup>[257]</sup></a> and that it was used + in subscript form in the <i>Kitāb al-Fihrist</i><a name="NtA_258" + href="#Nt_258"><sup>[258]</sup></a> in the tenth century, and as late as + the sixteenth century,<a name="NtA_259" + href="#Nt_259"><sup>[259]</sup></a> although in this case probably under + Arabic influence, we are forced to believe that this form may also have + been of Hindu origin.</p> + + <p>The fact seems to be that, as already stated,<a name="NtA_260" + href="#Nt_260"><sup>[260]</sup></a> the Arabs did not immediately adopt + the Hindu zero, because it resembled their 5; they used the superscript + dot as serving their purposes fairly well; they may, indeed, have carried + this to the west and have added it to the ġobār forms already + there, just as they transmitted it to the Persians. Furthermore, the Arab + and Hebrew scholars of Northern Africa in the tenth century knew these + numerals as Indian forms, for a commentary on the <i><span + class="special" title="Sefer Yesirah">Sēfer + Yeṣīrāh</span></i> by Abū Sahl ibn Tamim + (probably composed at Kairwān, c. 950) speaks of "the Indian + arithmetic known under the name of <i>ġobār</i> or dust + calculation."<a name="NtA_261" href="#Nt_261"><sup>[261]</sup></a> All + this suggests that the Arabs may very <!-- Page 68 --><span + class="pagenum"><a name="page68"></a>[68]</span>likely have known the + ġobār forms before the numerals reached them again in 773.<a + name="NtA_262" href="#Nt_262"><sup>[262]</sup></a> The term + "ġobār numerals" was also used without any reference to the + peculiar use of dots.<a name="NtA_263" + href="#Nt_263"><sup>[263]</sup></a> In this connection it is worthy of + mention that the Algerians employed two different forms of numerals in + manuscripts even of the fourteenth century,<a name="NtA_264" + href="#Nt_264"><sup>[264]</sup></a> and that the Moroccans of to-day + employ the European forms instead of the present Arabic.</p> + + <p>The Indian use of subscript dots to indicate the tens, hundreds, + thousands, etc., is established by a passage in the <i>Kitāb + al-Fihrist</i><a name="NtA_265" href="#Nt_265"><sup>[265]</sup></a> (987 + <span class="scac">A.D.</span>) in which the writer discusses the written + language of the people of India. Notwithstanding the importance of this + reference for the early history of the numerals, it has not been + mentioned by previous writers on this subject. The numeral forms given + are those which have usually been called Indian,<a name="NtA_266" + href="#Nt_266"><sup>[266]</sup></a> in opposition to ġobār. + In this document the dots are placed below the characters, instead of + being superposed as described above. The significance was the same.</p> + + <p>In form these ġobār numerals resemble our own much more + closely than the Arab numerals do. They varied more or less, but were + substantially as follows:</p> + +<p><!-- Page 69 --><span class="pagenum"><a name="page69"></a>[69]</span></p> + +<table class="nobctr"> +<tr><td>1 <a name="NtA_267" href="#Nt_267"><sup>[267]</sup></a></td><td><a href="images/075a.png"><img src="images/075a.png" class="middle" style="height:4.5ex" alt="numerals" /></a></td></tr> +<tr><td>2 <a name="NtA_268" href="#Nt_268"><sup>[268]</sup></a></td><td><a href="images/075b.png"><img src="images/075b.png" class="middle" style="height:4.5ex" alt="numerals" /></a></td></tr> +<tr><td>3 <a name="NtA_269" href="#Nt_269"><sup>[269]</sup></a></td><td><a href="images/075c.png"><img src="images/075c.png" class="middle" style="height:4.5ex" alt="numerals" /></a></td></tr> +<tr><td>4 <a name="NtA_270" href="#Nt_270"><sup>[270]</sup></a></td><td><a href="images/075d.png"><img src="images/075d.png" class="middle" style="height:4.5ex" alt="numerals" /></a></td></tr> +<tr><td>5 <a name="NtA_271" href="#Nt_271"><sup>[271]</sup></a></td><td><a href="images/075e.png"><img src="images/075e.png" class="middle" style="height:4.5ex" alt="numerals" /></a></td></tr> +<tr><td>6 <a href="#Nt_271"><sup>[271]</sup></a></td><td><a href="images/075f.png"><img src="images/075f.png" class="middle" style="height:4.5ex" alt="numerals" /></a></td></tr> +</table> + + <p>The question of the possible influence of the Egyptian demotic and + hieratic ordinal forms has been so often suggested that it seems well to + introduce them at this point, for comparison with the ġobār + forms. They would as appropriately be used in connection with the Hindu + forms, and the evidence of a relation of the first three with all these + systems is apparent. The only further resemblance is in the Demotic 4 and + in the 9, so that the statement that the Hindu forms in general came from + <!-- Page 70 --><span class="pagenum"><a + name="page70"></a>[70]</span>this source has no foundation. The first + four Egyptian cardinal numerals<a name="NtA_272" + href="#Nt_272"><sup>[272]</sup></a> resemble more the modern Arabic.</p> + + <div class="figleft" style="width:20%;"> + <a href="images/076a.png"><img style="width:100%" src="images/076a.png" + alt="Demotic and Hieratic Ordinals" title="Demotic and Hieratic Ordinals" /></a> + <span class="sc">Demotic and Hieratic Ordinals</span> + </div> + <p>This theory of the very early introduction of the numerals into Europe + fails in several points. In the first place the early Western forms are + not known; in the second place some early Eastern forms are like the + ġobār, as is seen in the third line on p. <a + href="#page69">69</a>, where the forms are from a manuscript written at + Shiraz about 970 <span class="scac">A.D.</span>, and in which some + western Arabic forms, e.g. <a href="images/076b.png"><img + src="images/076b.png" class="middle" style="height:2ex" alt="symbol" + /></a> for 2, are also used. Probably most significant of all is the fact + that the ġobār 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 ġobār numerals. We now have to consider the question + as to whether Boethius knew these ġobār forms, or forms akin + to them.</p> + + <p>This large question<a name="NtA_273" + href="#Nt_273"><sup>[273]</sup></a> suggests several minor ones: (1) Who + was Boethius? (2) Could he have known these numerals? (3) Is there any + positive or strong circumstantial evidence that he did know them? (4) + What are the probabilities in the case?</p> + +<p><!-- Page 71 --><span class="pagenum"><a name="page71"></a>[71]</span></p> + + <p>First, who was Boethius,—Divus<a name="NtA_274" + href="#Nt_274"><sup>[274]</sup></a> Boethius as he was called in the + Middle Ages? Anicius Manlius Severinus Boethius<a name="NtA_275" + href="#Nt_275"><sup>[275]</sup></a> was born at Rome c. 475. He was a + member of the distinguished family of the Anicii,<a name="NtA_276" + href="#Nt_276"><sup>[276]</sup></a> which had for some time before his + birth been Christian. Early left an orphan, the tradition is that he was + taken to Athens at about the age of ten, and that he remained there + eighteen years.<a name="NtA_277" href="#Nt_277"><sup>[277]</sup></a> He + married Rusticiana, daughter of the senator Symmachus, and this union of + two such powerful families allowed him to move in the highest circles.<a + name="NtA_278" href="#Nt_278"><sup>[278]</sup></a> Standing strictly for + the right, and against all iniquity at court, he became the object of + hatred on the part of all the unscrupulous element near the throne, and + his bold defense of the ex-consul Albinus, unjustly accused of treason, + led to his imprisonment at Pavia<a name="NtA_279" + href="#Nt_279"><sup>[279]</sup></a> and his execution in 524.<a + name="NtA_280" href="#Nt_280"><sup>[280]</sup></a> Not many generations + after his death, the period being one in which historical criticism was + at its lowest ebb, the church found it profitable to look upon his + execution as a martyrdom.<a name="NtA_281" + href="#Nt_281"><sup>[281]</sup></a> He was <!-- Page 72 --><span + class="pagenum"><a name="page72"></a>[72]</span>accordingly looked upon + as a saint,<a name="NtA_282" href="#Nt_282"><sup>[282]</sup></a> his + bones were enshrined,<a name="NtA_283" + href="#Nt_283"><sup>[283]</sup></a> and as a natural consequence his + books were among the classics in the church schools for a thousand + years.<a name="NtA_284" href="#Nt_284"><sup>[284]</sup></a> It is + pathetic, however, to think of the medieval student trying to extract + mental nourishment from a work so abstract, so meaningless, so + unnecessarily complicated, as the arithmetic of Boethius.</p> + + <p>He was looked upon by his contemporaries and immediate successors as a + master, for Cassiodorus<a name="NtA_285" + href="#Nt_285"><sup>[285]</sup></a> (c. 490-c. 585 <span + class="scac">A.D.</span>) says to him: "Through your translations the + music of Pythagoras and the astronomy of Ptolemy are read by those of + Italy, and the arithmetic of Nicomachus and the geometry of Euclid are + known to those of the West."<a name="NtA_286" + href="#Nt_286"><sup>[286]</sup></a> Founder of the medieval + scholasticism, <!-- Page 73 --><span class="pagenum"><a + name="page73"></a>[73]</span>distinguishing the trivium and quadrivium,<a + name="NtA_287" href="#Nt_287"><sup>[287]</sup></a> writing the only + classics of his time, Gibbon well called him "the last of the Romans whom + Cato or Tully could have acknowledged for their countryman."<a + name="NtA_288" href="#Nt_288"><sup>[288]</sup></a></p> + + <p>The second question relating to Boethius is this: Could he possibly + have known the Hindu numerals? In view of the relations that will be + shown to have existed between the East and the West, there can only be an + affirmative answer to this question. The numerals had existed, without + the zero, for several centuries; they had been well known in India; there + had been a continued interchange of thought between the East and West; + and warriors, ambassadors, scholars, and the restless trader, all had + gone back and forth, by land or more frequently by sea, between the + Mediterranean lands and the centers of Indian commerce and culture. + Boethius could very well have learned one or more forms of Hindu numerals + from some traveler or merchant.</p> + + <p>To justify this statement it is necessary to speak more fully of these + relations between the Far East and Europe. It is true that we have no + records of the interchange of learning, in any large way, between eastern + Asia and central Europe in the century preceding the time of Boethius. + But it is one of the mistakes of scholars to believe that they are the + sole transmitters of knowledge. <!-- Page 74 --><span class="pagenum"><a + name="page74"></a>[74]</span>As a matter of fact there is abundant reason + for believing that Hindu numerals would naturally have been known to the + Arabs, and even along every trade route to the remote west, long before + the zero entered to make their place-value possible, and that the + characters, the methods of calculating, the improvements that took place + from time to time, the zero when it appeared, and the customs as to + solving business problems, would all have been made known from generation + to generation along these same trade routes from the Orient to the + Occident. It must always be kept in mind that it was to the tradesman and + the wandering scholar that the spread of such learning was due, rather + than to the school man. Indeed, Avicenna<a name="NtA_289" + href="#Nt_289"><sup>[289]</sup></a> (980-1037 <span + class="scac">A.D.</span>) in a short biography of himself relates that + when his people were living at Bokhāra his father sent him to the + house of a grocer to learn the Hindu art of reckoning, in which this + grocer (oil dealer, possibly) was expert. Leonardo of Pisa, too, had a + similar training.</p> + + <p>The whole question of this spread of mercantile knowledge along the + trade routes is so connected with the ġobār numerals, the + Boethius question, Gerbert, Leonardo of Pisa, and other names and events, + that a digression for its consideration now becomes necessary.<a + name="NtA_290" href="#Nt_290"><sup>[290]</sup></a></p> + +<p><!-- Page 75 --><span class="pagenum"><a name="page75"></a>[75]</span></p> + + <p>Even in very remote times, before the Hindu numerals were sculptured + in the cave of Nānā Ghāt, there were trade relations + between Arabia and India. Indeed, long before the Aryans went to India + the great Turanian race had spread its civilization from the + Mediterranean to the Indus.<a name="NtA_291" + href="#Nt_291"><sup>[291]</sup></a> At a much later period the Arabs were + the intermediaries between Egypt and Syria on the west, and the farther + Orient.<a name="NtA_292" href="#Nt_292"><sup>[292]</sup></a> In the sixth + century <span class="scac">B.C.</span>, Hecatæus,<a name="NtA_293" + href="#Nt_293"><sup>[293]</sup></a> the father of geography, was + acquainted not only with the Mediterranean lands but with the countries + as far as the Indus,<a name="NtA_294" href="#Nt_294"><sup>[294]</sup></a> + and in Biblical times there were regular triennial voyages to India. + Indeed, the story of Joseph bears witness to the caravan trade from + India, across Arabia, and on to the banks of the Nile. About the same + time as Hecatæus, Scylax, a Persian admiral under Darius, from Caryanda + on the coast of Asia Minor, traveled to <!-- Page 76 --><span + class="pagenum"><a name="page76"></a>[76]</span>northwest India and wrote + upon his ventures.<a name="NtA_295" href="#Nt_295"><sup>[295]</sup></a> + He induced the nations along the Indus to acknowledge the Persian + supremacy, and such number systems as there were in these lands would + naturally have been known to a man of his attainments.</p> + + <p>A century after Scylax, Herodotus showed considerable knowledge of + India, speaking of its cotton and its gold,<a name="NtA_296" + href="#Nt_296"><sup>[296]</sup></a> telling how Sesostris<a + name="NtA_297" href="#Nt_297"><sup>[297]</sup></a> fitted out ships to + sail to that country, and mentioning the routes to the east. These routes + were generally by the Red Sea, and had been followed by the + Phœnicians and the Sabæans, and later were taken by the Greeks and + Romans.<a name="NtA_298" href="#Nt_298"><sup>[298]</sup></a></p> + + <p>In the fourth century <span class="scac">B.C.</span> the West and East + came into very close relations. As early as 330, Pytheas of Massilia + (Marseilles) had explored as far north as the northern end of the British + Isles and the coasts of the German Sea, while Macedon, in close touch + with southern France, was also sending her armies under Alexander<a + name="NtA_299" href="#Nt_299"><sup>[299]</sup></a> through Afghanistan as + far east as the Punjab.<a name="NtA_300" + href="#Nt_300"><sup>[300]</sup></a> Pliny tells us that Alexander the + Great employed surveyors to measure <!-- Page 77 --><span + class="pagenum"><a name="page77"></a>[77]</span>the roads of India; and + one of the great highways is described by Megasthenes, who in 295 <span + class="scac">B.C.</span>, as the ambassador of Seleucus, resided at <span + class="special" + title="Pataliputra">Pātalīpuṭra</span>, the present + Patna.<a name="NtA_301" href="#Nt_301"><sup>[301]</sup></a></p> + + <p>The Hindus also learned the art of coining from the Greeks, or + possibly from the Chinese, and the stores of Greco-Hindu coins still + found in northern India are a constant source of historical + information.<a name="NtA_302" href="#Nt_302"><sup>[302]</sup></a> The + Rāmāyana speaks of merchants traveling in great caravans and + embarking by sea for foreign lands.<a name="NtA_303" + href="#Nt_303"><sup>[303]</sup></a> Ceylon traded with Malacca and Siam, + and Java was colonized by Hindu traders, so that mercantile knowledge was + being spread about the Indies during all the formative period of the + numerals.</p> + + <p>Moreover the results of the early Greek invasion were embodied by + Dicæarchus of Messana (about 320 <span class="scac">B.C.</span>) in a map + that long remained a standard. Furthermore, Alexander did not allow his + influence on the East to cease. He divided India into three satrapies,<a + name="NtA_304" href="#Nt_304"><sup>[304]</sup></a> placing Greek + governors over two of them and leaving a Hindu ruler in charge of the + third, and in Bactriana, a part of Ariana or ancient Persia, he left + governors; and in these the western civilization was long in evidence. + Some of the Greek and Roman metrical and astronomical terms <!-- Page 78 + --><span class="pagenum"><a name="page78"></a>[78]</span>found their way, + doubtless at this time, into the Sanskrit language.<a name="NtA_305" + href="#Nt_305"><sup>[305]</sup></a> Even as late as from the second to + the fifth centuries <span class="scac">A.D.</span>, Indian coins showed + the Hellenic influence. The Hindu astronomical terminology reveals the + same relationship to western thought, for Varāha-Mihira (6th + century <span class="scac">A.D.</span>), a contemporary of <span + class="special" title="Aryabhata">Āryabhaṭa</span>, entitled + a work of his the <i><span class="special" + title="Brhat-Samhita">Bṛhat-Saṃhitā</span></i>, a + literal translation of <span title="megalê suntaxis" class="grk" + >μεγάλη + σύνταξις</span> of Ptolemy;<a + name="NtA_306" href="#Nt_306"><sup>[306]</sup></a> and in various ways is + this interchange of ideas apparent.<a name="NtA_307" + href="#Nt_307"><sup>[307]</sup></a> It could not have been at all unusual + for the ancient Greeks to go to India, for Strabo lays down the route, + saying that all who make the journey start from Ephesus and traverse + Phrygia and Cappadocia before taking the direct road.<a name="NtA_308" + href="#Nt_308"><sup>[308]</sup></a> The products of the East were always + finding their way to the West, the Greeks getting their ginger<a + name="NtA_309" href="#Nt_309"><sup>[309]</sup></a> from Malabar, as the + Phœnicians had long before brought gold from Malacca.</p> + + <p>Greece must also have had early relations with China, for there is a + notable similarity between the Greek and Chinese life, as is shown in + their houses, their domestic customs, their marriage ceremonies, the + public story-tellers, the puppet shows which Herodotus says were + introduced from Egypt, the street jugglers, the games of dice,<a + name="NtA_310" href="#Nt_310"><sup>[310]</sup></a> the game of + finger-guessing,<a name="NtA_311" href="#Nt_311"><sup>[311]</sup></a> the + water clock, the <!-- Page 79 --><span class="pagenum"><a + name="page79"></a>[79]</span>music system, the use of the myriad,<a + name="NtA_312" href="#Nt_312"><sup>[312]</sup></a> the calendars, and in + many other ways.<a name="NtA_313" href="#Nt_313"><sup>[313]</sup></a> In + passing through the suburbs of Peking to-day, on the way to the Great + Bell temple, one is constantly reminded of the semi-Greek architecture of + Pompeii, so closely does modern China touch the old classical + civilization of the Mediterranean. The Chinese historians tell us that + about 200 <span class="scac">B.C.</span> their arms were successful in + the far west, and that in 180 <span class="scac">B.C.</span> an + ambassador went to Bactria, then a Greek city, and reported that Chinese + products were on sale in the markets there.<a name="NtA_314" + href="#Nt_314"><sup>[314]</sup></a> There is also a noteworthy + resemblance between certain Greek and Chinese words,<a name="NtA_315" + href="#Nt_315"><sup>[315]</sup></a> showing that in remote times there + must have been more or less interchange of thought.</p> + + <p>The Romans also exchanged products with the East. Horace says, "A busy + trader, you hasten to the farthest Indies, flying from poverty over sea, + over crags, over fires."<a name="NtA_316" + href="#Nt_316"><sup>[316]</sup></a> The products of the Orient, spices + and jewels from India, frankincense from Persia, and silks from China, + being more in demand than the exports from the Mediterranean lands, the + balance of trade was against the West, and thus Roman coin found its way + eastward. In 1898, for example, a number of Roman coins dating from 114 + <span class="scac">B.C.</span> to Hadrian's time were found at + Paklī, a part of the Hazāra district, sixteen miles north of + Abbottābād,<a name="NtA_317" + href="#Nt_317"><sup>[317]</sup></a> and numerous similar discoveries have + been made from time to time.</p> + +<p><!-- Page 80 --><span class="pagenum"><a name="page80"></a>[80]</span></p> + + <p>Augustus speaks of envoys received by him from India, a thing never + before known,<a name="NtA_318" href="#Nt_318"><sup>[318]</sup></a> and it + is not improbable that he also received an embassy from China.<a + name="NtA_319" href="#Nt_319"><sup>[319]</sup></a> Suetonius (first + century <span class="scac">A.D.</span>) speaks in his history of these + relations,<a name="NtA_320" href="#Nt_320"><sup>[320]</sup></a> as do + several of his contemporaries,<a name="NtA_321" + href="#Nt_321"><sup>[321]</sup></a> and Vergil<a name="NtA_322" + href="#Nt_322"><sup>[322]</sup></a> tells of Augustus doing battle in + Persia. In Pliny's time the trade of the Roman Empire with Asia amounted + to a million and a quarter dollars a year, a sum far greater relatively + then than now,<a name="NtA_323" href="#Nt_323"><sup>[323]</sup></a> while + by the time of Constantine Europe was in direct communication with the + Far East.<a name="NtA_324" href="#Nt_324"><sup>[324]</sup></a></p> + + <p>In view of these relations it is not beyond the range of possibility + that proof may sometime come to light to show that the Greeks and Romans + knew something of the <!-- Page 81 --><span class="pagenum"><a + name="page81"></a>[81]</span>number system of India, as several writers + have maintained.<a name="NtA_325" href="#Nt_325"><sup>[325]</sup></a></p> + + <p>Returning to the East, there are many evidences of the spread of + knowledge in and about India itself. In the third century <span + class="scac">B.C.</span> Buddhism began to be a connecting medium of + thought. It had already permeated the Himalaya territory, had reached + eastern Turkestan, and had probably gone thence to China. Some centuries + later (in 62 <span class="scac">A.D.</span>) the Chinese emperor sent an + ambassador to India, and in 67 <span class="scac">A.D.</span> a Buddhist + monk was invited to China.<a name="NtA_326" + href="#Nt_326"><sup>[326]</sup></a> Then, too, in India itself + Aśoka, whose name has already been mentioned in this work, extended + the boundaries of his domains even into Afghanistan, so that it was + entirely possible for the numerals of the Punjab to have worked their way + north even at that early date.<a name="NtA_327" + href="#Nt_327"><sup>[327]</sup></a></p> + + <p>Furthermore, the influence of Persia must not be forgotten in + considering this transmission of knowledge. In the fifth century the + Persian medical school at Jondi-Sapur admitted both the Hindu and the + Greek doctrines, and Firdusī tells us that during the brilliant + reign of <!-- Page 82 --><span class="pagenum"><a + name="page82"></a>[82]</span>Khosrū I,<a name="NtA_328" + href="#Nt_328"><sup>[328]</sup></a> the golden age of Pahlavī + literature, the Hindu game of chess was introduced into Persia, at a time + when wars with the Greeks were bringing prestige to the Sassanid + dynasty.</p> + + <p>Again, not far from the time of Boethius, in the sixth century, the + Egyptian monk Cosmas, in his earlier years as a trader, made journeys to + Abyssinia and even to India and Ceylon, receiving the name + <i>Indicopleustes</i> (the Indian traveler). His map (547 <span + class="scac">A.D.</span>) shows some knowledge of the earth from the + Atlantic to India. Such a man would, with hardly a doubt, have observed + every numeral system used by the people with whom he sojourned,<a + name="NtA_329" href="#Nt_329"><sup>[329]</sup></a> and whether or not he + recorded his studies in permanent form he would have transmitted such + scraps of knowledge by word of mouth.</p> + + <p>As to the Arabs, it is a mistake to feel that their activities began + with Mohammed. Commerce had always been held in honor by them, and the + Qoreish<a name="NtA_330" href="#Nt_330"><sup>[330]</sup></a> had annually + for many generations sent caravans bearing the spices and textiles of + Yemen to the shores of the Mediterranean. In the fifth century they + traded by sea with India and even with China, and <span class="special" + title="Hira">Ḥira</span> was an emporium for the wares of the + East,<a name="NtA_331" href="#Nt_331"><sup>[331]</sup></a> so that any + numeral system of any part of the trading world could hardly have + remained isolated.</p> + + <p>Long before the warlike activity of the Arabs, Alexandria had become + the great market-place of the world. From this center caravans traversed + Arabia to Hadramaut, where they met ships from India. Others went north + to Damascus, while still others made their way <!-- Page 83 --><span + class="pagenum"><a name="page83"></a>[83]</span>along the southern shores + of the Mediterranean. Ships sailed from the isthmus of Suez to all the + commercial ports of Southern Europe and up into the Black Sea. Hindus + were found among the merchants<a name="NtA_332" + href="#Nt_332"><sup>[332]</sup></a> who frequented the bazaars of + Alexandria, and Brahmins were reported even in Byzantium.</p> + + <p>Such is a very brief résumé of the evidence showing that the numerals + of the Punjab and of other parts of India as well, and indeed those of + China and farther Persia, of Ceylon and the Malay peninsula, might well + have been known to the merchants of Alexandria, and even to those of any + other seaport of the Mediterranean, in the time of Boethius. The + Brāhmī numerals would not have attracted the attention of + scholars, for they had no zero so far as we know, and therefore they were + no better and no worse than those of dozens of other systems. If Boethius + was attracted to them it was probably exactly as any one is naturally + attracted to the bizarre or the mystic, and he would have mentioned them + in his works only incidentally, as indeed they are mentioned in the + manuscripts in which they occur.</p> + + <p>In answer therefore to the second question, Could Boethius have known + the Hindu numerals? the reply must be, without the slightest doubt, that + he could easily have known them, and that it would have been strange if a + man of his inquiring mind did not pick up many curious bits of + information of this kind even though he never thought of making use of + them.</p> + + <p>Let us now consider the third question, Is there any positive or + strong circumstantial evidence that Boethius did know these numerals? The + question is not new, <!-- Page 84 --><span class="pagenum"><a + name="page84"></a>[84]</span>nor is it much nearer being answered than it + was over two centuries ago when Wallis (1693) expressed his doubts about + it<a name="NtA_333" href="#Nt_333"><sup>[333]</sup></a> soon after + Vossius (1658) had called attention to the matter.<a name="NtA_334" + href="#Nt_334"><sup>[334]</sup></a> Stated briefly, there are three works + on mathematics attributed to Boethius:<a name="NtA_335" + href="#Nt_335"><sup>[335]</sup></a> (1) the arithmetic, (2) a work on + music, and (3) the geometry.<a name="NtA_336" + href="#Nt_336"><sup>[336]</sup></a></p> + + <p>The genuineness of the arithmetic and the treatise on music is + generally recognized, but the geometry, which contains the Hindu numerals + with the zero, is under suspicion.<a name="NtA_337" + href="#Nt_337"><sup>[337]</sup></a> There are plenty of supporters of the + idea that Boethius knew the numerals and included them in this book,<a + name="NtA_338" href="#Nt_338"><sup>[338]</sup></a> and on the other hand + there are as many who <!-- Page 85 --><span class="pagenum"><a + name="page85"></a>[85]</span>feel that the geometry, or at least the part + mentioning the numerals, is spurious.<a name="NtA_339" + href="#Nt_339"><sup>[339]</sup></a> The argument of those who deny the + authenticity of the particular passage in question may briefly be stated + thus:</p> + + <p>1. The falsification of texts has always been the subject of + complaint. It was so with the Romans,<a name="NtA_340" + href="#Nt_340"><sup>[340]</sup></a> it was common in the Middle Ages,<a + name="NtA_341" href="#Nt_341"><sup>[341]</sup></a> and it is much more + prevalent <!-- Page 86 --><span class="pagenum"><a + name="page86"></a>[86]</span>to-day than we commonly think. We have but + to see how every hymn-book compiler feels himself authorized to change at + will the classics of our language, and how unknown editors have mutilated + Shakespeare, to see how much more easy it was for medieval scribes to + insert or eliminate paragraphs without any protest from critics.<a + name="NtA_342" href="#Nt_342"><sup>[342]</sup></a></p> + + <p>2. If Boethius had known these numerals he would have mentioned them + in his arithmetic, but he does not do so.<a name="NtA_343" + href="#Nt_343"><sup>[343]</sup></a></p> + + <p>3. If he had known them, and had mentioned them in any of his works, + his contemporaries, disciples, and successors would have known and + mentioned them. But neither Capella (c. 475)<a name="NtA_344" + href="#Nt_344"><sup>[344]</sup></a> nor any of the numerous medieval + writers who knew the works of Boethius makes any reference to the + system.<a name="NtA_345" href="#Nt_345"><sup>[345]</sup></a></p> + +<p><!-- Page 87 --><span class="pagenum"><a name="page87"></a>[87]</span></p> + + <p>4. The passage in question has all the appearance of an interpolation + by some scribe. Boethius is speaking of angles, in his work on geometry, + when the text suddenly changes to a discussion of classes of numbers.<a + name="NtA_346" href="#Nt_346"><sup>[346]</sup></a> This is followed by a + chapter in explanation of the abacus,<a name="NtA_347" + href="#Nt_347"><sup>[347]</sup></a> in which are described those numeral + forms which are called <i>apices</i> or <i>caracteres</i>.<a + name="NtA_348" href="#Nt_348"><sup>[348]</sup></a> The forms<a + name="NtA_349" href="#Nt_349"><sup>[349]</sup></a> of these characters + vary in different manuscripts, but in general are about as shown on page + <a href="#page88">88</a>. They are commonly written with the 9 at the + left, decreasing to the unit at the right, numerous writers stating that + this was because they were derived from Semitic sources in which the + direction of writing is the opposite of our own. This practice continued + until the sixteenth century.<a name="NtA_350" + href="#Nt_350"><sup>[350]</sup></a> The writer then leaves the subject + entirely, using the Roman numerals for the rest of his discussion, a + proceeding so foreign to the method of Boethius as to be inexplicable on + the hypothesis of authenticity. Why should such a scholarly writer have + given them with no mention of their origin or use? Either he would have + mentioned some historical interest attaching to them, or he would have + used them in some discussion; he certainly would not have left the + passage as it is.</p> + +<p><!-- Page 88 --><span class="pagenum"><a name="page88"></a>[88]</span></p> + +<h3><span class="sc">Forms of the Numerals, Largely from Works on the Abacus</span><a name="NtA_351" href="#Nt_351"><sup>[351]</sup></a></h3> + +<table class="nobctr"> +<tr><td> </td><td><a href="images/094.png"><img src="images/094.png" class="middle" style="height:2.25ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">a <a name="NtA_352" href="#Nt_352"><sup>[352]</sup></a></td><td><a href="images/094a.png"><img src="images/094a.png" class="middle" style="height:5.4ex" alt="1 2 3 4 5 6 7 8 9 0" /></a></td></tr> +<tr><td valign="middle">b <a name="NtA_353" href="#Nt_353"><sup>[353]</sup></a></td><td><a href="images/094b.png"><img src="images/094b.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">c <a name="NtA_354" href="#Nt_354"><sup>[354]</sup></a></td><td><a href="images/094c.png"><img src="images/094c.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">d <a name="NtA_355" href="#Nt_355"><sup>[355]</sup></a></td><td><a href="images/094d.png"><img src="images/094d.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">e <a name="NtA_356" href="#Nt_356"><sup>[356]</sup></a></td><td><a href="images/094e.png"><img src="images/094e.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">f <a name="NtA_357" href="#Nt_357"><sup>[357]</sup></a></td><td><a href="images/094f.png"><img src="images/094f.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">g <a name="NtA_358" href="#Nt_358"><sup>[358]</sup></a></td><td><a href="images/094g.png"><img src="images/094g.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">h <a name="NtA_359" href="#Nt_359"><sup>[359]</sup></a></td><td><a href="images/094h.png"><img src="images/094h.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr> +<tr><td valign="middle">i <a name="NtA_360" href="#Nt_360"><sup>[360]</sup></a></td><td><a href="images/094i.png"><img src="images/094i.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td></tr> +</table> + +<p><!-- Page 89 --><span class="pagenum"><a name="page89"></a>[89]</span></p> + + <p>Sir E. Clive Bayley has added<a name="NtA_361" + href="#Nt_361"><sup>[361]</sup></a> a further reason for believing them + spurious, namely that the 4 is not of the Nānā Ghāt + type, but of the Kabul form which the Arabs did not receive until 776;<a + name="NtA_362" href="#Nt_362"><sup>[362]</sup></a> so that it is not + likely, even if the characters were known in Europe in the time of + Boethius, that this particular form was recognized. It is worthy of + mention, also, that in the six abacus forms from the chief manuscripts as + given by Friedlein,<a name="NtA_363" href="#Nt_363"><sup>[363]</sup></a> + each contains some form of zero, which symbol probably originated in + India about this time or later. It could hardly have reached Europe so + soon.</p> + + <p>As to the fourth question, Did Boethius probably know the numerals? It + seems to be a fair conclusion, according to our present evidence, that + (1) Boethius might very easily have known these numerals without the + zero, but, (2) there is no reliable evidence that he did know them. And + just as Boethius might have come in contact with them, so any other + inquiring mind might have done so either in his time or at any time + before they definitely appeared in the tenth century. These centuries, + five in number, represented the darkest of the Dark Ages, and even if + these numerals were occasionally met and studied, no trace of them would + be likely to show itself in the <!-- Page 90 --><span class="pagenum"><a + name="page90"></a>[90]</span>literature of the period, unless by chance + it should get into the writings of some man like Alcuin. As a matter of + fact, it was not until the ninth or tenth century that there is any + tangible evidence of their presence in Christendom. They were probably + known to merchants here and there, but in their incomplete state they + were not of sufficient importance to attract any considerable + attention.</p> + + <p>As a result of this brief survey of the evidence several conclusions + seem reasonable: (1) commerce, and travel for travel's sake, never died + out between the East and the West; (2) merchants had every opportunity of + knowing, and would have been unreasonably stupid if they had not known, + the elementary number systems of the peoples with whom they were trading, + but they would not have put this knowledge in permanent written form; (3) + wandering scholars would have known many and strange things about the + peoples they met, but they too were not, as a class, writers; (4) there + is every reason a priori for believing that the ġobār + numerals would have been known to merchants, and probably to some of the + wandering scholars, long before the Arabs conquered northern Africa; (5) + the wonder is not that the Hindu-Arabic numerals were known about 1000 + <span class="scac">A.D.</span>, and that they were the subject of an + elaborate work in 1202 by Fibonacci, but rather that more extended + manuscript evidence of their appearance before that time has not been + found. That they were more or less known early in the Middle Ages, + certainly to many merchants of Christian Europe, and probably to several + scholars, but without the zero, is hardly to be doubted. The lack of + documentary evidence is not at all strange, in view of all of the + circumstances.</p> + +<hr class="full" > + +<p><!-- Page 91 --><span class="pagenum"><a name="page91"></a>[91]</span></p> + +<h3>CHAPTER VI</h3> + +<p class="cenhead">THE DEVELOPMENT OF THE NUMERALS +AMONG THE ARABS</p> + + <p>If the numerals had their origin in India, as seems most probable, + when did the Arabs come to know of them? It is customary to say that it + was due to the influence of Mohammedanism that learning spread through + Persia and Arabia; and so it was, in part. But learning was already + respected in these countries long before Mohammed appeared, and commerce + flourished all through this region. In Persia, for example, the reign of + Khosrū Nuśīrwān,<a name="NtA_364" + href="#Nt_364"><sup>[364]</sup></a> the great contemporary of Justinian + the law-maker, was characterized not only by an improvement in social and + economic conditions, but by the cultivation of letters. Khosrū + fostered learning, inviting to his court scholars from Greece, and + encouraging the introduction of culture from the West as well as from the + East. At this time Aristotle and Plato were translated, and portions of + the <i>Hito-padēśa</i>, or Fables of Pilpay, were rendered + from the Sanskrit into Persian. All this means that some three centuries + before the great intellectual ascendancy of Bagdad a similar fostering of + learning was taking place in Persia, and under pre-Mohammedan + influences.</p> + +<p><!-- Page 92 --><span class="pagenum"><a name="page92"></a>[92]</span></p> + + <p>The first definite trace that we have of the introduction of the Hindu + system into Arabia dates from 773 <span class="scac">A.D.</span>,<a + name="NtA_365" href="#Nt_365"><sup>[365]</sup></a> when an Indian + astronomer visited the court of the caliph, bringing with him + astronomical tables which at the caliph's command were translated into + Arabic by Al-Fazārī.<a name="NtA_366" + href="#Nt_366"><sup>[366]</sup></a> Al-Khowārazmī and <span + class="special" title="Habash">Ḥabash</span> (<span + class="special" title="Ahmed ibn `Abdallah">Aḥmed ibn + ‛Abdallāh</span>, died c. 870) based their well-known tables + upon the work of Al-Fāzarī. It may be asserted as highly + probable that the numerals came at the same time as the tables. They were + certainly known a few decades later, and before 825 <span + class="scac">A.D.</span>, about which time the original of the + <i>Algoritmi de numero Indorum</i> was written, as that work makes no + pretense of being the first work to treat of the Hindu numerals.</p> + + <p>The three writers mentioned cover the period from the end of the + eighth to the end of the ninth century. While the historians + Al-Maś‛ūdī and Al-Bīrūnī + follow quite closely upon the men mentioned, it is well to note again the + Arab writers on Hindu arithmetic, contemporary with + Al-Khowārazmī, who were mentioned in chapter I, viz. + Al-Kindī, Sened ibn ‛Alī, and <span class="special" + title="Al-Sufi">Al-Ṣūfī</span>.</p> + + <p>For over five hundred years Arabic writers and others continued to + apply to works on arithmetic the name "Indian." In the tenth century such + writers are <span class="special" title="`Abdallah ibn al-Hasan" + >‛Abdallāh ibn al-Ḥasan</span>, Abū + 'l-Qāsim<a name="NtA_367" href="#Nt_367"><sup>[367]</sup></a> (died + 987 <span class="scac">A.D.</span>) of Antioch, and <span class="special" + title="Mohammed ibn `Abdallah, Abu Nasr">Moḥammed ibn + ‛Abdallāh, Abū Naṣr</span><a name="NtA_368" + href="#Nt_368"><sup>[368]</sup></a> (c. 982), of Kalwādā near + Bagdad. Others of the same period or <!-- Page 93 --><span + class="pagenum"><a name="page93"></a>[93]</span>earlier (since they are + mentioned in the <i>Fihrist</i>,<a name="NtA_369" + href="#Nt_369"><sup>[369]</sup></a> 987 <span class="scac">A.D.</span>), + who explicitly use the word "Hindu" or "Indian," are <span + class="special" title="Sinan ibn al-Fath">Sinān ibn + al-Fatḥ</span><a name="NtA_370" + href="#Nt_370"><sup>[370]</sup></a> of <span class="special" + title="Harran">Ḥarrān</span>, and Ahmed ibn ‛Omar, + al-Karābīsī.<a name="NtA_371" + href="#Nt_371"><sup>[371]</sup></a> In the eleventh century come + Al-Bīrūnī<a name="NtA_372" + href="#Nt_372"><sup>[372]</sup></a> (973-1048) and <span class="special" + title="`Ali ibn Ahmed, Abu 'l-Hasan">‛Ali ibn Aḥmed, + Abū 'l-Ḥasan</span>, Al-Nasawī<a name="NtA_373" + href="#Nt_373"><sup>[373]</sup></a> (c. 1030). The following century + brings similar works by <span class="special" title="Ishaq ibn Yusuf al-Sardafi" + >Ishāq ibn Yūsuf al-Ṣardafī</span><a + name="NtA_374" href="#Nt_374"><sup>[374]</sup></a> and + Samū'īl ibn <span class="special" + title="Yahya">Yaḥyā</span> ibn ‛Abbās + al-Maġrebī al-Andalusī<a name="NtA_375" + href="#Nt_375"><sup>[375]</sup></a> (c. 1174), and in the thirteenth + century are ‛Abdallatīf ibn Yūsuf ibn <span + class="special" title="Mohammed">Moḥammed</span>, Muwaffaq + al-Dīn Abū <span class="special" + title="Mohammed">Moḥammed</span> al-Baġdādī<a + name="NtA_376" href="#Nt_376"><sup>[376]</sup></a> (c. 1231), and Ibn + al-Bannā.<a name="NtA_377" href="#Nt_377"><sup>[377]</sup></a></p> + + <p>The Greek monk Maximus Planudes, writing in the first half of the + fourteenth century, followed the Arabic usage in calling his work + <i>Indian Arithmetic</i>.<a name="NtA_378" + href="#Nt_378"><sup>[378]</sup></a> There were numerous other Arabic + writers upon arithmetic, as that subject occupied one of the high places + among the sciences, but most of them did not feel it necessary to refer + to the origin of the symbols, the knowledge of which might well have been + taken for granted.</p> + +<p><!-- Page 94 --><span class="pagenum"><a name="page94"></a>[94]</span></p> + + <p>One document, cited by Woepcke,<a name="NtA_379" + href="#Nt_379"><sup>[379]</sup></a> is of special interest since it shows + at an early period, 970 <span class="scac">A.D.</span>, the use of the + ordinary Arabic forms alongside the ġobār. The title of the + work is <i>Interesting and Beautiful Problems on Numbers</i> copied by + <span class="special" title="Ahmed ibn Mohammed ibn `Abdaljalil" + >Aḥmed ibn Moḥammed ibn ‛Abdaljalīl</span>, + Abū Sa‛īd, al-Sijzī,<a name="NtA_380" + href="#Nt_380"><sup>[380]</sup></a> (951-1024) from a work by a priest + and physician, <span class="special" + title="Nazif">Naẓīf</span> ibn Yumn,<a name="NtA_381" + href="#Nt_381"><sup>[381]</sup></a> al-Qass (died c. 990). Suter does not + mention this work of <span class="special" + title="Nazif">Naẓīf</span>.</p> + + <p>The second reason for not ascribing too much credit to the purely Arab + influence is that the Arab by himself never showed any intellectual + strength. What took place after <span class="special" + title="Mohammed">Moḥammed</span> 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.<a + name="NtA_382" href="#Nt_382"><sup>[382]</sup></a> The Arab influence + came not from its purity, but from its intermingling with an influence + more cultured if less virile.</p> + + <p>As a result of this interactivity among peoples of diverse interests + and powers, Mohammedanism was to the world from the eighth to the + thirteenth century what Rome and Athens and the Italo-Hellenic influence + generally had <!-- Page 95 --><span class="pagenum"><a + name="page95"></a>[95]</span>been to the ancient civilization. "If they + did not possess the spirit of invention which distinguished the Greeks + and the Hindus, if they did not show the perseverance in their + observations that characterized the Chinese astronomers, they at least + possessed the virility of a new and victorious people, with a desire to + understand what others had accomplished, and a taste which led them with + equal ardor to the study of algebra and of poetry, of philosophy and of + language."<a name="NtA_383" href="#Nt_383"><sup>[383]</sup></a></p> + + <p>It was in 622 <span class="scac">A.D.</span> that <span + class="special" title="Mohammed">Moḥammed</span> fled from Mecca, + and within a century from that time the crescent had replaced the cross + in Christian Asia, in Northern Africa, and in a goodly portion of Spain. + The Arab empire was an ellipse of learning with its foci at Bagdad and + Cordova, and its rulers not infrequently took pride in demanding + intellectual rather than commercial treasure as the result of conquest.<a + name="NtA_384" href="#Nt_384"><sup>[384]</sup></a></p> + + <p>It was under these influences, either pre-Mohammedan or later, that + the Hindu numerals found their way to the North. If they were known + before <span class="special" title="Mohammed">Moḥammed</span>'s + time, the proof of this fact is now lost. This much, however, is known, + that in the eighth century they were taken to Bagdad. It was early in + that century that the Mohammedans obtained their first foothold in + northern India, thus foreshadowing an epoch of supremacy that endured + with varied fortunes until after the golden age of Akbar the Great + (1542-1605) and Shah Jehan. They also conquered Khorassan and + Afghanistan, so that the learning and the commercial customs of India at + once found easy <!-- Page 96 --><span class="pagenum"><a + name="page96"></a>[96]</span>access to the newly-established schools and + the bazaars of Mesopotamia and western Asia. The particular paths of + conquest and of commerce were either by way of the Khyber Pass and + through Kabul, Herat and Khorassan, or by sea through the strait of Ormuz + to Basra (Busra) at the head of the Persian Gulf, and thence to Bagdad. + As a matter of fact, one form of Arabic numerals, the one now in use by + the Arabs, is attributed to the influence of Kabul, while the other, + which eventually became our numerals, may very likely have reached Arabia + by the other route. It is in Bagdad,<a name="NtA_385" + href="#Nt_385"><sup>[385]</sup></a> Dār al-Salām—"the + Abode of Peace," that our special interest in the introduction of the + numerals centers. Built upon the ruins of an ancient town by <span + class="special" title="Al-Mansur">Al-Manṣūr</span><a + name="NtA_386" href="#Nt_386"><sup>[386]</sup></a> in the second half of + the eighth century, it lies in one of those regions where the converging + routes of trade give rise to large cities.<a name="NtA_387" + href="#Nt_387"><sup>[387]</sup></a> Quite as well of Bagdad as of Athens + might Cardinal Newman have said:<a name="NtA_388" + href="#Nt_388"><sup>[388]</sup></a></p> + + <p>"What it lost in conveniences of approach, it gained in its + neighborhood to the traditions of the mysterious East, and in the + loveliness of the region in which it lay. Hither, then, as to a sort of + ideal land, where all archetypes of the great and the fair were found in + substantial being, and all departments of truth explored, and all + diversities of intellectual power exhibited, where taste and philosophy + were majestically enthroned as in a royal court, where there was no + sovereignty but that of mind, and no nobility but that of genius, where + professors were <!-- Page 97 --><span class="pagenum"><a + name="page97"></a>[97]</span>rulers, and princes did homage, thither + flocked continually from the very corners of the <i>orbis terrarum</i> + the many-tongued generation, just rising, or just risen into manhood, in + order to gain wisdom." For here it was that <span class="special" + title="Al-Mansur">Al-Manṣūr</span> and Al-Māmūn + and Hārūn al-Rashīd (Aaron the Just) made for a time + the world's center of intellectual activity in general and in the domain + of mathematics in particular.<a name="NtA_389" + href="#Nt_389"><sup>[389]</sup></a> It was just after the <i>Sindhind</i> + was brought to Bagdad that <span class="special" + title="Mohammed">Moḥammed</span> ibn Mūsā + al-Khowārazmī, whose name has already been mentioned,<a + name="NtA_390" href="#Nt_390"><sup>[390]</sup></a> was called to that + city. He was the most celebrated mathematician of his time, either in the + East or West, writing treatises on arithmetic, the sundial, the + astrolabe, chronology, geometry, and algebra, and giving through the + Latin transliteration of his name, <i>algoritmi</i>, the name of algorism + to the early arithmetics using the new Hindu numerals.<a name="NtA_391" + href="#Nt_391"><sup>[391]</sup></a> Appreciating at once the value of the + position system so recently brought from India, he wrote an arithmetic + based upon these numerals, and this was translated into Latin in the time + of Adelhard of Bath (c. 1180), although possibly by his contemporary + countryman Robert Cestrensis.<a name="NtA_392" + href="#Nt_392"><sup>[392]</sup></a> This translation was found in + Cambridge and was published by Boncompagni in 1857.<a name="NtA_393" + href="#Nt_393"><sup>[393]</sup></a></p> + + <p>Contemporary with Al-Khowārazmī, and working also under + Al-Māmūn, was a Jewish astronomer, <span class="special" + title="Abu 'l-Teiyib">Abū 'l-Ṭeiyib</span>, <!-- Page 98 + --><span class="pagenum"><a name="page98"></a>[98]</span>Sened ibn + ‛Alī, who is said to have adopted the Mohammedan religion at + the caliph's request. He also wrote a work on Hindu arithmetic,<a + name="NtA_394" href="#Nt_394"><sup>[394]</sup></a> so that the subject + must have been attracting considerable attention at that time. Indeed, + the struggle to have the Hindu numerals replace the Arabic did not cease + for a long time thereafter. ‛Alī ibn <span class="special" + title="Ahmed">Aḥmed</span> al-Nasawī, in his arithmetic of + c. 1025, tells us that the symbolism of number was still unsettled in his + day, although most people preferred the strictly Arabic forms.<a + name="NtA_395" href="#Nt_395"><sup>[395]</sup></a></p> + + <p>We thus have the numerals in Arabia, in two forms: one the form now + used there, and the other the one used by Al-Khowārazmī. The + question then remains, how did this second form find its way into Europe? + and this question will be considered in the next chapter.</p> + +<hr class="full" > + +<p><!-- Page 99 --><span class="pagenum"><a name="page99"></a>[99]</span></p> + +<h3>CHAPTER VII</h3> + +<p class="cenhead">THE DEFINITE INTRODUCTION OF THE NUMERALS +INTO EUROPE</p> + + <p>It being doubtful whether Boethius ever knew the Hindu numeral forms, + certainly without the zero in any case, it becomes necessary now to + consider the question of their definite introduction into Europe. From + what has been said of the trade relations between the East and the West, + and of the probability that it was the trader rather than the scholar who + carried these numerals from their original habitat to various commercial + centers, it is evident that we shall never know when they first made + their inconspicuous entrance into Europe. Curious customs from the East + and from the tropics,—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.</p> + + <p>There were many opportunities for such knowledge to reach Spain and + Italy. In the first place the Moors went into Spain as helpers of a + claimant of the throne, and <!-- Page 100 --><span class="pagenum"><a + name="page100"></a>[100]</span>remained as conquerors. The power of the + Goths, who had held Spain for three centuries, was shattered at the + battle of Jerez de la Frontera in 711, and almost immediately the Moors + became masters of Spain and so remained for five hundred years, and + masters of Granada for a much longer period. Until 850 the Christians + were absolutely free as to religion and as to holding political office, + so that priests and monks were not infrequently skilled both in Latin and + Arabic, acting as official translators, and naturally reporting directly + or indirectly to Rome. There was indeed at this time a complaint that + Christian youths cultivated too assiduously a love for the literature of + the Saracen, and married too frequently the daughters of the infidel.<a + name="NtA_396" href="#Nt_396"><sup>[396]</sup></a> It is true that this + happy state of affairs was not permanent, but while it lasted the + learning and the customs of the East must have become more or less the + property of Christian Spain. At this time the ġobār numerals + were probably in that country, and these may well have made their way + into Europe from the schools of Cordova, Granada, and Toledo.</p> + + <p>Furthermore, there was abundant opportunity for the numerals of the + East to reach Europe through the journeys of travelers and ambassadors. + It was from the records of Suleimān the Merchant, a well-known Arab + trader of the ninth century, that part of the story of Sindbād the + Sailor was taken.<a name="NtA_397" href="#Nt_397"><sup>[397]</sup></a> + Such a merchant would have been particularly likely to know the numerals + of the people whom he met, and he is a type of man that may well have + taken such symbols to European markets. A little later, <!-- Page 101 + --><span class="pagenum"><a name="page101"></a>[101]</span><span + class="special" title="Abu 'l-Hasan">Abū 'l-Ḥasan</span> + ‛Alī al-Mas‛ūdī (d. 956) of Bagdad + traveled to the China Sea on the east, at least as far south as Zanzibar, + and to the Atlantic on the west,<a name="NtA_398" + href="#Nt_398"><sup>[398]</sup></a> and he speaks of the nine figures + with which the Hindus reckoned.<a name="NtA_399" + href="#Nt_399"><sup>[399]</sup></a></p> + + <p>There was also a Bagdad merchant, one Abū 'l-Qāsim + ‛Obeidallāh ibn <span class="special" + title="Ahmed">Aḥmed</span>, better known by his Persian name <span + class="special" title="Ibn Khordadbeh">Ibn + Khordāḍbeh</span>,<a name="NtA_400" + href="#Nt_400"><sup>[400]</sup></a> who wrote about 850 <span + class="scac">A.D.</span> a work entitled <i>Book of Roads and + Provinces</i><a name="NtA_401" href="#Nt_401"><sup>[401]</sup></a> in + which the following graphic account appears:<a name="NtA_402" + href="#Nt_402"><sup>[402]</sup></a> "The Jewish merchants speak Persian, + Roman (Greek and Latin), Arabic, French, Spanish, and Slavic. They travel + from the West to the East, and from the East to the West, sometimes by + land, sometimes by sea. They take ship from France on the Western Sea, + and they voyage to Farama (near the ruins of the ancient Pelusium); there + they transfer their goods to caravans and go by land to Colzom (on the + Red Sea). They there reëmbark on the Oriental (Red) Sea and go to Hejaz + and to Jiddah, and thence to the Sind, India, and China. Returning, they + bring back the products of the oriental lands.... These journeys are also + made by land. The merchants, leaving France and Spain, cross to Tangier + and thence pass through the African provinces and Egypt. They then go to + Ramleh, visit Damascus, Kufa, Bagdad, and Basra, penetrate into Ahwaz, + Fars, Kerman, Sind, and thus reach India and China." Such travelers, + about 900 <span class="scac">A.D.</span>, must necessarily have spread + abroad a knowledge of all number <!-- Page 102 --><span + class="pagenum"><a name="page102"></a>[102]</span>systems used in + recording prices or in the computations of the market. There is an + interesting witness to this movement, a cruciform brooch now in the + British Museum. It is English, certainly as early as the eleventh + century, but it is inlaid with a piece of paste on which is the + Mohammedan inscription, in Kufic characters, "There is no God but God." + How did such an inscription find its way, perhaps in the time of Alcuin + of York, to England? And if these Kufic characters reached there, then + why not the numeral forms as well?</p> + + <p>Even in literature of the better class there appears now and then some + stray proof of the important fact that the great trade routes to the far + East were never closed for long, and that the customs and marks of trade + endured from generation to generation. The <i>Gulistān</i> of the + Persian poet Sa‛dī<a name="NtA_403" + href="#Nt_403"><sup>[403]</sup></a> contains such a passage:</p> + + <p>"I met a merchant who owned one hundred and forty camels, and fifty + slaves and porters.... He answered to me: 'I want to carry sulphur of + Persia to China, which in that country, as I hear, bears a high price; + and thence to take Chinese ware to Roum; and from Roum to load up with + brocades for Hind; and so to trade Indian steel (<i>pûlab</i>) to Halib. + From Halib I will convey its glass to Yeman, and carry the painted cloths + of Yeman back to Persia.'"<a name="NtA_404" + href="#Nt_404"><sup>[404]</sup></a> On the other hand, these men were not + of the learned class, nor would they preserve in treatises any knowledge + that they might have, although this knowledge would occasionally reach + the ears of the learned as bits of curious information.</p> + +<p><!-- Page 103 --><span class="pagenum"><a name="page103"></a>[103]</span></p> + + <p>There were also ambassadors passing back and forth from time to time, + between the East and the West, and in particular during the period when + these numerals probably began to enter Europe. Thus Charlemagne (c. 800) + sent emissaries to Bagdad just at the time of the opening of the + mathematical activity there.<a name="NtA_405" + href="#Nt_405"><sup>[405]</sup></a> And with such ambassadors must have + gone the adventurous scholar, inspired, as Alcuin says of Archbishop + Albert of York (766-780),<a name="NtA_406" + href="#Nt_406"><sup>[406]</sup></a> to seek the learning of other lands. + Furthermore, the Nestorian communities, established in Eastern Asia and + in India at this time, were favored both by the Persians and by their + Mohammedan conquerors. The Nestorian Patriarch of Syria, Timotheus + (778-820), sent missionaries both to India and to China, and a bishop was + appointed for the latter field. Ibn Wahab, who traveled to China in the + ninth century, found images of Christ and the apostles in the Emperor's + court.<a name="NtA_407" href="#Nt_407"><sup>[407]</sup></a> Such a + learned body of men, knowing intimately the countries in which they + labored, could hardly have failed to make strange customs known as they + returned to their home stations. Then, too, in Alfred's time (849-901) + emissaries went <!-- Page 104 --><span class="pagenum"><a + name="page104"></a>[104]</span>from England as far as India,<a + name="NtA_408" href="#Nt_408"><sup>[408]</sup></a> and generally in the + Middle Ages groceries came to Europe from Asia as now they come from the + colonies and from America. Syria, Asia Minor, and Cyprus furnished sugar + and wool, and India yielded her perfumes and spices, while rich + tapestries for the courts and the wealthy burghers came from Persia and + from China.<a name="NtA_409" href="#Nt_409"><sup>[409]</sup></a> Even in + the time of Justinian (c. 550) there seems to have been a silk trade with + China, which country in turn carried on commerce with Ceylon,<a + name="NtA_410" href="#Nt_410"><sup>[410]</sup></a> and reached out to + Turkestan where other merchants transmitted the Eastern products + westward. In the seventh century there was a well-defined commerce + between Persia and India, as well as between Persia and Constantinople.<a + name="NtA_411" href="#Nt_411"><sup>[411]</sup></a> The Byzantine + <i>commerciarii</i> were stationed at the outposts not merely as customs + officers but as government purchasing agents.<a name="NtA_412" + href="#Nt_412"><sup>[412]</sup></a></p> + + <p>Occasionally there went along these routes of trade men of real + learning, and such would surely have carried the knowledge of many + customs back and forth. Thus at a period when the numerals are known to + have been partly understood in Italy, at the opening of the eleventh + century, one Constantine, an African, traveled from Italy through a great + part of Africa and Asia, even on to India, for the purpose of learning + the sciences of the Orient. He spent thirty-nine years in travel, having + been hospitably received in Babylon, and upon his return he was welcomed + with great honor at Salerno.<a name="NtA_413" + href="#Nt_413"><sup>[413]</sup></a></p> + + <p>A very interesting illustration of this intercourse also appears in + the tenth century, when the son of Otto I <!-- Page 105 --><span + class="pagenum"><a name="page105"></a>[105]</span>(936-973) married a + princess from Constantinople. This monarch was in touch with the Moors of + Spain and invited to his court numerous scholars from abroad,<a + name="NtA_414" href="#Nt_414"><sup>[414]</sup></a> and his intercourse + with the East as well as the West must have brought together much of the + learning of each.</p> + + <p>Another powerful means for the circulation of mysticism and + philosophy, and more or less of culture, took its start just before the + conversion of Constantine (c. 312), in the form of Christian pilgrim + travel. This was a feature peculiar to the zealots of early Christianity, + found in only a slight degree among their Jewish predecessors in the + annual pilgrimage to Jerusalem, and almost wholly wanting in other + pre-Christian peoples. Chief among these early pilgrims were the two + Placentians, John and Antonine the Elder (c. 303), who, in their + wanderings to Jerusalem, seem to have started a movement which culminated + centuries later in the crusades.<a name="NtA_415" + href="#Nt_415"><sup>[415]</sup></a> In 333 a Bordeaux pilgrim compiled + the first Christian guide-book, the <i>Itinerary from Bordeaux to + Jerusalem</i>,<a name="NtA_416" href="#Nt_416"><sup>[416]</sup></a> and + from this time on the holy pilgrimage never entirely ceased.</p> + + <p>Still another certain route for the entrance of the numerals into + Christian Europe was through the pillaging and trading carried on by the + Arabs on the northern shores of the Mediterranean. As early as 652 <span + class="scac">A.D.</span>, in the thirtieth year of the Hejira, the + Mohammedans descended upon the shores of Sicily and took much spoil. + Hardly had the wretched Constans given place to the <!-- Page 106 + --><span class="pagenum"><a name="page106"></a>[106]</span>young + Constantine IV when they again attacked the island and plundered ancient + Syracuse. Again in 827, under Asad, they ravaged the coasts. Although at + this time they failed to conquer Syracuse, they soon held a good part of + the island, and a little later they successfully besieged the city. + Before Syracuse fell, however, they had plundered the shores of Italy, + even to the walls of Rome itself; and had not Leo IV, in 849, repaired + the neglected fortifications, the effects of the Moslem raid of that year + might have been very far-reaching. <span class="special" title="Ibn Khordadbeh" + >Ibn Khordāḍbeh</span>, who left Bagdad in the latter part + of the ninth century, gives a picture of the great commercial activity at + that time in the Saracen city of Palermo. In this same century they had + established themselves in Piedmont, and in 906 they pillaged Turin.<a + name="NtA_417" href="#Nt_417"><sup>[417]</sup></a> On the Sorrento + peninsula the traveler who climbs the hill to the beautiful Ravello sees + still several traces of the Arab architecture, reminding him of the fact + that about 900 <span class="scac">A.D.</span> Amalfi was a commercial + center of the Moors.<a name="NtA_418" href="#Nt_418"><sup>[418]</sup></a> + Not only at this time, but even a century earlier, the artists of + northern India sold their wares at such centers, and in the courts both + of Hārūn al-Rashīd and of Charlemagne.<a name="NtA_419" + href="#Nt_419"><sup>[419]</sup></a> Thus the Arabs dominated the + Mediterranean Sea long before Venice</p> + + <div class="poem"> + <div class="stanza"> + <p class="i12">"held the gorgeous East in fee</p> + <p>And was the safeguard of the West,"</p> + </div> + </div> + <p>and long before Genoa had become her powerful rival.<a name="NtA_420" + href="#Nt_420"><sup>[420]</sup></a></p> + +<p><!-- Page 107 --><span class="pagenum"><a name="page107"></a>[107]</span></p> + + <p>Only a little later than this the brothers Nicolo and Maffeo Polo + entered upon their famous wanderings.<a name="NtA_421" + href="#Nt_421"><sup>[421]</sup></a> Leaving Constantinople in 1260, they + went by the Sea of Azov to Bokhara, and thence to the court of Kublai + Khan, penetrating China, and returning by way of Acre in 1269 with a + commission which required them to go back to China two years later. This + time they took with them Nicolo's son Marco, the historian of the + journey, and went across the plateau of Pamir; they spent about twenty + years in China, and came back by sea from China to Persia.</p> + + <p>The ventures of the Poli were not long unique, however: the thirteenth + century had not closed before Roman missionaries and the merchant Petrus + de Lucolongo had penetrated China. Before 1350 the company of + missionaries was large, converts were numerous, churches and Franciscan + convents had been organized in the East, travelers were appealing for the + truth of their accounts to the "many" persons in Venice who had been in + China, Tsuan-chau-fu had a European merchant community, and Italian trade + and travel to China was a thing that occupied two chapters of a + commercial handbook.<a name="NtA_422" + href="#Nt_422"><sup>[422]</sup></a></p> + +<p><!-- Page 108 --><span class="pagenum"><a name="page108"></a>[108]</span></p> + + <p>It is therefore reasonable to conclude that in the Middle Ages, as in + the time of Boethius, it was a simple matter for any inquiring scholar to + become acquainted with such numerals of the Orient as merchants may have + used for warehouse or price marks. And the fact that Gerbert seems to + have known only the forms of the simplest of these, not comprehending + their full significance, seems to prove that he picked them up in just + this way.</p> + + <p>Even if Gerbert did not bring his knowledge of the Oriental numerals + from Spain, he may easily have obtained them from the marks on merchant's + goods, had he been so inclined. Such knowledge was probably obtainable in + various parts of Italy, though as parts of mere mercantile knowledge the + forms might soon have been lost, it needing the pen of the scholar to + preserve them. Trade at this time was not stagnant. During the eleventh + and twelfth centuries the Slavs, for example, had very great commercial + interests, their trade reaching to Kiev and Novgorod, and thence to the + East. Constantinople was a great clearing-house of commerce with the + Orient,<a name="NtA_423" href="#Nt_423"><sup>[423]</sup></a> and the + Byzantine merchants must have been entirely familiar with the various + numerals of the Eastern peoples. In the eleventh century the Italian town + of Amalfi established a factory<a name="NtA_424" + href="#Nt_424"><sup>[424]</sup></a> in Constantinople, and had trade + relations with Antioch and Egypt. Venice, as early as the ninth century, + had a valuable trade with Syria and Cairo.<a name="NtA_425" + href="#Nt_425"><sup>[425]</sup></a> Fifty years after Gerbert died, in + the time of Cnut, the Dane and the Norwegian pushed their commerce far + beyond the northern seas, both by caravans through Russia to the Orient, + and by their venturesome barks which <!-- Page 109 --><span + class="pagenum"><a name="page109"></a>[109]</span>sailed through the + Strait of Gibraltar into the Mediterranean.<a name="NtA_426" + href="#Nt_426"><sup>[426]</sup></a> Only a little later, probably before + 1200 <span class="scac">A.D.</span>, a clerk in the service of Thomas à + Becket, present at the latter's death, wrote a life of the martyr, to + which (fortunately for our purposes) he prefixed a brief eulogy of the + city of London.<a name="NtA_427" href="#Nt_427"><sup>[427]</sup></a> This + clerk, William Fitz Stephen by name, thus speaks of the British + capital:</p> + + <div class="poem"> + <div class="stanza"> + <p>Aurum mittit Arabs: species et thura Sabæus:</p> + <p>Arma Sythes: oleum palmarum divite sylva</p> + <p>Pingue solum Babylon: Nilus lapides pretiosos:</p> + <p>Norwegi, Russi, varium grisum, sabdinas:</p> + <p>Seres, purpureas vestes: Galli, sua vina.</p> + </div> + </div> + <p>Although, as a matter of fact, the Arabs had no gold to send, and the + Scythians no arms, and Egypt no precious stones save only the turquoise, + the Chinese (<i>Seres</i>) may have sent their purple vestments, and the + north her sables and other furs, and France her wines. At any rate the + verses show very clearly an extensive foreign trade.</p> + + <p>Then there were the Crusades, which in these times brought the East in + touch with the West. The spirit of the Orient showed itself in the songs + of the troubadours, and the <i>baudekin</i>,<a name="NtA_428" + href="#Nt_428"><sup>[428]</sup></a> the canopy of Bagdad,<a + name="NtA_429" href="#Nt_429"><sup>[429]</sup></a> became common in the + churches of Italy. In Sicily and in Venice the textile industries of the + East found place, and made their way even to the Scandinavian + peninsula.<a name="NtA_430" href="#Nt_430"><sup>[430]</sup></a></p> + + <p>We therefore have this state of affairs: There was abundant + intercourse between the East and West for <!-- Page 110 --><span + class="pagenum"><a name="page110"></a>[110]</span>some centuries before + the Hindu numerals appear in any manuscripts in Christian Europe. The + numerals must of necessity have been known to many traders in a country + like Italy at least as early as the ninth century, and probably even + earlier, but there was no reason for preserving them in treatises. + Therefore when a man like Gerbert made them known to the scholarly + circles, he was merely describing what had been familiar in a small way + to many people in a different walk of life.</p> + + <p>Since Gerbert<a name="NtA_431" href="#Nt_431"><sup>[431]</sup></a> was + for a long time thought to have been the one to introduce the numerals + into Italy,<a name="NtA_432" href="#Nt_432"><sup>[432]</sup></a> a brief + sketch of this unique character is proper. Born of humble parents,<a + name="NtA_433" href="#Nt_433"><sup>[433]</sup></a> this remarkable man + became the counselor and companion of kings, and finally wore the papal + tiara as Sylvester II, from 999 until his death in 1003.<a name="NtA_434" + href="#Nt_434"><sup>[434]</sup></a> He was early brought under the + influence of the monks at Aurillac, and particularly of Raimund, who had + been a pupil of Odo of Cluny, and there in due time he himself took holy + orders. He visited Spain in about 967 in company with Count Borel,<a + name="NtA_435" href="#Nt_435"><sup>[435]</sup></a> remaining there three + years, <!-- Page 111 --><span class="pagenum"><a + name="page111"></a>[111]</span>and studying under Bishop Hatto of Vich,<a + name="NtA_436" href="#Nt_436"><sup>[436]</sup></a> a city in the province + of Barcelona,<a name="NtA_437" href="#Nt_437"><sup>[437]</sup></a> then + entirely under Christian rule. Indeed, all of Gerbert's testimony is as + to the influence of the Christian civilization upon his education. Thus + he speaks often of his study of Boethius,<a name="NtA_438" + href="#Nt_438"><sup>[438]</sup></a> so that if the latter knew the + numerals Gerbert would have learned them from him.<a name="NtA_439" + href="#Nt_439"><sup>[439]</sup></a> If Gerbert had studied in any Moorish + schools he would, under the decree of the emir Hishām (787-822), + have been obliged to know Arabic, which would have taken most of his + three years in Spain, and of which study we have not the slightest hint + in any of his letters.<a name="NtA_440" + href="#Nt_440"><sup>[440]</sup></a> On the other hand, Barcelona was the + only Christian province in immediate touch with the Moorish civilization + at that time.<a name="NtA_441" href="#Nt_441"><sup>[441]</sup></a> + Furthermore we know that earlier in the same century King Alonzo of + Asturias (d. 910) confided the education of his son Ordoño to the Arab + scholars of the court of the <!-- Page 112 --><span class="pagenum"><a + name="page112"></a>[112]</span>wālī of Saragossa,<a + name="NtA_442" href="#Nt_442"><sup>[442]</sup></a> so that there was more + or less of friendly relation between Christian and Moor.</p> + + <p>After his three years in Spain, Gerbert went to Italy, about 970, + where he met Pope John XIII, being by him presented to the emperor Otto + I. Two years later (972), at the emperor's request, he went to Rheims, + where he studied philosophy, assisting to make of that place an + educational center; and in 983 he became abbot at Bobbio. The next year + he returned to Rheims, and became archbishop of that diocese in 991. For + political reasons he returned to Italy in 996, became archbishop of + Ravenna in 998, and the following year was elected to the papal chair. + Far ahead of his age in wisdom, he suffered as many such scholars have + even in times not so remote by being accused of heresy and witchcraft. As + late as 1522, in a biography published at Venice, it is related that by + black art he attained the papacy, after having given his soul to the + devil.<a name="NtA_443" href="#Nt_443"><sup>[443]</sup></a> Gerbert was, + however, interested in astrology,<a name="NtA_444" + href="#Nt_444"><sup>[444]</sup></a> although this was merely the + astronomy of that time and was such a science as any learned man would + wish to know, even as to-day we wish to be reasonably familiar with + physics and chemistry.</p> + + <p>That Gerbert and his pupils knew the ġobār numerals is a + fact no longer open to controversy.<a name="NtA_445" + href="#Nt_445"><sup>[445]</sup></a> Bernelinus and Richer<a + name="NtA_446" href="#Nt_446"><sup>[446]</sup></a> call them by the + well-known name of <!-- Page 113 --><span class="pagenum"><a + name="page113"></a>[113]</span>"caracteres," a word used by Radulph of + Laon in the same sense a century later.<a name="NtA_447" + href="#Nt_447"><sup>[447]</sup></a> It is probable that Gerbert was the + first to describe these ġobār numerals in any scientific way + in Christian Europe, but without the zero. If he knew the latter he + certainly did not understand its use.<a name="NtA_448" + href="#Nt_448"><sup>[448]</sup></a></p> + + <p>The question still to be settled is as to where he found these + numerals. That he did not bring them from Spain is the opinion of a + number of careful investigators.<a name="NtA_449" + href="#Nt_449"><sup>[449]</sup></a> This is thought to be the more + probable because most of the men who made Spain famous for learning lived + after Gerbert was there. Such were Ibn Sīnā (Avicenna) who + lived at the beginning, and Gerber of Seville who flourished in the + middle, of the eleventh century, and Abū Roshd (Averroës) who lived + at the end of the twelfth.<a name="NtA_450" + href="#Nt_450"><sup>[450]</sup></a> Others hold that his proximity to + <!-- Page 114 --><span class="pagenum"><a + name="page114"></a>[114]</span>the Arabs for three years makes it + probable that he assimilated some of their learning, in spite of the fact + that the lines between Christian and Moor at that time were sharply + drawn.<a name="NtA_451" href="#Nt_451"><sup>[451]</sup></a> Writers fail, + however, to recognize that a commercial numeral system would have been + more likely to be made known by merchants than by scholars. The itinerant + peddler knew no forbidden pale in Spain, any more than he has known one + in other lands. If the ġobār 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 + ġobār, make it all the more probable that it was through the + small tradesman of the Moors that this versatile scholar derived his + knowledge. Moreover the part of the geometry bearing his name, and that + seems unquestionably his, shows the Arab influence, proving that he at + least came into contact with the transplanted Oriental learning, even + though imperfectly.<a name="NtA_452" href="#Nt_452"><sup>[452]</sup></a> + There was also the persistent Jewish merchant trading with both peoples + then as now, always alive to the acquiring of useful knowledge, and it + would be very natural for a man like Gerbert to welcome learning from + such a source.</p> + + <p>On the other hand, the two leading sources of information as to the + life of Gerbert reveal practically nothing to show that he came within + the Moorish sphere of influence during his sojourn in Spain. These + sources <!-- Page 115 --><span class="pagenum"><a + name="page115"></a>[115]</span>are his letters and the history written by + Richer. Gerbert was a master of the epistolary art, and his exalted + position led to the preservation of his letters to a degree that would + not have been vouchsafed even by their classic excellence.<a + name="NtA_453" href="#Nt_453"><sup>[453]</sup></a> Richer was a monk at + St. Remi de Rheims, and was doubtless a pupil of Gerbert. The latter, + when archbishop of Rheims, asked Richer to write a history of his times, + and this was done. The work lay in manuscript, entirely forgotten until + Pertz discovered it at Bamberg in 1833.<a name="NtA_454" + href="#Nt_454"><sup>[454]</sup></a> The work is dedicated to Gerbert as + archbishop of Rheims,<a name="NtA_455" + href="#Nt_455"><sup>[455]</sup></a> and would assuredly have testified to + such efforts as he may have made to secure the learning of the Moors.</p> + + <p>Now it is a fact that neither the letters nor this history makes any + statement as to Gerbert's contact with the Saracens. The letters do not + speak of the Moors, of the Arab numerals, nor of Cordova. Spain is not + referred to by that name, and only one Spanish scholar is mentioned. In + one of his letters he speaks of Joseph Ispanus,<a name="NtA_456" + href="#Nt_456"><sup>[456]</sup></a> or Joseph Sapiens, but who this + Joseph the Wise of Spain may have been we do not know. Possibly <!-- Page + 116 --><span class="pagenum"><a name="page116"></a>[116]</span>it was he + who contributed the morsel of knowledge so imperfectly assimilated by the + young French monk.<a name="NtA_457" href="#Nt_457"><sup>[457]</sup></a> + Within a few years after Gerbert's visit two young Spanish monks of + lesser fame, and doubtless with not that keen interest in mathematical + matters which Gerbert had, regarded the apparently slight knowledge which + they had of the Hindu numeral forms as worthy of somewhat permanent + record<a name="NtA_458" href="#Nt_458"><sup>[458]</sup></a> in + manuscripts which they were transcribing. The fact that such knowledge + had penetrated to their modest cloisters in northern Spain—the one + Albelda or Albaida—indicates that it was rather widely + diffused.</p> + + <p>Gerbert's treatise <i>Libellus de numerorum divisione</i><a + name="NtA_459" href="#Nt_459"><sup>[459]</sup></a> is characterized by + Chasles as "one of the most obscure documents in the history of + science."<a name="NtA_460" href="#Nt_460"><sup>[460]</sup></a> The most + complete information in regard to this and the other mathematical works + of Gerbert is given by Bubnov,<a name="NtA_461" + href="#Nt_461"><sup>[461]</sup></a> who considers this work to be + genuine.<a name="NtA_462" href="#Nt_462"><sup>[462]</sup></a></p> + +<p><!-- Page 117 --><span class="pagenum"><a name="page117"></a>[117]</span></p> + + <p>So little did Gerbert appreciate these numerals that in his works + known as the <i>Regula de abaco computi</i> and the <i>Libellus</i> he + makes no use of them at all, employing only the Roman forms.<a + name="NtA_463" href="#Nt_463"><sup>[463]</sup></a> Nevertheless + Bernelinus<a name="NtA_464" href="#Nt_464"><sup>[464]</sup></a> refers to + the nine ġobār characters.<a name="NtA_465" + href="#Nt_465"><sup>[465]</sup></a> These Gerbert had marked on a + thousand <i>jetons</i> or counters,<a name="NtA_466" + href="#Nt_466"><sup>[466]</sup></a> using the latter on an abacus which + he had a sign-maker prepare for him.<a name="NtA_467" + href="#Nt_467"><sup>[467]</sup></a> Instead of putting eight counters in + say the tens' column, Gerbert would put a single counter marked 8, and so + for the other places, leaving the column empty where we would place a + zero, but where he, lacking the zero, had no counter to place. These + counters he possibly called <i>caracteres</i>, a name which adhered also + to the figures themselves. It is an interesting speculation to consider + whether these <i>apices</i>, as they are called in the Boethius + interpolations, were in any way suggested by those Roman jetons generally + known in numismatics as <i>tesserae</i>, and bearing the figures I-XVI, + the sixteen referring to the number of <i>assi</i> in a + <i>sestertius</i>.<a name="NtA_468" href="#Nt_468"><sup>[468]</sup></a> + The <!-- Page 118 --><span class="pagenum"><a + name="page118"></a>[118]</span>name <i>apices</i> adhered to the + Hindu-Arabic numerals until the sixteenth century.<a name="NtA_469" + href="#Nt_469"><sup>[469]</sup></a></p> + + <p>To the figures on the <i>apices</i> were given the names Igin, andras, + ormis, arbas, quimas, calctis or caltis, zenis, temenias, celentis, + sipos,<a name="NtA_470" href="#Nt_470"><sup>[470]</sup></a> the origin + and meaning of which still remain a mystery. The Semitic origin of + several of the words seems probable. <i>Wahud</i>, <i>thaneine</i>, <!-- + Page 119 --><span class="pagenum"><a + name="page119"></a>[119]</span><i>thalata</i>, <i>arba</i>, <i>kumsa</i>, + <i>setta</i>, <i>sebba</i>, <i>timinia</i>, <i>taseud</i> are given by + the Rev. R. Patrick<a name="NtA_471" href="#Nt_471"><sup>[471]</sup></a> + as the names, in an Arabic dialect used in Morocco, for the numerals from + one to nine. Of these the words for four, five, and eight are strikingly + like those given above.</p> + + <p>The name <i>apices</i> was not, however, a common one in later times. + <i>Notae</i> was more often used, and it finally gave the name to + notation.<a name="NtA_472" href="#Nt_472"><sup>[472]</sup></a> Still more + common were the names <i>figures</i>, <i>ciphers</i>, <i>signs</i>, + <i>elements</i>, and <i>characters</i>.<a name="NtA_473" + href="#Nt_473"><sup>[473]</sup></a></p> + + <p>So little effect did the teachings of Gerbert have in making known the + new numerals, that O'Creat, who lived a century later, a friend and pupil + of Adelhard <!-- Page 120 --><span class="pagenum"><a + name="page120"></a>[120]</span>of Bath, used the zero with the Roman + characters, in contrast to Gerbert's use of the ġobār forms + without the zero.<a name="NtA_474" href="#Nt_474"><sup>[474]</sup></a> + O'Creat uses three forms for zero, o, ō, and <span title="t" class="grk" + >τ</span>, 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. <span class="grk">τ</span>. <span class="grk">τ</span> for + 1200, I. O. VIII. IX for 1089, and I. IIII. IIII. <span + class="grk">τ</span><span class="grk">τ</span><span + class="grk">τ</span><span class="grk">τ</span> for the square of + 1200.</p> + + <p>The period from the time of Gerbert until after the appearance of + Leonardo's monumental work may be called the period of the abacists. Even + for many years after the appearance early in the twelfth century of the + books explaining the Hindu art of reckoning, there was strife between the + abacists, the advocates of the abacus, and the algorists, those who + favored the new numerals. The words <i>cifra</i> and <i>algorismus + cifra</i> were used with a somewhat derisive significance, indicative of + absolute uselessness, as indeed the zero is useless on an abacus in which + the value of any unit is given by the column which it occupies.<a + name="NtA_475" href="#Nt_475"><sup>[475]</sup></a> So Gautier de Coincy + (1177-1236) in a work on the miracles of Mary says:</p> + + <div class="poem"> + <div class="stanza"> + <p>A horned beast, a sheep,</p> + <p>An algorismus-cipher,</p> + <p>Is a priest, who on such a feast day</p> + <p>Does not celebrate the holy Mother.<a name="NtA_476" href="#Nt_476"><sup>[476]</sup></a></p> + </div> + </div> + <p>So the abacus held the field for a long time, even against the new + algorism employing the new numerals. <!-- Page 121 --><span + class="pagenum"><a name="page121"></a>[121]</span>Geoffrey Chaucer<a + name="NtA_477" href="#Nt_477"><sup>[477]</sup></a> describes in <i>The + Miller's Tale</i> the clerk with</p> + + <div class="poem"> + <div class="stanza"> + <p class="hg3">"His Almageste and bokes grete and smale,</p> + <p>His astrelabie, longinge for his art,</p> + <p>His augrim-stones layen faire apart</p> + <p>On shelves couched at his beddes heed."</p> + </div> + </div> + <p>So, too, in Chaucer's explanation of the astrolabe,<a name="NtA_478" + href="#Nt_478"><sup>[478]</sup></a> written for his son Lewis, the number + of degrees is expressed on the instrument in Hindu-Arabic numerals: "Over + the whiche degrees ther ben noumbres of augrim, that devyden thilke same + degrees fro fyve to fyve," and "... the nombres ... ben writen in + augrim," meaning in the way of the algorism. Thomas Usk about 1387 + writes:<a name="NtA_479" href="#Nt_479"><sup>[479]</sup></a> "a sypher in + augrim have no might in signification of it-selve, yet he yeveth power in + signification to other." So slow and so painful is the assimilation of + new ideas.</p> + + <p>Bernelinus<a name="NtA_480" href="#Nt_480"><sup>[480]</sup></a> states + that the abacus is a well-polished board (or table), which is covered + with blue sand and used by geometers in drawing geometrical figures. We + have previously mentioned the fact that the Hindus also performed + mathematical computations in the sand, although there is no evidence to + show that they had any column abacus.<a name="NtA_481" + href="#Nt_481"><sup>[481]</sup></a> For the purposes of computation, + Bernelinus continues, the board is divided into thirty vertical columns, + three of which are reserved for fractions. Beginning with the units + columns, each set of <!-- Page 122 --><span class="pagenum"><a + name="page122"></a>[122]</span>three columns (<i>lineae</i> is the word + which Bernelinus uses) is grouped together by a semicircular arc placed + above them, while a smaller arc is placed over the units column and + another joins the tens and hundreds columns. Thus arose the designation + <i>arcus pictagore</i><a name="NtA_482" + href="#Nt_482"><sup>[482]</sup></a> or sometimes simply <i>arcus</i>.<a + name="NtA_483" href="#Nt_483"><sup>[483]</sup></a> The operations of + addition, subtraction, and multiplication upon this form of the abacus + required little explanation, although they were rather extensively + treated, especially the multiplication of different orders of numbers. + But the operation of division was effected with some difficulty. For the + explanation of the method of division by the use of the complementary + difference,<a name="NtA_484" href="#Nt_484"><sup>[484]</sup></a> long the + stumbling-block in the way of the medieval arithmetician, the reader is + referred to works on the history of mathematics<a name="NtA_485" + href="#Nt_485"><sup>[485]</sup></a> and to works relating particularly to + the abacus.<a name="NtA_486" href="#Nt_486"><sup>[486]</sup></a></p> + + <p>Among the writers on the subject may be mentioned Abbo<a + name="NtA_487" href="#Nt_487"><sup>[487]</sup></a> of Fleury (c. 970), + Heriger<a name="NtA_488" href="#Nt_488"><sup>[488]</sup></a> of Lobbes or + Laubach <!-- Page 123 --><span class="pagenum"><a + name="page123"></a>[123]</span>(c. 950-1007), and Hermannus Contractus<a + name="NtA_489" href="#Nt_489"><sup>[489]</sup></a> (1013-1054), all of + whom employed only the Roman numerals. Similarly Adelhard of Bath (c. + 1130), in his work <i>Regulae Abaci</i>,<a name="NtA_490" + href="#Nt_490"><sup>[490]</sup></a> gives no reference to the new + numerals, although it is certain that he knew them. Other writers on the + abacus who used some form of Hindu numerals were Gerland<a name="NtA_491" + href="#Nt_491"><sup>[491]</sup></a> (first half of twelfth century) and + Turchill<a name="NtA_492" href="#Nt_492"><sup>[492]</sup></a> (c. 1200). + For the forms used at this period the reader is referred to the plate on + page <a href="#page88">88</a>.</p> + + <p>After Gerbert's death, little by little the scholars of Europe came to + know the new figures, chiefly through the introduction of Arab learning. + The Dark Ages had passed, although arithmetic did not find another + advocate as prominent as Gerbert for two centuries. Speaking of this + great revival, Raoul Glaber<a name="NtA_493" + href="#Nt_493"><sup>[493]</sup></a> (985-c. 1046), a monk of the great + Benedictine abbey of Cluny, of the eleventh century, says: "It was as + though the world had arisen and tossed aside the worn-out garments of + ancient time, and wished to apparel itself in a white robe of churches." + And with this activity in religion came a corresponding interest in other + lines. Algorisms began to appear, and knowledge from the outside world + found <!-- Page 124 --><span class="pagenum"><a + name="page124"></a>[124]</span>interested listeners. Another Raoul, or + Radulph, to whom we have referred as Radulph of Laon,<a name="NtA_494" + href="#Nt_494"><sup>[494]</sup></a> a teacher in the cloister school of + his city, and the brother of Anselm of Laon<a name="NtA_495" + href="#Nt_495"><sup>[495]</sup></a> the celebrated theologian, wrote a + treatise on music, extant but unpublished, and an arithmetic which Nagl + first published in 1890.<a name="NtA_496" + href="#Nt_496"><sup>[496]</sup></a> The latter work, preserved to us in a + parchment manuscript of seventy-seven leaves, contains a curious mixture + of Roman and ġobār numerals, the former for expressing large + results, the latter for practical calculation. These ġobār + "caracteres" include the sipos (zero), <a href="images/130a.png"><img + src="images/130a.png" class="middle" style="height:2ex" alt="Symbol" + /></a>, of which, however, Radulph did not know the full significance; + showing that at the opening of the twelfth century the system was still + uncertain in its status in the church schools of central France.</p> + + <p>At the same time the words <i>algorismus</i> and <i>cifra</i> were + coming into general use even in non-mathematical literature. Jordan <a + name="NtA_497" href="#Nt_497"><sup>[497]</sup></a> cites numerous + instances of such use from the works of Alanus ab Insulis<a + name="NtA_498" href="#Nt_498"><sup>[498]</sup></a> (Alain de Lille), + Gautier de Coincy (1177-1236), and others.</p> + + <p>Another contributor to arithmetic during this interesting period was a + prominent Spanish Jew called variously John of Luna, John of Seville, + Johannes Hispalensis, Johannes Toletanus, and Johannes Hispanensis de + Luna.<a name="NtA_499" href="#Nt_499"><sup>[499]</sup></a> <!-- Page 125 + --><span class="pagenum"><a name="page125"></a>[125]</span>His date is + rather closely fixed by the fact that he dedicated a work to Raimund who + was archbishop of Toledo between 1130 and 1150.<a name="NtA_500" + href="#Nt_500"><sup>[500]</sup></a> His interests were chiefly in the + translation of Arabic works, especially such as bore upon the + Aristotelian philosophy. From the standpoint of arithmetic, however, the + chief interest centers about a manuscript entitled <i>Joannis Hispalensis + liber Algorismi de Practica Arismetrice</i> which Boncompagni found in + what is now the <i>Bibliothèque nationale</i> at Paris. Although this + distinctly lays claim to being Al-Khowārazmī's work,<a + name="NtA_501" href="#Nt_501"><sup>[501]</sup></a> the evidence is + altogether against the statement,<a name="NtA_502" + href="#Nt_502"><sup>[502]</sup></a> but the book is quite as valuable, + since it represents the knowledge of the time in which it was written. It + relates to the operations with integers and sexagesimal fractions, + including roots, and contains no applications.<a name="NtA_503" + href="#Nt_503"><sup>[503]</sup></a></p> + + <p>Contemporary with John of Luna, and also living in Toledo, was Gherard + of Cremona,<a name="NtA_504" href="#Nt_504"><sup>[504]</sup></a> who has + sometimes been identified, but erroneously, with Gernardus,<a + name="NtA_505" href="#Nt_505"><sup>[505]</sup></a> the <!-- Page 126 + --><span class="pagenum"><a name="page126"></a>[126]</span>author of a + work on algorism. He was a physician, an astronomer, and a mathematician, + translating from the Arabic both in Italy and in Spain. In arithmetic he + was influential in spreading the ideas of algorism.</p> + + <p>Four Englishmen—Adelhard of Bath (c. 1130), Robert of Chester + (Robertus Cestrensis, c. 1143), William Shelley, and Daniel Morley + (1180)—are known<a name="NtA_506" + href="#Nt_506"><sup>[506]</sup></a> to have journeyed to Spain in the + twelfth century for the purpose of studying mathematics and Arabic. + Adelhard of Bath made translations from Arabic into Latin of + Al-Khowārazmī's astronomical tables<a name="NtA_507" + href="#Nt_507"><sup>[507]</sup></a> and of Euclid's Elements,<a + name="NtA_508" href="#Nt_508"><sup>[508]</sup></a> while Robert of + Chester is known as the translator of Al-Khowārazmī's + algebra.<a name="NtA_509" href="#Nt_509"><sup>[509]</sup></a> There is no + reason to doubt that all of these men, and others, were familiar with the + numerals which the Arabs were using.</p> + + <p>The earliest trace we have of computation with Hindu numerals in + Germany is in an Algorismus of 1143, now in the Hofbibliothek in + Vienna.<a name="NtA_510" href="#Nt_510"><sup>[510]</sup></a> It is bound + in with a <!-- Page 127 --><span class="pagenum"><a + name="page127"></a>[127]</span><i>Computus</i> by the same author and + bearing the date given. It contains chapters "De additione," "De + diminutione," "De mediatione," "De divisione," and part of a chapter on + multiplication. The numerals are in the usual medieval forms except the 2 + which, as will be seen from the illustration,<a name="NtA_511" + href="#Nt_511"><sup>[511]</sup></a> is somewhat different, and the 3, + which takes the peculiar shape <a href="images/133a.png"><img + src="images/133a.png" class="middle" style="height:2ex" alt="Symbol" + /></a>, a form characteristic of the twelfth century.</p> + + <p>It was about the same time that the <i>Sefer ha-Mispar</i>,<a + name="NtA_512" href="#Nt_512"><sup>[512]</sup></a> the Book of Number, + appeared in the Hebrew language. The author, Rabbi Abraham ibn Meïr ibn + Ezra,<a name="NtA_513" href="#Nt_513"><sup>[513]</sup></a> was born in + Toledo (c. 1092). In 1139 he went to Egypt, Palestine, and the Orient, + spending also some years in Italy. Later he lived in southern France and + in England. He died in 1167. The probability is that he acquired his + knowledge of the Hindu arithmetic<a name="NtA_514" + href="#Nt_514"><sup>[514]</sup></a> in his native town of Toledo, but it + is also likely that the knowledge of other systems which he acquired on + travels increased his appreciation of this one. We have mentioned the + fact that he used the first letters of the Hebrew alphabet, <span + lang="he" class="heb" title="A B G D H W Z CH T`" ><bdo dir="rtl">א + ב ג ד ה ו ז ח + ט</bdo></span>, for the numerals 9 8 7 6 5 4 3 2 1, and a circle + for the zero. The quotation in the note given below shows that he knew of + the Hindu origin; but in his manuscript, although he set down the Hindu + forms, he used the above nine Hebrew letters with place value for all + computations.</p> + +<hr class="full" > + +<p><!-- Page 128 --><span class="pagenum"><a name="page128"></a>[128]</span></p> + +<h3>CHAPTER VIII</h3> + +<p class="cenhead">THE SPREAD OF THE NUMERALS IN EUROPE</p> + + <p>Of all the medieval writers, probably the one most influential in + introducing the new numerals to the scholars of Europe was Leonardo + Fibonacci, of Pisa.<a name="NtA_515" href="#Nt_515"><sup>[515]</sup></a> + This remarkable man, the most noteworthy mathematical genius of the + Middle Ages, was born at Pisa about 1175.<a name="NtA_516" + href="#Nt_516"><sup>[516]</sup></a></p> + + <p>The traveler of to-day may cross the Via Fibonacci on his way to the + Campo Santo, and there he may see at the end of the long corridor, across + the quadrangle, the statue of Leonardo in scholars garb. Few towns have + honored a mathematician more, and few mathematicians have so distinctly + honored their birthplace. Leonardo was born in the golden age of this + city, the period of its commercial, religious, and intellectual + prosperity.<a name="NtA_517" href="#Nt_517"><sup>[517]</sup></a> <!-- + Page 129 --><span class="pagenum"><a + name="page129"></a>[129]</span>Situated practically at the mouth of the + Arno, Pisa formed with Genoa and Venice the trio of the greatest + commercial centers of Italy at the opening of the thirteenth century. + Even before Venice had captured the Levantine trade, Pisa had close + relations with the East. An old Latin chronicle relates that in 1005 + "Pisa was captured by the Saracens," that in the following year "the + Pisans overthrew the Saracens at Reggio," and that in 1012 "the Saracens + came to Pisa and destroyed it." The city soon recovered, however, sending + no fewer than a hundred and twenty ships to Syria in 1099,<a + name="NtA_518" href="#Nt_518"><sup>[518]</sup></a> founding a merchant + colony in Constantinople a few years later,<a name="NtA_519" + href="#Nt_519"><sup>[519]</sup></a> and meanwhile carrying on an + interurban warfare in Italy that seemed to stimulate it to great + activity.<a name="NtA_520" href="#Nt_520"><sup>[520]</sup></a> A writer + of 1114 tells us that at that time there were many heathen + people—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,<a + name="NtA_521" href="#Nt_521"><sup>[521]</sup></a> that the genius of + Leonardo appears as one of the surprises of history, warning us again + that "we should draw no horoscope; that we should expect little, for what + we expect will not come to pass."<a name="NtA_522" + href="#Nt_522"><sup>[522]</sup></a></p> + + <p>Leonardo's father was one William,<a name="NtA_523" + href="#Nt_523"><sup>[523]</sup></a> and he had a brother named + Bonaccingus,<a name="NtA_524" href="#Nt_524"><sup>[524]</sup></a> but + nothing further is <!-- Page 130 --><span class="pagenum"><a + name="page130"></a>[130]</span>known of his family. As to Fibonacci, most + writers<a name="NtA_525" href="#Nt_525"><sup>[525]</sup></a> have assumed + that his father's name was Bonaccio,<a name="NtA_526" + href="#Nt_526"><sup>[526]</sup></a> whence <i>filius Bonaccii</i>, or + Fibonacci. Others<a name="NtA_527" href="#Nt_527"><sup>[527]</sup></a> + believe that the name, even in the Latin form of <i>filius Bonaccii</i> + as used in Leonardo's work, was simply a general one, like our Johnson or + Bronson (Brown's son); and the only contemporary evidence that we have + bears out this view. As to the name Bigollo, used by Leonardo, some have + thought it a self-assumed one meaning blockhead, a term that had been + applied to him by the commercial world or possibly by the university + circle, and taken by him that he might prove what a blockhead could do. + Milanesi,<a name="NtA_528" href="#Nt_528"><sup>[528]</sup></a> however, + has shown that the word Bigollo (or Pigollo) was used in Tuscany to mean + a traveler, and was naturally assumed by one who had studied, as Leonardo + had, in foreign lands.</p> + + <p>Leonardo's father was a commercial agent at Bugia, the modern + Bougie,<a name="NtA_529" href="#Nt_529"><sup>[529]</sup></a> the ancient + Saldae on the coast of Barbary,<a name="NtA_530" + href="#Nt_530"><sup>[530]</sup></a> a royal capital under the Vandals and + again, a century before Leonardo, under the Beni Hammad. It had one of + the best harbors on the coast, sheltered as it is by Mt. Lalla Guraia,<a + name="NtA_531" href="#Nt_531"><sup>[531]</sup></a> and at the close of + the twelfth century it was a center of African commerce. It was here that + Leonardo was taken as a child, and here he went to school to a Moorish + master. When he reached the years of young manhood he started on a tour + of the Mediterranean Sea, and visited Egypt, Syria, Greece, Sicily, and + Provence, meeting with scholars as well as with <!-- Page 131 --><span + class="pagenum"><a name="page131"></a>[131]</span>merchants, and imbibing + a knowledge of the various systems of numbers in use in the centers of + trade. All these systems, however, he says he counted almost as errors + compared with that of the Hindus.<a name="NtA_532" + href="#Nt_532"><sup>[532]</sup></a> Returning to Pisa, he wrote his + <i>Liber Abaci</i><a name="NtA_533" href="#Nt_533"><sup>[533]</sup></a> + in 1202, rewriting it in 1228.<a name="NtA_534" + href="#Nt_534"><sup>[534]</sup></a> In this work the numerals are + explained and are used in the usual computations of business. Such a + treatise was not destined to be popular, however, because it was too + advanced for the mercantile class, and too novel for the conservative + university circles. Indeed, at this time mathematics had only slight + place in the newly established universities, as witness the oldest known + statute of the Sorbonne at Paris, dated 1215, where the subject is + referred to only in an incidental way.<a name="NtA_535" + href="#Nt_535"><sup>[535]</sup></a> The period was one of great + commercial activity, and on this very <!-- Page 132 --><span + class="pagenum"><a name="page132"></a>[132]</span>account such a book + would attract even less attention than usual.<a name="NtA_536" + href="#Nt_536"><sup>[536]</sup></a></p> + + <p>It would now be thought that the western world would at once adopt the + new numerals which Leonardo had made known, and which were so much + superior to anything that had been in use in Christian Europe. The + antagonism of the universities would avail but little, it would seem, + against such an improvement. It must be remembered, however, that there + was great difficulty in spreading knowledge at this time, some two + hundred and fifty years before printing was invented. "Popes and princes + and even great religious institutions possessed far fewer books than many + farmers of the present age. The library belonging to the Cathedral Church + of San Martino at Lucca in the ninth century contained only nineteen + volumes of abridgments from ecclesiastical commentaries."<a + name="NtA_537" href="#Nt_537"><sup>[537]</sup></a> Indeed, it was not + until the early part of the fifteenth century that Palla degli Strozzi + took steps to carry out the project that had been in the mind of + Petrarch, the founding of a public library. It was largely by word of + mouth, therefore, that this early knowledge had to be transmitted. + Fortunately the presence of foreign students in Italy at this time made + this transmission feasible. (If human nature was the same then as now, it + is not impossible that the very opposition of the faculties to the works + of Leonardo led the students to investigate <!-- Page 133 --><span + class="pagenum"><a name="page133"></a>[133]</span>them the more + zealously.) At Vicenza in 1209, for example, there were Bohemians, Poles, + Frenchmen, Burgundians, Germans, and Spaniards, not to speak of + representatives of divers towns of Italy; and what was true there was + also true of other intellectual centers. The knowledge could not fail to + spread, therefore, and as a matter of fact we find numerous bits of + evidence that this was the case. Although the bankers of Florence were + forbidden to use these numerals in 1299, and the statutes of the + university of Padua required stationers to keep the price lists of books + "non per cifras, sed per literas claros,"<a name="NtA_538" + href="#Nt_538"><sup>[538]</sup></a> the numerals really made much headway + from about 1275 on.</p> + + <p>It was, however, rather exceptional for the common people of Germany + to use the Arabic numerals before the sixteenth century, a good witness + to this fact being the popular almanacs. Calendars of 1457-1496<a + name="NtA_539" href="#Nt_539"><sup>[539]</sup></a> have generally the + Roman numerals, while Köbel's calendar of 1518 gives the Arabic forms as + subordinate to the Roman. In the register of the Kreuzschule at Dresden + the Roman forms were used even until 1539.</p> + + <p>While not minimizing the importance of the scientific work of Leonardo + of Pisa, we may note that the more popular treatises by Alexander de + Villa Dei (c. 1240 <span class="scac">A.D.</span>) and John of Halifax + (Sacrobosco, c. 1250 <span class="scac">A.D.</span>) were much more + widely used, and doubtless contributed more to the spread of the numerals + among the common people.</p> + +<p><!-- Page 134 --><span class="pagenum"><a name="page134"></a>[134]</span></p> + + <p>The <i>Carmen de Algorismo</i><a name="NtA_540" + href="#Nt_540"><sup>[540]</sup></a> of Alexander de Villa Dei was written + in verse, as indeed were many other textbooks of that time. That it was + widely used is evidenced by the large number of manuscripts<a + name="NtA_541" href="#Nt_541"><sup>[541]</sup></a> extant in European + libraries. Sacrobosco's <i>Algorismus</i>,<a name="NtA_542" + href="#Nt_542"><sup>[542]</sup></a> in which some lines from the Carmen + are quoted, enjoyed a wide popularity as a textbook for university + instruction.<a name="NtA_543" href="#Nt_543"><sup>[543]</sup></a> The + work was evidently written with this end in view, as numerous + commentaries by university lecturers are found. Probably the most widely + used of these was that of Petrus de Dacia<a name="NtA_544" + href="#Nt_544"><sup>[544]</sup></a> written in 1291. These works throw an + interesting light upon the method of instruction in mathematics in use in + the universities from the thirteenth even to the sixteenth century. + Evidently the text was first read and copied by students.<a + name="NtA_545" href="#Nt_545"><sup>[545]</sup></a> Following this came + line by line an exposition of the text, such as is given in Petrus de + Dacia's commentary.</p> + + <p>Sacrobosco's work is of interest also because it was probably due to + the extended use of this work that the <!-- Page 135 --><span + class="pagenum"><a name="page135"></a>[135]</span>term <i>Arabic + numerals</i> became common. In two places there is mention of the + inventors of this system. In the introduction it is stated that this + science of reckoning was due to a philosopher named Algus, whence the + name <i>algorismus</i>,<a name="NtA_546" + href="#Nt_546"><sup>[546]</sup></a> and in the section on numeration + reference is made to the Arabs as the inventors of this science.<a + name="NtA_547" href="#Nt_547"><sup>[547]</sup></a> While some of the + commentators, Petrus de Dacia<a name="NtA_548" + href="#Nt_548"><sup>[548]</sup></a> among them, knew of the Hindu origin, + most of them undoubtedly took the text as it stood; and so the Arabs were + credited with the invention of the system.</p> + + <p>The first definite trace that we have of an algorism in the French + language is found in a manuscript written about 1275.<a name="NtA_549" + href="#Nt_549"><sup>[549]</sup></a> This interesting leaf, for the part + on algorism consists of a single folio, was noticed by the Abbé + Lebœuf as early as 1741,<a name="NtA_550" + href="#Nt_550"><sup>[550]</sup></a> and by Daunou in 1824.<a + name="NtA_551" href="#Nt_551"><sup>[551]</sup></a> It then seems to have + been lost in the multitude of Paris manuscripts; for although Chasles<a + name="NtA_552" href="#Nt_552"><sup>[552]</sup></a> relates his vain + search for it, it was not rediscovered until 1882. In that year M. Ch. + Henry found it, and to his care we owe our knowledge of the interesting + manuscript. The work is anonymous and is devoted almost entirely to + geometry, only <!-- Page 136 --><span class="pagenum"><a + name="page136"></a>[136]</span>two pages (one folio) relating to + arithmetic. In these the forms of the numerals are given, and a very + brief statement as to the operations, it being evident that the writer + himself had only the slightest understanding of the subject.</p> + + <p>Once the new system was known in France, even thus superficially, it + would be passed across the Channel to England. Higden,<a name="NtA_553" + href="#Nt_553"><sup>[553]</sup></a> writing soon after the opening of the + fourteenth century, speaks of the French influence at that time and for + some generations preceding:<a name="NtA_554" + href="#Nt_554"><sup>[554]</sup></a> "For two hundred years children in + scole, agenst the usage and manir of all other nations beeth compelled + for to leave hire own language, and for to construe hir lessons and hire + thynges in Frensche.... Gentilmen children beeth taught to speke Frensche + from the tyme that they bith rokked in hir cradell; and uplondissche men + will likne himself to gentylmen, and fondeth with greet besynesse for to + speke Frensche."</p> + + <p>The question is often asked, why did not these new numerals attract + more immediate attention? Why did they have to wait until the sixteenth + century to be generally used in business and in the schools? In reply it + may be said that in their elementary work the schools always wait upon + the demands of trade. That work which pretends to touch the life of the + people must come reasonably near doing so. Now the computations of + business until about 1500 did not demand the new figures, for two + reasons: First, cheap paper was not known. Paper-making of any kind was + not introduced into Europe until <!-- Page 137 --><span + class="pagenum"><a name="page137"></a>[137]</span>the twelfth century, + and cheap paper is a product of the nineteenth. Pencils, too, of the + modern type, date only from the sixteenth century. In the second place, + modern methods of operating, particularly of multiplying and dividing + (operations of relatively greater importance when all measures were in + compound numbers requiring reductions at every step), were not yet + invented. The old plan required the erasing of figures after they had + served their purpose, an operation very simple with counters, since they + could be removed. The new plan did not as easily permit this. Hence we + find the new numerals very tardily admitted to the counting-house, and + not welcomed with any enthusiasm by teachers.<a name="NtA_555" + href="#Nt_555"><sup>[555]</sup></a></p> + + <p>Aside from their use in the early treatises on the new art of + reckoning, the numerals appeared from time to time in the dating of + manuscripts and upon monuments. The oldest definitely dated European + document known <!-- Page 138 --><span class="pagenum"><a + name="page138"></a>[138]</span>to contain the numerals is a Latin + manuscript,<a name="NtA_556" href="#Nt_556"><sup>[556]</sup></a> the + Codex Vigilanus, written in the Albelda Cloister not far from Logroño in + Spain, in 976 <span class="scac">A.D.</span> The nine characters (of + ġobār type), without the zero, are given as an addition to + the first chapters of the third book of the <i>Origines</i> by Isidorus + of Seville, in which the Roman numerals are under discussion. Another + Spanish copy of the same work, of 992 <span class="scac">A.D.</span>, + contains the numerals in the corresponding section. The writer ascribes + an Indian origin to them in the following words: "Item de figuris + arithmeticę. 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 ġobār characters follow. Some of the abacus forms<a + name="NtA_557" href="#Nt_557"><sup>[557]</sup></a> previously given are + doubtless also of the tenth century. The earliest Arabic documents + containing the numerals are two manuscripts of 874 and 888 <span + class="scac">A.D.</span><a name="NtA_558" + href="#Nt_558"><sup>[558]</sup></a> They appear about a century later in + a work<a name="NtA_559" href="#Nt_559"><sup>[559]</sup></a> written at + Shiraz in 970 <span class="scac">A.D.</span> There is also an early trace + of their use on a pillar recently discovered in a church apparently + destroyed as early as the tenth century, not far from the Jeremias + Monastery, in Egypt. <!-- Page 139 --><span class="pagenum"><a + name="page139"></a>[139]</span>A graffito in Arabic on this pillar has + the date 349 <span class="scac">A.H.</span>, which corresponds to 961 + <span class="scac">A.D.</span><a name="NtA_560" + href="#Nt_560"><sup>[560]</sup></a> For the dating of Latin documents the + Arabic forms were used as early as the thirteenth century.<a + name="NtA_561" href="#Nt_561"><sup>[561]</sup></a></p> + + <p>On the early use of these numerals in Europe the only scientific study + worthy the name is that made by Mr. G. F. Hill of the British Museum.<a + name="NtA_562" href="#Nt_562"><sup>[562]</sup></a> From his + investigations it appears that the earliest occurrence of a date in these + numerals on a coin is found in the reign of Roger of Sicily in 1138.<a + name="NtA_563" href="#Nt_563"><sup>[563]</sup></a> Until recently it was + thought that the earliest such date was 1217 <span + class="scac">A.D.</span> for an Arabic piece and 1388 for a Turkish + one.<a name="NtA_564" href="#Nt_564"><sup>[564]</sup></a> Most of the + seals and medals containing dates that were at one time thought to be + very early have been shown by Mr. Hill to be of relatively late + workmanship. There are, however, in European manuscripts, numerous + instances of the use of these numerals before the twelfth century. + Besides the example in the Codex Vigilanus, another of the tenth century + has been found in the St. Gall MS. now in the University Library at + Zürich, the forms differing materially from those in the Spanish + codex.</p> + + <p>The third specimen in point of time in Mr. Hill's list is from a + Vatican MS. of 1077. The fourth and fifth specimens are from the Erlangen + MS. of Boethius, of the same <!-- Page 140 --><span class="pagenum"><a + name="page140"></a>[140]</span>(eleventh) century, and the sixth and + seventh are also from an eleventh-century MS. of Boethius at Chartres. + These and other early forms are given by Mr. Hill in this table, which is + reproduced with his kind permission.</p> + +<h3><span class="sc">Earliest Manuscript Forms</span></h3> + + <div class="figcenter" style="width:50%;"> + <a href="images/146a.png"><img style="width:100%" src="images/146a.png" + alt="Earliest Manuscript Forms" title="Earliest Manuscript Forms" /></a> + </div> + <p>This is one of more than fifty tables given in Mr. Hill's valuable + paper, and to this monograph students <!-- Page 141 --><span + class="pagenum"><a name="page141"></a>[141]</span>are referred for + details as to the development of number-forms in Europe from the tenth to + the sixteenth century. It is of interest to add that he has found that + among the earliest dates of European coins or medals in these numerals, + after the Sicilian one already mentioned, are the following: Austria, + 1484; Germany, 1489 (Cologne); Switzerland, 1424 (St. Gall); Netherlands, + 1474; France, 1485; Italy, 1390.<a name="NtA_565" + href="#Nt_565"><sup>[565]</sup></a></p> + + <p>The earliest English coin dated in these numerals was struck in + 1551,<a name="NtA_566" href="#Nt_566"><sup>[566]</sup></a> although there + is a Scotch piece of 1539.<a name="NtA_567" + href="#Nt_567"><sup>[567]</sup></a> In numbering pages of a printed book + these numerals were first used in a work of Petrarch's published at + Cologne in 1471.<a name="NtA_568" href="#Nt_568"><sup>[568]</sup></a> The + date is given in the following form in the <i>Biblia Pauperum</i>,<a + name="NtA_569" href="#Nt_569"><sup>[569]</sup></a> a block-book of + 1470,</p> + + <div class="figcenter" style="width:15%;"> + <a href="images/147a.png"><img style="width:100%" src="images/147a.png" + alt="Numerals 1470" title="Numerals 1470" /></a> + </div> + <p>while in another block-book which possibly goes back to c. 1430<a + name="NtA_570" href="#Nt_570"><sup>[570]</sup></a> the numerals appear in + several illustrations, with forms as follows:</p> + + <div class="figcenter" style="width:36%;"> + <a href="images/147b.png"><img style="width:100%" src="images/147b.png" + alt="Numerals" title="Numerals" /></a> + </div> + <p>Many printed works anterior to 1471 have pages or chapters numbered by + hand, but many of these numerals are <!-- Page 142 --><span + class="pagenum"><a name="page142"></a>[142]</span>of date much later than + the printing of the work. Other works were probably numbered directly + after printing. Thus the chapters 2, 3, 4, 5, 6 in a book of 1470<a + name="NtA_571" href="#Nt_571"><sup>[571]</sup></a> are numbered as + follows: Capitulem <a href="images/148a.png"><img src="images/148a.png" + class="middle" style="height:1.5ex" alt="Symbol 2" /></a>m.,... <a + href="images/148b.png"><img src="images/148b.png" class="middle" + style="height:1.5ex" alt="Symbol 3" /></a>m.,... 4m.,... v,... vi, and + followed by Roman numerals. This appears in the body of the text, in + spaces left by the printer to be filled in by hand. Another book<a + name="NtA_572" href="#Nt_572"><sup>[572]</sup></a> of 1470 has pages + numbered by hand with a mixture of Roman and Hindu numerals, thus,</p> + +<table class="nobctr"> +<tr><td valign="middle"><a href="images/148c.png"><img src="images/148c.png" class="middle" style="height:3.5ex" alt="Symbols" /></a></td><td valign="middle">for 125</td> +<td valign="middle"><a href="images/148e.png"><img src="images/148e.png" class="middle" style="height:3.5ex" alt="Symbols" /></a></td><td valign="middle">for 150</td></tr> +<tr><td valign="middle"><a href="images/148d.png"><img src="images/148d.png" class="middle" style="height:3.5ex" alt="Symbols" /></a></td><td valign="middle">for 147</td> +<td valign="middle"><a href="images/148f.png"><img src="images/148f.png" class="middle" style="height:3.5ex" alt="Symbols" /></a></td><td valign="middle">for 202</td></tr> +</table> + + <p>As to monumental inscriptions,<a name="NtA_573" + href="#Nt_573"><sup>[573]</sup></a> there was once thought to be a + gravestone at Katharein, near Troppau, with the date 1007, and one at + Biebrich of 1299. There is no doubt, however, of one at Pforzheim of 1371 + and one at Ulm of 1388.<a name="NtA_574" + href="#Nt_574"><sup>[574]</sup></a> Certain numerals on Wells Cathedral + have been assigned to the thirteenth century, but they are undoubtedly + considerably later.<a name="NtA_575" + href="#Nt_575"><sup>[575]</sup></a></p> + + <p>The table on page 143 will serve to supplement that from Mr. Hill's + work.<a name="NtA_576" href="#Nt_576"><sup>[576]</sup></a></p> + +<p><!-- Page 143 --><span class="pagenum"><a name="page143"></a>[143]</span></p> + +<h3><span class="sc">Early Manuscript Forms</span></h3> + +<table class="nobctr"> +<tr><td> </td><td><a href="images/149.png"><img src="images/149.png" class="middle" style="height:4.5ex" alt="1 2 3 4 5 6 7 8 9 0" /></a></td><td> </td></tr> +<tr><td valign="middle">a <a name="NtA_577" href="#Nt_577"><sup>[577]</sup></a></td><td><a href="images/149a.png"><img src="images/149a.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> Twelfth century <span class="scac">A.D.</span></td></tr> +<tr><td valign="middle">b <a name="NtA_578" href="#Nt_578"><sup>[578]</sup></a></td><td><a href="images/149b.png"><img src="images/149b.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> 1197 <span class="scac">A.D.</span></td></tr> +<tr><td valign="middle">c <a name="NtA_579" href="#Nt_579"><sup>[579]</sup></a></td><td><a href="images/149c.png"><img src="images/149c.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> 1275 <span class="scac">A.D.</span></td></tr> +<tr><td valign="middle">d <a name="NtA_580" href="#Nt_580"><sup>[580]</sup></a></td><td><a href="images/149d.png"><img src="images/149d.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> c. 1294 <span class="scac">A.D.</span></td></tr> +<tr><td valign="middle">e <a name="NtA_581" href="#Nt_581"><sup>[581]</sup></a></td><td><a href="images/149e.png"><img src="images/149e.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> c. 1303 <span class="scac">A.D.</span></td></tr> +<tr><td valign="middle">f <a name="NtA_582" href="#Nt_582"><sup>[582]</sup></a></td><td><a href="images/149f.png"><img src="images/149f.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> c. 1360 <span class="scac">A.D.</span></td></tr> +<tr><td valign="middle">g <a name="NtA_583" href="#Nt_583"><sup>[583]</sup></a></td><td><a href="images/149g.png"><img src="images/149g.png" class="middle" style="height:4.5ex" alt="Numerals" /></a></td><td valign="middle"> c. 1442 <span class="scac">A.D.</span></td></tr> +</table> + +<p><!-- Page 144 --><span class="pagenum"><a name="page144"></a>[144]</span></p> + + <div class="figleft" style="width:15%;"> + <a href="images/150a.png"><img style="width:100%" src="images/150a.png" + alt="Powers of 2." title="Powers of 2." /></a> + </div> + <p>For the sake of further comparison, three illustrations from works in + Mr. Plimpton's library, reproduced from the <i>Rara Arithmetica</i>, may + be considered. The first is from a Latin manuscript on arithmetic,<a + name="NtA_584" href="#Nt_584"><sup>[584]</sup></a> of which the original + was written at Paris in 1424 by Rollandus, a Portuguese physician, who + prepared the work at the command of John of Lancaster, Duke of Bedford, + at one time Protector of England and Regent of France, to whom the work + is dedicated. The figures show the successive powers of 2. The second + illustration is from Luca da Firenze's <i>Inprencipio darte + dabacho</i>,<a name="NtA_585" href="#Nt_585"><sup>[585]</sup></a> c. + 1475, and the third is from an anonymous manuscript<a name="NtA_586" + href="#Nt_586"><sup>[586]</sup></a> of about 1500.</p> + + <div class="figcenter" style="width:50%;"> + <a href="images/150b.png"><img style="width:100%" src="images/150b.png" + alt="Numerals." title="Numerals." /></a> + </div> + <p>As to the forms of the numerals, fashion played a leading part until + printing was invented. This tended to fix these forms, although in + writing there is still a great variation, as witness the French 5 and the + German 7 and 9. Even in printing there is not complete uniformity, <!-- + Page 145 --><span class="pagenum"><a name="page145"></a>[145]</span>and + it is often difficult for a foreigner to distinguish between the 3 and 5 + of the French types.</p> + + <div class="figcenter" style="width:60%;"> + <a href="images/150c.png"><img style="width:100%" src="images/150c.png" + alt="Numerals." title="Numerals." /></a> + </div> + <p>As to the particular numerals, the following are some of the forms to + be found in the later manuscripts and in the early printed books.</p> + + <p>1. In the early printed books "one" was often i, perhaps to save + types, just as some modern typewriters use the same character for l and + 1.<a name="NtA_587" href="#Nt_587"><sup>[587]</sup></a> In the + manuscripts the "one" appears in such forms as<a name="NtA_588" + href="#Nt_588"><sup>[588]</sup></a></p> + + <div class="figcenter" style="width:36%;"> + <a href="images/151a.png"><img style="width:100%" src="images/151a.png" + alt="Variations of 1." title="Variations of 1." /></a> + </div> + <p>2. "Two" often appears as z in the early printed books, 12 appearing + as iz.<a name="NtA_589" href="#Nt_589"><sup>[589]</sup></a> In the + medieval manuscripts the following forms are common:<a name="NtA_590" + href="#Nt_590"><sup>[590]</sup></a></p> + + <div class="figcenter" style="width:50%;"> + <a href="images/151b.png"><img style="width:100%" src="images/151b.png" + alt="Variations of 2." title="Variations of 2." /></a> + </div> +<p><!-- Page 146 --><span class="pagenum"><a name="page146"></a>[146]</span></p> + + <p>It is evident, from the early traces, that it is merely a cursive form + for the primitive <a href="images/033f.png"><img src="images/033f.png" + class="middle" style="height:1.5ex" alt="2 horizontal strokes" /></a>, + just as 3 comes from <a href="images/033h.png"><img src="images/033h.png" + class="middle" style="height:1.5ex" alt="3 horizontal strokes" /></a>, as + in the Nānā Ghāt inscriptions.</p> + + <p>3. "Three" usually had a special type in the first printed books, + although occasionally it appears as <a href="images/152a.png"><img + src="images/152a.png" class="middle" style="height:2ex" alt="Symbol" + /></a>.<a name="NtA_591" href="#Nt_591"><sup>[591]</sup></a> In the + medieval manuscripts it varied rather less than most of the others. The + following are common forms:<a name="NtA_592" + href="#Nt_592"><sup>[592]</sup></a></p> + + <div class="figcenter" style="width:48%;"> + <a href="images/152b.png"><img style="width:100%" src="images/152b.png" + alt="Variations of 3." title="Variations of 3." /></a> + </div> + <p>4. "Four" has changed greatly; and one of the first tests as to the + age of a manuscript on arithmetic, and the place where it was written, is + the examination of this numeral. Until the time of printing the most + common form was <a href="images/152c.png"><img src="images/152c.png" + class="middle" style="height:2ex" alt="Symbol" /></a>, although the + Florentine manuscript of Leonard of Pisa's work has the form <a + href="images/152d.png"><img src="images/152d.png" class="middle" + style="height:2ex" alt="Symbol" /></a>;<a name="NtA_593" + href="#Nt_593"><sup>[593]</sup></a> but the manuscripts show that the + Florentine arithmeticians and astronomers rather early began to + straighten the first of these forms up to forms like <a + href="images/152e.png"><img src="images/152e.png" class="middle" + style="height:2ex" alt="Symbol" /></a><a name="NtA_594" + href="#Nt_594"><sup>[594]</sup></a> and <a href="images/152f.png"><img + src="images/152f.png" class="middle" style="height:2ex" alt="Symbol" + /></a><a href="#Nt_594"><sup>[594]</sup></a> or <a + href="images/152g.png"><img src="images/152g.png" class="middle" + style="height:2ex" alt="Symbol" /></a>,<a name="NtA_595" + href="#Nt_595"><sup>[595]</sup></a> more closely resembling our own. The + first printed books generally used our present form<a name="NtA_596" + href="#Nt_596"><sup>[596]</sup></a> with the closed top <a + href="images/152h.png"><img src="images/152h.png" class="middle" + style="height:2ex" alt="Symbol" /></a>, the open top used in writing ( <a + href="images/152i.png"><img src="images/152i.png" class="middle" + style="height:2ex" alt="Symbol" /></a>) being <!-- Page 147 --><span + class="pagenum"><a name="page147"></a>[147]</span>purely modern. The + following are other forms of the four, from various manuscripts:<a + name="NtA_597" href="#Nt_597"><sup>[597]</sup></a></p> + + <div class="figcenter" style="width:50%;"> + <a href="images/153a.png"><img style="width:100%" src="images/153a.png" + alt="Variations of 4." title="Variations of 4." /></a> + </div> + <p>5. "Five" also varied greatly before the time of printing. The + following are some of the forms:<a name="NtA_598" + href="#Nt_598"><sup>[598]</sup></a></p> + + <div class="figcenter" style="width:42%;"> + <a href="images/153b.png"><img style="width:100%" src="images/153b.png" + alt="Variations of 5." title="Variations of 5." /></a> + </div> + <p>6. "Six" has changed rather less than most of the others. The chief + variation has been in the slope of the top, as will be seen in the + following:<a name="NtA_599" href="#Nt_599"><sup>[599]</sup></a></p> + + <div class="figcenter" style="width:30%;"> + <a href="images/153c.png"><img style="width:100%" src="images/153c.png" + alt="Variations of 6." title="Variations of 6." /></a> + </div> + <p>7. "Seven," like "four," has assumed its present erect form only since + the fifteenth century. In medieval times it appeared as follows:<a + name="NtA_600" href="#Nt_600"><sup>[600]</sup></a></p> + + <div class="figcenter" style="width:52%;"> + <a href="images/153d.png"><img style="width:100%" src="images/153d.png" + alt="Variations of 7." title="Variations of 7." /></a> + </div> +<p><!-- Page 148 --><span class="pagenum"><a name="page148"></a>[148]</span></p> + + <p>8. "Eight," like "six," has changed but little. In medieval times + there are a few variants of interest as follows:<a name="NtA_601" + href="#Nt_601"><sup>[601]</sup></a></p> + + <div class="figcenter" style="width:20%;"> + <a href="images/154a.png"><img style="width:100%" src="images/154a.png" + alt="Variations of 8." title="Variations of 8." /></a> + </div> + <p>In the sixteenth century, however, there was manifested a tendency to + write it <a href="images/154b.png"><img src="images/154b.png" + class="middle" style="height:1.5ex" alt="Symbol" /></a>.<a name="NtA_602" + href="#Nt_602"><sup>[602]</sup></a></p> + + <p>9. "Nine" has not varied as much as most of the others. Among the + medieval forms are the following:<a name="NtA_603" + href="#Nt_603"><sup>[603]</sup></a></p> + + <div class="figcenter" style="width:45%;"> + <a href="images/154c.png"><img style="width:100%" src="images/154c.png" + alt="Variations of 9." title="Variations of 9." /></a> + </div> + <p>0. The shape of the zero also had a varied history. The following are + common medieval forms:<a name="NtA_604" + href="#Nt_604"><sup>[604]</sup></a></p> + + <div class="figcenter" style="width:30%;"> + <a href="images/154d.png"><img style="width:100%" src="images/154d.png" + alt="Variations of 0." title="Variations of 0." /></a> + </div> + <p>The explanation of the place value was a serious matter to most of the + early writers. If they had been using an abacus constructed like the + Russian chotü, and had placed this before all learners of the positional + system, there would have been little trouble. But the medieval <!-- Page + 149 --><span class="pagenum"><a + name="page149"></a>[149]</span>line-reckoning, where the lines stood for + powers of 10 and the spaces for half of such powers, did not lend itself + to this comparison. Accordingly we find such labored explanations as the + following, from <i>The Crafte of Nombrynge</i>:</p> + + <p>"Euery of these figuris bitokens hym selfe & no more, yf he stonde + in the first place of the rewele....</p> + + <p>"If it stonde in the secunde place of the rewle, he betokens ten tymes + hym selfe, as this figure 2 here 20 tokens ten tyme hym selfe, that is + twenty, for he hym selfe betokens tweyne, & 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....</p> + + <p>"Nil cifra significat sed dat signare sequenti. Expone this verse. A + cifre tokens noȝt, 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...."<a name="NtA_605" href="#Nt_605"><sup>[605]</sup></a></p> + + <p>It would seem that a system that was thus used for dating documents, + coins, and monuments, would have been generally adopted much earlier than + it was, particularly in those countries north of Italy where it did not + come into general use until the sixteenth century. This, however, has + been the fate of many inventions, as witness our neglect of logarithms + and of contracted processes to-day.</p> + + <p>As to Germany, the fifteenth century saw the rise of the new + symbolism; the sixteenth century saw it slowly <!-- Page 150 --><span + class="pagenum"><a name="page150"></a>[150]</span>gain the mastery; the + seventeenth century saw it finally conquer the system that for two + thousand years had dominated the arithmetic of business. Not a little of + the success of the new plan was due to Luther's demand that all learning + should go into the vernacular.<a name="NtA_606" + href="#Nt_606"><sup>[606]</sup></a></p> + + <p>During the transition period from the Roman to the Arabic numerals, + various anomalous forms found place. For example, we have in the + fourteenth century c<span class="grk">α</span> for 104;<a + name="NtA_607" href="#Nt_607"><sup>[607]</sup></a> 1000. 300. 80 et 4 for + 1384;<a name="NtA_608" href="#Nt_608"><sup>[608]</sup></a> and in a + manuscript of the fifteenth century 12901 for 1291.<a name="NtA_609" + href="#Nt_609"><sup>[609]</sup></a> In the same century m. cccc. 8II + appears for 1482,<a name="NtA_610" href="#Nt_610"><sup>[610]</sup></a> + while M<sup>o</sup>CCCC<sup>o</sup>50 (1450) and MCCCCXL6 (1446) are used + by Theodoricus Ruffi about the same time.<a name="NtA_611" + href="#Nt_611"><sup>[611]</sup></a> To the next century belongs the form + 1vojj for 1502. Even in Sfortunati's <i>Nuovo lume</i><a name="NtA_612" + href="#Nt_612"><sup>[612]</sup></a> the use of ordinals is quite + confused, the propositions on a single page being numbered "tertia," "4," + and "V."</p> + + <p>Although not connected with the Arabic numerals in any direct way, the + medieval astrological numerals may here be mentioned. These are given by + several early writers, but notably by Noviomagus (1539),<a name="NtA_613" + href="#Nt_613"><sup>[613]</sup></a> as follows<a name="NtA_614" + href="#Nt_614"><sup>[614]</sup></a>:</p> + + <div class="figcenter" style="width:52%;"> + <a href="images/156a.png"><img style="width:100%" src="images/156a.png" + alt="Astrological numerals." title="Astrological numerals." /></a> + </div> +<p><!-- Page 151 --><span class="pagenum"><a name="page151"></a>[151]</span></p> + + <p>Thus we find the numerals gradually replacing the Roman forms all over + Europe, from the time of Leonardo of Pisa until the seventeenth century. + But in the Far East to-day they are quite unknown in many countries, and + they still have their way to make. In many parts of India, among the + common people of Japan and China, in Siam and generally about the Malay + Peninsula, in Tibet, and among the East India islands, the natives still + adhere to their own numeral forms. Only as Western civilization is making + its way into the commercial life of the East do the numerals as used by + us find place, save as the Sanskrit forms appear in parts of India. It is + therefore with surprise that the student of mathematics comes to realize + how modern are these forms so common in the West, how limited is their + use even at the present time, and how slow the world has been and is in + adopting such a simple device as the Hindu-Arabic numerals.</p> + +<hr class="full" > + +<p><!-- Page 153 --><span class="pagenum"><a name="page153"></a>[153]</span></p> + +<h3>INDEX</h3> + + <p><i>Transcriber's note: many of the entries refer to footnotes linked + from the page numbers given.</i></p> + + <div class="poem"> + <div class="stanza"> + <p>Abbo of Fleury, <a href="#page122">122</a></p> + <p><span class="special" title="`Abdallah ibn al-Hasan">‛Abdallāh ibn al-Ḥasan</span>, <a href="#page92">92</a></p> + <p>‛Abdallatīf ibn Yūsuf, <a href="#page93">93</a></p> + <p>‛Abdalqādir ibn ‛Alī al-Sakhāwī, <a href="#page6">6</a></p> + <p>Abenragel, <a href="#page34">34</a></p> + <p>Abraham ibn Meïr ibn Ezra, <i>see</i> Rabbi ben Ezra</p> + <p><span class="special" title="Abu `Ali al-Hosein ibn Sina">Abū ‛Alī al-Ḥosein ibn Sīnā</span>, <a href="#page74">74</a></p> + <p><span class="special" title="Abu 'l-Hasan">Abū 'l-Ḥasan</span>, <a href="#page93">93</a>, <a href="#page100">100</a></p> + <p>Abū 'l-Qāsim, <a href="#page92">92</a></p> + <p><span class="special" title="Abu 'l-Teiyib">Abū 'l-Ṭeiyib</span>, <a href="#page97">97</a></p> + <p><span class="special" title="Abu Nasr">Abū Naṣr</span>, <a href="#page92">92</a></p> + <p>Abū Roshd, <a href="#page113">113</a></p> + <p>Abu Sahl Dunash ibn Tamim, <a href="#page65">65</a>, <a href="#page67">67</a></p> + <p>Adelhard of Bath, <a href="#page5">5</a>, <a href="#page55">55</a>, <a href="#page97">97</a>, <a href="#page119">119</a>, <a href="#page123">123</a>, <a href="#page126">126</a></p> + <p>Adhemar of Chabanois, <a href="#page111">111</a></p> + <p><span class="special" title="Ahmed al-Nasawi">Aḥmed al-Nasawī</span>, <a href="#page98">98</a></p> + <p><span class="special" title="Ahmed ibn `Abdallah">Aḥmed ibn ‛Abdallāh</span>, <a href="#page9">9</a>, <a href="#page92">92</a></p> + <p><span class="special" title="Ahmed ibn Mohammed">Aḥmed ibn Moḥammed</span>, <a href="#page94">94</a></p> + <p><span class="special" title="Ahmed ibn `Omar">Aḥmed ibn ‛Omar</span>, <a href="#page93">93</a></p> + <p><span class="special" title="Aksaras">Akṣaras</span>, <a href="#page32">32</a></p> + <p>Alanus ab Insulis, <a href="#page124">124</a></p> + <p>Al-Baġdādī, <a href="#page93">93</a></p> + <p>Al-Battānī, <a href="#page54">54</a></p> + <p>Albelda (Albaida) MS., <a href="#page116">116</a></p> + <p>Albert, J., <a href="#page62">62</a></p> + <p>Albert of York, <a href="#page103">103</a></p> + <p>Al-Bīrūnī, <a href="#page6">6</a>, <a href="#page41">41</a>, <a href="#page49">49</a>, <a href="#page65">65</a>, <a href="#page92">92</a>, <a href="#page93">93</a></p> + <p>Alcuin, <a href="#page103">103</a></p> + <p>Alexander the Great, <a href="#page76">76</a></p> + <p>Alexander de Villa Dei, <a href="#page11">11</a>, <a href="#page133">133</a></p> + <p>Alexandria, <a href="#page64">64</a>, <a href="#page82">82</a></p> + <p>Al-Fazārī, <a href="#page92">92</a></p> + <p>Alfred, <a href="#page103">103</a></p> + <p>Algebra, etymology, <a href="#page5">5</a></p> + <p>Algerian numerals, <a href="#page68">68</a></p> + <p>Algorism, <a href="#page97">97</a></p> + <p>Algorismus, <a href="#page124">124</a>, <a href="#page126">126</a>, <a href="#page135">135</a></p> + <p>Algorismus cifra, <a href="#page120">120</a></p> + <p><span class="special" title="Al-Hassar">Al-Ḥaṣṣār</span>, <a href="#page65">65</a></p> + <p>‛Alī ibn Abī Bekr, <a href="#page6">6</a></p> + <p><span class="special" title="`Ali ibn Ahmed">‛Alī ibn Aḥmed</span>, <a href="#page93">93</a>, <a href="#page98">98</a></p> + <p>Al-Karābīsī, <a href="#page93">93</a></p> + <p>Al-Khowārazmī, <a href="#page4">4</a>, <a href="#page9">9</a>, <a href="#page10">10</a>, <a href="#page92">92</a>, <a href="#page97">97</a>, <a href="#page98">98</a>, <a href="#page125">125</a>, <a href="#page126">126</a></p> + <p>Al-Kindī, <a href="#page10">10</a>, <a href="#page92">92</a></p> + <p>Almagest, <a href="#page54">54</a></p> + <p>Al-Maġrebī, <a href="#page93">93</a></p> + <p><span class="special" title="Al-Mahalli">Al-Maḥallī</span>, <a href="#page6">6</a></p> + <p>Al-Māmūn, <a href="#page10">10</a>, <a href="#page97">97</a></p> + <p><span class="special" title="Al-Mansur">Al-Manṣūr</span>, <a href="#page96">96</a>, <a href="#page97">97</a></p> + <p>Al-Mas‛ūdī, <a href="#page7">7</a>, <a href="#page92">92</a></p> + <p>Al-Nadīm, <a href="#page9">9</a></p> + <p>Al-Nasawī, <a href="#page93">93</a>, <a href="#page98">98</a></p> + <p>Alphabetic numerals, <a href="#page39">39</a>, <a href="#page40">40</a>, <a href="#page43">43</a></p> + <p>Al-Qāsim, <a href="#page92">92</a></p> + <p>Al-Qass, <a href="#page94">94</a></p> + <p>Al-Sakhāwī, <a href="#page6">6</a></p> + <p><span class="special" title="Al-Sardafi">Al-Ṣardafī</span>, <a href="#page93">93</a></p> + <p>Al-Sijzī, <a href="#page94">94</a></p> + <p>Al-Sūfī, <a href="#page10">10</a>, <a href="#page92">92</a></p> + <p>Ambrosoli, <a href="#page118">118</a></p> + <p><span class="special" title="Ankapalli">Aṅkapalli</span>, <a href="#page43">43</a></p> + <p>Apices, <a href="#page87">87</a>, <a href="#page117">117</a>, <a href="#page118">118</a></p> + <p>Arabs, <a href="#page91">91</a>-<a href="#page98">98</a></p> + <p>Arbuthnot, <a href="#page141">141</a></p> +<!-- Page 154 --><span class="pagenum"><a name="page154"></a>[154]</span> + <p>Archimedes, <a href="#page15">15</a>, <a href="#page16">16</a></p> + <p>Arcus Pictagore, <a href="#page122">122</a></p> + <p>Arjuna, <a href="#page15">15</a></p> + <p>Arnold, E., <a href="#page15">15</a>, <a href="#page102">102</a></p> + <p>Ars memorandi, <a href="#page141">141</a></p> + <p><span class="special" title="Aryabhata">Āryabhaṭa</span>, <a href="#page39">39</a>, <a href="#page43">43</a>, <a href="#page44">44</a></p> + <p>Aryan numerals, <a href="#page19">19</a></p> + <p>Aschbach, <a href="#page134">134</a></p> + <p>Ashmole, <a href="#page134">134</a></p> + <p>Aśoka, <a href="#page19">19</a>, <a href="#page20">20</a>, <a href="#page22">22</a>, <a href="#page81">81</a></p> + <p><span class="special" title="As-sifr">Aṣ-ṣifr</span>, <a href="#page57">57</a>, <a href="#page58">58</a></p> + <p>Astrological numerals, <a href="#page150">150</a></p> + <p>Atharva-Veda, <a href="#page48">48</a>, <a href="#page49">49</a>, <a href="#page55">55</a></p> + <p>Augustus, <a href="#page80">80</a></p> + <p>Averroës, <a href="#page113">113</a></p> + <p>Avicenna, <a href="#page58">58</a>, <a href="#page74">74</a>, <a href="#page113">113</a></p> + </div> + + <div class="stanza"> + <p>Babylonian numerals, <a href="#page28">28</a></p> + <p>Babylonian zero, <a href="#page51">51</a></p> + <p>Bacon, R., <a href="#page131">131</a></p> + <p>Bactrian numerals, <a href="#page19">19</a>, <a href="#page30">30</a></p> + <p>Bæda, <a href="#page2">2</a>, <a href="#page72">72</a></p> + <p>Bagdad, <a href="#page4">4</a>, <a href="#page96">96</a></p> + <p><span class="special" title="Bakhsali">Bakhṣālī</span> manuscript, <a href="#page43">43</a>, <a href="#page49">49</a>, <a href="#page52">52</a>, <a href="#page53">53</a></p> + <p>Ball, C. J., <a href="#page35">35</a></p> + <p>Ball, W. W. R., <a href="#page36">36</a>, <a href="#page131">131</a></p> + <p><span class="special" title="Bana">Bāṇa</span>, <a href="#page44">44</a></p> + <p>Barth, A., <a href="#page39">39</a></p> + <p>Bayang inscriptions, <a href="#page39">39</a></p> + <p>Bayer, <a href="#page33">33</a></p> + <p>Bayley, E. C., <a href="#page19">19</a>, <a href="#page23">23</a>, <a href="#page30">30</a>, <a href="#page32">32</a>, <a href="#page52">52</a>, <a href="#page89">89</a></p> + <p>Beazley, <a href="#page75">75</a></p> + <p>Bede, <i>see</i> Bæda</p> + <p>Beldomandi, <a href="#page137">137</a></p> + <p>Beloch, J., <a href="#page77">77</a></p> + <p>Bendall, <a href="#page25">25</a>, <a href="#page52">52</a></p> + <p>Benfey, T., <a href="#page26">26</a></p> + <p>Bernelinus, <a href="#page88">88</a>, <a href="#page112">112</a>, <a href="#page117">117</a>, <a href="#page121">121</a></p> + <p>Besagne, <a href="#page128">128</a></p> + <p>Besant, W., <a href="#page109">109</a></p> + <p>Bettino, <a href="#page36">36</a></p> + <p>Bhandarkar, <a href="#page18">18</a>, <a href="#page47">47</a>, <a href="#page49">49</a></p> + <p>Bhāskara, <a href="#page53">53</a>, <a href="#page55">55</a></p> + <p>Biernatzki, <a href="#page32">32</a></p> + <p>Biot, <a href="#page32">32</a></p> + <p>Björnbo, A. A., <a href="#page125">125</a>, <a href="#page126">126</a></p> + <p>Blassière, <a href="#page119">119</a></p> + <p>Bloomfield, <a href="#page48">48</a></p> + <p>Blume, <a href="#page85">85</a></p> + <p>Boeckh, <a href="#page62">62</a></p> + <p>Boehmer, <a href="#page143">143</a></p> + <p>Boeschenstein, <a href="#page119">119</a></p> + <p>Boethius, <a href="#page63">63</a>, <a href="#page70">70</a>-<a href="#page73">73</a>, <a href="#page83">83</a>-<a href="#page90">90</a></p> + <p>Boissière, <a href="#page63">63</a></p> + <p>Bombelli, <a href="#page81">81</a></p> + <p>Bonaini, <a href="#page128">128</a></p> + <p>Boncompagni, <a href="#page5">5</a>, <a href="#page6">6</a>, <a href="#page10">10</a>, <a href="#page48">48</a>, <a href="#page49">49</a>, <a href="#page123">123</a>, <a href="#page125">125</a></p> + <p>Borghi, <a href="#page59">59</a></p> + <p>Borgo, <a href="#page119">119</a></p> + <p>Bougie, <a href="#page130">130</a></p> + <p>Bowring, J., <a href="#page56">56</a></p> + <p>Brahmagupta, <a href="#page52">52</a></p> + <p><span class="special" title="Brahmanas">Brāhmaṇas</span>, <a href="#page12">12</a>, <a href="#page13">13</a></p> + <p>Brāhmī, <a href="#page19">19</a>, <a href="#page20">20</a>, <a href="#page31">31</a>, <a href="#page83">83</a></p> + <p>Brandis, J., <a href="#page54">54</a></p> + <p><span class="special" title="Brhat-Samhita">Bṛhat-Saṃhita</span>, <a href="#page39">39</a>, <a href="#page44">44</a>, <a href="#page78">78</a></p> + <p>Brockhaus, <a href="#page43">43</a></p> + <p>Bubnov, <a href="#page65">65</a>, <a href="#page84">84</a>, <a href="#page110">110</a>, <a href="#page116">116</a></p> + <p>Buddha, education of, <a href="#page15">15</a>, <a href="#page16">16</a></p> + <p>Büdinger, <a href="#page110">110</a></p> + <p>Bugia, <a href="#page130">130</a></p> + <p>Bühler, G., <a href="#page15">15</a>, <a href="#page19">19</a>, <a href="#page22">22</a>, <a href="#page31">31</a>, <a href="#page44">44</a>, <a href="#page49">49</a></p> + <p>Burgess, <a href="#page25">25</a></p> + <p>Bürk, <a href="#page13">13</a></p> + <p>Burmese numerals, <a href="#page36">36</a></p> + <p>Burnell, A. C., <a href="#page18">18</a>, <a href="#page40">40</a></p> + <p>Buteo, <a href="#page61">61</a></p> + </div> + + <div class="stanza"> + <p>Calandri, <a href="#page59">59</a>, <a href="#page81">81</a></p> + <p>Caldwell, R., <a href="#page19">19</a></p> + <p>Calendars, <a href="#page133">133</a></p> + <p>Calmet, <a href="#page34">34</a></p> + <p>Cantor, M., <a href="#page5">5</a>, <a href="#page13">13</a>, <a href="#page30">30</a>, <a href="#page43">43</a>, <a href="#page84">84</a></p> +<!-- Page 155 --><span class="pagenum"><a name="page155"></a>[155]</span> + <p>Capella, <a href="#page86">86</a></p> + <p>Cappelli, <a href="#page143">143</a></p> + <p>Caracteres, <a href="#page87">87</a>, <a href="#page113">113</a>, <a href="#page117">117</a>, <a href="#page119">119</a></p> + <p>Cardan, <a href="#page119">119</a></p> + <p>Carmen de Algorismo, <a href="#page11">11</a>, <a href="#page134">134</a></p> + <p>Casagrandi, <a href="#page132">132</a></p> + <p>Casiri, <a href="#page8">8</a>, <a href="#page10">10</a></p> + <p>Cassiodorus, <a href="#page72">72</a></p> + <p>Cataldi, <a href="#page62">62</a></p> + <p>Cataneo, <a href="#page3">3</a></p> + <p>Caxton, <a href="#page143">143</a>, <a href="#page146">146</a></p> + <p>Ceretti, <a href="#page32">32</a></p> + <p>Ceylon numerals, <a href="#page36">36</a></p> + <p>Chalfont, F. H., <a href="#page28">28</a></p> + <p>Champenois, <a href="#page60">60</a></p> + <p>Characters, <i>see</i> Caracteres</p> + <p>Charlemagne, <a href="#page103">103</a></p> + <p>Chasles, <a href="#page54">54</a>, <a href="#page60">60</a>, <a href="#page85">85</a>, <a href="#page116">116</a>, <a href="#page122">122</a>, <a href="#page135">135</a></p> + <p>Chassant, L. A., <a href="#page142">142</a></p> + <p>Chaucer, <a href="#page121">121</a></p> + <p>Chiarini, <a href="#page145">145</a>, <a href="#page146">146</a></p> + <p>Chiffre, <a href="#page58">58</a></p> + <p>Chinese numerals, <a href="#page28">28</a>, <a href="#page56">56</a></p> + <p>Chinese zero, <a href="#page56">56</a></p> + <p>Cifra, <a href="#page120">120</a>, <a href="#page124">124</a></p> + <p>Cipher, <a href="#page58">58</a></p> + <p>Circulus, <a href="#page58">58</a>, <a href="#page60">60</a></p> + <p>Clichtoveus, <a href="#page61">61</a>, <a href="#page119">119</a>, <a href="#page145">145</a></p> + <p>Codex Vigilanus, <a href="#page138">138</a></p> + <p>Codrington, O., <a href="#page139">139</a></p> + <p>Coins dated, <a href="#page141">141</a></p> + <p>Colebrooke, <a href="#page8">8</a>, <a href="#page26">26</a>, <a href="#page46">46</a>, <a href="#page53">53</a></p> + <p>Constantine, <a href="#page104">104</a>, <a href="#page105">105</a></p> + <p>Cosmas, <a href="#page82">82</a></p> + <p>Cossali, <a href="#page5">5</a></p> + <p>Counters, <a href="#page117">117</a></p> + <p>Courteille, <a href="#page8">8</a></p> + <p>Coxe, <a href="#page59">59</a></p> + <p>Crafte of Nombrynge, <a href="#page11">11</a>, <a href="#page87">87</a>, <a href="#page149">149</a></p> + <p>Crusades, <a href="#page109">109</a></p> + <p>Cunningham, A., <a href="#page30">30</a>, <a href="#page75">75</a></p> + <p>Curtze, <a href="#page55">55</a>, <a href="#page59">59</a>, <a href="#page126">126</a>, <a href="#page134">134</a></p> + <p>Cyfra, <a href="#page55">55</a></p> + </div> + + <div class="stanza"> + <p>Dagomari, <a href="#page146">146</a></p> + <p>D'Alviella, <a href="#page15">15</a></p> + <p>Dante, <a href="#page72">72</a></p> + <p>Dasypodius, <a href="#page33">33</a>, <a href="#page67">67</a>, <a href="#page63">63</a></p> + <p>Daunou, <a href="#page135">135</a></p> + <p>Delambre, <a href="#page54">54</a></p> + <p>Devanāgarī, <a href="#page7">7</a></p> + <p>Devoulx, A., <a href="#page68">68</a></p> + <p>Dhruva, <a href="#page49">49</a></p> + <p>Dicæarchus of Messana, <a href="#page77">77</a></p> + <p>Digits, <a href="#page119">119</a></p> + <p>Diodorus Siculus, <a href="#page76">76</a></p> + <p>Du Cange, <a href="#page62">62</a></p> + <p>Dumesnil, <a href="#page36">36</a></p> + <p>Dutt, R. C., <a href="#page12">12</a>, <a href="#page15">15</a>, <a href="#page18">18</a>, <a href="#page75">75</a></p> + <p>Dvivedī, <a href="#page44">44</a></p> + </div> + + <div class="stanza"> + <p>East and West, relations, <a href="#page73">73</a>-<a href="#page81">81</a>, <a href="#page100">100</a>-<a href="#page109">109</a></p> + <p>Egyptian numerals, <a href="#page27">27</a></p> + <p>Eisenlohr, <a href="#page28">28</a></p> + <p>Elia Misrachi, <a href="#page57">57</a></p> + <p>Enchiridion Algorismi, <a href="#page58">58</a></p> + <p>Eneström, <a href="#page5">5</a>, <a href="#page48">48</a>, <a href="#page59">59</a>, <a href="#page97">97</a>, <a href="#page125">125</a>, <a href="#page128">128</a></p> + <p>Europe, numerals in, <a href="#page63">63</a>, <a href="#page99">99</a>, <a href="#page128">128</a>, <a href="#page136">136</a></p> + <p>Eusebius Caesariensis, <a href="#page142">142</a></p> + <p>Euting, <a href="#page21">21</a></p> + <p>Ewald, P., <a href="#page116">116</a></p> + </div> + + <div class="stanza"> + <p>Fazzari, <a href="#page53">53</a>, <a href="#page54">54</a></p> + <p>Fibonacci, <i>see</i> Leonardo of Pisa</p> + <p>Figura nihili, <a href="#page58">58</a></p> + <p>Figures, <a href="#page119">119</a>. <i>See</i> numerals.</p> + <p>Fihrist, <a href="#page67">67</a>, <a href="#page68">68</a>, <a href="#page93">93</a></p> + <p>Finaeus, <a href="#page57">57</a></p> + <p>Firdusī, <a href="#page81">81</a></p> + <p>Fitz Stephen, W., <a href="#page109">109</a></p> + <p>Fleet, J. C., <a href="#page19">19</a>, <a href="#page20">20</a>, <a href="#page49">49</a></p> +<!-- Page 156 --><span class="pagenum"><a name="page156"></a>[156]</span> + <p>Florus, <a href="#page80">80</a></p> + <p>Flügel, G., <a href="#page68">68</a></p> + <p>Francisco de Retza, <a href="#page142">142</a></p> + <p>François, <a href="#page58">58</a></p> + <p>Friedlein, G., <a href="#page84">84</a>, <a href="#page113">113</a>, <a href="#page116">116</a>, <a href="#page122">122</a></p> + <p>Froude, J. A., <a href="#page129">129</a></p> + </div> + + <div class="stanza"> + <p>Gandhāra, <a href="#page19">19</a></p> + <p>Garbe, <a href="#page48">48</a></p> + <p>Gasbarri, <a href="#page58">58</a></p> + <p>Gautier de Coincy, <a href="#page120">120</a>, <a href="#page124">124</a></p> + <p>Gemma Frisius, <a href="#page2">2</a>, <a href="#page3">3</a>, <a href="#page119">119</a></p> + <p>Gerber, <a href="#page113">113</a></p> + <p>Gerbert, <a href="#page108">108</a>, <a href="#page110">110</a>-<a href="#page120">120</a>, <a href="#page122">122</a></p> + <p>Gerhardt, C. I., <a href="#page43">43</a>, <a href="#page56">56</a>, <a href="#page93">93</a>, <a href="#page118">118</a></p> + <p>Gerland, <a href="#page88">88</a>, <a href="#page123">123</a></p> + <p>Gherard of Cremona, <a href="#page125">125</a></p> + <p>Gibbon, <a href="#page72">72</a></p> + <p>Giles, H. A., <a href="#page79">79</a></p> + <p>Ginanni, <a href="#page81">81</a></p> + <p>Giovanni di Danti, <a href="#page58">58</a></p> + <p>Glareanus, <a href="#page4">4</a>, <a href="#page119">119</a></p> + <p>Gnecchi, <a href="#page71">71</a>, <a href="#page117">117</a></p> + <p>Ġobār numerals, <a href="#page65">65</a>, <a href="#page100">100</a>, <a href="#page112">112</a>, <a href="#page124">124</a>, <a href="#page138">138</a></p> + <p>Gow, J., <a href="#page81">81</a></p> + <p>Grammateus, <a href="#page61">61</a></p> + <p>Greek origin, <a href="#page33">33</a></p> + <p>Green, J. R., <a href="#page109">109</a></p> + <p>Greenwood, I., <a href="#page62">62</a>, <a href="#page119">119</a></p> + <p>Guglielmini, <a href="#page128">128</a></p> + <p>Gulistān, <a href="#page102">102</a></p> + <p>Günther, S., <a href="#page131">131</a></p> + <p>Guyard, S., <a href="#page82">82</a></p> + </div> + + <div class="stanza"> + <p><span class="special" title="Habash">Ḥabash</span>, <a href="#page9">9</a>, <a href="#page92">92</a></p> + <p>Hager, J. (G.), <a href="#page28">28</a>, <a href="#page32">32</a></p> + <p>Halliwell, <a href="#page59">59</a>, <a href="#page85">85</a></p> + <p>Hankel, <a href="#page93">93</a></p> + <p>Hārūn al-Rashīd, <a href="#page97">97</a>, <a href="#page106">106</a></p> + <p>Havet, <a href="#page110">110</a></p> + <p>Heath, T. L., <a href="#page125">125</a></p> + <p>Hebrew numerals, <a href="#page127">127</a></p> + <p>Hecatæus, <a href="#page75">75</a></p> + <p>Heiberg, J. L., <a href="#page55">55</a>, <a href="#page85">85</a>, <a href="#page148">148</a></p> + <p>Heilbronner, <a href="#page5">5</a></p> + <p>Henry, C., <a href="#page5">5</a>, <a href="#page31">31</a>, <a href="#page55">55</a>, <a href="#page87">87</a>, <a href="#page120">120</a>, <a href="#page135">135</a></p> + <p>Heriger, <a href="#page122">122</a></p> + <p>Hermannus Contractus, <a href="#page123">123</a></p> + <p>Herodotus, <a href="#page76">76</a>, <a href="#page78">78</a></p> + <p>Heyd, <a href="#page75">75</a></p> + <p>Higden, <a href="#page136">136</a></p> + <p>Hill, G. F., <a href="#page52">52</a>, <a href="#page139">139</a>, <a href="#page142">142</a></p> + <p>Hillebrandt, A., <a href="#page15">15</a>, <a href="#page74">74</a></p> + <p>Hilprecht, H. V., <a href="#page28">28</a></p> + <p>Hindu forms, early, <a href="#page12">12</a></p> + <p>Hindu number names, <a href="#page42">42</a></p> + <p>Hodder, <a href="#page62">62</a></p> + <p>Hoernle, <a href="#page43">43</a>, <a href="#page49">49</a></p> + <p>Holywood, <i>see</i> Sacrobosco</p> + <p>Hopkins, E. W., <a href="#page12">12</a></p> + <p>Horace, <a href="#page79">79</a>, <a href="#page80">80</a></p> + <p><span class="special" title="Hosein ibn Mohammed al-Mahalli">Ḥosein ibn Moḥammed al-Maḥallī</span>, <a href="#page6">6</a></p> + <p>Hostus, M., <a href="#page56">56</a></p> + <p>Howard, H. H., <a href="#page29">29</a></p> + <p>Hrabanus Maurus, <a href="#page72">72</a></p> + <p>Huart, <a href="#page7">7</a></p> + <p>Huet, <a href="#page33">33</a></p> + <p>Hugo, H., <a href="#page57">57</a></p> + <p>Humboldt, A. von, <a href="#page62">62</a></p> + <p>Huswirt, <a href="#page58">58</a></p> + </div> + + <div class="stanza"> + <p>Iamblichus, <a href="#page81">81</a></p> + <p>Ibn Abī Ya‛qūb, <a href="#page9">9</a></p> + <p>Ibn al-Adamī, <a href="#page92">92</a></p> + <p>Ibn al-Bannā, <a href="#page93">93</a></p> + <p><span class="special" title="Ibn Khordadbeh">Ibn Khordāḍbeh</span>, <a href="#page101">101</a>, <a href="#page106">106</a></p> + <p>Ibn Wahab, <a href="#page103">103</a></p> + <p>India, history of, <a href="#page14">14</a></p> + <p class="i2">writing in, <a href="#page18">18</a></p> + <p>Indicopleustes, <a href="#page83">83</a></p> + <p>Indo-Bactrian numerals, <a href="#page19">19</a></p> +<!-- Page 157 --><span class="pagenum"><a name="page157"></a>[157]</span> + <p>Indrājī, <a href="#page23">23</a></p> + <p><span class="special" title="Ishaq ibn Yusuf al-Sardafi">Isḥāq ibn Yūsuf al-Ṣardafī</span>, <a href="#page93">93</a></p> + </div> + + <div class="stanza"> + <p>Jacob of Florence, <a href="#page57">57</a></p> + <p>Jacquet, E., <a href="#page38">38</a></p> + <p>Jamshid, <a href="#page56">56</a></p> + <p>Jehan Certain, <a href="#page59">59</a></p> + <p>Jetons, <a href="#page58">58</a>, <a href="#page117">117</a></p> + <p>Jevons, F. B., <a href="#page76">76</a></p> + <p>Johannes Hispalensis, <a href="#page48">48</a>, <a href="#page88">88</a>, <a href="#page124">124</a></p> + <p>John of Halifax, <i>see</i> Sacrobosco</p> + <p>John of Luna, <i>see</i> Johannes Hispalensis</p> + <p>Jordan, L., <a href="#page58">58</a>, <a href="#page124">124</a></p> + <p>Joseph Ispanus (Joseph Sapiens), <a href="#page115">115</a></p> + <p>Justinian, <a href="#page104">104</a></p> + </div> + + <div class="stanza"> + <p>Kále, M. R., <a href="#page26">26</a></p> + <p>Karabacek, <a href="#page56">56</a></p> + <p>Karpinski, L. C., <a href="#page126">126</a>, <a href="#page134">134</a>, <a href="#page138">138</a></p> + <p>Kātyāyana, <a href="#page39">39</a></p> + <p>Kaye, C. R., <a href="#page6">6</a>, <a href="#page16">16</a>, <a href="#page43">43</a>, <a href="#page46">46</a>, <a href="#page121">121</a></p> + <p>Keane, J., <a href="#page75">75</a>, <a href="#page82">82</a></p> + <p>Keene, H. G., <a href="#page15">15</a></p> + <p>Kern, <a href="#page44">44</a></p> + <p><span class="special" title="Kharosthi">Kharoṣṭhī</span>, <a href="#page19">19</a>, <a href="#page20">20</a></p> + <p>Khosrū, <a href="#page82">82</a>, <a href="#page91">91</a></p> + <p>Kielhorn, F., <a href="#page46">46</a>, <a href="#page47">47</a></p> + <p>Kircher, A., <a href="#page34">34</a></p> + <p>Kitāb al-Fihrist, <i>see</i> Fihrist</p> + <p>Kleinwächter, <a href="#page32">32</a></p> + <p>K<span class="over">l</span>os, <a href="#page62">62</a></p> + <p>Köbel, <a href="#page4">4</a>, <a href="#page58">58</a>, <a href="#page60">60</a>, <a href="#page119">119</a>, <a href="#page123">123</a></p> + <p>Krumbacher, K., <a href="#page57">57</a></p> + <p>Kuckuck, <a href="#page62">62</a>, <a href="#page133">133</a></p> + <p>Kugler, F. X., <a href="#page51">51</a></p> + </div> + + <div class="stanza"> + <p>Lachmann, <a href="#page85">85</a></p> + <p>Lacouperie, <a href="#page33">33</a>, <a href="#page35">35</a></p> + <p>Lalitavistara, <a href="#page15">15</a>, <a href="#page17">17</a></p> + <p>Lami, G., <a href="#page57">57</a></p> + <p>La Roche, <a href="#page61">61</a></p> + <p>Lassen, <a href="#page39">39</a></p> + <p><span class="special" title="Latyayana">Lāṭyāyana</span>, <a href="#page39">39</a></p> + <p>Lebœuf, <a href="#page135">135</a></p> + <p>Leonardo of Pisa, <a href="#page5">5</a>, <a href="#page10">10</a>, <a href="#page57">57</a>, <a href="#page64">64</a>, <a href="#page74">74</a>, <a href="#page120">120</a>, <a href="#page128">128</a>-<a href="#page133">133</a></p> + <p>Lethaby, W. R., <a href="#page142">142</a></p> + <p>Levi, B., <a href="#page13">13</a></p> + <p>Levias, <a href="#page3">3</a></p> + <p>Libri, <a href="#page73">73</a>, <a href="#page85">85</a>, <a href="#page95">95</a></p> + <p>Light of Asia, <a href="#page16">16</a></p> + <p>Luca da Firenze, <a href="#page144">144</a></p> + <p>Lucas, <a href="#page128">128</a></p> + </div> + + <div class="stanza"> + <p>Mahābhārata, <a href="#page18">18</a></p> + <p>Mahāvīrācārya, <a href="#page53">53</a></p> + <p>Malabar numerals, <a href="#page36">36</a></p> + <p>Malayalam numerals, <a href="#page36">36</a></p> + <p>Mannert, <a href="#page81">81</a></p> + <p>Margarita Philosophica, <a href="#page146">146</a></p> + <p>Marie, <a href="#page78">78</a></p> + <p>Marquardt, J., <a href="#page85">85</a></p> + <p>Marshman, J. C., <a href="#page17">17</a></p> + <p>Martin, T. H., <a href="#page30">30</a>, <a href="#page62">62</a>, <a href="#page85">85</a>, <a href="#page113">113</a></p> + <p>Martines, D. C., <a href="#page58">58</a></p> + <p>Māshāllāh, <a href="#page3">3</a></p> + <p>Maspero, <a href="#page28">28</a></p> + <p>Mauch, <a href="#page142">142</a></p> + <p>Maximus Planudes, <a href="#page2">2</a>, <a href="#page57">57</a>, <a href="#page66">66</a>, <a href="#page93">93</a>, <a href="#page120">120</a></p> + <p>Megasthenes, <a href="#page77">77</a></p> + <p>Merchants, <a href="#page114">114</a></p> + <p>Meynard, <a href="#page8">8</a></p> + <p>Migne, <a href="#page87">87</a></p> + <p>Mikami, Y., <a href="#page56">56</a></p> + <p>Milanesi, <a href="#page128">128</a></p> + <p><span class="special" title="Mohammed ibn `Abdallah">Moḥammed ibn ‛Abdallāh</span>, <a href="#page92">92</a></p> + <p><span class="special" title="Mohammed ibn Ahmed">Moḥammed ibn Aḥmed</span>, <a href="#page6">6</a></p> + <p><span class="special" title="Mohammed ibn `Ali `Abdi">Moḥammed ibn ‛Alī ‛Abdī</span>, <a href="#page8">8</a></p> + <p><span class="special" title="Mohammed ibn Musa">Moḥammed ibn Mūsā</span>, <i>see</i> Al-Khowārazmī</p> + <p>Molinier, <a href="#page123">123</a></p> + <p>Monier-Williams, <a href="#page17">17</a></p> +<!-- Page 158 --><span class="pagenum"><a name="page158"></a>[158]</span> + <p>Morley, D., <a href="#page126">126</a></p> + <p>Moroccan numerals, <a href="#page68">68</a>, <a href="#page119">119</a></p> + <p>Mortet, V., <a href="#page11">11</a></p> + <p>Moseley, C. B., <a href="#page33">33</a></p> + <p><span class="special" title="Motahhar ibn Tahir">Moṭahhar ibn Ṭāhir</span>, <a href="#page7">7</a></p> + <p>Mueller, A., <a href="#page68">68</a></p> + <p>Mumford, J. K., <a href="#page109">109</a></p> + <p>Muwaffaq al-Dīn, <a href="#page93">93</a></p> + </div> + + <div class="stanza"> + <p>Nabatean forms, <a href="#page21">21</a></p> + <p>Nallino, <a href="#page4">4</a>, <a href="#page54">54</a>, <a href="#page55">55</a></p> + <p>Nagl, A., <a href="#page55">55</a>, <a href="#page110">110</a>, <a href="#page113">113</a>, <a href="#page126">126</a></p> + <p>Nānā Ghāt inscriptions, <a href="#page20">20</a>, <a href="#page22">22</a>, <a href="#page23">23</a>, <a href="#page40">40</a></p> + <p>Narducci, <a href="#page123">123</a></p> + <p>Nasik cave inscriptions, <a href="#page24">24</a></p> + <p><span class="special" title="Nazif ibn Yumn">Naẓīf ibn Yumn</span>, <a href="#page94">94</a></p> + <p>Neander, A., <a href="#page75">75</a></p> + <p>Neophytos, <a href="#page57">57</a>, <a href="#page62">62</a></p> + <p>Neo-Pythagoreans, <a href="#page64">64</a></p> + <p>Nesselmann, <a href="#page58">58</a></p> + <p>Newman, Cardinal, <a href="#page96">96</a></p> + <p>Newman, F. W., <a href="#page131">131</a></p> + <p>Nöldeke, Th., <a href="#page91">91</a></p> + <p>Notation, <a href="#page61">61</a></p> + <p>Note, <a href="#page61">61</a>, <a href="#page119">119</a></p> + <p>Noviomagus, <a href="#page45">45</a>, <a href="#page61">61</a>, <a href="#page119">119</a>, <a href="#page150">150</a></p> + <p>Null, <a href="#page61">61</a></p> + <p>Numerals,</p> + <p class="i2">Algerian, <a href="#page68">68</a></p> + <p class="i2">astrological, <a href="#page150">150</a></p> + <p class="i2">Brāhmī, <a href="#page19">19</a>-<a href="#page22">22</a>, <a href="#page83">83</a></p> + <p class="i2">early ideas of origin, <a href="#page1">1</a></p> + <p class="i2">Hindu, <a href="#page26">26</a></p> + <p class="i2">Hindu, classified, <a href="#page19">19</a>, <a href="#page38">38</a></p> + <p class="i2"><span class="special" title="Kharosthi">Kharoṣṭhī</span>, <a href="#page19">19</a>-<a href="#page22">22</a></p> + <p class="i2">Moroccan, <a href="#page68">68</a></p> + <p class="i2">Nabatean, <a href="#page21">21</a></p> + <p class="i2">origin, <a href="#page27">27</a>, <a href="#page30">30</a>, <a href="#page31">31</a>, <a href="#page37">37</a></p> + <p class="i2">supposed Arabic origin, <a href="#page2">2</a></p> + <p class="i2">supposed Babylonian origin, <a href="#page28">28</a></p> + <p class="i2">supposed Chaldean and Jewish origin, <a href="#page3">3</a></p> + <p class="i2">supposed Chinese origin, <a href="#page28">28</a>, <a href="#page32">32</a></p> + <p class="i2">supposed Egyptian origin, <a href="#page27">27</a>, <a href="#page30">30</a>, <a href="#page69">69</a>, <a href="#page70">70</a></p> + <p class="i2">supposed Greek origin, <a href="#page33">33</a></p> + <p class="i2">supposed Phœnician origin, <a href="#page32">32</a></p> + <p class="i2">tables of, <a href="#page22">22</a>-<a href="#page27">27</a>, <a href="#page36">36</a>, <a href="#page48">48</a>, <a href="#page49">49</a>, <a href="#page69">69</a>, <a href="#page88">88</a>, <a href="#page140">140</a>, <a href="#page143">143</a>, <a href="#page145">145</a>-<a href="#page148">148</a></p> + </div> + + <div class="stanza"> + <p>O'Creat, <a href="#page5">5</a>, <a href="#page55">55</a>, <a href="#page119">119</a>, <a href="#page120">120</a></p> + <p>Olleris, <a href="#page110">110</a>, <a href="#page113">113</a></p> + <p>Oppert, G., <a href="#page14">14</a>, <a href="#page75">75</a></p> + </div> + + <div class="stanza"> + <p>Pali, <a href="#page22">22</a></p> + <p>Pañcasiddhāntikā, <a href="#page44">44</a></p> + <p>Paravey, <a href="#page32">32</a>, <a href="#page57">57</a></p> + <p><span class="special" title="Pataliputra">Pātalīpuṭra</span>, <a href="#page77">77</a></p> + <p>Patna, <a href="#page77">77</a></p> + <p>Patrick, R., <a href="#page119">119</a></p> + <p>Payne, E. J., <a href="#page106">106</a></p> + <p>Pegolotti, <a href="#page107">107</a></p> + <p>Peletier, <a href="#page2">2</a>, <a href="#page62">62</a></p> + <p>Perrot, <a href="#page80">80</a></p> + <p>Persia, <a href="#page66">66</a>, <a href="#page91">91</a>, <a href="#page107">107</a></p> + <p>Pertz, <a href="#page115">115</a></p> + <p>Petrus de Dacia, <a href="#page59">59</a>, <a href="#page61">61</a>, <a href="#page62">62</a></p> + <p>Pez, P. B., <a href="#page117">117</a></p> + <p>"Philalethes," <a href="#page75">75</a></p> + <p>Phillips, G., <a href="#page107">107</a></p> + <p>Picavet, <a href="#page105">105</a></p> + <p>Pichler, F., <a href="#page141">141</a></p> + <p>Pihan, A. P., <a href="#page36">36</a></p> + <p>Pisa, <a href="#page128">128</a></p> + <p>Place value, <a href="#page26">26</a>, <a href="#page42">42</a>, <a href="#page46">46</a>, <a href="#page48">48</a></p> + <p>Planudes, <i>see</i> Maximus Planudes</p> + <p>Plimpton, G. A., <a href="#page56">56</a>, <a href="#page59">59</a>, <a href="#page85">85</a>, <a href="#page143">143</a>, <a href="#page144">144</a>, <a href="#page145">145</a>, <a href="#page148">148</a></p> + <p>Pliny, <a href="#page76">76</a></p> + <p>Polo, N. and M., <a href="#page107">107</a></p> +<!-- Page 159 --><span class="pagenum"><a name="page159"></a>[159]</span> + <p>Prändel, J. G., <a href="#page54">54</a></p> + <p>Prinsep, J., <a href="#page20">20</a>, <a href="#page31">31</a></p> + <p>Propertius, <a href="#page80">80</a></p> + <p>Prosdocimo de' Beldomandi, <a href="#page137">137</a></p> + <p>Prou, <a href="#page143">143</a></p> + <p>Ptolemy, <a href="#page54">54</a>, <a href="#page78">78</a></p> + <p>Putnam, <a href="#page103">103</a></p> + <p>Pythagoras, <a href="#page63">63</a></p> + <p>Pythagorean numbers, <a href="#page13">13</a></p> + <p>Pytheas of Massilia, <a href="#page76">76</a></p> + </div> + + <div class="stanza"> + <p>Rabbi ben Ezra, <a href="#page60">60</a>, <a href="#page127">127</a></p> + <p>Radulph of Laon, <a href="#page60">60</a>, <a href="#page113">113</a>, <a href="#page118">118</a>, <a href="#page124">124</a></p> + <p>Raets, <a href="#page62">62</a></p> + <p>Rainer, <i>see</i> Gemma Frisius</p> + <p>Rāmāyana, <a href="#page18">18</a></p> + <p>Ramus, <a href="#page2">2</a>, <a href="#page41">41</a>, <a href="#page60">60</a>, <a href="#page61">61</a></p> + <p>Raoul Glaber, <a href="#page123">123</a></p> + <p>Rapson, <a href="#page77">77</a></p> + <p>Rauhfuss, <i>see</i> Dasypodius</p> + <p>Raumer, K. von, <a href="#page111">111</a></p> + <p>Reclus, E., <a href="#page14">14</a>, <a href="#page96">96</a>, <a href="#page130">130</a></p> + <p>Recorde, <a href="#page3">3</a>, <a href="#page58">58</a></p> + <p>Reinaud, <a href="#page67">67</a>, <a href="#page74">74</a>, <a href="#page80">80</a></p> + <p>Reveillaud, <a href="#page36">36</a></p> + <p>Richer, <a href="#page110">110</a>, <a href="#page112">112</a>, <a href="#page115">115</a></p> + <p>Riese, A., <a href="#page119">119</a></p> + <p>Robertson, <a href="#page81">81</a></p> + <p>Robertus Cestrensis, <a href="#page97">97</a>, <a href="#page126">126</a></p> + <p>Rodet, <a href="#page5">5</a>, <a href="#page44">44</a></p> + <p>Roediger, J., <a href="#page68">68</a></p> + <p>Rollandus, <a href="#page144">144</a></p> + <p>Romagnosi, <a href="#page81">81</a></p> + <p>Rosen, F., <a href="#page5">5</a></p> + <p>Rotula, <a href="#page60">60</a></p> + <p>Rudolff, <a href="#page85">85</a></p> + <p>Rudolph, <a href="#page62">62</a>, <a href="#page67">67</a></p> + <p>Ruffi, <a href="#page150">150</a></p> + </div> + + <div class="stanza"> + <p>Sachau, <a href="#page6">6</a></p> + <p>Sacrobosco, <a href="#page3">3</a>, <a href="#page58">58</a>, <a href="#page133">133</a></p> + <p>Sacy, S. de, <a href="#page66">66</a>, <a href="#page70">70</a></p> + <p>Sa‛dī, <a href="#page102">102</a></p> + <p>Śaka inscriptions, <a href="#page20">20</a></p> + <p><span class="special" title="Samu'il ibn Yahya">Samū'īl ibn Yaḥyā</span>, <a href="#page93">93</a></p> + <p>Śāradā characters, <a href="#page55">55</a></p> + <p>Savonne, <a href="#page60">60</a></p> + <p>Scaliger, J. C., <a href="#page73">73</a></p> + <p>Scheubel, <a href="#page62">62</a></p> + <p>Schlegel, <a href="#page12">12</a></p> + <p>Schmidt, <a href="#page133">133</a></p> + <p>Schonerus, <a href="#page87">87</a>, <a href="#page119">119</a></p> + <p>Schroeder, L. von, <a href="#page13">13</a></p> + <p>Scylax, <a href="#page75">75</a></p> + <p>Sedillot, <a href="#page8">8</a>, <a href="#page34">34</a></p> + <p>Senart, <a href="#page20">20</a>, <a href="#page24">24</a>, <a href="#page25">25</a></p> + <p>Sened ibn ‛Alī, <a href="#page10">10</a>, <a href="#page98">98</a></p> + <p>Sfortunati, <a href="#page62">62</a>, <a href="#page150">150</a></p> + <p>Shelley, W., <a href="#page126">126</a></p> + <p>Siamese numerals, <a href="#page36">36</a></p> + <p>Siddhānta, <a href="#page8">8</a>, <a href="#page18">18</a></p> + <p><span class="special" title="Sifr">Ṣifr</span>, <a href="#page57">57</a></p> + <p>Sigsboto, <a href="#page55">55</a></p> + <p>Sihāb al-Dīn, <a href="#page67">67</a></p> + <p>Silberberg, <a href="#page60">60</a></p> + <p>Simon, <a href="#page13">13</a></p> + <p><span class="special" title="Sinan ibn al-Fath">Sinān ibn al-Fatḥ</span>, <a href="#page93">93</a></p> + <p>Sindbad, <a href="#page100">100</a></p> + <p>Sindhind, <a href="#page97">97</a></p> + <p>Sipos, <a href="#page60">60</a></p> + <p>Sirr, H. C., <a href="#page75">75</a></p> + <p>Skeel, C. A., <a href="#page74">74</a></p> + <p>Smith, D. E., <a href="#page11">11</a>, <a href="#page17">17</a>, <a href="#page53">53</a>, <a href="#page86">86</a>, <a href="#page141">141</a>, <a href="#page143">143</a></p> + <p>Smith, V. A., <a href="#page20">20</a>, <a href="#page35">35</a>, <a href="#page46">46</a>, <a href="#page47">47</a></p> + <p>Smith, Wm., <a href="#page75">75</a></p> + <p><span class="special" title="Smrti">Smṛti</span>, <a href="#page17">17</a></p> + <p>Spain, <a href="#page64">64</a>, <a href="#page65">65</a>, <a href="#page100">100</a></p> + <p>Spitta-Bey, <a href="#page5">5</a></p> + <p>Sprenger, <a href="#page94">94</a></p> + <p>Śrautasūtra, <a href="#page39">39</a></p> + <p>Steffens, F., <a href="#page116">116</a></p> + <p>Steinschneider, <a href="#page5">5</a>, <a href="#page57">57</a>, <a href="#page65">65</a>, <a href="#page66">66</a>, <a href="#page98">98</a>, <a href="#page126">126</a></p> + <p>Stifel, <a href="#page62">62</a></p> +<!-- Page 160 --><span class="pagenum"><a name="page160"></a>[160]</span> + <p>Subandhus, <a href="#page44">44</a></p> + <p>Suetonius, <a href="#page80">80</a></p> + <p>Suleimān, <a href="#page100">100</a></p> + <p>Śūnya, <a href="#page43">43</a>, <a href="#page53">53</a>, <a href="#page57">57</a></p> + <p>Suter, <a href="#page5">5</a>, <a href="#page9">9</a>, <a href="#page68">68</a>, <a href="#page69">69</a>, <a href="#page93">93</a>, <a href="#page116">116</a>, <a href="#page131">131</a></p> + <p>Sūtras, <a href="#page13">13</a></p> + <p>Sykes, P. M., <a href="#page75">75</a></p> + <p>Sylvester II, <i>see</i> Gerbert</p> + <p>Symonds, J. A., <a href="#page129">129</a></p> + </div> + + <div class="stanza"> + <p>Tannery, P., <a href="#page62">62</a>, <a href="#page84">84</a>, <a href="#page85">85</a></p> + <p>Tartaglia, <a href="#page4">4</a>, <a href="#page61">61</a></p> + <p>Taylor, I., <a href="#page19">19</a>, <a href="#page30">30</a></p> + <p>Teca, <a href="#page55">55</a>, <a href="#page61">61</a></p> + <p>Tennent, J. E., <a href="#page75">75</a></p> + <p>Texada, <a href="#page60">60</a></p> + <p>Theca, <a href="#page58">58</a>, <a href="#page61">61</a></p> + <p>Theophanes, <a href="#page64">64</a></p> + <p>Thibaut, G., <a href="#page12">12</a>, <a href="#page13">13</a>, <a href="#page16">16</a>, <a href="#page44">44</a>, <a href="#page47">47</a></p> + <p>Tibetan numerals, <a href="#page36">36</a></p> + <p>Timotheus, <a href="#page103">103</a></p> + <p>Tonstall, C., <a href="#page3">3</a>, <a href="#page61">61</a></p> + <p>Trenchant, <a href="#page60">60</a></p> + <p>Treutlein, <a href="#page5">5</a>, <a href="#page63">63</a>, <a href="#page123">123</a></p> + <p>Trevisa, <a href="#page136">136</a></p> + <p>Treviso arithmetic, <a href="#page145">145</a></p> + <p>Trivium and quadrivium, <a href="#page73">73</a></p> + <p>Tsin, <a href="#page56">56</a></p> + <p>Tunis, <a href="#page65">65</a></p> + <p>Turchill, <a href="#page88">88</a>, <a href="#page118">118</a>, <a href="#page123">123</a></p> + <p>Turnour, G., <a href="#page75">75</a></p> + <p>Tziphra, <a href="#page57">57</a>, <a href="#page62">62</a></p> + <p><span title="tziphra" class="grk">τζίφρα</span>, <a href="#page55">55</a>, <a href="#page57">57</a>, <a href="#page62">62</a></p> + <p>Tzwivel, <a href="#page61">61</a>, <a href="#page118">118</a>, <a href="#page145">145</a></p> + </div> + + <div class="stanza"> + <p>Ujjain, <a href="#page32">32</a></p> + <p>Unger, <a href="#page133">133</a></p> + <p>Upanishads, <a href="#page12">12</a></p> + <p>Usk, <a href="#page121">121</a></p> + </div> + + <div class="stanza"> + <p>Valla, G., <a href="#page61">61</a></p> + <p>Van der Schuere, <a href="#page62">62</a></p> + <p>Varāha-Mihira, <a href="#page39">39</a>, <a href="#page44">44</a>, <a href="#page78">78</a></p> + <p>Vāsavadattā, <a href="#page44">44</a></p> + <p>Vaux, Carra de, <a href="#page9">9</a>, <a href="#page74">74</a></p> + <p>Vaux, W. S. W., <a href="#page91">91</a></p> + <p><span class="special" title="Vedangas">Vedāṅgas</span>, <a href="#page17">17</a></p> + <p>Vedas, <a href="#page12">12</a>, <a href="#page15">15</a>, <a href="#page17">17</a></p> + <p>Vergil, <a href="#page80">80</a></p> + <p>Vincent, A. J. H., <a href="#page57">57</a></p> + <p>Vogt, <a href="#page13">13</a></p> + <p>Voizot, P., <a href="#page36">36</a></p> + <p>Vossius, <a href="#page4">4</a>, <a href="#page76">76</a>, <a href="#page81">81</a>, <a href="#page84">84</a></p> + </div> + + <div class="stanza"> + <p>Wallis, <a href="#page3">3</a>, <a href="#page62">62</a>, <a href="#page84">84</a>, <a href="#page116">116</a></p> + <p>Wappler, E., <a href="#page54">54</a>, <a href="#page126">126</a></p> + <p>Wäschke, H., <a href="#page2">2</a>, <a href="#page93">93</a></p> + <p>Wattenbach, <a href="#page143">143</a></p> + <p>Weber, A., <a href="#page31">31</a></p> + <p>Weidler, I. F., <a href="#page34">34</a>, <a href="#page66">66</a></p> + <p>Weidler, I. F. and G. I., <a href="#page63">63</a>, <a href="#page66">66</a></p> + <p>Weissenborn, <a href="#page85">85</a>, <a href="#page110">110</a></p> + <p>Wertheim, G., <a href="#page57">57</a>, <a href="#page61">61</a></p> + <p>Whitney, W. D., <a href="#page13">13</a></p> + <p>Wilford, F., <a href="#page75">75</a></p> + <p>Wilkens, <a href="#page62">62</a></p> + <p>Wilkinson, J. G., <a href="#page70">70</a></p> + <p>Willichius, <a href="#page3">3</a></p> + <p>Woepcke, <a href="#page3">3</a>, <a href="#page6">6</a>, <a href="#page42">42</a>, <a href="#page63">63</a>, <a href="#page64">64</a>, <a href="#page65">65</a>, <a href="#page67">67</a>, <a href="#page69">69</a>, <a href="#page70">70</a>, <a href="#page94">94</a>, <a href="#page113">113</a>, <a href="#page138">138</a></p> + <p>Wolack, G., <a href="#page54">54</a></p> + <p>Woodruff, C. E., <a href="#page32">32</a></p> + <p>Word and letter numerals, <a href="#page38">38</a>, <a href="#page44">44</a></p> + <p>Wüstenfeld, <a href="#page74">74</a></p> + </div> + + <div class="stanza"> + <p>Yule, H., <a href="#page107">107</a></p> + </div> + + <div class="stanza"> + <p>Zephirum, <a href="#page57">57</a>, <a href="#page58">58</a></p> + <p>Zephyr, <a href="#page59">59</a></p> + <p>Zepiro, <a href="#page58">58</a></p> + <p>Zero, <a href="#page26">26</a>, <a href="#page38">38</a>, <a href="#page40">40</a>, <a href="#page43">43</a>, <a href="#page45">45</a>, <a href="#page49">49</a>, <a href="#page51">51</a>-<a href="#page62">62</a>, <a href="#page67">67</a></p> + <p>Zeuero, <a href="#page58">58</a></p> + </div> + </div> +<hr class="full" > + +<h3>ANNOUNCEMENTS</h3> + +<hr class="full" > + +<h3>WENTWORTH'S</h3> + +<h2>COLLEGE ALGEBRA</h2> + +<p class="cenhead">REVISED EDITION</p> + +<p class="cenhead">12mo. Half morocco. 530 pages. List price, $1.50; mailing price, $1.65</p> + +<hr class="short" > + + <p>This book is a thorough revision of the author's "College Algebra." + Some chapters of the old edition have been wholly rewritten, and the + other chapters have been rewritten in part and greatly improved. The + order of topics has been changed to a certain extent; the plan is to have + each chapter as complete in itself as possible, so that the teacher may + vary the order of succession at his discretion.</p> + + <p>As the name implies, the work is intended for colleges and scientific + schools. The first part is simply a review of the principles of algebra + preceding Quadratic Equations, with just enough examples to illustrate + and enforce these principles. By this brief treatment of the first + chapters sufficient space is allowed, without making the book cumbersome, + for a full discussion of Quadratic Equations, The Binomial Theorem, + Choice Chance, Series, Determinants, and the General Properties of + Equations.</p> + + <p>Every effort has been made to present in the clearest light each + subject discussed, and to give in matter and methods the best training in + algebraic analysis at present attainable.</p> + +<p class="cenhead">ADDITIONAL PUBLICATIONS</p> + +<p class="cenhead">By G. A. 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It is arranged in harmony with the leading courses of study that + include algebra in the curriculum of the grades.</p> + + <p>The relation of algebra to arithmetic is emphasized, the subject is + treated topically, and each important point is touched at least twice. + The book begins by showing the uses of algebra, employing such practical + applications as are within the pupil's range of knowledge. When an + interest has thus been awakened in the subject, the fundamental + operations are presented with the simple explanations necessary to make + the student independent of dogmatic rules. Throughout the book abundant + oral and written drill exercises are provided. The work includes linear + equations with two unknown quantities, and easy quadratics.</p> + + <p>The leading features may be summarized as follows: (1) an arrangement + in harmony with existing courses of study; (2) a presentation designed to + awaken the interest of the pupils; (3) a topical arrangement for each + half year, every important topic being repeated; (4) simplicity of + explanations; (5) development of the relation of algebra to arithmetic + both in theory and in applications; (6) emphasis laid on the importance + of oral as well as written algebra.</p> + +<hr class="full" > + +<h2>GINN & COMPANY <span class="sc">Publishers</span></h2> + +<hr class="full" > + +<h2>BOOKS FOR TEACHERS</h2> + +<hr class="short" > + +<table class="nobctr"> +<tr><td> </td><td align="right">List price</td></tr> +<tr><td>Allen: Civics and Health</td><td align="right">$1.25</td></tr> +<tr><td>Brigham: Geographic Influences in American History</td><td align="right">1.25</td></tr> +<tr><td>Channing and Hart: Guide to the Study of American History</td><td align="right">2.00</td></tr> +<tr><td>Hall: Aspects of Child Life and Education</td><td align="right">1.50</td></tr> +<tr><td>Hodge: Nature Study and Life</td><td align="right">1.50</td></tr> +<tr><td>Johnson: Education by Plays and Games</td><td align="right">.90</td></tr> +<tr><td>Johnson: What to Do at Recess</td><td align="right">.25</td></tr> +<tr><td>Kern: Among Country Schools</td><td align="right">1.25</td></tr> +<tr><td>Mace: Method in History</td><td align="right">1.00</td></tr> +<tr><td>MacVicar: The Principles of Education</td><td align="right">.60</td></tr> +<tr><td>Moral Training in the Public Schools</td><td align="right">1.25</td></tr> +<tr><td>Prince: Courses of Studies and Methods of Teaching</td><td align="right">.75</td></tr> +<tr><td>Scott: Social Education</td><td align="right">1.25</td></tr> +<tr><td>Tompkins: Philosophy of School Management</td><td align="right">.75</td></tr> +<tr><td>Tompkins: Philosophy of Teaching</td><td align="right">.75</td></tr> +<tr><td>Wiltse: Place of the Story in Early Education, and Other Essays. 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[Bede, + <i>De computo dialogus</i> (doubtfully assigned to him), <i>Opera + omnia</i>, Paris, 1862, Vol. I, p. 650.]</p> + + <p>"Alii referunt ad Phœnices inventores arithmeticæ, propter + eandem commerciorum caussam: Alii ad Indos: Ioannes de Sacrobosco, cujus + sepulchrum est Lutetiæ in comitio Maturinensi, refert ad Arabes." [Ramus, + <i>Arithmeticæ libri dvo</i>, Basel, 1569, p. 112.]</p> + + <p>Similar notes are given by Peletarius in his commentary on the + arithmetic of Gemma Frisius (1563 ed., fol. 77), and in his own work + (1570 Lyons ed., p. 14): "La valeur des Figures commence au coste dextre + tirant vers le coste senestre: au rebours de notre maniere d'escrire par + ce que la premiere prattique est venue des Chaldees: ou des Pheniciens, + qui ont été les premiers traffiquers de marchandise."</p> + + <p><a name="Nt_2" href="#NtA_2">[2]</a> Maximus Planudes (c. 1330) states + that "the nine symbols come from the Indians." [Wäschke's German + translation, Halle, 1878, p. 3.] Willichius speaks of the "Zyphræ + Indicæ," in his <i>Arithmeticæ libri tres</i> (Strasburg, 1540, p. 93), + and Cataneo of "le noue figure de gli Indi," in his <i>Le pratiche delle + dve prime mathematiche</i> (Venice, 1546, fol. 1). Woepcke is not + correct, therefore, in saying ("Mémoire sur la propagation des chiffres + indiens," hereafter referred to as <i>Propagation</i> [<i>Journal + Asiatique</i>, Vol. I (6), 1863, p. 34]) that Wallis (<i>A Treatise on + Algebra, both historical and practical</i>, London, 1685, p. 13, and + <i>De algebra tractatus</i>, Latin edition in his <i>Opera omnia</i>, + 1693, Vol. II, p. 10) was one of the first to give the Hindu origin.</p> + + <p><a name="Nt_3" href="#NtA_3">[3]</a> From the 1558 edition of <i>The + Grovnd of Artes</i>, fol. C, 5. Similarly Bishop Tonstall writes: "Qui a + Chaldeis primum in finitimos, deinde in omnes pene gentes fluxit.... + Numerandi artem a Chaldeis esse profectam: qui dum scribunt, a dextra + incipiunt, et in leuam progrediuntur." [<i>De arte supputandi</i>, + London, 1522, fol. B, 3.] Gemma Frisius, the great continental rival of + Recorde, had the same idea: "Primùm autem appellamus dexterum locum, eo + quòd haec ars vel à Chaldæis, vel ab Hebræis ortum habere credatur, qui + etiam eo ordine scribunt"; but this refers more evidently to the Arabic + numerals. [<i>Arithmeticæ practicæ methodvs facilis</i>, Antwerp, 1540, + fol. 4 of the 1563 ed.] Sacrobosco (c. 1225) mentions the same thing. + Even the modern Jewish writers claim that one of their scholars, + Māshāllāh (c. 800), introduced them to the Mohammedan + world. [C. Levias, <i>The Jewish Encyclopedia</i>, New York, 1905, Vol. + IX, p. 348.]</p> + + <p><a name="Nt_4" href="#NtA_4">[4]</a> "... & que esto fu trouato di + fare da gli Arabi con diece figure." [<i>La prima parte del general + trattato di nvmeri, et misvre</i>, Venice, 1556, fol. 9 of the 1592 + edition.]</p> + + <p><a name="Nt_5" href="#NtA_5">[5]</a> "Vom welchen Arabischen auch disz + Kunst entsprungen ist." [<i>Ain nerv geordnet Rechenbiechlin</i>, + Augsburg, 1514, fol. 13 of the 1531 edition. The printer used the letters + <i>rv</i> for <i>w</i> in "new" in the first edition, as he had no + <i>w</i> of the proper font.]</p> + + <p><a name="Nt_6" href="#NtA_6">[6]</a> Among them Glareanus: + "Characteres simplices sunt nouem significatiui, ab Indis usque, siue + Chaldæis asciti .1.2.3.4.5.6.7.8.9. Est item unus .0 circulus, qui nihil + significat." [<i>De VI. Arithmeticae practicae speciebvs</i>, Paris, + 1539, fol. 9 of the 1543 edition.]</p> + + <p><a name="Nt_7" href="#NtA_7">[7]</a> "Barbarische oder gemeine + Ziffern." [Anonymous, <i>Das Einmahl Eins cum notis variorum</i>, + Dresden, 1703, p. 3.] So Vossius (<i>De universae matheseos natura et + constitutione liber</i>, Amsterdam, 1650, p. 34) calls them "Barbaras + numeri notas." The word at that time was possibly synonymous with + Arabic.</p> + + <p><a name="Nt_8" href="#NtA_8">[8]</a> His full name was + ‛Abū ‛Abdallāh <span class="special" + title="Mohammed">Moḥammed</span> ibn Mūsā + al-Khowārazmī. He was born in Khowārezm, "the + lowlands," the country about the present Khiva and bordering on the Oxus, + and lived at Bagdad under the caliph al-Māmūn. He died + probably between 220 and 230 of the Mohammedan era, that is, between 835 + and 845 <span class="scac">A.D.</span>, although some put the date as + early as 812. The best account of this great scholar may be found in an + article by C. Nallino, "<span class="special" + title="Al-Huwarizmi">Al-Ḫuwārizmī</span>" in the + <i>Atti della R. Accad. dei Lincei</i>, Rome, 1896. See also + <i>Verhandlungen des 5. Congresses der Orientalisten</i>, Berlin, 1882, + Vol. II, p. 19; W. Spitta-Bey in the <i>Zeitschrift der deutschen + Morgenländ. Gesellschaft</i>, Vol. XXXIII, p. 224; Steinschneider in the + <i>Zeitschrift der deutschen Morgenländ. Gesellschaft</i>, Vol. L, p. + 214; Treutlein in the <i>Abhandlungen zur Geschichte der Mathematik</i>, + Vol. I, p. 5; Suter, "Die Mathematiker und Astronomen der Araber und ihre + Werke," <i>Abhandlungen zur Geschichte der Mathematik</i>, Vol. X, + Leipzig, 1900, p. 10, and "Nachträge," in Vol. XIV, p. 158; Cantor, + <i>Geschichte der Mathematik</i>, Vol. I, 3d ed., pp. 712-733 etc.; F. + Woepcke in <i>Propagation</i>, p. 489. So recently has he become known + that Heilbronner, writing in 1742, merely mentions him as "Ben-Musa, + inter Arabes celebris Geometra, scripsit de figuris planis & + sphericis." [<i>Historia matheseos universæ</i>, Leipzig, 1742, p. + 438.]</p> + + <p>In this work most of the Arabic names will be transliterated + substantially as laid down by Suter in his work <i>Die Mathematiker</i> + etc., except where this violates English pronunciation. The scheme of + pronunciation of oriental names is set forth in the preface.</p> + + <p><a name="Nt_9" href="#NtA_9">[9]</a> Our word <i>algebra</i> is from + the title of one of his works, Al-jabr wa'l-muqābalah, Completion + and Comparison. The work was translated into English by F. Rosen, London, + 1831, and treated in <i>L'Algèbre d'al-Khārizmi et les méthodes + indienne et grecque</i>, Léon Rodet, Paris, 1878, extract from the + <i>Journal Asiatique</i>. For the derivation of the word <i>algebra</i>, + see Cossali, <i>Scritti Inediti</i>, pp. 381-383, Rome, 1857; Leonardo's + <i>Liber Abbaci</i> (1202), p. 410, Rome, 1857; both published by B. + Boncompagni. "Almuchabala" also was used as a name for algebra.</p> + + <p><a name="Nt_10" href="#NtA_10">[10]</a> This learned scholar, teacher + of O'Creat who wrote the <i>Helceph</i> ("<i>Prologus N. Ocreati in + Helceph ad Adelardum Batensem magistrum suum</i>"), studied in Toledo, + learned Arabic, traveled as far east as Egypt, and brought from the + Levant numerous manuscripts for study and translation. See Henry in the + <i>Abhandlungen zur Geschichte der Mathematik</i>, Vol. III, p. 131; + Woepcke in <i>Propagation</i>, p. 518.</p> + + <p><a name="Nt_11" href="#NtA_11">[11]</a> The title is <i>Algoritmi de + numero Indorum</i>. That he did not make this translation is asserted by + Eneström in the <i>Bibliotheca Mathematica</i>, Vol. I (3), p. 520.</p> + + <p><a name="Nt_12" href="#NtA_12">[12]</a> Thus he speaks "de numero + indorum per .IX. literas," and proceeds: "Dixit algoritmi: Cum uidissem + yndos constituisse .IX. literas in uniuerso numero suo, propter + dispositionem suam quam posuerunt, uolui patefacere de opera quod fit per + eas aliquid quod esset leuius discentibus, si deus uoluerit." + [Boncompagni, <i>Trattati d'Aritmetica</i>, Rome, 1857.] Discussed by F. + Woepcke, <i>Sur l'introduction de l'arithmétique indienne en + Occident</i>, Rome, 1859.</p> + + <p><a name="Nt_13" href="#NtA_13">[13]</a> Thus in a commentary by + ‛Alī ibn Abī Bekr ibn al-Jamāl <span + class="special" title="al-Ansari">al-Anṣārī</span> + al-Mekkī on a treatise on ġobār arithmetic (explained + later) called <i>Al-murshidah</i>, found by Woepcke in Paris + (<i>Propagation</i>, p. 66), there is mentioned the fact that there are + "nine Indian figures" and "a second kind of Indian figures ... although + these are the figures of the ġobār writing." So in a + commentary by <span class="special" title="Hosein ibn Mohammed al-Mahalli" + >Ḥosein ibn Moḥammed al-Maḥallī</span> (died + in 1756) on the <i><span class="special" title="Mokhtasar fi`ilm el-hisab" + >Mokhtaṣar fī‛ilm el-ḥisāb</span></i> + (Extract from Arithmetic) by ‛Abdalqādir ibn + ‛Alī al-Sakhāwī (died c. 1000) it is related + that "the preface treats of the forms of the figures of Hindu signs, such + as were established by the Hindu nation." [Woepcke, <i>Propagation</i>, + p. 63.]</p> + + <p><a name="Nt_14" href="#NtA_14">[14]</a> See also Woepcke, + <i>Propagation</i>, p. 505. The origin is discussed at much length by G. + R. Kaye, "Notes on Indian Mathematics.—Arithmetical Notation," + <i>Journ. and Proc. of the Asiatic Soc. of Bengal</i>, Vol. III, 1907, p. + 489.</p> + + <p><a name="Nt_15" href="#NtA_15">[15]</a> <i>Alberuni's India</i>, + Arabic version, London, 1887; English translation, ibid., 1888.</p> + + <p><a name="Nt_16" href="#NtA_16">[16]</a> <i>Chronology of Ancient + Nations</i>, London, 1879. Arabic and English versions, by C. E. + Sachau.</p> + + <p><a name="Nt_17" href="#NtA_17">[17]</a> <i>India</i>, Vol. I, chap. + xvi.</p> + + <p><a name="Nt_18" href="#NtA_18">[18]</a> The Hindu name for the symbols + of the decimal place system.</p> + + <p><a name="Nt_19" href="#NtA_19">[19]</a> Sachau's English edition of + the <i>Chronology</i>, p. 64.</p> + + <p><a name="Nt_20" href="#NtA_20">[20]</a> <i>Littérature arabe</i>, Cl. + Huart, Paris, 1902.</p> + + <p><a name="Nt_21" href="#NtA_21">[21]</a> Huart, <i>History of Arabic + Literature</i>, English ed., New York, 1903, p. 182 seq.</p> + + <p><a name="Nt_22" href="#NtA_22">[22]</a> + Al-Mas‛ūdī's <i>Meadows of Gold</i>, translated in + part by Aloys Sprenger, London, 1841; <i>Les prairies d'or</i>, trad. par + C. Barbier de Meynard et Pavet de Courteille, Vols. I to IX, Paris, + 1861-1877.</p> + + <p><a name="Nt_23" href="#NtA_23">[23]</a> <i>Les prairies d'or</i>, Vol. + VIII, p. 289 seq.</p> + + <p><a name="Nt_24" href="#NtA_24">[24]</a> <i>Essays</i>, Vol. II, p. + 428.</p> + + <p><a name="Nt_25" href="#NtA_25">[25]</a> Loc. cit., p. 504.</p> + + <p><a name="Nt_26" href="#NtA_26">[26]</a> <i>Matériaux pour servir à + l'histoire comparée des sciences mathématiques chez les Grecs et les + Orientaux</i>, 2 vols., Paris, 1845-1849, pp. 438-439.</p> + + <p><a name="Nt_27" href="#NtA_27">[27]</a> He made an exception, however, + in favor of the numerals, loc. cit., Vol. II, p. 503.</p> + + <p><a name="Nt_28" href="#NtA_28">[28]</a> <i>Bibliotheca Arabico-Hispana + Escurialensis</i>, Madrid, 1760-1770, pp. 426-427.</p> + + <p><a name="Nt_29" href="#NtA_29">[29]</a> The author, <span + class="special" title="Ibn al-Qifti">Ibn al-Qifṭī</span>, + flourished <span class="scac">A.D.</span> 1198 [Colebrooke, loc. cit., + note Vol. II, p. 510].</p> + + <p><a name="Nt_30" href="#NtA_30">[30]</a> "Liber Artis Logisticae à + Mohamado Ben Musa <i>Alkhuarezmita</i> exornatus, qui ceteros omnes + brevitate methodi ac facilitate praestat, Indorum que in praeclarissimis + inventis ingenium & acumen ostendit." [Casiri, loc. cit., p. + 427.]</p> + + <p><a name="Nt_31" href="#NtA_31">[31]</a> Maçoudi, <i>Le livre de + l'avertissement et de la révision</i>. Translation by B. Carra de Vaux, + Paris, 1896.</p> + + <p><a name="Nt_32" href="#NtA_32">[32]</a> Verifying the hypothesis of + Woepcke, <i>Propagation</i>, that the Sindhind included a treatment of + arithmetic.</p> + + <p><a name="Nt_33" href="#NtA_33">[33]</a> <span class="special" + title="Ahmed ibn `Abdallah">Aḥmed ibn + ‛Abdallāh</span>, Suter, <i>Die Mathematiker</i>, etc., p. + 12.</p> + + <p><a name="Nt_34" href="#NtA_34">[34]</a> <i>India</i>, Vol. II, p. + 15.</p> + + <p><a name="Nt_35" href="#NtA_35">[35]</a> See H. Suter, "Das + Mathematiker-Verzeichniss im Fihrist," <i>Abhandlungen zur Geschichte der + Mathematik</i>, Vol. VI, Leipzig, 1892. For further references to early + Arabic writers the reader is referred to H. Suter, <i>Die Mathematiker + und Astronomen der Araber und ihre Werke</i>. Also "Nachträge und + Berichtigungen" to the same (<i>Abhandlungen</i>, Vol. XIV, 1902, pp. + 155-186).</p> + + <p><a name="Nt_36" href="#NtA_36">[36]</a> Suter, loc. cit., note 165, + pp. 62-63.</p> + + <p><a name="Nt_37" href="#NtA_37">[37]</a> "Send Ben Ali,... tùm + arithmetica scripta maximè celebrata, quae publici juris fecit." [Loc. + cit., p. 440.]</p> + + <p><a name="Nt_38" href="#NtA_38">[38]</a> <i>Scritti di Leonardo + Pisano</i>, Vol. I, <i>Liber Abbaci</i> (1857); Vol. II, <i>Scritti</i> + (1862); published by Baldassarre Boncompagni, Rome. Also <i>Tre Scritti + Inediti</i>, and <i>Intorno ad Opere di Leonardo Pisano</i>, Rome, + 1854.</p> + + <p><a name="Nt_39" href="#NtA_39">[39]</a> "Ubi ex mirabili magisterio in + arte per novem figuras indorum introductus" etc. In another place, as a + heading to a separate division, he writes, "De cognitione novem figurarum + yndorum" etc. "Novem figure indorum he sunt 9 8 7 6 5 4 3 2 1."</p> + + <p><a name="Nt_40" href="#NtA_40">[40]</a> See <i>An Ancient English + Algorism</i>, by David Eugene Smith, in <i>Festschrift Moritz Cantor</i>, + Leipzig, 1909. See also Victor Mortet, "Le plus ancien traité francais + d'algorisme," <i>Bibliotheca Mathematica</i>, Vol. IX (3), pp. 55-64.</p> + + <p><a name="Nt_41" href="#NtA_41">[41]</a> These are the two opening + lines of the <i>Carmen de Algorismo</i> that the anonymous author is + explaining. They should read as follows:</p> + + <div class="poem"> + <div class="stanza"> + <p>Haec algorismus ars praesens dicitur, in qua</p> + <p>Talibus Indorum fruimur bis quinque figuris.</p> + </div> + </div> + <p>What follows is the translation.</p> + + <p><a name="Nt_42" href="#NtA_42">[42]</a> Thibaut, <i>Astronomie, + Astrologie und Mathematik</i>, Strassburg, 1899.</p> + + <p><a name="Nt_43" href="#NtA_43">[43]</a> Gustave Schlegel, + <i>Uranographie chinoise ou preuves directes que l'astronomie primitive + est originaire de la Chine, et qu'elle a été empruntée par les anciens + peuples occidentaux à la sphère chinoise; ouvrage accompagné d'un atlas + céleste chinois et grec</i>, The Hague and Leyden, 1875.</p> + + <p><a name="Nt_44" href="#NtA_44">[44]</a> E. W. Hopkins, <i>The + Religions of India</i>, Boston, 1898, p. 7.</p> + + <p><a name="Nt_45" href="#NtA_45">[45]</a> R. C. Dutt, <i>History of + India</i>, London, 1906.</p> + + <p><a name="Nt_46" href="#NtA_46">[46]</a> W. D. Whitney, <i>Sanskrit + Grammar</i>, 3d ed., Leipzig, 1896.</p> + + <p><a name="Nt_47" href="#NtA_47">[47]</a> "Das + Āpastamba-Śulba-Sūtra," <i>Zeitschrift der deutschen + Morgenländischen Gesellschaft</i>, Vol. LV, p. 543, and Vol. LVI, p. + 327.</p> + + <p><a name="Nt_48" href="#NtA_48">[48]</a> <i>Geschichte der Math.</i>, + Vol. I, 2d ed., p. 595.</p> + + <p><a name="Nt_49" href="#NtA_49">[49]</a> L. von Schroeder, + <i>Pythagoras und die Inder</i>, Leipzig, 1884; H. Vogt, "Haben die alten + Inder den Pythagoreischen Lehrsatz und das Irrationale gekannt?" + <i>Bibliotheca Mathematica</i>, Vol. VII (3), pp. 6-20; A. Bürk, loc. + cit.; Max Simon, <i>Geschichte der Mathematik im Altertum</i>, Berlin, + 1909, pp. 137-165; three Sūtras are translated in part by Thibaut, + <i>Journal of the Asiatic Society of Bengal</i>, 1875, and one appeared + in <i>The Pandit</i>, 1875; Beppo Levi, "Osservazioni e congetture sopra + la geometria degli indiani," <i>Bibliotheca Mathematica</i>, Vol. IX (3), + 1908, pp. 97-105.</p> + + <p><a name="Nt_50" href="#NtA_50">[50]</a> Loc. cit.; also <i>Indiens + Literatur und Cultur</i>, Leipzig, 1887.</p> + + <p><a name="Nt_51" href="#NtA_51">[51]</a> It is generally agreed that + the name of the river Sindhu, corrupted by western peoples to Hindhu, + Indos, Indus, is the root of Hindustan and of India. Reclus, <i>Asia</i>, + English ed., Vol. III, p. 14.</p> + + <p><a name="Nt_52" href="#NtA_52">[52]</a> See the comments of Oppert, + <i>On the Original Inhabitants of <span class="special" + title="Bharatavarsa">Bharatavarṣa</span> or India</i>, London, + 1893, p. 1.</p> + + <p><a name="Nt_53" href="#NtA_53">[53]</a> A. Hillebrandt, + <i>Alt-Indien</i>, Breslau, 1899, p. 111. Fragmentary records relate that + Khāravela, king of <span class="special" + title="Kalinga">Kaliṅga</span>, learned as a boy + <i>lekhā</i> (writing), <i><span class="special" + title="ganana">gaṇanā</span></i> (reckoning), and + <i>rūpa</i> (arithmetic applied to monetary affairs and + mensuration), probably in the 5th century <span class="scac">B.C.</span> + [Bühler, <i>Indische Palaeographie</i>, Strassburg, 1896, p. 5.]</p> + + <p><a name="Nt_54" href="#NtA_54">[54]</a> R. C. Dutt, <i>A History of + Civilization in Ancient India</i>, London, 1893, Vol. I, p. 174.</p> + + <p><a name="Nt_55" href="#NtA_55">[55]</a> The Buddha. The date of his + birth is uncertain. Sir Edwin Arnold put it c. 620 <span + class="scac">B.C.</span></p> + + <p><a name="Nt_56" href="#NtA_56">[56]</a> I.e. 100·10<sup>7</sup>.</p> + + <p><a name="Nt_57" href="#NtA_57">[57]</a> There is some uncertainty + about this limit.</p> + + <p><a name="Nt_58" href="#NtA_58">[58]</a> This problem deserves more + study than has yet been given it. A beginning may be made with Comte + Goblet d'Alviella, <i>Ce que l'Inde doit à la Grèce</i>, Paris, 1897, and + H. G. Keene's review, "The Greeks in India," in the <i>Calcutta + Review</i>, Vol. CXIV, 1902, p. 1. See also F. Woepeke, + <i>Propagation</i>, p. 253; G. R. Kaye, loc. cit., p. 475 seq., and "The + Source of Hindu Mathematics," <i>Journal of the Royal Asiatic + Society</i>, July, 1910, pp. 749-760; G. Thibaut, <i>Astronomie, + Astrologie und Mathematik</i>, pp. 43-50 and 76-79. It will be discussed + more fully in Chapter VI.</p> + + <p><a name="Nt_59" href="#NtA_59">[59]</a> I.e. to 100,000. The lakh is + still the common large unit in India, like the myriad in ancient Greece + and the million in the West.</p> + + <p><a name="Nt_60" href="#NtA_60">[60]</a> This again suggests the + <i>Psammites</i>, or <i>De harenae numero</i> as it is called in the 1544 + edition of the <i>Opera</i> of Archimedes, a work in which the great + Syracusan proposes to show to the king "by geometric proofs which you can + follow, that the numbers which have been named by us ... are sufficient + to exceed not only the number of a sand-heap as large as the whole earth, + but one as large as the universe." For a list of early editions of this + work see D. E. Smith, <i>Rara Arithmetica</i>, Boston, 1909, p. 227.</p> + + <p><a name="Nt_61" href="#NtA_61">[61]</a> I.e. the Wise.</p> + + <p><a name="Nt_62" href="#NtA_62">[62]</a> Sir Monier Monier-Williams, + <i>Indian Wisdom</i>, 4th ed., London, 1893, pp. 144, 177. See also J. C. + Marshman, <i>Abridgment of the History of India</i>, London, 1893, p. + 2.</p> + + <p><a name="Nt_63" href="#NtA_63">[63]</a> For a list and for some + description of these works see R. C. Dutt, <i>A History of Civilization + in Ancient India</i>, Vol. II, p. 121.</p> + + <p><a name="Nt_64" href="#NtA_64">[64]</a> Professor Ramkrishna Gopal + Bhandarkar fixes the date as the fifth century <span + class="scac">B.C.</span> ["Consideration of the Date of the + Mahābhārata," in the <i>Journal of the Bombay Branch of the + R. A. Soc.</i>, Bombay, 1873, Vol. X, p. 2.].</p> + + <p><a name="Nt_65" href="#NtA_65">[65]</a> Marshman, loc. cit., p. 2.</p> + + <p><a name="Nt_66" href="#NtA_66">[66]</a> A. C. Burnell, <i>South Indian + Palæography</i>, 2d ed., London, 1878, p. 1, seq.</p> + + <p><a name="Nt_67" href="#NtA_67">[67]</a> This extensive subject of + palpable arithmetic, essentially the history of the abacus, deserves to + be treated in a work by itself.</p> + + <p><a name="Nt_68" href="#NtA_68">[68]</a> The following are the leading + sources of information upon this subject: G. Bühler, <i>Indische + Palaeographie</i>, particularly chap. vi; A. C. Burnell, <i>South Indian + Palæography</i>, 2d ed., London, 1878, where tables of the various Indian + numerals are given in Plate XXIII; E. C. Bayley, "On the Genealogy of + Modern Numerals," <i>Journal of the Royal Asiatic Society</i>, Vol. XIV, + part 3, and Vol. XV, part 1, and reprint, London, 1882; I. Taylor, in + <i>The Academy</i>, January 28, 1882, with a repetition of his argument + in his work <i>The Alphabet</i>, London, 1883, Vol. II, p. 265, based on + Bayley; G. R. Kaye, loc. cit., in some respects one of the most critical + articles thus far published; J. C. Fleet, <i>Corpus inscriptionum + Indicarum</i>, London, 1888, Vol. III, with facsimiles of many Indian + inscriptions, and <i>Indian Epigraphy</i>, Oxford, 1907, reprinted from + the <i>Imperial Gazetteer of India</i>, Vol. II, pp. 1-88, 1907; G. + Thibaut, loc. cit., <i>Astronomie</i> etc.; R. Caldwell, <i>Comparative + Grammar of the Dravidian Languages</i>, London, 1856, p. 262 seq.; and + <i>Epigraphia Indica</i> (official publication of the government of + India), Vols. I-IX. Another work of Bühler's, <i>On the Origin of the + Indian Brāhma Alphabet</i>, is also of value.</p> + + <p><a name="Nt_69" href="#NtA_69">[69]</a> The earliest work on the + subject was by James Prinsep, "On the Inscriptions of Piyadasi or + Aśoka," etc., <i>Journal of the Asiatic Society of Bengal</i>, + 1838, following a preliminary suggestion in the same journal in 1837. See + also "Aśoka Notes," by V. A. Smith, <i>The Indian Antiquary</i>, + Vol. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, pp. 151-159, June, 1909; + <i>The Early History of India</i>, 2d ed., Oxford, 1908, p. 154; J. F. + Fleet, "The Last Words of Aśoka," <i>Journal of the Royal Asiatic + Society</i>, October, 1909, pp. 981-1016; E. Senart, <i>Les inscriptions + de Piyadasi</i>, 2 vols., Paris, 1887.</p> + + <p><a name="Nt_70" href="#NtA_70">[70]</a> For a discussion of the minor + details of this system, see Bühler, loc. cit., p. 73.</p> + + <p><a name="Nt_71" href="#NtA_71">[71]</a> Julius Euting, <i>Nabatäische + Inschriften aus Arabien</i>, Berlin, 1885, pp. 96-97, with a table of + numerals.</p> + + <p><a name="Nt_72" href="#NtA_72">[72]</a> For the five principal + theories see Bühler, loc. cit., p. 10.</p> + + <p><a name="Nt_73" href="#NtA_73">[73]</a> Bayley, loc. cit., reprint p. + 3.</p> + + <p><a name="Nt_74" href="#NtA_74">[74]</a> Bühler, loc. cit.; + <i>Epigraphia Indica</i>, Vol. III, p. 134; <i>Indian Antiquary</i>, Vol. + VI, p. 155 seq., and Vol. X, p. 107.</p> + + <p><a name="Nt_75" href="#NtA_75">[75]</a> Pandit Bhagavānlāl + Indrājī, "On Ancient Nāgāri Numeration; from an + Inscription at Nāneghāt," <i>Journal of the Bombay Branch of + the Royal Asiatic Society</i>, 1876, Vol. XII, p. 404.</p> + + <p><a name="Nt_76" href="#NtA_76">[76]</a> Ib., p. 405. He gives also a + plate and an interpretation of each numeral.</p> + + <p><a name="Nt_77" href="#NtA_77">[77]</a> These may be compared with + Bühler's drawings, loc. cit.; with Bayley, loc. cit., p. 337 and plates; + and with Bayley's article in the <i>Encyclopædia Britannica</i>, 9th ed., + art. "Numerals."</p> + + <p><a name="Nt_78" href="#NtA_78">[78]</a> E. Senart, "The Inscriptions + in the Caves at Nasik," <i>Epigraphia Indica</i>, Vol. VIII, pp. 59-96; + "The Inscriptions in the Cave at Karle," <i>Epigraphia Indica</i>, Vol. + VII, pp. 47-74; Bühler, <i>Palaeographie</i>, Tafel IX.</p> + + <p><a name="Nt_79" href="#NtA_79">[79]</a> See Fleet, loc. cit. See also + T. Benfey, <i>Sanskrit Grammar</i>, London, 1863, p. 217; M. R. Kále, + <i>Higher Sanskrit Grammar</i>, 2d ed., Bombay, 1898, p. 110, and other + authorities as cited.</p> + + <p><a name="Nt_80" href="#NtA_80">[80]</a> <span class="special" + title="Kharosthi">Kharoṣṭhī</span> numerals, + Aśoka inscriptions, c. 250 <span class="scac">B.C.</span> Senart, + <i>Notes d'épigraphie indienne</i>. Given by Bühler, loc. cit., Tafel + I.</p> + + <p><a name="Nt_81" href="#NtA_81">[81]</a> Same, Śaka inscriptions, + probably of the first century <span class="scac">B.C.</span> Senart, loc. + cit.; Bühler, loc. cit.</p> + + <p><a name="Nt_82" href="#NtA_82">[82]</a> Brāhmī numerals, + Aśoka inscriptions, c. 250 <span class="scac">B.C.</span> <i>Indian + Antiquary</i>, Vol. VI, p. 155 seq.</p> + + <p><a name="Nt_83" href="#NtA_83">[83]</a> Same, Nānā + Ghāt inscriptions, c. 150 <span class="scac">B.C.</span> + Bhagavānlāl Indrājī, <i>On Ancient + Nāgarī Numeration</i>, loc. cit. Copied from a squeeze of the + original.</p> + + <p><a name="Nt_84" href="#NtA_84">[84]</a> Same, Nasik inscription, c. + 100 <span class="scac">B.C.</span> Burgess, <i>Archeological Survey + Report, Western India</i>; Senart, <i>Epigraphia Indica</i>, Vol. VII, + pp. 47-79, and Vol. VIII, pp. 59-96.</p> + + <p><a name="Nt_85" href="#NtA_85">[85]</a> <span class="special" + title="Ksatrapa">Kṣatrapa</span> coins, c. 200 <span + class="scac">A.D.</span> <i>Journal of the Royal Asiatic Society</i>, + 1890, p. 639.</p> + + <p><a name="Nt_86" href="#NtA_86">[86]</a> <span class="special" + title="Kusana">Kuṣana</span> inscriptions, c. 150 <span + class="scac">A.D.</span> <i>Epigraphia Indica</i>, Vol. I, p. 381, and + Vol. II, p. 201.</p> + + <p><a name="Nt_87" href="#NtA_87">[87]</a> Gupta Inscriptions, c. 300 + <span class="scac">A.D.</span> to 450 <span class="scac">A.D.</span> + Fleet, loc. cit., Vol. III.</p> + + <p><a name="Nt_88" href="#NtA_88">[88]</a> Valhabī, c. 600 <span + class="scac">A.D.</span> <i>Corpus</i>, Vol. III.</p> + + <p><a name="Nt_89" href="#NtA_89">[89]</a> Bendall's Table of Numerals, + in <i>Cat. Sansk. Budd. MSS.</i>, British Museum.</p> + + <p><a name="Nt_90" href="#NtA_90">[90]</a> <i>Indian Antiquary</i>, Vol. + XIII, 120; <i>Epigraphia Indica</i>, Vol. III, 127 ff.</p> + + <p><a name="Nt_91" href="#NtA_91">[91]</a> Fleet, loc. cit.</p> + + <p><a name="Nt_92" href="#NtA_92">[92]</a> Bayley, loc. cit., p. 335.</p> + + <p><a name="Nt_93" href="#NtA_93">[93]</a> From a copper plate of 493 + <span class="scac">A.D.</span>, found at + Kārītalāī, Central India. [Fleet, loc. cit., + Plate XVI.] It should be stated, however, that many of these copper + plates, being deeds of property, have forged dates so as to give the + appearance of antiquity of title. On the other hand, as Colebrooke long + ago pointed out, a successful forgery has to imitate the writing of the + period in question, so that it becomes evidence well worth considering, + as shown in Chapter III.</p> + + <p><a name="Nt_94" href="#NtA_94">[94]</a> From a copper plate of 510 + <span class="scac">A.D.</span>, found at Majhgawāin, Central India. + [Fleet, loc. cit., Plate XIV.]</p> + + <p><a name="Nt_95" href="#NtA_95">[95]</a> From an inscription of 588 + <span class="scac">A.D.</span>, found at Bōdh-Gayā, Bengal + Presidency. [Fleet, loc. cit., Plate XXIV.]</p> + + <p><a name="Nt_96" href="#NtA_96">[96]</a> From a copper plate of 571 + <span class="scac">A.D.</span>, found at Māliyā, Bombay + Presidency. [Fleet, loc. cit., Plate XXIV.]</p> + + <p><a name="Nt_97" href="#NtA_97">[97]</a> From a <span class="special" + title="Bijayagadh">Bijayagaḍh</span> pillar inscription of 372 + <span class="scac">A.D.</span> [Fleet, loc. cit., Plate XXXVI, C.]</p> + + <p><a name="Nt_98" href="#NtA_98">[98]</a> From a copper plate of 434 + <span class="scac">A.D.</span> [<i>Indian Antiquary</i>, Vol. I, p. + 60.]</p> + + <p><a name="Nt_99" href="#NtA_99">[99]</a> Gadhwa inscription, c. 417 + <span class="scac">A.D.</span> [Fleet, loc. cit., Plate IV, D.]</p> + + <p><a name="Nt_100" href="#NtA_100">[100]</a> + Kārītalāī plate of 493 <span + class="scac">A.D.</span>, referred to above.</p> + + <p><a name="Nt_101" href="#NtA_101">[101]</a> It seems evident that the + Chinese four, curiously enough called "eight in the mouth," is only a + cursive <a href="images/034b.png"><img src="images/034b.png" + class="middle" style="height:1.5ex" alt="4 vertical strokes" /></a>.</p> + + <p><a name="Nt_102" href="#NtA_102">[102]</a> Chalfont, F. H., <i>Memoirs + of the Carnegie Museum</i>, Vol. IV, no. 1; J. Hager, <i>An Explanation + of the Elementary Characters of the Chinese</i>, London, 1801.</p> + + <p><a name="Nt_103" href="#NtA_103">[103]</a> H. V. Hilprecht, + <i>Mathematical, Metrological and Chronological Tablets from the Temple + Library at Nippur</i>, Vol. XX, part I, of Series A, Cuneiform Texts + Published by the Babylonian Expedition of the University of Pennsylvania, + 1906; A. Eisenlohr, <i>Ein altbabylonischer Felderplan</i>, Leipzig, + 1906; Maspero, <i>Dawn of Civilization</i>, p. 773.</p> + + <p><a name="Nt_104" href="#NtA_104">[104]</a> Sir H. H. Howard, "On the + Earliest Inscriptions from Chaldea," <i>Proceedings of the Society of + Biblical Archæology</i>, XXI, p. 301, London, 1899.</p> + + <p><a name="Nt_105" href="#NtA_105">[105]</a> For a bibliography of the + principal hypotheses of this nature see Bühler, loc. cit., p. 77. Bühler + (p. 78) feels that of all these hypotheses that which connects the + Brāhmī with the Egyptian numerals is the most plausible, + although he does not adduce any convincing proof. Th. Henri Martin, "Les + signes numéraux et l'arithmétique chez les peuples de l'antiquité et du + moyen âge" (being an examination of Cantor's <i>Mathematische Beiträge + zum Culturleben der Völker</i>), <i>Annali di matematica pura ed + applicata</i>, Vol. V, Rome, 1864, pp. 8, 70. Also, same author, + "Recherches nouvelles sur l'origine de notre système de numération + écrite," <i>Revue Archéologique</i>, 1857, pp. 36, 55. See also the + tables given later in this work.</p> + + <p><a name="Nt_106" href="#NtA_106">[106]</a> <i>Journal of the Royal + Asiatic Society, Bombay Branch</i>, Vol. XXIII.</p> + + <p><a name="Nt_107" href="#NtA_107">[107]</a> Loc. cit., reprint, Part I, + pp. 12, 17. Bayley's deductions are generally regarded as + unwarranted.</p> + + <p><a name="Nt_108" href="#NtA_108">[108]</a> <i>The Alphabet</i>; + London, 1883, Vol. II, pp. 265, 266, and <i>The Academy</i> of Jan. 28, + 1882.</p> + + <p><a name="Nt_109" href="#NtA_109">[109]</a> Taylor, <i>The + Alphabet</i>, loc. cit., table on p. 266.</p> + + <p><a name="Nt_110" href="#NtA_110">[110]</a> Bühler, <i>On the Origin of + the Indian Brāhma Alphabet</i>, Strassburg, 1898, footnote, pp. 52, + 53.</p> + + <p><a name="Nt_111" href="#NtA_111">[111]</a> Albrecht Weber, <i>History + of Indian Literature</i>, English ed., Boston, 1878, p. 256: "The Indian + figures from 1-9 are abbreviated forms of the initial letters of the + numerals themselves...: the zero, too, has arisen out of the first letter + of the word <i><span class="special" + title="sunya">ṣunya</span></i> (empty) (it occurs even in + Piñgala). It is the decimal place value of these figures which gives them + significance." C. Henry, "Sur l'origine de quelques notations + mathématiques," <i>Revue Archéologique</i>, June and July, 1879, attempts + to derive the Boethian forms from the initials of Latin words. See also + J. Prinsep, "Examination of the Inscriptions from Girnar in Gujerat, and + Dhauli in Cuttach," <i>Journal of the Asiatic Society of Bengal</i>, + 1838, especially Plate XX, p. 348; this was the first work on the + subject.</p> + + <p><a name="Nt_112" href="#NtA_112">[112]</a> Bühler, + <i>Palaeographie</i>, p. 75, gives the list, with the list of letters (p. + 76) corresponding to the number symbols.</p> + + <p><a name="Nt_113" href="#NtA_113">[113]</a> For a general discussion of + the connection between the numerals and the different kinds of alphabets, + see the articles by U. Ceretti, "Sulla origine delle cifre numerali + moderne," <i>Rivista di fisica, matematica e scienze naturali</i>, Pisa + and Pavia, 1909, anno X, numbers 114, 118, 119, and 120, and continuation + in 1910.</p> + + <p><a name="Nt_114" href="#NtA_114">[114]</a> This is one of Bühler's + hypotheses. See Bayley, loc. cit., reprint p. 4; a good bibliography of + original sources is given in this work, p. 38.</p> + + <p><a name="Nt_115" href="#NtA_115">[115]</a> Loc. cit., reprint, part I, + pp. 12, 17. See also Burnell, loc. cit., p. 64, and tables in plate + XXIII.</p> + + <p><a name="Nt_116" href="#NtA_116">[116]</a> This was asserted by G. + Hager (<i>Memoria sulle cifre arabiche</i>, Milan, 1813, also published + in <i>Fundgruben des Orients</i>, Vienna, 1811, and in <i>Bibliothèque + Britannique</i>, Geneva, 1812). See also the recent article by Major + Charles E. Woodruff, "The Evolution of Modern Numerals from Tally Marks," + <i>American Mathematical Monthly</i>, August-September, 1909. Biernatzki, + "Die Arithmetik der Chinesen," <i>Crelle's Journal für die reine und + angewandte Mathematik</i>, Vol. LII, 1857, pp. 59-96, also asserts the + priority of the Chinese claim for a place system and the zero, but upon + the flimsiest authority. Ch. de Paravey, <i>Essai sur l'origine unique et + hiéroglyphique des chiffres et des lettres de tous les peuples</i>, + Paris, 1826; G. Kleinwächter, "The Origin of the Arabic Numerals," + <i>China Review</i>, Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp. + 28-30; Biot, "Note sur la connaissance que les Chinois ont eue de la + valeur de position des chiffres," <i>Journal Asiatique</i>, 1839, pp. + 497-502. A. Terrien de Lacouperie, "The Old Numerals, the Counting-Rods + and the Swan-Pan in China," <i>Numismatic Chronicle</i>, Vol. III (3), + pp. 297-340, and Crowder B. Moseley, "Numeral Characters: Theory of + Origin and Development," <i>American Antiquarian</i>, Vol. XXII, pp. + 279-284, both propose to derive our numerals from Chinese characters, in + much the same way as is done by Major Woodruff, in the article above + cited.</p> + + <p><a name="Nt_117" href="#NtA_117">[117]</a> The Greeks, probably + following the Semitic custom, used nine letters of the alphabet for the + numerals from 1 to 9, then nine others for 10 to 90, and further letters + to represent 100 to 900. As the ordinary Greek alphabet was insufficient, + containing only twenty-four letters, an alphabet of twenty-seven letters + was used.</p> + + <p><a name="Nt_118" href="#NtA_118">[118]</a> <i>Institutiones + mathematicae</i>, 2 vols., Strassburg, 1593-1596, a somewhat rare work + from which the following quotation is taken:</p> + + <p>"<i>Quis est harum Cyphrarum autor?</i></p> + + <p>"A quibus hae usitatae syphrarum notae sint inventae: hactenus + incertum fuit: meo tamen iudicio, quod exiguum esse fateor: a graecis + librarijs (quorum olim magna fuit copia) literae Graecorum quibus veteres + Graeci tamquam numerorum notis sunt usi: fuerunt corruptae. vt ex his + licet videre.</p> + + <p>"Graecorum Literae corruptae.</p> + + <div class="figright" style="width:20%;"> + <a href="images/039a.png"><img style="width:100%" src="images/039a.png" + alt="Graecorum Literae corruptae." title="Graecorum Literae corruptae." /></a> + </div> + <p><i>"Sed qua ratione graecorum literae ita fuerunt corruptae?</i></p> + + <p>"Finxerunt has corruptas Graecorum literarum notas: vel abiectione vt + in nota binarij numeri, vel additione vt in ternarij, vel inuersione vt + in septenarij, numeri nota, nostrae notae, quibus hodie utimur: ab his + sola differunt elegantia, vt apparet."</p> + + <p>See also Bayer, <i>Historia regni Graecorum Bactriani</i>, St. + Petersburg, 1788, pp. 129-130, quoted by Martin, <i>Recherches + nouvelles</i>, etc., loc. cit.</p> + + <p><a name="Nt_119" href="#NtA_119">[119]</a> P. D. Huet, <i>Demonstratio + evangelica</i>, Paris, 1769, note to p. 139 on p. 647: "Ab Arabibus vel + ab Indis inventas esse, non vulgus eruditorum modo, sed doctissimi quique + ad hanc diem arbitrati sunt. Ego vero falsum id esse, merosque esse + Graecorum characteres aio; à librariis Graecae linguae ignaris + interpolatos, et diuturna scribendi consuetudine corruptos. Nam primum 1 + apex fuit, seu virgula, nota <span title="monados" class="grk" + >μονάδος</span>. 2, est ipsum + <span class="grk">β</span> extremis suis truncatum. <span + class="grk">γ</span>, si in sinistram partem inclinaveris & + cauda mutilaveris & sinistrum cornu sinistrorsum flexeris, fiet 3. + Res ipsa loquitur 4 ipsissimum esse <span class="grk">Δ</span>, + cujus crus sinistrum erigitur <span title="kata katheton" class="grk" + >κατὰ + κάθετον</span>, & infra + basim descendit; basis vero ipsa ultra crus producta eminet. Vides quam 5 + simile sit <span title="tôi" class="grk">τῷ</span> <a + href="images/040a.png"><img src="images/040a.png" class="middle" + style="height:2ex" alt="epsilon" /></a>; infimo tantum semicirculo, qui + sinistrorsum patebat, dextrorsum converso. <span title="episêmon bau" class="grk" + >ἐπίσημον + βαῦ</span> quod ita notabatur <a + href="images/040b.png"><img src="images/040b.png" class="middle" + style="height:2ex" alt="digamma" /></a>, rotundato ventre, pede detracto, + peperit <span title="to" class="grk">τὸ</span> 6. Ex <span + class="grk">Ζ</span> basi sua mutilato, ortum est <span title="to" class="grk" + >τὸ</span> 7. Si <span class="grk">Η</span> inflexis + introrsum apicibus in rotundiorem & commodiorem formam mutaveris, + exurget <span title="to" class="grk">τὸ</span> 8. At 9 + ipsissimum est <a href="images/040c.png"><img src="images/040c.png" + class="middle" style="height:2ex" alt="alt theta" /></a>."</p> + + <p>I. Weidler, <i>Spicilegium observationum ad historiam notarum + numeralium</i>, Wittenberg, 1755, derives them from the Hebrew letters; + Dom Augustin Calmet, "Recherches sur l'origine des chiffres + d'arithmétique," <i>Mémoires pour l'histoire des sciences et des beaux + arts</i>, Trévoux, 1707 (pp. 1620-1635, with two plates), derives the + current symbols from the Romans, stating that they are relics of the + ancient "Notae Tironianae." These "notes" were part of a system of + shorthand invented, or at least perfected, by Tiro, a slave who was freed + by Cicero. L. A. Sedillot, "Sur l'origine de nos chiffres," <i>Atti dell' + Accademia pontificia dei nuovi Lincei</i>, Vol. XVIII, 1864-1865, pp. + 316-322, derives the Arabic forms from the Roman numerals.</p> + + <p><a name="Nt_120" href="#NtA_120">[120]</a> Athanasius Kircher, + <i>Arithmologia sive De abditis Numerorum, mysterijs qua origo, + antiquitas & fabrica Numerorum exponitur</i>, Rome, 1665.</p> + + <p><a name="Nt_121" href="#NtA_121">[121]</a> See Suter, <i>Die + Mathematiker und Astronomen der Araber</i>, p. 100.</p> + + <p><a name="Nt_122" href="#NtA_122">[122]</a> "Et hi numeri sunt numeri + Indiani, a Brachmanis Indiae Sapientibus ex figura circuli secti + inuenti."</p> + + <p><a name="Nt_123" href="#NtA_123">[123]</a> V. A. Smith, <i>The Early + History of India</i>, Oxford, 2d ed., 1908, p. 333.</p> + + <p><a name="Nt_124" href="#NtA_124">[124]</a> C. J. Ball, "An Inscribed + Limestone Tablet from Sippara," <i>Proceedings of the Society of Biblical + Archæology</i>, Vol. XX, p. 25 (London, 1898). Terrien de Lacouperie + states that the Chinese used the circle for 10 before the beginning of + the Christian era. [<i>Catalogue of Chinese Coins</i>, London, 1892, p. + xl.]</p> + + <p><a name="Nt_125" href="#NtA_125">[125]</a> For a purely fanciful + derivation from the corresponding number of strokes, see W. W. R. Ball, + <i>A Short Account of the History of Mathematics</i>, 1st ed., London, + 1888, p. 147; similarly J. B. Reveillaud, <i>Essai sur les chiffres + arabes</i>, Paris, 1883; P. Voizot, "Les chiffres arabes et leur + origine," <i>La Nature</i>, 1899, p. 222; G. Dumesnil, "De la forme des + chiffres usuels," <i>Annales de l'université de Grenoble</i>, 1907, Vol. + XIX, pp. 657-674, also a note in <i>Revue Archéologique</i>, 1890, Vol. + XVI (3), pp. 342-348; one of the earliest references to a possible + derivation from points is in a work by Bettino entitled <i>Apiaria + universae philosophiae mathematicae in quibus paradoxa et noua + machinamenta ad usus eximios traducta, et facillimis demonstrationibus + confirmata</i>, Bologna, 1545, Vol. II, Apiarium XI, p. 5.</p> + + <p><a name="Nt_126" href="#NtA_126">[126]</a> <i>Alphabetum Barmanum</i>, + Romae, <span class="scac">MDCCLXXVI</span>, p. 50. The 1 is evidently + Sanskrit, and the 4, 7, and possibly 9 are from India.</p> + + <p><a name="Nt_127" href="#NtA_127">[127]</a> <i>Alphabetum + Grandonico-Malabaricum</i>, Romae, <span class="scac">MDCCLXXII</span>, + p. 90. The zero is not used, but the symbols for 10, 100, and so on, are + joined to the units to make the higher numbers.</p> + + <p><a name="Nt_128" href="#NtA_128">[128]</a> <i>Alphabetum + Tangutanum</i>, Romae, <span class="scac">MDCCLXXIII</span>, p. 107. In a + Tibetan MS. in the library of Professor Smith, probably of the eighteenth + century, substantially these forms are given.</p> + + <p><a name="Nt_129" href="#NtA_129">[129]</a> Bayley, loc. cit., plate + II. Similar forms to these here shown, and numerous other forms found in + India, as well as those of other oriental countries, are given by A. P. + Pihan, <i>Exposé des signes de numération usités chez les peuples + orientaux anciens et modernes</i>, Paris, 1860.</p> + + <p><a name="Nt_130" href="#NtA_130">[130]</a> Bühler, loc. cit., p. 80; + J. F. Fleet, <i>Corpus inscriptionum Indicarum</i>, Vol. III, Calcutta, + 1888. Lists of such words are given also by Al-Bīrūnī + in his work <i>India</i>; by Burnell, loc. cit.; by E. Jacquet, "Mode + d'expression symbolique des nombres employé par les Indiens, les + Tibétains et les Javanais," <i>Journal Asiatique</i>, Vol. XVI, Paris, + 1835.</p> + + <p><a name="Nt_131" href="#NtA_131">[131]</a> This date is given by + Fleet, loc. cit., Vol. III, p. 73, as the earliest epigraphical instance + of this usage in India proper.</p> + + <p><a name="Nt_132" href="#NtA_132">[132]</a> Weber, <i>Indische + Studien</i>, Vol. VIII, p. 166 seq.</p> + + <p><a name="Nt_133" href="#NtA_133">[133]</a> <i>Journal of the Royal + Asiatic Society</i>, Vol. I (<span class="scac">N.S.</span>), p. 407.</p> + + <p><a name="Nt_134" href="#NtA_134">[134]</a> VIII, 20, 21.</p> + + <p><a name="Nt_135" href="#NtA_135">[135]</a> Th. H. Martin, <i>Les + signes numéraux</i> ..., Rome, 1864; Lassen, <i>Indische + Alterthumskunde</i>, Vol. II, 2d ed., Leipzig and London, 1874, p. + 1153.</p> + + <p><a name="Nt_136" href="#NtA_136">[136]</a> But see Burnell, loc. cit., + and Thibaut, <i>Astronomie, Astrologie und Mathematik</i>, p. 71.</p> + + <p><a name="Nt_137" href="#NtA_137">[137]</a> A. Barth, "Inscriptions + Sanscrites du Cambodge," in the <i>Notices et extraits des Mss. de la + Bibliothèque nationale</i>, Vol. XXVII, Part I, pp. 1-180, 1885; see also + numerous articles in <i>Journal Asiatique</i>, by Aymonier.</p> + + <p><a name="Nt_138" href="#NtA_138">[138]</a> Bühler, loc. cit., p. + 82.</p> + + <p><a name="Nt_139" href="#NtA_139">[139]</a> Loc. cit., p. 79.</p> + + <p><a name="Nt_140" href="#NtA_140">[140]</a> Bühler, loc. cit., p. 83. + The Hindu astrologers still use an alphabetical system of numerals. + [Burnell, loc. cit., p. 79.]</p> + + <p><a name="Nt_141" href="#NtA_141">[141]</a> Well could Ramus say, + "Quicunq; autem fuerit inventor decem notarum laudem magnam meruit."</p> + + <p><a name="Nt_142" href="#NtA_142">[142]</a> Al-Bīrūnī + gives lists.</p> + + <p><a name="Nt_143" href="#NtA_143">[143]</a> <i>Propagation</i>, loc. + cit., p. 443.</p> + + <p><a name="Nt_144" href="#NtA_144">[144]</a> See the quotation from + <i>The Light of Asia</i> in Chapter II, p. 16.</p> + + <p><a name="Nt_145" href="#NtA_145">[145]</a> The nine ciphers were + called <i><span class="special" title="anka">aṅka</span></i>.</p> + + <p><a name="Nt_146" href="#NtA_146">[146]</a> "Zur Geschichte des + indischen Ziffernsystems," <i>Zeitschrift für die Kunde des + Morgenlandes</i>, Vol. IV, 1842, pp. 74-83.</p> + + <p><a name="Nt_147" href="#NtA_147">[147]</a> It is found in the <span + class="special" title="Bakhsali">Bakhṣālī</span> MS. + of an elementary arithmetic which Hoernle placed, at first, about the + beginning of our era, but the date is much in question. G. Thibaut, loc. + cit., places it between 700 and 900 <span class="scac">A.D.</span>; + Cantor places the body of the work about the third or fourth century + <span class="scac">A.D.</span>, <i>Geschichte der Mathematik</i>, Vol. I + (3), p. 598.</p> + + <p><a name="Nt_148" href="#NtA_148">[148]</a> For the opposite side of + the case see G. R. Kaye, "Notes on Indian Mathematics, No. 2.—<span + class="special" title="Aryabhata">Āryabhaṭa</span>," + <i>Journ. and Proc. of the Asiatic Soc. of Bengal</i>, Vol. IV, 1908, pp. + 111-141.</p> + + <p><a name="Nt_149" href="#NtA_149">[149]</a> He used one of the + alphabetic systems explained above. This ran up to 10<sup>18</sup> and + was not difficult, beginning as follows:</p> + + <div class="figcenter" style="width:30%;"> + <a href="images/049a.png"><img style="width:100%" src="images/049a.png" + alt="The system of Aryabhata" title="The system of Aryabhata" /></a> + </div> + <p>the same letter (<i>ka</i>) appearing in the successive consonant + forms, <i>ka</i>, <i>kha</i>, <i>ga</i>, <i>gha</i>, etc. See C. I. + Gerhardt, <i>Über die Entstehung und Ausbreitung des dekadischen + Zahlensystems</i>, Programm, p. 17, Salzwedel, 1853, and <i>Études + historiques sur l'arithmétique de position</i>, Programm, p. 24, Berlin, + 1856; E. Jacquet, <i>Mode d'expression symbolique des nombres</i>, loc. + cit., p. 97; L. Rodet, "Sur la véritable signification de la notation + numérique inventée par Āryabhata," <i>Journal Asiatique</i>, Vol. + XVI (7), pp. 440-485. On the two <span class="special" + title="Aryabhatas">Āryabhaṭas</span> see Kaye, <i>Bibl. + Math.</i>, Vol. X (3), p. 289.</p> + + <p><a name="Nt_150" href="#NtA_150">[150]</a> Using <i>kha</i>, a synonym + of <i>śūnya</i>. [Bayley, loc. cit., p. 22, and L. Rodet, + <i>Journal Asiatique</i>, Vol. XVI (7), p. 443.]</p> + + <p><a name="Nt_151" href="#NtA_151">[151]</a> Varāha-Mihira, + <i>Pañcasiddhāntikā</i>, translated by G. Thibaut and M. S. + Dvivedī, Benares, 1889; see Bühler, loc. cit., p. 78; Bayley, loc. + cit., p. 23.</p> + + <p><a name="Nt_152" href="#NtA_152">[152]</a> <i><span class="special" + title="Brhat Samhita">Bṛhat Saṃhitā</span></i>, + translated by Kern, <i>Journal of the Royal Asiatic Society</i>, + 1870-1875.</p> + + <p><a name="Nt_153" href="#NtA_153">[153]</a> It is stated by Bühler in a + personal letter to Bayley (loc. cit., p. 65) that there are hundreds of + instances of this usage in the <i><span class="special" title="Brhat Samhita" + >Bṛhat Saṃhitā</span></i>. The system was also used + in the <i>Pañcasiddhāntikā</i> as early as 505 <span + class="scac">A.D.</span> [Bühler, <i>Palaeographie</i>, p. 80, and Fleet, + <i>Journal of the Royal Asiatic Society</i>, 1910, p. 819.]</p> + + <p><a name="Nt_154" href="#NtA_154">[154]</a> Cantor, <i>Geschichte der + Mathematik</i>, Vol. I (3), p. 608.</p> + + <p><a name="Nt_155" href="#NtA_155">[155]</a> Bühler, loc. cit., p. + 78.</p> + + <p><a name="Nt_156" href="#NtA_156">[156]</a> Bayley, p. 38.</p> + + <p><a name="Nt_157" href="#NtA_157">[157]</a> Noviomagus, in his <i>De + numeris libri duo</i>, Paris, 1539, confesses his ignorance as to the + origin of the zero, but says: "D. Henricus Grauius, vir Graecè & + Hebraicè eximè doctus, Hebraicam originem ostendit," adding that Valla + "Indis Orientalibus gentibus inventionem tribuit."</p> + + <p><a name="Nt_158" href="#NtA_158">[158]</a> See <i>Essays</i>, Vol. II, + pp. 287 and 288.</p> + + <p><a name="Nt_159" href="#NtA_159">[159]</a> Vol. XXX, p. 205 seqq.</p> + + <p><a name="Nt_160" href="#NtA_160">[160]</a> Loc. cit., p. 284 seqq.</p> + + <p><a name="Nt_161" href="#NtA_161">[161]</a> Colebrooke, loc. cit., p. + 288.</p> + + <p><a name="Nt_162" href="#NtA_162">[162]</a> Loc. cit., p. 78.</p> + + <p><a name="Nt_163" href="#NtA_163">[163]</a> Hereafter, unless expressly + stated to the contrary, we shall use the word "numerals" to mean numerals + with place value.</p> + + <p><a name="Nt_164" href="#NtA_164">[164]</a> "The Gurjaras of + Rājputāna and Kanauj," in <i>Journal of the Royal Asiatic + Society</i>, January and April, 1909.</p> + + <p><a name="Nt_165" href="#NtA_165">[165]</a> Vol. IX, 1908, p. 248.</p> + + <p><a name="Nt_166" href="#NtA_166">[166]</a> <i>Epigraphia Indica</i>, + Vol. IX, pp. 193 and 198.</p> + + <p><a name="Nt_167" href="#NtA_167">[167]</a> <i>Epigraphia Indica</i>, + Vol. IX, p. 1.</p> + + <p><a name="Nt_168" href="#NtA_168">[168]</a> Loc. cit., p. 71.</p> + + <p><a name="Nt_169" href="#NtA_169">[169]</a> Thibaut, p. 71.</p> + + <p><a name="Nt_170" href="#NtA_170">[170]</a> "Est autem in aliquibus + figurarum istaram apud multos diuersitas. Quidam enim septimam hanc + figuram representant," etc. [Boncompagni, <i>Trattati</i>, p. 28.] + Eneström has shown that very likely this work is incorrectly attributed + to Johannes Hispalensis. [<i>Bibliotheca Mathematica</i>, Vol. IX (3), p. + 2.]</p> + + <p><a name="Nt_171" href="#NtA_171">[171]</a> <i>Indische + Palaeographie</i>, Tafel IX.</p> + + <p><a name="Nt_172" href="#NtA_172">[172]</a> Edited by Bloomfield and + Garbe, Baltimore, 1901, containing photographic reproductions of the + manuscript.</p> + + <p><a name="Nt_173" href="#NtA_173">[173]</a> <span class="special" + title="Bakhsali">Bakhṣālī</span> MS. See page 43; + Hoernle, R., <i>The Indian Antiquary</i>, Vol. XVII, pp. 33-48, 1 plate; + Hoernle, <i>Verhandlungen des VII. Internationalen + Orientalisten-Congresses, Arische Section</i>, Vienna, 1888, "On the + Bakshālī Manuscript," pp. 127-147, 3 plates; Bühler, loc. + cit.</p> + + <p><a name="Nt_174" href="#NtA_174">[174]</a> 3, 4, 6, from H. H. Dhruva, + "Three Land-Grants from Sankheda," <i>Epigraphia Indica</i>, Vol. II, pp. + 19-24 with plates; date 595 <span class="scac">A.D.</span> 7, 1, 5, from + Bhandarkar, "Daulatabad Plates," <i>Epigraphia Indica</i>, Vol. IX, part + V; date c. 798 <span class="scac">A.D.</span></p> + + <p><a name="Nt_175" href="#NtA_175">[175]</a> 8, 7, 2, from "Buckhala + Inscription of Nagabhatta," Bhandarkar, <i>Epigraphia Indica</i>, Vol. + IX, part V; date 815 <span class="scac">A.D.</span> 5 from "The Morbi + Copper-Plate," Bhandarkar, <i>The Indian Antiquary</i>, Vol. II, pp. + 257-258, with plate; date 804 <span class="scac">A.D.</span> See Bühler, + loc. cit.</p> + + <p><a name="Nt_176" href="#NtA_176">[176]</a> 8 from the above Morbi + Copper-Plate. 4, 5, 7, 9, and 0, from "Asni Inscription of Mahipala," + <i>The Indian Antiquary</i>, Vol. XVI, pp. 174-175; inscription is on red + sandstone, date 917 <span class="scac">A.D.</span> See Bühler.</p> + + <p><a name="Nt_177" href="#NtA_177">[177]</a> 8, 9, 4, from "Rashtrakuta + Grant of Amoghavarsha," J. F. Fleet, <i>The Indian Antiquary</i>, Vol. + XII, pp. 263-272; copper-plate grant of date c. 972 <span + class="scac">A.D.</span> See Bühler. 7, 3, 5, from "Torkhede Copper-Plate + Grant of the Time of Govindaraja of Gujerat," Fleet, <i>Epigraphia + Indica</i>, Vol. III, pp. 53-58. See Bühler.</p> + + <p><a name="Nt_178" href="#NtA_178">[178]</a> From "A Copper-Plate Grant + of King Tritochanapâla Chanlukya of <span class="special" + title="Latadesa">Lāṭadeśa</span>," H.H. Dhruva, + <i>Indian Antiquary</i>, Vol. XII, pp. 196-205; date 1050 <span + class="scac">A.D.</span> See Bühler.</p> + + <p><a name="Nt_179" href="#NtA_179">[179]</a> Burnell, A. C., <i>South + Indian Palæography</i>, plate XXIII, Telugu-Canarese numerals of the + eleventh century. See Bühler.</p> + + <p><a name="Nt_180" href="#NtA_180">[180]</a> From a manuscript of the + second half of the thirteenth century, reproduced in "Della vita e delle + opere di Leonardo Pisano," Baldassare Boncompagni, Rome, 1852, in <i>Atti + dell' Accademia Pontificia dei nuovi Lincei</i>, anno V.</p> + + <p><a name="Nt_181" href="#NtA_181">[181]</a> From a fourteenth-century + manuscript, as reproduced in <i>Della vita</i> etc., Boncompagni, loc. + cit.</p> + + <p><a name="Nt_182" href="#NtA_182">[182]</a> From a Tibetan MS. in the + library of D. E. Smith.</p> + + <p><a name="Nt_183" href="#NtA_183">[183]</a> From a Tibetan block-book + in the library of D. E. Smith.</p> + + <p><a name="Nt_184" href="#NtA_184">[184]</a> Śāradā + numerals from <i>The Kashmirian Atharva-Veda, reproduced by + chromophotography from the manuscript in the University Library at + Tübingen</i>, Bloomfield and Garbe, Baltimore, 1901. Somewhat similar + forms are given under "Numération Cachemirienne," by Pihan, <i>Exposé</i> + etc., p. 84.</p> + + <p><a name="Nt_185" href="#NtA_185">[185]</a> Franz X. Kugler, <i>Die + Babylonische Mondrechnung</i>, Freiburg i. Br., 1900, in the numerous + plates at the end of the book; practically all of these contain the + symbol to which reference is made. Cantor, <i>Geschichte</i>, Vol. I, p. + 31.</p> + + <p><a name="Nt_186" href="#NtA_186">[186]</a> F. X. Kugler, <i>Sternkunde + und Sterndienst in Babel</i>, I. Buch, from the beginnings to the time of + Christ, Münster i. Westfalen, 1907. It also has numerous tables + containing the above zero.</p> + + <p><a name="Nt_187" href="#NtA_187">[187]</a> From a letter to D. E. + Smith, from G. F. Hill of the British Museum. See also his monograph "On + the Early Use of Arabic Numerals in Europe," in <i>Archæologia</i>, Vol. + LXII (1910), p. 137.</p> + + <p><a name="Nt_188" href="#NtA_188">[188]</a> R. Hoernle, "The + Bakshālī Manuscript," <i>Indian Antiquary</i>, Vol. XVII, pp. + 33-48 and 275-279, 1888; Thibaut, <i>Astronomie, Astrologie und + Mathematik</i>, p. 75; Hoernle, <i>Verhandlungen</i>, loc. cit., p. + 132.</p> + + <p><a name="Nt_189" href="#NtA_189">[189]</a> Bayley, loc. cit., Vol. XV, + p. 29. Also Bendall, "On a System of Numerals used in South India," + <i>Journal of the Royal Asiatic Society</i>, 1896, pp. 789-792.</p> + + <p><a name="Nt_190" href="#NtA_190">[190]</a> V. A. Smith, <i>The Early + History of India</i>, 2d ed., Oxford, 1908, p. 14.</p> + + <p><a name="Nt_191" href="#NtA_191">[191]</a> Colebrooke, <i>Algebra, + with Arithmetic and Mensuration, from the Sanskrit of Brahmegupta and + Bháscara</i>, London, 1817, pp. 339-340.</p> + + <p><a name="Nt_192" href="#NtA_192">[192]</a> Ibid., p. 138.</p> + + <p><a name="Nt_193" href="#NtA_193">[193]</a> D. E. Smith, in the + <i>Bibliotheca Mathematica</i>, Vol. IX (3), pp. 106-110.</p> + + <p><a name="Nt_194" href="#NtA_194">[194]</a> As when we use three dots + (...).</p> + + <p><a name="Nt_195" href="#NtA_195">[195]</a> "The Hindus call the nought + explicitly <i>śūnyabindu</i> 'the dot marking a blank,' and + about 500 <span class="scac">A.D.</span> they marked it by a simple dot, + which latter is commonly used in inscriptions and MSS. in order to mark a + blank, and which was later converted into a small circle." [Bühler, <i>On + the Origin of the Indian Alphabet</i>, p. 53, note.]</p> + + <p><a name="Nt_196" href="#NtA_196">[196]</a> Fazzari, <i>Dell' origine + delle parole zero e cifra</i>, Naples, 1903.</p> + + <p><a name="Nt_197" href="#NtA_197">[197]</a> E. Wappler, "Zur Geschichte + der Mathematik im 15. Jahrhundert," in the <i>Zeitschrift für Mathematik + und Physik</i>, Vol. XLV, <i>Hist.-lit. Abt.</i>, p. 47. The manuscript + is No. C. 80, in the Dresden library.</p> + + <p><a name="Nt_198" href="#NtA_198">[198]</a> J. G. Prändel, <i>Algebra + nebst ihrer literarischen Geschichte</i>, p. 572, Munich, 1795.</p> + + <p><a name="Nt_199" href="#NtA_199">[199]</a> See the table, p. 23. Does + the fact that the early European arithmetics, following the Arab custom, + always put the 0 after the 9, suggest that the 0 was derived from the old + Hindu symbol for 10?</p> + + <p><a name="Nt_200" href="#NtA_200">[200]</a> Bayley, loc. cit., p. 48. + From this fact Delambre (<i>Histoire de l'astronomie ancienne</i>) + inferred that Ptolemy knew the zero, a theory accepted by Chasles, + <i>Aperçu historique sur l'origine et le développement des méthodes en + géométrie</i>, 1875 ed., p. 476; Nesselmann, however, showed (<i>Algebra + der Griechen</i>, 1842, p. 138), that Ptolemy merely used <span title="o" class="grk" + >ο</span> for <span title="ouden" class="grk" + >οὐδὲν</span>, with no notion of zero. See + also G. Fazzari, "Dell' origine delle parole zero e cifra," + <i>Ateneo</i>, Anno I, No. 11, reprinted at Naples in 1903, where the use + of the point and the small cross for zero is also mentioned. Th. H. + Martin, <i>Les signes numéraux</i> etc., reprint p. 30, and J. Brandis, + <i>Das Münz-, Mass- und Gewichtswesen in Vorderasien bis auf Alexander + den Grossen</i>, Berlin, 1866, p. 10, also discuss this usage of <span + title="o" class="grk">ο</span>, without the notion of place + value, by the Greeks.</p> + + <p><a name="Nt_201" href="#NtA_201">[201]</a> <i>Al-Battānī + sive Albatenii opus astronomicum</i>. Ad fidem codicis escurialensis + arabice editum, latine versum, adnotationibus instructum a Carolo + Alphonso Nallino, 1899-1907. Publicazioni del R. Osservatorio di Brera in + Milano, No. XL.</p> + + <p><a name="Nt_202" href="#NtA_202">[202]</a> Loc. cit., Vol. II, p. + 271.</p> + + <p><a name="Nt_203" href="#NtA_203">[203]</a> C. Henry, "Prologus N. + Ocreati in Helceph ad Adelardum Batensem magistrum suum," <i>Abhandlungen + zur Geschichte der Mathematik</i>, Vol. III, 1880.</p> + + <p><a name="Nt_204" href="#NtA_204">[204]</a> Max. Curtze, "Ueber eine + Algorismus-Schrift des XII. Jahrhunderts," <i>Abhandlungen zur Geschichte + der Mathematik</i>, Vol. VIII, 1898, pp. 1-27; Alfred Nagl, "Ueber eine + Algorismus-Schrift des XII. Jahrhunderts und über die Verbreitung der + indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," + <i>Zeitschrift für Mathematik und Physik, Hist.-lit. Abth.</i>, Vol. + XXXIV, pp. 129-146 and 161-170, with one plate.</p> + + <p><a name="Nt_205" href="#NtA_205">[205]</a> "Byzantinische Analekten," + <i>Abhandlungen zur Geschichte der Mathematik</i>, Vol. IX, pp. + 161-189.</p> + + <p><a name="Nt_206" href="#NtA_206">[206]</a> <a + href="images/061d.png"><img src="images/061d.png" class="middle" + style="height:2ex" alt="symbol" /></a> or <a href="images/061e.png"><img + src="images/061e.png" class="middle" style="height:2ex" alt="symbol" + /></a> for 0. <a href="images/061d.png"><img src="images/061d.png" + class="middle" style="height:2ex" alt="symbol" /></a> also used for 5. <a + href="images/061f.png"><img src="images/061f.png" class="middle" + style="height:2ex" alt="symbols" /></a> for 13. [Heiberg, loc. cit.]</p> + + <p><a name="Nt_207" href="#NtA_207">[207]</a> Gerhardt, <i>Études + historiques sur l'arithmétique de position</i>, Berlin, 1856, p. 12; J. + Bowring, <i>The Decimal System in Numbers, Coins, & Accounts</i>, + London, 1854, p. 33.</p> + + <p><a name="Nt_208" href="#NtA_208">[208]</a> Karabacek, <i>Wiener + Zeitschrift für die Kunde des Morgenlandes</i>, Vol. XI, p. 13; <i>Führer + durch die Papyrus-Ausstellung Erzherzog Rainer</i>, Vienna, 1894, p. + 216.</p> + + <p><a name="Nt_209" href="#NtA_209">[209]</a> In the library of G. A. + Plimpton, Esq.</p> + + <p><a name="Nt_210" href="#NtA_210">[210]</a> Cantor, <i>Geschichte</i>, + Vol. I (3), p. 674; Y. Mikami, "A Remark on the Chinese Mathematics in + Cantor's Geschichte der Mathematik," <i>Archiv der Mathematik und + Physik</i>, Vol. XV (3), pp. 68-70.</p> + + <p><a name="Nt_211" href="#NtA_211">[211]</a> Of course the earlier + historians made innumerable guesses as to the origin of the word + <i>cipher</i>. E.g. Matthew Hostus, <i>De numeratione emendata</i>, + Antwerp, 1582, p. 10, says: "Siphra vox Hebræam originem sapit refértque: + & ut docti arbitrantur, à verbo saphar, quod Ordine numerauit + significat. Unde Sephar numerus est: hinc Siphra (vulgo corruptius). Etsi + verò gens Iudaica his notis, quæ hodie Siphræ vocantur, usa non fuit: + mansit tamen rei appellatio apud multas gentes." Dasypodius, + <i>Institutiones mathematicae</i>, Vol. I, 1593, gives a large part of + this quotation word for word, without any mention of the source. + Hermannus Hugo, <i>De prima scribendi origine</i>, Trajecti ad Rhenum, + 1738, pp. 304-305, and note, p. 305; Karl Krumbacher, "Woher stammt das + Wort Ziffer (Chiffre)?", <i>Études de philologie néo-grecque</i>, Paris, + 1892.</p> + + <p><a name="Nt_212" href="#NtA_212">[212]</a> Bühler, loc. cit., p. 78 + and p. 86.</p> + + <p><a name="Nt_213" href="#NtA_213">[213]</a> Fazzari, loc. cit., p. 4. + So Elia Misrachi (1455-1526) in his posthumous <i>Book of Number</i>, + Constantinople, 1534, explains <i>sifra</i> as being Arabic. See also + Steinschneider, <i>Bibliotheca Mathematica</i>, 1893, p. 69, and G. + Wertheim, <i>Die Arithmetik des Elia Misrachi</i>, Programm, Frankfurt, + 1893.</p> + + <p><a name="Nt_214" href="#NtA_214">[214]</a> "Cum his novem figuris, et + cum hoc signo 0, quod arabice zephirum appellatur, scribitur quilibet + numerus."</p> + + <p><a name="Nt_215" href="#NtA_215">[215]</a> <span title="tziphra" class="grk" + >τζίφρα</span>, a form also used by + Neophytos (date unknown, probably c. 1330). It is curious that Finaeus + (1555 ed., f. 2) used the form <i>tziphra</i> throughout. A. J. H. + Vincent ["Sur l'origine de nos chiffres," <i>Notices et Extraits des + MSS.</i>, Paris, 1847, pp. 143-150] says: "Ce cercle fut nommé par les + uns, <i>sipos, rota, galgal</i> ...; par les autres <i>tsiphra</i> (de + <span lang="he" class="heb" title="TSPR" ><bdo + dir="rtl">צפר</bdo></span>, <i>couronne</i> ou + <i>diadème</i>) ou <i>ciphra</i> (de <span lang="he" class="heb" + title="SPR" ><bdo dir="rtl">ספר</bdo></span>, + <i>numération</i>)." Ch. de Paravey, <i>Essai sur l'origine unique et + hiéroglyphique des chiffres et des lettres de tous les peuples</i>, + Paris, 1826, p. 165, a rather fanciful work, gives "vase, vase arrondi et + fermé par un couvercle, qui est le symbole de la 10<sup>e</sup> Heure, <a + href="images/063a.png"><img src="images/063a.png" class="middle" + style="height:2ex" alt="symbol" /></a>," among the Chinese; also + "Tsiphron Zéron, ou tout à fait vide en arabe, <span title="tziphra" class="grk" + >τζίφρα</span> en grec ... d'où chiffre + (qui dérive plutôt, suivant nous, de l'Hébreu <i>Sepher</i>, + compter.")</p> + + <p><a name="Nt_216" href="#NtA_216">[216]</a> "Compilatus a Magistro + Jacobo de Florentia apud montem pesalanum," and described by G. Lami in + his <i>Catalogus codicum manuscriptorum qui in bibliotheca Riccardiana + Florentiæ adservantur</i>. See Fazzari, loc. cit., p. 5.</p> + + <p><a name="Nt_217" href="#NtA_217">[217]</a> "Et doveto sapere chel + zeuero per se solo non significa nulla ma è potentia di fare significare, + ... Et decina o centinaia o migliaia non si puote scrivere senza questo + segno 0. la quale si chiama zeuero." [Fazzari, loc. cit., p. 5.]</p> + + <p><a name="Nt_218" href="#NtA_218">[218]</a> Ibid., p. 6.</p> + + <p><a name="Nt_219" href="#NtA_219">[219]</a> Avicenna (980-1036), + translation by Gasbarri et François, "più il punto (gli Arabi adoperavano + il punto in vece dello zero il cui segno 0 in arabo si chiama + <i>zepiro</i> donde il vocabolo zero), che per sè stesso non esprime + nessun numero." This quotation is taken from D. C. Martines, <i>Origine e + progressi dell' aritmetica</i>, Messina, 1865.</p> + + <p><a name="Nt_220" href="#NtA_220">[220]</a> Leo Jordan, "Materialien + zur Geschichte der arabischen Zahlzeichen in Frankreich," <i>Archiv für + Kulturgeschichte</i>, Berlin, 1905, pp. 155-195, gives the following two + schemes of derivation, (1) "zefiro, zeviro, zeiro, zero," (2) "zefiro, + zefro, zevro, zero."</p> + + <p><a name="Nt_221" href="#NtA_221">[221]</a> Köbel (1518 ed., f. A_4) + speaks of the numerals in general as "die der gemain man Zyfer nendt." + Recorde (<i>Grounde of Artes</i>, 1558 ed., f. B_6) says that the zero is + "called priuatly a Cyphar, though all the other sometimes be likewise + named."</p> + + <p><a name="Nt_222" href="#NtA_222">[222]</a> "Decimo X 0 theca, circul<a + href="images/064a.png"><img src="images/064a.png" class="middle" + style="height:2ex" alt="us" /></a> cifra sive figura nihili + appelat′." [<i>Enchiridion Algorismi</i>, Cologne, 1501.] Later, + "quoniam de integris tam in cifris quam in proiectilibus,"—the word + <i>proiectilibus</i> referring to markers "thrown" and used on an abacus, + whence the French <i>jetons</i> and the English expression "to + <i>cast</i> an account."</p> + + <p><a name="Nt_223" href="#NtA_223">[223]</a> "Decima vero o dicitur + teca, circulus, vel cyfra vel figura nichili." [Maximilian Curtze, + <i>Petri Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco + commentarius, una cum Algorismo ipso</i>, Copenhagen, 1897, p. 2.] Curtze + cites five manuscripts (fourteenth and fifteenth centuries) of Dacia's + commentary in the libraries at Erfurt, Leipzig, and Salzburg, in addition + to those given by Eneström, <i>Öfversigt af Kongl. Vetenskaps-Akademiens + Förhandlingar</i>, 1885, pp. 15-27, 65-70; 1886, pp. 57-60.</p> + + <p><a name="Nt_224" href="#NtA_224">[224]</a> Curtze, loc. cit., p. <span + class="scac">VI</span>.</p> + + <p><a name="Nt_225" href="#NtA_225">[225]</a> <i>Rara Mathematica</i>, + London, 1841, chap, i, "Joannis de Sacro-Bosco Tractatus de Arte + Numerandi."</p> + + <p><a name="Nt_226" href="#NtA_226">[226]</a> Smith, <i>Rara + Arithmetica</i>, Boston, 1909.</p> + + <p><a name="Nt_227" href="#NtA_227">[227]</a> In the 1484 edition, Borghi + uses the form "çefiro: ouero nulla:" while in the 1488 edition he uses + "zefiro: ouero nulla," and in the 1540 edition, f. 3, appears "Chiamata + zero, ouero nulla." Woepcke asserted that it first appeared in Calandri + (1491) in this sentence: "Sono dieci le figure con le quali ciascuno + numero si può significare: delle quali n'è una che si chiama zero: et per + se sola nulla significa." (f. 4). [See <i>Propagation</i>, p. 522.]</p> + + <p><a name="Nt_228" href="#NtA_228">[228]</a> Boncompagni + <i>Bulletino</i>, Vol. XVI, pp. 673-685.</p> + + <p><a name="Nt_229" href="#NtA_229">[229]</a> Leo Jordan, loc. cit. In + the <i>Catalogue of MSS., Bibl. de l'Arsenal</i>, Vol. III, pp. 154-156, + this work is No. 2904 (184 S.A.F.), Bibl. Nat., and is also called + <i>Petit traicté de algorisme</i>.</p> + + <p><a name="Nt_230" href="#NtA_230">[230]</a> Texada (1546) says that + there are "nueue letros yvn zero o cifra" (f. 3).</p> + + <p><a name="Nt_231" href="#NtA_231">[231]</a> Savonne (1563, 1751 ed., f. + 1): "Vne ansi formee (o) qui s'appelle nulle, & entre marchans zero," + showing the influence of Italian names on French mercantile customs. + Trenchant (Lyons, 1566, 1578 ed., p. 12) also says: "La derniere qui + s'apele nulle, ou zero;" but Champenois, his contemporary, writing in + Paris in 1577 (although the work was not published until 1578), uses + "cipher," the Italian influence showing itself less in this center of + university culture than in the commercial atmosphere of Lyons.</p> + + <p><a name="Nt_232" href="#NtA_232">[232]</a> Thus Radulph of Laon (c. + 1100): "Inscribitur in ultimo ordine et figura <a + href="images/066a.png"><img src="images/066a.png" class="middle" + style="height:3ex" alt="symbol" /></a> sipos nomine, quae, licet numerum + nullum signitet, tantum ad alia quaedam utilis, ut insequentibus + declarabitur." ["Der Arithmetische Tractat des Radulph von Laon," + <i>Abhandlungen zur Geschichte der Mathematik</i>, Vol. V, p. 97, from a + manuscript of the thirteenth century.] Chasles (<i>Comptes rendus</i>, t. + 16, 1843, pp. 1393, 1408) calls attention to the fact that Radulph did + not know how to use the zero, and he doubts if the sipos was really + identical with it. Radulph says: "... figuram, cui sipos nomen est <a + href="images/066b.png"><img src="images/066b.png" class="middle" + style="height:3ex" alt="symbol" /></a> in motum rotulae formatam nullius + numeri significatione inscribi solere praediximus," and thereafter uses + <i>rotula</i>. He uses the sipos simply as a kind of marker on the + abacus.</p> + + <p><a name="Nt_233" href="#NtA_233">[233]</a> Rabbi ben Ezra (1092-1168) + used both <span lang="he" class="heb" title="GLGL" ><bdo + dir="rtl">גלגל</bdo></span>, <i>galgal</i> (the + Hebrew for <i>wheel</i>), and <span lang="he" class="heb" title="SPR'" + ><bdo dir="rtl">ספרא</bdo></span>, <i>sifra</i>. + See M. Steinschneider, "Die Mathematik bei den Juden," in <i>Bibliotheca + Mathematica</i>, 1893, p. 69, and Silberberg, <i>Das Buch der Zahl des R. + Abraham ibn Esra</i>, Frankfurt a. M., 1895, p. 96, note 23; in this work + the Hebrew letters are used for numerals with place value, having the + zero.</p> + + <p><a name="Nt_234" href="#NtA_234">[234]</a> E.g., in the + twelfth-century <i>Liber aligorismi</i> (see Boncompagni's + <i>Trattati</i>, II, p. 28). So Ramus (<i>Libri II</i>, 1569 ed., p. 1) + says: "Circulus quæ nota est ultima: nil per se significat." (See also + the Schonerus ed. of Ramus, 1586, p. 1.)</p> + + <p><a name="Nt_235" href="#NtA_235">[235]</a> "Und wirt das ringlein o. + die Ziffer genant die nichts bedeut." [Köbel's <i>Rechenbuch</i>, 1549 + ed., f. 10, and other editions.]</p> + + <p><a name="Nt_236" href="#NtA_236">[236]</a> I.e. "circular figure," our + word <i>notation</i> having come from the medieval <i>nota</i>. Thus + Tzwivel (1507, f. 2) says: "Nota autem circularis .o. per se sumpta nihil + vsus habet. alijs tamen adiuncta earum significantiam et auget et ordinem + permutat quantum quo ponit ordinem. vt adiuncta note binarij hoc modo 20 + facit eam significare bis decem etc." Also (ibid., f. 4), "figura + circularis," "circularis nota." Clichtoveus (1503 ed., f. <span + class="scac">XXXVII</span>) calls it "nota aut circularis o," "circularis + nota," and "figura circularis." Tonstall (1522, f. B_3) says of it: + "Decimo uero nota ad formam <a href="images/067a.png"><img + src="images/067a.png" class="middle" style="height:2ex" alt="symbol" + /></a> litteræ circulari figura est: quam alij circulum, uulgus cyphram + uocat," and later (f. C_4) speaks of the "circulos." Grammateus, in his + <i>Algorismus de integris</i> (Erfurt, 1523, f. A_2), speaking of the + nine significant figures, remarks: "His autem superadditur decima figura + circularis ut 0 existens que ratione sua nihil significat." Noviomagus + (<i>De Numeris libri II</i>, Paris, 1539, chap. xvi, "De notis numerorum, + quas zyphras vocant") calls it "circularis nota, quam ex his solam, alij + sipheram, Georgius Valla zyphram."</p> + + <p><a name="Nt_237" href="#NtA_237">[237]</a> Huswirt, as above. Ramus + (<i>Scholae mathematicae</i>, 1569 ed., p. 112) discusses the name + interestingly, saying: "Circulum appellamus cum multis, quam alii thecam, + alii figuram nihili, alii figuram privationis, seu figuram nullam vocant, + alii ciphram, cùm tamen hodie omnes hæ notæ vulgò ciphræ nominentur, + & 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."</p> + + <p><a name="Nt_238" href="#NtA_238">[238]</a> "Quare autem aliis + nominibus vocetur, non dicit auctor, quia omnia alia nomina habent + rationem suae lineationis sive figurationis. Quia rotunda est, dicitur + haec figura teca ad similitudinem tecae. Teca enim est ferrum figurae + rotundae, quod ignitum solet in quibusdam regionibus imprimi fronti vel + maxillae furis seu latronum." [Loc. cit., p. 26.] But in Greek + <i>theca</i> (<a href="images/067b.png"><img src="images/067b.png" + class="middle" style="height:2ex" alt="THEKE" /></a>, <span title="thêkê" class="grk" + >θήκη</span>) is a place to put something, a + receptacle. If a vacant column, e.g. in the abacus, was so called, the + initial might have given the early forms <a href="images/067c.png"><img + src="images/067c.png" class="middle" style="height:2ex" alt="symbol" + /></a> and <a href="images/067d.png"><img src="images/067d.png" + class="middle" style="height:2ex" alt="symbol" /></a> for the zero.</p> + + <p><a name="Nt_239" href="#NtA_239">[239]</a> Buteo, <i>Logistica</i>, + Lyons, 1559. See also Wertheim in the <i>Bibliotheca Mathematica</i>, + 1901, p. 214.</p> + + <p><a name="Nt_240" href="#NtA_240">[240]</a> "0 est appellee chiffre ou + nulle ou figure de nulle valeur." [La Roche, <i>L'arithmétique</i>, + Lyons, 1520.]</p> + + <p><a name="Nt_241" href="#NtA_241">[241]</a> "Decima autem figura nihil + uocata," "figura nihili (quam etiam cifram uocant)." [Stifel, + <i>Arithmetica integra</i>, 1544, f. 1.]</p> + + <p><a name="Nt_242" href="#NtA_242">[242]</a> "Zifra, & Nulla uel + figura Nihili." [Scheubel, 1545, p. 1 of ch. 1.] <i>Nulla</i> is also + used by Italian writers. Thus Sfortunati (1545 ed., f. 4) says: "et la + decima nulla & e chiamata questa decima zero;" Cataldi (1602, p. 1): + "La prima, che è o, si chiama nulla, ouero zero, ouero niente." It also + found its way into the Dutch arithmetics, e.g. Raets (1576, 1580 ed., f. + A_3): "Nullo dat ist niet;" Van der Schuere (1600, 1624 ed., f. 7); + Wilkens (1669 ed., p. 1). In Germany Johann Albert (Wittenberg, 1534) and + Rudolff (1526) both adopted the Italian <i>nulla</i> and popularized it. + (See also Kuckuck, <i>Die Rechenkunst im sechzehnten Jahrhundert</i>, + Berlin, 1874, p. 7; Günther, <i>Geschichte</i>, p. 316.)</p> + + <p><a name="Nt_243" href="#NtA_243">[243]</a> "La dixième s'appelle + chifre vulgairement: les vns l'appellant zero: nous la pourrons appeller + vn Rien." [Peletier, 1607 ed., p. 14.]</p> + + <p><a name="Nt_244" href="#NtA_244">[244]</a> It appears in the Polish + arithmetic of K<span class="over">l</span>os (1538) as <i>cyfra</i>. "The + Ciphra 0 augmenteth places, but of himselfe signifieth not," Digges, + 1579, p. 1. Hodder (10th ed., 1672, p. 2) uses only this word (cypher or + cipher), and the same is true of the first native American arithmetic, + written by Isaac Greenwood (1729, p. 1). Petrus de Dacia derives + <i>cyfra</i> from circumference. "Vocatur etiam cyfra, quasi circumfacta + vel circumferenda, quod idem est, quod circulus non habito respectu ad + centrum." [Loc. cit., p. 26.]</p> + + <p><a name="Nt_245" href="#NtA_245">[245]</a> <i>Opera mathematica</i>, + 1695, Oxford, Vol. I, chap. ix, <i>Mathesis universalis</i>, "De figuris + numeralibus," pp. 46-49; Vol. II, <i>Algebra</i>, p. 10.</p> + + <p><a name="Nt_246" href="#NtA_246">[246]</a> Martin, <i>Origine de notre + système de numération écrite</i>, note 149, p. 36 of reprint, spells + <span title="tsiphra" class="grk" + >τσίφρα</span> from Maximus Planudes, + citing Wallis as an authority. This is an error, for Wallis gives the + correct form as above.</p> + + <p>Alexander von Humboldt, "Über die bei verschiedenen Völkern üblichen + Systeme von Zahlzeichen und über den Ursprung des Stellenwerthes in den + indischen Zahlen," Crelle's <i>Journal für reine und angewandte + Mathematik</i>, Vol. IV, 1829, called attention to the work <span + title="arithmoi Indikoi" class="grk" + >ἀριθμοὶ + Ἰνδικοί</span> of the monk + Neophytos, supposed to be of the fourteenth century. In this work the + forms <span title="tzuphra" class="grk" + >τζύφρα</span> and <span title="tzumphra" class="grk" + >τζύμφρα</span> appear. See also Boeckh, + <i>De abaco Graecorum</i>, Berlin, 1841, and Tannery, "Le Scholie du + moine Néophytos," <i>Revue Archéologique</i>, 1885, pp. 99-102. Jordan, + loc. cit., gives from twelfth and thirteenth century manuscripts the + forms <i>cifra</i>, <i>ciffre</i>, <i>chifras</i>, and <i>cifrus</i>. Du + Cange, <i>Glossarium mediae et infimae Latinitatis</i>, Paris, 1842, + gives also <i>chilerae</i>. Dasypodius, <i>Institutiones + Mathematicae</i>, Strassburg, 1593-1596, adds the forms <i>zyphra</i> and + <i>syphra</i>. Boissière, <i>L'art d'arythmetique contenant toute + dimention, tres-singulier et commode, tant pour l'art militaire que + autres calculations</i>, Paris, 1554: "Puis y en a vn autre dict zero + lequel ne designe nulle quantité par soy, ains seulement les loges + vuides."</p> + + <p><a name="Nt_247" href="#NtA_247">[247]</a> <i>Propagation</i>, pp. 27, + 234, 442. Treutlein, "Das Rechnen im 16. Jahrhundert," <i>Abhandlungen + zur Geschichte der Mathematik</i>, Vol. I, p. 5, favors the same view. It + is combated by many writers, e.g. A. C. Burnell, loc. cit., p. 59. Long + before Woepcke, I. F. and G. I. Weidler, <i>De characteribus numerorum + vulgaribus et eorum aetatibus</i>, Wittenberg, 1727, asserted the + possibility of their introduction into Greece by Pythagoras or one of his + followers: "Potuerunt autem ex oriente, uel ex phoenicia, ad graecos + traduci, uel Pythagorae, uel eius discipulorum auxilio, cum aliquis eo, + proficiendi in literis causa, iter faceret, et hoc quoque inuentum + addisceret."</p> + + <p><a name="Nt_248" href="#NtA_248">[248]</a> E.g., they adopted the + Greek numerals in use in Damascus and Syria, and the Coptic in Egypt. + Theophanes (758-818 <span class="scac">A.D.</span>), + <i>Chronographia</i>, Scriptores Historiae Byzantinae, Vol. XXXIX, + Bonnae, 1839, p. 575, relates that in 699 <span class="scac">A.D.</span> + the caliph Walīd forbade the use of the Greek language in the + bookkeeping of the treasury of the caliphate, but permitted the use of + the Greek alphabetic numerals, since the Arabs had no convenient number + notation: <span title="kai ekôluse graphesthai Hellênisti tous dêmosious tôn logothesiôn kôdikas, all' Arabiois auta parasêmainesthai, chôris tôn psêphôn, epeidê adunaton têi ekeinôn glôssêi monada ê duada ê triada ê oktô hêmisu ê tria graphesthai; dio kai heôs sêmeron eisin sun autois notarioi Christianoi." class="grk" + >καὶ + ἐκώλυσε + γράφεσθαι + Ἑλληνιστὶ + τοὺς + δημοσίους + τῶν + λογοθεσίων + κώδικας, + ἀλλ' + Ἀραβίοις + αὐτὰ + παρασημαίνεσθαι, + χωρὶς τῶν + ψήφων, ἐπειδὴ + ἀδύνατον τῇ + ἐκείνων + γλώσσῃ + μονάδα ἢ + δυάδα ἢ + τριάδα ἢ + ὀκτὼ ἥμισυ ἢ + τρία + γράφεσθαι· + διὸ καὶ ἕως + σήμερόν + εἰσιν σὺν + αὐτοῖς + νοτάριοι + Χριστιανοί.</span> + The importance of this contemporaneous document was pointed out by + Martin, loc. cit. Karabacek, "Die Involutio im arabischen Schriftwesen," + Vol. CXXXV of <i>Sitzungsberichte d. phil.-hist. Classe d. k. Akad. d. + Wiss.</i>, Vienna, 1896, p. 25, gives an Arabic date of 868 <span + class="scac">A.D.</span> in Greek letters.</p> + + <p><a name="Nt_249" href="#NtA_249">[249]</a> <i>The Origin and History + of Our Numerals</i> (in Russian), Kiev, 1908; <i>The Independence of + European Arithmetic</i> (in Russian), Kiev.</p> + + <p><a name="Nt_250" href="#NtA_250">[250]</a> Woepcke, loc. cit., pp. + 462, 262.</p> + + <p><a name="Nt_251" href="#NtA_251">[251]</a> Woepcke, loc. cit., p. 240. + <i><span class="special" + title="Hisab-al-Gobar">Ḥisāb-al-Ġobār</span></i>, + by an anonymous author, probably Abū Sahl Dunash ibn Tamim, is + given by Steinschneider, "Die Mathematik bei den Juden," <i>Bibliotheca + Mathematica</i>, 1896, p. 26.</p> + + <p><a name="Nt_252" href="#NtA_252">[252]</a> Steinschneider in the + <i>Abhandlungen</i>, Vol. III, p. 110.</p> + + <p><a name="Nt_253" href="#NtA_253">[253]</a> See his <i>Grammaire + arabe</i>, Vol. I, Paris, 1810, plate VIII; Gerhardt, <i>Études</i>, pp. + 9-11, and <i>Entstehung</i> etc., p. 8; I. F. Weidler, <i>Spicilegium + observationum ad historiam notarum numeralium pertinentium</i>, + Wittenberg, 1755, speaks of the "figura cifrarum Saracenicarum" as being + different from that of the "characterum Boethianorum," which are similar + to the "vulgar" or common numerals; see also Humboldt, loc. cit.</p> + + <p><a name="Nt_254" href="#NtA_254">[254]</a> Gerhardt mentions it in his + <i>Entstehung</i> etc., p. 8; Woepcke, <i>Propagation</i>, states that + these numerals were used not for calculation, but very much as we use + Roman numerals. These superposed dots are found with both forms of + numerals (<i>Propagation</i>, pp. 244-246).</p> + + <p><a name="Nt_255" href="#NtA_255">[255]</a> Gerhardt (<i>Études</i>, p. + 9) from a manuscript in the Bibliothèque Nationale. The numeral forms are + <a href="images/072c.png"><img src="images/072c.png" class="middle" + style="height:2ex" alt="symbols" /></a>, 20 being indicated by <a + href="images/072d.png"><img src="images/072d.png" class="middle" + style="height:2.2ex" alt="symbol with dot" /></a> and 200 by <a + href="images/072e.png"><img src="images/072e.png" class="middle" + style="height:2.2ex" alt="symbol with 2 dots" /></a>. This scheme of zero + dots was also adopted by the Byzantine Greeks, for a manuscript of + Planudes in the Bibliothèque Nationale has numbers like <a + href="images/072f.png"><img src="images/072f.png" class="middle" + style="height:2ex" alt="pi alpha with 4 dots" /></a> for 8,100,000,000. + See Gerhardt, <i>Études</i>, p. 19. Pihan, <i>Exposé</i> etc., p. 208, + gives two forms, Asiatic and Maghrebian, of "Ghobār" numerals.</p> + + <p><a name="Nt_256" href="#NtA_256">[256]</a> See Chap. IV.</p> + + <p><a name="Nt_257" href="#NtA_257">[257]</a> Possibly as early as the + third century <span class="scac">A.D.</span>, but probably of the eighth + or ninth. See Cantor, I (3), p. 598.</p> + + <p><a name="Nt_258" href="#NtA_258">[258]</a> Ascribed by the Arabic + writer to India.</p> + + <p><a name="Nt_259" href="#NtA_259">[259]</a> See Woepcke's description + of a manuscript in the Chasles library, "Recherches sur l'histoire des + sciences mathématiques chez les orientaux," <i>Journal Asiatique</i>, IV + (5), 1859, p. 358, note.</p> + + <p><a name="Nt_260" href="#NtA_260">[260]</a> P. 56.</p> + + <p><a name="Nt_261" href="#NtA_261">[261]</a> Reinaud, <i>Mémoire sur + l'Inde</i>, p. 399. In the fourteenth century one Sihāb + al-Dīn wrote a work on which, a scholiast to the Bodleian + manuscript remarks: "The science is called Algobar because the inventor + had the habit of writing the figures on a tablet covered with sand." + [Gerhardt, <i>Études, </i>p. 11, note.]</p> + + <p><a name="Nt_262" href="#NtA_262">[262]</a> Gerhardt, <i>Entstehung + </i>etc., p. 20.</p> + + <p><a name="Nt_263" href="#NtA_263">[263]</a> H. Suter, "Das Rechenbuch + des <span class="special" title="Abu Zakarija el-Hassar">Abū + Zakarījā el-Ḥaṣṣār</span>," + <i>Bibliotheca Mathematica</i>, Vol. II (3), p. 15.</p> + + <p><a name="Nt_264" href="#NtA_264">[264]</a> A. Devoulx, "Les chiffres + arabes," <i>Revue Africaine</i>, Vol. XVI, pp. 455-458.</p> + + <p><a name="Nt_265" href="#NtA_265">[265]</a> <i>Kitāb + al-Fihrist</i>, G. Flügel, Leipzig, Vol. I, 1871, and Vol. II, 1872. This + work was published after Professor Flügel's death by J. Roediger and A. + Mueller. The first volume contains the Arabic text and the second volume + contains critical notes upon it.</p> + + <p><a name="Nt_266" href="#NtA_266">[266]</a> Like those of line 5 in the + illustration on page <a href="#page69">69</a>.</p> + + <p><a name="Nt_267" href="#NtA_267">[267]</a> Woepcke, <i>Recherches sur + l'histoire des sciences mathématiques chez les orientaux</i>, loc. cit.; + <i>Propagation, </i>p. 57.</p> + + <p><a name="Nt_268" href="#NtA_268">[268]</a> <span class="special" + title="Al-Hassar's">Al-Ḥaṣṣār's</span> forms, + Suter, <i>Bibliotheca Mathematica</i>, Vol. II (3), p. 15.</p> + + <p><a name="Nt_269" href="#NtA_269">[269]</a> Woepcke, <i>Sur une donnée + historique</i>, etc., loc. cit. The name <i>ġobār</i> is not + used in the text. The manuscript from which these are taken is the oldest + (970 <span class="scac">A.D.</span>) Arabic document known to contain all + of the numerals.</p> + + <p><a name="Nt_270" href="#NtA_270">[270]</a> Silvestre de Sacy, loc. + cit. He gives the ordinary modern Arabic forms, calling them + <i>Indien</i>.</p> + + <p><a name="Nt_271" href="#NtA_271">[271]</a> Woepcke, "Introduction au + calcul Gobārī et Hawāī," <i>Atti dell' accademia + pontificia dei nuovi Lincei</i>, Vol. XIX. The adjective applied to the + forms in 5 is <i>gobārī</i> and to those in 6 + <i>indienne</i>. This is the direct opposite of Woepcke's use of these + adjectives in the <i>Recherches sur l'histoire</i> cited above, in which + the ordinary Arabic forms (like those in row 5) are called + <i>indiens</i>.</p> + + <p>These forms are usually written from right to left.</p> + + <p><a name="Nt_272" href="#NtA_272">[272]</a> J. G. Wilkinson, <i>The + Manners and Customs of the Ancient Egyptians</i>, revised by S. Birch, + London, 1878, Vol. II, p. 493, plate XVI.</p> + + <p><a name="Nt_273" href="#NtA_273">[273]</a> There is an extensive + literature on this "Boethius-Frage." The reader who cares to go fully + into it should consult the various volumes of the <i>Jahrbuch über die + Fortschritte der Mathematik</i>.</p> + + <p><a name="Nt_274" href="#NtA_274">[274]</a> This title was first + applied to Roman emperors in posthumous coins of Julius Cæsar. + Subsequently the emperors assumed it during their own lifetimes, thus + deifying themselves. See F. Gnecchi, <i>Monete romane</i>, 2d ed., Milan, + 1900, p. 299.</p> + + <p><a name="Nt_275" href="#NtA_275">[275]</a> This is the common spelling + of the name, although the more correct Latin form is Boëtius. See + Harper's <i>Dict. of Class. Lit. and Antiq.</i>, New York, 1897, Vol. I, + p. 213. There is much uncertainty as to his life. A good summary of the + evidence is given in the last two editions of the <i>Encyclopædia + Britannica</i>.</p> + + <p><a name="Nt_276" href="#NtA_276">[276]</a> His father, Flavius Manlius + Boethius, was consul in 487.</p> + + <p><a name="Nt_277" href="#NtA_277">[277]</a> There is, however, no good + historic evidence of this sojourn in Athens.</p> + + <p><a name="Nt_278" href="#NtA_278">[278]</a> His arithmetic is dedicated + to Symmachus: "Domino suo patricio Symmacho Boetius." [Friedlein ed., p. + 3.]</p> + + <p><a name="Nt_279" href="#NtA_279">[279]</a> It was while here that he + wrote <i>De consolatione philosophiae</i>.</p> + + <p><a name="Nt_280" href="#NtA_280">[280]</a> It is sometimes given as + 525.</p> + + <p><a name="Nt_281" href="#NtA_281">[281]</a> There was a medieval + tradition that he was executed because of a work on the Trinity.</p> + + <p><a name="Nt_282" href="#NtA_282">[282]</a> Hence the <i>Divus</i> in + his name.</p> + + <p><a name="Nt_283" href="#NtA_283">[283]</a> Thus Dante, speaking of his + burial place in the monastery of St. Pietro in Ciel d'Oro, at Pavia, + says:</p> + + <div class="poem"> + <div class="stanza"> + <p class="i4">"The saintly soul, that shows</p> + <p>The world's deceitfulness, to all who hear him,</p> + <p>Is, with the sight of all the good that is,</p> + <p>Blest there. The limbs, whence it was driven, lie</p> + <p>Down in Cieldauro; and from martyrdom</p> + <p>And exile came it here."—<i>Paradiso</i>, Canto X.</p> + </div> + </div> + <p><a name="Nt_284" href="#NtA_284">[284]</a> Not, however, in the + mercantile schools. The arithmetic of Boethius would have been about the + last book to be thought of in such institutions. While referred to by + Bæda (672-735) and Hrabanus Maurus (c. 776-856), it was only after + Gerbert's time that the <i>Boëtii de institutione arithmetica libri + duo</i> was really a common work.</p> + + <p><a name="Nt_285" href="#NtA_285">[285]</a> Also spelled + Cassiodorius.</p> + + <p><a name="Nt_286" href="#NtA_286">[286]</a> As a matter of fact, + Boethius could not have translated any work by Pythagoras on music, + because there was no such work, but he did make the theories of the + Pythagoreans known. Neither did he translate Nicomachus, although he + embodied many of the ideas of the Greek writer in his own arithmetic. + Gibbon follows Cassiodorus in these statements in his <i>Decline and Fall + of the Roman Empire</i>, chap. xxxix. Martin pointed out with + positiveness the similarity of the first book of Boethius to the first + five books of Nicomachus. [<i>Les signes numéraux</i> etc., reprint, p. + 4.]</p> + + <p><a name="Nt_287" href="#NtA_287">[287]</a> The general idea goes back + to Pythagoras, however.</p> + + <p><a name="Nt_288" href="#NtA_288">[288]</a> J. C. Scaliger in his + <i>Poëtice</i> also said of him: "Boethii Severini ingenium, eruditio, + ars, sapientia facile provocat omnes auctores, sive illi Graeci sint, + sive Latini" [Heilbronner, <i>Hist. math. univ.</i>, p. 387]. Libri, + speaking of the time of Boethius, remarks: "Nous voyons du temps de + Théodoric, les lettres reprendre une nouvelle vie en Italie, les écoles + florissantes et les savans honorés. Et certes les ouvrages de Boëce, de + Cassiodore, de Symmaque, surpassent de beaucoup toutes les productions du + siècle précédent." [<i>Histoire des mathématiques</i>, Vol. I, p. + 78.]</p> + + <p><a name="Nt_289" href="#NtA_289">[289]</a> Carra de Vaux, + <i>Avicenne</i>, Paris, 1900; Woepcke, <i>Sur l'introduction</i>, etc.; + Gerhardt, <i>Entstehung</i> etc., p. 20. Avicenna is a corruption from + Ibn Sīnā, as pointed out by Wüstenfeld, <i>Geschichte der + arabischen Aerzte und Naturforscher</i>, Göttingen, 1840. His full name + is <span class="special" title="Abu `Ali al-Hosein ibn Sina">Abū + ‛Alī al-Ḥosein ibn Sīnā</span>. For notes + on Avicenna's arithmetic, see Woepcke, <i>Propagation</i>, p. 502.</p> + + <p><a name="Nt_290" href="#NtA_290">[290]</a> On the early travel between + the East and the West the following works may be consulted: A. + Hillebrandt, <i>Alt-Indien</i>, containing "Chinesische Reisende in + Indien," Breslau, 1899, p. 179; C. A. Skeel, <i>Travel in the First + Century after Christ</i>, Cambridge, 1901, p. 142; M. Reinaud, "Relations + politiques et commerciales de l'empire romain avec l'Asie orientale," in + the <i>Journal Asiatique</i>, Mars-Avril, 1863, Vol. I (6), p. 93; + Beazley, <i>Dawn of Modern Geography, a History of Exploration and + Geographical Science from the Conversion of the Roman Empire to <span + class="scac">A.D.</span> 1420</i>, London, 1897-1906, 3 vols.; Heyd, + <i>Geschichte des Levanthandels im Mittelalter</i>, Stuttgart, 1897; J. + Keane, <i>The Evolution of Geography</i>, London, 1899, p. 38; A. + Cunningham, <i>Corpus inscriptionum Indicarum</i>, Calcutta, 1877, Vol. + I; A. Neander, <i>General History of the Christian Religion and + Church</i>, 5th American ed., Boston, 1855, Vol. III, p. 89; R. C. Dutt, + <i>A History of Civilization in Ancient India</i>, Vol. II, Bk. V, chap, + ii; E. C. Bayley, loc. cit., p. 28 et seq.; A. C. Burnell, loc. cit., p. + 3; J. E. Tennent, <i>Ceylon</i>, London, 1859, Vol. I, p. 159; Geo. + Turnour, <i>Epitome of the History of Ceylon</i>, London, n.d., preface; + "Philalethes," <i>History of Ceylon</i>, London, 1816, chap, i; H. C. + Sirr, <i>Ceylon and the Cingalese</i>, London, 1850, Vol. I, chap. ix. On + the Hindu knowledge of the Nile see F. Wilford, <i>Asiatick + Researches</i>, Vol. III, p. 295, Calcutta, 1792.</p> + + <p><a name="Nt_291" href="#NtA_291">[291]</a> G. Oppert, <i>On the + Ancient Commerce of India</i>, Madras, 1879, p. 8.</p> + + <p><a name="Nt_292" href="#NtA_292">[292]</a> Gerhardt, <i>Études</i> + etc., pp. 8, 11.</p> + + <p><a name="Nt_293" href="#NtA_293">[293]</a> See Smith's <i>Dictionary + of Greek and Roman Biography and Mythology</i>.</p> + + <p><a name="Nt_294" href="#NtA_294">[294]</a> P. M. Sykes, <i>Ten + Thousand Miles in Persia, or Eight Years in Irán</i>, London, 1902, p. + 167. Sykes was the first European to follow the course of Alexander's + army across eastern Persia.</p> + + <p><a name="Nt_295" href="#NtA_295">[295]</a> Bühler, <i>Indian + Brāhma Alphabet</i>, note, p. 27; <i>Palaeographie</i>, p. 2; + <i>Herodoti Halicarnassei historia</i>, Amsterdam, 1763, Bk. IV, p. 300; + Isaac Vossius, <i>Periplus Scylacis Caryandensis</i>, 1639. It is + doubtful whether the work attributed to Scylax was written by him, but in + any case the work dates back to the fourth century <span + class="scac">B.C.</span> See Smith's <i>Dictionary of Greek and Roman + Biography</i>.</p> + + <p><a name="Nt_296" href="#NtA_296">[296]</a> Herodotus, Bk. III.</p> + + <p><a name="Nt_297" href="#NtA_297">[297]</a> Rameses II(?), the + <i>Sesoosis</i> of Diodorus Siculus.</p> + + <p><a name="Nt_298" href="#NtA_298">[298]</a> <i>Indian Antiquary</i>, + Vol. I, p. 229; F. B. Jevons, <i>Manual of Greek Antiquities</i>, London, + 1895, p. 386. On the relations, political and commercial, between India + and Egypt c. 72 <span class="scac">B.C.</span>, under Ptolemy Auletes, + see the <i>Journal Asiatique</i>, 1863, p. 297.</p> + + <p><a name="Nt_299" href="#NtA_299">[299]</a> Sikandar, as the name still + remains in northern India.</p> + + <p><a name="Nt_300" href="#NtA_300">[300]</a> <i>Harper's Classical + Dict.</i>, New York, 1897, Vol. I, p. 724; F. B. Jevons, loc. cit., p. + 389; J. C. Marshman, <i>Abridgment of the History of India</i>, chaps. i + and ii.</p> + + <p><a name="Nt_301" href="#NtA_301">[301]</a> Oppert, loc. cit., p. 11. + It was at or near this place that the first great Indian mathematician, + <span class="special" title="Aryabhata">Āryabhaṭa</span>, + was born in 476 <span class="scac">A.D.</span></p> + + <p><a name="Nt_302" href="#NtA_302">[302]</a> Bühler, + <i>Palaeographie</i>, p. 2, speaks of Greek coins of a period anterior to + Alexander, found in northern India. More complete information may be + found in <i>Indian Coins</i>, by E. J. Rapson, Strassburg, 1898, pp. + 3-7.</p> + + <p><a name="Nt_303" href="#NtA_303">[303]</a> Oppert, loc. cit., p. 14; + and to him is due other similar information.</p> + + <p><a name="Nt_304" href="#NtA_304">[304]</a> J. Beloch, <i>Griechische + Geschichte</i>, Vol. III, Strassburg, 1904, pp. 30-31.</p> + + <p><a name="Nt_305" href="#NtA_305">[305]</a> E.g., the denarius, the + words for hour and minute (<span title="hôra, lepton" class="grk" + >ὥρα, λεπτόν</span>), + and possibly the signs of the zodiac. [R. Caldwell, <i>Comparative + Grammar of the Dravidian Languages</i>, London, 1856, p. 438.] On the + probable Chinese origin of the zodiac see Schlegel, loc. cit.</p> + + <p><a name="Nt_306" href="#NtA_306">[306]</a> Marie, Vol. II, p. 73; R. + Caldwell, loc. cit.</p> + + <p><a name="Nt_307" href="#NtA_307">[307]</a> A. Cunningham, loc. cit., + p. 50.</p> + + <p><a name="Nt_308" href="#NtA_308">[308]</a> C. A. J. Skeel, + <i>Travel</i>, loc. cit., p. 14.</p> + + <p><a name="Nt_309" href="#NtA_309">[309]</a> <i>Inchiver</i>, from + <i>inchi</i>, "the green root." [<i>Indian Antiquary</i>, Vol. I, p. + 352.]</p> + + <p><a name="Nt_310" href="#NtA_310">[310]</a> In China dating only from + the second century <span class="scac">A.D.</span>, however.</p> + + <p><a name="Nt_311" href="#NtA_311">[311]</a> The Italian + <i>morra</i>.</p> + + <p><a name="Nt_312" href="#NtA_312">[312]</a> J. Bowring, <i>The Decimal + System</i>, London, 1854, p. 2.</p> + + <p><a name="Nt_313" href="#NtA_313">[313]</a> H. A. Giles, lecture at + Columbia University, March 12, 1902, on "China and Ancient Greece."</p> + + <p><a name="Nt_314" href="#NtA_314">[314]</a> Giles, loc. cit.</p> + + <p><a name="Nt_315" href="#NtA_315">[315]</a> E.g., the names for grape, + radish (<i>la-po</i>, <span title="rhaphê" class="grk" + >ῥάφη</span>), water-lily (<i>si-kua</i>, "west + gourds"; <span title="sikua" class="grk" + >σικύα</span>, "gourds"), are much alike. + [Giles, loc. cit.]</p> + + <p><a name="Nt_316" href="#NtA_316">[316]</a> <i>Epistles</i>, I, 1, + 45-46. On the Roman trade routes, see Beazley, loc. cit., Vol. I, p. + 179.</p> + + <p><a name="Nt_317" href="#NtA_317">[317]</a> <i>Am. Journ. of + Archeol.</i>, Vol. IV, p. 366.</p> + + <p><a name="Nt_318" href="#NtA_318">[318]</a> M. Perrot gives this + conjectural restoration of his words: "Ad me ex India regum legationes + saepe missi sunt numquam antea visae apud quemquam principem Romanorum." + [M. Reinaud, "Relations politiques et commerciales de l'empire romain + avec l'Asie orientale," <i>Journ. Asiat.</i>, Vol. I (6), p. 93.]</p> + + <p><a name="Nt_319" href="#NtA_319">[319]</a> Reinaud, loc. cit., p. 189. + Florus, II, 34 (IV, 12), refers to it: "Seres etiam habitantesque sub + ipso sole Indi, cum gemmis et margaritis elephantes quoque inter munera + trahentes nihil magis quam longinquitatem viae imputabant." Horace shows + his geographical knowledge by saying: "Not those who drink of the deep + Danube shall now break the Julian edicts; not the Getae, not the Seres, + nor the perfidious Persians, nor those born on the river Tanaïs." + [<i>Odes</i>, Bk. IV, Ode 15, 21-24.]</p> + + <p><a name="Nt_320" href="#NtA_320">[320]</a> "Qua virtutis + moderationisque fama Indos etiam ac Scythas auditu modo cognitos pellexit + ad amicitiam suam populique Romani ultro per legatos petendam." [Reinaud, + loc. cit., p. 180.]</p> + + <p><a name="Nt_321" href="#NtA_321">[321]</a> Reinaud, loc. cit., p. + 180.</p> + + <p><a name="Nt_322" href="#NtA_322">[322]</a> <i>Georgics</i>, II, + 170-172. So Propertius (<i>Elegies</i>, III, 4):</p> + + <div class="poem"> + <div class="stanza"> + <p>Arma deus Caesar dites meditatur ad Indos</p> + <p class="i2">Et freta gemmiferi findere classe maris.</p> + </div> + </div> + <p>"The divine Cæsar meditated carrying arms against opulent India, and + with his ships to cut the gem-bearing seas."</p> + + <p><a name="Nt_323" href="#NtA_323">[323]</a> Heyd, loc. cit., Vol. I, p. + 4.</p> + + <p><a name="Nt_324" href="#NtA_324">[324]</a> Reinaud, loc. cit., p. + 393.</p> + + <p><a name="Nt_325" href="#NtA_325">[325]</a> The title page of Calandri + (1491), for example, represents Pythagoras with these numerals before + him. [Smith, <i>Rara Arithmetica</i>, p. 46.] Isaacus Vossius, + <i>Observationes ad Pomponium Melam de situ orbis</i>, 1658, maintained + that the Arabs derived these numerals from the west. A learned + dissertation to this effect, but deriving them from the Romans instead of + the Greeks, was written by Ginanni in 1753 (<i>Dissertatio mathematica + critica de numeralium notarum minuscularum origine</i>, Venice, 1753). + See also Mannert, <i>De numerorum quos arabicos vocant vera origine + Pythagorica</i>, Nürnberg, 1801. Even as late as 1827 Romagnosi (in his + supplement to <i>Ricerche storiche sull' India</i> etc., by Robertson, + Vol. II, p. 580, 1827) asserted that Pythagoras originated them. [R. + Bombelli, <i>L'antica numerazione italica</i>, Rome, 1876, p. 59.] Gow + (<i>Hist. of Greek Math.</i>, p. 98) thinks that Iamblichus must have + known a similar system in order to have worked out certain of his + theorems, but this is an unwarranted deduction from the passage + given.</p> + + <p><a name="Nt_326" href="#NtA_326">[326]</a> A. Hillebrandt, + <i>Alt-Indien</i>, p. 179.</p> + + <p><a name="Nt_327" href="#NtA_327">[327]</a> J. C. Marshman, loc. cit., + chaps. i and ii.</p> + + <p><a name="Nt_328" href="#NtA_328">[328]</a> He reigned 631-579 <span + class="scac">A.D.</span>; called Nuśīrwān, <i>the holy + one</i>.</p> + + <p><a name="Nt_329" href="#NtA_329">[329]</a> J. Keane, <i>The Evolution + of Geography</i>, London, 1899, p. 38.</p> + + <p><a name="Nt_330" href="#NtA_330">[330]</a> The Arabs who lived in and + about Mecca.</p> + + <p><a name="Nt_331" href="#NtA_331">[331]</a> S. Guyard, in <i>Encyc. + Brit.</i>, 9th ed., Vol. XVI, p. 597.</p> + + <p><a name="Nt_332" href="#NtA_332">[332]</a> Oppert, loc. cit., p. + 29.</p> + + <p><a name="Nt_333" href="#NtA_333">[333]</a> "At non credendum est id in + Autographis contigisse, aut vetustioribus Codd. MSS." [Wallis, <i>Opera + omnia</i>, Vol. II, p. 11.]</p> + + <p><a name="Nt_334" href="#NtA_334">[334]</a> In <i>Observationes ad + Pomponium Melam de situ orbis</i>. The question was next taken up in a + large way by Weidler, loc. cit., <i>De characteribus</i> etc., 1727, and + in <i>Spicilegium</i> etc., 1755.</p> + + <p><a name="Nt_335" href="#NtA_335">[335]</a> The best edition of these + works is that of G. Friedlein, <i>Anicii Manlii Torquati Severini Boetii + de institutione arithmetica libri duo, de institutione musica libri + quinque. Accedit geometria quae fertur Boetii</i>.... Leipzig.... <span + class="scac">MDCCCLXVII</span>.</p> + + <p><a name="Nt_336" href="#NtA_336">[336]</a> See also P. Tannery, "Notes + sur la pseudo-géometrie de Boèce," in <i>Bibliotheca Mathematica</i>, + Vol. I (3), p. 39. This is not the geometry in two books in which are + mentioned the numerals. There is a manuscript of this pseudo-geometry of + the ninth century, but the earliest one of the other work is of the + eleventh century (Tannery), unless the Vatican codex is of the tenth + century as Friedlein (p. 372) asserts.</p> + + <p><a name="Nt_337" href="#NtA_337">[337]</a> Friedlein feels that it is + partly spurious, but he says: "Eorum librorum, quos Boetius de geometria + scripsisse dicitur, investigare veram inscriptionem nihil aliud esset + nisi operam et tempus perdere." [Preface, p. v.] N. Bubnov in the Russian + <i>Journal of the Ministry of Public Instruction</i>, 1907, in an article + of which a synopsis is given in the <i>Jahrbuch über die Fortschritte der + Mathematik</i> for 1907, asserts that the geometry was written in the + eleventh century.</p> + + <p><a name="Nt_338" href="#NtA_338">[338]</a> The most noteworthy of + these was for a long time Cantor (<i>Geschichte</i>, Vol. I., 3d ed., pp. + 587-588), who in his earlier days even believed that Pythagoras had known + them. Cantor says (<i>Die römischen Agrimensoren</i>, Leipzig, 1875, p. + 130): "Uns also, wir wiederholen es, ist die Geometrie des Boetius echt, + dieselbe Schrift, welche er nach Euklid bearbeitete, von welcher ein + Codex bereits in Jahre 821 im Kloster Reichenau vorhanden war, von + welcher ein anderes Exemplar im Jahre 982 zu Mantua in die Hände + Gerbert's gelangte, von welcher mannigfache Handschriften noch heute + vorhanden sind." But against this opinion of the antiquity of MSS. + containing these numerals is the important statement of P. Tannery, + perhaps the most critical of modern historians of mathematics, that none + exists earlier than the eleventh century. See also J. L. Heiberg in + <i>Philologus, Zeitschrift f. d. klass. Altertum</i>, Vol. XLIII, p. + 508.</p> + + <p>Of Cantor's predecessors, Th. H. Martin was one of the most prominent, + his argument for authenticity appearing in the <i>Revue Archéologique</i> + for 1856-1857, and in his treatise <i>Les signes numéraux</i> etc. See + also M. Chasles, "De la connaissance qu'ont eu les anciens d'une + numération décimale écrite qui fait usage de neuf chiffres prenant les + valeurs de position," <i>Comptes rendus</i>, Vol. VI, pp. 678-680; "Sur + l'origine de notre système de numération," <i>Comptes rendus</i>, Vol. + VIII, pp. 72-81; and note "Sur le passage du premier livre de la + géométrie de Boèce, relatif à un nouveau système de numération," in his + work <i>Aperçu historique sur l'origine et le devéloppement des méthodes + en géométrie</i>, of which the first edition appeared in 1837.</p> + + <p><a name="Nt_339" href="#NtA_339">[339]</a> J. L. Heiberg places the + book in the eleventh century on philological grounds, <i>Philologus</i>, + loc. cit.; Woepcke, in <i>Propagation</i>, p. 44; Blume, Lachmann, and + Rudorff, <i>Die Schriften der römischen Feldmesser</i>, Berlin, 1848; + Boeckh, <i>De abaco graecorum</i>, Berlin, 1841; Friedlein, in his + Leipzig edition of 1867; Weissenborn, <i>Abhandlungen</i>, Vol. II, p. + 185, his <i>Gerbert</i>, pp. 1, 247, and his <i>Geschichte der Einführung + der jetzigen Ziffern in Europa durch Gerbert</i>, Berlin, 1892, p. 11; + Bayley, loc. cit., p. 59; Gerhardt, <i>Études</i>, p. 17, <i>Entstehung + und Ausbreitung</i>, p. 14; Nagl, <i>Gerbert</i>, p. 57; Bubnov, loc. + cit. See also the discussion by Chasles, Halliwell, and Libri, in the + <i>Comptes rendus</i>, 1839, Vol. IX, p. 447, and in Vols. VIII, XVI, + XVII of the same journal.</p> + + <p><a name="Nt_340" href="#NtA_340">[340]</a> J. Marquardt, <i>La vie + privée des Romains</i>, Vol. II (French trans.), p. 505, Paris, 1893.</p> + + <p><a name="Nt_341" href="#NtA_341">[341]</a> In a Plimpton manuscript of + the arithmetic of Boethius of the thirteenth century, for example, the + Roman numerals are all replaced by the Arabic, and the same is true in + the first printed edition of the book. (See Smith's <i>Rara + Arithmetica</i>, pp. 434, 25-27.) D. E. Smith also copied from a + manuscript of the arithmetic in the Laurentian library at Florence, of + 1370, the following forms, <a href="images/092a.png"><img + src="images/092a.png" class="middle" style="height:2.5ex" alt="Forged + numerals" /></a> which, of course, are interpolations. An interesting + example of a forgery in ecclesiastical matters is in the charter said to + have been given by St. Patrick, granting indulgences to the benefactors + of Glastonbury, dated "In nomine domini nostri Jhesu Christi Ego + Patricius humilis servunculus Dei anno incarnationis ejusdem ccccxxx." + Now if the Benedictines are right in saying that Dionysius Exiguus, a + Scythian monk, first arranged the Christian chronology c. 532 <span + class="scac">A.D.</span>, this can hardly be other than spurious. See + Arbuthnot, loc. cit., p. 38.</p> + + <p><a name="Nt_342" href="#NtA_342">[342]</a> Halliwell, in his <i>Rara + Mathematica, </i>p. 107, states that the disputed passage is not in a + manuscript belonging to Mr. Ames, nor in one at Trinity College. See also + Woepcke, in <i>Propagation</i>, pp. 37 and 42. It was the evident + corruption of the texts in such editions of Boethius as those of Venice, + 1499, Basel, 1546 and 1570, that led Woepcke to publish his work <i>Sur + l'introduction de l'arithmétique indienne en Occident</i>.</p> + + <p><a name="Nt_343" href="#NtA_343">[343]</a> They are found in none of + the very ancient manuscripts, as, for example, in the ninth-century (?) + codex in the Laurentian library which one of the authors has examined. It + should be said, however, that the disputed passage was written after the + arithmetic, for it contains a reference to that work. See the Friedlein + ed., p. 397.</p> + + <p><a name="Nt_344" href="#NtA_344">[344]</a> Smith, <i>Rara + Arithmetica</i>, p. 66.</p> + + <p><a name="Nt_345" href="#NtA_345">[345]</a> J. L. Heiberg, + <i>Philologus</i>, Vol. XLIII, p. 507.</p> + + <p><a name="Nt_346" href="#NtA_346">[346]</a> "Nosse autem huius artis + dispicientem, quid sint digiti, quid articuli, quid compositi, quid + incompositi numeri." [Friedlein ed., p. 395.]</p> + + <p><a name="Nt_347" href="#NtA_347">[347]</a> <i>De ratione abaci.</i> In + this he describes "quandam formulam, quam ob honorem sui praeceptoris + mensam Pythagoream nominabant ... a posterioribus appellabatur abacus." + This, as pictured in the text, is the common Gerbert abacus. In the + edition in Migne's <i>Patrologia Latina</i>, Vol. LXIII, an ordinary + multiplication table (sometimes called Pythagorean abacus) is given in + the illustration.</p> + + <p><a name="Nt_348" href="#NtA_348">[348]</a> "Habebant enim diverse + formatos apices vel caracteres." See the reference to Gerbert on p. + 117.</p> + + <p><a name="Nt_349" href="#NtA_349">[349]</a> C. Henry, "Sur l'origine de + quelques notations mathématiques," <i>Revue Archéologique</i>, 1879, + derives these from the initial letters used as abbreviations for the + names of the numerals, a theory that finds few supporters.</p> + + <p><a name="Nt_350" href="#NtA_350">[350]</a> E.g., it appears in + Schonerus, <i>Algorithmus Demonstratus</i>, Nürnberg, 1534, f. A4. In + England it appeared in the earliest English arithmetical manuscript + known, <i>The Crafte of Nombrynge</i>: "¶ fforthermore ye most + vndirstonde that in this craft ben vsid teen figurys, as here bene writen + for ensampul, <a href="images/093a.png"><img src="images/093a.png" + class="middle" style="height:2ex" alt="Numerals" /></a> ... in the quych + we vse teen figurys of Inde. Questio. ¶ why ten fyguris of Inde? Solucio. + for as I have sayd afore thei were fonde fyrst in Inde of a kynge of that + Cuntre, that was called Algor." See Smith, <i>An Early English + Algorism</i>, loc. cit.</p> + + <p><a name="Nt_351" href="#NtA_351">[351]</a> Friedlein ed., p. 397.</p> + + <p><a name="Nt_352" href="#NtA_352">[352]</a> Carlsruhe codex of + Gerlando.</p> + + <p><a name="Nt_353" href="#NtA_353">[353]</a> Munich codex of + Gerlando.</p> + + <p><a name="Nt_354" href="#NtA_354">[354]</a> Carlsruhe codex of + Bernelinus.</p> + + <p><a name="Nt_355" href="#NtA_355">[355]</a> Munich codex of + Bernelinus.</p> + + <p><a name="Nt_356" href="#NtA_356">[356]</a> Turchill, c. 1200.</p> + + <p><a name="Nt_357" href="#NtA_357">[357]</a> Anon. MS., thirteenth + century, Alexandrian Library, Rome.</p> + + <p><a name="Nt_358" href="#NtA_358">[358]</a> Twelfth-century Boethius, + Friedlein, p. 396.</p> + + <p><a name="Nt_359" href="#NtA_359">[359]</a> Vatican codex, tenth + century, Boethius.</p> + + <p><a name="Nt_360" href="#NtA_360">[360]</a> a, h, i, are from the + Friedlein ed.; the original in the manuscript from which a is taken + contains a zero symbol, as do all of the six plates given by Friedlein. + b-e from the Boncompagni <i>Bulletino</i>, Vol. X, p. 596; f ibid., Vol. + XV, p. 186; g <i>Memorie della classe di sci., Reale Acc. dei Lincei</i>, + An. CCLXXIV (1876-1877), April, 1877. A twelfth-century arithmetician, + possibly John of Luna (Hispalensis, of Seville, c. 1150), speaks of the + great diversity of these forms even in his day, saying: "Est autem in + aliquibus figuram istarum apud multos diuersitas. Quidam enim septimam + hanc figuram representant <a href="images/094j.png"><img + src="images/094j.png" class="middle" style="height:2.5ex" alt="Symbol" + /></a> alii autem sic <a href="images/094k.png"><img + src="images/094k.png" class="middle" style="height:2.5ex" alt="Symbol" + /></a>, uel sic <a href="images/094l.png"><img src="images/094l.png" + class="middle" style="height:2.5ex" alt="Symbol" /></a>. Quidam vero + quartam sic <a href="images/094m.png"><img src="images/094m.png" + class="middle" style="height:2.5ex" alt="Symbol" /></a>." [Boncompagni, + <i>Trattati</i>, Vol. II, p. 28.]</p> + + <p><a name="Nt_361" href="#NtA_361">[361]</a> Loc. cit., p. 59.</p> + + <p><a name="Nt_362" href="#NtA_362">[362]</a> Ibid., p. 101.</p> + + <p><a name="Nt_363" href="#NtA_363">[363]</a> Loc. cit., p. 396.</p> + + <p><a name="Nt_364" href="#NtA_364">[364]</a> Khosrū I, who began + to reign in 531 <span class="scac">A.D.</span> See W. S. W Vaux, + <i>Persia, </i>London, 1875, p. 169; Th. Nöldeke, <i>Aufsätze zur + persichen Geschichte</i>, Leipzig, 1887, p. 113, and his article in the + ninth edition of the <i>Encyclopædia Britannica</i>.</p> + + <p><a name="Nt_365" href="#NtA_365">[365]</a> Colebrooke, <i>Essays</i>, + Vol. II, p. 504, on the authority of Ibn al-Adamī, astronomer, in a + work published by his continuator Al-Qāsim in 920 <span + class="scac">A.D.</span>; Al-Bīrūnī, <i>India, </i>Vol. + II, p. 15.</p> + + <p><a name="Nt_366" href="#NtA_366">[366]</a> H. Suter, <i>Die + Mathematiker</i> etc., pp. 4-5, states that Al-Fazārī died + between 796 and 806.</p> + + <p><a name="Nt_367" href="#NtA_367">[367]</a> Suter, loc. cit., p. + 63.</p> + + <p><a name="Nt_368" href="#NtA_368">[368]</a> Suter, loc. cit., p. + 74.</p> + + <p><a name="Nt_369" href="#NtA_369">[369]</a> Suter, <i>Das + Mathematiker-Verzeichniss im Fihrist</i>. The references to Suter, unless + otherwise stated, are to his later work <i>Die Mathematiker und + Astronomen der Araber</i> etc.</p> + + <p><a name="Nt_370" href="#NtA_370">[370]</a> Suter, <i>Fihrist</i>, p. + 37, no date.</p> + + <p><a name="Nt_371" href="#NtA_371">[371]</a> Suter, <i>Fihrist</i>, p. + 38, no date.</p> + + <p><a name="Nt_372" href="#NtA_372">[372]</a> Possibly late tenth, since + he refers to one arithmetical work which is entitled <i>Book of the + Cyphers</i> in his <i>Chronology</i>, English ed., p. 132. Suter, <i>Die + Mathematiker</i> etc., pp. 98-100, does not mention this work; see the + <i>Nachträge und Berichtigungen</i>, pp. 170-172.</p> + + <p><a name="Nt_373" href="#NtA_373">[373]</a> Suter, pp. 96-97.</p> + + <p><a name="Nt_374" href="#NtA_374">[374]</a> Suter, p. 111.</p> + + <p><a name="Nt_375" href="#NtA_375">[375]</a> Suter, p. 124. As the name + shows, he came from the West.</p> + + <p><a name="Nt_376" href="#NtA_376">[376]</a> Suter, p. 138.</p> + + <p><a name="Nt_377" href="#NtA_377">[377]</a> Hankel, <i>Zur Geschichte + der Mathematik</i>, p. 256, refers to him as writing on the Hindu art of + reckoning; Suter, p. 162.</p> + + <p><a name="Nt_378" href="#NtA_378">[378]</a> <span title="Psêphophoria kat' Indous" class="grk" + >Ψηφοφορία + κατ' Ἰνδούς</span>, + Greek ed., C. I. Gerhardt, Halle, 1865; and German translation, <i>Das + Rechenbuch des Maximus Planudes</i>, H. Wäschke, Halle, 1878.</p> + + <p><a name="Nt_379" href="#NtA_379">[379]</a> "Sur une donnée historique + relative à l'emploi des chiffres indiens par les Arabes," Tortolini's + <i>Annali di scienze mat. e fis.</i>, 1855.</p> + + <p><a name="Nt_380" href="#NtA_380">[380]</a> Suter, p. 80.</p> + + <p><a name="Nt_381" href="#NtA_381">[381]</a> Suter, p. 68.</p> + + <p><a name="Nt_382" href="#NtA_382">[382]</a> Sprenger also calls + attention to this fact, in the <i>Zeitschrift d. deutschen morgenländ. + Gesellschaft</i>, Vol. XLV, p. 367.</p> + + <p><a name="Nt_383" href="#NtA_383">[383]</a> Libri, <i>Histoire des + mathématiques</i>, Vol. I, p. 147.</p> + + <p><a name="Nt_384" href="#NtA_384">[384]</a> "Dictant la paix à + l'empereur de Constantinople, l'Arabe victorieux demandait des manuscrits + et des savans." [Libri, loc. cit., p. 108.]</p> + + <p><a name="Nt_385" href="#NtA_385">[385]</a> Persian <i>bagadata</i>, + "God-given."</p> + + <p><a name="Nt_386" href="#NtA_386">[386]</a> One of the Abbassides, the + (at least pretended) descendants of ‛Al-Abbās, uncle and + adviser of <span class="special" + title="Mohammed">Moḥammed</span>.</p> + + <p><a name="Nt_387" href="#NtA_387">[387]</a> E. Reclus, <i>Asia</i>, + American ed., N. Y., 1891, Vol. IV, p. 227.</p> + + <p><a name="Nt_388" href="#NtA_388">[388]</a> <i>Historical Sketches</i>, + Vol. III, chap. iii.</p> + + <p><a name="Nt_389" href="#NtA_389">[389]</a> On its prominence at that + period see Villicus, p. 70.</p> + + <p><a name="Nt_390" href="#NtA_390">[390]</a> See pp. 4-5.</p> + + <p><a name="Nt_391" href="#NtA_391">[391]</a> Smith, D. E., in the + <i>Cantor Festschrift</i>, 1909, note pp. 10-11. See also F. Woepcke, + <i>Propagation</i>.</p> + + <p><a name="Nt_392" href="#NtA_392">[392]</a> Eneström, in <i>Bibliotheca + Mathematica</i>, Vol. I (3), p. 499; Cantor, <i>Geschichte</i>, Vol. I + (3), p. 671.</p> + + <p><a name="Nt_393" href="#NtA_393">[393]</a> Cited in Chapter I. It + begins: "Dixit algoritmi: laudes deo rectori nostro atque defensori + dicamus dignas." It is devoted entirely to the fundamental operations and + contains no applications.</p> + + <p><a name="Nt_394" href="#NtA_394">[394]</a> M. Steinschneider, "Die + Mathematik bei den Juden," <i>Bibliotheca Mathematica</i>, Vol. VIII (2), + p. 99. See also the reference to this writer in Chapter I.</p> + + <p><a name="Nt_395" href="#NtA_395">[395]</a> Part of this work has been + translated from a Leyden MS. by F. Woepcke, <i>Propagation</i>, and more + recently by H. Suter, <i>Bibliotheca Mathematica</i>, Vol. VII (3), pp. + 113-119.</p> + + <p><a name="Nt_396" href="#NtA_396">[396]</a> A. Neander, <i>General + History of the Christian Religion and Church</i>, 5th American ed., + Boston, 1855, Vol. III, p. 335.</p> + + <p><a name="Nt_397" href="#NtA_397">[397]</a> Beazley, loc. cit., Vol. I, + p. 49.</p> + + <p><a name="Nt_398" href="#NtA_398">[398]</a> Beazley, loc. cit., Vol. I, + pp. 50, 460.</p> + + <p><a name="Nt_399" href="#NtA_399">[399]</a> See pp. <a + href="#page7">7</a>-<a href="#page8">8</a>.</p> + + <p><a name="Nt_400" href="#NtA_400">[400]</a> The name also appears as + <span class="special" title="Mohammed">Moḥammed</span> + Abū'l-Qāsim, and Ibn Hauqal. Beazley, loc. cit., Vol. I, p. + 45.</p> + + <p><a name="Nt_401" href="#NtA_401">[401]</a> <i>Kitāb + al-masālik wa'l-mamālik.</i></p> + + <p><a name="Nt_402" href="#NtA_402">[402]</a> Reinaud, <i>Mém. sur + l'Inde</i>; in Gerhardt, <i>Études</i>, p. 18.</p> + + <p><a name="Nt_403" href="#NtA_403">[403]</a> Born at Shiraz in 1193. He + himself had traveled from India to Europe.</p> + + <p><a name="Nt_404" href="#NtA_404">[404]</a> <i>Gulistan</i> (<i>Rose + Garden</i>), Gateway the third, XXII. Sir Edwin Arnold's translation, N. + Y., 1899, p. 177.</p> + + <p><a name="Nt_405" href="#NtA_405">[405]</a> Cunningham, loc. cit., p. + 81.</p> + + <p><a name="Nt_406" href="#NtA_406">[406]</a> Putnam, <i>Books</i>, Vol. + I, p. 227:</p> + + <div class="poem"> + <div class="stanza"> + <p class="hg3">"Non semel externas peregrino tramite terras</p> + <p>Jam peragravit ovans, sophiae deductus amore,</p> + <p>Si quid forte novi librorum seu studiorum</p> + <p>Quod secum ferret, terris reperiret in illis.</p> + <p>Hic quoque Romuleum venit devotus ad urbem."</p> + </div> + </div> + <p>("More than once he has traveled joyfully through remote regions and + by strange roads, led on by his zeal for knowledge and seeking to + discover in foreign lands novelties in books or in studies which he could + take back with him. And this zealous student journeyed to the city of + Romulus.")</p> + + <p><a name="Nt_407" href="#NtA_407">[407]</a> A. Neander, <i>General + History of the Christian Religion and Church</i>, 5th American ed., + Boston, 1855, Vol. III, p. 89, note 4; Libri, <i>Histoire</i>, Vol. I, p. + 143.</p> + + <p><a name="Nt_408" href="#NtA_408">[408]</a> Cunningham, loc. cit., p. + 81.</p> + + <p><a name="Nt_409" href="#NtA_409">[409]</a> Heyd, loc. cit., Vol. I, p. + 4.</p> + + <p><a name="Nt_410" href="#NtA_410">[410]</a> Ibid., p. 5.</p> + + <p><a name="Nt_411" href="#NtA_411">[411]</a> Ibid., p. 21.</p> + + <p><a name="Nt_412" href="#NtA_412">[412]</a> Ibid., p. 23.</p> + + <p><a name="Nt_413" href="#NtA_413">[413]</a> Libri, <i>Histoire</i>, + Vol. I, p. 167.</p> + + <p><a name="Nt_414" href="#NtA_414">[414]</a> Picavet, <i>Gerbert, un + pape philosophe, d'après l'histoire et d'après la légende</i>, Paris, + 1897, p. 19.</p> + + <p><a name="Nt_415" href="#NtA_415">[415]</a> Beazley, loc. cit., Vol. I, + chap, i, and p. 54 seq.</p> + + <p><a name="Nt_416" href="#NtA_416">[416]</a> Ibid., p. 57.</p> + + <p><a name="Nt_417" href="#NtA_417">[417]</a> Libri, <i>Histoire</i>, + Vol. I, p. 110, n., citing authorities, and p. 152.</p> + + <p><a name="Nt_418" href="#NtA_418">[418]</a> Possibly the old tradition, + "Prima dedit nautis usum magnetis Amalphis," is true so far as it means + the modern form of compass card. See Beazley, loc. cit., Vol. II, p. + 398.</p> + + <p><a name="Nt_419" href="#NtA_419">[419]</a> R. C. Dutt, loc. cit., Vol. + II, p. 312.</p> + + <p><a name="Nt_420" href="#NtA_420">[420]</a> E. J. Payne, in <i>The + Cambridge Modern History</i>, London, 1902, Vol. I, chap. i.</p> + + <p><a name="Nt_421" href="#NtA_421">[421]</a> Geo. Phillips, "The + Identity of Marco Polo's Zaitun with Changchau, in T'oung pao," + <i>Archives pour servir à l'étude de l'histoire de l'Asie orientale</i>, + Leyden, 1890, Vol. I, p. 218. W. Heyd, <i>Geschichte des Levanthandels im + Mittelalter</i>, Vol. II, p. 216.</p> + + <p>The Palazzo dei Poli, where Marco was born and died, still stands in + the Corte del Milione, in Venice. The best description of the Polo + travels, and of other travels of the later Middle Ages, is found in C. R. + Beazley's <i>Dawn of Modern Geography</i>, Vol. III, chap, ii, and Part + II.</p> + + <p><a name="Nt_422" href="#NtA_422">[422]</a> Heyd, loc. cit., Vol. II, + p. 220; H. Yule, in <i>Encyclopædia Britannica</i>, 9th (10th) or 11th + ed., article "China." The handbook cited is Pegolotti's <i>Libro di + divisamenti di paesi</i>, chapters i-ii, where it is implied that $60,000 + would be a likely amount for a merchant going to China to invest in his + trip.</p> + + <p><a name="Nt_423" href="#NtA_423">[423]</a> Cunningham, loc. cit., p. + 194.</p> + + <p><a name="Nt_424" href="#NtA_424">[424]</a> I.e. a commission + house.</p> + + <p><a name="Nt_425" href="#NtA_425">[425]</a> Cunningham, loc. cit., p. + 186.</p> + + <p><a name="Nt_426" href="#NtA_426">[426]</a> J. R. Green, <i>Short + History of the English People</i>, New York, 1890, p. 66.</p> + + <p><a name="Nt_427" href="#NtA_427">[427]</a> W. Besant, <i>London</i>, + New York, 1892, p. 43.</p> + + <p><a name="Nt_428" href="#NtA_428">[428]</a> <i>Baldakin</i>, + <i>baldekin</i>, <i>baldachino</i>.</p> + + <p><a name="Nt_429" href="#NtA_429">[429]</a> Italian + <i>Baldacco</i>.</p> + + <p><a name="Nt_430" href="#NtA_430">[430]</a> J. K. Mumford, <i>Oriental + Rugs</i>, New York, 1901, p. 18.</p> + + <p><a name="Nt_431" href="#NtA_431">[431]</a> Or Girbert, the Latin forms + <i>Gerbertus</i> and <i>Girbertus</i> appearing indifferently in the + documents of his time.</p> + + <p><a name="Nt_432" href="#NtA_432">[432]</a> See, for example, J. C. + Heilbronner, <i>Historia matheseos universæ</i>, p. 740.</p> + + <p><a name="Nt_433" href="#NtA_433">[433]</a> "Obscuro loco natum," as an + old chronicle of Aurillac has it.</p> + + <p><a name="Nt_434" href="#NtA_434">[434]</a> N. Bubnov, <i>Gerberti + postea Silvestri II papae opera mathematica</i>, Berlin, 1899, is the + most complete and reliable source of information; Picavet, loc. cit., + <i>Gerbert</i> etc.; Olleris, <i>Œuvres de Gerbert</i>, Paris, + 1867; Havet, <i>Lettres de Gerbert</i>, Paris, 1889 ; H. Weissenborn, + <i>Gerbert; Beiträge zur Kenntnis der Mathematik des Mittelalters</i>, + Berlin, 1888, and <i>Zur Geschichte der Einführung der jetzigen Ziffern + in Europa durch Gerbert</i>, Berlin, 1892; Büdinger, <i>Ueber Gerberts + wissenschaftliche und politische Stellung</i>, Cassel, 1851; Richer, + "Historiarum liber III," in Bubnov, loc. cit., pp. 376-381; Nagl, + <i>Gerbert und die Rechenkunst des 10. Jahrhunderts</i>, Vienna, + 1888.</p> + + <p><a name="Nt_435" href="#NtA_435">[435]</a> Richer tells of the visit + to Aurillac by Borel, a Spanish nobleman, just as Gerbert was entering + into young manhood. He relates how affectionately the abbot received him, + asking if there were men in Spain well versed in the arts. Upon Borel's + reply in the affirmative, the abbot asked that one of his young men might + accompany him upon his return, that he might carry on his studies + there.</p> + + <p><a name="Nt_436" href="#NtA_436">[436]</a> Vicus Ausona. Hatto also + appears as Atton and Hatton.</p> + + <p><a name="Nt_437" href="#NtA_437">[437]</a> This is all that we know of + his sojourn in Spain, and this comes from his pupil Richer. The stories + told by Adhemar of Chabanois, an apparently ignorant and certainly + untrustworthy contemporary, of his going to Cordova, are unsupported. + (See e.g. Picavet, p. 34.) Nevertheless this testimony is still accepted: + K. von Raumer, for example (<i>Geschichte der Pädagogik</i>, 6th ed., + 1890, Vol. I, p. 6), says "Mathematik studierte man im Mittelalter bei + den Arabern in Spanien. Zu ihnen gieng Gerbert, nachmaliger Pabst + Sylvester II."</p> + + <p><a name="Nt_438" href="#NtA_438">[438]</a> Thus in a letter to + Aldaberon he says: "Quos post repperimus speretis, id est VIII volumina + Boeti de astrologia, praeclarissima quoque figurarum geometriæ, aliaque + non minus admiranda" (Epist. 8). Also in a letter to Rainard (Epist. + 130), he says: "Ex tuis sumptibus fac ut michi scribantur M. Manlius + (Manilius in one MS.) de astrologia."</p> + + <p><a name="Nt_439" href="#NtA_439">[439]</a> Picavet, loc. cit., p. + 31.</p> + + <p><a name="Nt_440" href="#NtA_440">[440]</a> Picavet, loc. cit., p. + 36.</p> + + <p><a name="Nt_441" href="#NtA_441">[441]</a> Havet, loc. cit., p. + vii.</p> + + <p><a name="Nt_442" href="#NtA_442">[442]</a> Picavet, loc. cit., p. + 37.</p> + + <p><a name="Nt_443" href="#NtA_443">[443]</a> "Con sinistre arti + conseguri la dignita del Pontificato.... Lasciato poi l' abito, e 'l + monasterio, e datosi tutto in potere del diavolo." [Quoted in Bombelli, + <i>L'antica numerazione Italica</i>, Rome, 1876, p. 41 n.]</p> + + <p><a name="Nt_444" href="#NtA_444">[444]</a> He writes from Rheims in + 984 to one Lupitus, in Barcelona, saying: "Itaque librum de astrologia + translatum a te michi petenti dirige," presumably referring to some + Arabic treatise. [Epist. no. 24 of the Havet collection, p. 19.]</p> + + <p><a name="Nt_445" href="#NtA_445">[445]</a> See Bubnov, loc. cit., p. + x.</p> + + <p><a name="Nt_446" href="#NtA_446">[446]</a> Olleris, loc. cit., p. 361, + l. 15, for Bernelinus; and Bubnov, loc. cit., p. 381, l. 4, for + Richer.</p> + + <p><a name="Nt_447" href="#NtA_447">[447]</a> Woepcke found this in a + Paris MS. of Radulph of Laon, c. 1100. [<i>Propagation</i>, p. 246.] "Et + prima quidem trium spaciorum superductio unitatis caractere inscribitur, + qui chaldeo nomine dicitur igin." See also Alfred Nagl, "Der + arithmetische Tractat des Radulph von Laon" (<i>Abhandlungen zur + Geschichte der Mathematik</i>, Vol. V, pp. 85-133), p. 97.</p> + + <p><a name="Nt_448" href="#NtA_448">[448]</a> Weissenborn, loc. cit., p. + 239. When Olleris (<i>Œuvres de Gerbert</i>, Paris, 1867, p. cci) + says, "C'est à lui et non point aux Arabes, que l'Europe doit son système + et ses signes de numération," he exaggerates, since the evidence is all + against his knowing the place value. Friedlein emphasizes this in the + <i>Zeitschrift für Mathematik und Physik</i>, Vol. XII (1867), + <i>Literaturzeitung</i>, p. 70: "Für das <i>System</i> unserer Numeration + ist die <i>Null</i> das wesentlichste Merkmal, und diese kannte Gerbert + nicht. Er selbst schrieb alle Zahlen mit den römischen Zahlzeichen und + man kann ihm also nicht verdanken, was er selbst nicht kannte."</p> + + <p><a name="Nt_449" href="#NtA_449">[449]</a> E.g., Chasles, Büdinger, + Gerhardt, and Richer. So Martin (<i>Recherches nouvelles</i> etc.) + believes that Gerbert received them from Boethius or his followers. See + Woepcke, <i>Propagation</i>, p. 41.</p> + + <p><a name="Nt_450" href="#NtA_450">[450]</a> Büdinger, loc. cit., p. 10. + Nevertheless, in Gerbert's time one <span class="special" + title="Al-Mansur">Al-Manṣūr</span>, governing Spain under + the name of Hishām (976-1002), called from the Orient + Al-Beġānī to teach his son, so that scholars were + recognized. [Picavet, p. 36.]</p> + + <p><a name="Nt_451" href="#NtA_451">[451]</a> Weissenborn, loc. cit., p. + 235.</p> + + <p><a name="Nt_452" href="#NtA_452">[452]</a> Ibid., p. 234.</p> + + <p><a name="Nt_453" href="#NtA_453">[453]</a> These letters, of the + period 983-997, were edited by Havet, loc. cit., and, less completely, by + Olleris, loc. cit. Those touching mathematical topics were edited by + Bubnov, loc. cit., pp. 98-106.</p> + + <p><a name="Nt_454" href="#NtA_454">[454]</a> He published it in the + <i>Monumenta Germaniae historica</i>, "Scriptores," Vol. III, and at + least three other editions have since appeared, viz. those by Guadet in + 1845, by Poinsignon in 1855, and by Waitz in 1877.</p> + + <p><a name="Nt_455" href="#NtA_455">[455]</a> Domino ac beatissimo Patri + Gerberto, Remorum archiepiscopo, Richerus Monchus, Gallorum congressibus + in volumine regerendis, imperii tui, pater sanctissime Gerberte, + auctoritas seminarium dedit.</p> + + <p><a name="Nt_456" href="#NtA_456">[456]</a> In epistle 17 (Havet + collection) he speaks of the "De multiplicatione et divisione numerorum + libellum a Joseph Ispano editum abbas Warnerius" (a person otherwise + unknown). In epistle 25 he says: "De multiplicatione et divisione + numerorum, Joseph Sapiens sententias quasdam edidit."</p> + + <p><a name="Nt_457" href="#NtA_457">[457]</a> H. Suter, "Zur Frage über + den Josephus Sapiens," <i>Bibliotheca Mathematica</i>, Vol. VIII (2), p. + 84; Weissenborn, <i>Einführung</i>, p. 14; also his <i>Gerbert</i>; M. + Steinschneider, in <i>Bibliotheca Mathematica</i>, 1893, p. 68. Wallis + (<i>Algebra</i>, 1685, chap. 14) went over the list of Spanish Josephs + very carefully, but could find nothing save that "Josephus Hispanus seu + Josephus sapiens videtur aut Maurus fuisse aut alius quis in + Hispania."</p> + + <p><a name="Nt_458" href="#NtA_458">[458]</a> P. Ewald, <i>Mittheilungen, + Neues Archiv d. Gesellschaft für ältere deutsche Geschichtskunde</i>, + Vol. VIII, 1883, pp. 354-364. One of the manuscripts is of 976 <span + class="scac">A.D.</span> and the other of 992 <span + class="scac">A.D.</span> See also Franz Steffens, <i>Lateinische + Paläographie</i>, Freiburg (Schweiz), 1903, pp. xxxix-xl. The forms are + reproduced in the plate on page 140.</p> + + <p><a name="Nt_459" href="#NtA_459">[459]</a> It is entitled + <i>Constantino suo Gerbertus scolasticus</i>, because it was addressed to + Constantine, a monk of the Abbey of Fleury. The text of the letter to + Constantine, preceding the treatise on the Abacus, is given in the + <i>Comptes rendus</i>, Vol. XVI (1843), p. 295. This book seems to have + been written c. 980 <span class="scac">A.D.</span> [Bubnov, loc. cit., p. + 6.]</p> + + <p><a name="Nt_460" href="#NtA_460">[460]</a> "Histoire de + l'Arithmétique," <i>Comptes rendus</i>, Vol. XVI (1843), pp. 156, + 281.</p> + + <p><a name="Nt_461" href="#NtA_461">[461]</a> Loc. cit., <i>Gerberti + Opera</i> etc.</p> + + <p><a name="Nt_462" href="#NtA_462">[462]</a> Friedlein thought it + spurious. See <i>Zeitschrift für Mathematik und Physik</i>, Vol. XII + (1867), Hist.-lit. suppl., p. 74. It was discovered in the library of the + Benedictine monastry of St. Peter, at Salzburg, and was published by + Peter Bernhard Pez in 1721. Doubt was first cast upon it in the Olleris + edition (<i>Œuvres de Gerbert</i>). See Weissenborn, + <i>Gerbert</i>, pp. 2, 6, 168, and Picavet, p. 81. Hock, Cantor, and Th. + Martin place the composition of the work at c. 996 when Gerbert was in + Germany, while Olleris and Picavet refer it to the period when he was at + Rheims.</p> + + <p><a name="Nt_463" href="#NtA_463">[463]</a> Picavet, loc. cit., p. + 182.</p> + + <p><a name="Nt_464" href="#NtA_464">[464]</a> Who wrote after Gerbert + became pope, for he uses, in his preface, the words, "a domino pape + Gerberto." He was quite certainly not later than the eleventh century; we + do not have exact information about the time in which he lived.</p> + + <p><a name="Nt_465" href="#NtA_465">[465]</a> Picavet, loc. cit., p. 182. + Weissenborn, <i>Gerbert</i>, p. 227. In Olleris, <i>Liber Abaci</i> (of + Bernelinus), p. 361.</p> + + <p><a name="Nt_466" href="#NtA_466">[466]</a> Richer, in Bubnov, loc. + cit., p. 381.</p> + + <p><a name="Nt_467" href="#NtA_467">[467]</a> Weissenborn, + <i>Gerbert</i>, p. 241.</p> + + <p><a name="Nt_468" href="#NtA_468">[468]</a> Writers on numismatics are + quite uncertain as to their use. See F. Gnecchi, <i>Monete Romane</i>, 2d + ed., Milan, 1900, cap. XXXVII. For pictures of old Greek tesserae of + Sarmatia, see S. Ambrosoli, <i>Monete Greche</i>, Milan, 1899, p. + 202.</p> + + <p><a name="Nt_469" href="#NtA_469">[469]</a> Thus Tzwivel's arithmetic + of 1507, fol. 2, v., speaks of the ten figures as "characteres sive + numerorum apices a diuo Seuerino Boetio."</p> + + <p><a name="Nt_470" href="#NtA_470">[470]</a> Weissenborn uses + <i>sipos</i> for 0. It is not given by Bernelinus, and appears in Radulph + of Laon, in the twelfth century. See Günther's <i>Geschichte</i>, p. 98, + n.; Weissenborn, p. 11; Pihan, <i>Exposé</i> etc., pp. xvi-xxii.</p> + + <p>In Friedlein's <i>Boetius</i>, p. 396, the plate shows that all of the + six important manuscripts from which the illustrations are taken contain + the symbol, while four out of five which give the words use the word + <i>sipos</i> for 0. The names appear in a twelfth-century anonymous + manuscript in the Vatican, in a passage beginning</p> + + <div class="poem"> + <div class="stanza"> + <p>Ordine primigeno sibi nomen possidet igin.</p> + <p>Andras ecce locum mox uendicat ipse secundum</p> + <p>Ormis post numeros incompositus sibi primus.</p> + </div> + </div> + <p>[Boncompagni <i>Buttetino</i>, XV, p. 132.] Turchill (twelfth century) + gives the names Igin, andras, hormis, arbas, quimas, caletis, zenis, + temenias, celentis, saying: "Has autem figuras, ut donnus [dominus] + Gvillelmus Rx testatur, a pytagoricis habemus, nomina uero ab arabibus." + (Who the William R. was is not known. Boncompagni <i>Bulletino</i> XV, p. + 136.) Radulph of Laon (d. 1131) asserted that they were Chaldean + (<i>Propagation</i>, p. 48 n.). A discussion of the whole question is + also given in E. C. Bayley, loc. cit. Huet, writing in 1679, asserted + that they were of Semitic origin, as did Nesselmann in spite of his + despair over ormis, calctis, and celentis; see Woepcke, + <i>Propagation</i>, p. 48. The names were used as late as the fifteenth + century, without the zero, but with the superscript dot for 10's, two + dots for 100's, etc., as among the early Arabs. Gerhardt mentions having + seen a fourteenth or fifteenth century manuscript in the Bibliotheca + Amploniana with the names "Ingnin, andras, armis, arbas, quinas, calctis, + zencis, zemenias, zcelentis," and the statement "Si unum punctum super + ingnin ponitur, X significat.... Si duo puncta super ... figuras + superponunter, fiet decuplim illius quod cum uno puncto significabatur," + in <i>Monatsberichte der K. P. Akad. d. Wiss.</i>, Berlin, 1867, p. + 40.</p> + + <p><a name="Nt_471" href="#NtA_471">[471]</a> <i>A chart of ten numerals + in 200 tongues</i>, by Rev. R. Patrick, London, 1812.</p> + + <p><a name="Nt_472" href="#NtA_472">[472]</a> "Numeratio figuralis est + cuiusuis numeri per notas, et figuras numerates descriptio." + [Clichtoveus, edition of c. 1507, fol. C ii, v.] "Aristoteles enim uoces + rerum <span title="sumbola" class="grk" + >σύμβολα</span> uocat: id + translatum, sonat notas." [Noviomagus, <i>De Numeris Libri II</i>, cap. + vi.] "Alphabetum decem notarum." [Schonerus, notes to Ramus, 1586, p. 3 + seq.] Richer says: "novem numero notas omnem numerum significantes." + [Bubnov, loc. cit., p. 381.]</p> + + <p><a name="Nt_473" href="#NtA_473">[473]</a> "Il y a dix Characteres, + autrement Figures, Notes, ou Elements." [Peletier, edition of 1607, p. + 13.] "Numerorum notas alij figuras, alij signa, alij characteres uocant." + [Glareanus, 1545 edition, f. 9, r.] "Per figuras (quas zyphras uocant) + assignationem, quales sunt hæ notulæ, 1. 2. 3. 4...." [Noviomagus, <i>De + Numeris Libri II</i>, cap. vi.] Gemma Frisius also uses <i>elementa</i> + and Cardan uses <i>literae</i>. In the first arithmetic by an American + (Greenwood, 1729) the author speaks of "a few Arabian <i>Charecters</i> + or Numeral Figures, called <i>Digits</i>" (p. 1), and as late as 1790, in + the third edition of J. J. Blassière's arithmetic (1st ed. 1769), the + name <i>characters</i> is still in use, both for "de Latynsche en de + Arabische" (p. 4), as is also the term "Cyfferletters" (p. 6, n.). + <i>Ziffer</i>, the modern German form of cipher, was commonly used to + designate any of the nine figures, as by Boeschenstein and Riese, + although others, like Köbel, used it only for the zero. So <i>zifre</i> + appears in the arithmetic by Borgo, 1550 ed. In a Munich codex of the + twelfth century, attributed to Gerland, they are called <i>characters</i> + only: "Usque ad VIIII. enim porrigitur omnis numerus et qui supercrescit + eisdem designator Karacteribus." [Boncompagni <i>Bulletino</i>, Vol. X. + p. 607.]</p> + + <p><a name="Nt_474" href="#NtA_474">[474]</a> The title of his work is + <i>Prologus N. Ocreati in Helceph</i> (Arabic <i>al-qeif</i>, + investigation or memoir) <i>ad Adelardum Batensem magistrum suum</i>. The + work was made known by C. Henry, in the <i>Zeitschrift für Mathematik und + Physik</i>, Vol. XXV, p. 129, and in the <i>Abhandlungen zur Geschichte + der Mathematik</i>, Vol. III; Weissenborn, <i>Gerbert</i>, p. 188.</p> + + <p><a name="Nt_475" href="#NtA_475">[475]</a> The zero is indicated by a + vacant column.</p> + + <p><a name="Nt_476" href="#NtA_476">[476]</a> Leo Jordan, loc. cit., p. + 170. "Chifre en augorisme" is the expression used, while a century later + "giffre en argorisme" and "cyffres d'augorisme" are similarly used.</p> + + <p><a name="Nt_477" href="#NtA_477">[477]</a> <i>The Works of Geoffrey + Chaucer</i>, edited by W. W. Skeat, Vol. IV, Oxford, 1894, p. 92.</p> + + <p><a name="Nt_478" href="#NtA_478">[478]</a> Loc. cit., Vol. III, pp. + 179 and 180.</p> + + <p><a name="Nt_479" href="#NtA_479">[479]</a> In Book II, chap, vii, of + <i>The Testament of Love</i>, printed with Chaucer's Works, loc. cit., + Vol. VII, London, 1897.</p> + + <p><a name="Nt_480" href="#NtA_480">[480]</a> <i>Liber Abacci</i>, + published in Olleris, <i>Œuvres de Gerbert</i>, pp. 357-400.</p> + + <p><a name="Nt_481" href="#NtA_481">[481]</a> G. R. Kaye, "The Use of the + Abacus in Ancient India," <i>Journal and Proceedings of the Asiatic + Society of Bengal</i>, 1908, pp. 293-297.</p> + + <p><a name="Nt_482" href="#NtA_482">[482]</a> <i>Liber Abbaci</i>, by + Leonardo Pisano, loc. cit., p. 1.</p> + + <p><a name="Nt_483" href="#NtA_483">[483]</a> Friedlein, "Die + Entwickelung des Rechnens mit Columnen," <i>Zeitschrift für Mathematik + und Physik</i>, Vol. X, p. 247.</p> + + <p><a name="Nt_484" href="#NtA_484">[484]</a> The divisor 6 or 16 being + increased by the difference 4, to 10 or 20 respectively.</p> + + <p><a name="Nt_485" href="#NtA_485">[485]</a> E.g. Cantor, Vol. I, p. + 882.</p> + + <p><a name="Nt_486" href="#NtA_486">[486]</a> Friedlein, loc. cit.; + Friedlein, "Gerbert's Regeln der Division" and "Das Rechnen mit Columnen + vor dem 10. Jahrhundert," <i>Zeitschrift für Mathematik und Physik</i>, + Vol. IX; Bubnov, loc. cit., pp. 197-245; M. Chasles, "Histoire de + l'arithmétique. Recherches des traces du système de l'abacus, après que + cette méthode a pris le nom d'Algorisme.—Preuves qu'à toutes les + époques, jusq'au <span class="scac">XVI</span><sup>e</sup> siècle, on a + su que l'arithmétique vulgaire avait pour origine cette méthode + ancienne," <i>Comptes rendus</i>, Vol. XVII, pp. 143-154, also "Règles de + l'abacus," <i>Comptes rendus</i>, Vol. XVI, pp. 218-246, and "Analyse et + explication du traité de Gerbert," <i>Comptes rendus</i>, Vol. XVI, pp. + 281-299.</p> + + <p><a name="Nt_487" href="#NtA_487">[487]</a> Bubnov, loc. cit., pp. + 203-204, "Abbonis abacus."</p> + + <p><a name="Nt_488" href="#NtA_488">[488]</a> "Regulae de numerorum abaci + rationibus," in Bubnov, loc. cit., pp. 205-225.</p> + + <p><a name="Nt_489" href="#NtA_489">[489]</a> P. Treutlein, "Intorno ad + alcuni scritti inediti relativi al calcolo dell' abaco," <i>Bulletino di + bibliografia e di storia delle scienze matematiche e fisiche</i>, Vol. X, + pp. 589-647.</p> + + <p><a name="Nt_490" href="#NtA_490">[490]</a> "Intorno ad uno scritto + inedito di Adelhardo di Bath intitolato 'Regulae Abaci,'" B. Boncompagni, + in his <i>Bulletino</i>, Vol. XIV, pp. 1-134.</p> + + <p><a name="Nt_491" href="#NtA_491">[491]</a> Treutlein, loc. cit.; + Boncompagni, "Intorno al Tractatus de Abaco di Gerlando," + <i>Bulletino</i>, Vol. X, pp. 648-656.</p> + + <p><a name="Nt_492" href="#NtA_492">[492]</a> E. Narducci, "Intorno a due + trattati inediti d'abaco contenuti in due codici Vaticani del secolo + XII," Boncompagni <i>Bulletino</i>, Vol. XV, pp. 111-162.</p> + + <p><a name="Nt_493" href="#NtA_493">[493]</a> See Molinier, <i>Les + sources de l'histoire de France</i>, Vol. II, Paris, 1902, pp. 2, 3.</p> + + <p><a name="Nt_494" href="#NtA_494">[494]</a> Cantor, <i>Geschichte</i>, + Vol. I, p. 762. A. Nagl in the <i>Abhandlungen zur Geschichte der + Mathematik</i>, Vol. V, p. 85.</p> + + <p><a name="Nt_495" href="#NtA_495">[495]</a> 1030-1117.</p> + + <p><a name="Nt_496" href="#NtA_496">[496]</a> <i>Abhandlungen zur + Geschichte der Mathematik</i>, Vol. V, pp. 85-133. The work begins + "Incipit Liber Radulfi laudunensis de abaco."</p> + + <p><a name="Nt_497" href="#NtA_497">[497]</a> <i>Materialien zur + Geschichte der arabischen Zahlzeichen in Frankreich</i>, loc. cit.</p> + + <p><a name="Nt_498" href="#NtA_498">[498]</a> Who died in 1202.</p> + + <p><a name="Nt_499" href="#NtA_499">[499]</a> Cantor, <i>Geschichte</i>, + Vol. I (3), pp. 800-803; Boncompagni, <i>Trattati</i>, Part II. M. + Steinschneider ("Die Mathematik bei den Juden," <i>Bibliotheca + Mathematica</i>, Vol. X (2), p. 79) ingeniously derives another name by + which he is called (Abendeuth) from Ibn Daūd (Son of David). See + also <i>Abhandlungen</i>, Vol. III, p. 110.</p> + + <p><a name="Nt_500" href="#NtA_500">[500]</a> John is said to have died + in 1157.</p> + + <p><a name="Nt_501" href="#NtA_501">[501]</a> For it says, "Incipit + prologus in libro alghoarismi de practica arismetrice. Qui editus est a + magistro Johanne yspalensi." It is published in full in the second part + of Boncompagni's <i>Trattati d'aritmetica</i>.</p> + + <p><a name="Nt_502" href="#NtA_502">[502]</a> Possibly, indeed, the + meaning of "libro alghoarismi" is not "to Al-Khowārazmī's + book," but "to a book of algorism." John of Luna says of it: "Hoc idem + est illud etiam quod ... alcorismus dicere videtur." [<i>Trattati</i>, p. + 68.]</p> + + <p><a name="Nt_503" href="#NtA_503">[503]</a> For a résumé, see Cantor, + Vol. I (3), pp. 800-803. As to the author, see Eneström in the + <i>Bibliotheca Mathematica</i>, Vol. VI (3), p. 114, and Vol. IX (3), p. + 2.</p> + + <p><a name="Nt_504" href="#NtA_504">[504]</a> Born at Cremona (although + some have asserted at Carmona, in Andalusia) in 1114; died at Toledo in + 1187. Cantor, loc. cit.; Boncompagni, <i>Atti d. R. Accad. d. n. + Lincei</i>, 1851.</p> + + <p><a name="Nt_505" href="#NtA_505">[505]</a> See <i>Abhandlungen zur + Geschichte der Mathematik</i>, Vol. XIV, p. 149; <i>Bibliotheca + Mathematica</i>, Vol. IV (3), p. 206. Boncompagni had a + fourteenth-century manuscript of his work, <i>Gerardi Cremonensis artis + metrice practice</i>. See also T. L. Heath, <i>The Thirteen Books of + Euclid's Elements</i>, 3 vols., Cambridge, 1908, Vol. I, pp. 92-94 ; A. + A. Björnbo, "Gerhard von Cremonas Übersetzung von Alkwarizmis Algebra und + von Euklids Elementen," <i>Bibliotheca Mathematica</i>, Vol. VI (3), pp. + 239-248.</p> + + <p><a name="Nt_506" href="#NtA_506">[506]</a> Wallis, <i>Algebra</i>, + 1685, p. 12 seq.</p> + + <p><a name="Nt_507" href="#NtA_507">[507]</a> Cantor, <i>Geschichte</i>, + Vol. I (3), p. 906; A. A. Björnbo, "Al-Chwārizmī's + trigonometriske Tavler," <i>Festskrift til H. G. Zeuthen</i>, Copenhagen, + 1909, pp. 1-17.</p> + + <p><a name="Nt_508" href="#NtA_508">[508]</a> Heath, loc. cit., pp. + 93-96.</p> + + <p><a name="Nt_509" href="#NtA_509">[509]</a> M. Steinschneider, + <i>Zeitschrift der deutschen morgenländischen Gesellschaft</i>, Vol. XXV, + 1871, p. 104, and <i>Zeitschrift für Mathematik und Physik</i>, Vol. XVI, + 1871, pp. 392-393; M. Curtze, <i>Centralblatt für Bibliothekswesen</i>, + 1899, p. 289; E. Wappler, <i>Zur Geschichte der deutschen Algebra im 15. + Jahrhundert</i>, Programm, Zwickau, 1887; L. C. Karpinski, "Robert of + Chester's Translation of the Algebra of Al-Khowārazmī," + <i>Bibliotheca Mathematica</i>, Vol. XI (3), p. 125. He is also known as + Robertus Retinensis, or Robert of Reading.</p> + + <p><a name="Nt_510" href="#NtA_510">[510]</a> Nagl, A., "Ueber eine + Algorismus-Schrift des XII. Jahrhunderts und über die Verbreitung der + indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," + in the <i>Zeitschrift für Mathematik und Physik, Hist.-lit. Abth.</i>, + Vol. XXXIV, p. 129. Curtze, <i>Abhandlungen zur Geschichte der + Mathematik</i>, Vol. VIII, pp. 1-27.</p> + + <p><a name="Nt_511" href="#NtA_511">[511]</a> See line <i>a</i> in the + plate on p. <a href="#page143">143</a>.</p> + + <p><a name="Nt_512" href="#NtA_512">[512]</a> <i>Sefer ha-Mispar, Das + Buch der Zahl, ein hebräisch-arithmetisches Werk des R. Abraham ibn + Esra</i>, Moritz Silberberg, Frankfurt a. M., 1895.</p> + + <p><a name="Nt_513" href="#NtA_513">[513]</a> Browning's "Rabbi ben + Ezra."</p> + + <p><a name="Nt_514" href="#NtA_514">[514]</a> "Darum haben auch die + Weisen Indiens all ihre Zahlen durch neun bezeichnet und Formen für die 9 + Ziffern gebildet." [<i>Sefer ha-Mispar</i>, loc. cit., p. 2.]</p> + + <p><a name="Nt_515" href="#NtA_515">[515]</a> F. Bonaini, "Memoria unica + sincrona di Leonardo Fibonacci," Pisa, 1858, republished in 1867, and + appearing in the <i>Giornale Arcadico</i>, Vol. CXCVII (N.S. LII); + Gaetano Milanesi, <i>Documento inedito e sconosciuto a Lionardo + Fibonacci</i>, Roma, 1867; Guglielmini, <i>Elogio di Lionardo Pisano</i>, + Bologna, 1812, p. 35; Libri, <i>Histoire des sciences mathématiques</i>, + Vol. II, p. 25; D. Martines, <i>Origine e progressi dell' aritmetica</i>, + Messina, 1865, p. 47; Lucas, in Boncompagni <i>Bulletino</i>, Vol. X, pp. + 129, 239; Besagne, ibid., Vol. IX, p. 583; Boncompagni, three works as + cited in Chap. I; G. Eneström, "Ueber zwei angebliche mathematische + Schulen im christlichen Mittelalter," <i>Bibliotheca Mathematica</i>, + Vol. VIII (3), pp. 252-262; Boncompagni, "Della vita e delle opere di + Leonardo Pisano," loc. cit.</p> + + <p><a name="Nt_516" href="#NtA_516">[516]</a> The date is purely + conjectural. See the <i>Bibliotheca Mathematica</i>, Vol. IV (3), p. + 215.</p> + + <p><a name="Nt_517" href="#NtA_517">[517]</a> An old chronicle relates + that in 1063 Pisa fought a great battle with the Saracens at Palermo, + capturing six ships, one being "full of wondrous treasure," and this was + devoted to building the cathedral.</p> + + <p><a name="Nt_518" href="#NtA_518">[518]</a> Heyd, loc. cit., Vol. I, p. + 149.</p> + + <p><a name="Nt_519" href="#NtA_519">[519]</a> Ibid., p. 211.</p> + + <p><a name="Nt_520" href="#NtA_520">[520]</a> J. A. Symonds, + <i>Renaissance in Italy. The Age of Despots.</i> New York, 1883, p. + 62.</p> + + <p><a name="Nt_521" href="#NtA_521">[521]</a> Symonds, loc. cit., p. + 79.</p> + + <p><a name="Nt_522" href="#NtA_522">[522]</a> J. A. Froude, <i>The + Science of History</i>, London, 1864. "Un brevet d'apothicaire n'empêcha + pas Dante d'être le plus grand poète de l'Italie, et ce fut un petit + marchand de Pise qui donna l'algèbre aux Chrétiens." [Libri, + <i>Histoire</i>, Vol. I, p. xvi.]</p> + + <p><a name="Nt_523" href="#NtA_523">[523]</a> A document of 1226, found + and published in 1858, reads: "Leonardo bigollo quondam Guilielmi."</p> + + <p><a name="Nt_524" href="#NtA_524">[524]</a> "Bonaccingo germano + suo."</p> + + <p><a name="Nt_525" href="#NtA_525">[525]</a> E.g. Libri, Guglielmini, + Tiraboschi.</p> + + <p><a name="Nt_526" href="#NtA_526">[526]</a> Latin, + <i>Bonaccius</i>.</p> + + <p><a name="Nt_527" href="#NtA_527">[527]</a> Boncompagni and + Milanesi.</p> + + <p><a name="Nt_528" href="#NtA_528">[528]</a> Reprint, p. 5.</p> + + <p><a name="Nt_529" href="#NtA_529">[529]</a> Whence the French name for + candle.</p> + + <p><a name="Nt_530" href="#NtA_530">[530]</a> Now part of Algiers.</p> + + <p><a name="Nt_531" href="#NtA_531">[531]</a> E. Reclus, <i>Africa</i>, + New York, 1893, Vol. II, p. 253.</p> + + <p><a name="Nt_532" href="#NtA_532">[532]</a> "Sed hoc totum et + algorismum atque arcus pictagore quasi errorem computavi respectu modi + indorum." Woepcke, <i>Propagation</i> etc., regards this as referring to + two different systems, but the expression may very well mean algorism as + performed upon the Pythagorean arcs (or table).</p> + + <p><a name="Nt_533" href="#NtA_533">[533]</a> "Book of the Abacus," this + term then being used, and long afterwards in Italy, to mean merely the + arithmetic of computation.</p> + + <p><a name="Nt_534" href="#NtA_534">[534]</a> "Incipit liber Abaci a + Leonardo filio Bonacci compositus anno 1202 et correctus ab eodem anno + 1228." Three MSS. of the thirteenth century are known, viz. at Milan, at + Siena, and in the Vatican library. The work was first printed by + Boncompagni in 1857.</p> + + <p><a name="Nt_535" href="#NtA_535">[535]</a> I.e. in relation to the + quadrivium. "Non legant in festivis diebus, nisi Philosophos et + rhetoricas et quadrivalia et barbarismum et ethicam, si placet." Suter, + <i>Die Mathematik auf den Universitäten des Mittelalters</i>, Zürich, + 1887, p. 56. Roger Bacon gives a still more gloomy view of Oxford in his + time in his <i>Opus minus</i>, in the <i>Rerum Britannicarum medii aevi + scriptores</i>, London, 1859, Vol. I, p. 327. For a picture of Cambridge + at this time consult F. W. Newman, <i>The English Universities, + translated from the German of V. A. Huber</i>, London, 1843, Vol. I, p. + 61; W. W. R. Ball, <i>History of Mathematics at Cambridge</i>, 1889; S. + Günther, <i>Geschichte des mathematischen Unterrichts im deutschen + Mittelalter bis zum Jahre 1525</i>, Berlin, 1887, being Vol. III of + <i>Monumenta Germaniae paedagogica</i>.</p> + + <p><a name="Nt_536" href="#NtA_536">[536]</a> On the commercial activity + of the period, it is known that bills of exchange passed between Messina + and Constantinople in 1161, and that a bank was founded at Venice in + 1170, the Bank of San Marco being established in the following year. The + activity of Pisa was very manifest at this time. Heyd, loc. cit., Vol. + II, p. 5; V. Casagrandi, <i>Storia e cronologia</i>, 3d ed., Milan, 1901, + p. 56.</p> + + <p><a name="Nt_537" href="#NtA_537">[537]</a> J. A. Symonds, loc. cit., + Vol. II, p. 127.</p> + + <p><a name="Nt_538" href="#NtA_538">[538]</a> I. Taylor, <i>The + Alphabet</i>, London, 1883, Vol. II, p. 263.</p> + + <p><a name="Nt_539" href="#NtA_539">[539]</a> Cited by Unger's History, + p. 15. The Arabic numerals appear in a Regensburg chronicle of 1167 and + in Silesia in 1340. See Schmidt's <i>Encyclopädie der Erziehung</i>, Vol. + VI, p. 726; A. Kuckuk, "Die Rechenkunst im sechzehnten Jahrhundert," + <i>Festschrift zur dritten Säcularfeier des Berlinischen Gymnasiums zum + grauen Kloster</i>, Berlin, 1874, p. 4.</p> + + <p><a name="Nt_540" href="#NtA_540">[540]</a> The text is given in + Halliwell, <i>Rara Mathematica</i>, London, 1839.</p> + + <p><a name="Nt_541" href="#NtA_541">[541]</a> Seven are given in + Ashmole's <i>Catalogue of Manuscripts in the Oxford Library</i>, + 1845.</p> + + <p><a name="Nt_542" href="#NtA_542">[542]</a> Maximilian Curtze, <i>Petri + Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco + commentarius, una cum Algorismo ipso</i>, Copenhagen, 1897; L. C. + Karpinski, "Jordanus Nemorarius and John of Halifax," <i>American + Mathematical Monthly</i>, Vol. XVII, pp. 108-113.</p> + + <p><a name="Nt_543" href="#NtA_543">[543]</a> J. Aschbach, <i>Geschichte + der Wiener Universität im ersten Jahrhunderte ihres Bestehens</i>, Wien, + 1865, p. 93.</p> + + <p><a name="Nt_544" href="#NtA_544">[544]</a> Curtze, loc. cit., gives + the text.</p> + + <p><a name="Nt_545" href="#NtA_545">[545]</a> Curtze, loc. cit., found + some forty-five copies of the <i>Algorismus</i> in three libraries of + Munich, Venice, and Erfurt (Amploniana). Examination of two manuscripts + from the Plimpton collection and the Columbia library shows such marked + divergence from each other and from the text published by Curtze that the + conclusion seems legitimate that these were students' lecture notes. The + shorthand character of the writing further confirms this view, as it + shows that they were written largely for the personal use of the + writers.</p> + + <p><a name="Nt_546" href="#NtA_546">[546]</a> "Quidam philosophus edidit + nomine Algus, unde et Algorismus nuncupatur." [Curtze, loc. cit., p. + 1.]</p> + + <p><a name="Nt_547" href="#NtA_547">[547]</a> "Sinistrorsum autera + scribimus in hac arte more arabico sive iudaico, huius scientiae + inventorum." [Curtze, loc. cit., p. 7.] The Plimpton manuscript omits the + words "sive iudaico."</p> + + <p><a name="Nt_548" href="#NtA_548">[548]</a> "Non enim omnis numerus per + quascumque figuras Indorum repraesentatur, sed tantum determinatus per + determinatam, ut 4 non per 5,..." [Curtze, loc. cit., p. 25.]</p> + + <p><a name="Nt_549" href="#NtA_549">[549]</a> C. Henry, "Sur les deux + plus anciens traités français d'algorisme et de géométrie," Boncompagni + <i>Bulletino</i>, Vol. XV, p. 49; Victor Mortet, "Le plus ancien traité + français d'algorisme," loc. cit.</p> + + <p><a name="Nt_550" href="#NtA_550">[550]</a> <i>L'État des sciences en + France, depute la mort du Roy Robert, arrivée en 1031, jusqu'à celle de + Philippe le Bel, arrivée en 1314</i>, Paris, 1741.</p> + + <p><a name="Nt_551" href="#NtA_551">[551]</a> <i>Discours sur l'état des + lettres en France au XIII<sup>e</sup> siecle</i>, Paris, 1824.</p> + + <p><a name="Nt_552" href="#NtA_552">[552]</a> <i>Aperçu historique</i>, + Paris, 1876 ed., p. 464.</p> + + <p><a name="Nt_553" href="#NtA_553">[553]</a> Ranulf Higden, a native of + the west of England, entered St. Werburgh's monastery at Chester in 1299. + He was a Benedictine monk and chronicler, and died in 1364. His + <i>Polychronicon</i>, a history in seven books, was printed by Caxton in + 1480.</p> + + <p><a name="Nt_554" href="#NtA_554">[554]</a> Trevisa's translation, + Higden having written in Latin.</p> + + <p><a name="Nt_555" href="#NtA_555">[555]</a> An illustration of this + feeling is seen in the writings of Prosdocimo de' Beldomandi (b. c. + 1370-1380, d. 1428): "Inveni in quam pluribus libris algorismi nuncupatis + mores circa numeros operandi satis varios atque diversos, qui licet boni + existerent atque veri erant, tamen fastidiosi, tum propter ipsarum + regularum multitudinem, tum propter earum deleationes, tum etiam propter + ipsarum operationum probationes, utrum si bone fuerint vel ne. Erant et + etiam isti modi interim fastidiosi, quod si in aliquo calculo astroloico + error contigisset, calculatorem operationem suam a capite incipere + oportebat, dato quod error suus adhuc satis propinquus existeret; et hoc + propter figuras in sua operatione deletas. Indigebat etiam calculator + semper aliquo lapide vel sibi conformi, super quo scribere atque + faciliter delere posset figuras cum quibus operabatur in calculo suo. Et + quia haec omnia satis fastidiosa atque laboriosa mihi visa sunt, disposui + libellum edere in quo omnia ista abicerentur: qui etiam algorismus sive + liber de numeris denominari poterit. Scias tamen quod in hoc libello + ponere non intendo nisi ea quae ad calculum necessaria sunt, alia quae in + aliis libris practice arismetrice tanguntur, ad calculum non necessaria, + propter brevitatem dimitendo." [Quoted by A. Nagl, <i>Zeitschrift für + Mathematik und Physik, Hist.-lit. Abth.</i>, Vol. XXXIV, p. 143; Smith, + <i>Rara Arithmetica</i>, p. 14, in facsimile.]</p> + + <p><a name="Nt_556" href="#NtA_556">[556]</a> P. Ewald, loc. cit.; Franz + Steffens, <i>Lateinische Paläographie</i>, pp. xxxix-xl. We are indebted + to Professor J. M. Burnam for a photograph of this rare manuscript.</p> + + <p><a name="Nt_557" href="#NtA_557">[557]</a> See the plate of forms on + p. <a href="#page88">88</a>.</p> + + <p><a name="Nt_558" href="#NtA_558">[558]</a> Karabacek, loc. cit., p. + 56; Karpinski, "Hindu Numerals in the Fihrist," <i>Bibliotheca + Mathematica</i>, Vol. XI (3), p. 121.</p> + + <p><a name="Nt_559" href="#NtA_559">[559]</a> Woepcke, "Sur une donnée + historique," etc., loc. cit., and "Essai d'une restitution de travaux + perdus d'Apollonius sur les quantités irrationnelles, d'après des + indications tirées d'un manuscrit arabe," <i>Tome XIV des Mémoires + présentés par divers savants à l'Académie des sciences</i>, Paris, 1856, + note, pp. 6-14.</p> + + <p><a name="Nt_560" href="#NtA_560">[560]</a> <i>Archeological Report of + the Egypt Exploration Fund for 1908-1909</i>, London, 1910, p. 18.</p> + + <p><a name="Nt_561" href="#NtA_561">[561]</a> There was a set of + astronomical tables in Boncompagni's library bearing this date: "Nota + quod anno d<span class="over">n</span>i <span class="over">n</span>ri ihû + x<span class="over">p</span>i. 1264. perfecto." See Narducci's + <i>Catalogo</i>, p. 130.</p> + + <p><a name="Nt_562" href="#NtA_562">[562]</a> "On the Early use of Arabic + Numerals in Europe," read before the Society of Antiquaries April 14, + 1910, and published in <i>Archæologia</i> in the same year.</p> + + <p><a name="Nt_563" href="#NtA_563">[563]</a> Ibid., p. 8, n. The date is + part of an Arabic inscription.</p> + + <p><a name="Nt_564" href="#NtA_564">[564]</a> O. Codrington, <i>A Manual + of Musalman Numismatics</i>, London, 1904.</p> + + <p><a name="Nt_565" href="#NtA_565">[565]</a> See Arbuthnot, <i>The + Mysteries of Chronology</i>, London, 1900, pp. 75, 78, 98; F. Pichler, + <i>Repertorium der steierischen Münzkunde</i>, Grätz, 1875, where the + claim is made of an Austrian coin of 1458; <i>Bibliotheca + Mathematica</i>, Vol. X (2), p. 120, and Vol. XII (2), p. 120. There is a + Brabant piece of 1478 in the collection of D. E. Smith.</p> + + <p><a name="Nt_566" href="#NtA_566">[566]</a> A specimen is in the + British Museum. [Arbuthnot, p. 79.]</p> + + <p><a name="Nt_567" href="#NtA_567">[567]</a> Ibid., p. 79.</p> + + <p><a name="Nt_568" href="#NtA_568">[568]</a> <i>Liber de Remediis + utriusque fortunae Coloniae.</i></p> + + <p><a name="Nt_569" href="#NtA_569">[569]</a> Fr. Walthern et Hans + Hurning, Nördlingen.</p> + + <p><a name="Nt_570" href="#NtA_570">[570]</a> <i>Ars Memorandi</i>, one + of the oldest European block-books.</p> + + <p><a name="Nt_571" href="#NtA_571">[571]</a> Eusebius Caesariensis, + <i>De praeparatione evangelica</i>, Venice, Jenson, 1470. The above + statement holds for copies in the Astor Library and in the Harvard + University Library.</p> + + <p><a name="Nt_572" href="#NtA_572">[572]</a> Francisco de Retza, + <i>Comestorium vitiorum</i>, Nürnberg, 1470. The copy referred to is in + the Astor Library.</p> + + <p><a name="Nt_573" href="#NtA_573">[573]</a> See Mauch, "Ueber den + Gebrauch arabischer Ziffern und die Veränderungen derselben," <i>Anzeiger + für Kunde der deutschen Vorzeit</i>, 1861, columns 46, 81, 116, 151, 189, + 229, and 268; Calmet, <i>Recherches sur l'origine des chiffres + d'arithmétique</i>, plate, loc. cit.</p> + + <p><a name="Nt_574" href="#NtA_574">[574]</a> Günther, <i>Geschichte</i>, + p. 175, n.; Mauch, loc. cit.</p> + + <p><a name="Nt_575" href="#NtA_575">[575]</a> These are given by W. R. + Lethaby, from drawings by J. T. Irvine, in the <i>Proceedings of the + Society of Antiquaries</i>, 1906, p. 200.</p> + + <p><a name="Nt_576" href="#NtA_576">[576]</a> There are some + ill-tabulated forms to be found in J. Bowring, <i>The Decimal System</i>, + London, 1854, pp. 23, 25, and in L. A. Chassant, <i>Dictionnaire des + abréviations latines et françaises ... du moyen âge</i>, Paris, <span + class="scac">MDCCCLXVI</span>, p. 113. The best sources we have at + present, aside from the Hill monograph, are P. Treutlein, <i>Geschichte + unserer Zahlzeichen</i>, Karlsruhe, 1875; Cantor's <i>Geschichte</i>, + Vol. I, table; M. Prou, <i>Manuel de paléographie latine et + française</i>, 2d ed., Paris, 1892, p. 164; A. Cappelli, <i>Dizionario di + abbreviature latine ed italiane</i>, Milan, 1899. An interesting early + source is found in the rare Caxton work of 1480, <i>The Myrrour of the + World</i>. In Chap. X is a cut with the various numerals, the chapter + beginning "The fourth scyence is called arsmetrique." Two of the fifteen + extant copies of this work are at present in the library of Mr. J. P. + Morgan, in New York.</p> + + <p><a name="Nt_577" href="#NtA_577">[577]</a> From the twelfth-century + manuscript on arithmetic, Curtze, loc. cit., <i>Abhandlungen</i>, and + Nagl, loc. cit. The forms are copied from Plate VII in <i>Zeitschrift für + Mathematik und Physik</i>, Vol. XXXIV.</p> + + <p><a name="Nt_578" href="#NtA_578">[578]</a> From the Regensburg + chronicle. Plate containing some of these numerals in <i>Monumenta + Germaniae historica</i>, "Scriptores" Vol. XVII, plate to p. 184; + Wattenbach, <i>Anleitung zur lateinischen Palaeographie</i>, Leipzig, + 1886, p. 102; Boehmer, <i>Fontes rerum Germanicarum</i>, Vol. III, + Stuttgart, 1852, p. lxv.</p> + + <p><a name="Nt_579" href="#NtA_579">[579]</a> French Algorismus of 1275; + from an unpublished photograph of the original, in the possession of D. + E. Smith. See also p. 135.</p> + + <p><a name="Nt_580" href="#NtA_580">[580]</a> From a manuscript of + Boethius c. 1294, in Mr. Plimpton's library. Smith, <i>Rara + Arithmetica</i>, Plate I.</p> + + <p><a name="Nt_581" href="#NtA_581">[581]</a> Numerals in a 1303 + manuscript in Sigmaringen, copied from Wattenbach, loc. cit., p. 102.</p> + + <p><a name="Nt_582" href="#NtA_582">[582]</a> From a manuscript, Add. + Manuscript 27,589, British Museum, 1360 <span class="scac">A.D.</span> + The work is a computus in which the date 1360 appears, assigned in the + British Museum catalogue to the thirteenth century.</p> + + <p><a name="Nt_583" href="#NtA_583">[583]</a> From the copy of + Sacrabosco's <i>Algorismus</i> in Mr. Plimpton's library. Date c. 1442. + See Smith, <i>Rara Arithmetica</i>, p. 450.</p> + + <p><a name="Nt_584" href="#NtA_584">[584]</a> See <i>Rara + Arithmetica</i>, pp. 446-447.</p> + + <p><a name="Nt_585" href="#NtA_585">[585]</a> Ibid., pp. 469-470.</p> + + <p><a name="Nt_586" href="#NtA_586">[586]</a> Ibid., pp. 477-478.</p> + + <p><a name="Nt_587" href="#NtA_587">[587]</a> The i is used for "one" in + the Treviso arithmetic (1478), Clichtoveus (c. 1507 ed., where both i and + j are so used), Chiarini (1481), Sacrobosco (1488 ed.), and Tzwivel (1507 + ed., where jj and jz are used for 11 and 12). This was not universal, + however, for the <i>Algorithmus linealis</i> of c. 1488 has a special + type for 1. In a student's notebook of lectures taken at the University + of Würzburg in 1660, in Mr. Plimpton's library, the ones are all in the + form of i.</p> + + <p><a name="Nt_588" href="#NtA_588">[588]</a> Thus the date <a + href="images/151c.png"><img src="images/151c.png" class="middle" + style="height:2ex" alt="Numerals 1580" /></a>, for 1580, appears in a MS. + in the Laurentian library at Florence. The second and the following five + characters are taken from Cappelli's <i>Dizionario</i>, p. 380, and are + from manuscripts of the twelfth, thirteenth, fourteenth, sixteenth, + seventeenth, and eighteenth centuries, respectively.</p> + + <p><a name="Nt_589" href="#NtA_589">[589]</a> E.g. Chiarini's work of + 1481; Clichtoveus (c. 1507).</p> + + <p><a name="Nt_590" href="#NtA_590">[590]</a> The first is from an + algorismus of the thirteenth century, in the Hannover Library. [See + Gerhardt, "Ueber die Entstehung und Ausbreitung des dekadischen + Zahlensystems," loc. cit., p. 28.] The second character is from a French + algorismus, c. 1275. [Boncompagni <i>Bulletino</i>, Vol. XV, p. 51.] The + third and the following sixteen characters are given by Cappelli, loc. + cit., and are from manuscripts of the twelfth (1), thirteenth (2), + fourteenth (7), fifteenth (3), sixteenth (1), seventeenth (2), and + eighteenth (1) centuries, respectively.</p> + + <p><a name="Nt_591" href="#NtA_591">[591]</a> Thus Chiarini (1481) has <a + href="images/152j.png"><img src="images/152j.png" class="middle" + style="height:1.5ex" alt="Symbol" /></a> for 23.</p> + + <p><a name="Nt_592" href="#NtA_592">[592]</a> The first of these is from + a French algorismus, c. 1275. The second and the following eight + characters are given by Cappelli, loc. cit., and are from manuscripts of + the twelfth (2), thirteenth, fourteenth, fifteenth (3), seventeenth, and + eighteenth centuries, respectively.</p> + + <p><a name="Nt_593" href="#NtA_593">[593]</a> See Nagl, loc. cit.</p> + + <p><a name="Nt_594" href="#NtA_594">[594]</a> Hannover algorismus, + thirteenth century.</p> + + <p><a name="Nt_595" href="#NtA_595">[595]</a> See the Dagomari + manuscript, in <i>Rara Arithmetica</i>, pp. 435, 437-440.</p> + + <p><a name="Nt_596" href="#NtA_596">[596]</a> But in the woodcuts of the + <i>Margarita Philosophica</i> (1503) the old forms are used, although the + new ones appear in the text. In Caxton's <i>Myrrour of the World</i> + (1480) the old form is used.</p> + + <p><a name="Nt_597" href="#NtA_597">[597]</a> Cappelli, loc. cit. They + are partly from manuscripts of the tenth, twelfth, thirteenth (3), + fourteenth (7), fifteenth (6), and eighteenth centuries, respectively. + Those in the third line are from Chassant's <i>Dictionnaire</i>, p. 113, + without mention of dates.</p> + + <p><a name="Nt_598" href="#NtA_598">[598]</a> The first is from the + Hannover algorismus, thirteenth century. The second is taken from the + Rollandus manuscript, 1424. The others in the first two lines are from + Cappelli, twelfth (3), fourteenth (6), fifteenth (13) centuries, + respectively. The third line is from Chassant, loc. cit., p. 113, no + mention of dates.</p> + + <p><a name="Nt_599" href="#NtA_599">[599]</a> The first of these forms is + from the Hannover algorismus, thirteenth century. The following are from + Cappelli, fourteenth (3), fifteenth, sixteenth (2), and eighteenth + centuries, respectively.</p> + + <p><a name="Nt_600" href="#NtA_600">[600]</a> The first of these is taken + from the Hannover algorismus, thirteenth century. The following forms are + from Cappelli, twelfth, thirteenth, fourteenth (5), fifteenth (2), + seventeenth, and eighteenth centuries, respectively.</p> + + <p><a name="Nt_601" href="#NtA_601">[601]</a> All of these are given by + Cappelli, thirteenth, fourteenth, fifteenth (2), and sixteenth centuries, + respectively.</p> + + <p><a name="Nt_602" href="#NtA_602">[602]</a> Smith, <i>Rara + Arithmetica</i>, p. 489. This is also seen in several of the Plimpton + manuscripts, as in one written at Ancona in 1684. See also Cappelli, loc. + cit.</p> + + <p><a name="Nt_603" href="#NtA_603">[603]</a> French algorismus, c. 1275, + for the first of these forms. Cappelli, thirteenth, fourteenth, fifteenth + (3), and seventeenth centuries, respectively. The last three are taken + from <i>Byzantinische Analekten</i>, J. L. Heiberg, being forms of the + fifteenth century, but not at all common. <a href="images/154e.png"><img + src="images/154e.png" class="middle" style="height:1.8ex" alt="Symbol: + Qoppa" /></a> was the old Greek symbol for 90.</p> + + <p><a name="Nt_604" href="#NtA_604">[604]</a> For the first of these the + reader is referred to the forms ascribed to Boethius, in the illustration + on p. <a href="#page88">88</a>; for the second, to Radulph of Laon, see + p. <a href="#Nt_232">60</a>. The third is used occasionally in the + Rollandus (1424) manuscript, in Mr. Plimpton's library. The remaining + three are from Cappelli, fourteenth (2) and seventeenth centuries.</p> + + <p><a name="Nt_605" href="#NtA_605">[605]</a> Smith, <i>An Early English + Algorism</i>.</p> + + <p><a name="Nt_606" href="#NtA_606">[606]</a> Kuckuck, p. 5.</p> + + <p><a name="Nt_607" href="#NtA_607">[607]</a> A. Cappelli, loc. cit., p. + 372.</p> + + <p><a name="Nt_608" href="#NtA_608">[608]</a> Smith, <i>Rara + Arithmetica</i>, p. 443.</p> + + <p><a name="Nt_609" href="#NtA_609">[609]</a> Curtze, <i>Petri Philomeni + de Dacia</i> etc., p. <span class="scac">IX</span>.</p> + + <p><a name="Nt_610" href="#NtA_610">[610]</a> Cappelli, loc. cit., p. + 376.</p> + + <p><a name="Nt_611" href="#NtA_611">[611]</a> Curtze, loc. cit., pp. + <span class="scac">VIII-IX</span>, note.</p> + + <p><a name="Nt_612" href="#NtA_612">[612]</a> Edition of 1544-1545, f. + 52.</p> + + <p><a name="Nt_613" href="#NtA_613">[613]</a> <i>De numeris libri II</i>, + 1544 ed., cap. <span class="scac">XV</span>. Heilbronner, loc. cit., p. + 736, also gives them, and compares this with other systems.</p> + + <p><a name="Nt_614" href="#NtA_614">[614]</a> Noviomagus says of them: + "De quibusdam Astrologicis, sive Chaldaicis numerorum notis.... Sunt + & aliæ quædam notæ, quibus Chaldaei & Astrologii quemlibet + numerum artificiose & arguté describunt, scitu periucundae, quas + nobis communicauit Rodolphus Paludanus Nouiomagus."</p> + +</div> + + + + + + + +<pre> + + + + + +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-h.htm or 22599-h.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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