{144}

[Ill.u.s.tration]

For the sake of further comparison, three ill.u.s.trations from works in Mr.

Plimpton's library, reproduced from the _Rara Arithmetica_, may be considered. The first is from a Latin ma.n.u.script on arithmetic,[584] of which the original was written at Paris in 1424 by Rollandus, a Portuguese physician, who prepared the work at the command of John of Lancaster, Duke of Bedford, at one time Protector of England and Regent of France, to whom the work is dedicated. The figures show the successive powers of 2. The second ill.u.s.tration is from Luca da Firenze's _Inprencipio darte dabacho_,[585] c. 1475, and the third is from an anonymous ma.n.u.script[586]

of about 1500.

[Ill.u.s.tration]

As to the forms of the numerals, fashion played a leading part until printing was invented. This tended to fix these forms, although in writing there is still a great variation, as witness the French 5 and the German 7 and 9. Even in printing there is not complete uniformity, {145} and it is often difficult for a foreigner to distinguish between the 3 and 5 of the French types.

[Ill.u.s.tration]

As to the particular numerals, the following are some of the forms to be found in the later ma.n.u.scripts and in the early printed books.

1. In the early printed books "one" was often i, perhaps to save types, just as some modern typewriters use the same character for l and 1.[587] In the ma.n.u.scripts the "one" appears in such forms as[588]

[Ill.u.s.tration]

2. "Two" often appears as z in the early printed books, 12 appearing as iz.[589] In the medieval ma.n.u.scripts the following forms are common:[590]

[Ill.u.s.tration]

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It is evident, from the early traces, that it is merely a cursive form for the primitive [2 horizontal strokes], just as 3 comes from [3 horizontal strokes], as in the N[=a]n[=a] Gh[=a]t inscriptions.

3. "Three" usually had a special type in the first printed books, although occasionally it appears as [Symbol].[591] In the medieval ma.n.u.scripts it varied rather less than most of the others. The following are common forms:[592]

[Ill.u.s.tration]

4. "Four" has changed greatly; and one of the first tests as to the age of a ma.n.u.script on arithmetic, and the place where it was written, is the examination of this numeral. Until the time of printing the most common form was [Symbol], although the Florentine ma.n.u.script of Leonard of Pisa's work has the form [Symbol];[593] but the ma.n.u.scripts show that the Florentine arithmeticians and astronomers rather early began to straighten the first of these forms up to forms like [Symbol][594] and [Symbol][594]

or [Symbol],[595] more closely resembling our own. The first printed books generally used our present form[596] with the closed top [Symbol], the open top used in writing ( [Symbol]) being {147} purely modern. The following are other forms of the four, from various ma.n.u.scripts:[597]

[Ill.u.s.tration]

5. "Five" also varied greatly before the time of printing. The following are some of the forms:[598]

[Ill.u.s.tration]

6. "Six" has changed rather less than most of the others. The chief variation has been in the slope of the top, as will be seen in the following:[599]

[Ill.u.s.tration]

7. "Seven," like "four," has a.s.sumed its present erect form only since the fifteenth century. In medieval times it appeared as follows:[600]

[Ill.u.s.tration]

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8. "Eight," like "six," has changed but little. In medieval times there are a few variants of interest as follows:[601]

[Ill.u.s.tration]

In the sixteenth century, however, there was manifested a tendency to write it [Symbol].[602]

9. "Nine" has not varied as much as most of the others. Among the medieval forms are the following:[603]

[Ill.u.s.tration]

0. The shape of the zero also had a varied history. The following are common medieval forms:[604]

[Ill.u.s.tration]

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 chotu, and had placed this before all learners of the positional system, there would have been little trouble. But the medieval {149} line-reckoning, where the lines stood for powers of 10 and the s.p.a.ces for half of such powers, did not lend itself to this comparison. Accordingly we find such labored explanations as the following, from _The Crafte of Nombrynge_:

"Euery of these figuris bitokens hym selfe & no more, yf he stonde in the first place of the rewele....

"If it stonde in the secunde place of the rewle, he betokens ten tymes hym selfe, as this figure 2 here 20 tokens ten tyme hym selfe, that is twenty, for he hym selfe betokens tweyne, & ten tymes twene is twenty. And for he stondis on the lyft side & in the secunde place, he betokens ten tyme hym selfe. And so go forth....

"Nil cifra significat sed dat signare sequenti. Expone this verse. A cifre tokens no[gh]t, bot he makes the figure to betoken that comes after hym more than he shuld & he were away, as thus 10. here the figure of one tokens ten, & yf the cifre were away & no figure byfore hym he schuld token bot one, for than he schuld stonde in the first place...."[605]

It would seem that a system that was thus used for dating doc.u.ments, coins, and monuments, would have been generally adopted much earlier than it was, particularly in those countries north of Italy where it did not come into general use until the sixteenth century. This, however, has been the fate of many inventions, as witness our neglect of logarithms and of contracted processes to-day.

As to Germany, the fifteenth century saw the rise of the new symbolism; the sixteenth century saw it slowly {150} gain the mastery; the seventeenth century saw it finally conquer the system that for two thousand years had dominated the arithmetic of business. Not a little of the success of the new plan was due to Luther's demand that all learning should go into the vernacular.[606]

During the transition period from the Roman to the Arabic numerals, various anomalous forms found place. For example, we have in the fourteenth century c[alpha] for 104;[607] 1000. 300. 80 et 4 for 1384;[608] and in a ma.n.u.script of the fifteenth century 12901 for 1291.[609] In the same century m. cccc. 8II appears for 1482,[610] while M^oCCCC^o50 (1450) and MCCCCXL6 (1446) are used by Theodoricus Ruffi about the same time.[611] To the next century belongs the form 1vojj for 1502. Even in Sfortunati's _Nuovo lume_[612] the use of ordinals is quite confused, the propositions on a single page being numbered "tertia," "4," and "V."

Although not connected with the Arabic numerals in any direct way, the medieval astrological numerals may here be mentioned. These are given by several early writers, but notably by Noviomagus (1539),[613] as follows[614]:

[Ill.u.s.tration]

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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 j.a.pan 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.

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INDEX

_Transcriber's note: many of the entries refer to footnotes linked from the page numbers given._

Abbo of Fleury, 122 'Abdall[=a]h ibn al-[H.]asan, 92 'Abdallat[=i]f ibn Y[=u]suf, 93 'Abdalq[=a]dir ibn 'Al[=i] al-Sakh[=a]w[=i], 6 Abenragel, 34 Abraham ibn Mer ibn Ezra, _see_ Rabbi ben Ezra Ab[=u] 'Al[=i] al-[H.]osein ibn S[=i]n[=a], 74 Ab[=u] 'l-[H.]asan, 93, 100 Ab[=u] 'l-Q[=a]sim, 92 Ab[=u] 'l-[T.]eiyib, 97 Ab[=u] Na[s.]r, 92 Ab[=u] Roshd, 113 Abu Sahl Dunash ibn Tamim, 65, 67 Adelhard of Bath, 5, 55, 97, 119, 123, 126 Adhemar of Chabanois, 111 A[h.]med al-Nasaw[=i], 98 A[h.]med ibn 'Abdall[=a]h, 9, 92 A[h.]med ibn Mo[h.]ammed, 94 A[h.]med ibn 'Omar, 93 Ak[s.]aras, 32 Ala.n.u.s ab Insulis, 124 Al-Ba[.g]d[=a]d[=i], 93 Al-Batt[=a]n[=i], 54 Albelda (Albaida) MS., 116 Albert, J., 62 Albert of York, 103 Al-B[=i]r[=u]n[=i], 6, 41, 49, 65, 92, 93 Alcuin, 103 Alexander the Great, 76 Alexander de Villa Dei, 11, 133 Alexandria, 64, 82 Al-Faz[=a]r[=i], 92 Alfred, 103 Algebra, etymology, 5 Algerian numerals, 68 Algorism, 97 Algorismus, 124, 126, 135 Algorismus cifra, 120 Al-[H.]a[s.][s.][=a]r, 65 'Al[=i] ibn Ab[=i] Bekr, 6 'Al[=i] ibn A[h.]med, 93, 98 Al-Kar[=a]b[=i]s[=i], 93 Al-Khow[=a]razm[=i], 4, 9, 10, 92, 97, 98, 125, 126 Al-Kind[=i], 10, 92 Almagest, 54 Al-Ma[.g]reb[=i], 93 Al-Ma[h.]all[=i], 6 Al-M[=a]m[=u]n, 10, 97 Al-Man[s.][=u]r, 96, 97 Al-Mas'[=u]d[=i], 7, 92 Al-Nad[=i]m, 9 Al-Nasaw[=i], 93, 98 Alphabetic numerals, 39, 40, 43 Al-Q[=a]sim, 92 Al-Qa.s.s, 94 Al-Sakh[=a]w[=i], 6 Al-[S.]ardaf[=i], 93 Al-Sijz[=i], 94 Al-S[=u]f[=i], 10, 92 Ambrosoli, 118 A[.n]kapalli, 43 Apices, 87, 117, 118 Arabs, 91-98 Arbuthnot, 141 {154} Archimedes, 15, 16 Arcus Pictagore, 122 Arjuna, 15 Arnold, E., 15, 102 Ars memorandi, 141 [=A]ryabha[t.]a, 39, 43, 44 Aryan numerals, 19 Aschbach, 134 Ashmole, 134 A['s]oka, 19, 20, 22, 81 A[s.]-[s.]ifr, 57, 58 Astrological numerals, 150 Atharva-Veda, 48, 49, 55 Augustus, 80 Averroes, 113 Avicenna, 58, 74, 113

Babylonian numerals, 28 Babylonian zero, 51 Bacon, R., 131 Bactrian numerals, 19, 30 Baeda, 2, 72 Bagdad, 4, 96 Bakh[s.][=a]l[=i] ma.n.u.script, 43, 49, 52, 53 Ball, C. J., 35 Ball, W. W. R., 36, 131 B[=a][n.]a, 44 Barth, A., 39 Bayang inscriptions, 39 Bayer, 33 Bayley, E. C., 19, 23, 30, 32, 52, 89 Beazley, 75 Bede, _see_ Baeda Beldomandi, 137 Beloch, J., 77 Bendall, 25, 52 Benfey, T., 26 Bernelinus, 88, 112, 117, 121 Besagne, 128 Besant, W., 109 Bettino, 36 Bhandarkar, 18, 47, 49 Bh[=a]skara, 53, 55 Biernatzki, 32 Biot, 32 Bjornbo, A. A., 125, 126 Bla.s.siere, 119 Bloomfield, 48 Blume, 85 Boeckh, 62 Boehmer, 143 Boeschenstein, 119 Boethius, 63, 70-73, 83-90 Boissiere, 63 Bombelli, 81 Bonaini, 128 Boncompagni, 5, 6, 10, 48, 49, 123, 125 Borghi, 59 Borgo, 119 Bougie, 130 Bowring, J., 56 Brahmagupta, 52 Br[=a]hma[n.]as, 12, 13 Br[=a]hm[=i], 19, 20, 31, 83 Brandis, J., 54 B[r.]hat-Sa[m.]hita, 39, 44, 78 Brockhaus, 43 Bubnov, 65, 84, 110, 116 Buddha, education of, 15, 16 Budinger, 110 Bugia, 130 Buhler, G., 15, 19, 22, 31, 44, 49 Burgess, 25 Burk, 13 Burmese numerals, 36 Burnell, A. C., 18, 40 Buteo, 61