Since [=A]ryabha[t.]a gives our common method of extracting roots, it would seem that he may have known a decimal notation,[148] although he did not use the characters from which our numerals are derived.[149] Moreover, he frequently speaks of the {44} void.[150] If he refers to a symbol this would put the zero as far back as 500 A.D., but of course he may have referred merely to the concept of nothingness.

A little later, but also in the sixth century, Var[=a]ha-Mihira[151] wrote a work ent.i.tled _B[r.]hat Sa[m.]hit[=a]_[152] in which he frequently uses _['s][=u]nya_ in speaking of numerals, so that it has been thought that he was referring to a definite symbol. This, of course, would add to the probability that [=A]ryabha[t.]a was doing the same.

It should also be mentioned as a matter of interest, and somewhat related to the question at issue, that Var[=a]ha-Mihira used the word-system with place value[153] as explained above.

The first kind of alphabetic numerals and also the word-system (in both of which the place value is used) are plays upon, or variations of, position arithmetic, which would be most likely to occur in the country of its origin.[154]

At the opening of the next century (c. 620 A.D.) B[=a][n.]a[155] wrote of Subandhus's _V[=a]savadatt[=a]_ as a celebrated work, {45} and mentioned that the stars dotting the sky are here compared with zeros, these being points as in the modern Arabic system. On the other hand, a strong argument against any Hindu knowledge of the symbol zero at this time is the fact that about 700 A.D. the Arabs overran the province of Sind and thus had an opportunity of knowing the common methods used there for writing numbers.

And yet, when they received the complete system in 776 they looked upon it as something new.[156] Such evidence is not conclusive, but it tends to show that the complete system was probably not in common use in India at the beginning of the eighth century. On the other hand, we must bear in mind the fact that a traveler in Germany in the year 1700 would probably have heard or seen nothing of decimal fractions, although these were perfected a century before that date. The elite of the mathematicians may have known the zero even in [=A]ryabha[t.]a's time, while the merchants and the common people may not have grasped the significance of the novelty until a long time after. On the whole, the evidence seems to point to the west coast of India as the region where the complete system was first seen.[157] As mentioned above, traces of the numeral words with place value, which do not, however, absolutely require a decimal place-system of symbols, are found very early in Cambodia, as well as in India.

Concerning the earliest epigraphical instances of the use of the nine symbols, plus the zero, with place value, there {46} is some question.

Colebrooke[158] in 1807 warned against the possibility of forgery in many of the ancient copper-plate land grants. On this account Fleet, in the _Indian Antiquary_,[159] discusses at length this phase of the work of the epigraphists in India, holding that many of these forgeries were made about the end of the eleventh century. Colebrooke[160] takes a more rational view of these forgeries than does Kaye, who seems to hold that they tend to invalidate the whole Indian hypothesis. "But even where that may be suspected, the historical uses of a monument fabricated so much nearer to the times to which it a.s.sumes to belong, will not be entirely superseded.

The necessity of rendering the forged grant credible would compel a fabricator to adhere to history, and conform to established notions: and the tradition, which prevailed in his time, and by which he must be guided, would probably be so much nearer to the truth, as it was less remote from the period which it concerned."[161] Buhler[162] gives the copper-plate Gurjara inscription of Cedi-sa[m.]vat 346 (595 A.D.) as the oldest epigraphical use of the numerals[163] "in which the symbols correspond to the alphabet numerals of the period and the place." Vincent A. Smith[164]

quotes a stone inscription of 815 A.D., dated Sa[m.]vat 872. So F. Kielhorn in the _Epigraphia Indica_[165] gives a Pathari pillar inscription of Parabala, dated Vikrama-sa[m.]vat 917, which corresponds to 861 A.D., {47} and refers also to another copper-plate inscription dated Vikrama-sa[m.]vat 813 (756 A.D.). The inscription quoted by V. A. Smith above is that given by D. R. Bhandarkar,[166] and another is given by the same writer as of date Saka-sa[m.]vat 715 (798 A.D.), being incised on a pilaster.

Kielhorn[167] also gives two copper-plate inscriptions of the time of Mahendrapala of Kanauj, Valhab[=i]-sa[m.]vat 574 (893 A.D.) and Vikrama-sa[m.]vat 956 (899 A.D.). That there should be any inscriptions of date as early even as 750 A.D., would tend to show that the system was at least a century older. As will be shown in the further development, it was more than two centuries after the introduction of the numerals into Europe that they appeared there upon coins and inscriptions. While Thibaut[168]

does not consider it necessary to quote any specific instances of the use of the numerals, he states that traces are found from 590 A.D. on. "That the system now in use by all civilized nations is of Hindu origin cannot be doubted; no other nation has any claim upon its discovery, especially since the references to the origin of the system which are found in the nations of western Asia point unanimously towards India."[169]

The testimony and opinions of men like Buhler, Kielhorn, V. A. Smith, Bhandarkar, and Thibaut are ent.i.tled to the most serious consideration. As authorities on ancient Indian epigraphy no others rank higher. Their work is accepted by Indian scholars the world over, and their united judgment as to the rise of the system with a place value--that it took place in India as early as the {48} sixth century A.D.--must stand unless new evidence of great weight can be submitted to the contrary.

Many early writers remarked upon the diversity of Indian numeral forms.

Al-B[=i]r[=u]n[=i] was probably the first; noteworthy is also Johannes Hispalensis,[170] who gives the variant forms for seven and four. We insert on p. 49 a table of numerals used with place value. While the chief authority for this is Buhler,[171] several specimens are given which are not found in his work and which are of unusual interest.

The ['S][=a]rad[=a] forms given in the table use the circle as a symbol for 1 and the dot for zero. They are taken from the paging and text of _The Kashmirian Atharva-Veda_[172], of which the ma.n.u.script used is certainly four hundred years old. Similar forms are found in a ma.n.u.script belonging to the University of Tubingen. Two other series presented are from Tibetan books in the library of one of the authors.

For purposes of comparison the modern Sanskrit and Arabic numeral forms are added.

Sanskrit, [Ill.u.s.tration]

Arabic, [Ill.u.s.tration]

{49}

NUMERALS USED WITH PLACE VALUE

1 2 3 4 5 6 7 8 9 0 a[173] [Ill.u.s.tration]

b[174] [Ill.u.s.tration]

c[175] [Ill.u.s.tration]

d[176] [Ill.u.s.tration]

e[177] [Ill.u.s.tration]

f[178] [Ill.u.s.tration]

g[179] [Ill.u.s.tration]

h[180] [Ill.u.s.tration]

i[180] [Ill.u.s.tration]

j[181] [Ill.u.s.tration]

k[181] [Ill.u.s.tration]

l[182] [Ill.u.s.tration]

m[183] [Ill.u.s.tration]

n[184] [Ill.u.s.tration]

{51}

CHAPTER IV

THE SYMBOL ZERO

What has been said of the improved Hindu system with a place value does not touch directly the origin of a symbol for zero, although it a.s.sumes that such a symbol exists. The importance of such a sign, the fact that it is a prerequisite to a place-value system, and the further fact that without it the Hindu-Arabic numerals would never have dominated the computation system of the western world, make it proper to devote a chapter to its origin and history.

It was some centuries after the primitive Br[=a]hm[=i] and Kharo[s.][t.]h[=i] numerals had made their appearance in India that the zero first appeared there, although such a character was used by the Babylonians[185] in the centuries immediately preceding the Christian era.

The symbol is [Babylonian zero symbol] or [Babylonian zero symbol], and apparently it was not used in calculation. Nor does it always occur when units of any order are lacking; thus 180 is written [Babylonian numerals 180] with the meaning three sixties and no units, since 181 immediately following is [Babylonian numerals 181], three sixties and one unit.[186]

The main {52} use of this Babylonian symbol seems to have been in the fractions, 60ths, 3600ths, etc., and somewhat similar to the Greek use of [Greek: o], for [Greek: ouden], with the meaning _vacant_.

"The earliest undoubted occurrence of a zero in India is an inscription at Gwalior, dated Samvat 933 (876 A.D.). Where 50 garlands are mentioned (line 20), 50 is written [Gwalior numerals 50]. 270 (line 4) is written [Gwalior numerals 270]."[187] The Bakh[s.][=a]l[=i] Ma.n.u.script[188] probably antedates this, using the point or dot as a zero symbol. Bayley mentions a grant of Jaika Rashtrakuta of Bharuj, found at Okamandel, of date 738 A.D., which contains a zero, and also a coin with indistinct Gupta date 707 (897 A.D.), but the reliability of Bayley's work is questioned. As has been noted, the appearance of the numerals in inscriptions and on coins would be of much later occurrence than the origin and written exposition of the system. From the period mentioned the spread was rapid over all of India, save the southern part, where the Tamil and Malayalam people retain the old system even to the present day.[189]

Aside from its appearance in early inscriptions, there is still another indication of the Hindu origin of the symbol in the special treatment of the concept zero in the early works on arithmetic. Brahmagupta, who lived in Ujjain, the center of Indian astronomy,[190] in the early part {53} of the seventh century, gives in his arithmetic[191] a distinct treatment of the properties of zero. He does not discuss a symbol, but he shows by his treatment that in some way zero had acquired a special significance not found in the Greek or other ancient arithmetics. A still more scientific treatment is given by Bh[=a]skara,[192] although in one place he permits himself an unallowed liberty in dividing by zero. The most recently discovered work of ancient Indian mathematical lore, the Ganita-S[=a]ra-Sa[.n]graha[193] of Mah[=a]v[=i]r[=a]c[=a]rya (c. 830 A.D.), while it does not use the numerals with place value, has a similar discussion of the calculation with zero.

What suggested the form for the zero is, of course, purely a matter of conjecture. The dot, which the Hindus used to fill up lacunae in their ma.n.u.scripts, much as we indicate a break in a sentence,[194] would have been a more natural symbol; and this is the one which the Hindus first used[195] and which most Arabs use to-day. There was also used for this purpose a cross, like our X, and this is occasionally found as a zero symbol.[196] In the Bakh[s.][=a]l[=i] ma.n.u.script above mentioned, the word _['s][=u]nya_, with the dot as its symbol, is used to denote the unknown quant.i.ty, as well as to denote zero. An a.n.a.logous use of the {54} zero, for the unknown quant.i.ty in a proportion, appears in a Latin ma.n.u.script of some lectures by Gottfried Wolack in the University of Erfurt in 1467 and 1468.[197] The usage was noted even as early as the eighteenth century.[198]

The small circle was possibly suggested by the spurred circle which was used for ten.[199] It has also been thought that the omicron used by Ptolemy in his _Almagest_, to mark accidental blanks in the s.e.xagesimal system which he employed, may have influenced the Indian writers.[200] This symbol was used quite generally in Europe and Asia, and the Arabic astronomer Al-Batt[=a]n[=i][201] (died 929 A.D.) used a similar symbol in connection with the alphabetic system of numerals. The occasional use by Al-Batt[=a]n[=i] of the Arabic negative, _l[=a]_, to indicate the absence of minutes {55} (or seconds), is noted by Nallino.[202] Noteworthy is also the use of the [Circle] for unity in the ['S][=a]rad[=a] characters of the Kashmirian Atharva-Veda, the writing being at least 400 years old.

Bh[=a]skara (c. 1150) used a small circle above a number to indicate subtraction, and in the Tartar writing a redundant word is removed by drawing an oval around it. It would be interesting to know whether our score mark [score mark], read "four in the hole," could trace its pedigree to the same sources. O'Creat[203] (c. 1130), in a letter to his teacher, Adelhard of Bath, uses [Greek: t] for zero, being an abbreviation for the word _teca_ which we shall see was one of the names used for zero, although it could quite as well be from [Greek: tziphra]. More rarely O'Creat uses [circle with bar], applying the name _cyfra_ to both forms. Frater Sigsboto[204] (c. 1150) uses the same symbol. Other peculiar forms are noted by Heiberg[205] as being in use among the Byzantine Greeks in the fifteenth century. It is evident from the text that some of these writers did not understand the import of the new system.[206]

Although the dot was used at first in India, as noted above, the small circle later replaced it and continues in use to this day. The Arabs, however, did not adopt the {56} circle, since it bore some resemblance to the letter which expressed the number five in the alphabet system.[207] The earliest Arabic zero known is the dot, used in a ma.n.u.script of 873 A.D.[208] Sometimes both the dot and the circle are used in the same work, having the same meaning, which is the case in an Arabic MS., an abridged arithmetic of Jamshid,[209] 982 A.H. (1575 A.D.). As given in this work the numerals are [symbols]. The form for 5 varies, in some works becoming [symbol] or [symbol]; [symbol] is found in Egypt and [symbol] appears in some fonts of type. To-day the Arabs use the 0 only when, under European influence, they adopt the ordinary system. Among the Chinese the first definite trace of zero is in the work of Tsin[210] of 1247 A.D. The form is the circular one of the Hindus, and undoubtedly was brought to China by some traveler.

The name of this all-important symbol also demands some attention, especially as we are even yet quite undecided as to what to call it. We speak of it to-day as _zero, naught_, and even _cipher_; the telephone operator often calls it _O_, and the illiterate or careless person calls it _aught_. In view of all this uncertainty we may well inquire what it has been called in the past.[211]

{57}

As already stated, the Hindus called it _['s][=u]nya_, "void."[212] This pa.s.sed over into the Arabic as _a[s.]-[s.]ifr_ or _[s.]ifr_.[213] When Leonard of Pisa (1202) wrote upon the Hindu numerals he spoke of this character as _zephirum_.[214] Maximus Planudes (1330), writing under both the Greek and the Arabic influence, called it _tziphra_.[215] In a treatise on arithmetic written in the Italian language by Jacob of Florence[216]

{58} (1307) it is called _zeuero_,[217] while in an arithmetic of Giovanni di Danti of Arezzo (1370) the word appears as _ceuero_.[218] Another form is _zepiro_,[219] which was also a step from _zephirum_ to zero.[220]

Of course the English _cipher_, French _chiffre_, is derived from the same Arabic word, _a[s.]-[s.]ifr_, but in several languages it has come to mean the numeral figures in general. A trace of this appears in our word _ciphering_, meaning figuring or computing.[221] Johann Huswirt[222] uses the word with both meanings; he gives for the tenth character the four names _theca, circulus, cifra_, and _figura nihili_. In this statement Huswirt probably follows, as did many writers of that period, the _Algorismus_ of Johannes de Sacrobosco (c. 1250 A.D.), who was also known as John of Halifax or John of Holywood. The commentary of {59} Petrus de Dacia[223] (c. 1291 A.D.) on the _Algorismus vulgaris_ of Sacrobosco was also widely used. The widespread use of this Englishman's work on arithmetic in the universities of that time is attested by the large number[224] of MSS. from the thirteenth to the seventeenth century still extant, twenty in Munich, twelve in Vienna, thirteen in Erfurt, several in England given by Halliwell,[225] ten listed in c.o.xe's _Catalogue of the Oxford College Library_, one in the Plimpton collection,[226] one in the Columbia University Library, and, of course, many others.

From _a[s.]-[s.]ifr _has come _zephyr, cipher,_ and finally the abridged form _zero_. The earliest printed work in which is found this final form appears to be Calandri's arithmetic of 1491,[227] while in ma.n.u.script it appears at least as early as the middle of the fourteenth century.[228] It also appears in a work, _Le Kadran des marchans_, by Jehan {60} Certain,[229] written in 1485. This word soon became fairly well known in Spain[230] and France.[231] The medieval writers also spoke of it as the _sipos_,[232] and occasionally as the _wheel_,[233] _circulus_[234] (in German _das Ringlein_[235]), _circular {61} note_,[236] _theca_,[237] long supposed to be from its resemblance to the Greek theta, but explained by Petrus de Dacia as being derived from the name of the iron[238] used to brand thieves and robbers with a circular mark placed on the forehead or on the cheek. It was also called _omicron_[239] (the Greek _o_), being sometimes written or [Greek: ph] to distinguish it from the letter _o_.

It also went by the name _null_[240] (in the Latin books {62} _nihil_[241]

or _nulla_,[242] and in the French _rien_[243]), and very commonly by the name _cipher_.[244] Wallis[245] gives one of the earliest extended discussions of the various forms of the word, giving certain other variations worthy of note, as _ziphra_, _zifera_, _siphra_, _ciphra_, _tsiphra_, _tziphra,_ and the Greek [Greek: tziphra].[246]

{63}

CHAPTER V

THE QUESTION OF THE INTRODUCTION OF THE NUMERALS INTO EUROPE BY BOETHIUS

Just as we were quite uncertain as to the origin of the numeral forms, so too are we uncertain as to the time and place of their introduction into Europe. There are two general theories as to this introduction. The first is that they were carried by the Moors to Spain in the eighth or ninth century, and thence were transmitted to Christian Europe, a theory which will be considered later. The second, advanced by Woepcke,[247] is that they were not brought to Spain by the Moors, but that they were already in Spain when the Arabs arrived there, having reached the West through the Neo-Pythagoreans. There are two facts to support this second theory: (1) the forms of these numerals are characteristic, differing materially from those which were brought by Leonardo of Pisa from Northern Africa early in the thirteenth century (before 1202 A.D.); (2) they are essentially those which {64} tradition has so persistently a.s.signed to Boethius (c. 500 A.D.), and which he would naturally have received, if at all, from these same Neo-Pythagoreans or from the sources from which they derived them.

Furthermore, Woepcke points out that the Arabs on entering Spain (711 A.D.) would naturally have followed their custom of adopting for the computation of taxes the numerical systems of the countries they conquered,[248] so that the numerals brought from Spain to Italy, not having undergone the same modifications as those of the Eastern Arab empire, would have differed, as they certainly did, from those that came through Bagdad. The theory is that the Hindu system, without the zero, early reached Alexandria (say 450 A.D.), and that the Neo-Pythagorean love for the mysterious and especially for the Oriental led to its use as something bizarre and cabalistic; that it was then pa.s.sed 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.