Chapter 4 THE SYMBOL ZERO

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

It was some centuries after the primitive Brāhmī and Kharo??hī 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 or , and apparently it was not used in calculation. Nor does it always occur when units of any order are lacking; thus 180 is written with the meaning three sixties and no units, since 181 immediately following is , three sixties and one unit.[186] The main use of this Babylonian symbol seems to have been in the fractions, 60ths, 3600ths, etc., and somewhat similar to the Greek use of ο, for ο?δ?ν, 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 . 270 (line 4) is written ."[187] The Bakh?ālī Manuscript[188] probably antedates this, using the point or dot as a zero symbol. Bayley mentions a grant of Jaika Rashtrakúta of Bharuj, found at Okamandel, of date 738 A.D., which contains a zero, and also a coin with indistinct Gupta date 707 (897 A.D.), but the reliability of Bayley's work is questioned. As has been noted, the appearance of the numerals in inscriptions and on coins would be of much later occurrence than the origin and written exposition of the system. From the period mentioned the spread was rapid over all of India, save the southern part, where the Tamil and Malayalam people retain the old system even to the present day.[189]

Aside from its appearance in early inscriptions, there is still another indication of the Hindu origin of the symbol in the special treatment of the concept zero in the early works on arithmetic. Brahmagupta, who lived in Ujjain, the center of Indian astronomy,[190] in the early part 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 Bhāskara,[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-Sāra-Sa?graha[193] of Mahāvīrācārya (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 lacun? in their manuscripts, much as we indicate a break in a sentence,[194] would have been a more natural symbol; and this is the one which the Hindus first used[195] and which most Arabs use to-day. There was also used for this purpose a cross, like our X, and this is occasionally found as a zero symbol.[196] In the Bakh?ālī manuscript above mentioned, the word ?ūnya, with the dot as its symbol, is used to denote the unknown quantity, as well as to denote zero. An analogous use of the zero, for the unknown quantity in a proportion, appears in a Latin manuscript of some lectures by Gottfried Wolack in the University of Erfurt in 1467 and 1468.[197] The usage was noted even as early as the eighteenth century.[198]

The small circle was possibly suggested by the spurred circle which was used for ten.[199] It has also been thought that the omicron used by Ptolemy in his Almagest, to mark accidental blanks in the sexagesimal system which he employed, may have influenced the Indian writers.[200] This symbol was used quite generally in Europe and Asia, and the Arabic astronomer Al-Battānī[201] (died 929 A.D.) used a similar symbol in connection with the alphabetic system of numerals. The occasional use by Al-Battānī of the Arabic negative, lā, to indicate the absence of minutes (or seconds), is noted by Nallino.[202] Noteworthy is also the use of the for unity in the ?āradā 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 , 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 τ 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 τζ?φρα. More rarely O'Creat uses , 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 circle, since it bore some resemblance to the letter which expressed the number five in the alphabet system.[207] The earliest Arabic zero known is the dot, used in a manuscript of 873 A.D.[208] Sometimes both the dot and the circle are used in the same work, having the same meaning, which is the case in an Arabic MS., an abridged arithmetic of Jamshid,[209] 982 A.H. (1575 A.D.). As given in this work the numerals are . The form for 5 varies, in some works becoming or ; is found in Egypt and 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]

As already stated, the Hindus called it ?ūnya, "void."[212] This passed over into the Arabic as a?-?ifr or ?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] (1307) it is called zeuero,[217] while in an arithmetic of Giovanni di Danti of Arezzo (1370) the word appears as ?euero.[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?-?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 Petrus de Dacia[223] (c. 1291 A.D.) on the Algorismus vulgaris of Sacrobosco was also widely used. The widespread use of this Englishman's work on arithmetic in the universities of that time is attested by the large number[224] of MSS. from the thirteenth to the seventeenth century still extant, twenty in Munich, twelve in Vienna, thirteen in Erfurt, several in England given by Halliwell,[225] ten listed in Coxe's Catalogue of the Oxford College Library, one in the Plimpton collection,[226] one in the Columbia University Library, and, of course, many others.

From a?-?ifr has come zephyr, cipher, and finally the abridged form zero. The earliest printed work in which is found this final form appears to be Calandri's arithmetic of 1491,[227] while in manuscript it appears at least as early as the middle of the fourteenth century.[228] It also appears in a work, Le Kadran des marchans, by Jehan 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 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 φ to distinguish it from the letter o. It also went by the name null[240] (in the Latin books 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 τζ?φρα.[246]

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