CHAPTER IV
THE SYMBOL ZERO
What has been said of the improved Hindu system with a place value does not touch directly the origin of a symbol for zero, although it assumes that such a symbol exists. The importance of such a sign, the fact that it is a prerequisite to a place-value system, and the further fact that without it the Hindu-Arabic numerals would never have dominated the computation system of the western world, make it proper to devote a chapter to its origin and history.
It was some centuries after the primitive Brāhmī and Kharoṣṭhī 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
"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
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-Saṅgraha[[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 lacunæ in their manuscripts, much as we indicate a break in a sentence,[[194]] would have been a more natural symbol; and this is the one which the Hindus first used[[195]] and which most Arabs use to-day. There was also used for this purpose a cross, like our X, and this is occasionally found as a zero symbol.[[196]] In the Bakhṣālī manuscript above mentioned, the word śūnya, 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
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 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 śūnya, "void."[[212]] This passed over into the Arabic as aṣ-ṣifr or ṣifr.[[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 çeuero.[[218]] Another form is zepiro,[[219]] which was also a step from zephirum to zero.[[220]]
Of course the English cipher, French chiffre, is derived from the same Arabic word, aṣ-ṣifr, 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 aṣ-ṣifr 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 õ 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 τζίφρα.[[246]]