The Hindu-Arabic Numerals c8: Chapter IV. The Symbol Zero

The Hindu-Arabic Numerals c8

Language: English

Chapter IV. The Symbol Zero

c8.2

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.

c8.3

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 Babylonians¹ in the centuries immediately preceding the Christian era. The symbol is [Figure:] or [Figure:], and apparently it was not used in calculation. Nor does it always occur when units of any order are lacking; thus 180 is written [Figure:] with the meaning three sixties and no units, since 181 immediately following is written [Figure:], three sixties and one unit.² 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 *o*, for οὐδέν, with the meaning *vacant*.

c8.4

¹ Franz X. Kugler, *Die Babylonische Mondrechnung*, 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, *Geschichte*, Vol. I, p. 31.

c8.5

² F. X. Kugler, *Sternkunde und Sterndienst in Babel*, I. Buch, from the beginnings to the time of Christ, Münster i. Westfalen, 1907. It also has numerous tables containing the above zero.

c8.6

"The earliest undoubted occurrence of a zero in India is an inscription at Gwalior, dated Saṃvat 933 (876 A.D.). Where 50 garlands are mentioned (line 20), 50 is written [Figure:]. 270 (line 4) is written [Figure:]." ¹ The Bakhṣālī Manuscript ² 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.³

c8.7

¹ 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 *Archæologia*, Vol. LXII (1910), p. 137.

c8.8

² R. Hoernle, "The Bakshālī Manuscript," *Indian Antiquary*, Vol. XVII, pp. 33-48 and 275-279, 1888; Thibaut, *Astronomie, Astrologie und Mathematik*, p. 75; Hoernle, *Verhandlungen*, loc. cit., p. 132.

c8.9

³ Bayley, loc. cit., Vol. XV, p. 29. Also Bendall, "On a System of Numerals used in South India," *Journal of the Royal Asiatic Society*, 1896, pp. 789-792.

c8.10

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,⁴ in the early part of the seventh century, gives in his arithmetic¹ 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,² 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-Saṅgraha³ 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.

c8.11

⁴ V. A. Smith, *The Early History of India*, 2d ed., Oxford, 1908, p. 14.

c8.12

¹ Colebrooke, *Algebra, with Arithmetic and Mensuration, from the Sanskrit of Brahmegupta and Bháscara*, London, 1817, pp. 339–340.

c8.13

² Ibid., p. 138.

c8.14

³ D. E. Smith, in the *Bibliotheca Mathematica*, Vol. IX (3), pp. 106–110.

c8.15

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,⁴ would have been a more natural symbol; and this is the one which the Hindus first used⁵ 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.⁶ 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 1468.¹ The usage was noted even as early as the eighteenth century.²

c8.16

⁴ As when we use three dots (...).

c8.17

⁵ "The Hindus call the nought explicitly *śūnyabindu* 'the dot marking a blank,' and about 500 A.D. 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, *On the Origin of the Indian Alphabet*, p. 53, note.]

c8.18

⁶ Fazzari, *Dell' origine delle parole zero e cifra*, Naples, 1903.

c8.19

¹ E. Wappler, "Zur Geschichte der Mathematik im 15. Jahrhundert," in the *Zeitschrift für Mathematik und Physik*, Vol. XLV, *Hist.-lit. Abt.*, p. 47. The manuscript is No. C. 80, in the Dresden library.

c8.20

² J. G. Prändel, *Algebra nebst ihrer literarischen Geschichte*, p. 572, Munich, 1795.

c8.21

The small circle was possibly suggested by the spurred circle which was used for ten.³ 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.⁴ This symbol was used quite generally in Europe and Asia, and the Arabic astronomer Al-Battānī⁵ (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.¹ Noteworthy is also the use of the ⚬ 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 ④, read “four in the hole,” could trace its pedigree to the same sources. O’Creat² (*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 τζιφρα. More rarely O’Creat uses o, applying the name *cyfra* to both forms. Frater Sigsboto³ (*c.* 1150) uses the same symbol. Other peculiar forms are noted by Heiberg⁴ 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.⁵

c8.22

³ 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?

c8.23

⁴ Bayley, loc. cit., p. 48. From this fact Delambre (*Histoire de l'astronomie ancienne*) inferred that Ptolemy knew the zero, a theory accepted by Chasles, *Aperçu historique sur l'origine et le développement des méthodes en géométrie*, 1875 ed., p. 476; Nesselmann, however, showed (*Algebra der Griechen*, 1842, p. 138), that Ptolemy merely used o for οὐδὲν, with no notion of zero. See also G. Fazzari, "Dell' origine delle parole zero e cifra," *Ateneo*, 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, *Les signes numéraux etc.*, reprint p. 30, and J. Brandis, *Das Münz-, Mass- und Gewichtswesen in Vorderasien bis auf Alexander den Grossen*, Berlin, 1866, p. 10, also discuss this usage of o, without the notion of place value, by the Greeks.

c8.24

⁵ *Al-Battānī sive Albatenii opus astronomicum*. 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.

c8.25

¹ Loc. cit., Vol. II, p. 271.

c8.26

² C. Henry, “Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum,” *Abhandlungen zur Geschichte der Mathematik*, Vol. III, 1880.

c8.27

³ Max. Curtze, “Ueber eine Algorismus-Schrift des XII. Jahrhunderts,” *Abhandlungen zur Geschichte der Mathematik*, 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,” *Zeitschrift für Mathematik und Physik, Hist.-lit. Abth.*, Vol. XXXIV, pp. 129–146 and 161–170, with one plate.

c8.28

⁴ “Byzantinische Analekten,” *Abhandlungen zur Geschichte der Mathematik*, Vol. IX, pp. 161–189.

c8.29

⁵ Ч or ၄ for 0. Ч also used for 5. I μ. for 13. [Heiberg, loc. cit.]

c8.30

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.¹ The earliest Arabic zero known is the dot, used in a manuscript of 873 a.d.² 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 Jamshīd,³ 982 a.h. (1575 a.d.). As given in this work the numerals are ۹ ٨ ٧ ६ ۵ ۴ ۳ ۲ ۱ ۰. The form for 5 varies, in some works becoming 𐹣 or 𐹢; 𐹡 is found in Egypt and د 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 Tsin⁴ of 1247 a.d. The form is the circular one of the Hindus, and undoubtedly was brought to China by some traveler.

c8.31

¹ Gerhardt, *Études historiques sur l'arithmétique de position*, Berlin, 1856, p. 12; J. Bowring, *The Decimal System in Numbers, Coins, & Accounts*, London, 1854, p. 33.

c8.32

² Karabacek, *Wiener Zeitschrift für die Kunde des Morgenlandes*, Vol. XI, p. 13; *Führer durch die Papyrus-Ausstellung Erzherzog Rainer*, Vienna, 1894, p. 216.

c8.33

³ In the library of G. A. Plimpton, Esq.

c8.34

⁴ Cantor, *Geschichte*, Vol. I (3), p. 674; Y. Mikami, "A Remark on the Chinese Mathematics in Cantor's Geschichte der Mathematik," *Archiv der Mathematik und Physik*, Vol. XV (3), pp. 68-70.

c8.35

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.⁵

c8.36

⁵ Of course the earlier historians made innumerable guesses as to the origin of the word *cipher*. E.g. Matthew Hostus, *De numeratione* *emendata*, 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, *Institutiones mathematicae*, Vol. I, 1593, gives a large part of this quotation word for word, without any mention of the source. Hermannus Hugo, *De prima scribendi origine*, Trajecti ad Rhenum, 1738, pp. 304–305, and note, p. 305; Karl Krumbacher, "Woher stammt das Wort Ziffer (Chiffre)?", *Études de philologie néo-grecque*, Paris, 1892.

c8.37

As already stated, the Hindus called it *śūnya*, "void."¹ This passed over into the Arabic as *aṣ-ṣifr* or *ṣifr*.² When Leonard of Pisa (1202) wrote upon the Hindu numerals he spoke of this character as *zephirum*.³ Maximus Planudes (1330), writing under both the Greek and the Arabic influence, called it *tziphra*.⁴ In a treatise on arithmetic written in the Italian language by Jacob of Florence⁵ (1307) it is called *zeuero*,¹ while in an arithmetic of Giovanni di Danti of Arezzo (1370) the word appears as *çeuero*.² Another form is *zepiro*,³ which was also a step from *zephirum* to zero.⁴

c8.38

¹ Bühler, loc. cit., p. 78 and p. 86.

c8.39

² Fazzari, loc. cit., p. 4. So Elia Misrachi (1455–1526) in his posthumous *Book of Number*, Constantinople, 1534, explains *sifra* as being Arabic. See also Steinschneider, *Bibliotheca Mathematica*, 1893, p. 69, and G. Wertheim, *Die Arithmetik des Elia Misrachi*, Programm, Frankfurt, 1893.

c8.40

³ "Cum his novem figuris, et cum hoc signo 0, quod arabice zephirum appellatur, scribitur quilibet numerus."

c8.41

⁴ τζίφρα, a form also used by Neophytos (date unknown, probably c. 1330). It is curious that Finaeus (1555 ed., f. 2) used the form *tziphra* throughout. A. J. H. Vincent ["Sur l'origine de nos chiffres," *Notices et Extraits des MSS.*, Paris, 1847, pp. 143–150] says: "Ce cercle fut nommé par les uns, *sipos*, *rota*, *galgal* . . .; par les autres *tsiphra* (de צפר, *couronne* ou *diadème*) ou *ciphra* (de ספר, *numération*)." Ch. de Paravey, *Essai sur l'origine unique et hiéroglyphique des chiffres et des lettres de tous les peuples*, Paris, 1826, p. 165, a rather fanciful work, gives "vase, vase arrondi et fermé par un couvercle, qui est le symbole de la 10^e Heure, ठ," among the Chinese; also "Tsiphron Zéron, ou tout à fait vide en arabe, τζίφρα en grec . . . d'où chiffre (qui dérive plutôt, suivant nous, de l'Hébreu *Sepher*, compter.")

c8.42

⁵ "Compilatus a Magistro Jacobo de Florentia apud montem pesalanum," and described by G. Lami in his *Catalogus codicum manuscriptorum qui in bibliotheca Riccardiana Florentiæ adservantur*. See Fazzari, loc. cit., p. 5.

c8.43

¹ "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.]

c8.44

² Ibid., p. 6.

c8.45

³ 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 zepiro donde il vocabolo zero), che per sè stesso non esprime nessun numero." This quotation is taken from D. C. Martines, *Origine e progressi dell' aritmetica*, Messina, 1865.

c8.46

⁴ Leo Jordan, "Materialien zur Geschichte der arabischen Zahlzeichen in Frankreich," *Archiv für Kulturgeschichte*, Berlin, 1905, pp. 155–195, gives the following two schemes of derivation, (1) "zefiro, zeviro, zeiro, zero," (2) "zefiro, zefro, zevro, zero."

c8.47

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.⁵ Johann Huswirt⁶ 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 Dacia¹ (*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 number² 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,³ ten listed in Coxe's *Catalogue of the Oxford College Library*, one in the Plimpton collection,⁴ one in the Columbia University Library, and, of course, many others.

c8.48

⁵ Köbel (1518 ed., f. A_4) speaks of the numerals in general as "die der gemain man Zyfer nendt." Recorde (*Grounde of Artes*, 1558 ed., f. B_6) says that the zero is "called priuatly a Cyphar, though all the other sometimes be likewise named."

c8.49

⁶ "Decimo X 0 theca, circulꝰ cifra sive figura nihili appellat'." [*Enchiridion Algorismi*, Cologne, 1501.] Later, "quoniam de integris tam in cifris quam in proiectilibus,"—the word *proiectilibus* referring to markers "thrown" and used on an abacus, whence the French *jetons* and the English expression "to cast an account."

c8.50

¹ "Decima vero o dicitur teca, circulus, vel cyfra vel figura nichili." [Maximilian Curtze, *Petri Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco commentarius, una cum Algorismo ipso*, 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, *Öfversigt af Kongl. Vetenskaps-Akademiens Förhandlingar*, 1885, pp. 15–27, 65–70; 1886, pp. 57–60.

c8.51

² Curtze, *loc. cit.*, p. vi.

c8.52

³ *Rara Mathematica*, London, 1841, chap. i, "Joannis de Sacro-Bosco Tractatus de Arte Numerandi."

c8.53

⁴ Smith, *Rara Arithmetica*, Boston, 1909.

c8.54

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 1491,⁵ while in manuscript it appears at least as early as the middle of the fourteenth century.⁶ It also appears in a work, *Le Kadran des marchans*, by Jehan Certain,¹ written in 1485. This word soon became fairly well known in Spain² and France.³ The medieval writers also spoke of it as the *sipos*,⁴ and occasionally as the *wheel*,⁵ *circulus*⁶ (in German das *Ringlein*⁷), *circular* *note*,¹ *theca*,² 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 iron³ used to brand thieves and robbers with a circular mark placed on the forehead or on the cheek. It was also called *omicron*⁴ (the Greek *o*), being sometimes written o or φ to distinguish it from the letter *o*. It also went by the name *null*⁵ (in the Latin books *nihil*¹ or *nulla*,² and in the French *rien*³), and very commonly by the name *cipher*.⁴ Wallis⁵ 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 τζιφρα.⁶

c8.55

⁵ 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 *Propagation*, p. 522.]

c8.56

⁶ Boncompagni *Bulletino*, Vol. XVI, pp. 673–685.

c8.57

¹ Leo Jordan, loc. cit. In the *Catalogue of MSS., Bibl. de l'Arsenal*, Vol. III, pp. 154–155, this work is No. 2904 (184 S.A.F.), Bibl. Nat., and is also called *Petit traicté de algorisme*.

c8.58

² Texada (1546) says that there are "nueue letros y vn zero o cifra" (f. 3).

c8.59

³ Savonne (1563, 1751 ed., f. 1): "Vne ansi formee (0) 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.

c8.60

⁴ Thus Radulph of Laon (c. 1100): "Inscribitur in ultimo ordine et figura [Figure:] sipos nomine, quae, licet numerum nullum signitet, tantum ad alia quaedam utilis, ut insequentibus declarabitur." ["Der Arithmetische Tractat des Radulph von Laon," *Abhandlungen zur Geschichte der Mathematik*, Vol. V, p. 97, from a manuscript of the thirteenth century.] Chasles (*Comptes rendus*, 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 [Figure:] in motum rotulae formatam nullius numeri significatione inscribi solere praediximus," and thereafter uses rotula. He uses the sipos simply as a kind of marker on the abacus.

c8.61

⁵ Rabbi ben Ezra (1092–1168) used both גלגל, *galgal* (the Hebrew for *wheel*), and ספרא, *sifra*. See M. Steinschneider, "Die Mathematik bei den Juden," in *Bibliotheca Mathematica*, 1893, p. 69, and Silberberg, *Das Buch der Zahl des R. Abraham ibn Esra*, Frankfurt a. M., 1895, p. 96, note 23; in this work the Hebrew letters are used for numerals with place value, having the zero.

c8.62

⁶ E.g., in the twelfth-century *Liber algorismi* (see Boncompagni's *Trattati*, II, p. 28). So Ramus (*Libri II*, 1569 ed., p. 1) says: "Circulus quæ nota est ultima: nil per se significat." (See also the Schonerus ed. of Ramus, 1586, p. 1.)

c8.63

⁷ "Und wirt das ringlein o. die Ziffer genant die nichts bedeut." [Köbel's *Rechenbuch*, 1549 ed., f. 10, and other editions.]

c8.64

¹ *I.e.* “circular figure,” our word *notation* having come from the medieval *nota*. 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. xxxvii) calls it “nota aut circularis o,” “circularis nota,” and “figura circularis.” Tonstall (1522, f. B₃) says of it: “Decimo uero nota ad formam .O. litteræ circulari figura est: quam alij circulum, uulgus cyphram uocat,” and later (f. C₄) speaks of the “circulos.” Grammateus, in his *Algorismus de integris* (Erfurt, 1523, f. A₂), speaking of the nine significant figures, remarks: “His autem superadditur decima figura circularis ut 0 existens que ratione sua nihil significat.” Noviomagus (*De Numeris libri II*, Paris, 1539, chap. xvi, “De notis numerorum, quas zyphras vocant”) calls it “circularis nota, quam ex his solam, alij sipheram, Georgius Valla zyphram.”

c8.65

² Huswirt, as above. Ramus (*Scholae mathematicae*, 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.”

c8.66

³ “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 quibus-dam regionibus imprimi fronti vel maxillae furis seu latronum.” [Loc. cit., p. 26.] But in Greek *theca* (ΘΗΚΗ, θη′κη) is a place to put some-thing, a receptacle. If a vacant column, e.g. in the abacus, was so called, the initial might have given the early forms ⊙ and ø for the zero.

c8.67

⁴ Buteo, *Logistica*, Lyons, 1559. See also Wertheim in the *Biblio-theca Mathematica*, 1901, p. 214.

c8.68

⁵ “O est appellee chiffre ou nulle ou figure de nulle valeur.” [La Roche, *L'arithmétique*, Lyons, 1520.]

c8.69

¹ "Decima autem figura nihil uocata," "figura nihili (quam etiam cifram uocant)." [Stifel, *Arithmetica integra*, 1544, f. 1.]

c8.70

² "Zifra, & Nulla uel figura Nihili." [Scheubel, 1545, p. 1 of ch. 1.] *Nulla* 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 0, si chiama nulla, ouero zero, ouero niente." It also found its way into the Dutch arithmetics, e.g. Raets (1576, 1580 ed., f. A₃): "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 *nulla* and popularized it. (See also Kuckuck, *Die Rechenkunst im sechzehnten Jahrhundert*, Berlin, 1874, p. 7; Günther, *Geschichte*, p. 316.)

c8.71

³ "La dixième s'appelle chifre vulgairement: les vns l'appellant zero: nous la pourrons appeller vn Rien." [Peletier, 1607 ed., p. 14.]

c8.72

⁴ It appears in the Polish arithmetic of Kłos (1538) as *cyfra*. "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 *cyfra* from *circumference*. "Vocatur etiam cyfra, quasi circumfacta vel circumferenda, quod idem est, quod circulus non habito respectu ad centrum." [Loc. cit., p. 26.]

c8.73

⁵ *Opera mathematica*, 1695, Oxford, Vol. I, chap. ix, *Mathesis universalis*, "De figuris numeralibus," pp. 46–49; Vol. II, *Algebra*, p. 10.

c8.74

⁶ Martin, *Origine de notre système de numération écrite*, note 149, p. 36 of reprint, spells τσιφρα from Maximus Planudes, citing Wallis as an authority. This is an error, for Wallis gives the correct form as above. Alexander von Humboldt, "Über die bei verschiedenen Völkern üblichen Systeme von Zahlzeichen und über den Ursprung des Stellenwerthes in den indischen Zahlen," Crelle's *Journal für reine und angewandte Mathematik*, Vol. IV, 1829, called attention to the work αριθι' Iνδιι of the monk Neophytos, supposed to be of the fourteenth century. In this work the forms τζυφρα and τζυμφρα appear. See also Boeckh, *De abaco Graecorum*, Berlin, 1841, and Tannery, "Le Scholie du moine Néophytos," *Revue Archéologique*, 1885, pp. 99–102. Jordan, loc. cit., gives from twelfth and thirteenth century manuscripts the forms *cifra*, *ciffre*, *chifras*, and *cifrus*. Du Cange, *Glossarium mediae et infimae Latinitatis*, Paris, 1842, gives also *chilerae*. Dasypodius, *Institutiones Mathematicae*, Strassburg, 1593–1596, adds the forms *zyphra* and *syphra*. Boissière, *L'art d'arythmetique contenant toute dimention, tres-singulier et commode, tant pour l'art militaire que autres calculations*, Paris, 1554: "Puis y en a vn autre dict zero lequel ne designe nulle quantité par soy, ains seulement les loges vuides,"

Linked texts (5)

Gaṇitasārasaṅgraha — Mahāvīra (mathematician), Sanskrit
Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit
Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Atharvavedasaṃhitā — Atharvan & Aṅgiras, Sanskrit
Bakhshālī Manuscript — Sanskrit