The Hindu-Arabic Numerals Part 16

[223] "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 c.u.m Algorismo ipso_, Copenhagen, 1897, p. 2.] Curtze cites five ma.n.u.scripts (fourteenth and fifteenth centuries) of Dacia's commentary in the libraries at Erfurt, Leipzig, and Salzburg, in addition to those given by Enestrom, _ofversigt af Kongl.

Vetenskaps-Akademiens Forhandlingar_, 1885, pp. 15-27, 65-70; 1886, pp.

57-60.

[224] Curtze, loc. cit., p. VI.

[225] _Rara Mathematica_, London, 1841, chap, i, "Joannis de Sacro-Bosco Tractatus de Arte Numerandi."

[226] Smith, _Rara Arithmetica_, Boston, 1909.

[227] In the 1484 edition, Borghi uses the form "cefiro: 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 a.s.serted that it first appeared in Calandri (1491) in this sentence: "Sono dieci le figure con le quali ciascuno numero si pu significare: delle quali n'e una che si chiama zero: et per se sola nulla significa." (f. 4). [See _Propagation_, p. 522.]

[228] Boncompagni _Bulletino_, Vol. XVI, pp. 673-685.

[229] Leo Jordan, loc. cit. In the _Catalogue of MSS., Bibl. de l'a.r.s.enal_, Vol. III, pp. 154-156, this work is No. 2904 (184 S.A.F.), Bibl. Nat., and is also called _Pet.i.t traicte de algorisme_.

[230] Texada (1546) says that there are "nueue letros yvn zero o cifra" (f.

3).

[231] Savonne (1563, 1751 ed., f. 1): "Vne ansi formee (o) 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.

[232] Thus Radulph of Laon (c. 1100): "Inscribitur in ultimo ordine et figura [symbol] 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 ma.n.u.script 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 [symbol] 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.

[233] Rabbi ben Ezra (1092-1168) used both [Hebrew: GLGL], _galgal_ (the Hebrew for _wheel_), and [Hebrew: SPR'], _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.

[234] E.g., in the twelfth-century _Liber aligorismi_ (see Boncompagni's _Trattati_, II, p. 28). So Ramus (_Libri II_, 1569 ed., p. 1) says: "Circulus quae nota est ultima: nil per se significat." (See also the Schonerus ed. of Ramus, 1586, p. 1.)

[235] "Und wirt das ringlein o. die Ziffer genant die nichts bedeut."

[Kobel's _Rechenbuch_, 1549 ed., f. 10, and other editions.]

[236] 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.

x.x.xVII) calls it "nota aut circularis o," "circularis nota," and "figura circularis." Tonstall (1522, f. B_3) says of it: "Decimo uero nota ad formam [symbol] litterae circulari figura est: quam alij circulum, uulgus cyphram uocat," and later (f. C_4) speaks of the "circulos." Grammateus, in his _Algorismus de integris_ (Erfurt, 1523, f. A_2), 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."

[237] Huswirt, as above. Ramus (_Scholae mathematicae_, 1569 ed., p. 112) discusses the name interestingly, saying: "Circulum appellamus c.u.m multis, quam alii thecam, alii figuram nihili, alii figuram privationis, seu figuram nullam vocant, alii ciphram, c.u.m tamen hodie omnes hae notae vulg ciphrae 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."

[238] "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 quibusdam regionibus imprimi fronti vel maxillae furis seu latronum." [Loc. cit., p. 26.] But in Greek _theca_ ([THEKE], [Greek: theke]) is a place to put something, a receptacle. If a vacant column, e.g. in the abacus, was so called, the initial might have given the early forms [symbol] and [symbol] for the zero.

[239] Buteo, _Logistica_, Lyons, 1559. See also Wertheim in the _Bibliotheca Mathematica_, 1901, p. 214.

[240] "0 est appellee chiffre ou nulle ou figure de nulle valeur." [La Roche, _L'arithmetique_, Lyons, 1520.]

[241] "Decima autem figura nihil uocata," "figura nihili (quam etiam cifram uocant)." [Stifel, _Arithmetica integra_, 1544, f. 1.]

[242] "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 e o, si chiama nulla, ouero zero, ouero niente." It also found its way into the Dutch arithmetics, e.g. Raets (1576, 1580 ed., f. A_3): "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; Gunther, _Geschichte_, p. 316.)

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

[244] It appears in the Polish arithmetic of K[=l]os (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 circ.u.mference. "Vocatur etiam cyfra, quasi circ.u.mfacta vel circ.u.mferenda, quod idem est, quod circulus non habito respectu ad centrum." [Loc. cit., p. 26.]

[245] _Opera mathematica_, 1695, Oxford, Vol. I, chap. ix, _Mathesis universalis_, "De figuris numeralibus," pp. 46-49; Vol. II, _Algebra_, p.

10.

[246] Martin, _Origine de notre systeme de numeration ecrite_, note 149, p.

36 of reprint, spells [Greek: tsiphra] from Maximus Planudes, citing Wallis as an authority. This is an error, for Wallis gives the correct form as above.

Alexander von Humboldt, "uber die bei verschiedenen Volkern ublichen Systeme von Zahlzeichen und uber den Ursprung des Stellenwerthes in den indischen Zahlen," Crelle's _Journal fur reine und angewandte Mathematik_, Vol. IV, 1829, called attention to the work [Greek: arithmoi Indikoi] of the monk Neophytos, supposed to be of the fourteenth century. In this work the forms [Greek: tzuphra] and [Greek: tzumphra] appear. See also Boeckh, _De abaco Graecorum_, Berlin, 1841, and Tannery, "Le Scholie du moine Neophytos," _Revue Archeologique_, 1885, pp. 99-102. Jordan, loc. cit., gives from twelfth and thirteenth century ma.n.u.scripts the forms _cifra_, _ciffre_, _chifras_, and _cifrus_. Du Cange, _Glossarium mediae et infimae Latinitatis_, Paris, 1842, gives also _chilerae_. Dasypodius, _Inst.i.tutiones Mathematicae_, Stra.s.sburg, 1593-1596, adds the forms _zyphra_ and _syphra_. Boissiere, _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 quant.i.te par soy, ains seulement les loges vuides."

[247] _Propagation_, pp. 27, 234, 442. Treutlein, "Das Rechnen im 16.

Jahrhundert," _Abhandlungen zur Geschichte der Mathematik_, Vol. I, p. 5, favors the same view. It is combated by many writers, e.g. A. C. Burnell, loc. cit., p. 59. Long before Woepcke, I. F. and G. I. Weidler, _De characteribus numerorum vulgaribus et eorum aetatibus_, Wittenberg, 1727, a.s.serted the possibility of their introduction into Greece by Pythagoras or one of his followers: "Potuerunt autem ex oriente, uel ex phoenicia, ad graecos traduci, uel Pythagorae, uel eius discipulorum auxilio, c.u.m aliquis eo, proficiendi in literis causa, iter faceret, et hoc quoque inuentum addisceret."

[248] E.g., they adopted the Greek numerals in use in Damascus and Syria, and the Coptic in Egypt. Theophanes (758-818 A.D.), _Chronographia_, Scriptores Historiae Byzantinae, Vol. x.x.xIX, Bonnae, 1839, p. 575, relates that in 699 A.D. the caliph Wal[=i]d forbade the use of the Greek language in the bookkeeping of the treasury of the caliphate, but permitted the use of the Greek alphabetic numerals, since the Arabs had no convenient number notation: [Greek: kai ekoluse graphesthai h.e.l.lenisti tous demosious ton logothesion kodikas, all' Arabiois auta parasemainesthai, choris ton psephon, epeide adunaton tei ekeinon glossei monada e duada e triada e okto hemisu e tria graphesthai; dio kai heos semeron eisin sun autois notarioi Christianoi.] The importance of this contemporaneous doc.u.ment was pointed out by Martin, loc. cit. Karabacek, "Die Involutio im arabischen Schriftwesen," Vol. Cx.x.xV of _Sitzungsberichte d. phil.-hist. Cla.s.se d. k.

Akad. d. Wiss._, Vienna, 1896, p. 25, gives an Arabic date of 868 A.D. in Greek letters.

[249] _The Origin and History of Our Numerals_ (in Russian), Kiev, 1908; _The Independence of European Arithmetic_ (in Russian), Kiev.

[250] Woepcke, loc. cit., pp. 462, 262.

[251] Woepcke, loc. cit., p. 240. _[H.]is[=a]b-al-[.G]ob[=a]r_, by an anonymous author, probably Ab[=u] Sahl Dunash ibn Tamim, is given by Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca Mathematica_, 1896, p. 26.

[252] Steinschneider in the _Abhandlungen_, Vol. III, p. 110.

[253] See his _Grammaire arabe_, Vol. I, Paris, 1810, plate VIII; Gerhardt, _etudes_, pp. 9-11, and _Entstehung_ etc., p. 8; I. F. Weidler, _Spicilegium observationum ad historiam notarum numeralium pertinentium_, Wittenberg, 1755, speaks of the "figura cifrarum Saracenicarum" as being different from that of the "characterum Boethianorum," which are similar to the "vulgar" or common numerals; see also Humboldt, loc. cit.

[254] Gerhardt mentions it in his _Entstehung_ etc., p. 8; Woepcke, _Propagation_, states that these numerals were used not for calculation, but very much as we use Roman numerals. These superposed dots are found with both forms of numerals (_Propagation_, pp. 244-246).

[255] Gerhardt (_etudes_, p. 9) from a ma.n.u.script in the Bibliotheque Nationale. The numeral forms are [symbols], 20 being indicated by [symbol with dot] and 200 by [symbol with 2 dots]. This scheme of zero dots was also adopted by the Byzantine Greeks, for a ma.n.u.script of Planudes in the Bibliotheque Nationale has numbers like [pi alpha with 4 dots] for 8,100,000,000. See Gerhardt, _etudes_, p. 19. Pihan, _Expose_ etc., p. 208, gives two forms, Asiatic and Maghrebian, of "Ghob[=a]r" numerals.

[256] See Chap. IV.

[257] Possibly as early as the third century A.D., but probably of the eighth or ninth. See Cantor, I (3), p. 598.

[258] Ascribed by the Arabic writer to India.

[259] See Woepcke's description of a ma.n.u.script in the Chasles library, "Recherches sur l'histoire des sciences mathematiques chez les orientaux,"

_Journal Asiatique_, IV (5), 1859, p. 358, note.

[260] P. 56.

[261] Reinaud, _Memoire sur l'Inde_, p. 399. In the fourteenth century one Sih[=a]b al-D[=i]n wrote a work on which, a scholiast to the Bodleian ma.n.u.script remarks: "The science is called Algobar because the inventor had the habit of writing the figures on a tablet covered with sand." [Gerhardt, _etudes, _p. 11, note.]