[180] From a ma.n.u.script of the second half of the thirteenth century, reproduced in "Della vita e delle opere di Leonardo Pisano," Balda.s.sare Boncompagni, Rome, 1852, in _Atti dell' Accademia Pontificia dei nuovi Lincei_, anno V.

[181] From a fourteenth-century ma.n.u.script, as reproduced in _Della vita_ etc., Boncompagni, loc. cit.

[182] From a Tibetan MS. in the library of D. E. Smith.

[183] From a Tibetan block-book in the library of D. E. Smith.

[184] ['S][=a]rad[=a] numerals from _The Kashmirian Atharva-Veda, reproduced by chromophotography from the ma.n.u.script in the University Library at Tubingen_, Bloomfield and Garbe, Baltimore, 1901. Somewhat similar forms are given under "Numeration Cachemirienne," by Pihan, _Expose_ etc., p. 84.

[185] 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.

[186] F. X. Kugler, _Sternkunde und Sterndienst in Babel_, I. Buch, from the beginnings to the time of Christ, Munster i. Westfalen, 1907. It also has numerous tables containing the above zero.

[187] 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 _Archaeologia_, Vol. LXII (1910), p. 137.

[188] R. h.o.e.rnle, "The Baksh[=a]l[=i] Ma.n.u.script," _Indian Antiquary_, Vol.

XVII, pp. 33-48 and 275-279, 1888; Thibaut, _Astronomie, Astrologie und Mathematik_, p. 75; h.o.e.rnle, _Verhandlungen_, loc. cit., p. 132.

[189] 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.

[190] V. A. Smith, _The Early History of India_, 2d ed., Oxford, 1908, p.

14.

[191] Colebrooke, _Algebra, with Arithmetic and Mensuration, from the Sanskrit of Brahmegupta and Bhascara_, London, 1817, pp. 339-340.

[192] Ibid., p. 138.

[193] D. E. Smith, in the _Bibliotheca Mathematica_, Vol. IX (3), pp.

106-110.

[194] As when we use three dots (...).

[195] "The Hindus call the nought explicitly _['s][=u]nyabindu_ '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." [Buhler, _On the Origin of the Indian Alphabet_, p. 53, note.]

[196] Fazzari, _Dell' origine delle parole zero e cifra_, Naples, 1903.

[197] E. Wappler, "Zur Geschichte der Mathematik im 15. Jahrhundert," in the _Zeitschrift fur Mathematik und Physik_, Vol. XLV, _Hist.-lit. Abt._, p. 47. The ma.n.u.script is No. C. 80, in the Dresden library.

[198] J. G. Prandel, _Algebra nebst ihrer literarischen Geschichte_, p.

572, Munich, 1795.

[199] 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?

[200] 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, _Apercu historique sur l'origine et le developpement des methodes en geometrie_, 1875 ed., p. 476; Nesselmann, however, showed (_Algebra der Griechen_, 1842, p. 138), that Ptolemy merely used [Greek: o]

for [Greek: ouden], 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 numeraux_ etc., reprint p. 30, and J. Brandis, _Das Munz-, Ma.s.s- und Gewichtswesen in Vorderasien bis auf Alexander den Grossen_, Berlin, 1866, p. 10, also discuss this usage of [Greek: o], without the notion of place value, by the Greeks.

[201] _Al-Batt[=a]n[=i] sive Albatenii opus astronomic.u.m_. 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.

[202] Loc. cit., Vol. II, p. 271.

[203] C. Henry, "Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum," _Abhandlungen zur Geschichte der Mathematik_, Vol. III, 1880.

[204] 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 uber die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," _Zeitschrift fur Mathematik und Physik, Hist.-lit.

Abth._, Vol. x.x.xIV, pp. 129-146 and 161-170, with one plate.

[205] "Byzantinische a.n.a.lekten," _Abhandlungen zur Geschichte der Mathematik_, Vol. IX, pp. 161-189.

[206] [symbol] or [symbol] for 0. [symbol] also used for 5. [symbols] for 13. [Heiberg, loc. cit.]

[207] Gerhardt, _etudes historiques sur l'arithmetique de position_, Berlin, 1856, p. 12; J. Bowring, _The Decimal System in Numbers, Coins, & Accounts_, London, 1854, p. 33.

[208] Karabacek, _Wiener Zeitschrift fur die Kunde des Morgenlandes_, Vol.

XI, p. 13; _Fuhrer durch die Papyrus-Ausstellung Erzherzog Rainer_, Vienna, 1894, p. 216.

[209] In the library of G. A. Plimpton, Esq.

[210] 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.

[211] 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 Hebraeam originem sapit refertque: & ut docti arbitrantur, a verbo saphar, quod Ordine numerauit significat. Unde Sephar numerus est: hinc Siphra (vulgo corruptius). Etsi ver gens Iudaica his notis, quae hodie Siphrae vocantur, usa non fuit: mansit tamen rei appellatio apud multas gentes." Dasypodius, _Inst.i.tutiones 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)?", _etudes de philologie neo-grecque_, Paris, 1892.

[212] Buhler, loc. cit., p. 78 and p. 86.

[213] 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.

[214] "c.u.m his novem figuris, et c.u.m hoc signo 0, quod arabice zephirum appellatur, scribitur quilibet numerus."

[215] [Greek: tziphra], 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 nomme par les uns, _sipos, rota, galgal_ ...; par les autres _tsiphra_ (de [Hebrew: TSPR], _couronne_ ou _diademe_) ou _ciphra_ (de [Hebrew: SPR], _numeration_)." Ch. de Paravey, _Essai sur l'origine unique et hieroglyphique des chiffres et des lettres de tous les peuples_, Paris, 1826, p. 165, a rather fanciful work, gives "vase, vase arrondi et ferme par un couvercle, qui est le symbole de la 10^e Heure, [symbol]," among the Chinese; also "Tsiphron Zeron, ou tout a fait vide en arabe, [Greek: tziphra] en grec ... d'ou chiffre (qui derive plutot, suivant nous, de l'Hebreu _Sepher_, compter.")

[216] "Compilatus a Magistro Jacobo de Florentia apud montem pesalanum,"

and described by G. Lami in his _Catalogus codic.u.m ma.n.u.scriptorum qui in bibliotheca Riccardiana Florentiae adservantur_. See Fazzari, loc. cit., p.

5.

[217] "Et doveto sapere chel zeuero per se solo non significa nulla ma e 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.]

[218] Ibid., p. 6.

[219] Avicenna (980-1036), translation by Gasbarri et Francois, "piu 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 se stesso non esprime nessun numero." This quotation is taken from D. C. Martines, _Origine e progressi dell' aritmetica_, Messina, 1865.

[220] Leo Jordan, "Materialien zur Geschichte der arabischen Zahlzeichen in Frankreich," _Archiv fur Kulturgeschichte_, Berlin, 1905, pp. 155-195, gives the following two schemes of derivation, (1) "zefiro, zeviro, zeiro, zero," (2) "zefiro, zefro, zevro, zero."

[221] Kobel (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."

[222] "Decimo X 0 theca, circul[us] cifra sive figura nihili appelat'."

[_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."