[135] Th. H. Martin, _Les signes numeraux_ ..., Rome, 1864; Lassen,
_Indische Alterthumskunde_, Vol. II, 2d ed., Leipzig and London, 1874, p.
1153.
[136] But see Burnell, loc. cit., and Thibaut, _Astronomie, Astrologie und
Mathematik_, p. 71.
[137] A. Barth, "Inscriptions Sanscrites du Cambodge," in the _Notices et
extraits des Mss. de la Bibliotheque nationale_, Vol. XXVII, Part I, pp.
1-180, 1885; see also numerous articles in _Journal Asiatique_, by
Aymonier.
[138] Buehler, loc. cit., p. 82.
[139] Loc. cit., p. 79.
[140] Buehler, loc. cit., p. 83. The Hindu astrologers still use an
alphabetical system of numerals. [Burnell, loc. cit., p. 79.]
[141] Well could Ramus say, "Quicunq; autem fuerit inventor decem notarum
laudem magnam meruit."
[142] Al-B[=i]r[=u]n[=i] gives lists.
[143] _Propagation_, loc. cit., p. 443.
[144] See the quotation from _The Light of Asia_ in Chapter II, p. 16.
[145] The nine ciphers were called _a[.n]ka_.
[146] "Zur Geschichte des indischen Ziffernsystems," _Zeitschrift fuer die
Kunde des Morgenlandes_, Vol. IV, 1842, pp. 74-83.
[147] It is found in the Bakh[s.][=a]l[=i] MS. of an elementary arithmetic
which Hoernle placed, at first, about the beginning of our era, but the
date is much in question. G. Thibaut, loc. cit., places it between 700 and
900 A.D.; Cantor places the body of the work about the third or fourth
century A.D., _Geschichte der Mathematik_, Vol. I (3), p. 598.
[148] For the opposite side of the case see G. R. Kaye, "Notes on Indian
Mathematics, No. 2.--[=A]ryabha[t.]a," _Journ. and Proc. of the Asiatic
Soc. of Bengal_, Vol. IV, 1908, pp. 111-141.
[149] He used one of the alphabetic systems explained above. This ran up to
10^{18} and was not difficult, beginning as follows:
[Illustration]
the same letter (_ka_) appearing in the successive consonant forms, _ka_,
_kha_, _ga_, _gha_, etc. See C. I. Gerhardt, _Ueber die Entstehung und
Ausbreitung des dekadischen Zahlensystems_, Programm, p. 17, Salzwedel,
1853, and _Etudes historiques sur l'arithmetique de position_, Programm, p.
24, Berlin, 1856; E. Jacquet, _Mode d'expression symbolique des nombres_,
loc. cit., p. 97; L. Rodet, "Sur la veritable signification de la notation
numerique inventee par [=A]ryabhata," _Journal Asiatique_, Vol. XVI (7),
pp. 440-485. On the two [=A]ryabha[t.]as see Kaye, _Bibl. Math._, Vol. X
(3), p. 289.
[150] Using _kha_, a synonym of _['s][=u]nya_. [Bayley, loc. cit., p. 22,
and L. Rodet, _Journal Asiatique_, Vol. XVI (7), p. 443.]
[151] Var[=a]ha-Mihira, _Pancasiddh[=a]ntik[=a]_, translated by G. Thibaut
and M. S. Dvived[=i], Benares, 1889; see Buehler, loc. cit., p. 78; Bayley,
loc. cit., p. 23.
[152] _B[r.]hat Sa[m.]hit[=a]_, translated by Kern, _Journal of the Royal
Asiatic Society_, 1870-1875.
[153] It is stated by Buehler in a personal letter to Bayley (loc. cit., p.
65) that there are hundreds of instances of this usage in the _B[r.]hat
Sa[m.]hit[=a]_. The system was also used in the _Pancasiddh[=a]ntik[=a]_ as
early as 505 A.D. [Buehler, _Palaeographie_, p. 80, and Fleet, _Journal of
the Royal Asiatic Society_, 1910, p. 819.]
[154] Cantor, _Geschichte der Mathematik_, Vol. I (3), p. 608.
[155] Buehler, loc. cit., p. 78.
[156] Bayley, p. 38.
[157] Noviomagus, in his _De numeris libri duo_, Paris, 1539, confesses his
ignorance as to the origin of the zero, but says: "D. Henricus Grauius, vir
Graece & Hebraice exime doctus, Hebraicam originem ostendit," adding that
Valla "Indis Orientalibus gentibus inventionem tribuit."
[158] See _Essays_, Vol. II, pp. 287 and 288.
[159] Vol. XXX, p. 205 seqq.
[160] Loc. cit., p. 284 seqq.
[161] Colebrooke, loc. cit., p. 288.
[162] Loc. cit., p. 78.
[163] Hereafter, unless expressly stated to the contrary, we shall use the
word "numerals" to mean numerals with place value.
[164] "The Gurjaras of R[=a]jput[=a]na and Kanauj," in _Journal of the
Royal Asiatic Society_, January and April, 1909.
[165] Vol. IX, 1908, p. 248.
[166] _Epigraphia Indica_, Vol. IX, pp. 193 and 198.
[167] _Epigraphia Indica_, Vol. IX, p. 1.
[168] Loc. cit., p. 71.
[169] Thibaut, p. 71.
[170] "Est autem in aliquibus figurarum istaram apud multos diuersitas.
Quidam enim septimam hanc figuram representant," etc. [Boncompagni,
_Trattati_, p. 28.] Enestroem has shown that very likely this work is
incorrectly attributed to Johannes Hispalensis. [_Bibliotheca Mathematica_,
Vol. IX (3), p. 2.]
[171] _Indische Palaeographie_, Tafel IX.
[172] Edited by Bloomfield and Garbe, Baltimore, 1901, containing
photographic reproductions of the manuscript.
[173] Bakh[s.][=a]l[=i] MS. See page 43; Hoernle, R., _The Indian
Antiquary_, Vol. XVII, pp. 33-48, 1 plate; Hoernle, _Verhandlungen des VII.
Internationalen Orientalisten-Congresses, Arische Section_, Vienna, 1888,
"On the Baksh[=a]l[=i] Manuscript," pp. 127-147, 3 plates; Buehler, loc.
cit.
[174] 3, 4, 6, from H. H. Dhruva, "Three Land-Grants from Sankheda,"
_Epigraphia Indica_, Vol. II, pp. 19-24 with plates; date 595 A.D. 7, 1, 5,
from Bhandarkar, "Daulatabad Plates," _Epigraphia Indica_, Vol. IX, part V;
date c. 798 A.D.
[175] 8, 7, 2, from "Buckhala Inscription of Nagabhatta," Bhandarkar,
_Epigraphia Indica_, Vol. IX, part V; date 815 A.D. 5 from "The Morbi
Copper-Plate," Bhandarkar, _The Indian Antiquary_, Vol. II, pp. 257-258,
with plate; date 804 A.D. See Buehler, loc. cit.
[176] 8 from the above Morbi Copper-Plate. 4, 5, 7, 9, and 0, from "Asni
Inscription of Mahipala," _The Indian Antiquary_, Vol. XVI, pp. 174-175;
inscription is on red sandstone, date 917 A.D. See Buehler.
[177] 8, 9, 4, from "Rashtrakuta Grant of Amoghavarsha," J. F. Fleet, _The
Indian Antiquary_, Vol. XII, pp. 263-272; copper-plate grant of date c. 972
A.D. See Buehler. 7, 3, 5, from "Torkhede Copper-Plate Grant of the Time of
Govindaraja of Gujerat," Fleet, _Epigraphia Indica_, Vol. III, pp. 53-58.
See Buehler.
[178] From "A Copper-Plate Grant of King Tritochanapala Chanlukya of
L[=a][t.]ade['s]a," H.H. Dhruva, _Indian Antiquary_, Vol. XII, pp. 196-205;
date 1050 A.D. See Buehler.
[179] Burnell, A. C., _South Indian Palaeography_, plate XXIII,
Telugu-Canarese numerals of the eleventh century. See Buehler.
[180] From a manuscript of the second half of the thirteenth century,
reproduced in "Della vita e delle opere di Leonardo Pisano," Baldassare
Boncompagni, Rome, 1852, in _Atti dell' Accademia Pontificia dei nuovi
Lincei_, anno V.
[181] From a fourteenth-century manuscript, 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 manuscript in the University
Library at Tuebingen_, 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, Muenster 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. Hoernle, "The Baksh[=a]l[=i] 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.
[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." [Buehler, _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 fuer Mathematik und Physik_, Vol. XLV, _Hist.-lit. Abt._,
p. 47. The manuscript is No. C. 80, in the Dresden library.
[198] J. G. Praendel, _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 Muenz-, Mass- 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 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.
[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 ueber
die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im
christl. Abendlande," _Zeitschrift fuer Mathematik und Physik, Hist.-lit.
Abth._, Vol. XXXIV, pp. 129-146 and 161-170, with one plate.
[205] "Byzantinische Analekten," _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 fuer die Kunde des Morgenlandes_, Vol.
XI, p. 13; _Fuehrer 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
vero gens Iudaica his notis, quae hodie Siphrae 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)?", _Etudes de
philologie neo-grecque_, Paris, 1892.
[212] Buehler, 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] "Cum his novem figuris, et cum 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 codicum manuscriptorum 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 fuer Kulturgeschichte_, Berlin, 1905, pp. 155-195,
gives the following two schemes of derivation, (1) "zefiro, zeviro, zeiro,
zero," (2) "zefiro, zefro, zevro, zero."
[221] Koebel (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."
[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 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 Enestroem, _Oefversigt af Kongl.
Vetenskaps-Akademiens Foerhandlingar_, 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 asserted that
it first appeared in Calandri (1491) in this sentence: "Sono dieci le
figure con le quali ciascuno numero si puo 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'Arsenal_,
Vol. III, pp. 154-156, this work is No. 2904 (184 S.A.F.), Bibl. Nat., and
is also called _Petit 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 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 [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."
[Koebel'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.
XXXVII) 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 cum multis,
quam alii thecam, alii figuram nihili, alii figuram privationis, seu
figuram nullam vocant, alii ciphram, cum tamen hodie omnes hae notae vulgo
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; Guenther, _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 circumference. "Vocatur etiam cyfra, quasi circumfacta vel
circumferenda, 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, "Ueber die bei verschiedenen Voelkern ueblichen
Systeme von Zahlzeichen und ueber den Ursprung des Stellenwerthes in den
indischen Zahlen," Crelle's _Journal fuer 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 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_. 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 quantite 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,
asserted 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, cum 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. XXXIX, 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 Hellenisti 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 document was pointed
out by Martin, loc. cit. Karabacek, "Die Involutio im arabischen
Schriftwesen," Vol. CXXXV of _Sitzungsberichte d. phil.-hist. Classe 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 manuscript 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 manuscript 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 manuscript 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
manuscript 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.]
[262] Gerhardt, _Entstehung _etc., p. 20.
[263] H. Suter, "Das Rechenbuch des Ab[=u] Zakar[=i]j[=a]
el-[H.]a[s.][s.][=a]r," _Bibliotheca Mathematica_, Vol. II (3), p. 15.
[264] A. Devoulx, "Les chiffres arabes," _Revue Africaine_, Vol. XVI, pp.
455-458.
[265] _Kit[=a]b al-Fihrist_, G. Fluegel, Leipzig, Vol. I, 1871, and Vol. II,
1872. This work was published after Professor Fluegel's death by J. Roediger
and A. Mueller. The first volume contains the Arabic text and the second
volume contains critical notes upon it.
[266] Like those of line 5 in the illustration on page 69.
[267] Woepcke, _Recherches sur l'histoire des sciences mathematiques chez
les orientaux_, loc. cit.; _Propagation, _p. 57.
[268] Al-[H.]a[s.][s.][=a]r's forms, Suter, _Bibliotheca Mathematica_, Vol.
II (3), p. 15.
[269] Woepcke, _Sur une donnee historique_, etc., loc. cit. The name
_[.g]ob[=a]r_ is not used in the text. The manuscript from which these are
taken is the oldest (970 A.D.) Arabic document known to contain all of the
numerals.
[270] Silvestre de Sacy, loc. cit. He gives the ordinary modern Arabic
forms, calling them _Indien_.
[271] Woepcke, "Introduction au calcul Gob[=a]r[=i] et Haw[=a][=i]," _Atti
dell' accademia pontificia dei nuovi Lincei_, Vol. XIX. The adjective
applied to the forms in 5 is _gob[=a]r[=i]_ and to those in 6 _indienne_.
This is the direct opposite of Woepcke's use of these adjectives in the
_Recherches sur l'histoire_ cited above, in which the ordinary Arabic forms
(like those in row 5) are called _indiens_.
These forms are usually written from right to left.
[272] J. G. Wilkinson, _The Manners and Customs of the Ancient Egyptians_,
revised by S. Birch, London, 1878, Vol. II, p. 493, plate XVI.
[273] There is an extensive literature on this "Boethius-Frage." The reader
who cares to go fully into it should consult the various volumes of the
_Jahrbuch ueber die Fortschritte der Mathematik_.
[274] This title was first applied to Roman emperors in posthumous coins of
Julius Caesar. Subsequently the emperors assumed it during their own
lifetimes, thus deifying themselves. See F. Gnecchi, _Monete romane_, 2d
ed., Milan, 1900, p. 299.
[275] This is the common spelling of the name, although the more correct
Latin form is Boetius. See Harper's _Dict. of Class. Lit. and Antiq._, New
York, 1897, Vol. I, p. 213. There is much uncertainty as to his life. A
good summary of the evidence is given in the last two editions of the
_Encyclopaedia Britannica_.
[276] His father, Flavius Manlius Boethius, was consul in 487.
[277] There is, however, no good historic evidence of this sojourn in
Athens.
[278] His arithmetic is dedicated to Symmachus: "Domino suo patricio
Symmacho Boetius." [Friedlein ed., p. 3.]
[279] It was while here that he wrote _De consolatione philosophiae_.
[280] It is sometimes given as 525.
[281] There was a medieval tradition that he was executed because of a work
on the Trinity.
[282] Hence the _Divus_ in his name.
[283] Thus Dante, speaking of his burial place in the monastery of St.
Pietro in Ciel d'Oro, at Pavia, says:
"The saintly soul, that shows
The world's deceitfulness, to all who hear him,
Is, with the sight of all the good that is,
Blest there. The limbs, whence it was driven, lie
Down in Cieldauro; and from martyrdom
And exile came it here."--_Paradiso_, Canto X.
[284] Not, however, in the mercantile schools. The arithmetic of Boethius
would have been about the last book to be thought of in such institutions.
While referred to by Baeda (672-735) and Hrabanus Maurus (c. 776-856), it
was only after Gerbert's time that the _Boetii de institutione arithmetica
libri duo_ was really a common work.
[285] Also spelled Cassiodorius.
[286] As a matter of fact, Boethius could not have translated any work by
Pythagoras on music, because there was no such work, but he did make the
theories of the Pythagoreans known. Neither did he translate Nicomachus,
although he embodied many of the ideas of the Greek writer in his own
arithmetic. Gibbon follows Cassiodorus in these statements in his _Decline
and Fall of the Roman Empire_, chap. xxxix. Martin pointed out with
positiveness the similarity of the first book of Boethius to the first five
books of Nicomachus. [_Les signes numeraux_ etc., reprint, p. 4.]
[287] The general idea goes back to Pythagoras, however.
[288] J. C. Scaliger in his _Poetice_ also said of him: "Boethii Severini
ingenium, eruditio, ars, sapientia facile provocat omnes auctores, sive
illi Graeci sint, sive Latini" [Heilbronner, _Hist. math. univ._, p. 387].
Libri, speaking of the time of Boethius, remarks: "Nous voyons du temps de
Theodoric, les lettres reprendre une nouvelle vie en Italie, les ecoles
florissantes et les savans honores. Et certes les ouvrages de Boece, de
Cassiodore, de Symmaque, surpassent de beaucoup toutes les productions du
siecle precedent." [_Histoire des mathematiques_, Vol. I, p. 78.]
[289] Carra de Vaux, _Avicenne_, Paris, 1900; Woepcke, _Sur
l'introduction_, etc.; Gerhardt, _Entstehung_ etc., p. 20. Avicenna is a
corruption from Ibn S[=i]n[=a], as pointed out by Wuestenfeld, _Geschichte
der arabischen Aerzte und Naturforscher_, Goettingen, 1840. His full name is
Ab[=u] `Al[=i] al-[H.]osein ibn S[=i]n[=a]. For notes on Avicenna's
arithmetic, see Woepcke, _Propagation_, p. 502.
[290] On the early travel between the East and the West the following works
may be consulted: A. Hillebrandt, _Alt-Indien_, containing "Chinesische
Reisende in Indien," Breslau, 1899, p. 179; C. A. Skeel, _Travel in the
First Century after Christ_, Cambridge, 1901, p. 142; M. Reinaud,
"Relations politiques et commerciales de l'empire romain avec l'Asie
orientale," in the _Journal Asiatique_, Mars-Avril, 1863, Vol. I (6), p.
93; Beazley, _Dawn of Modern Geography, a History of Exploration and
Geographical Science from the Conversion of the Roman Empire to A.D. 1420_,
London, 1897-1906, 3 vols.; Heyd, _Geschichte des Levanthandels im
Mittelalter_, Stuttgart, 1897; J. Keane, _The Evolution of Geography_,
London, 1899, p. 38; A. Cunningham, _Corpus inscriptionum Indicarum_,
Calcutta, 1877, Vol. I; A. Neander, _General History of the Christian
Religion and Church_, 5th American ed., Boston, 1855, Vol. III, p. 89; R.
C. Dutt, _A History of Civilization in Ancient India_, Vol. II, Bk. V,
chap, ii; E. C. Bayley, loc. cit., p. 28 et seq.; A. C. Burnell, loc. cit.,
p. 3; J. E. Tennent, _Ceylon_, London, 1859, Vol. I, p. 159; Geo. Turnour,
_Epitome of the History of Ceylon_, London, n.d., preface; "Philalethes,"
_History of Ceylon_, London, 1816, chap, i; H. C. Sirr, _Ceylon and the
Cingalese_, London, 1850, Vol. I, chap. ix. On the Hindu knowledge of the
Nile see F. Wilford, _Asiatick Researches_, Vol. III, p. 295, Calcutta,
1792.
[291] G. Oppert, _On the Ancient Commerce of India_, Madras, 1879, p. 8.
[292] Gerhardt, _Etudes_ etc., pp. 8, 11.
[293] See Smith's _Dictionary of Greek and Roman Biography and Mythology_.
[294] P. M. Sykes, _Ten Thousand Miles in Persia, or Eight Years in Iran_,
London, 1902, p. 167. Sykes was the first European to follow the course of
Alexander's army across eastern Persia.
[295] Buehler, _Indian Br[=a]hma Alphabet_, note, p. 27; _Palaeographie_, p.
2; _Herodoti Halicarnassei historia_, Amsterdam, 1763, Bk. IV, p. 300;
Isaac Vossius, _Periplus Scylacis Caryandensis_, 1639. It is doubtful
whether the work attributed to Scylax was written by him, but in any case
the work dates back to the fourth century B.C. See Smith's _Dictionary of
Greek and Roman Biography_.
[296] Herodotus, Bk. III.
[297] Rameses II(?), the _Sesoosis_ of Diodorus Siculus.
[298] _Indian Antiquary_, Vol. I, p. 229; F. B. Jevons, _Manual of Greek
Antiquities_, London, 1895, p. 386. On the relations, political and
commercial, between India and Egypt c. 72 B.C., under Ptolemy Auletes, see
the _Journal Asiatique_, 1863, p. 297.
[299] Sikandar, as the name still remains in northern India.
[300] _Harper's Classical Dict._, New York, 1897, Vol. I, p. 724; F. B.
Jevons, loc. cit., p. 389; J. C. Marshman, _Abridgment of the History of
India_, chaps. i and ii.
[301] Oppert, loc. cit., p. 11. It was at or near this place that the first
great Indian mathematician, [=A]ryabha[t.]a, was born in 476 A.D.
[302] Buehler, _Palaeographie_, p. 2, speaks of Greek coins of a period
anterior to Alexander, found in northern India. More complete information
may be found in _Indian Coins_, by E. J. Rapson, Strassburg, 1898, pp. 3-7.
[303] Oppert, loc. cit., p. 14; and to him is due other similar
information.
[304] J. Beloch, _Griechische Geschichte_, Vol. III, Strassburg, 1904, pp.
30-31.
[305] E.g., the denarius, the words for hour and minute ([Greek: hora,
lepton]), and possibly the signs of the zodiac. [R. Caldwell, _Comparative
Grammar of the Dravidian Languages_, London, 1856, p. 438.] On the probable
Chinese origin of the zodiac see Schlegel, loc. cit.
[306] Marie, Vol. II, p. 73; R. Caldwell, loc. cit.
[307] A. Cunningham, loc. cit., p. 50.
[308] C. A. J. Skeel, _Travel_, loc. cit., p. 14.
[309] _Inchiver_, from _inchi_, "the green root." [_Indian Antiquary_, Vol.
I, p. 352.]
[310] In China dating only from the second century A.D., however.
[311] The Italian _morra_.
[312] J. Bowring, _The Decimal System_, London, 1854, p. 2.
[313] H. A. Giles, lecture at Columbia University, March 12, 1902, on
"China and Ancient Greece."
[314] Giles, loc. cit.
[315] E.g., the names for grape, radish (_la-po_, [Greek: rhaphe]),
water-lily (_si-kua_, "west gourds"; [Greek: sikua], "gourds"), are much
alike. [Giles, loc. cit.]
[316] _Epistles_, I, 1, 45-46. On the Roman trade routes, see Beazley, loc.
cit., Vol. I, p. 179.
[317] _Am. Journ. of Archeol._, Vol. IV, p. 366.
[318] M. Perrot gives this conjectural restoration of his words: "Ad me ex
India regum legationes saepe missi sunt numquam antea visae apud quemquam
principem Romanorum." [M. Reinaud, "Relations politiques et commerciales de
l'empire romain avec l'Asie orientale," _Journ. Asiat._, Vol. I (6), p.
93.]
[319] Reinaud, loc. cit., p. 189. Florus, II, 34 (IV, 12), refers to it:
"Seres etiam habitantesque sub ipso sole Indi, cum gemmis et margaritis
elephantes quoque inter munera trahentes nihil magis quam longinquitatem
viae imputabant." Horace shows his geographical knowledge by saying: "Not
those who drink of the deep Danube shall now break the Julian edicts; not
the Getae, not the Seres, nor the perfidious Persians, nor those born on
the river Tanais." [_Odes_, Bk. IV, Ode 15, 21-24.]
[320] "Qua virtutis moderationisque fama Indos etiam ac Scythas auditu modo
cognitos pellexit ad amicitiam suam populique Romani ultro per legatos
petendam." [Reinaud, loc. cit., p. 180.]
[321] Reinaud, loc. cit., p. 180.
[322] _Georgics_, II, 170-172. So Propertius (_Elegies_, III, 4):
Arma deus Caesar dites meditatur ad Indos
Et freta gemmiferi findere classe maris.
"The divine Caesar meditated carrying arms against opulent India, and with
his ships to cut the gem-bearing seas."
[323] Heyd, loc. cit., Vol. I, p. 4.
[324] Reinaud, loc. cit., p. 393.
[325] The title page of Calandri (1491), for example, represents Pythagoras
with these numerals before him. [Smith, _Rara Arithmetica_, p. 46.] Isaacus
Vossius, _Observationes ad Pomponium Melam de situ orbis_, 1658, maintained
that the Arabs derived these numerals from the west. A learned dissertation
to this effect, but deriving them from the Romans instead of the Greeks,
was written by Ginanni in 1753 (_Dissertatio mathematica critica de
numeralium notarum minuscularum origine_, Venice, 1753). See also Mannert,
_De numerorum quos arabicos vocant vera origine Pythagorica_, Nuernberg,
1801. Even as late as 1827 Romagnosi (in his supplement to _Ricerche
storiche sull' India_ etc., by Robertson, Vol. II, p. 580, 1827) asserted
that Pythagoras originated them. [R. Bombelli, _L'antica numerazione
italica_, Rome, 1876, p. 59.] Gow (_Hist. of Greek Math._, p. 98) thinks
that Iamblichus must have known a similar system in order to have worked
out certain of his theorems, but this is an unwarranted deduction from the
passage given.
[326] A. Hillebrandt, _Alt-Indien_, p. 179.
[327] J. C. Marshman, loc. cit., chaps. i and ii.
[328] He reigned 631-579 A.D.; called Nu['s][=i]rw[=a]n, _the holy one_.
[329] J. Keane, _The Evolution of Geography_, London, 1899, p. 38.
[330] The Arabs who lived in and about Mecca.
[331] S. Guyard, in _Encyc. Brit._, 9th ed., Vol. XVI, p. 597.
[332] Oppert, loc. cit., p. 29.
[333] "At non credendum est id in Autographis contigisse, aut vetustioribus
Codd. MSS." [Wallis, _Opera omnia_, Vol. II, p. 11.]
[334] In _Observationes ad Pomponium Melam de situ orbis_. The question was
next taken up in a large way by Weidler, loc. cit., _De characteribus_
etc., 1727, and in _Spicilegium_ etc., 1755.
[335] The best edition of these works is that of G. Friedlein, _Anicii
Manlii Torquati Severini Boetii de institutione arithmetica libri duo, de
institutione musica libri quinque. Accedit geometria quae fertur
Boetii_.... Leipzig.... MDCCCLXVII.
[336] See also P. Tannery, "Notes sur la pseudo-geometrie de Boece," in
_Bibliotheca Mathematica_, Vol. I (3), p. 39. This is not the geometry in
two books in which are mentioned the numerals. There is a manuscript of
this pseudo-geometry of the ninth century, but the earliest one of the
other work is of the eleventh century (Tannery), unless the Vatican codex
is of the tenth century as Friedlein (p. 372) asserts.
[337] Friedlein feels that it is partly spurious, but he says: "Eorum
librorum, quos Boetius de geometria scripsisse dicitur, investigare veram
inscriptionem nihil aliud esset nisi operam et tempus perdere." [Preface,
p. v.] N. Bubnov in the Russian _Journal of the Ministry of Public
Instruction_, 1907, in an article of which a synopsis is given in the
_Jahrbuch ueber die Fortschritte der Mathematik_ for 1907, asserts that the
geometry was written in the eleventh century.
[338] The most noteworthy of these was for a long time Cantor
(_Geschichte_, Vol. I., 3d ed., pp. 587-588), who in his earlier days even
believed that Pythagoras had known them. Cantor says (_Die roemischen
Agrimensoren_, Leipzig, 1875, p. 130): "Uns also, wir wiederholen es, ist
die Geometrie des Boetius echt, dieselbe Schrift, welche er nach Euklid
bearbeitete, von welcher ein Codex bereits in Jahre 821 im Kloster
Reichenau vorhanden war, von welcher ein anderes Exemplar im Jahre 982 zu
Mantua in die Haende Gerbert's gelangte, von welcher mannigfache
Handschriften noch heute vorhanden sind." But against this opinion of the
antiquity of MSS. containing these numerals is the important statement of
P. Tannery, perhaps the most critical of modern historians of mathematics,
that none exists earlier than the eleventh century. See also J. L. Heiberg
in _Philologus, Zeitschrift f. d. klass. Altertum_, Vol. XLIII, p. 508.
Of Cantor's predecessors, Th. H. Martin was one of the most prominent, his
argument for authenticity appearing in the _Revue Archeologique_ for
1856-1857, and in his treatise _Les signes numeraux_ etc. See also M.
Chasles, "De la connaissance qu'ont eu les anciens d'une numeration
decimale ecrite qui fait usage de neuf chiffres prenant les valeurs de
position," _Comptes rendus_, Vol. VI, pp. 678-680; "Sur l'origine de notre
systeme de numeration," _Comptes rendus_, Vol. VIII, pp. 72-81; and note
"Sur le passage du premier livre de la geometrie de Boece, relatif a un
nouveau systeme de numeration," in his work _Apercu historique sur
l'origine et le developpement des methodes en geometrie_, of which the
first edition appeared in 1837.
[339] J. L. Heiberg places the book in the eleventh century on philological
grounds, _Philologus_, loc. cit.; Woepcke, in _Propagation_, p. 44; Blume,
Lachmann, and Rudorff, _Die Schriften der roemischen Feldmesser_, Berlin,
1848; Boeckh, _De abaco graecorum_, Berlin, 1841; Friedlein, in his Leipzig
edition of 1867; Weissenborn, _Abhandlungen_, Vol. II, p. 185, his
_Gerbert_, pp. 1, 247, and his _Geschichte der Einfuehrung der jetzigen
Ziffern in Europa durch Gerbert_, Berlin, 1892, p. 11; Bayley, loc. cit.,
p. 59; Gerhardt, _Etudes_, p. 17, _Entstehung und Ausbreitung_, p. 14;
Nagl, _Gerbert_, p. 57; Bubnov, loc. cit. See also the discussion by
Chasles, Halliwell, and Libri, in the _Comptes rendus_, 1839, Vol. IX, p.
447, and in Vols. VIII, XVI, XVII of the same journal.
[340] J. Marquardt, _La vie privee des Romains_, Vol. II (French trans.),
p. 505, Paris, 1893.
[341] In a Plimpton manuscript of the arithmetic of Boethius of the
thirteenth century, for example, the Roman numerals are all replaced by the
Arabic, and the same is true in the first printed edition of the book. (See
Smith's _Rara Arithmetica_, pp. 434, 25-27.) D. E. Smith also copied from a
manuscript of the arithmetic in the Laurentian library at Florence, of
1370, the following forms, [Forged numerals
[342] Halliwell, in his _Rara Mathematica, _p. 107, states that the
disputed passage is not in a manuscript belonging to Mr. Ames, nor in one
at Trinity College. See also Woepcke, in _Propagation_, pp. 37 and 42. It
was the evident corruption of the texts in such editions of Boethius as
those of Venice, 1499, Basel, 1546 and 1570, that led Woepcke to publish
his work _Sur l'introduction de l'arithmetique indienne en Occident_.
[343] They are found in none of the very ancient manuscripts, as, for
example, in the ninth-century (?) codex in the Laurentian library which one
of the authors has examined. It should be said, however, that the disputed
passage was written after the arithmetic, for it contains a reference to
that work. See the Friedlein ed., p. 397.
[344] Smith, _Rara Arithmetica_, p. 66.
[345] J. L. Heiberg, _Philologus_, Vol. XLIII, p. 507.
[346] "Nosse autem huius artis dispicientem, quid sint digiti, quid
articuli, quid compositi, quid incompositi numeri." [Friedlein ed., p.
395.]
[347] _De ratione abaci._ In this he describes "quandam formulam, quam ob
honorem sui praeceptoris mensam Pythagoream nominabant ... a posterioribus
appellabatur abacus." This, as pictured in the text, is the common Gerbert
abacus. In the edition in Migne's _Patrologia Latina_, Vol. LXIII, an
ordinary multiplication table (sometimes called Pythagorean abacus) is
given in the illustration.
[348] "Habebant enim diverse formatos apices vel caracteres." See the
reference to Gerbert on p. 117.
[349] C. Henry, "Sur l'origine de quelques notations mathematiques," _Revue
Archeologique_, 1879, derives these from the initial letters used as
abbreviations for the names of the numerals, a theory that finds few
supporters.
[350] E.g., it appears in Schonerus, _Algorithmus Demonstratus_, Nuernberg,
1534, f. A4. In England it appeared in the earliest English arithmetical
manuscript known, _The Crafte of Nombrynge_: "¶ fforthermore ye most
vndirstonde that in this craft ben vsid teen figurys, as here bene writen
for ensampul, [Numerals] ... in the quych we vse teen figurys of Inde.
Questio. ¶ why ten fyguris of Inde? Solucio. for as I have sayd afore thei
were fonde fyrst in Inde of a kynge of that Cuntre, that was called Algor."
See Smith, _An Early English Algorism_, loc. cit.
[351] Friedlein ed., p. 397.
[352] Carlsruhe codex of Gerlando.
[353] Munich codex of Gerlando.
[354] Carlsruhe codex of Bernelinus.
[355] Munich codex of Bernelinus.
[356] Turchill, c. 1200.
[357] Anon. MS., thirteenth century, Alexandrian Library, Rome.
[358] Twelfth-century Boethius, Friedlein, p. 396.
[359] Vatican codex, tenth century, Boethius.
[360] a, h, i, are from the Friedlein ed.; the original in the manuscript
from which a is taken contains a zero symbol, as do all of the six plates
given by Friedlein. b-e from the Boncompagni _Bulletino_, Vol. X, p. 596; f
ibid., Vol. XV, p. 186; g _Memorie della classe di sci., Reale Acc. dei
Lincei_, An. CCLXXIV (1876-1877), April, 1877. A twelfth-century
arithmetician, possibly John of Luna (Hispalensis, of Seville, c. 1150),
speaks of the great diversity of these forms even in his day, saying: "Est
autem in aliquibus figuram istarum apud multos diuersitas. Quidam enim
septimam hanc figuram representant [Symbol] alii autem sic [Symbol], uel
sic [Symbol]. Quidam vero quartam sic [Symbol]." [Boncompagni, _Trattati_,
Vol. II, p. 28.]
[361] Loc. cit., p. 59.
[362] Ibid., p. 101.
[363] Loc. cit., p. 396.
[364] Khosr[=u] I, who began to reign in 531 A.D. See W. S. W Vaux,
_Persia, _London, 1875, p. 169; Th. Noeldeke, _Aufsaetze zur persichen
Geschichte_, Leipzig, 1887, p. 113, and his article in the ninth edition of
the _Encyclopaedia Britannica_.
[365] Colebrooke, _Essays_, Vol. II, p. 504, on the authority of Ibn
al-Adam[=i], astronomer, in a work published by his continuator Al-Q[=a]sim
in 920 A.D.; Al-B[=i]r[=u]n[=i], _India, _Vol. II, p. 15.
[366] H. Suter, _Die Mathematiker_ etc., pp. 4-5, states that
Al-Faz[=a]r[=i] died between 796 and 806.
[367] Suter, loc. cit., p. 63.
[368] Suter, loc. cit., p. 74.
[369] Suter, _Das Mathematiker-Verzeichniss im Fihrist_. The references to
Suter, unless otherwise stated, are to his later work _Die Mathematiker und
Astronomen der Araber_ etc.
[370] Suter, _Fihrist_, p. 37, no date.
[371] Suter, _Fihrist_, p. 38, no date.
[372] Possibly late tenth, since he refers to one arithmetical work which
is entitled _Book of the Cyphers_ in his _Chronology_, English ed., p. 132.
Suter, _Die Mathematiker_ etc., pp. 98-100, does not mention this work; see
the _Nachtraege und Berichtigungen_, pp. 170-172.
[373] Suter, pp. 96-97.
[374] Suter, p. 111.
[375] Suter, p. 124. As the name shows, he came from the West.
[376] Suter, p. 138.
[377] Hankel, _Zur Geschichte der Mathematik_, p. 256, refers to him as
writing on the Hindu art of reckoning; Suter, p. 162.
[378] [Greek: Psephophoria kat' Indous], Greek ed., C. I. Gerhardt, Halle,
1865; and German translation, _Das Rechenbuch des Maximus Planudes_, H.
Waeschke, Halle, 1878.
[379] "Sur une donnee historique relative a l'emploi des chiffres indiens
par les Arabes," Tortolini's _Annali di scienze mat. e fis._, 1855.
[380] Suter, p. 80.
[381] Suter, p. 68.
[382] Sprenger also calls attention to this fact, in the _Zeitschrift d.
deutschen morgenlaend. Gesellschaft_, Vol. XLV, p. 367.
[383] Libri, _Histoire des mathematiques_, Vol. I, p. 147.
[384] "Dictant la paix a l'empereur de Constantinople, l'Arabe victorieux
demandait des manuscrits et des savans." [Libri, loc. cit., p. 108.]
[385] Persian _bagadata_, "God-given."
[386] One of the Abbassides, the (at least pretended) descendants of
`Al-Abb[=a]s, uncle and adviser of Mo[h.]ammed.
[387] E. Reclus, _Asia_, American ed., N. Y., 1891, Vol. IV, p. 227.
[388] _Historical Sketches_, Vol. III, chap. iii.
[389] On its prominence at that period see Villicus, p. 70.
[390] See pp. 4-5.
[391] Smith, D. E., in the _Cantor Festschrift_, 1909, note pp. 10-11. See
also F. Woepcke, _Propagation_.
[392] Enestroem, in _Bibliotheca Mathematica_, Vol. I (3), p. 499; Cantor,
_Geschichte_, Vol. I (3), p. 671.
[393] Cited in Chapter I. It begins: "Dixit algoritmi: laudes deo rectori
nostro atque defensori dicamus dignas." It is devoted entirely to the
fundamental operations and contains no applications.
[394] M. Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca
Mathematica_, Vol. VIII (2), p. 99. See also the reference to this writer
in Chapter I.
[395] Part of this work has been translated from a Leyden MS. by F.
Woepcke, _Propagation_, and more recently by H. Suter, _Bibliotheca
Mathematica_, Vol. VII (3), pp. 113-119.
[396] A. Neander, _General History of the Christian Religion and Church_,
5th American ed., Boston, 1855, Vol. III, p. 335.
[397] Beazley, loc. cit., Vol. I, p. 49.
[398] Beazley, loc. cit., Vol. I, pp. 50, 460.
[399] See pp. 7-8.
[400] The name also appears as Mo[h.]ammed Ab[=u]'l-Q[=a]sim, and Ibn
Hauqal. Beazley, loc. cit., Vol. I, p. 45.
[401] _Kit[=a]b al-mas[=a]lik wa'l-mam[=a]lik._
[402] Reinaud, _Mem. sur l'Inde_; in Gerhardt, _Etudes_, p. 18.
[403] Born at Shiraz in 1193. He himself had traveled from India to Europe.
[404] _Gulistan_ (_Rose Garden_), Gateway the third, XXII. Sir Edwin
Arnold's translation, N. Y., 1899, p. 177.
[405] Cunningham, loc. cit., p. 81.
[406] Putnam, _Books_, Vol. I, p. 227:
"Non semel externas peregrino tramite terras
Jam peragravit ovans, sophiae deductus amore,
Si quid forte novi librorum seu studiorum
Quod secum ferret, terris reperiret in illis.
Hic quoque Romuleum venit devotus ad urbem."
("More than once he has traveled joyfully through remote regions and by
strange roads, led on by his zeal for knowledge and seeking to discover in
foreign lands novelties in books or in studies which he could take back
with him. And this zealous student journeyed to the city of Romulus.")
[407] A. Neander, _General History of the Christian Religion and Church_,
5th American ed., Boston, 1855, Vol. III, p. 89, note 4; Libri, _Histoire_,
Vol. I, p. 143.
[408] Cunningham, loc. cit., p. 81.
[409] Heyd, loc. cit., Vol. I, p. 4.
[410] Ibid., p. 5.
[411] Ibid., p. 21.
[412] Ibid., p. 23.
[413] Libri, _Histoire_, Vol. I, p. 167.
[414] Picavet, _Gerbert, un pape philosophe, d'apres l'histoire et d'apres
la legende_, Paris, 1897, p. 19.
[415] Beazley, loc. cit., Vol. I, chap, i, and p. 54 seq.
[416] Ibid., p. 57.
[417] Libri, _Histoire_, Vol. I, p. 110, n., citing authorities, and p.
152.
[418] Possibly the old tradition, "Prima dedit nautis usum magnetis
Amalphis," is true so far as it means the modern form of compass card. See
Beazley, loc. cit., Vol. II, p. 398.
[419] R. C. Dutt, loc. cit., Vol. II, p. 312.
[420] E. J. Payne, in _The Cambridge Modern History_, London, 1902, Vol. I,
chap. i.
[421] Geo. Phillips, "The Identity of Marco Polo's Zaitun with Changchau,
in T'oung pao," _Archives pour servir a l'etude de l'histoire de l'Asie
orientale_, Leyden, 1890, Vol. I, p. 218. W. Heyd, _Geschichte des
Levanthandels im Mittelalter_, Vol. II, p. 216.
The Palazzo dei Poli, where Marco was born and died, still stands in the
Corte del Milione, in Venice. The best description of the Polo travels, and
of other travels of the later Middle Ages, is found in C. R. Beazley's
_Dawn of Modern Geography_, Vol. III, chap, ii, and Part II.
[422] Heyd, loc. cit., Vol. II, p. 220; H. Yule, in _Encyclopaedia
Britannica_, 9th (10th) or 11th ed., article "China." The handbook cited is
Pegolotti's _Libro di divisamenti di paesi_, chapters i-ii, where it is
implied that $60,000 would be a likely amount for a merchant going to China
to invest in his trip.
[423] Cunningham, loc. cit., p. 194.
[424] I.e. a commission house.
[425] Cunningham, loc. cit., p. 186.
[426] J. R. Green, _Short History of the English People_, New York, 1890,
p. 66.
[427] W. Besant, _London_, New York, 1892, p. 43.
[428] _Baldakin_, _baldekin_, _baldachino_.
[429] Italian _Baldacco_.
[430] J. K. Mumford, _Oriental Rugs_, New York, 1901, p. 18.
[431] Or Girbert, the Latin forms _Gerbertus_ and _Girbertus_ appearing
indifferently in the documents of his time.
[432] See, for example, J. C. Heilbronner, _Historia matheseos universae_,
p. 740.
[433] "Obscuro loco natum," as an old chronicle of Aurillac has it.
[434] N. Bubnov, _Gerberti postea Silvestri II papae opera mathematica_,
Berlin, 1899, is the most complete and reliable source of information;
Picavet, loc. cit., _Gerbert_ etc.; Olleris, _Oeuvres de Gerbert_, Paris,
1867; Havet, _Lettres de Gerbert_, Paris, 1889 ; H. Weissenborn, _Gerbert;
Beitraege zur Kenntnis der Mathematik des Mittelalters_, Berlin, 1888, and
_Zur Geschichte der Einfuehrung der jetzigen Ziffern in Europa durch
Gerbert_, Berlin, 1892; Buedinger, _Ueber Gerberts wissenschaftliche und
politische Stellung_, Cassel, 1851; Richer, "Historiarum liber III," in
Bubnov, loc. cit., pp. 376-381; Nagl, _Gerbert und die Rechenkunst des 10.
Jahrhunderts_, Vienna, 1888.
[435] Richer tells of the visit to Aurillac by Borel, a Spanish nobleman,
just as Gerbert was entering into young manhood. He relates how
affectionately the abbot received him, asking if there were men in Spain
well versed in the arts. Upon Borel's reply in the affirmative, the abbot
asked that one of his young men might accompany him upon his return, that
he might carry on his studies there.
[436] Vicus Ausona. Hatto also appears as Atton and Hatton.
[437] This is all that we know of his sojourn in Spain, and this comes from
his pupil Richer. The stories told by Adhemar of Chabanois, an apparently
ignorant and certainly untrustworthy contemporary, of his going to Cordova,
are unsupported. (See e.g. Picavet, p. 34.) Nevertheless this testimony is
still accepted: K. von Raumer, for example (_Geschichte der Paedagogik_, 6th
ed., 1890, Vol. I, p. 6), says "Mathematik studierte man im Mittelalter bei
den Arabern in Spanien. Zu ihnen gieng Gerbert, nachmaliger Pabst Sylvester
II."
[438] Thus in a letter to Aldaberon he says: "Quos post repperimus
speretis, id est VIII volumina Boeti de astrologia, praeclarissima quoque
figurarum geometriae, aliaque non minus admiranda" (Epist. 8). Also in a
letter to Rainard (Epist. 130), he says: "Ex tuis sumptibus fac ut michi
scribantur M. Manlius (Manilius in one MS.) de astrologia."
[439] Picavet, loc. cit., p. 31.
[440] Picavet, loc. cit., p. 36.
[441] Havet, loc. cit., p. vii.
[442] Picavet, loc. cit., p. 37.
[443] "Con sinistre arti conseguri la dignita del Pontificato.... Lasciato
poi l' abito, e 'l monasterio, e datosi tutto in potere del diavolo."
[Quoted in Bombelli, _L'antica numerazione Italica_, Rome, 1876, p. 41 n.]
[444] He writes from Rheims in 984 to one Lupitus, in Barcelona, saying:
"Itaque librum de astrologia translatum a te michi petenti dirige,"
presumably referring to some Arabic treatise. [Epist. no. 24 of the Havet
collection, p. 19.]
[445] See Bubnov, loc. cit., p. x.
[446] Olleris, loc. cit., p. 361, l. 15, for Bernelinus; and Bubnov, loc.
cit., p. 381, l. 4, for Richer.
[447] Woepcke found this in a Paris MS. of Radulph of Laon, c. 1100.
[_Propagation_, p. 246.] "Et prima quidem trium spaciorum superductio
unitatis caractere inscribitur, qui chaldeo nomine dicitur igin." See also
Alfred Nagl, "Der arithmetische Tractat des Radulph von Laon"
(_Abhandlungen zur Geschichte der Mathematik_, Vol. V, pp. 85-133), p. 97.
[448] Weissenborn, loc. cit., p. 239. When Olleris (_Oeuvres de Gerbert_,
Paris, 1867, p. cci) says, "C'est a lui et non point aux Arabes, que
l'Europe doit son systeme et ses signes de numeration," he exaggerates,
since the evidence is all against his knowing the place value. Friedlein
emphasizes this in the _Zeitschrift fuer Mathematik und Physik_, Vol. XII
(1867), _Literaturzeitung_, p. 70: "Fuer das _System_ unserer Numeration ist
die _Null_ das wesentlichste Merkmal, und diese kannte Gerbert nicht. Er
selbst schrieb alle Zahlen mit den roemischen Zahlzeichen und man kann ihm
also nicht verdanken, was er selbst nicht kannte."
[449] E.g., Chasles, Buedinger, Gerhardt, and Richer. So Martin (_Recherches
nouvelles_ etc.) believes that Gerbert received them from Boethius or his
followers. See Woepcke, _Propagation_, p. 41.
[450] Buedinger, loc. cit., p. 10. Nevertheless, in Gerbert's time one
Al-Man[s.][=u]r, governing Spain under the name of Hish[=a]m (976-1002),
called from the Orient Al-Be[.g][=a]n[=i] to teach his son, so that
scholars were recognized. [Picavet, p. 36.]
[451] Weissenborn, loc. cit., p. 235.
[452] Ibid., p. 234.
[453] These letters, of the period 983-997, were edited by Havet, loc.
cit., and, less completely, by Olleris, loc. cit. Those touching
mathematical topics were edited by Bubnov, loc. cit., pp. 98-106.
[454] He published it in the _Monumenta Germaniae historica_, "Scriptores,"
Vol. III, and at least three other editions have since appeared, viz. those
by Guadet in 1845, by Poinsignon in 1855, and by Waitz in 1877.
[455] Domino ac beatissimo Patri Gerberto, Remorum archiepiscopo, Richerus
Monchus, Gallorum congressibus in volumine regerendis, imperii tui, pater
sanctissime Gerberte, auctoritas seminarium dedit.
[456] In epistle 17 (Havet collection) he speaks of the "De multiplicatione
et divisione numerorum libellum a Joseph Ispano editum abbas Warnerius" (a
person otherwise unknown). In epistle 25 he says: "De multiplicatione et
divisione numerorum, Joseph Sapiens sententias quasdam edidit."
[457] H. Suter, "Zur Frage ueber den Josephus Sapiens," _Bibliotheca
Mathematica_, Vol. VIII (2), p. 84; Weissenborn, _Einfuehrung_, p. 14; also
his _Gerbert_; M. Steinschneider, in _Bibliotheca Mathematica_, 1893, p.
68. Wallis (_Algebra_, 1685, chap. 14) went over the list of Spanish
Josephs very carefully, but could find nothing save that "Josephus Hispanus
seu Josephus sapiens videtur aut Maurus fuisse aut alius quis in Hispania."
[458] P. Ewald, _Mittheilungen, Neues Archiv d. Gesellschaft fuer aeltere
deutsche Geschichtskunde_, Vol. VIII, 1883, pp. 354-364. One of the
manuscripts is of 976 A.D. and the other of 992 A.D. See also Franz
Steffens, _Lateinische Palaeographie_, Freiburg (Schweiz), 1903, pp.
xxxix-xl. The forms are reproduced in the plate on page 140.
[459] It is entitled _Constantino suo Gerbertus scolasticus_, because it
was addressed to Constantine, a monk of the Abbey of Fleury. The text of
the letter to Constantine, preceding the treatise on the Abacus, is given
in the _Comptes rendus_, Vol. XVI (1843), p. 295. This book seems to have
been written c. 980 A.D. [Bubnov, loc. cit., p. 6.]
[460] "Histoire de l'Arithmetique," _Comptes rendus_, Vol. XVI (1843), pp.
156, 281.
[461] Loc. cit., _Gerberti Opera_ etc.
[462] Friedlein thought it spurious. See _Zeitschrift fuer Mathematik und
Physik_, Vol. XII (1867), Hist.-lit. suppl., p. 74. It was discovered in
the library of the Benedictine monastry of St. Peter, at Salzburg, and was
published by Peter Bernhard Pez in 1721. Doubt was first cast upon it in
the Olleris edition (_Oeuvres de Gerbert_). See Weissenborn, _Gerbert_, pp.
2, 6, 168, and Picavet, p. 81. Hock, Cantor, and Th. Martin place the
composition of the work at c. 996 when Gerbert was in Germany, while
Olleris and Picavet refer it to the period when he was at Rheims.
[463] Picavet, loc. cit., p. 182.
[464] Who wrote after Gerbert became pope, for he uses, in his preface, the
words, "a domino pape Gerberto." He was quite certainly not later than the
eleventh century; we do not have exact information about the time in which
he lived.
[465] Picavet, loc. cit., p. 182. Weissenborn, _Gerbert_, p. 227. In
Olleris, _Liber Abaci_ (of Bernelinus), p. 361.
[466] Richer, in Bubnov, loc. cit., p. 381.
[467] Weissenborn, _Gerbert_, p. 241.
[468] Writers on numismatics are quite uncertain as to their use. See F.
Gnecchi, _Monete Romane_, 2d ed., Milan, 1900, cap. XXXVII. For pictures of
old Greek tesserae of Sarmatia, see S. Ambrosoli, _Monete Greche_, Milan,
1899, p. 202.
[469] Thus Tzwivel's arithmetic of 1507, fol. 2, v., speaks of the ten
figures as "characteres sive numerorum apices a diuo Seuerino Boetio."
[470] Weissenborn uses _sipos_ for 0. It is not given by Bernelinus, and
appears in Radulph of Laon, in the twelfth century. See Guenther's
_Geschichte_, p. 98, n.; Weissenborn, p. 11; Pihan, _Expose_ etc., pp.
xvi-xxii.
In Friedlein's _Boetius_, p. 396, the plate shows that all of the six
important manuscripts from which the illustrations are taken contain the
symbol, while four out of five which give the words use the word _sipos_
for 0. The names appear in a twelfth-century anonymous manuscript in the
Vatican, in a passage beginning
Ordine primigeno sibi nomen possidet igin.
Andras ecce locum mox uendicat ipse secundum
Ormis post numeros incompositus sibi primus.
[Boncompagni _Buttetino_, XV, p. 132.] Turchill (twelfth century) gives the
names Igin, andras, hormis, arbas, quimas, caletis, zenis, temenias,
celentis, saying: "Has autem figuras, ut donnus [dominus] Gvillelmus Rx
testatur, a pytagoricis habemus, nomina uero ab arabibus." (Who the William
R. was is not known. Boncompagni _Bulletino_ XV, p. 136.) Radulph of Laon
(d. 1131) asserted that they were Chaldean (_Propagation_, p. 48 n.). A
discussion of the whole question is also given in E. C. Bayley, loc. cit.
Huet, writing in 1679, asserted that they were of Semitic origin, as did
Nesselmann in spite of his despair over ormis, calctis, and celentis; see
Woepcke, _Propagation_, p. 48. The names were used as late as the fifteenth
century, without the zero, but with the superscript dot for 10's, two dots
for 100's, etc., as among the early Arabs. Gerhardt mentions having seen a
fourteenth or fifteenth century manuscript in the Bibliotheca Amploniana
with the names "Ingnin, andras, armis, arbas, quinas, calctis, zencis,
zemenias, zcelentis," and the statement "Si unum punctum super ingnin
ponitur, X significat.... Si duo puncta super ... figuras superponunter,
fiet decuplim illius quod cum uno puncto significabatur," in
_Monatsberichte der K. P. Akad. d. Wiss._, Berlin, 1867, p. 40.
[471] _A chart of ten numerals in 200 tongues_, by Rev. R. Patrick, London,
1812.
[472] "Numeratio figuralis est cuiusuis numeri per notas, et figuras
numerates descriptio." [Clichtoveus, edition of c. 1507, fol. C ii, v.]
"Aristoteles enim uoces rerum [Greek: sumbola] uocat: id translatum, sonat
notas." [Noviomagus, _De Numeris Libri II_, cap. vi.] "Alphabetum decem
notarum." [Schonerus, notes to Ramus, 1586, p. 3 seq.] Richer says: "novem
numero notas omnem numerum significantes." [Bubnov, loc. cit., p. 381.]
[473] "Il y a dix Characteres, autrement Figures, Notes, ou Elements."
[Peletier, edition of 1607, p. 13.] "Numerorum notas alij figuras, alij
signa, alij characteres uocant." [Glareanus, 1545 edition, f. 9, r.] "Per
figuras (quas zyphras uocant) assignationem, quales sunt hae notulae, 1. 2.
3. 4...." [Noviomagus, _De Numeris Libri II_, cap. vi.] Gemma Frisius also
uses _elementa_ and Cardan uses _literae_. In the first arithmetic by an
American (Greenwood, 1729) the author speaks of "a few Arabian _Charecters_
or Numeral Figures, called _Digits_" (p. 1), and as late as 1790, in the
third edition of J. J. Blassiere's arithmetic (1st ed. 1769), the name
_characters_ is still in use, both for "de Latynsche en de Arabische" (p.
4), as is also the term "Cyfferletters" (p. 6, n.). _Ziffer_, the modern
German form of cipher, was commonly used to designate any of the nine
figures, as by Boeschenstein and Riese, although others, like Koebel, used
it only for the zero. So _zifre_ appears in the arithmetic by Borgo, 1550
ed. In a Munich codex of the twelfth century, attributed to Gerland, they
are called _characters_ only: "Usque ad VIIII. enim porrigitur omnis
numerus et qui supercrescit eisdem designator Karacteribus." [Boncompagni
_Bulletino_, Vol. X. p. 607.]
[474] The title of his work is _Prologus N. Ocreati in Helceph_ (Arabic
_al-qeif_, investigation or memoir) _ad Adelardum Batensem magistrum suum_.
The work was made known by C. Henry, in the _Zeitschrift fuer Mathematik und
Physik_, Vol. XXV, p. 129, and in the _Abhandlungen zur Geschichte der
Mathematik_, Vol. III; Weissenborn, _Gerbert_, p. 188.
[475] The zero is indicated by a vacant column.
[476] Leo Jordan, loc. cit., p. 170. "Chifre en augorisme" is the
expression used, while a century later "giffre en argorisme" and "cyffres
d'augorisme" are similarly used.
[477] _The Works of Geoffrey Chaucer_, edited by W. W. Skeat, Vol. IV,
Oxford, 1894, p. 92.
[478] Loc. cit., Vol. III, pp. 179 and 180.
[479] In Book II, chap, vii, of _The Testament of Love_, printed with
Chaucer's Works, loc. cit., Vol. VII, London, 1897.
[480] _Liber Abacci_, published in Olleris, _Oeuvres de Gerbert_, pp.
357-400.
[481] G. R. Kaye, "The Use of the Abacus in Ancient India," _Journal and
Proceedings of the Asiatic Society of Bengal_, 1908, pp. 293-297.
[482] _Liber Abbaci_, by Leonardo Pisano, loc. cit., p. 1.
[483] Friedlein, "Die Entwickelung des Rechnens mit Columnen," _Zeitschrift
fuer Mathematik und Physik_, Vol. X, p. 247.
[484] The divisor 6 or 16 being increased by the difference 4, to 10 or 20
respectively.
[485] E.g. Cantor, Vol. I, p. 882.
[486] Friedlein, loc. cit.; Friedlein, "Gerbert's Regeln der Division" and
"Das Rechnen mit Columnen vor dem 10. Jahrhundert," _Zeitschrift fuer
Mathematik und Physik_, Vol. IX; Bubnov, loc. cit., pp. 197-245; M.
Chasles, "Histoire de l'arithmetique. Recherches des traces du systeme de
l'abacus, apres que cette methode a pris le nom d'Algorisme.--Preuves qu'a
toutes les epoques, jusq'au XVI^e siecle, on a su que l'arithmetique
vulgaire avait pour origine cette methode ancienne," _Comptes rendus_, Vol.
XVII, pp. 143-154, also "Regles de l'abacus," _Comptes rendus_, Vol. XVI,
pp. 218-246, and "Analyse et explication du traite de Gerbert," _Comptes
rendus_, Vol. XVI, pp. 281-299.
[487] Bubnov, loc. cit., pp. 203-204, "Abbonis abacus."
[488] "Regulae de numerorum abaci rationibus," in Bubnov, loc. cit., pp.
205-225.
[489] P. Treutlein, "Intorno ad alcuni scritti inediti relativi al calcolo
dell' abaco," _Bulletino di bibliografia e di storia delle scienze
matematiche e fisiche_, Vol. X, pp. 589-647.
[490] "Intorno ad uno scritto inedito di Adelhardo di Bath intitolato
'Regulae Abaci,'" B. Boncompagni, in his _Bulletino_, Vol. XIV, pp. 1-134.
[491] Treutlein, loc. cit.; Boncompagni, "Intorno al Tractatus de Abaco di
Gerlando," _Bulletino_, Vol. X, pp. 648-656.
[492] E. Narducci, "Intorno a due trattati inediti d'abaco contenuti in due
codici Vaticani del secolo XII," Boncompagni _Bulletino_, Vol. XV, pp.
111-162.
[493] See Molinier, _Les sources de l'histoire de France_, Vol. II, Paris,
1902, pp. 2, 3.
[494] Cantor, _Geschichte_, Vol. I, p. 762. A. Nagl in the _Abhandlungen
zur Geschichte der Mathematik_, Vol. V, p. 85.
[495] 1030-1117.
[496] _Abhandlungen zur Geschichte der Mathematik_, Vol. V, pp. 85-133. The
work begins "Incipit Liber Radulfi laudunensis de abaco."
[497] _Materialien zur Geschichte der arabischen Zahlzeichen in
Frankreich_, loc. cit.
[498] Who died in 1202.
[499] Cantor, _Geschichte_, Vol. I (3), pp. 800-803; Boncompagni,
_Trattati_, Part II. M. Steinschneider ("Die Mathematik bei den Juden,"
_Bibliotheca Mathematica_, Vol. X (2), p. 79) ingeniously derives another
name by which he is called (Abendeuth) from Ibn Da[=u]d (Son of David). See
also _Abhandlungen_, Vol. III, p. 110.
[500] John is said to have died in 1157.
[501] For it says, "Incipit prologus in libro alghoarismi de practica
arismetrice. Qui editus est a magistro Johanne yspalensi." It is published
in full in the second part of Boncompagni's _Trattati d'aritmetica_.
[502] Possibly, indeed, the meaning of "libro alghoarismi" is not "to
Al-Khow[=a]razm[=i]'s book," but "to a book of algorism." John of Luna says
of it: "Hoc idem est illud etiam quod ... alcorismus dicere videtur."
[_Trattati_, p. 68.]
[503] For a resume, see Cantor, Vol. I (3), pp. 800-803. As to the author,
see Enestroem in the _Bibliotheca Mathematica_, Vol. VI (3), p. 114, and
Vol. IX (3), p. 2.
[504] Born at Cremona (although some have asserted at Carmona, in
Andalusia) in 1114; died at Toledo in 1187. Cantor, loc. cit.; Boncompagni,
_Atti d. R. Accad. d. n. Lincei_, 1851.
[505] See _Abhandlungen zur Geschichte der Mathematik_, Vol. XIV, p. 149;
_Bibliotheca Mathematica_, Vol. IV (3), p. 206. Boncompagni had a
fourteenth-century manuscript of his work, _Gerardi Cremonensis artis
metrice practice_. See also T. L. Heath, _The Thirteen Books of Euclid's
Elements_, 3 vols., Cambridge, 1908, Vol. I, pp. 92-94 ; A. A. Bjoernbo,
"Gerhard von Cremonas Uebersetzung von Alkwarizmis Algebra und von Euklids
Elementen," _Bibliotheca Mathematica_, Vol. VI (3), pp. 239-248.
[506] Wallis, _Algebra_, 1685, p. 12 seq.
[507] Cantor, _Geschichte_, Vol. I (3), p. 906; A. A. Bjoernbo,
"Al-Chw[=a]rizm[=i]'s trigonometriske Tavler," _Festskrift til H. G.
Zeuthen_, Copenhagen, 1909, pp. 1-17.
[508] Heath, loc. cit., pp. 93-96.
[509] M. Steinschneider, _Zeitschrift der deutschen morgenlaendischen
Gesellschaft_, Vol. XXV, 1871, p. 104, and _Zeitschrift fuer Mathematik und
Physik_, Vol. XVI, 1871, pp. 392-393; M. Curtze, _Centralblatt fuer
Bibliothekswesen_, 1899, p. 289; E. Wappler, _Zur Geschichte der deutschen
Algebra im 15. Jahrhundert_, Programm, Zwickau, 1887; L. C. Karpinski,
"Robert of Chester's Translation of the Algebra of Al-Khow[=a]razm[=i],"
_Bibliotheca Mathematica_, Vol. XI (3), p. 125. He is also known as
Robertus Retinensis, or Robert of Reading.
[510] Nagl, A., "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und
ueber die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im
christl. Abendlande," in the _Zeitschrift fuer Mathematik und Physik,
Hist.-lit. Abth._, Vol. XXXIV, p. 129. Curtze, _Abhandlungen zur Geschichte
der Mathematik_, Vol. VIII, pp. 1-27.
[511] See line _a_ in the plate on p. 143.
[512] _Sefer ha-Mispar, Das Buch der Zahl, ein hebraeisch-arithmetisches
Werk des R. Abraham ibn Esra_, Moritz Silberberg, Frankfurt a. M., 1895.
[513] Browning's "Rabbi ben Ezra."
[514] "Darum haben auch die Weisen Indiens all ihre Zahlen durch neun
bezeichnet und Formen fuer die 9 Ziffern gebildet." [_Sefer ha-Mispar_, loc.
cit., p. 2.]
[515] F. Bonaini, "Memoria unica sincrona di Leonardo Fibonacci," Pisa,
1858, republished in 1867, and appearing in the _Giornale Arcadico_, Vol.
CXCVII (N.S. LII); Gaetano Milanesi, _Documento inedito e sconosciuto a
Lionardo Fibonacci_, Roma, 1867; Guglielmini, _Elogio di Lionardo Pisano_,
Bologna, 1812, p. 35; Libri, _Histoire des sciences mathematiques_, Vol.
II, p. 25; D. Martines, _Origine e progressi dell' aritmetica_, Messina,
1865, p. 47; Lucas, in Boncompagni _Bulletino_, Vol. X, pp. 129, 239;
Besagne, ibid., Vol. IX, p. 583; Boncompagni, three works as cited in Chap.
I; G. Enestroem, "Ueber zwei angebliche mathematische Schulen im
christlichen Mittelalter," _Bibliotheca Mathematica_, Vol. VIII (3), pp.
252-262; Boncompagni, "Della vita e delle opere di Leonardo Pisano," loc.
cit.
[516] The date is purely conjectural. See the _Bibliotheca Mathematica_,
Vol. IV (3), p. 215.
[517] An old chronicle relates that in 1063 Pisa fought a great battle with
the Saracens at Palermo, capturing six ships, one being "full of wondrous
treasure," and this was devoted to building the cathedral.
[518] Heyd, loc. cit., Vol. I, p. 149.
[519] Ibid., p. 211.
[520] J. A. Symonds, _Renaissance in Italy. The Age of Despots._ New York,
1883, p. 62.
[521] Symonds, loc. cit., p. 79.
[522] J. A. Froude, _The Science of History_, London, 1864. "Un brevet
d'apothicaire n'empecha pas Dante d'etre le plus grand poete de l'Italie,
et ce fut un petit marchand de Pise qui donna l'algebre aux Chretiens."
[Libri, _Histoire_, Vol. I, p. xvi.]
[523] A document of 1226, found and published in 1858, reads: "Leonardo
bigollo quondam Guilielmi."
[524] "Bonaccingo germano suo."
[525] E.g. Libri, Guglielmini, Tiraboschi.
[526] Latin, _Bonaccius_.
[527] Boncompagni and Milanesi.
[528] Reprint, p. 5.
[529] Whence the French name for candle.
[530] Now part of Algiers.
[531] E. Reclus, _Africa_, New York, 1893, Vol. II, p. 253.
[532] "Sed hoc totum et algorismum atque arcus pictagore quasi errorem
computavi respectu modi indorum." Woepcke, _Propagation_ etc., regards this
as referring to two different systems, but the expression may very well
mean algorism as performed upon the Pythagorean arcs (or table).
[533] "Book of the Abacus," this term then being used, and long afterwards
in Italy, to mean merely the arithmetic of computation.
[534] "Incipit liber Abaci a Leonardo filio Bonacci compositus anno 1202 et
correctus ab eodem anno 1228." Three MSS. of the thirteenth century are
known, viz. at Milan, at Siena, and in the Vatican library. The work was
first printed by Boncompagni in 1857.
[535] I.e. in relation to the quadrivium. "Non legant in festivis diebus,
nisi Philosophos et rhetoricas et quadrivalia et barbarismum et ethicam, si
placet." Suter, _Die Mathematik auf den Universitaeten des Mittelalters_,
Zuerich, 1887, p. 56. Roger Bacon gives a still more gloomy view of Oxford
in his time in his _Opus minus_, in the _Rerum Britannicarum medii aevi
scriptores_, London, 1859, Vol. I, p. 327. For a picture of Cambridge at
this time consult F. W. Newman, _The English Universities, translated from
the German of V. A. Huber_, London, 1843, Vol. I, p. 61; W. W. R. Ball,
_History of Mathematics at Cambridge_, 1889; S. Guenther, _Geschichte des
mathematischen Unterrichts im deutschen Mittelalter bis zum Jahre 1525_,
Berlin, 1887, being Vol. III of _Monumenta Germaniae paedagogica_.
[536] On the commercial activity of the period, it is known that bills of
exchange passed between Messina and Constantinople in 1161, and that a bank
was founded at Venice in 1170, the Bank of San Marco being established in
the following year. The activity of Pisa was very manifest at this time.
Heyd, loc. cit., Vol. II, p. 5; V. Casagrandi, _Storia e cronologia_, 3d
ed., Milan, 1901, p. 56.
[537] J. A. Symonds, loc. cit., Vol. II, p. 127.
[538] I. Taylor, _The Alphabet_, London, 1883, Vol. II, p. 263.
[539] Cited by Unger's History, p. 15. The Arabic numerals appear in a
Regensburg chronicle of 1167 and in Silesia in 1340. See Schmidt's
_Encyclopaedie der Erziehung_, Vol. VI, p. 726; A. Kuckuk, "Die Rechenkunst
im sechzehnten Jahrhundert," _Festschrift zur dritten Saecularfeier des
Berlinischen Gymnasiums zum grauen Kloster_, Berlin, 1874, p. 4.
[540] The text is given in Halliwell, _Rara Mathematica_, London, 1839.
[541] Seven are given in Ashmole's _Catalogue of Manuscripts in the Oxford
Library_, 1845.
[542] Maximilian Curtze, _Petri Philomeni de Dacia in Algorismum Vulgarem
Johannis de Sacrobosco commentarius, una cum Algorismo ipso_, Copenhagen,
1897; L. C. Karpinski, "Jordanus Nemorarius and John of Halifax," _American
Mathematical Monthly_, Vol. XVII, pp. 108-113.
[543] J. Aschbach, _Geschichte der Wiener Universitaet im ersten
Jahrhunderte ihres Bestehens_, Wien, 1865, p. 93.
[544] Curtze, loc. cit., gives the text.
[545] Curtze, loc. cit., found some forty-five copies of the _Algorismus_
in three libraries of Munich, Venice, and Erfurt (Amploniana). Examination
of two manuscripts from the Plimpton collection and the Columbia library
shows such marked divergence from each other and from the text published by
Curtze that the conclusion seems legitimate that these were students'
lecture notes. The shorthand character of the writing further confirms this
view, as it shows that they were written largely for the personal use of
the writers.
[546] "Quidam philosophus edidit nomine Algus, unde et Algorismus
nuncupatur." [Curtze, loc. cit., p. 1.]
[547] "Sinistrorsum autera scribimus in hac arte more arabico sive iudaico,
huius scientiae inventorum." [Curtze, loc. cit., p. 7.] The Plimpton
manuscript omits the words "sive iudaico."
[548] "Non enim omnis numerus per quascumque figuras Indorum
repraesentatur, sed tantum determinatus per determinatam, ut 4 non per
5,..." [Curtze, loc. cit., p. 25.]
[549] C. Henry, "Sur les deux plus anciens traites francais d'algorisme et
de geometrie," Boncompagni _Bulletino_, Vol. XV, p. 49; Victor Mortet, "Le
plus ancien traite francais d'algorisme," loc. cit.
[550] _L'Etat des sciences en France, depute la mort du Roy Robert, arrivee
en 1031, jusqu'a celle de Philippe le Bel, arrivee en 1314_, Paris, 1741.
[551] _Discours sur l'etat des lettres en France au XIII^e siecle_, Paris,
1824.
[552] _Apercu historique_, Paris, 1876 ed., p. 464.
[553] Ranulf Higden, a native of the west of England, entered St.
Werburgh's monastery at Chester in 1299. He was a Benedictine monk and
chronicler, and died in 1364. His _Polychronicon_, a history in seven
books, was printed by Caxton in 1480.
[554] Trevisa's translation, Higden having written in Latin.
[555] An illustration of this feeling is seen in the writings of Prosdocimo
de' Beldomandi (b. c. 1370-1380, d. 1428): "Inveni in quam pluribus libris
algorismi nuncupatis mores circa numeros operandi satis varios atque
diversos, qui licet boni existerent atque veri erant, tamen fastidiosi, tum
propter ipsarum regularum multitudinem, tum propter earum deleationes, tum
etiam propter ipsarum operationum probationes, utrum si bone fuerint vel
ne. Erant et etiam isti modi interim fastidiosi, quod si in aliquo calculo
astroloico error contigisset, calculatorem operationem suam a capite
incipere oportebat, dato quod error suus adhuc satis propinquus existeret;
et hoc propter figuras in sua operatione deletas. Indigebat etiam
calculator semper aliquo lapide vel sibi conformi, super quo scribere atque
faciliter delere posset figuras cum quibus operabatur in calculo suo. Et
quia haec omnia satis fastidiosa atque laboriosa mihi visa sunt, disposui
libellum edere in quo omnia ista abicerentur: qui etiam algorismus sive
liber de numeris denominari poterit. Scias tamen quod in hoc libello ponere
non intendo nisi ea quae ad calculum necessaria sunt, alia quae in aliis
libris practice arismetrice tanguntur, ad calculum non necessaria, propter
brevitatem dimitendo." [Quoted by A. Nagl, _Zeitschrift fuer Mathematik und
Physik, Hist.-lit. Abth._, Vol. XXXIV, p. 143; Smith, _Rara Arithmetica_,
p. 14, in facsimile.]
[556] P. Ewald, loc. cit.; Franz Steffens, _Lateinische Palaeographie_, pp.
xxxix-xl. We are indebted to Professor J. M. Burnam for a photograph of
this rare manuscript.
[557] See the plate of forms on p. 88.
[558] Karabacek, loc. cit., p. 56; Karpinski, "Hindu Numerals in the
Fihrist," _Bibliotheca Mathematica_, Vol. XI (3), p. 121.
[559] Woepcke, "Sur une donnee historique," etc., loc. cit., and "Essai
d'une restitution de travaux perdus d'Apollonius sur les quantites
irrationnelles, d'apres des indications tirees d'un manuscrit arabe," _Tome
XIV des Memoires presentes par divers savants a l'Academie des sciences_,
Paris, 1856, note, pp. 6-14.
[560] _Archeological Report of the Egypt Exploration Fund for 1908-1909_,
London, 1910, p. 18.
[561] There was a set of astronomical tables in Boncompagni's library
bearing this date: "Nota quod anno d[=n]i [=n]ri ihu x[=p]i. 1264.
perfecto." See Narducci's _Catalogo_, p. 130.
[562] "On the Early use of Arabic Numerals in Europe," read before the
Society of Antiquaries April 14, 1910, and published in _Archaeologia_ in
the same year.
[563] Ibid., p. 8, n. The date is part of an Arabic inscription.
[564] O. Codrington, _A Manual of Musalman Numismatics_, London, 1904.
[565] See Arbuthnot, _The Mysteries of Chronology_, London, 1900, pp. 75,
78, 98; F. Pichler, _Repertorium der steierischen Muenzkunde_, Graetz, 1875,
where the claim is made of an Austrian coin of 1458; _Bibliotheca
Mathematica_, Vol. X (2), p. 120, and Vol. XII (2), p. 120. There is a
Brabant piece of 1478 in the collection of D. E. Smith.
[566] A specimen is in the British Museum. [Arbuthnot, p. 79.]
[567] Ibid., p. 79.
[568] _Liber de Remediis utriusque fortunae Coloniae._
[569] Fr. Walthern et Hans Hurning, Noerdlingen.
[570] _Ars Memorandi_, one of the oldest European block-books.
[571] Eusebius Caesariensis, _De praeparatione evangelica_, Venice, Jenson,
1470. The above statement holds for copies in the Astor Library and in the
Harvard University Library.
[572] Francisco de Retza, _Comestorium vitiorum_, Nuernberg, 1470. The copy
referred to is in the Astor Library.
[573] See Mauch, "Ueber den Gebrauch arabischer Ziffern und die
Veraenderungen derselben," _Anzeiger fuer Kunde der deutschen Vorzeit_, 1861,
columns 46, 81, 116, 151, 189, 229, and 268; Calmet, _Recherches sur
l'origine des chiffres d'arithmetique_, plate, loc. cit.
[574] Guenther, _Geschichte_, p. 175, n.; Mauch, loc. cit.
[575] These are given by W. R. Lethaby, from drawings by J. T. Irvine, in
the _Proceedings of the Society of Antiquaries_, 1906, p. 200.
[576] There are some ill-tabulated forms to be found in J. Bowring, _The
Decimal System_, London, 1854, pp. 23, 25, and in L. A. Chassant,
_Dictionnaire des abreviations latines et francaises ... du moyen age_,
Paris, MDCCCLXVI, p. 113. The best sources we have at present, aside from
the Hill monograph, are P. Treutlein, _Geschichte unserer Zahlzeichen_,
Karlsruhe, 1875; Cantor's _Geschichte_, Vol. I, table; M. Prou, _Manuel de
paleographie latine et francaise_, 2d ed., Paris, 1892, p. 164; A.
Cappelli, _Dizionario di abbreviature latine ed italiane_, Milan, 1899. An
interesting early source is found in the rare Caxton work of 1480, _The
Myrrour of the World_. In Chap. X is a cut with the various numerals, the
chapter beginning "The fourth scyence is called arsmetrique." Two of the
fifteen extant copies of this work are at present in the library of Mr. J.
P. Morgan, in New York.
[577] From the twelfth-century manuscript on arithmetic, Curtze, loc. cit.,
_Abhandlungen_, and Nagl, loc. cit. The forms are copied from Plate VII in
_Zeitschrift fuer Mathematik und Physik_, Vol. XXXIV.
[578] From the Regensburg chronicle. Plate containing some of these
numerals in _Monumenta Germaniae historica_, "Scriptores" Vol. XVII, plate
to p. 184; Wattenbach, _Anleitung zur lateinischen Palaeographie_, Leipzig,
1886, p. 102; Boehmer, _Fontes rerum Germanicarum_, Vol. III, Stuttgart,
1852, p. lxv.
[579] French Algorismus of 1275; from an unpublished photograph of the
original, in the possession of D. E. Smith. See also p. 135.
[580] From a manuscript of Boethius c. 1294, in Mr. Plimpton's library.
Smith, _Rara Arithmetica_, Plate I.
[581] Numerals in a 1303 manuscript in Sigmaringen, copied from Wattenbach,
loc. cit., p. 102.
[582] From a manuscript, Add. Manuscript 27,589, British Museum, 1360 A.D.
The work is a computus in which the date 1360 appears, assigned in the
British Museum catalogue to the thirteenth century.
[583] From the copy of Sacrabosco's _Algorismus_ in Mr. Plimpton's library.
Date c. 1442. See Smith, _Rara Arithmetica_, p. 450.
[584] See _Rara Arithmetica_, pp. 446-447.
[585] Ibid., pp. 469-470.
[586] Ibid., pp. 477-478.
[587] The i is used for "one" in the Treviso arithmetic (1478), Clichtoveus
(c. 1507 ed., where both i and j are so used), Chiarini (1481), Sacrobosco
(1488 ed.), and Tzwivel (1507 ed., where jj and jz are used for 11 and 12).
This was not universal, however, for the _Algorithmus linealis_ of c. 1488
has a special type for 1. In a student's notebook of lectures taken at the
University of Wuerzburg in 1660, in Mr. Plimpton's library, the ones are all
in the form of i.
[588] Thus the date [Numerals 1580], for 1580, appears in a MS. in the
Laurentian library at Florence. The second and the following five
characters are taken from Cappelli's _Dizionario_, p. 380, and are from
manuscripts of the twelfth, thirteenth, fourteenth, sixteenth, seventeenth,
and eighteenth centuries, respectively.
[589] E.g. Chiarini's work of 1481; Clichtoveus (c. 1507).
[590] The first is from an algorismus of the thirteenth century, in the
Hannover Library. [See Gerhardt, "Ueber die Entstehung und Ausbreitung des
dekadischen Zahlensystems," loc. cit., p. 28.] The second character is from
a French algorismus, c. 1275. [Boncompagni _Bulletino_, Vol. XV, p. 51.]
The third and the following sixteen characters are given by Cappelli, loc.
cit., and are from manuscripts of the twelfth (1), thirteenth (2),
fourteenth (7), fifteenth (3), sixteenth (1), seventeenth (2), and
eighteenth (1) centuries, respectively.
[591] Thus Chiarini (1481) has [Symbol] for 23.
[592] The first of these is from a French algorismus, c. 1275. The second
and the following eight characters are given by Cappelli, loc. cit., and
are from manuscripts of the twelfth (2), thirteenth, fourteenth, fifteenth
(3), seventeenth, and eighteenth centuries, respectively.
[593] See Nagl, loc. cit.
[594] Hannover algorismus, thirteenth century.
[595] See the Dagomari manuscript, in _Rara Arithmetica_, pp. 435, 437-440.
[596] But in the woodcuts of the _Margarita Philosophica_ (1503) the old
forms are used, although the new ones appear in the text. In Caxton's
_Myrrour of the World_ (1480) the old form is used.
[597] Cappelli, loc. cit. They are partly from manuscripts of the tenth,
twelfth, thirteenth (3), fourteenth (7), fifteenth (6), and eighteenth
centuries, respectively. Those in the third line are from Chassant's
_Dictionnaire_, p. 113, without mention of dates.
[598] The first is from the Hannover algorismus, thirteenth century. The
second is taken from the Rollandus manuscript, 1424. The others in the
first two lines are from Cappelli, twelfth (3), fourteenth (6), fifteenth
(13) centuries, respectively. The third line is from Chassant, loc. cit.,
p. 113, no mention of dates.
[599] The first of these forms is from the Hannover algorismus, thirteenth
century. The following are from Cappelli, fourteenth (3), fifteenth,
sixteenth (2), and eighteenth centuries, respectively.
[600] The first of these is taken from the Hannover algorismus, thirteenth
century. The following forms are from Cappelli, twelfth, thirteenth,
fourteenth (5), fifteenth (2), seventeenth, and eighteenth centuries,
respectively.
[601] All of these are given by Cappelli, thirteenth, fourteenth, fifteenth
(2), and sixteenth centuries, respectively.
[602] Smith, _Rara Arithmetica_, p. 489. This is also seen in several of
the Plimpton manuscripts, as in one written at Ancona in 1684. See also
Cappelli, loc. cit.
[603] French algorismus, c. 1275, for the first of these forms. Cappelli,
thirteenth, fourteenth, fifteenth (3), and seventeenth centuries,
respectively. The last three are taken from _Byzantinische Analekten_, J.
L. Heiberg, being forms of the fifteenth century, but not at all common.
[Symbol: Qoppa] was the old Greek symbol for 90.
[604] For the first of these the reader is referred to the forms ascribed
to Boethius, in the illustration on p. 88; for the second, to Radulph of
Laon, see p. 60. The third is used occasionally in the Rollandus (1424)
manuscript, in Mr. Plimpton's library. The remaining three are from
Cappelli, fourteenth (2) and seventeenth centuries.
[605] Smith, _An Early English Algorism_.
[606] Kuckuck, p. 5.
[607] A. Cappelli, loc. cit., p. 372.
[608] Smith, _Rara Arithmetica_, p. 443.
[609] Curtze, _Petri Philomeni de Dacia_ etc., p. IX.
[610] Cappelli, loc. cit., p. 376.
[611] Curtze, loc. cit., pp. VIII-IX, note.
[612] Edition of 1544-1545, f. 52.
[613] _De numeris libri II_, 1544 ed., cap. XV. Heilbronner, loc. cit., p.
736, also gives them, and compares this with other systems.
[614] Noviomagus says of them: "De quibusdam Astrologicis, sive Chaldaicis
numerorum notis.... Sunt & aliae quaedam notae, quibus Chaldaei & Astrologii
quemlibet numerum artificiose & argute describunt, scitu periucundae, quas
nobis communicauit Rodolphus Paludanus Nouiomagus."