tables in plate XXIII.

[116] This was asserted by G. Hager (_Memoria sulle cifre arabiche_, Milan,
1813, also published in _Fundgruben des Orients_, Vienna, 1811, and in
_Bibliotheque Britannique_, Geneva, 1812). See also the recent article by
Major Charles E. Woodruff, "The Evolution of Modern Numerals from Tally
Marks," _American Mathematical Monthly_, August-September, 1909.
Biernatzki, "Die Arithmetik der Chinesen," _Crelle's Journal fuer die reine
und angewandte Mathematik_, Vol. LII, 1857, pp. 59-96, also asserts the
priority of the Chinese claim for a place system and the zero, but upon the
flimsiest authority. Ch. de Paravey, _Essai sur l'origine unique et
hieroglyphique des chiffres et des lettres de tous les peuples_, Paris,
1826; G. Kleinwaechter, "The Origin of the Arabic Numerals," _China Review_,
Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp. 28-30; Biot, "Note sur la
connaissance que les Chinois ont eue de la valeur de position des
chiffres," _Journal Asiatique_, 1839, pp. 497-502. A. Terrien de
Lacouperie, "The Old Numerals, the Counting-Rods and the Swan-Pan in
China," _Numismatic Chronicle_, Vol. III (3), pp. 297-340, and Crowder B.
Moseley, "Numeral Characters: Theory of Origin and Development," _American
Antiquarian_, Vol. XXII, pp. 279-284, both propose to derive our numerals
from Chinese characters, in much the same way as is done by Major Woodruff,
in the article above cited.

[117] The Greeks, probably following the Semitic custom, used nine letters
of the alphabet for the numerals from 1 to 9, then nine others for 10 to
90, and further letters to represent 100 to 900. As the ordinary Greek
alphabet was insufficient, containing only twenty-four letters, an alphabet
of twenty-seven letters was used.

[118] _Institutiones mathematicae_, 2 vols., Strassburg, 1593-1596, a
somewhat rare work from which the following quotation is taken:

"_Quis est harum Cyphrarum autor?_

"A quibus hae usitatae syphrarum notae sint inventae: hactenus incertum
fuit: meo tamen iudicio, quod exiguum esse fateor: a graecis librarijs
(quorum olim magna fuit copia) literae Graecorum quibus veteres Graeci
tamquam numerorum notis sunt usi: fuerunt corruptae. vt ex his licet
videre.

"Graecorum Literae corruptae.

[Illustration]

_"Sed qua ratione graecorum literae ita fuerunt corruptae?_

"Finxerunt has corruptas Graecorum literarum notas: vel abiectione vt in
nota binarij numeri, vel additione vt in ternarij, vel