P. HISTORY OF ' AATHEMATICAL - School of Mathematics

HINDU-ARABIC NUMERALS
the old letter-numerals, mainly, no doubt, in daily intercourse." At
the close of one of the Byzantine manuscripts there is a table of
numerals containing an imitation of the Old Attic numerals. The table
gives also the Hindu-Arabic numerals, but apparently without recognition
of the principle of local value; in writing 80, the 0 is placed over
the 8. This procedure is probably due to the ignorance of the scribe.
85. A manuscript' of the twelfth century, in Latin, contains the
symbol t for 3 which Curtze and Nag12 declare to have been found
only in the twelfth century. According to Curtze, the foregoing
strange symbol for 3 is simply the symbol for tertia used in the notation
for sexagksirnal fractions which receive much attention in this
manuscript.
86. Recently the variations in form of our numerals have been summarized
as follows: "The form3 of the numerals 1, 6, 8 and 9 has not
varied much among the [medieval] Arabs nor among the Christians
of the Occident; the numerals of the Arabs of the Occident for 2,3 and
5 have forms offering some analogy to ours (the 3 and 5 are originally
reversed, as well among the Christians as among the Arabs of the
Occident); but the form of 4 and that of 7 have greatly modified
themselves. The numerals 5, 6, 7, 8 of the Arabs of the Orient differ
distinctly from those of the Arabs of the Occident (Gobar numerals).
For five one still writes 5 and 3 ." The use of i for 1 occurs in the first
printed arithmetic (Treviso, 1478), presumably because in this early
stage of printing there was no type for 1. Thus, 9,341 was printed
9342'.
87. Many points of historical interest are contained in the following
quotations from the writings of Alexander von Humboldt.
Although over a century old, they still are valuable.
"In the Gobar4 the group signs are dots, that is zeroes, for in
India, Tibet and Persia the zeroes and dots are identical. The Gobar
symbols, which since the year 1818 have commanded my whole attention,
were discovered by my friend and teacher, Mr. Silvestre de
Sacy, in a manuscript from the Library of the old Abbey St. Germain
du PrCs. This great orientalist says: 'Le Gobar a un grand rapport
1 Algorithmus-MSS Clm 13021, fols. 27-29, of the Munich Staatsbibliothek.
Printed and explained by Maximilian Curtze, Abhandlungen zur Gesehiehte dm
Mathematik, Vol. VIII (Leipzig, 1898), p. 3-27.
ZZeitsehrijt fur Mathematik und Physik (Hiet. Litt. Abth.), Vol. XXXIV
(Leipzig, 1889), p. 134.
Emye. des Scien. math., Tome I, Vol. I (1904), p. 20, n. 105, 106.
4 Alexander von Humboldt, Crelle's Journal, Vol. IV (1829), p. 223, 224.