P. HISTORY OF ' AATHEMATICAL - School of Mathematics

34 A HISTORY OF MATmMATICAL NOTATIONS
parentheses. Through Priscian it is established that this notation is
at least aa old as 500 A.D.; probably it was much older, but it was not
widely used before the Middle Ages.
54. While the Hindu-Arabic numerals became generally known
in Europe. about 1275, the Roman numerals continued to hold a commanding
place. For example, the fourteenth-century banking-house
of Pemzzi in Florence-Compagnia Pemzzi-did not use Arabic
numerals in their account-books. Roman numerals were used, but
the larger amounts, the thousands of lira, were written out in words;
one finds, for instance, "lb. quindicimilia CXV / V d VI in fiorini"
for 15,115 lira 5 sol& 6 denari; the specification being made that, the
lira are lira a jiorino d'oro at 20 soldi and 12 denari. There appears
also a symbol much like 1, for th0usand.l
Nagl states also: "Specially characteristic is . . . . during all the
Middle Ages, the regular prolongation of the last I in the units, as
V I [= VI I, which had no other purpose than to prevent the subsequent
addition of a further unit."
55. In a book by H. Giraua Tarragones2 at Milan the Roman
numerals appear in the running text and are usually underlined; in
the title-page, the date has the horizontal line above the numerals.
The Roman four is I I I I. In the tables, columns of degrees and minutes
are headed "G.M."; of hour and minutes, "H.M." In the tables, the
Hindu-Arabic numerals appear; the five is printed 5, without the
usual upper stroke. The vitality of the Roman notation is illustrated
further by a German writer, Sebastian Frank, of the sixteenth century,
who uses Roman numerals in numbering the folios of his book
and in his statistics: "Zimmet kumpt von Zailon .CC .VN LX.
teutscher meil von Calicut weyter gelegen. . . . . Die Nagelin kummen
von Meluza / fur Calicut hinaussgelegen vij-c. vnd XL. deutscher
me~l."~ The two numbers given are 260 and 740 German miles. Peculiar
is the insertion of vnd. ("and"). Observe also the use of the
principle of multiplication in vij-c. (=700). In Jakob Kobel's
Rechenbiechlin (Augsburg, 1514), fractions appear in Roman numerals;
IIC
thus, -- stands for ti$.
11110. LX
'Alfred Nagl, Zeilsch~ift fur Mdhemalik und Physik, Vol. XXXIV (1889),
Historisch-literarische Abtheilung, p. 164.
Dos .Lib~os de Cosmog~aphie, compuestos nueuamente por Hieronymo
Giraua Tarragones (Milan, M.L).LVI).
Wellbhh / spiegel vnd bildlnis des gantzen Erdtbodens . . . . von Sebastiano
Franco Wo~demi . . . . (M.D. XXXIIII), fol.. ccxx.