Elementary arithmetic, with brief notices of its history... by Robert Potts.

INTRODUCTION.

7

position of considerable importance in the recording of dates, indexes,
and monumental inscriptions in Western Europe.
In the early period of the history of the Hellenic race, before their
wants and necessities called for the use of large numbers, the initial
letters of the names of numbers were employed to express the numbers
themselves. Thus, the letters, I, II, A H, X, M, being the initial letters of
the words"Ioe (for Ztc), I[ivre, Atha, 'Hectaro', XiLXoI, Mvplot, were employed
to express 1, 5, 10, 100, 1000, 10,000. The other numbers were
expressed by repeating or combining these six characters. Abbreviated combinations were also employed; as when any of these
numeral letters were written within the capital letter II, they denoted
a number five times as great, as A written within II denoted 50. This
method was probably the first step towards a system of numerical
notation, and, except for inscriptions, was superseded for the more perfect system formed with the letters of the alphabet. At a subsequent
period the Greeks employed the letters of their alphabet with three
supplementary symbols to express the first order of digits 1 to 9;
the second, 10 to 90; and the third, 100 to 900. The fourth order of
thousands was formed by subscribing an t to each of the units of the
first order. The fifth, sixth, &c., orders were formed by affixing
M. or Mv. for Mvlptot, 10,000, to each character of the first, second,
&c., orders.
Of the three supplementary symbols employed in the Greek
numerical notation, there being no letter corresponding to the Hebrew
vau, the character < is used for the number 6, and called &oitallov
/3a, indicatingvau. The other two symbols were 5 for 90, and ) for
900, the former called iriaerlov KOwTrC and the latter ErCiarpoo' acvTmr,
that is, indicating Koph, and indicating Tsadi. By combining these
symbols any other numbers could be expressed as oa denotes 41; va,
401; act, 4001; ascXr, 1234; XM.Mv, 370,000. Neither the order nor
the number of the characters had any effect in fixing the value of any
number intended to be expressed. The value of the same combination
of symbols is the same in whatever order they are placed; in general,
however, they were written from right to left, according to their
increasing value.
The oldest arithmetical writings of the Greeks which have descended
to modern times are to be found in the seventh, eighth, ninth, and
tenth books of the Elements of Euclid, who lived between B.c. 323
and 284. These four books contain complete treatises on numbers,
their properties, proportions, commensurable and incommensurable,
and their application to geometry. Diophantus, who has been placed
by some writers as early as A.D. 280, by others as contemporary with
the Emperor Julian, was the author of thirteen books on arithmetic
and algebra. Only six of the thirteen books are known to be extant.
The first four were translated into French by Simon Stevin, and the
other two by Albert Girard. The six books are printed in the collected works of Stevin, revised and augmented by his friend Albert
Girard in 1634.
The progress of astronomical and mathematical science occasioned
the necessity of larger numbers than could be expressed by the Greek
notation then in use. Archimedes, who lived between B.c. 287 and
212, improved and extended the Greek method of notation. In his
work entitled auiTrrnc or Arenarius, he proposed to express a number
which should exceed the number of the grains of sand that might be