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

8

INTRODUCTION.

contained in the sphere of the universe as conceived by Aristarchus.
He assumed a scale of numeration whose radix or base is a myriad of
myriads, or ten thousand times ten thousand, the number which then
formed the limit of the Greek numerical language. All numbers less
than this radix he called primary numbers, and the radix itself ho
made the unit of secondary numbers; he then proceeds to ternary,
quaternary, and other numbers of higher orders, forming successive
classes.
According to the Indian notation, the units of Archimedes, the
primary, secondary, ternary, quaternary, quinary, &c., orders of
numbers, will consist of 1; 1 with 8 ciphers; 1 with 16 ciphers; 1 with
24 ciphers; 1 with 32 ciphers, &c., respectively. And the unit of the
ten thousand times ten thousandth order would consist of 1 with
799999992 ciphers.
His method for determining the number of places in any required
number is the following. He supposes a series of numbers beginning
with unity, in continued proportion, and shows that the product of any
two terms of this series is equal to that term whose place reckoned
from the first, is less by unity than the sum of the two numbers which
indicate the places of the two terms; as the 7th term of the series
is equal to the product of the 3rd and 5th terms. He then assumes
the series 1, 10, 100, 1000, &c., in which each successive term increases
tenfold, or where the common ratio is 10. The first 8 terms of this
series (omitting the first term) are primary numbers; the next 8 terms,
secondary numbers; the third 8, ternary numbers, and so on; and tho
question is to determine that term in the series which is equal to the
product of any two assigned terms, or the term whose place is the
sum (less by unity) of the two numbers which indicate the places of
the two assigned terms. The classes themselves he calls octades, or'
periods of eight, from each class requiring eight symbols, or eight
places of figures of common notation, to express the numbers included
in each class. He then shows, without finding or assuming the number
itself, that the number requiring for its expression not more than eight
of these octades, or, in our notation, not exceeding 64 places of figures,
will exceed the number of the grains of sand in the sphere of
Aristarchus, each grain of sand being so small that 10,000 of them
are less than one seed of poppy.
Apollonius, about 240 B.c., adopted the plan of Archimedes of
classifying numbers, but instead of the octades of Archimedes, ho
adopted tetrads, reducing the radix from ten thousand times ten thousand, to ten thousand. The units after the first class he designated in
order, the single myriad, the double myriad, the treble myriad, &c.,
which he denoted by MC, Mf3, My, &c., respectively. His chief object,
however, appears to have been to simplify the process of multiplication, and to make the multiplication of all higher numbers dependent
on the product of any two of the first nine digits. The work of
Apollonius has perished, and even the name of its title is unknown.
It is highly probable that the substance of it was embodied in the first
two books of the mathematical collections of Pappus. Only a fragment of the latter portion of the second part is known to be extant, in
which are exhibited several examples of the method of Apollonius.
This fragment was published by Dr. Wallis with a Latin translation
and notes in 1688, and is reprinted in the third volume of his works,
pp. 595-614.