Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

ever will give the mode of measuring according to the order
of collocation. For the first term numbers according to the
first, i.e. according to itself, the first term which it numerates;
but it numbers the second, which it numerates, by the second,
the third, by the third; and the fourth by the fourth. When
the second however begins to measure, it measures the first
which it numerates according to the first; but it measures the
second which it numerates by itself, i.e. by the second term;
and the third by the third; and so of the rest. Thus 3 measures
9 by 3; 15 by 5; 21 by 7; 27 by 9, and so on. But 5 measures
15 by 3; 25 by 5; 35 by 7; 45 by 9, and so of the rest. If
therefore we direct our attention to the other terms, either
those that measure others, or that are themselves measured by
others, we shall find that there cannot be at one and the same
time a common measure of all of them, nor that all of them
at the same time measure any other number; but it will appear
that some of them may be measured by another number, so as
only to be numbered by one term; others, so as to be numbered
by many terms; and some, so as to have no other measure
than unity. Hence, those that receive no measure besides unity,
are said to be first and incomposite numbers; but those that
receive a certain measure besides unity, or are allotted the a p
pellation of a foreign part, these are said to be second and composite
numbers.
The third species however, which is of itself second and
composite, but when one number is compared to the other is
first and incomposite, is obtained by the following method:
The squares of the first and incomposite numbers, when compared
to each other will be found to have no common measure.
Thus the square of 3 is 9, and the square of 5 is 25.
These therefore have no common measure. Again, the square
of 5 is 25, and of 7 is 49: and these compared to each other
will be found to be incommensurable. For there is no com-