Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

mon measure of these except unity which is the generator and
mother of all these.
CHAPTER XIII.
On the method of discovering the commensurability, or incommensurability
of these n urn bers.
THE art however of discovering whether such numbers are
commensurable by some number besides unity, or whether
each of them is measured by unity alone, is as follows: Two
unequal numbers being given, it will be requisite to take the
less from the greater, and if that which remains is still greater
to take from it again the less, but if it is less to take it from
the remaining greater number. And this must be done till unity
at last prevents any farther subtraction, or till some odd
number necessarily effects this, if both the proposed numbers
are odd; and this number which remains will be the common
measure of both. Thus for instance, let there be two numbers
9 and 29, of which it is proposed to investigate the common
measure. From the greater let the less be taken, and there will
remain 20. From this therefore let the less be again taken, and
11 will remain. From this again take 9, and 2 will remain. Let
2 then be taken from 9, and 7 will remain. From 7 let 2 be
taken, and there will remain 5. Again, from 5 let 2 be taken,
and 3 will remain. And farther still, if 2 be taken from 3,
there will remain 1. And in the last place, if 1 be taken from
2, the subtraction will terminate in unity, which will alone be
the measure of the two numbers 9 and 29. These numbers
therefore, are called with reference to each other prime numbers.
Let it, however, be proposed to investigate the common
measure of two other numbers 21 and 9, that it may be ascer-