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

2
Or in other words, if the quotient arising from the first of four
numbers divided by the second, be equal to the quotient of the third
divided by the fourth, the four numbers constitute a proportion.
The definition of proportion has been more briefly defined to be
the equality of two ratios.
Thus the four numbers 12, 4, 18, 6 fulfil the conditions required,
that the first, 12, has to 4, the second, the same ratio as 18, the third, has
to 6, the fourth, and are proportionals;
so that 12 bears the same relation to 4 as 18 does to 6,
which is usually written thus: 12: 4: 18: 6.
This same relation holds good when the terms of each ratio are
concrete numbers, as 12 yards bear the same relation to 4 yards, as 18
shillings bear to 6 shillings.
The four terms of any proportion may have their order reversed
without affecting the proportionality of the numbers, as the fourth,
third, second, and first terms of a proportion may be made the first,
second, third, and fourth times terms.
Conversely. If four numbers be proportionals, the quotient of the first
'divided by the second is equal to that of the third divided by the fourth.
Thus if 12, 4, 18, 6 form a proportion,
which is usually expressed, 12: 4:: 18: 6,
Then the ratio 12: 4 is -1- or 3,
and the ratio 18: 6 is -A8 or 3,
whence follows the equality, 1- = 2 j.
It also appears from the equality of the two fractions, that if the
first term of a proportion be greater than the second, the third is
greater than the fourth; if equal, equal; and if less, less.
If four numbers, as 12, 4, 18, 6, be prop2ortionals, the product of the
extremes s equal to the product of the means.
Since 12, 4, 18, 6 are proportionals,
the ratios 1-2 and -l- are equal, or - -= 1-.
1Multiplyin g these equals by 4 x 6
Then 12 x 6 - 18 x 4,
or the product of the extremes, 12 and 6, is equal to the product of the
means, 18 and 4.
Hence it appears, that if the product of the means, 18 x 4, be
divided by 12, one of the extremes, the quotient, 6, will be the other
extreme; and if the product of the extremes, 12 x 6, be divided by 4, one
of the means, the quotient will be the other mean.'
1 Some writers assume the letter x to denote the required number in questions of
proportion, and deal with it as a number unknown, but to be determined by the
conditions of the question. Others reject the use of all general symbols in numerical computations, as such symbols belong rather to Algebra. Whichever may be
the correct view of the matter, there is no doubt that such an assumption affords
both facility and convenience in arithmetical reasonings, when it is made the first