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

1

RATIO AND PROPORTION.
ART. 1. Two numbers or two magnitudes of the same kind may be
compared in two ways; by considering, first, how muLch one of them is
greater than the other; and, secondly, how many times one contains,
or is contained in, the other. The former is called their arithmetical,
the latter the geometrical relation.' It is under this latter view that
the ratio and proportion of numbers and magnitudes are considered.
Ratio is defined to be the relation which exists between two
magnitudes of the same kind, or two numbers, with respect to quotity.
Thus, two lines, two areas, two volumes, two weights, or two abstract
numbers, can have a ratio to each other; and the comparison is made
by considering what multiple part or parts the first is of the second.
Hence the ratio of two magnitudes of the same kind is represented
by the quotient which arises from dividing the units of the first magnitude, called the antecedent, by the units of the second magnitude,
called the consequent of the ratio: as, for instances, the ratio of a
guinea to a crown, or 21s. to 5s., is 2? or 64, an abstract number
which denotes the number of times, whole or fractional, the antecedent contains the consequent.
It is also obvious that the ratio of any two concrete numbers of the
same kind is the same as the ratio of two abstract numbers, as the
quotients in both cases are equal.
When the antecedent of a ratio is equal to the consequent, the
ratio is called a ratio of equality: when the antecedent is greater
than, or less than, the consequent, the ratio is called a ratio of greater
or of less inequality.
2. In the seventh book of Euclid's Elements, the definition of proportion is thus expressed:
Fouer numnbers are proportionals, whzen the first is the same mult1ple of
the second, or the same part or parts of it as tlhe third is of the fourth:
1 The terms arithmetical ratio and geometrical ratio are arbitrary names which
do not define nor explain their meaning. The arithmetical relation of two numbers
is properly a difference, and may be a concrete or abstract number, of the same
nature as the two given numbers. The geometrical ratio is a quotient, always all
abstract number. By the ratio of two magnitudes in the latter sense, is meant their
'relative mcagnitiude, how often one contains the other; not their absolu'te magnitude,
how much one exceeds the other. Thus, although the absolute magnitude of 12 miles
and 1 mile is much greater than that of 12 inches and 1 inch, yet the relative
magnitude, or the ratio of the two former, is exactly the same as that of the latter
two; or in other words, 1 mile is as small a space in comparison of 12 miles as
I inch is in comparison of 1 foot.