The practical method of finding the ratio of two magnitudes is to measure them, and to divide the numerical result of one measurement by that of the other. But if two line-segments have a common measure, their ratio and their common measbe found by the following process:

ure may

CD

2 EB + FD= 8 GB 3 GB = 11 GB.

AB 2CD + EB = 22 GB + 4 GB = 26 GB.

.. GB is a common measure, and the ratio of CD to AB is by definition of ratio.

DEFINITIONS. Two magnitudes that have a common measure are said to be commensurable; if they have no common measure they are said to be incommensurable.

For example, two surfaces having areas 10 sq. in. and 15 sq. in. are said to be commensurable, there being the common measures 5 sq. in., 1 sq. in., 2.5 sq. in., etc. But if the length of one line is represented by √2, and the length of another by 1, then there is no common measure, and the lines are said to be incommensurable.

A ratio may therefore be an integer, or a fraction, or an irrational number such as √2.

For practical purposes all magnitudes may be looked upon as commensurable, since a unit of measure can be so taken that the remainder may be as small as we wish.

In the ratio a: b, a and b are called the terms of the ratio, the former, a, being called the antecedent, and the latter, b, the consequent.

If the ratio a b equals the ratio c : d, the four terms are said to form a proportion.

The four terms are also said to be in proportion. The terms a and b are also said to be proportional to c and d.

a

C

b d'

This equality of ratios is indicated by the symbols = a: b = c:d, or a/b = c/d, read "a is to b as c is to d." Instead of the parallel bars (=), the double colon (::) is also used in this connection as a sign of equality, the proportion being written a b::c: d.

The first and last terms of a proportion are called the extremes, and the other terms the means.

III. If a b c : d, then (1) a:cb: d, and (2) d: b=c : a,

IV. If a:b=c:d, then (1) ka:b=kc:d, and (2) a:kb=c:kd.

Hence, if four magnitudes are in proportion, and both antecedents or both consequents are multiplied by the same number, the magnitudes are still in proportion.

COROLLARY. If four magnitudes are in proportion, and all are multiplied by the same number, the results are in proportion.

V. If a b c : d, then (1) a+b: b=c+d =c+d: c, and (3) a + b: a b c±d:c

±1,

c+d

which prove

a+b c+d

5. Or, by subtracting first,

ab c = d

c+d C+

by dividing

bi.

b2

then

=

b1t b2.
ba

=

a2

b2

=

b3

NOTE. k may be an integral, fractional, or irrational number.

=

II

DEFINITIONS. If a b c x, x is called the fourth proportional to a, b, c.

COROLLARIES. 1. By three of the four terms of a proportion the other term is determined.

For if a b c : x, or x : b = c:a, or c: x = a: b, etc., it follows that ax = bc, whence x = bc/a, a fixed number.

For if, in the proof of cor. 1, a = c, then b = x.

If, in a proportion, the two means are equal, as in a:x=x:b, this common mean is called the mean proportional, or geometric mean between the two extremes.

COROLLARIES.

The mean proportional between two numbers equals the square root of their product.

The geometric mean between two lines equals the side of that square which equals their rectangle.

NOTE. Because the number representing the square units of area of a rectangle is the product of the two numbers representing the linear units in two adjacent sides, the expression product of two lines is commonly used for rectangle of two lines.