14 I. ON ANGLES

Two angles are said to be complementary if their sum is a right angle; supple-

mentary if their sum is equal to two right angles.

17. Angle Measure. The ratio of two quantities of the same kind is3 the

number which expresses how many times one of the quantities is contained in the

other. For instance, if, in dividing the segment AB into ﬁve equal parts, one of

these parts is contained exactly three times in the segment BC, then the ratio of

BC to AB is said to be equal to 3/5. If, on the other hand, a ﬁfth of AB is not

contained an exact number of times in BC, for example if it is contained in BC

more than three times, but less than four times, then 3/5 would be an approximate

value of the ratio

BC

AB

: it would be within one ﬁfth less than the value of the ratio

(the value 4/5 would be within 1/5 more).

The ratio of two quantities a, b of the same kind is equal to the ratio of two

other quantities a , b also of the same kind (but not necessarily of the same kind

as the ﬁrst two) if, for every n, the approximation to within 1/n of the ﬁrst ratio

is equal to the approximation within 1/n of the second ratio.

The measure of a quantity, relative to a quantity of the same kind chosen as

unit, is the ratio of the given quantity to the unit.

In addition we have the following properties (Le¸ cons d’Arithm´etique by J. Tan-

nery):

1◦.

Two quantities of the same kind, which have the same measure relative to

the same unit, are equal;

2◦.

The ratio of two quantities of the same kind is equal to the ratio of the

numbers which serve as their measures relative to the same unit;

3◦. The ratio of two numbers is the same as the quotient of the two numbers;

etc.

Theorem. In the same circle, or in equal circles, the ratio of two central angles

is equal to the ratio of their subtended arcs.

Figure 15

3See

Le¸ cons d’Arithm´ etique by J. Tannery, chapters X and XIII.