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

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