I. ON ANGLES 15 Consider4 (Fig. 15) two arcs AB, CD on circle O. Divide central angle COD into three equal parts (for example), and suppose that one of these thirds can fit four, but not five times, into the angle AOB the value (smaller by less than 1/3) of the ratio AOB COD is 4 3 . But dividing the angle COD into three equal parts, we have at the same time divided arc CD into three equal parts (13). If a third of COD fits four, but not five times, into angle AOB, it follows the same way that a third of arc CD fits four, but not five times, into arc AB. The values to within 1/3 of the two ratios are therefore equal, and similarly the values to within 1/n will be equal for any integer n. The theorem is therefore proved. Corollary. If we take as the unit of angle measure the central angle which intercepts a unit arc, then every central angle will have the same measure as the arc contained between its sides. This statement is exactly the same as the preceding one, since the measure of a quantity is its ratio to the unit. Supposing, as we will from now on, that on each circle we choose as unit arc an arc intercepted by a unit central angle, the preceding corollary can be stated in the abridged form: A central angle is measued by the arc contained between its sides. 18. The definitions reviewed above allow us to establish an important conven- tion. From now on, we will be able to suppose that all the quantities about which we reason have been measured with an appropriate unit chosen for each kind of quantity. Then, in all the equations we write, the quantities which appear on the two sides of an equality will represent not the quantities themselves, but rather their measures. This will allow us to write a number of equations which otherwise would have no meaning. For instance, we might equate quantities of different kinds, since we will be dealing only with the numbers which measure them, the meaning of which is perfectly clear. We will also be able to consider the product of any two quantities, since we can talk about the product of two numbers, etc. In fact, whenever we write the equality of two quantities of the same kind, this equality will have the same meaning as before, since the equality of these quantities is the same as the equality of their measures. Following this convention we can write: AOB = arc AB, where AB is an arc of a circle and O is the center. It is important however to insist on the fact that the preceding equality assumes that the unit of angle measure and the unit of arc measure have been chosen in the manner specified above. 4 This theorem becomes obvious if we consider the following proposition from arithmetic (Tannery, Le¸ cons d’Arithm´ etique: Two quantities are proportional if: 1◦ To any value of the first, there corresponds an equal value of the second, and 2◦ the sum of two values of the first corresponds to the sum of the corresponding values of the second. These two conditions hold here (13). The argument in the text does no more than reproduce, for this particular case, the proof of this general theorem of arithmetic.

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