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.

4This

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.