Now, since the differentials of (x) and of (a) are equal, the difference between these quantities must be constant; but when = 0, 0, and .. a0=0; hence, in that case, their difference = 0, and, therefore, their difference must always= 0; hence, (*) must always equal (a 0), and .·. 0 = ~ .

a

From whence it appears, that when the angle is given, the arc is proportional to the radius.

COR. 8. If we suppose (0) to be a right angle, we shall have the quadrantal arcs of different circles proportional to their radii; and, therefore, four times the quadrantal arcs of these circles, or their circumferences will be found to be in proportion to their radii.

PROP. XXXVI.

To express the sin 0 and cos 0, in terms of (0).

In the same manner it may be shewn, that

where the values of C, C', &c. may be determined

by actual division.

COR. 2. Hence, if (x) be an arc, whose radius is (a) and (y) and (x), its sine and cosine to that radius respectively, we have, since

and, consequently, if (t) = tan ≈ to radius (a), we

have

t=z-C. + C'.

&c.

a+

where C and C' are the same as before.

COR. 3. (Prop. 32. Cor. 1.), we have

which expressions are only imaginary.

COR. 4. If for (0) we write me, we have

(€

1

Consequently, (cos + √1 sin 0)m

=cos me√1. sin me.

PROP. XXXVII.

To find the value of an arc in terms of its sine.

Let y sin 0, then 0-sin-ly*; and since the differential coefficient of (y) considered as a function = cos 0 = √√√1 −y, the differential coefficient of

of (0)=

(0) considered as a function of

و

or

y

* The symbols sin-ly, cos-ly, tan-ly, &c. are used to represent the length of a circular arc, whose sine, cosine, tangent, &c. respectively=y to radius unity.