Now, to proceed to the limit, putting n = an indefinitely large positive integer, and thereby rendering dx less than any assignable quantity, we have, To find the Differential Coefficient of sin x with regard to x. Now by the seventh Lemma of the first section of Newton's Principia we know that the arc and the chord of any curve vanish in a ratio of equality: whence it follows that the ratio between the sine and the circular measure of an angle is ultimately unity. Hence, in the limit, To find the Differential Coefficient of sec x. 37. Putting y = sec x, we get y + dy = sec (x + dx), sec x Sy = sec (x + dx) To find the Differential Coefficient of cosec x. 38. Putting y = cosec x, y + dy = cosec (x + dx), we have бу = cosec (x + dx) – cosec x |