supposed to increase by the addition to it of the small angle –an element of angle–the height of , the sine, will be increased by a small element . The new height will be the sine of the new angle , or, stating it as an equation,
;
and subtracting from this the first equation gives
.
The quantity on the right-hand side is the difference between two sines, and books on trigonometry tell us how to work this out. For they tell us that if and are two different angles,
.
If, then, we put for one angle, and for the other, we may write
But if we regard as indefinitely small, then in the limit we may neglect by comparison with , and may also take as being the same as . The equation then becomes: