Page:The Construction of the Wonderful Canon of Logarithms.djvu/92

68 TRIGONOMETRICAL PROPOSITIONS. three times applied. Therefore substitute A for D and D for A, and the problem will be as follows:—

Given A D & the angle A with the angle D, to find the side B A.

This is the same throughout as problem 11, and is solved by applying the “Rule of Three” twice only.

Iven two sides & the contained angle, to find the third side.

From the half-versed sine of the sum of the sides subtract the half-versed sine of their difference; multiply the remainder by the half-versed sine of the contained angle; divide the product by radius; to this add the half-versed sine of the difference of the sides, and you have the half-versed sine of the required base.

Given the base and the adjacent angles, the vertical angle will be found by similar reasoning.

From the half-versed sine of the base subtract the half-versed sine of the difference of the sides-multiplied by radius; divide the remainder by the half-versed sine of the sum of the sides diminished by the half-versed sine of their difference, and the half-versed sine of the vertical angle will be produced.

Given the three angles, the sides will be found by similar reasoning.

Let