Page:The Construction of the Wonderful Canon of Logarithms.djvu/89

TRIGONOMETRICAL PROPOSITIONS. 65

4.Given the side A D, & the angles D & B, to find the side B D.

Multiply radius by the sine of the complement of D; divide by the tangent of the complement of A D, and you will obtain the tangent of the arc C D: then multiply the sine of C D by the tangent of D; divide the product by the tangent of B, and the sine of B C will result: add or subtract B C and C D, and you have B D.

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

Multiply radius by the sine of the complement of A D; divide by the tangent of the complement of D, and the tangent of the complement of C A D will be produced; whence we have C A D itself. Similarly multiply the sine of the complement of B by the sine of C A D; divide by the sine of the complement of D, and the sine of B A C will be produced; which being added to or subtracted from C A D, you will obtain the required angle B A D.

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

Multiply radius by the sine of the complement of D; divide by the tangent of the complement of A D, and the tangent of C D will be produced; its arc C D subtract from, or add to, the side B D, and you have B C: then multiply the sine of C D by the tangent of D; divide the product by the sine of B C, and you have the tangent of the angle B.

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

Multiply radius by the sine of the complement of D; divide the product by the tangent of the complement of A D, and the tangent of C D will be

produced;