Page:The Construction of the Wonderful Canon of Logarithms.djvu/98

74 TRIGONOMETRICAL PROPOSITIONS.

Again, add together the logarithm of the half difference, the logarithm of the complement of the half sum, and the logarithm of the tangent of half the base; subtract the logarithm of the sum and the logarithm of half radius, and you will have the second found.

Proceed as above with the first and second found, and you will obtain the sides.

Another way of the same.

MUltiply the secant of the complement of the sum of the angles at the base by the tangent of half the base.

Multiply the product by the sine of the greater angle at the base, and you will have the first found.

Multiply the same product by the sine of the less angle, and you will have the second found.

[d]

Then divide the sum of the first and second found by the square of radius, and you will have the tangent of half the sum of the sides.

Also subtract the less from the greater and you will have the tangent of half the difference of the sides.

Whence add the arcs corresponding to these two tangents, and the greater side will be obtained; subtract the less arc from the greater and you have the less side.

Of the five consecutive parts of a spherical triangle, given the three intermediate, to find both extremes by one operation and without the need of discriminating between the several cases.

(*)

OF the angles at the base, the sine of the half difference is to the sine of the half sum, as the sine of the difference is to a fourth which is the sum of the sines.

And