Page:The Construction of the Wonderful Canon of Logarithms.djvu/97

TRIGONOMETRICAL PROPOSITIONS. 73

by the half difference between the versed sines of the sum and difference of the sides, and you have as quotient the half-versed sine of the vertical angle sought for.

[c]Of five parts of a spherical triangle, given the three intermediate, to find the two extremes by a single operation. Or otherwise, given the base and adjacent angles, to find the two sides.

(*)

OF the angles at the base, write down the sum, half sum, difference and half difference, along with their logarithms.

Add together the logarithm of the half sum, the logarithm of the difference, and the logarithm of the tangent of half the base; subtract the logarithm of the sum and the logarithm of the half difference, and you will have the first found.

Then to the logarithm of the half difference add the logarithm of the tangent of half the base; subtract the logarithm of the half sum, and you will have the second found.

Look for the first and second found among the logarithms of tangents, since they are such, then add their arcs and you will have the greater side; again subtract the less arc from the greater and you will have the less side.

Another way of finding the sides.

ADd together the logarithm of the half sum of the angles at the base, the logarithm of the complement of the half difference, 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 first found.

Again,