72 TRIGONOMETRICAL PROPOSITIONS.
14449; their half difference is 8° 59’, and its logarithm 1856956. Add these logarithms and you have 1871405; subtract 693147 and there remains 1178258. The arc corresponding to this logarithm is 17° 56’, which arc we call the third found.
From the logarithm of the third found, subtract the logarithms of the given sides, namely 581260 and 312858, and there remains 283533; halve this and you have 141766 for the logarithm of the half vertical angle 60° 12’ 24½”. The whole vertical angle sought is therefore 120° 24’ 49”.
Another rule for finding the base by prosthapharesis.—
[Given the sides and vertical angle, to find the base.]
NOte the half difference between the versed sines of the sum and difference of the sides, and also the half-versed sine of the vertical angle. Look among the common sines for the values noted, and find the arcs corresponding to them in the table. Then write for the second found the half difference of the versed sines of the sum and difference of these arcs.
Also, as before, take for the first found the half-versed sine of the difference of the sides.
Add the first and second found, and you will obtain the half-versed sine of the base sought for.
Conversely—[given the sides and the base, to find the vertical angle.]
The first found will be, as before, the half-versed sine of the difference of the sides.
From the half-versed sine of the base subtract the first found and you will have the second found.
Multiply the latter by the square of radius; divide
