78 NOTES ON TRIGONOMETRICAL PROPOSITIONS.
rithms 581261 and 312858, and the logarithm of the versed sine of the given angle −409615. The sum of these three logarithms is 484504, which is the logarithm of the difference between the versed sine of the base and the versed sine of the difference of the sides.
Now the line corresponding to this logarithm, whether a versed sine or a common sine, is 6160057, and consequently this is the difference between the versed sine of the base and the versed sine of the difference of the sides. If to this you add the versed sine of the difference of the sides, that is 256300, the sum will be the versed sine of the base required, namely 6416357, and this subtracted from radius leaves the sine of the complement of the base, namely 3583643, which is the sine of 21°. Consequently the base required is 69°.
Conversely, given three sides, to find any angle.
If from the logarithm of the difference between the versed sine of the base and the versed sine of the difference of the sides you subtract the logarithms of the sides, the remainder will be the logarithm of the versed sine of the angle sought for.
As in the previous example, let the logarithms of the sides be 581261 and 312858. Subtract their sum, 894119, from the logarithm 484504, and the remainder will be the negative logarithm −409615, which gives the versed sine of the required angle 120° 24’ 49’.
Of five parts of a spherical triangle ]
This proposition appears to be identical with the one which ts inserted at the end, and distinguished like the former by (*). The latter proposition I consider much the superior. There are, however, three operations in it, the first two of which I throw into one, as they ave better combined. Thus:—
