70 TRIGONOMETRICAL PROPOSITIONS.
Let the arc be 39° 56’, to which corresponds the logarithm 443791, the sine being unknown. To the logarithm 443791 add 693147, the logarithm of half radius, and you have 1136938. Halve this logarithm and you have 568469. To this corresponds the arc 34° 30’, which being doubled gives 69° for the arc which was sought. This is the case since the sine of 39° 56’ and the versed sine of 69° are each equal, or nearly so, to 641800.
LEt the sides be 34° and 47°, and the contained angle 120° 24’ 49″. Half the contained angle is 60° 12’ 24½″, and its logarithm 141766. To the double of the latter, namely 283533, add the logarithms of the sides, namely 581260 and 312858, and the sum is 1177651. This sum is the logarithm of half the difference between the versed sine of the base and the versed sine of the difference of the sides; it is also the logarithm of the sine of the are 17° 56’, which are we call the “second found,” for that which follows is first found,
Halve the difference of the sides, namely 13°, and you have 6° 30’, the logarithm of which is 2178570. Double the latter and you have 4357140 for the logarithm of the half-versed sine of 13°; it is also the logarithm of the sine of the arc 0° 44’, which are we call the “first found.”
The sum of the two arcs is 18° 40’, the half sum 9° 20’, and their logarithms 1139241 and 1819061 respectively. Also the difference of the two arcs is 17° 12’, the half difference 8° 36’, and their logarithms 1218382 and 1900221 respectively.
