Page:The Construction of the Wonderful Canon of Logarithms.djvu/93

TRIGONOMETRICAL PROPOSITIONS. 69 Let the arcs be 38° 1’ and 77°. Their complements are 51° 59’ and 13°. The half sum of the complements is 32° 29’, the half difference 19° 29’, and the logarithms are 621656 and 1098014 respectively. Adding these, you have 1719670, from which, subtracting 693147, the logarithm of half radius, there will remain 1026523, the logarithm of 21°, or thereabout. Whence the sine of 21°, namely 358368, is equal to the difference of the sines of the arcs 77° and 38° 1’, which sines are 974370 and 615891, more or less.

Let the arc be 13°; its half is 6° 30’, of which the logarithm is 2178570. From double this, namely 4357140, subtract 693147, and there will remain 3663993. The arc corresponding to this is 1° 28’, and the number put for the sine is 25595; but this is also the versed sine of 13°.&#8258;

Let the arcs be 38° 1’ and 1° 28’; their sum is 39° 29’ and their difference 36° 33’, also the half sum is 19° 44’ and the half difference 18° 16’. Wherefore add the logarithm of the half sum, viz. 1085655, to the logarithm of the difference, viz. 518313, and you have 1603968; from this subtract the logarithm of the half difference, namely 1160177, and there will remain the logarithm 443791, to which correspond the are 39° 56’ and sine 641896. But this sine is equal, or nearly so, to the sum of the sines of 38° 1’ and 1° 28’, namely 615661 and 25595 respectively.

Let