VOL. LXVIII.] PHILOSOPHICAL TKANSACTIONS. 417 



sine, and the analogy for that purpose is this, as the base : to the perpendicular 

 :: I (radius) : the tangent required, which will therefore be found by barely di- 

 viding the given perpendicular by the base ; and if we find this number in its 

 proper column in a table of sines and tangents, on the same line with it, in the 

 column of sines, will be found the sine of the angle required. This seems to 

 be the easiest way of resolving all the triangles when computed separately. But 

 as the labour would be very great in performing so many hundreds of arithmetical 

 divisions, &c. either by logarithms, or by the natural numbers, instead of it, 

 the following method, proposed by the Hon. Mr. Cavendish, was adopted, 

 being a much more expeditious way of obtaining the sum of the sines required. 

 This method consists in finding, in a very easy manner, the diff^erence between 

 each tangent and its corresponding sine, from the given base and perpendicular, 

 and then, subtracting the sum of all the differences from the sum of the tan 

 gents, there remains the sum of the sines. Several advantages attend this me- 

 thod of proceeding : for, to find the tangents we need not divide every perpen- 

 dicular separately by its corresponding base, but add together all the perpendicu- 

 lars that are on the same line, and divide their sum by their common base, 

 which is the radius of the middle of the ring, and is placed on the same line 

 with them towards the right hand ; for thus we shall have little more than a 12th 

 part of the number of divisions to perform : also a great part of the tangents 

 are so small that they do not at all differ from their corresponding sines, in the 

 number of decimals that it is necessary to continue the computations to, in all 

 which cases the trouble of finding the difference is saved ; and those differences 

 which it is necessary to compute, are very readily found by inspection on a 

 peculiar kind of sliding rule, which was constructed for this purpose. By which 

 rule were computed all the differences which were necessary to be found, and 

 placed in their proper squares formed by the meeting of the horizontal and 

 vertical lines, or rings and sectoral spaces, in a following set of 1 6 tables, which 

 correspond to the foregoing set of l6, each to each, according to the number 

 of them. 



After this follow the 12 columns of differences before mentioned, which are 

 succeeded by one or more columns containing the sums of each line of these 

 differences, which sums being added together, their total is placed at the bottom 

 of them ; and this total is the sum of all the differences between the sines and 

 the tangents, and it is therefore subtracted from the total of the tangents in the 

 4th column, when there remains the sum of the sines as required. 



Having now obtained the sums of the sines for the several quadrants, the 

 next business is to collect them together, and deduct the negatives from the 

 affirmatives. And this may be done either for each observatory separately, or 



VOL. XIV, 3 H 



