1
SO -CALCULATIONS TO ASCERTAIN' THE TRACT 26.
16. Depressions below p in the s.e. quarter.
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It remains next to find the sines of the vertical angles, sub-tended by all the foregoing altitudes and depressions; sincethe sum of these sines is the thing we are in quest of. Now,each altitude, or depression, is the perpendicular of a right-angled triangle, of which the given radius, standing on thesame line with it, in the right-hand margin, is the base, orthe other side about the right angle; and by the resolutionof the right-angled triangle, for each perpendicular, the samenumber of corresponding sines will be found. But with suchdata, the tangent of the angle is much easier to be found,than the sine, and the analogy for that purpose is this, as thebase : to the perpendicular :: radius 1 : the tangent, whichwill therefore be found, by barely dividing the given perpen-dicular by the base ; and if we find this number in its propercolumn, in a table of sines and tangents, then on the sameline with it, in the column of sines, will be foutfd the sine ofthe angle required. This seems to be the easiest way of re-solving all the triangles, when computed separately. But asthe labour would be very great, in performing so many hun-dreds of arithmetical divisions, &c, either by logarithms, or