52

CALCULATIONS TO ASCERTAIN THE TRACT 26.

scale; look out any base, and its corresponding perpendicular,in their proper columns, that is, any radius and its correspond-ing altitude or depression, in the sixteen foregoing tables,without regarding the number of places they contain, andbring them to correspond ; then, if they consist of the samenumber of places, the lower index on the slider, or first co-lumn, or that answering to 1000, points to the true differencebetween the sine and tangent, in the last column ; but if thenumber of places in the base exceed that in the perpendicularby 1, the upper index 100 must be used. And in this man-ner were computed all the differences which were necessaryto be found, and placed in their proper squares, formed bythe meeting of the horizontal and vertical lines, or rings andsectoral spaces, in the following set of 16 tables, which cor-respond to the foregoing set of 16, each to each, accordingto the number of them, and marked at the tops with thenumbers 1, 2, 3, &c, to 12, for the sectoral spaces, and withthe number of the rings on the left-hand margin. Also, in thecolumn immediately after the number of the ring, are placedthe radii which formed the last column in the preceding tables;then, in the third column, are placed the sums of the altitudesand depressions, found in each line of the former tables; and,in the next column, the quotients found by dividing the num-bers in the third column by those in the second ; these quo-tients are the sums of the tangents belonging to each line orring; which being all added together, their total is placed atthe bottom of the column: after this follow the 12 columnsof differences before mentioned, which are succeeded by onemore column, containing the sums of each line of these dif-ferences which sums being added together, their total isplaced at the bottom of them ; and this total is the sum of allthe differences between the sines and the tangents, and it istherefore subtracted from the total of the tangents in thefourth column, when at last there remains the sum of the sinesas required.