TRACT 26.

MEAN DENSITY OF THE EARTH.

£ 1

by the natural numbers, instead of it, the following methodwas adopted, being a much more expeditious way of obtain-ing the sum of the sines required. This method consists infinding, in a very easy manner, the difference between eachtangent and its corresponding sine, from the given base andperpendicular, and then, subtracting the sum of all the differ-ences from the sum of the tangents, there remains the sum ofthe sines. Several advantages attend this method of proceed-ing: for, to find the tangents, we need not divide every per-pendicular separately by its corresponding base, but addtogether all the perpendiculars that are on the same line, anddivide their sum by the common base, which is the radius ofthe middle of the ring, and is placed on the same line withthem towards the right-hand ; for thus we shall have littlemore than a twelfth part of the number of divisions to per-form : also, a great part of the tangents are so small, thatthey do not at all differ from their corresponding sines, in thenumber of decimals that it is necessary to continue the com-putations to ; in all which cases therefore the trouble of find-ing the differences is saved ; and those differences, which itis necessary to compute, are very readily found by inspection,on a peculiar kind of sliding rule, which was constructed forthis purpose, and of which I shall here give a short descrip-tion.

This rule (the figure of which is represented pi. 3, fig. 2)consists of three columns ; one marked af or base, which ismoveable by sliding it up or down by the side of the othertwo, which are fixed; of these two, the one contains the per-pendicular altitudes or depressions, and the other the differ-ences between the sines and tangents, to the radius 1. Toconstruct the numbers on this rule; form a series of logarith-mic tangents in arithmetical progression, of which the firstterm is 9-000, and the common difference '025 ; take out froma table the corresponding natural tangents, and place themin the first and second columns of base and perpendicular,and the difference between the natural sine and natural tan-gent in the last column, marked Diff. To make use of this

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