142 PHILOSOPHICAL TRANSACTIONS. [ANNO 1781. 



quantity or quantities; then, having taken away the known quantities by the 

 common algebraic rules, observe the following ones. 



1st. When the equation is found to correspond with the sum or difference of 

 2 formulae in these tables, which are the sine and tangent, sine and cosine, or 

 cosine and tangent, of the same arc, by running the eye along the tables of 

 natural sines and tangents, find these 2 arcs, immediately following each other, 

 the sum or difference of the sine and tangent, sine and cosine, or cosine and 

 tangent, of which are one of them greater, and the other less, than the number 

 which constitutes the known side of the equation. Take the excess of one of 

 these sums or differences above, and what the other sum or difference wants of 

 the said given number, add these two errors together, and say, as the sum of 

 them is to 60", so is that error which belongs to the less arc to a number of 

 seconds; which being added to the less arc will give one, the sum or difference of 

 whose sine and tangent, sine and cosine, or cosine and tangent, is exactly equal 

 to the number which constitutes the known side of the equation. Of the arc, 

 thus found, let such a part be taken as the table in which the formula? are found 

 directs, and the natural sine, tangent, secant, or versed sine (as the case may 

 require) of this part, being multiplied by the value of r, if r be found in the 

 equation, will be the value of x sought. 



2d. When the equation happens to be the product or quotient of 2 formulas 

 which express the sine and cosine, sine and tangent, or cosine and tangent, of 

 the same arc, take the logarithm of the number which constitutes the known 

 side of the equation, and then follow exactly the directions given in the first 

 case, using the tables of logarithmic sines and tangents, instead of the tables of 

 natural ones. 



3d. If the equation, finally resulting from the resolution of any problem, 

 present itself in an expression which is composed of the sum or difference of 

 the sine, cosine, or tangent, of an arc, of which the unknown quantity is the 

 sine, cosine, tangent, or versed sine, and the sine, cosine, or tangent, of some 

 multiple of that arc, it will then be convenient to have 2 tables of sines and 

 tangents; and in running the eye along them to find the 2 arcs immediately 

 following each other, of which the sum or difference of the sine, cosine, or 

 tangent, of one of them, and the sine, cosine, or tangent, of some multiple 

 of it, may be less, and the sum or difference of the sine, cosine, or tangent, 

 of the other, and the sine, cosine, or tangent, of the same multiple of it, may 

 be greater than the number which constitutes the known side of the equation, 

 for every minute of a degree that the finger is moved over in one, it must be 

 moved over a number of minutes in the other, which is equal to the number of 

 times that the single arc is contained in the multiple one. When these 2 arcs 



