VOL. LXXI.] PHILOSOPHICAL TRANSACTIONS. 143 



are found, the operation will not differ so materially from that which is pointed 

 out in the first rule as to merit repetition. 



4th. If instead of the sum or difference of the sine, cosine, or tangent, of an 

 arc, and the sine, cosine, or tangent, of some multiple of it, the form of 

 the equation be such as to be constituted of the product of them, or the quo- 

 tient of one divided by the other, the last rule will still hold good, using only 

 the logarithmic sines and tangents instead of the natural ones, and comparing 

 the sum or difference of them, according as the equation is composed of the 

 product or quotient of the 2 factors, with the logarithm of the number which 

 constitutes the known side of the equation, instead of that number itself. 



5th. Sometimes the final equation will come out in expressions which are 

 constituted of the sum, difference, product, or quotient, of the sine, cosine, 

 or tangent, of some multiple of an arc, of which the unknown quantity is the 

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

 other multiple of the same arc. And in any of these cases it is manifest, that 

 the method of proceeding, in order to obtain one of the multiple arcs, and from 

 thence the single one, of which the unknown quantity is the sine, tangent, &c. 

 will not be greatly different from those which have been described in the 3d and 

 4th rules. The most material difference consists in this, that instead of pro- 

 ceeding minute by minute, according to the directions in the 3d rule to find the 

 single arc, it will now be most convenient to proceed in each table by as many 

 minutes at each step as are equal in number to the number of times which the 

 single arc is contained in the multiple ones respectively. 



6th. Equations will frequently occur in formulae which express the square, 

 cube, &c. of the sine, cosine, or tangent, of the multiple of some arc, of 

 which the unknown quantity is the sine, tangent, secant, or versed sine; or in 

 formula? which are expressive of the sum, difference, product, &c. of the sine, 

 cosine, or tangent, of an arc, and some power of the sine, cosine, or tangent, 

 of the same arc; or of some multiple of it, the unknown quantity being some 

 other trigonometrical line belonging to that arc. Or the equation may be 

 compounded of the sum, difference, product, &c. of the same, or different 

 powers of the sines, tangents, or cosines, of different multiples of an arc, the 

 unknown quantity being the sine, tangent, secant, or versed sine, of that arc. 

 In every one of these cases the tables will give the value of the unknown quan- 

 tity, and in most of them with great ease and expedition. The method which 

 is to be pursued in each case will readily present itself to a skilful analyst, who 

 attends carefully to what has been already said, and to the examples which 

 follow. 



4. The formulae in the 4 tables may be greatly varied by supposing x, 

 the unknown quantity, to be some part or parts of the sine, tangent, &c. as 



