VOL. LXXI.] PHILOSOPHICAL TRANSACTIONS. 141 



the way through an operation, every expression, even to the final equation, will 

 exhibit the whole process up to that step; and it will appear as clearly, how every 

 expression has been derived, as it does in that mode of analysis which was used 

 by the ancient geometricians; whereas, when the several original expressions 

 are melted down as it were into one mass, by the multitude Of actual addi- 

 tions, subtractions, multiplications, and divisions, which they generally undergo, 

 in a long algebraical process, conducted in the usual manner, it is impossible to 

 trace the smallest vsstige of the original quantities in the final equation, except 

 such as are represented by a single letter. Of course, however obvious the 

 several steps might be at the time when they were taken, every idea of them 

 must be totally lost in the result; and it will be utterly impossible to trace them 

 back again, in the manner they are done in the composition of a problem, the 

 solution of which has been investigated by the geometrical analysis. Let me 

 add, that it is to this cause we must attribute all that obscurity which the algebraic 

 mode of investigation has been so frequently charged with. 



Mr. W. then sets down, in 6 tables, the analytical expressions or forms of 

 the correspondent sines, tangents, and secants of arcs, and their multiples, as 

 far as the 6th, each of those being expressed in terms of the others; which tables 

 are to serve as formulae or theorems, with which to compare the forms of equa- 

 tions, when these occur in practice. 



Observations on the tables. — Each of the formulas in these tables may be con- 

 sidered as one side of an equation, involving the unknown quantity x to different 

 dimensions. In some of the formulae the odd powers of x are only found, in 

 others the even ones alone, and in others both; but they are all equally useful in 

 finding the value of the unknown quantity in affected equations which contain all 

 the powers of that quantity, as will plainly appear from the following considerations. 



1 . If, on bringing the solution of any problem to an equation with some 

 known quantity, it be found to correspond with any of the formulae in these 

 tables; or, if by any means it can be reduced to any of them; it is manifest, 

 that nothing remains to be done but to divide the known side of the equation by 

 the value of the quantity which is here denoted by r, and to seek for the quo- 

 tient in the tables of sines, cosines, or tangents, as the case may require, and 

 the value of the unknown quantity will be the sine, tangent, secant, or versed 

 sine, of a given part of that arc (according as the expression is found in the I st, 

 2d, 3d, or 4th table) multiplied by the value of r. 



2. If, as will more frequently happen, the final equation of an operation be 

 found equivalent to the sum, difference, product, or quotient, of some 2 or 

 more of these formulae; or to the sum, difference, product, or quotient, of some 

 2 or more of them multiplied or divided, increased or lessened, by some known 



