Mr. wal.es on the Refolntlon, &c. 455 
firft reduced to cubic ones in the manner which has been ex- 
plained by Descartes and others, and the fecond term then 
taken away* 
Since that time M. m audit it has (hewn how to find the 
roots of all the three forms of cubic equations, by means of 
the tables of lines, <&c. in his excellent Treadle of Trigono- 
metry. But none of thefe authors have attempted to refolve 
equations of more dimenfions than three, by thefe means, 
without firft reducing them to that number ; nor even thefe, 
before the fecond term, or that which involves the fquare of 
the unknown quantity, is taken away : whereas fuch reduc- 
tions will generally take up more time than is required to bring 
out the value of the unknown quantity by the following me- 
thod ; and, after all, frequently ferve no other purpofe but 
that of rendering the operation more intricate and troublefome. 
The truly ingenious Mr. landen, in his lucubrations/ 
publifhed in 17 55, has given a general method of refolving 
that cafe of cubic equations, by means of the tables of lines, 
where all the roots are real, without the trouble of taking 
away the fecond term of the equation : and Mr. simpson has 
ffiewn how to refolve equations of any dimenfions, by the fame 
means, provided thofe equations involve only the odd powers of 
the unknown quantity, and that the co-efficients ob ferve fuch 
a, law as will reft rain the equation to that form which is expref- 
live of the cofine of the multiple of an arc, of which the un- 
known quantity is the cofine. This was firft done, I believe, 
by John Bernoulli, and afterwards by Mr. Euler, in his 
Intro duel, ad Analyt . Infinlt. and Mr. de moivre, in his Mif- 
cell. Analyt . ; but the refolution of all equations of this form, 
as well as many others, is comprehended in the firft of the 
following obfervations. 
O o 0 2, The 
