140 PHILOSOPHICAL TRANSACTIONS. |_ANNO 1/81. 



method of resolving that case of cubic equations, by means of the tables of 

 sines, where all the roots are real, without the trouble of taking away the 2d 

 term of the equation; and Mr. Simpson has shown how to resolve equations of 

 any dimensions, by the same means, provided those equations involve only the 

 odd powers of the unknown quantity, and that the co-efficients observe such a 

 law as will restrain the equation to that form which is expressive of the cosine of 

 the multiple of an arc, of which the unknown quantity is the cosine. This 

 was first done, it seems, by John Bernoulli, and afterwards by Mr. Euler, in 

 his Introduct. ad Analyt. Infinit. and Mr. de Moivre, in his Miscell. Analyt. ; 

 but the resolution of all equations of this form, as well as many others, is com- 

 prehended in the first of the following observations. 



The first thought of extending the use of the tables of sines, tangents, and 

 secants, beyond the cases which have been already mentioned, occurred while I 

 was considering the problem which produced the equation given in this paper as 

 the 4th example. And it is remarkable, that the very same thought occurred to 

 Dr. Hutton about the same time, and in the resolution of the same problem ; 

 and we were not a little surprized, on comparing our solutions together, to find 

 that our ideas had taken so exactly the same turn ; and that both should have 

 stumbled on a thought, which, as far as either of us knew, had never presented 

 itself to any one before. Having since examined further into the matter, I have 

 the satisfaction to find, that the principle is very extensive, and that a great 

 number of equations, especially such as arise in the practice of geometry, 

 astronomy, and optics, may be resolved by it with great ease and expedition. 



But besides the facility with which the value of the unknown quantity is 

 brought out by means of the tables of sines, tangents, and secants, this method 

 of resolution has another considerable advantage over most others which have 

 been proposed, inasmuch as the first state of the equation, without any previous 

 reduction, is generally the best it can be in for resolution; and from which it 

 may most readily be discovered, how to separate it into such parts as express the 

 sine, or the tangent, or the secant of the arc of a circle; or into the sine, tan- 

 gent, or secant of some multiple of that arc, or of a part of it: and in the 

 doing of which consists the principal part of the business in question. It will 

 also be of some advantage to preserve the original substitutions as distinct as 

 possible, by using only the signs of the several operations which it may be neces- 

 sary to go through in bringing the solution of a problem to an equation, instead 

 of performing the operations themselves. 



Besides the advantages which this method of procedure affords to the mode of 

 solution now more particularly under consideration, it has so many others over 

 that which is commonly used, that I am much surprized the latter should ever 

 have been adopted. By preserving thus the original substitutions distinct, all 



