Qj'2, I'HILOSOl'HICAL TRANSACTIONS. [aNNO 1780. 



l-ab/ij discovered by Scipin Fetreus, of Bononia, or luhoever else was the first 

 Inventor of them. By F. Maseres, Esq., F. R. S. p. 221. 

 j4rt. 1. There is nothing more amusing, or more grateful to an inquisitive 

 mind, in the study of the sciences of Geometry and Algebra (for if we banish 

 from it the ridiculous mysteries arising from the supposition of negative quan- 

 tities, or quantities less than nothing, the latter may deserve the name of a 

 science as well as the former) than to contemplate the methods by which the 

 several ingenious and surprizing truths that are delivered in the books that treat 

 of tliem were first discovered. This we are sometimes enabled to do by the 

 authors themselves to whom we are indebted for these discoveries, who have 

 candidly informed their readers of the several steps, and sometimes of the acci- 

 dents, by which they have been led to them : but it also often happens, that the 

 authors of these discoveries have neglected to give their readers this satisfaction, 

 and have contented themselves with either barely delivering the propositions 

 they have found out, without any demonstrations, or with giving formal and 

 positive demonstrations of them, which command indeed the assent of the under- 

 standing to their truth, but afford no clue, to discover the train of reasoning by 

 which they were first found out; and consequently contribute but little to enable 

 the reader to make similar discoveries himself on the like subjects. This seems 

 to be the case with those ingenious rules for the resolution of certain cubic equa- 

 tions, which are usually known by the name of Cardan's rules. We are told to 

 make certain substitutions of some quantities for others, in these equations 

 ;r' -j- oar = /' and x' — qx = r, which are the objects of those rules, and certain 

 suppositions concerning the quantities so substituted; by doing which we find, 

 that those equations will be transformed into other equations which will involve 

 the 6th power of the unknown quantity contained in them, but which (though 

 of double the dimensions of the original equation x^ -]- qx = r and x^ — qx = r, 

 from which they were derived) will be more easy to resolve than those equations, 

 because they will contain only the 6th power and the cube of the unknown 

 quantity which is their root, and consequently will be of the same form as 

 quadratic equations; so that, by resolving them as quadratic equations, we may 

 obtain the value of the cube of the unknown quantity which is their root, and 

 afterwards, by extracting the cube-root of the said value, we may obtain the 

 value of the said root, or unknown quantity itself; and then at last, by the 

 relation of this last root to x, or the root of the original equation, (which 

 relation is derived from the suppositions that have been made in the preceding 

 transformations) we may determine the value of .r. And if we please to examine 

 the several steps of this process with sufficient attention, we may perceive, as we 

 go along, that all these substitutions are legitimate and practicable, or are founded 

 on possible suppositions; though I ( annot but observe, that the writers on 



