(JOO PHILOSOPHICAL TRANSACTIONS. [ANNO 1784. 



published in the Meditationes. For example : in finding approximates to the 

 roots of given equations, the convergency depends on how much the approximates 

 given arc more near to one root than to any other ; and consequently, when 2 or 

 more roots or values of an unknown quantity are nearly equal, different rules are 

 to be applied, which are improvements of the rule of false. This rule, and the 

 above-mentioned observations, were first given in the Meditationes Algebraicoe et 

 Analytics, with several other additions on similar subjects. 



Many more things concerning the summation of series, which depend on 

 fluxional, &c. equations, might be added ; but I shall conclude this paper with 

 congratulating myself, that some algebraical inventions published by me have 

 been since thought not unworthy of being published by some of the greatest ma- 

 thematicians of this or any other age. 



1st. In the year 1757, I sent to the Royal Society the first edition of my 

 Meditationes Algebraicse: they were printed and published in the years 1760 and 

 17&2>, with Properties of Curve Lines, under the title of Miscellanea Analytica, 

 and a copy of them sent to Mr. Euler in the beginning of the year 1 763, in which 

 was contained a resolution of algebraical equations, not inferior, on account of 

 its generality and facility, to any yet published, viz. y = a'^/p -j- h^/p 1 4- 

 c\/p 3 + . . .fyp*-'. This resolution was published by Mr. Euler in the Peter- 

 sburgh Acts for the year 1764. Whether Mr. Euler ever received my book, I 

 cannot pretend to say ; nor is it material : for the fact is, that it was published 

 by me in the year 1760 and 1 762, and first by Mr. Euler in the year ] 764. Mr. 

 de la Grange and Mr. Bezout have ascribed this resolution to Mr. Euler, as first 

 published in the year 1 764, not having seen I suppose my Miscell. Analyt. Mr. 

 Bezout found from it some new equations, of which the resolution is known, and 

 applied it to the reduction of equations : more new equations are given, and the 

 resolution rendered more easy by me in the Philos. Trans. 



2d. In the above-mentioned Miscell. Analyt. an equation is transformed into 

 another, of which the roots are the squares of the differences of the roots of the 

 given equation ; and it is asserted in that book, that if the co-efficients of the 

 terms of the resulting equations change continually from + to — and — to 4- , 

 the roots of the given equation are all possible, otherwise not; and in a paper, in- 

 serted by me in the Philos. Trans, for the year 176-1, in which is found from this 

 transformation, when there are none, 2 or 4 impossible roots, contained in an 

 algebraical equation of 4 or 5 dimensions ; it is observed, that there will be none 

 or 4, &c. impossible roots contained in the given equation, if the last term be 4- 

 or "— ; and 2, &c. on the contrary, if the last term be — or +. These obser- 

 vations and transformation have been since published and explained in the Berlin 

 Acts for the years 1767 and 17(38, by Mr. de la Grange. 



