VOL. LXXVII.] PHILOSOPHICAL TRANSACTIONS. 1Q3 



viz. ^(a + a'^-^O + ^{a-\-^'^^b)+ '^{a + y^-b) + &c. where 

 a, (3, y, &c. denote the 2w roots of *^ — 1 . The general principles of disco- 

 vering the cases in which this happens, have been given in the Meditationes Al- 

 gebraicae. 



The roots of the equation x^ ± 1=0 will be found from common algebra and 

 these principles, if h is not greater than 10; or, more generally, if A = 2' X 3" 

 X 4'" . . JO'S where /, /', /" . . /" denote any whole numbers: or, in general, the 

 roots of the above-mentioned equation, or even of the equation a? = ^{±i.± 

 M V — l)» ^^" ^^ found from tables of sines. The same principles may be ap- 

 plied to the discovery of the values of exponential irrational quantities. 



In the Miscel. Analy. was given, from a substitution invented by me, and not 

 similar to any before given, a resolution of equations, which contains the reso- 

 lutions of all equations before given, and from which the resolutions of some 

 equations, not before delivered, have been added. 



Part ii. — 1. Let an equation a = O, involving r unknown independent quan- 

 tities, be predicated of another equation containing the same quantities, and the 

 demonstration of it be required. 1st. Reduce both the equations to equations 

 involving independent quantities only; then reduce the two equations to one, so 

 that one of the above-mentioned quantities may be exterminated, and if there 

 results a self-evident equation, viz. a = a, or a — a = O, in which the corre- 

 spondent terms destroy each other respectively; then the first equation is justly 

 predicated of the second; that is, if the above-mentioned equations afford the 

 same value of the quantity exterminated, the proposition is true; otherwise not. 



Corol. From these principles can be demonstrated many propositions given by 

 Pappus and others. 



Exam. Let ad = 2ac = 2.r, de = a, and eb = bj where ad, de, and eb 

 are independent quantities; if ab X be = (2x -{- a •}- b) X 

 A c DEB Z) = CB X bd = (^ -f- a -}- Z)) (a -j- b), then will cb = x -{• 

 a -jr b : BT> = a -{- b :: AC X ce = a? X {x -\- a) : aj> X de = 

 2x X a. Hence can be deduced the two equations (b — a)x = a^ -{- ab and 

 2a X {x -\- a -{• b) = {a -{- b) X {x -\- a); reduce these two equations to one, 

 so as to exterminate x, and there results the self-evident equation (a — b) X 



^T — — (— a^ — ab) -{• a^ -\- ab = Oj and consequently the proposition is true. 



2. If s equations, involving t -\- r unknown and independent quantities, be 

 predicated of t equations involving the above-mentioned quantities; reduce the t 

 equations and one of the above-mentioned ^ equations to one, so that t unknown 

 quantities may be exterminated; and if there results a self-evident equation, then 

 the above-mentioned equation is justly predicated of the t equations. And in 

 the same manner we may reason concerning the remaining s — 1 equations. 



3. 1. If one equation is justly predicated of another, and in both the un- 



VOL. XVI. C c 



