t 71 ] 
XIV. On finding the Values of Algebraical Quantities by con- 
verging Seriefes , and demonjirating and extending Propofitions 
given by Pappus and others. By Edward Waring, F.R.S. 
Profejfor of Mathematics at Cambridge. 
Read February 8, 1787. 
S UPPOSE the roots of the equation x b =*=i =0 to be given* 
where h denotes any whole number or fraftion ; to find the 
roots or values of any given algebraical quantity, by con- 
verging infinite feriefes. 
1. Let the algebraical quantity be d/ (±A), then the roots 
1 r 
of the algebraical quantity will be A" x (ai-A\/ — ij. A" x 
I 
(( 3 +g v/ - 1), A n x^-fV -1), &c. where osH-a.v'-i, 
/ 3 +g,\/ — 1, y + j's/ -1, 6cc. are the roots of the equation 
x n ztz 1=0; it will be + 1 if it was — A, and — 1 if + A. 
2. Let the given algebraical quantity b e \Z ( — v 7 ( — A) =+= 
py z±r C — &c.), and a + \\/ - 1, ct' + A y/ - 1, 
a." g-A's/ - 1, &c. and F + Av/ — 1 be refpedlively one of the 
roots of the equations *"=♦= 1 = o, 1 = o, x^-^z 1 =0, &c. and 
II I 
,v r =p 1=0; fubftitute =tP = =±= A" B” of z±zC n od' rt &c. and 
1 1 ( i_ 
dt Q— A" x / z±=C w A // d=&c. In the firff place let P 
be greater Q, and ±;P be +P, then will (P^Q^ (— i)} r — 
‘ (P 
