f f 4 Dr. Waring: on 
Meditationes Algebraic ce to the Royal Society, and publifhed it in 
1760, and afterwards in 1762, with another part added, on the 
Properties of curves, under the title of Mifcellanea Analytical in 
all which was given the law of a feries for finding the fum of 
the powersof the roots of an equation from its co-efficients. That 
great mathematician M. Le Grange and myfelf printed about 
the fame time an obfervation known to me at the time that I 
printed the above-mentioned book, that the law of its powers 
and roots, if it proceeds in. infinitum , is the fame ; fronr 
which feries of mine,, with great fagacity, 1 VL. Le Grange 
found the law which Sir Isaac Newton’s reverfTon of feries 
obferves. In the Meditationes the law is given, and the feries 
is made to proceed according to the. dimenfions of at, &c. 
15. It is afferted in the Meditationes ^ that in moft equations 
of high dimenfions, unlefs purpofedly conftituted, the fum. of 
the terms which, from the given hypothefis,. become the 
greateft, being fuppofed =: o, only an approximate to the value 
Ax n of y in the refulting equation can by the common algebra 
be deduced. In this cafe the approximate to the quantity A is 
to be found fo near as the approximate value of the quantity 
fought requires ; or perhaps it is preferable to correct in every 
operation the approximate values of the quantities A, B, C, &c„ 
in the feries required AT" -f BT”+ r + CV+ ar + &c. 
In the equation the quantity %±e may be fubftitut^d for 
and from the equation refulting a feries expreffing the value of y 
may be found, proceeding either according to the dimenfions of 
the quantity z, or its reciprocals, according to the conditions of 
the problem. 
16. If there are more than one ( [n ) equations having («+ 1) 
unknown quantities ( [x,y , z 9 &c.) : in each of the equations fory. 
