[;8p] 
the reft be encreafed or diminiihed, either Arithmetically, 
or by mulcipliGatioii and divifion in a known Ratio} certainly 
regular Progreffions will arife, though as yet. we cannca^en- 
creafe the true roots of an Equation without as mucli di 
minifhing the Negative, nor can we multiply or divide the 
roots without we alter all of them, and confequentlv cannot 
reduce coefficients to fuch habitudes as are defirable. 
^4 It is a pleafant concinnity out of a root to raife a Re- 
folvend without raifing any of the Powers of the root, and 
at the fame time without a thorough binomial Divifion to 
deprefs the Equation a degree lower- 
EXAMPLE, 
Let the^Squation be a t 10 at da t 202i- — 1072. 
Let the root be 4» the resolvend is thus raifed by adding 
the coefficients as you go, and multiplying by the root, thus 
t4t I o-^==i4X4~'=-yd4^7— -62x4 —248''' 20—26 8x4— 1 071'. 
with the fame work the iEquation may be deprefled without 
Divifion. EXaMple. 
Let the Equation be as before, and place the root with 
the former products underneath refped:ively, the fumm is 
the depreiTed Equation. 
a4t ioa3 4 dat 2ca- -1071=^0 
t4 t5<j t24-8 ■t'1072. 
3 2 
The fum a4+ 14 a t i ^fr6 8a=i:o. that is divided bv a. 
3 2 
at 14a t 62ati($8=o. which is the under ^- 
quation fought found without Divifion 
2f It s conceived that all iEquations may be fo regulated as 
to be reduced to as many Arithmetical Progreffions otmultiph- 
ers in whole numbers,asthe^quation hath dimenfions, where- 
of one of the progreffions fhall be a Series of roots • hence the 
raifingRefolvendsby tentative work is rendred Logarithmeti- 
cal ForExample write down any 3 arithmeticalProgreffionsrc^/s:'. 
R H 
ixdx3"=i 8 7 1 fay the Rank H are the Refolvends or 
2x7X^^=^=^70 rHcm jenea Com^araticnis of a Cubick Equation, 
3x8x7 =~i(^8 r whofe roots are the Rank R. Thiscubick 
4x9x^=3^4^ quation is eafily attained out of the diiferen- 
fxi X "-^^50^. cesof the Rank R. for out of the Rank R in 
any Equation propofed raife feparately the relped:ive pow- 
ers (with regard to their Coefficients) and out of the three 
a 3 / ranks fo raifed compofe their refpedi ve ditierences, and 
ai.Vthey fhall be the fame with the differences of the rank 
a ^ of Refolvends or Homogenea Comfarat fonts here noted by H. 
