CO 
I. A Difconrfe concerning the Methods of Ap^ 
proximation in the ExtraSlion of Surd Roots. 
By John Wallis, S. T. D. and Savilian Pro^ 
' f^lf^^ G^^^^^^y Oxford. 
TH E feveral Methods of Approximation, which have 
. been mentioned of iate Years ^ for ExcracMng the 
Roots of Simple or Affe<5ted Equations^ gives me oc- 
cafion of (aying fomewhat of thac Subjed. 
It is agreed by all, (and, I think, demonftrated by the 
Greeks long ago) that if a Number propo(eci be not a true 
Square, ic is in vain for to hope for a juft Q^ucJrarick Root 
thereof, explicable by Rational Numbeis, Integers, or Fraded. 
And therefore, in fuch cafes, we raufl content our fclves with 
Approximations (fomewhat near the truth) without pretend- 
ing to Accuracy. 
And ib, for the Cubick Root, of what is not a perfed 
Cube. And the like for Superiour Powers. 
Now the Ancients (being aware of this) had their Me- 
thods of Approximacion in fiich cafes ; whereof feme have 
been derived down even to this day. Of which we fliall 
Ipeak more anon. 
But fince the Methods of Decimal Fradions (as they are 
now wont to be called) have come into Pradice, it hath 
been ufual to profecute fuch Extractions ( beyond the place 
of Unites) in the places of Decimal Parts, to what Accu- 
racy wepleafe; whereby the former Methods of Approaches 
have been (not fo much forgotten, as) negledred. 
Not that if fuch Approaches by Decimals were always the 
moft fpeedy, or the moft exad; (for no Man doubrs but that 
1 is a more Simple, and more Intelligible Notation of that 
Quantity, than o.i2j,orYi|| : And \^ not only a more Brief, 
but a more Accurate defignation of the Square Root or |, 
than 0.3 3 3 3 ; J, &c, ) But, becaule Fradions reduced to the 
Decimal form, are more convenient for fubfequent Opera- 
tions, when there is occaficn for a further Progreft. 
Mr. Newton s Method of Approximatioa for the Extrading 
Roots, even of Affeded Equations, I have given feme Ac- 
count 
