(4) 
.( which are not likely to influence the Figure due to the fixth 
.place of Decimal Parts of the Root fought:) this long Pro- 
cefs will bo much fliortned. 
And if we further confider, what Preparative Operations 
are to be made in fome of thofe other Methods, before wc 
come to the prefcribed Divifion for giving the Root defircd ; 
the Advantage (though confiderable ) will not be fo great as 
may at firft be apprehended ; efpecially as to AfTcifted Equa- 
tions, in which the Parodical Powers have great CoeflScients. 
As will (bon appear inPradice, if we come to apply them 
to particular cafes. 
But, without dilparaging thefe Methods (which are really 
confiderable, and well worthy encouragement) that which 
I here intend, is, to fhew the true Foundation of the Me- 
thods ufed by the Ancients, ( however fince negleded ) and 
the juft Improvement of them. Which though Anciently 
fcarce applyed beyond the C^uadratick, or perhaps the Cu- 
bick Root, (for with the Higher Powers they did not much 
trouble themfelves) yet are equally applicable (by due ad- 
juftments ) to the Superiour Powers alfo. 
I (hall begin with the Square Root : For which the Anci- 
ent Method is to this purpofe. 
From the propofed Non-quadrate (fuppofeN) fubtrad 
{in the ufual manner) the greateft Square in Integers therein 
contained (fuppofeAq.) The remainder (fuppofe B = 2 AE 
*f-Eq) is to be the Numerator of a FraAion, for defigning 
the near value of E (the remaining part of the Root (ought 
A-I-E=^N,) whofe Denominator or Divilbr is to be 2 A 
(the double Root of the fubtra6ted Square) or 2A4-1 
(that double Root increafed by i) the true value falling be- 
tween thefe two 5 fbmetime the one, fbmeiime the other , 
being neareft to the true value. But (for avoiding of Ne- 
gative Numbers) the latter is commonly dire^ed. 
This Method Monfieur De L'agny affirms to be more than 
aoo Years old : And it is fo 5 for I find it in Lucas Paccioluf 
( other wife called Lucas Burgo^ or Bur go SanSli Sepulcbri) 
Printed af Venice in the Year 1494 ( if not even fooner than 
fo, for I find there have been (everal Editions of it.) And 
how, much older than fo, I cannot tell : For he doth not 
deliver it as a new Invention of his own, but as a received 
Practice, and derived from the Moors or Arabs ^ from whom 
they 
