CO 
they had their Algorifm^ or Pradice of Arkhmetick by the • 
Ten Numeral Figures now in ufe. 
And it is continued down hitherto in Books of Pradical 
Arithmetick in all Languages, which teach the Extradion of 
the Square Root, and (therein) this Method of Approxi- 
mation, in cafe of a Non-quadrate. 
The true ground of the Rule is this: Aq being (by Con- 
ftrudion) the greateft Integer Square contained in 'tis 
evident that E muft be lefi than i ] (other wife not Aq, but 
the Square of A -j- 1, or fome greater than fo, would bQ the 
greateft Integer Square contained in N.) Now if the Re- 
mainder B = 2AE-)-Eqbe divided by 2 A, the Refult will 
be too great for E, ( the Divifor being too little \ for it rtiould 
be 2 A+E, to make the Quotient E.) But if (to redifis 
this) we diminifli the Quotient, by increafing the Divifor^ 
adding I to it, it then becomes too little; becaufe the Divifbr 
is now too big. For ( E being lefs than i ) 2 A-j- i is more 
than 2 A + E; and therefore too big. 
As for inftance; If the Non-quadrate propofed be N=3 5^, 
the greateft Integer Square therein contained is Aq = 4 ( the 
Square of A = 2:) which being (ubtrac^ed, leaves N — Aq 
= 5'^4i=:i = B= 2AE Eq. Which divided by 
2 A =4, gives \ : But divided by 2A4-i=4+x=5'* 
gives I, That too great, and this too Hide for E. And there- 
fore the true Root ( A +• E = V N ) is lefs than 2 ^ = 2.25-, 
but greater than 2 } = 2.2 : And this was Anciently thought 
an Approach near enough. 
If this Approach be not now thought near enough, the 
fame Procefs may be again repeated ; and that as oft as is 
thought neceffary. 
Take now for A, 2 ^ = 2.2, whofe Square is 4.84= Aq, 
( now confidered as an integer in the (econd place of Deci- 
mal Parts.) This fubtraded from 5'.oo, (or, which is the 
fame, 0 84 the excels of this Square above the former, from 
I which was then the remainder,) leaves a new remainder 
.16 2 
B=o.i6: which divided by 2 A = 44, S'^^^^"^"^ ^ 
= 0.03636+, too much. But divided by 2 A -^* ^ =^ 
•16 8 
it gives ^--^ = — = 0.0 3 y ^ ^ too little. The true Va- 
lue (between thefe two) being 2.2 56 pox'me^ whofe Square is^ 
4.999696. C If 
