ift. x + y x x + 2 = xx + zxy+y, is your Canon for Extracting the Square- Root.

*See the Table in Page 42.

2dly. The greateft Member of the SquareRoot of 4624 is 60 (for 70 x 70 4900); therefore x60, and xx=3600, which ta ken from 4624, leaves 1024 for a Refolvend, pu and the Coefficient of y in the fecond Term of your Canon is 2x =120 for your Divifor; by which Dividing the Refolvend, viz. 1024, the Quotient is 8y; therefore 2xy + 1034 is the Ablativum, which taken from the Refolvend, leaves o. Whence the Square Root of 4624 is 60 + 8 = 68..

2. Let it be required to Extract the Cube Root of 99252847. x + xxx + xxx + 1 = xxx † 3xxy † 3xyy +y® Canon.

Operation.

99252847 (400 +60 +3=463 the Root required.

3. Suppose it was rrequired to Extract the Biquadrat Root of 6612111747853987761.

**+x+x+x+y=x+4x3y + 6a2)",

+4x+y Canon.

If the given Power hath not fuch a Root as is required, you may notwithstanding find a Root nearer the truth, than any affigned in the following manner.

Suppofe it was required to find the Square Root of 2 very

near.

Operation.

2 (1+4+:01+004 +0002 + &c. = 124142 +

The common Method of Extracting the Square Root, CubeRoot, &c. of Numbers, is only an Abridgement of the foregoing Method, and is thus perform'd, viz.

Place the firft Point always over the Figure, which is in the Place of Units. Place alfo a Point over every other Figure, denoted by the Denominator of the Index of the Root to be Extracted; that is, if it be the Square Root, Cube Root, &c. that is to be Extracted, Point over the 2d. 3d. &c. Figure refpectively to the Left Hand; and if there be any Decimals in the given Number, to the Right Hand of the faid Figure, which is in the place of Units. And as many Points as there are over the places of whole Numbers, fo many places of whole Numbers must be in the Root, and the reft are Decimals.

Again any Binomial as x+y, being Involved into it felf, as often as there are Units in the Denominator (1 being the Numerator) of the Index of the Root to be Extracted, produces your Canon for Evolving. Then.

ift. By the Table of Powers in Page 42 (or otherwise), find the greateft Power that is contain'd in the firft Period towards the Left Hand; viz. the greatest Square, if it be the Square Root; the greatest Cube, if it be the Cube-Root, &c. that is to be Extracted; then having plac'd the Root x in the place affign'd for it, which is likewife call'd the Root, Subftract the faid Power from the faid Period.

After the Remainder, place the Figures in the next Period, and call that Number your Refolvend; call alfo the Value of the Coefficient of y in the fecond Term of your Canon, your Divisor. Then ask how oft the Divifor is contained in the Refolvend omitting all the Figures in the laft Period but the firft, the Numbery fet in the Root next after the Value of x; then find the Ablativum thus. Place the Figures which are the Value of the 24. Term of your Canon, fo as the laft of them may be under the 1ft of the laft mention'd Period, and the Values of the 3d. 4th. 5th. &c. Terms 1, 2, 3, &c. Places refpectively more to the Right Hand, than those of the 2d. Term; and the Sum of the Values of the 2d, 3d, 4th, 5th, &c. Terms plac'd as aforefaid, is the faid Ablativum, which take from your Refolvend. But here Note, that if the Ablativum thus found fhould be greater than the Re folvend, then the Value of y is too Great, and must be made Less.

After the Remainder, place the Figures in the next Period, and call that Number your Refolvend, and call the Figures plac'd in the Root; by which find the Value of the nexty, in like manner as before directed, and fo proceed till you have done with all your Period; and if afterwards there is a Remainder, place Cyphers after it, in order to find as many Decimal Figures as you please.

Examples.

Let it be required to Extract the Square Root of 6968, very

mear.

Operation.

6968 (83.4749

64 xx

Div. 16=2x) 568 Refolvend.

48 axy

9= yy
489 Ablativum.

Div. 166 — 1x) 79.00 Refolvend.

664 = 2xy
16=99

6656 Ablativum.

Div. 1668 2x) 12440,0 Refolvend.

1168092xytyy Ablativum.

The other Figures of the Root to the 11th Fig. Inclufive, may be found by Divifion. Thus.

166949.0) 784975,0 (47018

117179

00315

148

16

Whence the Square Root of 6968 is very near 83.474547. 018.

CHA P. III.

Evolution Mixt of Numbers, or the Method of Extracting the Roots of Adfeaed Equations; which Method is generally call'd the Rumeral Exegetis.

i.]F a + a be equal 1860990, 'tis required to find one of the
Values of a.
Suppofe x+y=a. Then
Suppofex+y=a.

Canon,

a3 = y2
x2+3x2 + 3x2 +31 }=1860990;

And ax + 7.