that, according to the Rule, S. 29, for finding any Term in Geometrical Progreffion, the laft or 24th Term of the Progreffion relating to this Queftion, must be (as is found, (§. 31.) 8388608. Wherefore, according to the Rule, (§. 32,) for finding the Sum of any Geometrical Progreffion; the Sum here requir'd must be this, viz. 16777215 Farthings, which by Reduction will be found equal to 174761. 5 s. 3 d. 3 q. much too great a Price for the best of Horses.

Nay, fuppofing only a Pin to be paid 34. for the firft Nail, and confequently for Further of the 24th Nail to be paid 16777215 Pins, the fames and the faid Pins to be worth no more than a Groat a Thousand; yet it will appear by Reduction, that, according to this Rate, the Price of an Horse fo fold, will amount to 279l. 12s. 4d, and upwards; a fufficient Price for the best of Horfes. And thus much for fuch Rules of Arithmetick, as relate to Proportion.

CHAP.

I.

A Root,

and a Power, what.

The Dimenfions of Powers,

how reckon'd.

CHA P. XII.

Of the Extraction of the square and

cube Root.

Br

Ya Root is fignified any Number or Algebraical Quantity, from which more or less multiplied into it felf do arife Products, which are diftinguish'd (from other Products arifing from the Multiplication of two different Numbers or Quantities one into the other) by the Name of Powers. Thus, 3 is faid to be the Root of 9, and 27, &c. because 3*39, and 3×3×3=27. So A is faid to be the Root of AA or A2, and of AAA or A', &c. because AA=AA, A*A*A AAA.

2

As often as any Power involves its Root, fo many Dimensions the faid Power is faid to be of. Thus, 9 (or 3×3) is faid to be a Power of two Dimenfions; 27 (i. e. 3×3×3) of three Dimenfions, c. So AA or A has Two, AAA or A' has three Dimenfions: Where it is alfo to be obferv'd by the Way, that in A, A3, &c. the vertical Figures are from their Use, call'd the Indices of the Dimensions.

Each

liar Names

Each Power is diftinguish'd by a pecu» 3. liar Name, according to the Number of The pecuits Dimensions. Thus, A' or 9, is cal- of Powers, led a fecundan Power, or a Power of whence the fecond Order, A' or 27, a tertian Power, or Power of the third Order,

taken.

metrical

The feveral Powers do alfo borrow o- 4. ther Names from Geometrical Quantities. Other Geo Thus, a Root is otherwife call'd by a Names of Geometrical Name, a Side; a fecundan the PowPower is otherwife call'd by a Geome- ers. trical Name, a Square; a tertian Power, a Cube, (*) &c. Hence, Aq is the fame as AA or A2, Ac the fame as AAA or A3, &c. And 9 is faid to be the Square, 27 the Cube of 3, which is call'd the Side of 9 and 27, &c.

5.

Under

It was obferv'd, Chap. 9. §. 13, that the two Examples of Algebraical Multi- A thorough plication therein contain'd are of great standing of Ufe, when thoroughly understood; for- Algebraiafmuch as thereby the Extraction of the cal Multiplication, fquare and cube Root is render'd more is of great easy to be apprehended, and confequent- Ve toly to be perform'd. Namely, the Pro- Extraction duct of A+E multiplied into A+E, of the (Example 7,) viz. Aq+2AE+Eq, fhews fquare the several Members or Parts, whereof

(*) See the following Names and Characters in Notes to Chap. 9. of my Latin Arithmetick.

confifts

wards the

Root.

6.

And also

to the Ex

traction

Root.

confifts the Square of any binomial Root, i. e. of any Root confifting of two Figures or Quantities. For Aq+2AE+Eq exprefs'd in Words, denotes thus much, viz. that the Square of every binomial Root is made up of (Aq, i. e.) the Square of (A, i. e.) the greater Figure or Quantity, and of (Eq, i. e.) the Square of (E, i e.) the leffer Figure or Quantity and alfo of (2AE, i. e.) the Double of the Product of both Figures or Quantities, multiplied one into the other.

In like manner, the Produc of Aq+2AE+Eq, multiplied again into the Root A+E, (Example 8,) viz. of the cube Ac+3 AqE+3AEq+Ec, fhews the feveral Members, whereof confifts the Cube of any binomial Root. For Ac+3AqE+3AEq+Ec,denotes in Words thus much, viz. that the Cube of every binomial Root is made up of (Ac, i. e.) the Cube of (A,) the greater Figure or Quantity, and of (Ec, i. e.) the Cube of (E,) the leffer Figure or Quantity; and alfo of (3 AqE, i. e.) the Triple of the Product of the Square of the greater Figure or Quantity, multiplied into the Leffer; and likewife of (3AEq, i. e.) the Triple of the Product of the lefs Figure or Quantity, multiplied into the

Greater.

From

Algebra.

From what has been faid, appears 7. fufficiently one Excellency of Algebraical One ExcelOperations; inafmuch as any Operation lency of rightly perform'd, ferves as a Rule for performing all other Operations of the like Kind; and that too fuch a Rule, as the Understanding may much fooner apprehend, it being exprefs'd by a few Symbols or Characters lying all in one View, than when the fame Rule is run out into a Multitude of Words; neceffary indeed to exprefs it, but yet Confounding, rather than informing the Understanding. For which Reafon I choose to teach the Manner of Extracting the fquare and cube Root, by the faid Algebraical Symbols, rather than by Words, in the following Part of this Chapter.

for Ex

Root;

gures or

Whereas, then it has been fhewn, how 8. any binomial Root, denoted by A+E, The Rule being multiplied into it felf produces a tracting Square, confifting of three feveral Mem- the fquare bers, denoted by Aq+2AE+Eq; it when it hence becomes not difficult, having the confifts but Square of any binomial Root, denoted of two Fi by Aq+2AE+Eq, to refolve it into its Quanti Root, or as it is commonly exprefs'd, to ties. extract its Root, denoted by A+E Namely, fuch a Square is refolv'd, first by taking out of it Aq, and taking A for the greater Quantity of the Root fought; and then dividing the next Member 2AE

K

of