of any

greffion.

Chap. XI Term by the Second, and from the Product, fubftra&t the Square of the first Geome- Term. Then by the fecond Term, lestrical Pro- fen'd by the Subtraction of the firft Term, divide the Remainder, and the Quotient will be the Sum of the Progreflion. For Instance: 2. 4. 8. 16. 32. are so many Terms of a Geometrical Progreffion; which added together, make 62. And so much they will be found to make by Working according to the Rule here laid down. viz. 32x4=128, and 128-4 124, and then 124 divided by (4—2, i. e.) 2 will give 62.

33.

Having thus fhewn, that the aforesaid An Exam- Rule holds true, I fhall now fhew, how ple of the by the fame Rule is folv'd that common faid Rule. Question, viz. what the Price of an Horse will amount to, fuppose he be fold at the Rate of a Farthing for the first Nail, two Farthings for the fecond Nail, four Farthings for the third Nail, and fo on, doubling the Price of each Nail, for as many Nails as there be in all his four Shoes, fuppofing the Nails to be in all 24, viz. fix in each Shoe. Now, according to the State of this Question, it is evident, that the first Term of the Geometrical Progression relating thereunto is One, and the Ratio of the feveral Terms in the faid Progreffion is 2, as in §. 29. whence it follows,

that

that, according to the Rule, §. 29, for Chap. XIfinding any Term in Geometrical Progreffion, the last 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 beft of Horses.

Nay, fuppofing only a Pin to be paid 34. for the firft Nail, and confequently for Further of the fame. the 24th Nail to be paid 16777215 Pins, and the faid Pins to be worth no more than a Groat a Thoufand; yet it will appear by Reduction, that, according to this Rate, the Price of an Horse fo fold, will amount to 2791. 12s. 4 d, and upwards; a fufficient Price for the best of Horfes. And thus much for fuch Rules of Arithmetick, as relate to Proportion.

CHAP.

Ch. XII.

I.

A Root,

and a Power, what.

2.

menfions

of Powers,

bow reckon'd.

CHA P. XII.

Of the Extraction of the fquare and

cube Root.

Y a Root is fignified any Number or

B Algebraic

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 3×3=9, and 3×3×3=27. So A is faid to be the Root of AA or A', and of AAA or A3. &c. becaufe A× A=AA, A× A×A=AAA.

2

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

Each

Each Power is diftinguifh'd by a pecu- Ch. XII. liar Name, according to the Number of 3. its Dimensions. Thus, A' or 9, is cal- The peculed a fecundan Power, or a Power of liarNames the second Order, A' or 27, a tertian of Powers, Power, or Power of the third Order, taken. &c.

whence

metrical

ers.

The feveral Powers do alfo borrow o- 4. ther Names from Geometrical Quantities. Other GeoThus, a Root is otherwife call'd by a Names of Geometrical Name, a Side; a fecundan the PowPower is otherwife call'd by a Geometrical Name, a Square; a tertian Power, a Cube, (*) &c. Hence, Ag is the fame as AA or A', 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.

A tho rough Un

It was obferv'd, Chap. 9. §. 13, that the two Examples of Algebraical Multiplication therein contain❜d are of great derstandUfe, when thoroughly understood; for- ing of Alafmuch as thereby the Extraction of the gebraical Multiplifquare and cube Root is render'd more cation, is eafy to be apprehended, and confequent- of great Ufe toly to be perform❜d. Namely, the Pro- wards the duct of A+E multiplied into A+E, Extracti (Example 7,) viz. Aq+2AE+Eq, fhews fquare the feveral Members or Parts, whereof Root.

confifts

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

on of the

Ch. XII confifts the Square of any binomial Root,
i. e. of any Root confifting of two Fi-
gures or Quantities. For A4+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 Quan-
tity, 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 Quanti-
ties, multiplied one into the other.

to the Ex

6. In like manner, the Product of And also Ag+2AE+E7, multiplied again intò traction of the Root A+E, (Example 8,) viz. the cube Ac+3A7E+3AE+Ec, fhews the fe

Root.

veral 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 (3AqE, i. e.) the Triple of the
Product of the Square of the greater
Figure or Quantity, multiplied into the
Leffer; and likewife of (3AE1, i. e.)
the Triple of the Product of the Square
of the lefs Figure or Quantity, multiplied
into the Greater.

From

H