A Treatise of Algebra in Two Books

Sect. 2.

The Extermination of two or more unknown Quantities, by three or more quations.

*In Sect. 1. we have difcours'd of taking away one unknown Quantity by two Equations: But, if there be two, three or four, &c. Quantities to be taken away, there must be three, four or five, &c. Equations. And then the Bufinefs may be done by Degrees: As for Inftance,

If bay3, a+y=e, and 5a=y + 3e, that a and e may be exterminated.

First take away one of the Quantities a or e, fuppofe a, by fubftituting for its Value ye÷b (found by the 1ft. Equation) in the 2d. and 3d. Equations, and then you'll have

+y=e, and y=y+3e. Now, by thefe two last Equations you may easily take away e, as is taught in Sect, 1.

Scholium.

ayy

Hitherto may be referred the Extermination of Surd Quan tities out of Equations, by making them equal to any Letters. As, if you have ay vay √: aa ay: = 2 + √ ayy; by writing for ay, w for √:aa ay:, and x for you'll have the Equations vw=2a+x, vv=ay, and ww= aa ay, and x3ayy; out of which taking away, by Degrees, v, w and x, there will refult an Equation intirely free from Surdity.

PART.

PART VIII.

Of Proportional Quantities, Arithmetical, Geometrical, and Muйi

cal.

CHA P. I.

Of Arithmetical Proportion.

Definition.

W Hen any Rank or Series of Numbers or Quantities do either Increase or Decrease by an equal Interval, or common Difference or Excefs, they are faid to be in Arithmetical Progreffion, or Proportion continued:

In the former there is a continual Increase; in the latter a continual Decrease by 2, which is the common Difference or Excefs.

And univerfally putting a for the first Term, and e for the common Excefs or Difference, the Terms will be

a, a te, a + 2e, a † ze, &c. Increafing.

a, a — e, a — 2e, a ze, &c. Decreafing.

But the moft Simple and Natural Progreffion is that which begins with o, as

O, e, 2e, 3e, 4e, 5e, &c. 0,-e, -2, - ze, -4, -5e, &c.

When it begins with any other Term (as a in the former Progreffions) it is really a Compound of two Progreffions; one of Equals (a, a, a, a, Ec) and the other of Proportionals (0, e, 26, 3e, &c.).

Buc

But when the firft Term exceeds, or is exceeded by, the fecond Term by the fame Number or Quantity, that the third exceeds, or is exceeded by, the fourth Term, but not by the fame that the second exceeds, or is exceeded by the third Term; then that is faid to be a Discontinued, or Disjunct Arithmetical Progreffion :

So{

are faid to be Disjun&

a, ate, at4e, a thes or Discontinued Arithmetical Proportions.

Now in order to the finding out how to refolve Questions concerning these Progreffions.

the Sum of all the Series, viz. of all the Terms.

Lemma.

The Sum of the Extreams (i. e. of the leaft and greatest Terms) is equal to the Sum of any two Means that are equally diftant from their Extreams (i. e. equal to the Sum of the leaft but one, and greatest but one, or leaft but two, and greatest but two, or leaft but three, and greatest but three, &c. Terms) of any Arithmetical Progreffion; and confequently, if the Number of Terms be odd, the double of the middle Term is equal to the Sum of the Extreams; or, if the Number of Terms be even, the Sum of the two middle Terms is equal to the Sum of the Extreams.

Demonftration.

Leaft Term aaty +Greatest Term y

Leaft Term but one a +6 +Greatest Term but one yes

a+ 2e

Leaft Term but two a +22=aty +Greatest Term but two y ·2e.

}

Leaft Term but three a+3=a+y +Greatest Term but three y -3e.

&c.

N.B. What follows relates to →→→

-&c. Q: E. D.

Scholium

Scholium I.

Whence it follows (and is very eafy to conceive) that, if the Sum of the two Extreams be Multiplyed into the Number of all the Terms in the Series, that Product will be dou ble the Sum of all the Series; that is to fay :a+y: xn=25.

Scholium 2.

In the foregoing {Decreafing Series, viz. {