each to each respectively) then will the fecond Term quite vanish out of the Equation, and be wanting, because the Affirmatives and Negatives do mutually deftroy each other. And Vice Verfa, whenever the fecond Term is wanting in any of thefe Equations, the Roots are thus equal, and have contrary Signs.

II. The Co-efficient of the Third Term is the Aggregate of all the Rectangles made by the Multiplication of every pair of the Roots (with their proper Signs) as often as they can be taken, which in a Cubick is three, in a Biquadratick is fix, in a 5th Power is ten, &c. according to the order of Triangular Numbers. Thus, in the third Term of the Cubick Equation before-mentioned, be bd + cd the Co-efficient is the Aggregate of the three Rectangles of the Roots b, c and d, taken by Pairs.

And here if all the Negative Rectangles, fecluding their Signs, are equal to all the Affirmative ones, they will deftroy one another, and fo the third Term will vanish or be wanting.

III. The Co-efficient of the fourth Term is the Aggregate of all the Solids made by the continual Multiplication of all the Ternaries, or every three Roots, with contrary Signs, &r. And fo on ad Infinitum.

IV. As in Quadraticks the abfolute Number, or Quantity given is always the Rectangle of the two Roots or Values of a, fo in Cubicks, 'tis always the Solid of all the three Roots, with their Signs changed, one into another; and in Biquadraticks, of all the four Roots; &c.

From this Method of Compofition of thofe Equations, with due Confideration, it will be evident that,

ift. The Affirmative Roots of any Equation are changed into Negatives, and the Negative Roots into Affirmatives, by changing the Signs of every other Term of the Equation, that is the Signs of the 2d, 4th, 6th, 8th, &c. Terms, or of the 1ft, 3d, 5th, 7th, &c. Thus the Signs of the Roots of this Equation a4 - a3 a2 1-7a 60 (in which the Values of a are 1, − 2, I + √2 and 1 -√2) are changed by writing it thus at a3 a2 - 7a

I,

-

a2+7a+6=0 (in each of which two laft

Values of a are 1, 2, I

6=
O, or thus

√2, and

And the Sigs of the Roots of this Equation as

a4

Equations the 1+ √ -2): pa2 + qa ተ

+ro are changed by writing it thus a3-pa+qa — r =0, or thus a3 + pa2 — qa + r = 0.

And 2dly, The Number of imaginary Roots in any Equation whatever, not including any imaginary Term in it, is even; viz. none, two, four, fix, or eight, &c.

PART

The Solution of Adfected Quadratick Equations.

ALL Adfected Quadratick Equations do (as hath been already

fhewn) fall under the Confideration of these four Forms

or Cafes, Viz.

a2 + pa—b=0}

When there happens to be more Terms in one of these kind of Equations than two, and the highest Power of the unknown Quantity is multiplied into fome known Co-efficient, you must reduce them by Divifion (as in Part VII. Chap. 1. Sect. 4.) and for the fractional Quautities that may arise by thofe Divifions substitute another whole Quantity.

For Inftance, let baa+caa by dividing by b+c, you'll have aa

de + cb; then, dccb

da

ca

d

+ c

btc

b + c

N. B. The Values of +P, +9 and -la are fuppofed each of them to be Affirmative Numbers. =

-kal be the new Equation equal to the other, and now fitted for a Solution. But here we will leave it, and return to our four preceeding Cafes; in order to folve each of which, it will be requifite to premise this

Lemma.

Half (d) the Difference of any two Quantities (z and y, whereof z is the greater) added to half (s) their Sum, is equal to the greater; but fubtracted from half their Sum, is equal to the Leffer of them.

*

It hath been already proved, that in each of

the four foregoing Forms or Cafes of Adfected *See Pag. 110. Quadratick Equations, the Sum (s) and Rec

tangle (r) of the two fought Roots are given, to find their Difference, and then the Roots themselves feverally.

Now, by the foregoing Lemma,

is+id = the greater of the fought Roots,
t

and sd the leffer of them;

And, by multiplying them by one another, you'll have

dd

=r

Then, by multiplying the last Equation by 4, you'll have -dd = 4r

That is, by Tranfpofition, ss 4r dd
And by Evolution: ss4r:

d

ss4 the greater of the

2

Now, by our Lemma, is →→

√:55-47:

fought Roots. And is --

2

the leffer of them:

*

Wherefore the Canon for finding the two Values of a in this ift Cafe, Viz. aa+pa¬ho is (because, as we have already prov'd, s = ~ P, and rb in this first Cafe) - p JPP + 4b Greater

2

Leffer, or Negative Value of a.

Alfo

Alfo the Canon for finding the two Values of a in this 2d

Cafe, viz.

Again, the Canon for finding the two Values of a in this 3d Form or Cafe, viz.

aaga+b=o is (becaufes is q, and rb in this 3d Cafe) 9± Greater Value of a; both √:99-4b:

2

Leffer

of which Values are Affirmative if 994; and, for that Reafon, this Equation is called Ambiguous; but if qq4h, then thofe Values are Imaginary: For, in this latter Cafe, the SquareRoot of a Negative Quantity is to be Extracted in order to find the Value of a: But a Negative Quantity hath no SquareRoot; for whether the Root be Affirmative or Negative, its Square will be Affirmative; wherefore fuch impoffible Equations are faid to have Imaginary Roots.

And the Canon for finding the Values of a in this 4th Cafe, viz. aa+ga + b = o, is (becaufe sq, and rhin this 4th Cafe) — 9± √99 — 4b =

2

Greater

Leffer

Value of a, both of which Values are either Negative or Imaginary; and therefore this last Cafe is not taken of notice by many Algebrists; and I (for Brevities fake) will infert it no more.

Tho' the foregoing Method of Refolving Adfected Quadratick Equations is what naturally follows from Mr. Harriot's Method of Compounding them, yet it is not the fame with his, which is a peculiar Way of his own; and that is, after tranfpofing h, by adding the Square of half the Co-efficient to each Part of the Equation, he thereby makes the unknown Part a compleat Square in Species.

Thus by Tranfpofing the known or given Quantity in the three firft foregoing Forms or Cafes, they will become

I. aa + pa=h = b

2. aa

3. aa

pa

2

qa=- b) Refpectively.

And thefe Mr. Harriot calls Canonical Equations.

Now