Hence all adfected Quadratick Equations, wherein the highest Term is 4, or the Square of the Quantity fought, are reducible to these four Forms.

And thefe Mr. Harriot properly calls Original Æquations, and from them fhews that every Quadratick Equation, wherein the higheft Term is a', or the Square of the Quantity fought, hath juft two real Roots according to the Dimentions of the higheft Power, as being made up by the Multiplication of two Simple Equations; and thefe Roots may be both Affirmative, (as in Cafe 3d.) or both Negative, (as in Cafe 4th.) or one Affirmative, and the other Negative (as in the 1st and 2d Cales); and sometimes they are equal to each other, and fometimes not; and the abfolute Quantity given be, is always the Rectangle of the two Roots b and c (or of the two Values of a); and if be hath a pofitive Sign, the two Roots have like Signs, bur, if a Negative one, unlike; and the Coefficient of a in the 24 Term, is always equal to the Sum of both the Roots with contrary Signs, to what they had at first before Transposition:

That borc, with its proper Sign, is one true Root of the foregoing Original Equations, will appear, by Subftituting b or c with its proper Sign, and its Square, for, and inftead of a and its Square, in any of the foregoing Original Equations.

To inftance what I here fay, I will make use of the first Original Equation, Namely

--

bxa-bc6.

Now if you Subftitute bb for 44, and b for a in the foregoing Equation, It will become

bb + c = bxb-bco.

Which is manifeftly true. For by Abbreviating this Æquation all its Terms are deftroyed, and it becomes o

Whence it is certain, that one of the Roots of (or Values of 4 in) this Æquation

aa + c = b x a

be

is b.

In the next Place, I fayc is the other Root of the faid Original Equation; for by Subftituting cc (i.c. the Square of

Q

၄)

-'c)

c) for aa, and for a in the faid Original Equation, it becomes:

In like manner you may prove b and c, with their proper Signs, to be the two Roots of any of the other three Original Quadratick Equations.

The 4 foregoing Æquations will become to thefe 4 following ones refpectively.

9

1

In the first of which Cafes, you have the Sum of the two Roots fought (); in the 2d (s)=p; in the 3d Cafe (s) and in the 4th Cafe (s): as alfo the Rectangle (r)b, in the 1st and 2d Cafes; but+bin the 3d and 4th Cafes, given in order to find their Difference d); and then each of the faid Roots.

The manner of finding all which, fhall be fhew'd in Part X.

Now forafmuch as fome Algebraifts define adfected Quadratick Equations in this manner, viz.

• When the Quantity fought is brought to an Equality with thofe that are known, and is at one Side of the Equation, in no more than two different Powers, whofe Indices are double to one another; thofe Equations are called adfected Quadratick Æquations.

We may fay by that Definition,

That a+ba + fg +fg

Or a + da3 + c

Or Universally as =±R is an adfected Quadratick Equation.

But

But thefe Equations having as many Roots, Real, or Imaginary, as there are Units in the Index of the higheft Power of the Root fought; and being not produc'd by the Multiplication of two Lateral Equations (as the foregoing Original Quadraticks are), can't be faid to be Original adfected Quadratick Equations; but (fince the manner of folving thofe, is of the fame Nature with that of folving the foregoing ones) may be call'd adfected Quadratick Equations.

CHA P. II.

Of the Origination of Cubick, &c. Equations.

R. Harriot fhews the Original of a Cubick Equation, to be deriv'd from three Lateral Equations, reducd first to the Form of Binomials, or Refiduals; and then Multiplyed continually into each other; or elfe from one Quadratick Multiplyed by a Lateral. Whence he deduces, that all Cubick. Equations have three Roots, Real or Imaginary, or as many as are the Dimenfions of its higheft Power, and no more.

Original Cubick Equations may be thus form'd.

ac

a

T

Then by Tranfpofition

And these three Refiduals Multiplyed con

tinually into each other, will produce this Equation.

In like manner he fhews the Derivation of Biquadratick Equa tions, to be from four Simple, or Lateral Equations reduc'd (as above) to the Form of Refiduals, or Binomials, and then continually Multiplyed into each other: or elfe from a Cubick into a Lateral: or from one. Quadratick into another: or from one Quadratick Multiplyed into two Laterals continually. Wherefore he faith, every Biquadratick will have four Roots, Real or Imaginary, agreeable to the Dimenfions of its higheft Power, and no more Thus, if the former Cubick be Multiplyed by af, this Biquadratick Equation will be produc'd, viz.

From which Original of thefe Equations, 'tis plain, that after you difcover the Value of any one Root, you may deprefs the Equation a Dimenfion lower, by dividing it by fuch Root reduc'd to the Form of a Refidual or Binomial as above.

Thus if you find that one Root, or one a of the laft foregoing Equation, is-f; then Divide the Equation by a + 1 = 0, and it will bring it down to a Cubick; and that Cubick being again Divided by abo, or —co, or a ➡d = o, will be deprefs'd into an Original Quadratick, &e. And this is fometimes of good Ufe to diffolve Compound Equations into their Components, as fhall be fhew'd further on.

C.

From this Method of Compofition of thefe Equations, 'tis allo apparent of what Members each of the Coefficients are made up.

For the Coefficient of the 2d Term is always the Aggregat or Sum of all the Roots, with contrary Signs. Thus in the Cubick Equation above-mention'd, the Coefficient of the ad

b

b

Term is - c And is the Coefficient of the 2d Term of

=d +f

+b

+ d

the above Biquadratick Equation; And in the former,

+ b

and in the latter Equation, is the Sum of all the Roots;

+d

f

whence what we have faid in the 1ft is manifeft.

Wherefore it follows, that if all the Negative Roots, fecluding their Signs, be equal to all the Affirmative ones; (tho not each to each refpectively) then will the 2d Term quite vanish out of the Equation, and be wanting, because the Affirmatives and Negatives do mutually deftroy each other. And vice verfa, when ever the 2d Term is wanting in any of thele Equa tions, the Roots are thus equal, and have contrary Signs.

II. The Coefficient of the 3d Term is the Aggregat of all the Rectangles made by the Multiplication of every Pair of the Roots (with contrary Signs) as often as they can be taken, which in a Cubick is 3, in a Biquadratick is 6, in a 5th Power is 10.

&c.

&c. according to the order of Triangular Numbers. Thus in the

+ bc

3d Term of the Cubick Equation before mention'd bd the

+od Coefficient is the Aggregat 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 3d Term will vanifh, or be wanting.

III. The Coefficient of the 4th Term is the Aggregat of all the Solids made by the continual Multiplication of all the Ternarys, or every three Roots with contrary Signs, &c. and so 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 So 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 these Equations, 'tis also Evident, that 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 faid Equation, that is the Signs of the 2d, 4th, 6th, 8th, &c. Terms, or of the 1st, 3d, 5th, 7th, &c. Thus the Signs of the Roots of this Equation, a 443 — 15a2 + 58a400, are chang'd by writing it thus, 4+ + 443 —— 15a2 — 58a-400; Or thus, at 4a3 + 15a2 +58a +40 ≈0. - And the Signs of the Roots of this Equation, a3 + pa2+qa +r = o, are chang'd by writing it thus, a3paga-ro; Or thus, Or thus, 4+ pa2 — qa+r=9.

I. Again, Since any Original Quadratick Equation may (by what was before faid) be Defign'd by the following one, viz.

In which Equation, the two Roots or Values of a are b P. and b 9: and fuppofe p not, and o; then b p will be the greatest Value of a, which I fuppofe is not imaginary; for if it was, the faid Quadratick Equation, when its known Quantities were defign'd by Figures, would have Imaginary Terms in it, (which I fuppofe neither that, nor any of the following ones has) or the other Value of abg, must be ima ginary too, and of confequence the Original Quadratick Equation propos'd, have no real Root. Thus if bp be by