Furthermore any Equation wherein is the Co-efficient of the higheft Power of a the Root fought and o one of the Parts thereof may be reprefented by a + ba"-1 +ca"-2,&c. b=c(for b, c, &c. being Symbols or Univerfal, may therefore be equal to any Affirmative or Negative Numbers or Quantities); which Equation, being Multiplyed by : ak:=0, will produce the Equation a"++baca", &c.

Fka”

bka”

-ba

ka" Ibka"-1cka"-2, &c.hko. Now 'tis plain that ifk be Subftituted for and inftead of a in the last Equation, the Refult will be "+1 +bk” ±ck”-1,&c.+ lk" ck", &c. bko, viz. oo wherefore one of the Values of a in the faid last Equation is k; confequently ako.

+

kn

Hence we have ftrong Grounds to fuppofe. 1. That any Equation whatever, wherein is the Co-efficient of the higheft Power of a the Root fought, and o one of the Parts thereof, is produc'd by Subtracting the feveral Values of a therein from a (Where Note, that to Subtract is the fame thing as to Add + And Equating each Remainder to o, and then Multiplying all thefe Equations together: 2. Confequently that a any fuch Value of a (0) will Divide any fuch Equation without leaving any Remainder: And, 3. That in fuch an Equation a hath fo many Valucs, real or imaginary, as there are Units in the Index of its higheft Power in that Equation.

But in order to put thefe three Suppofitions beyond all doubt, Br. I will now confider them more particularly in like manner (I prefume) as the Sagacious Author Mr. Harriot hath done.

CHA P. I.

The Origine of Quadratick Equations is thus deriv'd by Mr. Harriot.

a = br

Cafe 1. If {+}; then by Tranfpofition, a-b = 0 2;

and, by Multiplying one by the other, you will have aa + ca-ba-bco; that is an ic-b: xa - b c = 0.

Cafe

Cafe 2. If Sa

sa==br ra =+c

and, by Multiplication,

;

then by Tranfpofition, = c = 0}2 Sa+b=02

aa—ca + ba—beo; that is aa-:c-b:xa — bc = 0°

sa

Cafe 3. If {^=+}; then by Tranfpofition, {a_c=0}, {a

a =+cs

and, by Multiplication,

aa-ic + bxa + bco.

a=

b

a

If {a}; then by Tranfpofition,{a+b=oy

a=

and, by Multiplication,

aa+c+b:x a + bc = 0;

Hence all Quadratick Equations, wherein the highest Term is a or the Square of the Quantity fought, are reducible to thefe four Forms; viz.

And these Mr. Harriot properly calls original Equations, and from them fhews that every Quadratick Equation, wherein the higheft Term is a', or the Square of the Quantity fought, hath juft two Roots according to the Dimensions of the highest Power, as being made up by the Multiplication of two Lateral Equations: And these two Roots may be one of them Affirmative, and the otherNegative, or both of them Affirmative, or both Negative, or both Imaginary and fometimes they are equal to each other. (Here Note, when they are fo with contrary Signs, the Equation will become a Simple Quadratick) and fometimes not: And the abfolute Number be is always the Rectangle of the two Roots b and c (or of the two Values of a): And if it have a poffitive Sign, the two Roots have like Signs; but if a Negative one, unlike: And the Co-efficient of a in the fecond Term is always equal to the Sum of both the Roots with contrary Signs.

That

That b and c with their proper Signs, are the true Roots of the foregoing original Equations will appear by fubftituting b and c feverally with their proper Signs, and their Squares, for and inftead of a and its Square in each of them.

bc:

To inftance what I here fay, I will make use of the first original Equation, namely aa +: c- b: x a o; in which, if you fubftitute bb for aa, and b for a, it will become bb +: c - b : x b — bc o,which is manifeftly true; for by abbreviating this Equation all its Terms are deftroyed, and it becomes o=0: Whence it is certain that one of the Roots of (or VaJues of a in) this Equation aa +:0 b:xa bco is b. In the next place, I fay - C is the other Root of the faid original Equation; for by fubftituting cc (i. e. the Square of c) for aa and c for a in that original Equation, it will become ccc — b: x — c — be=0; that is, cc - cc + be =0, oro=0.

But in the Equation aa+:c b: xa-bc = 0, a can't be equal to any other Quantity befides -+band-c: For fuppofing 4 equal to any Quantity either greater or less than b, as b+z I fay b+z is

ха

= C

Demonftration.

a being, by Suppofition, = b+z, therefore aa +: c-b: beo will become bb + 2bz+zz + cb + cz - bb. bz-bco; that is, bzzz + cz=o: wherefore bz +co, by Dividing each part by z (Viz. by z confequently b+z = c. Q. E. D.

b+2=

0 = 0);

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

Now fuppofing cb, and ebp, as alfo c + b = q, C =q, and beb, the four precedent Equations will become equal to these four following ones refpectively.

In the first of which Cafes you have the Sum of the two Roots fought (s); in the 2d. sp; in the 3d Cafe, s +9, and in the 4th Cafe, s= : as alfo their Rectangle (r) = in the ift and 2d Cafes, but = + b in the 3d and 4th Cafes, given in order to find their Difference (d); and then each of the faid Rocrs.

The Method of finding all which shall be fhewed in Part X.

Now,

Now, forafmuch as fome Algebrifts define Adfected Quadratick Equations in this manner; Viz.

When the Quantity fought is brought to an Equality with "those that are known, and is at one fide of the Equation in << no more than two different Powers whofe Indices are double 66 to one another; thofe Equations are called Adfected Quadratick "Equations,

We may fay, by that Definition, that aa ± ba2 =±ƒ8,

n

Or a da3 = c, or univerfally a" ± sa=R is an Adfected Quadratick Equation.

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

CHA P. II.

Of the Drigine of Cubick &c. Equations.

MR. Harriot fhews the Original of a Cubick Equation to be derived from three Lateral Equations, reduced first to the form of Refiduals or Binomials, and then multipled together; or elfe from one Quadratick Multiplied by a Lateral. Whence he deduces, that all Cubick Equations have three Roots Real or Imaginary, or as many as are the Dimensions of its highest Power,

and no more.

Thus to form a Cubick Equation, let the three Roots be

In like manner he fhews the Derivation of Biquadratick Equations to be from four Lateral Equations reduced (as above) to the Form of Refiduals or Binomials, and then multiplied together; or elfe from a Cubick into a Lateral; or from one Quadratick into another; or from a Quadratick multiplied by two Laterals continually. Wherefore he justly infers, that every Biquadratick will have four Roots real or imaginary agreeable to the Dimensions of its higheft Power, and no more.

Thus if the former Čubick be multiplied by : a+ƒ:=0, This Biquadratick Equation will be produced, Viz.

From which Original of thefe Equations 'tis plain, that after you discover the value of any one Root, you may deprefs the Equation a Dimension lower, by dividing it by fuch Root reduced to the form of a Refidual or Binomial, as above.

Thus, if you find that one Root, or one a of the foregoing Equation is f; then divide the Equation by :a+f:(=0), and it will bring it down to a Cubick; and that Cubick being again divided by:ab: (=0), or: a-c: (= 0), or :a-d: (0) will be deprefs'd into an original Quadratick, &c. And this is fometimes of good ufe to diffolve Compound Equations into their Components, as fhall be fhewed further on.

From this Method of Compofition of these Equations 'tis also apparent of what Members each of the Co-efficients are made up.

For,

I. The Co-efficient of the fecond Term is always the Aggrerate or Sum of all the Roots with contrary Signs. Thus in the above Cubick Equation, the Co-efficient of the fecond Term is b. C d: And --- - C d+f is the Co-efficient of the fecond Term of the above Biquadratick Equation. But ed in the former; and b+c+d-ƒ in the latter Equation, is the Sum of all the Roots; whence what we have here faid is manifeft.

Wherefore it follows, that if all the Negative Roots, fecluding their Signs, be equal to all the Affirmative ones; (tho' not

each