PART V.

ALGEBRA.

OF EQUATIONS OF SEVERAL DIMENSIONS.

A GENERAL view of the nature, formation, and roots of

equations.

1. A simple equation is that which contains the unknown quantity in its first power only.

Thus ax+b=c.

2. A quadratic equation is that which contains the second power of the unknown quantity, and no power of it higher than the second.

3. A cubic equation is that which contains the third, and no, higher power of the unknown quantity.

Thus ax3-bx2+cx=d, or ax3 + bx2=c, or ax3-bx=c.

4. A biquadratic equation is that which contains the fourth, and no higher power of the unknown quantity.

Thus ax + bx3 — cx2+dx-e=o, &c.

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5. In like manner, an equation of the fifth degree is that which contains the fifth, and no higher power of the unknown quantity; an equation of the sixth degree contains the sixth power; one of the seventh degree the seventh power of the unknown quantity, &c. &c.

6. All equations above simple, which contain only one power of the unknown quantity, are called pure.

Thus ax2=b is a pure quadratic, ax3=b is a pure cubic, arb a pure biquadratic, &c.

7. All equations containing two or more different powers of the unknown quantity, are called affected or adfected equations. Thus ar-bx c is an adfected quadratic; ax3-bx2=c, and ar+bx c are adfected cubics; x1-x2+ax=b, and ax-bx3c, and ax-bx + cx2-dx+e=o, are adfected biquadratics.

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8. An equation is said to be of as many dimensions, as there are units in the index of the highest power of the unknown quantity contained in it.

Thus a quadratic is said to be an equation of two dimensions ; a cubic of three; a biquadratic of four, &c.

9. A complete equation is that which contains all the powers of the unknown quantity, from the highest (by which it is named) downwards.

Thus ax2 bx+c=o, is a complete quadratic; ax3-bx2+cx -do, is a complete cubic ; x-x3 — x2+x—a=o, a complete biquadratic, &c.

10. A deficient equation is that in which some of the inferior powers of the unknown quantity are wanting.

As ax3-bx2+c=o, a deficient cubic; ax-bx2+ cx-d=o, a deficient biquadratic, &c.

11. An equation is said to be arranged according to its dimensions, when the term containing the highest power of the unknown quantity stands first (on the left); that which contains the next highest, second; that which contains the next highest, third; and so on.

Thus the equation x3— axa + bx3 —- cx2+dx—e=o, is arranged according to its dimensions.

COR. Hence every complete equation of n dimensions will contain n+1 terms.

12. The last term of any equation being always a known quantity, is usually called the absolute term: and note, this last or absolute term may be either simple, or compound, consisting of several known quantities connected by the sign + or -; which together are considered as but one term.

13. The roots of an equation are the values of the unknown quantity (expressed in known terms) contained in that equation; hence, to find the roots is the same thing as to resolve the equation.

14. The roots of equations are either possible, or imaginary. Possible roots are such as can be accurately determined, or their values approximated to, by the known principles of Algebra.

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Thus √a,3 √a-b, *√√c, &c. are possible roots.

15. Imaginary or impossible roots are such as come under the form of an even root of a negative quantity, which cannot be determined by any known method of analysis.

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Thus √—a, *√—ab, ° √✅—d, &c. are impossible roots.

16. The limits of the roots of an equation are two quantities, one of which is greater than the greatest root; and the other, less than the least. The greater of these quantities is called the superior limit, and the less, the inferior limit. Also the limits of each particular root, are quantities which fall between it and the preceding and following roots.

17. The depression of an equation is the reducing it to another equation, of fewer dimensions than the given one

possesses.

18. The transformation of an equation is the changing it into another, differing in the form or magnitude of its roots from the given equation.

OF THE GENERATION OF EQUATIONS OF

SEVERAL DIMENSIONS.

19. If several simple equations involving the same unknown quantity be multiplied continually together, the product will form an equation of as many dimensions as there are simple equations employed ".

Thus, the product of two simple equations is a quadratic; the continued product of three simple equations is a cubic; that of four, a biquadratic; and so on to any number of dimensions.

For, let x be any variable unknown quantity, and let the given quantities a, b, c, d, &c. be its several values, so that x=a, x=b, x=c, x=d, &c. these by transposition become x-a=o, x—b=o, x—c=o, x-d=o, &c. if the continued product of these simple equations be taken, (viz. x-a.x-b.x-c.x-d. &c.) it will

This method of generating superior equations by the continual multiplication of inferior ones, was the invention of Mr. Thomas Harriot, a celebrated English mathematician and philosopher, and was first published at London in the year 1631, being ten years after the author's decease, by his friend, Walter Warner, in a folio work, entitled, Artis Analyticæ Praxis, ad Equationes Algebraicas nova, expedita, et generali methodo, resolvendas. By this excellent contrivance the relations of the roots and coefficients, and the whole mystery of equations, are completely develope, and their various relations and properties discovered at a single glance. See on this subject Sir Isaac Newton's Arithmetica Universalis, p. 256, 257. Maclaurin's Algebra, p. 139. &c. Hutton's Mathematical Dictionary, Vol. I. p. 90. Simpson's Algebra, p. 131. &c. Dr. Wallis's Algebra; Professor Vilant's Elements of Mathematical Analysis, p. 48. and various other writers. VOL. II.

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constitute an equation (=o) of as many dimensions as there are factors, or simple equations, employed in its composition: for example.

From the inspection of these equations it appears, that 20. The product of two simple equations is a quadratic. 21. The continual product of three simple equations, or of one quadratic and one simple equation, is a cubic.

22. The continual product of four simple equations, or of two quadratics, or of one cubic and one simple equation, is a biquadratic; and so on for higher equations ".

23. The coefficient of the first term or higher power in each equation is unity.

24. The coefficient of the second term in each, is the sum of the roots with their signs changed *.

Thus, in the quadratic, whose roots are+a and+b, the coeffi cient is a-b; in the cubic, whose roots are+a,+b, and+c, it

* It is in like manner evident, that the roots of the compounded equations will have not only the same roots with its component simple equations, but that its roots will have the same signs as those of the latter.

■ Hence, if the sum of the affirmative roots be equal to the sum of the negative roots, the coefficient of the second term will be 0; that is, the second term will vanish: and conversely, if in an equation the second term be wanting, the sum of the affirmative roots and the sum of the negative roots are equal.