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. Thus ax2 bxc. 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. 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 ar2=b is a pure quadratic, ax3=b is a pure cubic, art-b a pure biquadratic, &c. 7. All equations containing two or more different powers of the unknown quantity, are called affected or adfected equations. Thus ax2-bx=c is an adfected quadratic; ax3-bx2=c, and ax3+bx=c are adfected cubics; x1—x2+ax=b, and ax1 — bx3 =c, and ax1—bx3 +cx2-dx+e=o, are adfected biquadratics. 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 -d=o, is a complete cubic ; xa—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; ax1-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 x-ax2+ 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. Thus a,a-b, c, &c. are possible roots. 4 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. Thus a,ab, 6-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 developed, 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. constitute an equation (=o) of as many dimensions as there are factors, or simple equations, employed in its composition: for example. Let x-a=0 Be multip. into x-b=o The product is xo_a}x+ab=o, a quadratic. Multiplied into x-co The product is x3-a +ab -b x2+ac x—abc=o, a cubic. -c +bc Multiplied into x- d=o The product is x1. -α -b -C -d 23 +ab -abc +ac -abdx+abcd=o, a 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 coefficient 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. a 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. is-a-b-c; in the biquadratic, whose roots are+a,+b+c, and+d, it is-a-b-c-d, &c. 25. The coefficient of the third term in each, is the sum of all the products that can possibly arise by combining the roots, with their proper signs, two and two. Thus, in the cubic, the coefficient of the third term is+ab+ ac+be; in the biquadratic, it is+ab+ac+ad+be+bd+cd, &c. 26. The coefficient of the fourth term in each, is the sum of all the products that can possibly arise by combining the roots, with their signs changed, three by three. Thus, in the biquadratic, the coefficient of the fourth term is-abc-abd-acd-bcd. In like manner, in higher equations, the coefficient of the fifth term will be the sum of all the products of the roots, having their proper signs, combined four by four; that of the sixth term, the roots, with their signs changed, five by five, &c. 27. The last, or absolute term, is always the continued product of all the roots, having their signs changed. Thus, in the quadratic, whose roots are+a and+b, the last term is+ab (or-ax-b); in the cubic, the absolute term is -- abc (=—a× —b× —c); in the biquadratic, the absolute_term is+ abcd (ax-bx-cx-d), &c. 28. The first term is always positive, and some pure power of x. 28. B. The second term is some power of x multiplied into -a,-b,-c, &c. and since x is affirmative, and each of these quantities negative, it follows that the second term itself is negative, since+x-produces -. 29. The third term will be positive, for its coefficient being the sum of the products of every two of the negative quantities -a,-b,-c, &c. and (since -×-produces+) therefore these sums, multiplied by any power of x, (which is always positive,) will always give a positive result. 30. For like reasons the fourth term will be negative, the fifth positive, the sixth negative, and so on; that is, when the roots are all positive, the signs of the terms of the equation will be alternately positive and negative and conversely, when the signs of the terms of the equation are alternately+and all the roots will be positive. |