| XXX | (101) | XXX |
A L G E tiplied by each other are different, then other equations than powers are generated; which to refolve .into the fimple equations whence they are generated is a different operation from involution, and is what is called, the refolution of equations. But as evolution is performed by obferving and tracing back the fteps of involution; fo to difcover the rules for the refolution of equations, we muft carefully obferve their generation. Suppofe the unknown quantity to be x, and its values in any fimple equations to be a, b, c, d, &c. then thofe fimple equations, by bringing all the terms to one fide, become x—a—a, x—b=o, x—cr=o. See. And, the produ<5t of any two of thefe, as x—axx—b=o will give a quadratic equation, or an equation of two dimenfions. The produdt of any three of them, as x—a X x—b X x—c — o, will give a cubic equation, or one of three dimenfions. The produdt of any four of them will give a biquadratic equation, or one of four dimenfions, as x—ax.x—bxx—c x x — d—o. And, in general, “ in the equation produced, the higheft dimenfion of “ the unknown quantity will be equal to the number “ of fimple equations that are multiplied by each o“ ther.” When any equation, equivalent to this biquadratic x — tfXx — by. x — cX x — d—o, is prOpofed to be refolved, the whole difficulty confifts in finding the fimple equations x—a — o, x — b—o, x—c—o, x-—d—o, by whofe multiplication it is produced; for each of thefe fimple equations gives one of the values of x, and one folution of the propofed equation. For, if any-of the values of x, deduced from thofe fimple equations, be fubftituted' in the propofed' equation in place of x, then all the terms of that equation will vanilh, and the whole be found equal to nothing. Becaufe, when it is fuppofed that x—a, or x=b, or x=<r, ox x—d, then the produdt x—aXx—bXx — cXx-—does vaniffi, becaufe one of the fadtors is equal to nothing. There are therefore four fuppofitions that give aT—aXx — bX x—cXx—d = o according to the propofed equation; that is, there are four roots of the prepofed equation. And after the fame manner, “ any other - equation admits of as many “ folutions as there are fimple equations multiplied by “ one another that produce it, or, as many as there are “ units in the higheft dimenfion -of the unknown quanti“ ty in the propofed equation.” But as there are no oth r quantities whatfoever bolides thefe four ( -, b, c, d,) that fubftituted in. the produdt x—aXx — bXx — c X x—d, in the place of x, will make the produdt vanilh; therefore the equation x — «Xx — bXx — cXx — dz=o, cannot pofiibly have more than thefe four roots, and cannot admit of more folutions than four. If you fubftitute in that produdt a quantity neither equal to a, nor b, nor c, nor d, which fuppofe e, then fince neither e—a, e—b, e—-c, nor e—d is equal to nothing; their produdt e — a Xe— by e-—cXe—d cannot be equal to nothing, but niuft be fome real produdt: and therefore there is no fuppofition befide one of the fqrefaid four that gives a juft Vol. I No. 5. 3
B R A. 101 value of x according to the propofed equation. So that it can have no more than thefe four roots. And after the fame manner it appears, that “ no equation can “ have more roots than it contains dimenfions of the un“ known quantity.” To make all this ftillplainer by an example,4in numbers; fuppofe the equation to be refolved, to be x — iox3 -f35X1—50x-(-24 = o, and that you difcover that this equation is the fame with the produdt of x— 1 X x-—2X x — 3XX:—4, then you certainly infer that the four values of x are 1, 2, 3, 4 ; feeing any of thefe numbers placed for x makes that produdt, and confequently x4— iox3+35x1—50x4-24, equal to nothing, according to the propofed equation. And it is certain that there can be no other values of x befides thefe four: fince when you fubftitute any other number for x in thofe fadtors x—1, x—2, x—3, x—4, none of the fadtors vanifir; and therefore their produdt cannot be equal to nothing, according to the equation. It may be ufeful fotnetimes to confider equations as generated from others of an inferior fort befides fimple ones. Thus a cubic equation may be conceived as generated from the quadr -tic x*—/>x--q=o, and the fimple equation x—a—o, multiplied by each other ; whofe produdt ^ X
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may exprefs any cubic equation whofe roots are the quantity [a) the value of x in the fimple equation, and the two roots of the quadratic equation, viz. and 2 2 . as appears from Chap. 12. And, according as thefc roots are real or impcffible, two of the roots of the cubic equation are real or impojjihle. In' the dodtrine of involution, We ffiewed, that " the “ fquare of any quantity, pofitive or negative, is always “ pofitive;” and therefore “ the fquare root of a nega“ tive is impoffible or imaginary.” For example, 'the tfa' is either -b?, or —a ; but can neither be +a nor —a, but muft be imaginary. Hence is undeiftood, that “ a quadratic equation may have no impoffible “ expreffion in its coefficients ; and yet, when it is re“ folved into the fimple equations that produce it, they “ may involve impoffible expreffions.” Thus, the quadratic equation-x*-|-«1:=9 has no impoffible coefficient; but the fimple1 equations from which it is produced, viz. x-f-v/—a’ —o) and x—y/-—al—o, both, involve an imaginary quantity ; as the fquare —az is a real quantity, but its fquare root is imaginary. After the fame manner, a biquadratic equation, when refolved, may give foW fimple equations, each of which may give an impoffible value for the root: and the fame may be faid of any equation that Can be produced from quadratic equations only, that is, whofe dimenfions are of the even numbers. But, “ a cubic equation (which cannot be generated “ from quadratic equations only, but requires one fim“ pie equation befides to produce it) if none of its coef“ ficients are impoffible, 'will have, at leaft, one real “ root,” the fame with the root of the fimple equation Cc whence^
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