Page images

let AC, and let A= 1, then the function becomes (a+B)-(+) or (a+B) + (a+8) or 2 (x+8); and although the function admits of six different values, yet two and two are equal, changing the signs: for (a + b) — (x + d) {(v + d) — (a + B) }

[ocr errors]


[ocr errors]

The reducing equation, which contains these several values, is z' +8qz2 + (16q3 — 645) z-64r2; which, putting z= 4y, is the same equation with that which results from the method of Des Cartes, as indeed it must be.

Finding, by Cardan's rule, or by some equivalent method, the three roots (z', z", z",) of this cubic, we shall have four equations to resolve four unknown quantities; that is, the four roots a, ß, v, d, of the biquadratic. The four equations are ä + B = a+y= a+d=

[blocks in formation]

whence a, f, y, d, by the ordinary processes of elimination. This is one instance of the resolution of a biquadratic by finding the value of a function of the roots. Other functions may be assumed, such as

[merged small][ocr errors][subsumed]

and if we investigate the equation that gives the values of this function, (those values are only three,) it will appear to be the cubic that results from employing Tschirnau's method. We may also assume a function of the roots, as aß +28; and the cubic which will give three values of the function, which are aß +rd, ar + Cd, ad + By, will in fact be the equation that results in Ferrari's or Bombelli's method.

These previous observations are not unconnected with the matter and reasonings of the tract before us; the author of which, in his preface, speaks of the resolution of equations by investigating the values of certain functions of the roots, and in his work resolves a biquadratic by solving the equation which generally represents the sum of any two roots: this is, in fact, what is done in the second problem. In that problem, it is required to find four numbers x, y, z, u, such that

[blocks in formation]
[blocks in formation]

is required. Now, in the first place, the author destroys the second term so that, if A, B, C, D, are the new values of the unknown quantities in the transformed equation, A+B+ C+Do, he next puts A+B 2s, and then deduces an equation in terms of s, which is,

[ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

If the equation deprived of its second term by substitution be xGxFx + E, this bicubic equation is the same with that of Euler, and in fact with that of Des Cartes; as indeed it must be, since in each an equation is sought that contains the sums of any two roots of the original equation.-We see nothing new in this mode of solving a biquadratic equation.

The solution of the other problems is effected by the aid of the formula for the sum of the mth powers of the roots expressed in terms of the coefficients, p, q, r, &c, of an equation x" —px"1 + &c. Such a formula is given in Waring, p. 1. Meditationes Algebraica, and in Arbogast's Calcul des Dérivations. The author, indeed, in the simplest cases, does not immediately resort to the general formula, since that would be unneces sarily to increase the difficulty of the problem: thus, in quest. 3d, three numbers are required, of which the continual producta, the sum of their squares = b, and the sum of their cubesc. Let x, y, z, be the numbers, and let their sump; then

x1+y'+z3 or c= (x2+y2+z` ) (x+y+x) — { xy2+yx2+xz2+&c.} bp − {xy + yz + &c. } (x + y + z) + 3 xyz:

or c=

but 2xy + 2yz + &c, = (x + y + z)2— (x2 + y2+ x2)

[ocr errors][ocr errors][merged small]

Consequently, c = bp — (122). p + 3α

[ocr errors]

or p3-3 bp + 2 c —

·ba = 0;

whence, by Cardan's rule, p or x + y + z; and since

[merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][subsumed]

In like manner, the succeeding problem (4th) may be solved: but in the 5th, in which x, y, z, are required


[blocks in formation]

pm-mqpm-2 + mrpm-3 — &c.

[ocr errors]

and from this, with certain values of m, the numbers x, y, z, may be obtained.

In several other problems, by ingenious transformations, and by a dexterous use of the above theorem, the author deduces solutions.

In problem 16, the writer proposes to solve a biquadratic by the mediation of a cubic wanting its second term; and his rule is sufficiently plain and simple, but it is not essentially different from rules delivered by other authors. It may in fact be deduced from that of Waring: according to whom, if the biquadratic be

[ocr errors]
[ocr errors][ocr errors]

the reducing cubic is


8n' + 4q n2 + (8s — 4rp) n + 495 + 4p2 s — p2 = 0 ; and the four roots of the biquadratic are the four roots of the two quadratics

x2 + { p + √ (p2 + 2n + q) }x+ n =√i+”2

If now we put p=0, put 3 a instead of b for q, "" and c for -s, and transform the reducing cubic by substituting m + a, instead of 2n, we shall obtain the present author's reducing cubic; and by taking the roots of the quadratics, the roots will be found to be the same as those which he has given.

In problem 17, the author proposes to deduce, from Bombelli's rule of solving a biquadratic, that of Euler and Descartes. He takes a biquadratic with all its terms complete, and then gives what he means to be Bombelli's reducing cubic in fact, the same as that which we have just stated. Now we think that, in point of history, the author has fallen into an error. Bombelli has given no rule (we speak, however, with some hesitation, since we have not his book before us,) for solving equations of the form

[merged small][ocr errors][merged small]

the equations which he solves want the second term; and we mention this principally because Dr. Waring takes merit to himself, in his preface to the Meditationes Algebraica, for this extension of Ferrari's method, and because Mr. Baron Maseres has, in a separate and full discussion, shewn that such extension brings no practical advantages of more easy or more concise computation. This observation being made, we remark

Dd 3


that the present author shews fully what he proposes to manifest; viz. that Des Cartes's bicubic may be derived from Ferrari; and perhaps we shall not be deemed unpardonably to trespass on the rights and privileges of authors, if we, after our own manner, establish the same point.

[blocks in formation]

Suppose this to be formed from two quadratic factors; then, since the coefficient affecting x is equal to o, the second terms in the two quadratics must be 2x, and-zx, respectively. Now the third terms of the quadratics multiplied together must produces: they cannot be of the form 5, because then there would be no term to correspond with rx: but, since (a + b) (a—b) = a2-b2, it is plain that the third terms may be of this form

[ocr errors][merged small][merged small][merged small]
[ocr errors]

If we multiply these two together, and compare the coeff cients affected with the powers of x, we have the result

2 v — z2 = q, 2 ≈ ✓ (v2 — s) = r.

[merged small][merged small][merged small][merged small][ocr errors]

whence 8 3-4 g v2 — 8 s v + 45qr; which is Fer

rari's reducing cubic.

Again, since 2 v = q + z2 .•. 4 v2 = q2 + 2qz2 + z'÷

[merged small][merged small][merged small][ocr errors]

··· z^ + 2qz1 + (q3 — 45) z2 — g2 = 0,

which is Descartes's reducing bicubic. Hence, from the same resolution of a biquadratic into factors, are deduced the solutions of both Ferrari and Descartes.

It has already been shewn that, if ✔z, z", z", be the roots of the cubic,

23 +8qx' + (16 q2 —— 64 s) ≈ — 64 r2,

α + β

a + y ==±√3′′

a + d ==√2"

and a + By + d = 0;


hence a=

B =

[ocr errors]
[ocr errors]


Now Euler assumes x, the root of the biquadratic, to be √p + √q + √r, p, q, r, being roots of a cubic equation: consequently, this cubic must be the same if, instead of z, we put 4'z; in which case the former equation becomes

[ocr errors][merged small][merged small][merged small]

which is Euler's reducing equation.

[ocr errors][merged small][merged small][merged small]

In the preceding process and investigation, Ferrari's and Des Cartes's methods have perhaps been brought more nearly toge ther than they ever were before; and if the occasion were fit, or if we had not already in some degree transgressed the line of our duty, we could shew that the method of solving a biquadratic, by deducing an equation involving powers of the last terms of the component quadratics, (if such quadratics be x2±zx+k, x2 zx+g,) is reducible to the former methods: as also that the final equation, although of six dimensions, admits of a solution which, if true, overturns Dr. Waring's assertion that the solution of a biquadratic is in vain sought from such reducing equation.

If the author of the present tract should continue his researches, (and we hope that he will,) we advise him to adhere to a certain form in his equations, and not capriciously to substitute different letters for the coefficients of his second, third, &c. terms. Very much labour, and fruitless toil of memory and reference, would be saved, if writers on algebra would agree on a common form for equations. The usual way of writing a general equation is

[merged small][ocr errors][ocr errors]
[ocr errors]

rx-3+ &c.

and no possible good can arise from changing q into p, or p into a, or q into b. M. Lacroix, in his treatise, has most injudiciously used p sometimes for the sum of the roots, and sometimes for the sum of the rectangles of the roots: Mr. Wood, if we recollect rightly, always employs the same symbols to represent the coefficients of the same terms: Dr. Waring, too, in most instances, keeps to the form x" - pxn−1 +qx”—2 &c. but Euler is perfectly lawless in his representation of the coeffi cients. Let no one, in the pride of originality and invention, ridicule us for attention to these minuti. We are sure that no person who is versed in mathematical investigations will deem such attention unimportant; and we hope that, in a future edition of this work, the author will conform to the general usage, and adopt the conventional symbols, which

Dd 4


« PreviousContinue »