32. It is required to find two numbers whose product shall be 320, and the difference of their cubes to the cube of their difference, as 61 is to unity. Ans. 20 and 16.

33. The sum of 700 dollars was divided among four persons, A, B, C, and D, whose shares were in geometrical progression; and the difference between the greatest and least, was to the difference between the two means, as 37 to 12 What were all the several shares?

Ans. 108, 144, 192, and 256 dollars.

OF CUBIC EQUATIONS.

A cubic equation is that in which the unknown quantity rises to three dimensions; and, like quadratics, or those of the higher orders, is either simple or compound.

A simple or pure cubic equation is of the form

= C,

A compound cubic equation is of the form x3 + ax = = b, x3 + ax2 = b, or x3 + ax2 + bx in each of which the known quantities a, b, c, may be either + or

Or either of the two latter of these equations may be reduced to the same form as the first, by taking away its second term; which is done as follows:

RULE. Take some new unknown quantity, and subjoin to it a third part of the coefficient of the second term of the equation with its sign changed; then, if this sum, or difference, as it may happen to be, be substituted for the original unknown quantity and its powers in the proposed equation, there will arise an equation wanting its second term.

Note. The second term of any of the higher orders of equations may also be exterminated in a similar manner, by substituting for the unknown quantity some other unknown quantity, and the 4th, 5th, &c., part of the coefficient of its second term, with the sign changed, according as the equation is of the 4th, 5th, &c. power.*

* Equations may be transformed into a variety of other new equations, the principal of which are as follows:

--

1. The equation x4- -4x3-19x2106x-1200, the roots of which are 2, 3, 4, and -5, by changing the signs of the second and fourth terms, becomes x44x3-19x2-106x-120=0, the roots of which are 5, -2, -3, and -4.

2. The equation x3+x2- - 10x+8=0, is transformed, by assuming xy-4 into y3—11y2+30y=0, or y2-11y+30=0; the roots of which are greater than those of the former by 4.

3. The equation x3 — - 6x2+9x − 1 = 0, may be transformed into one

EXAMPLES.

1. It is required to exterminate the second term of the equation x3 + 3ax2 = b, or x3 + 3ax2 − b = 0.

Whence 23 - 3a2z + 2a3 — b = 0,

Or z3 3a2zb-2a3,

in which equation the second power (22), of the unknown quantity, is wanting.

2. Let the equation x3- 12x2 + 3x = − 16, be transformed

into another that shall want the second term.

which is an equation where 22, or the second term, is wanting,

as before.

=

10, be transformed into

3. Let the equation 3 6x2 another that shall want the second term.

Ans. 23 12z 26. 4. Let y3 - 15y2+81y = 243, be transformed into an equation that shall want the second term.

Ans. x+6x=88.

which shall want the third term, by assuming xy+e, and in the resulting equation, let 3e2-12e +9, or c2-4e+3=0, in which the values of e are 1 and 3; then assume xy+3, or y+1, and the resulting equation will be y3+3y2-1 = 0, an equation wanting the third

term.

-

1

4. The equation 6x2- ·11x2+6x-1= 0, by assuming x = may be transformed into y3-6y2+11y-6=0; the roots of which are to be reciprocals of the former.

y

5. The equation 3x3 — 13x2 + 14x+16=0, by assuming x = may

3

be transformed into y3—13y2+42y +144 = 0, the roots of which are three times those of the former.-ED.

ed into another that shall want the second term.

0, be

=

6. Let the equation x2+8x3 5x2+10x 4 transformed into another, that shall want the second term.

2x+1=

8. Let the equation 3x3 O, be transformed into another, whose roots are the reciprocals of the former.

Ans. y3 - 2y2 + 3 = 0.

9. Let the equation + 3x2 - 3x + 13 = 0, be transformed into another, in which the coefficient of the highest term shall be unity, and the remaining terms integers. Ans. y3y+ 12y3 162 y + 72 = 0.

OF THE SOLUTION OF CUBIC EQUATIONS.

RULE. Take away the second term of the equation when necessary, as directed in the preceding rule. Then, if the numeral coefficients of the given équation, or of that arising from the reduction above-mentioned, be substituted for a and b in either of the following formulæ, the result will give one of the roots, as required.*

* If, instead of the regular method of reducing a cubic equation of the general form

x3+ax2 + bx + c = 0,

to another, wanting the second term, as pointed out in the preceding article, there be put x=(y-a), we shall have, by substitution and reduction, y3 +(9b3a2) y = 9ab27c2a3; where, since the value of y can be determined by either of the formulæ given in this rule, the value of x will also be known, being x =(y-a). And if b=0, or the original equation be of the following form, x3+ax2+c=0, the reduced equation will be y3-3a2y —— 2a3—27c, where the value of y, being found as above, we shall have, as before, x=(y—a), which formulæ, it may be observed, are more convenient, in some cases, than those resulting from the preceding article; as the coefficients, thus ob tained, are always integers; whereas, by the former method, they are frequently fractions.

coefficient a, of

Where, it is to be observed, that when the

a3

the second term of the above equation, is negative, as

α

27'

also, in the formula, will be negative; and if the absolute

b

term b be negative, in the formula will also be negative;

2

It may likewise be remarked, that when the equation is of the form

* This method of solving cubic equations is usually ascribed to Cardan, a celebrated Italian analyst of the 16th century; but the authors of it were Scipio Ferrcus and Nicholas Tartalea, who discovered it about the same time, independently of each other, as is proved by Montucla, in his Histoire des Mathematiques, Vol. I. p. 568, and more at large in Hutton's Mathematical Dictionary, Art. Algebra.

'The rule above given, which is similar to that of Čardan, may be demonstrated as follows:

=- a.

Let the equation, whose root is required, be x3+ax = b, And assume y + z = x, and 3yx Then, by substituting these values in the given equation, we shall have y3+3y2z+3yz 2 +z3+a × (y +-≈) = y3+z3+3yz × (y + z) + ax (y + z) = 13+ 23 — a × (y + z) + a × (y + z) = b, or y3+z3=b. And if, from the square of this last equation, there be taken 4 times the cube of the equation yz -a, we shall have ye2y323+26

=

=b2+ a3, or y3 -- 23 = √ (b2 + 217 a3).

But the sum of this equation and y3 +z3=b, is 2y3=b+√(b2+ a3), and their difference is 223 = b — ✔(b3+

a3); whence y =3√

=

[1b + √ (163 + 27 a3)], and ≈ = 3√ [ {b — √ ( \b2 + 27 a3)]. From which it appears, that y+z, or its equal, is √ [13+ √(162 +27 a3)] + V° [16−√ (‡b2+27 a3)], which is the

theorem;

a3

and is greater than

b3
4'

or 4a3 greater than 2762 the solu

27

tion of it cannot be obtained by the above rule; as the question, in this instance, falls under what is usually called the Irreducible Case of cubic equations.*

1. Given 2x3

EXAMPLES.

12x2+36x44, to find the value of x.

Here x3- 6x2+18x

22, by dividing by 2. And, in order to exterminate the second term,

6 3

Put x z + = z + 2,

(+2)31⁄23 +622+12% +8

Then -6 (≈ + 2)2= 622-24z-24

18 (2+2)=

= = 22,

18% +36

Whence 23+ 6% +20= 22, or 23 + 6z= 2, And consequently, by substituting 6 for a and 2 for b, in the first formula; we shall have

It may here be farther observed as a remarkable circumstance in the history of this science, that the solution of the Irreducible Case above-mentioned, except by means of a table of sines, or by infinite series, has hitherto baffled the united efforts of the most celebrated Mathematicians in Europe; although it is well known that all the three roots of the equation are, in this case, real; whereas in those that are resolvable by the above formula, only one of the roots is real; so that, in fact, the rule is only applicable to such cubics as have two equal, or two impossible roots.

The reason why the assumptions, made in the note to the former part of this article with respect to the solution of the equation x3-ax=b, are found to fail in the case in question (and it does not appear that any other can be adopted) is, that the two auxiliary equations, 3yz =-a and y3+23=b, which in this case become 3yza, and y3+23=h, and y3+23=6, cannot take place together; being in

or y3z3 =

=

аз

27'

consistent with each other.

For the greatest product that can be formed of the two quantities y3+23 is when they are all equal to each other; or since y3+ 23 = 6, when each of these b; in which case their product is 62.

the two conditions are incompatible with each other; and conequently

the solution of the problem, upon that supposition, can only be obtained by imaginary quantities.

12.