The Project Gutenberg eBook #36640: Lectures on Elementary Mathematics

x³ + px + q = 0.

Suppose
x = y + z,

where y and z are two new unknown quantities, of which one consequently may be taken at pleasure and determined as we think most convenient. Substituting this value for x, we obtain the transformed equation
y³ + 3y²z + 3yz² + z³ + p(y + z) + q = 0.

Factoring the two terms 3y²z + 3yz² we get

3yz(y + z),
and the transformed equation may be written as follows:
y³ + z³ + (3yz + p)(y + z) + q = 0.

Putting the factor multiplying y +z equal to zero,—which is per- missible owing to the two undetermined quantities involved,—we shall have the two equations

3yz + p = 0

and
y³ + z³ + q = 0,

from which y and z can be determined. The means which most naturally suggests itself to this end is to take from the first equation the value of z,

z = − _3y ,
p

and to substitute it in the second equation, removing the frac- tions by multiplication. So proceeding, we obtain the following equation of the sixth degree in y, called the reduced equation,

y⁶ + qy³

_p3

− ₂₇ = 0,

which, since it contains two powers only of the unknown quan- tity, of which one is the square of the other, is resolvable after the manner of equations of the second degree and gives immediately

₃ q ^. q² p³
= − ₂ +

4 ⁺ 27 ^,

y

from which, by extracting the cube root, we get

.³ q

^. _q2 _p3

and finally,

y = − ₂ +

4 ⁺ 27 ^,

p

x = y + z = y − _3y .

This expression for x may be simplified by remarking that the product of y by the radical

4 ⁺ 27 ^,

.³ q
− ₂ −


^. _q2 _p3

supposing all the quantities under the sign to be multiplied to- gether, is

.3 ^{p3 p}

− ₂₇ = − ₃ .

The term ^p , accordingly, takes the form

3y

.³ q

^. _q2 _p3

and we have

− − ₂ −

4 ⁺ 27 ^,

.³ q
− ₂ +

x =

. _q2

p³ .³ q


^. _q2 _p3

an expression in which the square root underneath the cubic radical occurs in both its plus and minus forms and where con- sequently there can, on this score, be no occasion for ambiguity. This last expression is known as the Rule of Cardan, and there has hitherto been no method devised for the resolution of equations of the third degree which does not lead to it. Since cubic radicals naturally present but a single value, it was long thought that Cardan’s rule could give but one of the roots of the

4 ⁺ 27 ⁺

− ₂ −

4 ⁺ 27 ^,

equation, and that in order to find the two others we must have recourse to the original equation and divide it by x − a, a being the first root found. The resulting quotient being an equation

of the second degree may be resolved in the usual manner. The division in question is not only always possible, but it is also very easy to perform. For in the case we are considering the equation being

x³ + px + q = 0,

if a is one of the roots we shall have

a³ + pa + q = 0,

which subtracted from the preceding will give

x³ − a³ + p(x − a) = 0,

a quantity divisible by x− a and having as its resulting quotient

x² + ax + a² + p = 0;

so that the new equation which is to be resolved for finding the two other roots will be

x² + ax + a² + p = 0,

from which we have at once
x = − ₂ ±
a

.−p −

3a² ₄ .

I see by the Algebra of Clairaut, printed in 1746, and by D’Alembert’s article on the Irreducible Case in the first Ency- clopædia that the idea referred to prevailed even in that period. But it would be the height of injustice to algebra to accuse it of not yielding results which were possessed of all the generality of which the question was susceptible. The sole requisite is to be able to read the peculiar hand-writing of algebra, and we shall then be able to see in it everything which by its nature it can be made to contain. In the case which we are considering it was forgotten that every cube root may have three values, as every square root has two. For the extraction of the cube root of a for

example is merely equivalent to the resolution of the equation of the third degree x³ − a = 0. Making x = y ^√3 a, this last equation passes into the simpler form y³ − 1 = 0, which has the root y = 1.

Then dividing by y − 1 we have