The Project Gutenberg eBook #36640: Lectures on Elementary Mathematics

x³ − 3r²x + r²c = 0,

an equation which leads to the irreducible case since c is al- ways necessarily less than 2r, and which, owing to the two un- determined quantities r and c, may be taken as the type of all equations of this class. For, if we compare it with the general equation

x³ + px + q = 0,

we shall have

r = .−^p and c = − ^3q
3


p
so that by trisecting the arc corresponding to the chord c in a circle of the radius r we shall obtain at once the value of a root x, which will be the chord of the third part of that arc. Now, from the nature of a circle the same chord c corresponds not only to the arc s but (calling the entire circumference u) also to the arcs

u − s, 2u + s, 3u − s, . . . .

Also the arcs

u + s, 2u − s, 3u + s, . . .

have the same chord, but taken negatively, for on completing a full circumference the chords become zero and then negative, and they do not become positive again until the completion of the second circumference, as you may readily see. Therefore,

the values of x are not only the chord of the arc ^s

3

but also the

chords of the arcs _{u s}
−


₃ ,

2u + s

₃ ,

and these chords will be the three roots of the equation proposed. If we were to take the succeeding arcs which have the same chord c we should be led simply to the same roots, for the arc 3u
−


s would give the chord of ^{3u − s} , that is, of u− ^s , which we have

3 _s 3

already seen is the same as that of ₃ , and so with the rest.

Since in the irreducible case the coefficient p is necessarily

negative, the value of the given chord c will be positive or neg- ative according as q is positive or negative. In the first case,

3q

we take for s the arc subtended by the positive chord c = − _p .

The second case is reducible to the first by making x negative, whereby the sign of the last term is changed; so that if again we

take for s an arc subtended by the positive chord have simply to change the sign of the three roots.

3q

, we shall

p

Although the preceding discussion may be deemed sufficient

to dispel all doubts concerning the nature of the roots of equa- tions of the third degree, we propose adding to it a few reflex- ions concerning the method by which the roots are found. The method which we have propounded in the foregoing and which is commonly called Cardan’s method, although it seems to me that we owe it to Hudde, has been frequently criticised, and will doubtless always be criticised, for giving the roots in the irreducible case in an imaginary form, solely because a suppo- sition is here made which is contradictory to the nature of the equation. For the very gist of the method consists in its suppos- ing the unknown quantity equal to two undetermined quantities y + z, in order to enable us afterwards to separate the resulting equation