Geometry Simplified

The method of expressing the value of all the roots of cubic equations: and subsequently of all those that rise only to the biquadratic.

For the rest, this way of expressing the value of the roots by the ratio they have to the sides of certain cubes of which only the volume is known, is as simple as expressing them by the ratio they have to the chords of certain arcs, or portions of circles, whose triple is given.

So that all those cubic equations that cannot be expressed by Cardano’s rules can be expressed as much or more clearly by the method proposed here.

For if, for example, one thinks one knows the root of this equation, $x^3 = -px + q$, because one knows that it is composed of two lines, one of which is the side of a cube whose volume is $\frac{1}{2}q$ added to the side of a square whose volume is $\sqrt{\frac{q^2}{4} - \frac{p^3}{27}}$; and the other is the side of another cube whose volume is the difference between $\frac{1}{2}q$ and the side of that square whose volume is $\sqrt{\frac{q^2}{4} - \frac{p^3}{27}}$, which is all that one learns from Cardano’s rule.

There is no doubt that one knows the root of this one

$x^3 = -px + q$

as much or more distinctly by considering it inscribed in a circle whose radius is $\sqrt{\frac{p}{3}}$, and knowing that it is the chord of an arc whose triple has for its chord $\frac{3q}{p} \sqrt{\frac{3}{p}}$. Even these terms (Maire, p. 401) are much less cumbersome than the others, and they will be found much shorter if one wishes to use some particular symbol (AT VI, 475) to express these chords, just as one uses the symbol $\sqrt[3]{ }$ to express the side of cubes."

And we can also, following from this, express the roots of all equations up to the biquadratic (i.e., degree four) by the rules explained above. So much so that I do not know anything more to be desired in this matter. For indeed, the nature of these roots does not allow them to be expressed in simpler terms, nor to be determined by any construction that is at once more general and easier.

Why solid problems cannot be constructed without conic sections, nor those which are more complex without some other more complex curves.

It is true that I have not yet explained the reasons upon which I base myself in daring to assert whether a thing is possible or not. But if one considers how, by the method I use, everything that falls under the consideration of geometers is reduced to a single kind of problem—which is to find the value of the roots of some equation—one will understand that it is not difficult to make an enumeration of all the ways by which they can be found, sufficient to demonstrate that one has chosen the most general and the simplest.

And particularly regarding solid problems, which I have said cannot be constructed without using some curve more complex than the circle, this is something one can quite clearly see from the fact that they all reduce to two constructions: in one of them, it is necessary to have at the same time the two points which determine two mean proportionals between two given lines; and in the other, the two points which divide a given arc into three equal parts. For since the curvature of the circle depends only on a simple ratio of all its parts to the point that is its center, it can only be used to determine a single point between two extremes—as in finding a single mean proportional between two given straight lines, or dividing a given arc into two.

Whereas the curvature of conic sections, which always depends on two different factors, can be used to determine two distinct points.

But for the same reason, it is impossible that any of the problems that are of a degree more complex than the solid ones—such as those that presuppose the invention of four mean proportionals, or the division of an angle into five equal parts—can be constructed by any of the conic sections. That is why I believe I will do the best possible in this regard if I provide a general rule for constructing them, using the curve traced by the intersection of a parabola and a straight line, in the manner explained above. For I dare to assert that there is no simpler one in nature that can serve the same purpose. And you have seen how it follows immediately after the conic sections in that much-sought question of the ancients, whose solution teaches in order all the curved lines that ought to be accepted in geometry.