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.

All of geometery can be reduced to a single kind of problem of finding the value of the roots of some equation.

It is easy to enumerate all the ways by which they can be found.

Solid problems can only be constructed by using some curve more complex than the circle.

This is seen in 2 constructions:

1. This has 2 points which determine 2 mean proportionals between 2 given lines
2. This has 2 points which divide a given arc into 3 equal parts

A circle’s curvature depends only on a simple ratio of all its parts to its center.

This can only be used to determine a single point between 2 extremes, such as in:

• finding a single mean proportional between 2 given straight lines, or
• dividing a given arc into two.

Whereas the curvature of conic sections always depends on 2 different factors.

• This can be used to determine 2 distinct points.

These are a degree more complex than solid problems:

• the invention of 4 mean proportionals
• the division of an angle into 5 equal parts

These problems cannot be constructed [without?] by any of the conic sections.

I shall 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.

I assert that this is the simplest 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 should be accepted in geometry.