The Construction of Problems That Are Solid or More Than Solid

All curved lines that can be described by some regular motion should be accepted in geometry.

But this does not mean that we can use just any line indiscriminately for the construction of each problem.

Rather, care must always be taken to choose the simplest one to solve it.

The simplest includes:

• the one that are the easiest to draw.
• those that make the demonstration of the proposed problem easier
• primarily those that belong to the simplest kind that can serve to determine the quantity being sought.

Example concerning the invention of several mean proportionals

For example, I do not believe that there is any easier way to find as many mean proportionals as one wishes, nor one whose demonstration is more evident, than by using the curved lines described by the instrument XYZ explained above.

In order to find 2 mean proportionals between YA and YE, one need only draw a circle whose diameter is YE.

Because this circle intersects the curve AD at point D, YD is one of the mean proportionals sought. The demonstration is evident to the eye by simply applying this instrument to the line YD.

For as YA (or YB, which is equal to it) is to YC, so YC is to YD, and YD to YE.

In the same way, to find 4 mean proportionals between YA and YG, or to find six between YA and YN, one need only draw the circle YFG, which, intersecting AF at point F, determines the straight line YF, which is one of those four proportionals; or YHN, which, intersecting AH at point H, determines YH, one of the six—and so on for the others.

But since the curved line AD is of the second kind, and since two mean proportionals can be found using conic sections, which are of the first kind—and also because four or six mean proportionals can be found using lines that are not as complex in kind as AF and AH—it would be an error in geometry to use the more complex ones. And it is equally an error, on the other hand, to waste effort trying to construct a problem using a simpler kind of line than its nature allows.

The Nature of Equations

To prevent the above-mentioned errors, we need to know the nature of equations.

These are sums composed of several terms

• some known and some unknown
• One side equals the other, or rather, where all considered together are equal to zero.

It is often best to consider them in this way.

How many roots there can be in each equation?

In every equation, the number of different roots—that is, values of the unknown quantity—can be as many as the degree (or dimension) of that unknown quantity.

For example, if we suppose:

x = 2, or x − 2 = 0;

x = 3, or x − 3 = 0;

If we multiply these 2 equations:

x − 2 = 0 and x − 3 = 0

we get:

x² − 5x + 6 = 0,

x² = 5x − 6,

This is an equation in which the value of x is both 2 and 3.

If x − 4 = 0 is multiplied by x² − 5x + 6 = 0 we get:

x³ − 9x² + 26x − 24 = 0,

This is another equation in which x, being raised to the third power, has 3 values: 2, 3, and 4.

Which roots are false?

But it often happens that some of these roots are false or less than nothing.

For instance, if we suppose that x also designates the lack of a quantity that is 5, we write:

x + 5 = 0,

which, when multiplied by:

x³ − 9x² + 26x − 24 = 0,

gives:

x⁴ − 4x³ − 19x² + 106x − 120 = 0,

This is an equation with 4 roots:

• true ones 2, 3, and 4
• a false one 5.