Analytical geometry

Not even conical sections can satisfy all the needs of science and technology. It is therefore necessary to go beyond conical sections, and to consider even more complex curves.

A very effective way to represent these curves on the plane consists in using their equation.

If you trace two perpendicular lines on the plane, it is possible to individuate every point through its Cartesian coordinates, that more or less represent the distance of the point from the two axes. They are called thus in honour of R. Descartes (Cartesius) who first used them systematically to describe curves.

The two coordinates are usually indicated as x and y. To each value of the couple (x,y) corresponds a point on the plane. When x and y vary in every possible way, the corresponding point describes the whole plane. If, instead, the coordinates are subject to an equation, the point that they represent is forced to move along a curve, of which the equation constitutes the analytical representation.

For example, if one fixes x through the equation x=1, the corresponding curve is a vertical straight line; while the equation y=3 represents a horizontal line. More generally, a first-degree equation (that is, an equation in which the variables x and y appear at the power of one) represents a straight line, while a second-degree equation creates one of the conical sections (including the circumference).

One can study curves with a third-degree equation, or fourth-degree, or of a higher and higher degree. For these, we find in the Encyclopedie di Diderot e D'Alembert an universal machine that allows, adding layer upon layer, to trace curves of higher and higher degrees. The one that has been realised has three layers, and therefore traces third-degree curves. It is a rather heavy machine, whose complexity is mainly due to the need to reduce to the minimum the friction that would otherwise prevent it from functioning.