The Nature of Curved Lines

The ancients rightly observed that:

• among geometrical problems, some are planar, others solid, and others linear
• some can be constructed using only straight lines and circles
• others require, at a minimum, the use of conic sections
• others require the use of more complex curves

But I am surprised that they:

• did not distinguish different degrees among these more complex curves.
• chose to call them mechanical rather than geometrical.

If some machine is needed to draw them, then we should also reject circles and straight linessince we only draw them on paper using a compass and a ruler for machines.

Nor is it because the instruments used to trace the more complex curves, being more elaborate than the ruler and compass, are less precise; for by that reasoning they should be excluded from mechanics — where precision in craftsmanship is desired — rather than from geometry, which only seeks precision in reasoning, and which can certainly be as perfect concerning these curves as it is for the simpler ones.

The ancients were content with assuming they could draw:

• a straight line between 2 points
• a circle from a center passing through a given point

This is because they had no scruple in supposing, for the treatment of conic sections, that one could cut any given cone with any given plane.

Descartes’ Relativity

In order to trace all the curves that I introduce here, we only need to know that:

• 2 or more lines may be moved relative to each other
• their intersections define other lines

The amcients did not entirely accept conic sections in their geometry either.

I do not want to change the names already used.

• Geometrical means that which is precise and exact
• Mechanical means that which is not

Geometry is a science of the measurements of all bodies.

We should include both the more complex curves and the simpler ones so long as they are drawn:

• by continuous motion, or
• by several motions that follow one another, with the last are entirely governed by the previous ones

In this way one can always have an exact understanding of their measurements.

The curves more complex than the conic sections are the spiral, the quadratrix, and similar ones.

The ancient geometers rejected these because they belonged only to the mechanical arts.

I think they should be accepted because they are drawn by 2 separate motions, which have no measurable relationship between them.

• These belong to the class of curves that I want to accept.

The ancients later examined curves like the conchoid, the cissoid, and a few others because:

• they did not sufficiently study their properties
• they did not regard them as any more valuable than the earlier ones
• they knew very little about conic sections as the straightedge and compass remained unknown to them