On Euclid’s Definitions and Postulates

Posted by Brian PCF
Oct 20 2013

The structure of Euclid’s definitions and postulates assumes a movement from simple to complex. From a beginning of the point, “that which has no part” we finish the postulates with a vast (possibly infinite in our modern understanding of the word) ability to create and explore points, lines, plane surfaces, and perhaps an approximation of our entire physical universe.

The definitions seem to build on each other in a seemingly logical form. We start with the point and line, and then expand on what a point and a line can bring about: straight lines, surfaces, plane angles, figures, circles, parallel lines. Literally, we can create from this ad infinitum. A basic proposition of Euclid seems to be that we can represent anything with this system of points and lines.

Interestingly though, we define these basic structures in the first two definitions in our Euclidean universe not by their properties, but by that which they are not: a point is “that which has no part”, a line “breadthless length”. We create representations of our physical universe from these definitions, but they can’t exist in our physical universe. Something that has no part, or no breadth or no depth can’t exist in reality as we understand it based on our senses.

Moreover, everything that comes after these basic definitions is relative to one another. There is no existence beyond the point and line in and of itself, simply (or perhaps more complicated, as we’ll see) is the idea that there are fundamental properties based on these relative relationships.

The first relative definition is the straight line, “a line which lies evenly with the points on itself.” Let’s assume that this means that “evenly” here means an angle of zero as we would define it. Euclid requires a much more flexible definition here as he doesn’t want to define a line with an angle zero. It seems from his Definition 8, “a plane angle is the inclination to one another of two lines in a plane which meet one another and do not line in a straight line” that he wants to have the ability to put as many points or lines on a single straight line as he needs. So a seemingly simple idea, but with an infinite amount of complexity that we can build in as needed.

We continue to see complex and relative figures emerge. Even something seemingly simple like a boundary, “that which is an extremity of anything”, or that of a figure, “that which is contained by any boundary or boundaries” depends upon some other thing for its existence and is hence relative.

Right angles are a fascinating example of this idea of relativity in that they are not only created by another right angle (definition 10), but are all exactly the same no matter what (postulate 4). So a right angle can’t come into being as a singular right angle, it must be born as a twin.

Circles are the next figure that require a relative relationship. They cannot exist in and of themselves, but require a center. The center also requires a circle, otherwise it is just a point. This and the right angle are all strikingly similar to “the swerve” of Lucretius: atoms colliding with one another to create the entire physical universe. While Euclid is contriving his universe, that idea of collision between basic atoms or ideas is still fundamental to the Elements.

Parallel lines are very similar to the idea of the right angle. They need each other to exist and they are fundamentally identical in that they are “breadthless length” and they share a relative angle.

So while nothing can exist in Euclid’s universe without points and lines, those can only exist by seemingly simple definitions that speak more to what these things are not versus what they are. Moreover, everything built from these points and lines is defined by their relative relationship with one another. There are no new things in Euclid, only these basic ideas of points and lines and how they interact. So the baseline is simple, but from this we can grow the whole of mathematics and approximate everything we see in the physical world.

[Using the Heath translation]

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