A Brief Guide to the Euclidean Postulates Found in Euclid’s Elements

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Latin version of Euclid’s Elements, by Campanus of Novara

Euclid’s Postulates are slightly more complicated than his axioms, but serve a similar purpose.

These axioms and postulates, intended to be self-evident, are sufficient to prove the many propositions he made in his Elements.

Postulates, like axioms, are statements made without proofs.

The postulates might be considered slightly “weaker” than axioms: if some results derived from the axioms and postulates are inconsistent, mathematicians would sooner change postulates than axioms.

It happens that Euclid oriented his postulates toward geometry, where the axioms are more general in nature.

Euclid’s Five Postulates

According to Norman Swartz’s quotations from Sir Thomas Heath’s, The Elements of Euclid:

Let the following be postulated, or assumed to be true.

  1. One is able to draw a straight line (segment) from any point to any point.
  2. One is able to produce a finite straight line (segment) continuously in a straight line.
  3. One is able to describe a circle with any centre and distance.
  4. All right angles are equal to one another.
  5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the straight lines, if produced indefinitely, will meet on that side on which the angles are less that two right angles.

What Were Euclid’s Tools?

Compass Times Two image by Mike DeHaan

Use a compass to construct geometric figures: Image by Mike DeHaan

The first three postulates deal with constructing geometric figures. Euclid would have restricted himself to two specific tools: a compass and straight-edge, or a ruler with no markings on it.

This image shows two compasses. These tools allow you to draw circles, and also to copy a line segment’s length by putting the tips at each endpoint.

A straight-edge is simple a ruler, or “rule,” but without inch or centimetre markings. With a straight-edge, you can draw a straight line, but cannot measure anything. Euclid’s implicit tools are “paper and pencil,” since you need some method of marking a flat surface to construct, measure, and calculate geometrical figures.

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