Euclid was also famous for his postulates, propositions (statements) that are true without proof and deal with specific subject matter, such as the properties of geometric objects (for more information about postulates, see “Foundations of Mathematics”). Along with definitions, Euclid began his text Elements with five postulates. These postulates are as follows (some of which may seem obvious to us now, but in Euclid’s time they had yet to be formally recorded):

• It is possible to draw a straight line from any point to another point.
• It is possible to produce a finite straight line continuously in a straight line.
• It is possible to describe a circle with any center and radius.
• All right angles are equal to one another.
• Given any straight line and a point not on it, there “exists one and only one straight line which passes” through that point and never intersects the first line, no matter how far the lines are extended. Another way to say this is: One and only one line can be drawn through a point parallel to a given line. This is also called the parallel postulate.

Mathematicians first believed this last postulate could be derived from the first four, but they now consider it to be independent of the others. In fact, this postulate leads to Euclidean geometry, and eventually to many non-Euclidean geometries that are made possible by changing the assumption of this fifth postulate.

Like many early attempts at explaining mathematics, not all these postulates tell the entire geometric story. There were still a large number of gaps, many of which were gradually filled in over time.