Professor Norman Swartz
Simon Fraser University



This version is given by Sir Thomas Heath (1861-1940) in The Elements of Euclid. (1908)


  1. Things which are equal to the same thing are also equal to one another.

  2. If equals be added to equals, the wholes are equal.

  3. If equals be subtracted from equals, the remainders are equal.

  4. Things which coincide with one another are equal to one another.

  5. The whole is greater than the part.


Let the following be postulated:

  1. To draw a straight line from any point to any point.

  2. To produce a finite straight line continuously in a straight line.

  3. To describe a circle with any center and distance.

  4. That all right angles are equal to one another.

  5. That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which the angles are less that two right angles.*

* In 1795, John Playfair (1748-1819) offered an alternative version of the Fifth Postulate. This alternative version gives rise to the identical geometry as Euclid's. It is Playfair's version of the Fifth Postulate that often appears in discussions of Euclidean Geometry:

5'. Through a given point P not on a line L, there is one and only one line in the plane of P and L which does not meet L.

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