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


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Visual expression of Euclid’s postulates: Image by Albert1Is

Euclid’s First Postulate: a Line Segment between Points

Euclid’s first postulate states that any two points can be joined by a straight line segment. It does not say that there is only one such line; it merely says that a straight line can be drawn between any two points.

Euclid’s Second Postulate: Extend a Straight Line

Euclid’s second postulate allows that line segment to be extended farther in that same direction, so that it can reach any required distance. This could result in an infinitely long line.

Euclid’s Third Postulate is Central to Circles

The third postulate starts with an arbitrary line segment, and an arbitrary point, which is not necessarily on the line segment. First, use the compass to note the end points of the line segment, then, put the sharp spike of the compass on the arbitrary point, and finally, draw the circle with the same radius as the line segment.

Euclid’s Fourth Postulate: All Right Angles are Equal

Euclid was probably thinking of right angles as made by constructing one line perpendicular to another. Any two such right angles are “equal” to one another.

Euclid’s fourth postulate states that, “if x and y are both right angles, then x=y.”

This may be more profound if the angles are oriented differently: opening to the left or right, up or down, or towards some other direction.

Euclid did not measure angles in degrees or radians, and he did not use a protractor. Instead, he usually discusses “how many angles in a diagram add up to some number of right angles.” For example, in Book I, Proposition 13 basically states that when a straight line “stands on” another straight base line, the sum of the two angles on the base line adds up to two right angles.

Euclid’s Fifth Postulate: the Parallel Postulate

A Parallelogram for Parallel Lines image by Mike DeHaan

A parallelogram demonstrates parallel lines: Image by Mike DeHaan

Euclid’s fifth postulate is the longest, and is now called the “parallel postulate.”

The fifth postulate has been the subject of much debate and labour over the centuries -can it be proven from the other postulates and axioms? Eventually mathematicians realized that the fifth posulate defines plane geometry, the geometry for a flat surface, and it cannot be derived from the other Euclidean axioms.

This postulate’s explanation needs diagrams.

Consider this parallogram, with the interior angles on the base marked in green. The sum of those two interior angles is 180 degrees. The two blue slanted lines are parallel; they will never meet even if extended infinitely far.

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