Elements of Geometry: A Brief Guide to the Euclidean Axioms

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Ancient copy of Euclid’s Elements, currently housed at the University of Pennsylvania. Image courtesy of Penn University

Euclid’s Elements is a multi-volume compendium of geometry and math theorems derived from only five axioms and five postulates.

What are these axioms, and why are the Euclidean axioms important?

Euclid’s Five Axioms

The five Euclidean axioms are terse and plausible statements.

Sir Thomas Heath’s English translation of “The Elements of Euclid” gives the axioms as follows:

  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.”

Euclid’s First Axiom: Equality

Euclid’s first axiom asserts that equality is transitive. In algebraic logic, one would say “if x=y, and if y=z, then x=z.”

Euclid’s Second and Third Axioms: Addition or Subtraction of Equals

One reason why Euclid had to separate his axioms regarding addition and subtraction, is that Greek mathematics did not include negative numbers.

An algebraic version of Euclid’s second axiom would read “if x=y, and if a=b, then x + a = y + b.”

His third axiom would then be “if x=y, and if a=b, then x – a = y – b.”

Euclid’s Fourth Axiom: Coincidental Equality

The fourth axiom seems to be the most obvious reference to geometry. If two shapes “coincide,” then one fills out the exact shape and volume of the second.

Simple cases include angles that are equal, straight line segments of the same length, and triangles of the same size and shape.

Consider drawing a triangle, and then constructing a second triangle in a way that copies the angles and lengths from the first triangle. Then, cut out the second triangle and lay it over the first. If these triangles precisely overlap, then they “coincide,” and are equal to one another.

Euclid’s Fifth Axiom: Part of the Whole

Euclid’s fifth axiom states that “x + a > x.” To a modern mathematician, this would not be true if ‘a’ had the value “zero,” or if ‘a’ were a negative number. For example, if ‘a’ were a geometric shape with no area, such as a line that has no thickness, then adding a line segment “beside” the edge of a square, ‘x’, would not increase the area of the square.

A more complete formula to cover our modern sensibilities would be “if a > zero, then x+a > x.”

Students today still follow Euclid’s Axioms, although our tools now include measurements. Image by nkzs

The Importance of the Euclidean Axioms and Postulates

Euclid of Alexandria is credited with writing the Elements sometime around 300 BC. This was perhaps the first mathematical treatise to develop a vast body of math starting from simple definitions, axioms and postulates.

In Euclid’s Elements, we start with 23 definitions, then the postulates and axioms noted above. Euclid then provides multiple proofs and propositions.

Some of his propositions explain how to construct geometric figures, such as how to bisect a line segment, and in other cases, Euclid proves that a given construction has specific properties.

(For example, if a triangle has two angles that are equal to each other, then the sides opposite those angles are of the same length.) Although this early collection of geometry laws and information was created by Euclid thousands of years ago, Elements is still used as a foundation for today’s mathematicians.

References:

Douglass, C. Euclid. (2007). Math Open Reference. Accessed June 24, 2012.

Swartz, N. Axioms and Postulates of Euclid. Simon Fraser University. Accessed June 24, 2012.

Fitzpatrick, R. Euclid’s Elements of Geometry. (1885). University of Texas. Accessed July 4, 2012.

Weisstein, E. W. Euclid’s Postulates. (2012). MathWorld–A Wolfram Web Resource. Accessed June 24, 2012.

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