Geometric Figures: Construction and Properties
Often Euclid explains how to construct geometric figures. In other cases, Euclid proves that a given construction has specific properties.
Euclid’s Axioms are intended to be simple truths that his fellow philosophers would find impossible to refute. These could apply to almost any branch of mathematics.
Euclid’s Postulates are slightly more complicated than his axioms, but serve a similar purpose. They seem more specific to geometry than the axioms.
Along with his definitions, these axioms and postulates are sufficient to prove the many propositions he made in his Elements.
The Elements confines itself to the ten axioms and postulates. It splurges on 131 definitions, but then goes on to prove 465 propositions. These propositions include both geometric constructions and proofs.
Euclid would construct his geometric figures with a straightedge, which is an unmarked ruler, plus a compass.
A simple construction is to bisect a line segment, cutting it into two equal halves.
A “simple” proof, found in the fifth proposition, demonstrates that in any isosceles triangle the angle between the base and one equal side is equal to the base’s angle with the other equal side.
In book 13, Euclid demonstrates how to create a regular tetrahedron inside an arbitrarily large sphere. In the same proposition, he proves that the square that is based on the diameter of that sphere covers an area 1.5 times larger than the square of the length of a side of that tetrahedron.
The Importance of Axioms and Postulates
A modern view of the importance of starting with axioms and postulates, is that they provide the foundations for later conclusions or theorems. Once an axiom is accepted, then only a misuse of the rules of logic can invalidate conclusions drawn from that axiom.
It is possible that Euclid intended to improve upon the Socratic method, by stating his premise and assumptions at the beginning. Socrates tended to ask questions until his co-philosopher realized his own erroneous prejudices had led him into a contradiction.
The Importance of Euclid’s Elements
First, Euclid’s Elements formed a reference compendium for mathematics and geometry; it was an authoritative source for centuries.
Secondly, the process of setting out assumptions, and then working through the logic, is very consistent with the disciplines of logic, philosophy and rhetoric.
The methods used by Euclid in stating definitions and axioms, deriving formal proofs of theorems, and then building more complex theorems from those already demonstrated, continue to guide mathematicians to this day.
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.
Decoding Science. One article at a time.