What is a point? Euclid said: "A point is that which has no part." What is a line? "A breadthless length." While modern mathematics considers point, line, and plane as undefined primitive terms, Euclid built an entire universe upon them.
- Things which are equal to the same thing are equal to one another ($A = C \text{ and } B = C \implies A = B$).
- If equals are added to equals, the wholes are equal ($A = B \implies A + C = B + C$).
- If equals are subtracted from equals, the remainders are equal ($A = B \implies A - C = B - C$).
- Things which coincide with one another are equal to one another (Principle of Superposition).
- The whole is greater than the part ($A > B$ if $B$ is a proper part of $A$).
- Things which are double of the same things are equal to one another.
- Things which are halves of the same things are equal to one another.
Problem: If $AC = BD$, prove that $AB = CD$ when $A, B, C, D$ lie on a straight line in order.
Given: $AC = BD$.
From line: $AC = AB + BC$ and $BD = BC + CD$.
Substitute: $AB + BC = BC + CD$.
Subtract $BC$ from both sides (using Euclid's Axiom 3: equals subtracted from equals leave equals): $AB = CD$. Hence Proved.
Students often use the terms interchangeably.
Distinction: Axioms are universal assumptions applied throughout all branches of mathematics (algebra, arithmetic, calculus); Postulates are assumptions specific strictly to the study of geometry.
The American Declaration of Independence famously declares: "We hold these truths to be self-evident..." — Thomas Jefferson deliberately modeled the founding document of the United States on Euclid's axiomatic structure!