5.2.3. Postulates

Starting with his definitions, Euclid assumed certain properties, which were not to be proved. These assumptions are actually ‘obvious universal truths’. He divided them into two types: axioms and postulates.

Euclid began his studies in geometry by proposing definitions of terms like point, line, and plane. From these definitions, he made assumptions about the properties of these geometric concepts. These assumptions are not proven, as they are considered self-evident or obvious truths. He divided these assumptions into two categories: axioms, which are universal truths applicable across all areas of mathematics, and postulates, which are specifically related to geometry.

Think of axioms and postulates like the rules of a game. In a game of basketball, the concept that 'you cannot travel' (axiom) is universally accepted in all basketball games. Meanwhile, the rule that 'you must bounce the ball while running' (postulate) is specific to basketball. Just like you need to accept these rules to play, mathematicians accept axioms and postulates to explore geometry.

He used the term ‘postulate’ for the assumptions that were specific to geometry. Common notions (often called axioms), on the other hand, were assumptions used throughout mathematics and not specifically linked to geometry. Some of Euclid’s axioms, not in his order, are given below: (1) Things which are equal to the same thing are equal to one another. (2) If equals are added to equals, the wholes are equal. (3) If equals are subtracted from equals, the remainders are equal.

Axioms in Euclid’s work serve as foundational statements that are accepted without proof because they are intuitively obvious. For example, if two angles are both equal to a third angle, then those two angles must be equal to each other. This logic applies to magnitudes in geometry, such as lengths and areas. Axioms form the basis for further reasoning and proofs in geometric propositions.

Imagine you have three identical boxes. If Box A is the same height as Box B, and Box B is the same height as Box C, then, logically, Box A must also be the same height as Box C. This straightforward reasoning is similar to how axioms operate in mathematics.

Now let us discuss Euclid’s five postulates. They are: Postulate 1: A straight line may be drawn from any one point to any other point. Postulate 2: A terminated line can be produced indefinitely. Postulate 3: A circle can be drawn with any centre and any radius. Postulate 4: All right angles are equal to one another. Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

Euclid's five postulates form the backbone of his geometric principles. Each postulate represents a basic assumption about the nature of space and shapes. For instance, the first postulate states that you can always draw a straight line between any two points, implying that lines are fundamental in geometry. The second postulate indicates that line segments can be extended indefinitely, reinforcing the idea of lines as infinite. The complexity of the fifth postulate highlights the intricate nature of geometry compared to the straightforward nature of the first four.

Picture a straight line drawn on a piece of paper. Postulate 1 signifies that if you place two dots anywhere on the paper, you can always draw a line connecting them, no matter how far apart they are. Postulate 2 allows you to envision extending that line beyond what you initially drew – it's like imagining the tracks of a train continuing into the horizon without end.

After Euclid stated his postulates and axioms, he used them to prove other results. These statements that were proved are called propositions or theorems. Euclid deduced 465 propositions in a logical chain using his axioms, postulates, definitions and theorems proved earlier in the chain.

Once Euclid established his axioms and postulates, he employed them to prove numerous other statements in geometry, known as theorems. Theorems require systematic reasoning based on the previously established definitions and axioms. By building a logical structure based on these foundational elements, Euclid was able to derive a comprehensive system of geometry that has influenced mathematics for centuries.

Consider learning how to build a house. Initially, you need a solid foundation (axioms and postulates). Then, you can structure the walls and roof (theorems) based on those strong foundational ideas. Just like a well-built house, a well-structured mathematical framework supports various complex ideas.

--