5.2. Euclid’s Definitions, Axioms and Postulates

The Greek mathematicians of Euclid’s time thought of geometry as an abstract model of the world in which they lived. The notions of point, line, plane (or surface) and so on were derived from what was seen around them. From studies of the space and solids in the space around them, an abstract geometrical notion of a solid object was developed. A solid has shape, size, position, and can be moved from one place to another. Its boundaries are called surfaces. They separate one part of the space from another, and are said to have no thickness. The boundaries of the surfaces are curves or straight lines. These lines end in points. Consider the three steps from solids to points (solids-surfaces-lines-points).

This chunk discusses how ancient Greek mathematicians conceptualized geometry. They viewed it as a model representing their surroundings. In this view, solids are considered the most complex objects, possessing three dimensions: shape, size, and position. Surfaces are two-dimensional and separate spaces but have no thickness, while lines are one-dimensional. Finally, points are zero-dimensional and mark precise locations. Each step from solids to points represents a reduction in dimensions, helping one understand geometric relationships.

Think of a fruit like an apple. The apple itself represents a solid object (three dimensions). Its skin represents a surface (two dimensions). The edge of a slice of the apple constitutes a line (one dimension), while the spot where you place your finger is akin to a point (zero dimensions). This analogy helps visualize how these geometric concepts build upon one another.

Euclid summarised these statements as definitions. He began his exposition by listing 23 definitions in Book 1 of the ‘Elements’. A few of them are given below :

1. A point is that which has no part.
2. A line is breadthless length.
3. The ends of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
5. A surface is that which has length and breadth only.
6. The edges of a surface are lines.
7. A plane surface is a surface which lies evenly with the straight lines on itself.

In this chunk, we examine Euclid's definitions that form the basis of his geometric framework. For example, a point has no dimensions, a line is length without breadth, and a surface has both length and breadth but no thickness. Euclid's definitions are foundational because they define essential geometric terms, which help in constructing propositions. However, some definitions use terms that are not defined themselves, which can create a continuous loop of needing definitions. Therefore, certain terms like point, line, and surface are treated as undefined in modern mathematics.

Imagine trying to define a zero at a point on a number line. You could describe it as an absence of value, but then what is value? Similarly, a line, as defined by Euclid, is a path between points but doesn’t itself 'exist' in a physical sense. This resonates with how we often use abstract concepts in daily life, like time or distance, which can be represented but not physically touched.

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. 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.

This section delineates between axioms and postulates, terms that are often confused. Axioms are general mathematical statements considered universally true, while postulates are specific assumptions about geometric relationships. For instance, an axiom might state that equal quantities remain equal when added together. Postulates, like those related to drawing lines and circles in geometry, serve as starting points from which further geometric truths can be derived. Both are essential as they provide the foundational truths upon which Euclidean geometry is built.

Consider the rule that 'two parallel lines never meet.’ This is a postulate in geometry, shaping how we understand various geometric shapes and concepts. In contrast, 'if you add two even numbers, the result is even' acts like an axiom since it applies across different mathematical disciplines. In life, rules we take for granted, like traffic laws, serve the same purpose—they form a framework from which safe practices (like driving) can evolve.

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.

In this chunk, we outline Euclid's five fundamental postulates that lay the groundwork for his geometry. Each postulate expresses a simple truth about geometric relationships. For example, Postulate 1 translates into the idea that between any two points, there exists a straight line. Postulate 5 discusses the conditions under which two lines intersect based on angle measures—this postulate is notably more complex, guiding later developments in geometry.

Imagine trying to draw a map. Postulate 1 suggests that you can draw a road (or line) directly connecting two locations (points). Postulate 2 allows you to extend that road indefinitely. Postulate 3 is akin to saying you can create parks (circles) of various sizes centered on given spots. While many paths lead to intersections in real life, Postulate 5 ensures these paths (lines) relate under specific geometric rules, creating a framework for navigation.

After Euclid stated his postulates and axioms, he used them to prove other results. Then using these results, he proved some more results by applying deductive reasoning. The 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.

This chunk explains the process by which Euclid constructed a framework of mathematical logic using his definitions, postulates, and axioms. By assuming these foundational truths, he deduced new statements called propositions or theorems via logical reasoning. This cumulative method illustrates how advanced geometric truths can be systematically derived from basic principles, showcasing the power of deductive reasoning in mathematics.

Think of building a house. You need a solid foundation (axioms and postulates), then you build walls (theorems), and finally, you can decorate and furnish it (propositions) based upon the structure you’ve created. Each stage relies on the correctness and stability of the previous one, reflecting how mathematics constructs complex ideas from simple truths.

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