Introduction To Euclid's Geometry
The word 'geometry' originates from the Greek words 'geo' (earth) and 'metrein' (to measure). It arose from the practical need to measure land in ancient civilizations.
Euclid was a Greek mathematician who collected and systematically arranged all known geometric work into his famous treatise, 'Elements'. This work is divided into thirteen chapters, each called a book.
To avoid an endless chain of definitions, mathematicians accept certain geometric terms like 'point', 'line', and 'plane' as undefined. We can represent them intuitively but do not formally define them.
Euclid provided 23 definitions. Key examples include: a point is that which has no part, a line is a breadthless length, and a surface has length and breadth only.
Axioms (or common notions) are assumptions accepted as obvious universal truths without proof. They are used throughout mathematics, not just in geometry.
Things which are equal to the same thing are equal to one another. For example, if the area of triangle A equals the area of rectangle B, and the area of B equals the area of square C, then the area of A equals the area of C.
The whole is greater than the part. For example, if a quantity B is a part of another quantity A, it means .
Postulates are assumptions that are specific to geometry. Unlike axioms, which are general common notions, postulates are geometric truths accepted without proof.
A straight line may be drawn from any one point to any other point. This ensures the existence of a line between any two points.
Given two distinct points, there is a unique line that passes through them. This is an important assumption used frequently by Euclid.
A terminated line (what we call a line segment today) can be produced indefinitely. This means a line segment can be extended to form a line.
A circle can be drawn with any centre and any radius. This allows for the construction of circles of any size anywhere in a plane.
All right angles are equal to one another. This establishes a standard measure for angles, making a universal constant.
If a line falling on two straight lines makes the interior angles on the same side sum to less than two right angles (), then the two lines, if produced indefinitely, will meet on that side.
Theorems are statements that are proven using definitions, axioms, postulates, and previously proven statements through deductive reasoning.
Two distinct lines cannot have more than one point in common. This is proven by contradicting the axiom that a unique line passes through two distinct points.
If a point B lies on a line between points A and C, then the lengths add up: . This is justified by Axiom 4, which states that things which coincide are equal.