Euclid's Definitions, Axioms and Postulates
This page provides a foundational reference list derived from the work of the renowned ancient Greek mathematician, Euclid. Specifically, it presents the fundamental Definitions, Axioms, and Postulates as laid out in his monumental and enduring treatise, the Elements. Published around 300 BCE, 'Elements' is one of the most influential works in the history of mathematics, establishing a systematic and logical approach to geometry that served as the standard for over two thousand years. These foundational elements were Euclid's starting point, the bedrock upon which he constructed the entirety of his geometric system.
The initial part of this resource focuses on Euclid's Definitions. These were his attempt to describe the most basic, undefined, or intuitively understood objects and concepts that form the subject matter of geometry. He began with fundamental ideas like a point, which he defined somewhat circularly as "that which has no part," and a line, described as "breadthless length." Other crucial definitions included concepts such as a straight line, a surface, angles (plane, rectilinear), and geometric figures like the circle.
While these definitions might appear to rely on intuitive understanding rather than the stringent, purely axiomatic foundations preferred in modern mathematics, they were absolutely revolutionary for their time. They succeeded in establishing a vital common vocabulary and a conceptual framework that could be universally understood by readers and students. By clearly attempting to define his terms upfront, Euclid provided the essential linguistic and conceptual building blocks that would be manipulated and reasoned about in his subsequent propositions, theorems, axioms, and postulates.
Following the definitions, Euclid presented his Axioms and Postulates. Unlike definitions, which describe objects, axioms (also known as common notions) and postulates are statements assumed to be true without proof. Axioms are general truths applicable across mathematics (e.g., "Things which are equal to the same thing are also equal to one another"), while postulates are specific to geometry (e.g., the famous Parallel Postulate). These assumed truths, when combined with the defined terms, served as the permissible starting points for all subsequent logical deductions and proofs within Euclid's system. They represent the fundamental assumptions about space and quantity upon which Euclidean geometry is built.
Reviewing this collection of Euclid's initial definitions, axioms, and postulates offers profound insight into the foundational thinking of classical geometry and provides a unique perspective on the historical development of mathematical rigor. It showcases the ambitious attempt to build a complex system of knowledge from a minimum set of explicitly stated starting points. Understanding these foundational elements is key to appreciating how Euclidean geometry, which was the standard geometry taught in schools globally for centuries, was meticulously constructed from its most basic components. This resource serves as a direct and accessible reference to these historical cornerstones of mathematics.
Euclid's Definitions
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.
8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
9. And when the lines containing the angle are straight, the angle is called rectilinear.
10. When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
11. An obtuse angle is an angle greater than a right angle.
12. An acute angle is an angle less than a right angle.
13. A boundary is that which is an extremity of anything.
14. A figure is that which is contained by any boundary or boundaries.
15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.
16. And the point is called the center of the circle.
17. A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.
18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.
19. Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.
20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.
21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.
22. Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
Euclid's Axioms
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.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
6. Things which are double of the same things are equal to one another.
7. Things which are halves of the same things are equal to one another.
Euclid's Postulates
1 : A straight line may be drawn from any one point to any other point.
2 : A finite straight line may be produced continuously in a straight line.
3 : A circle may be described with any center and radius.
4 : All right angles are equal to one another.
5 : If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.