Euclid’s Definitions

Around 300 BCE, the Greek mathematician Euclid wrote a monumental work called the Elements, which organized all of Greek geometry into a logical system built from the ground up. Before proving any theorem, Euclid recognized that he needed a starting point — a collection of precise definitions that give meaning to the most basic geometric objects. These definitions describe what a point, a line, a surface, and other fundamental figures actually are, providing the shared vocabulary that every geometric argument relies on. Euclid's definitions are notable for their economy and clarity. He defined a point as 'that which has no part,' meaning a point has position but no length, width, or depth. A line is 'breadthless length,' existing in one dimension only. A straight line is 'a line which lies evenly with the points on itself.' A surface has length and breadth but no depth. These characterizations feel almost philosophical, and indeed ancient critics noted that some of them are difficult to use directly in proofs. Nevertheless, they anchored geometry in precise language and set the standard for mathematical definition-making for over two thousand years. Understanding Euclid's definitions matters for modern students because they reveal the deductive structure of mathematics: every proof must trace back to agreed-upon starting points. The definitions work hand-in-hand with Euclid's five postulates and five common notions to form the complete foundation of Euclidean geometry. Together, these elements allow the entire body of classical geometry — from the Pythagorean theorem to properties of circles — to be derived through pure logical reasoning without any further appeal to physical intuition.

Key takeaways
  • Definition 1 — Point
  • Definition 2 — Line
  • Definition 3 — Straight Line
  • Definition 4 — Surface
  • Definitions 15–18 — Circle and Related Terms
  • Definitions 20–22 — Triangles

Definition 1 — Point

A point is that which has no part. Formally, a point is a location in space with zero dimensions: no length, no width, no depth. We label points with capital letters such as A, B, or C. Points cannot be broken into smaller pieces; they are indivisible. In coordinate geometry this corresponds to an ordered pair (x, y) in the plane or (x, y, z) in space, but the point itself occupies no area or volume.

Definition 2 — Line

A line is breadthless length. It extends in exactly one dimension — along its length — but has no width or thickness. A line continues infinitely in both directions. When we draw a line on paper the drawn mark has some width, but the geometric line it represents has none. Two distinct points determine exactly one line passing through them, a principle Euclid stated in Postulate 1.

Definition 3 — Straight Line

A straight line is a line that lies evenly with the points on itself. This means there is no bending or curving at any point along it. Every portion of a straight line looks the same as every other portion. In modern terms, a straight line is the shortest path between any two of its points, and it can be expressed algebraically as ax + by = c for constants a, b, c (not both zero).

Definition 4 — Surface

A surface is that which has length and breadth only. It is two-dimensional: it has area but no thickness or depth. A flat surface (plane) extends infinitely in two directions. Like a line, the drawn representation of a plane has some thickness, but the geometric object does not. Three non-collinear points (points not all on the same line) determine exactly one plane.

A circle is a plane figure contained by one line (the circumference) such that all straight lines from a certain point inside the figure (the centre) to the circumference are equal to one another. That equal distance is the radius r. The diameter is any straight line drawn through the centre and terminated in both directions by the circumference, equal in length to 2r. These definitions underpin every theorem about circles in Books III and IV of the Elements.

Definitions 20–22 — Triangles

Euclid defined triangles by their side lengths. An equilateral triangle has all three sides equal. An isosceles triangle has exactly two sides equal. A scalene triangle has all three sides unequal. He also classified triangles by angles: a right-angled triangle contains one right angle (90°), an obtuse-angled triangle contains one obtuse angle (greater than 90°), and an acute-angled triangle has all angles acute (each less than 90°). These distinctions appear throughout the Elements' proofs.

The Key Equation

C=2πrC = 2\pi r
  • C — Circumference (length (e.g., cm)): The total length of the curved line bounding the circle — Euclid's 'one line' that encloses the circular figure.
  • \pi — Pi (dimensionless): The constant ratio of a circle's circumference to its diameter, approximately 3.14159. Euclid did not use this symbol but his work implicitly relied on this ratio.
  • r — Radius (length (e.g., cm)): The equal distance from the centre to any point on the circumference — the defining measurement in Euclid's Definition 15.

Worked Example

Problem. Using Euclid's definition of a circle (all radii are equal), prove that if two circles have the same radius r = 5 cm and their centres are 0 cm apart (they share the same centre), then the circles are identical.

r1=r2 and same centre  ⟹  C1=C2r_1 = r_2 \text{ and same centre} \implies C_1 = C_2

Answer. Two circles sharing the same centre and the same radius are identical — they occupy exactly the same set of points.

Figure
Diagram
Euclid's Basic Definitions IllustratedABLine segment ABLine (infinite)Ord = 2rCircle: centre O, radius rCBATriangle (scalene)• Point: no part (dimensionless)• Line: breadthless length (1D)• Circle: all radii equal• Triangle: three-sided polygon
Origins · 1 minEuclid of Alexandria and how this was discovered
EO
Euclid of Alexandria
-325–-265 · Greek (active in Alexandria, Egypt)

Euclid wrote the Elements (c. 300 BCE), a thirteen-book treatise that compiled and systematized Greek mathematics. It begins with 23 definitions, 5 postulates, and 5 common notions, then derives hundreds of propositions purely by logical deduction. The Elements remained the primary textbook for teaching geometry for more than 2,000 years and is the second most printed book in history after the Bible. Almost nothing is known about Euclid's personal life — we do not know his birthplace, his parents, or the exact dates of his birth and death. What survives is entirely his mathematics.

  • Euclid's Postulates — The five postulates are the second foundational layer of the Elements, building directly on these definitions to state the basic rules of geometric construction.
  • Deductive Reasoning in Geometry — Euclid's definitions are the starting vocabulary for deductive proofs; understanding how definitions anchor logical arguments is central to geometric reasoning.
  • Types of Triangles — Euclid's Definitions 20–22 classify triangles by side length and angle; these classifications are used throughout all triangle theorems and proofs.
  • Properties of Circles — Euclid's Definition 15 (the circle) is the foundation for all theorems about arcs, chords, tangents, and angles in circles studied in later units.
  • Undefined Terms in Geometry — Modern geometry recognizes that some terms (point, line, plane) cannot be defined without circular reasoning — a direct refinement of the challenge Euclid's definitions present.