Modern Geometry: Origins and the Axiomatic Method (Vocabulary Flashcards)

Vocabulary flashcards covering key terms, figures, and concepts related to the origins of geometry and the development of the axiomatic method.

Geometry

Originating from geo (earth) + metron (to measure); the branch of mathematics dealing with shapes, sizes, and relative positions; origins tied to measurements of land and space.

Geo + metrien

Greek roots for geometry: geo (earth) and metrien (to measure) — the basis of the word geometry.

Euclid

Ancient Greek mathematician who compiled the Elements and helped formalize the axiomatic method in geometry.

The Elements

Euclid’s compilation of 13 books on geometry and number theory; a foundational text that is a compilation rather than original research.

Axiomatic Method

A deductive framework using undefined terms, axioms/postulates, and logical reasoning to derive theorems.

Undefined Terms

Foundational terms (e.g., point, line) treated as primitive and not defined; essential for building the axiomatic system.

Axiom / Postulate

A statement assumed to be true without proof within a theory (postulates are a type of axiom used in Euclidean geometry).

Postulate I (Two points determine a unique line)

Euclid’s first postulate: through any two distinct points there is exactly one line.

Postulate II (Segment extension)

For any segment, there exists a point extending it such that the extended segment is congruent to a given segment.

Postulate III (Circle with center and radius)

For any point O and any point A ≠ O, there exists a circle centered at O with radius OA.

Postulate IV (Right angles are congruent)

All right angles are congruent to each other.

Parallel Postulate

Euclid’s fifth postulate: through a point not on a given line, there exists exactly one line parallel to the given line; long-standing controversy and foundational to non-Euclidean geometry.

Playfair’s Postulate

A reformulation of Euclid’s Parallel Postulate: through a point not on a line, there is exactly one line parallel to the given line.

Thales of Miletus

Often called the Father of deductive geometry; first to prove geometric theorems logically and connect geometry with other mathematics.

Pythagoras

Founder of a mathematical school; known for systematic geometry and the mystical, purified approach to mathematics.

Hypocrates of Chios

Early figure who wrote the first systematic geometry treatise.

Babylonians

Ancient civilization known for early Pythagorean theorem relationships and advanced arithmetic; pre-Euclid geometry knowledge.

Rhind Papyrus

Egyptian manuscript with approximate pi value: π ≈ (16/9)² ≈ 3.1604.

π (pi)

Ratio of a circle's circumference to its diameter; ancient approximations and later proof as a transcendental number.

Transcendental

A type of number not a root of any nonzero polynomial equation with rational coefficients (π is transcendental, per Lindemann, 1882).