Plane Geometry: Area of Triangle, Quadrangle, Circle (256 divided by 81 as approximation for pi)​

Spatial Geometry: Volume of various Bodies

Spatial Geometry: Surface Area of various Bodies

General rules for calculating Areas

"Theorem of Pythagoras"

Perimeter and Area of a Circle (pi equals 3)

General rules for calculating Volumes

Area of an Isosceles Triangle (Area=Product of basis and height, divided by 2)

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Perimeter and Area of a Circle (256 divided by 81 as approximation for pi)

Volume of a Cylinder (256 divided by 81 as approximation for pi)

Trigonometry: Slope of a Pyramid

Theorem of Thales

Intercept Theorem

Proof of the Theorem of Pythagoras

Triangle: Congruence Theorems

Triangle: In-Circle and Circumcircle

General Theory of Proportion (Irrational Numbers, Ratios, Ratio Equations)

Volume of a Pyramid = one third of the Volume of the corresponding Prism

Volume of a Cone = ​one third of the Volume of the Cylinder

Exhaustion Method

Axiomatization of Geometry

Geometric mean Theorem and Leg rule

Arbitrarily accurate Approximation of pi

Calculation of Areas and Volumes (Variation of Cavalieri's Principle)

Definition of Ellipse, Circle, Parbola, and Hyperbola by Conic Sections

The Apollonic Problem

The Heron Method

Heron's formula

Formula for the Surface Area of Pyramid, Cylinder, Sphere

Formula for the Volume of Pyramid, Cylinder, Sphere, Prism

Spherical Geometry

Chord Function (Forerunner of the Sine)

Theorem of Menelaus

Triangulation and Chord Tables (Forerunner of Trigonometry)

Theorem of Ptolemy about Chordal Quadrangles

Formulas for the Surface Area and Volume of Bodies of Rotation

Theorem of Pappos

Area theorem of Pappos (Pythagoras for arbitrary Triangles using Parallelograms instead of Squares)

Approximation of pi to 3.1416

A Basis for the Sine Function

Theorem of Brahmagupta about Chord Quadrangles

Formula of Brahmagupta for calculating the Area of Chord Quadrangles

​pi as a Number

Volume of Sphere and Cylinder as a Number, not as a Ratio

Introduction of the Sine

 Identity tangens = quotient of sine and cosine without the terms cosine and tangens

Law of sines with Proof

First use of the Tangent

Introduction of six Trigonometric Functions

Proposal: Define Trigonometric Functions via Unit Circle

Sine Theorem and Proof for general spherical Triangles

Summary and Translation of the Geometry of the Arabs

Formalization of Plane and Spherical Trigonometry

Construction of Figures with Compass and Ruler; Discussion of Proportion and Perspective

Definition of Trigonometric Functions via Right-Angled Triangle

Extensive Tables for Trigonometric Functions

Closed Formula for the Circle Constant pi

Foundation of Analytical Geometry

Theorem of Desargues

Fundamentals of Projective Geometry

Theorem of Pascal

Principle of Cavalieri

Concrete Algebraic Equations and a common Symbolism for Analytical Geometry

Theorem of Ceva

Exponential, Sine and Cosine Series

Founder of many symbols: pi for the Circle Constant, standard notation for the Triangle, i for the imaginary unit

Eulerian Polyhedron Theorem

Simplification and Modernization of the Elements of Euclid

First Ideas on Non-Euclidean Geometry

Gaussian Number Plane

Construction Criterium for regular n-edged Polygons

Fundamental Ideas of Projective Geometry: the Invariance of the Double Ratio, Projective Relations, the Principle of Duality, Poncelet's Principle of Continuity

Non-Euclidean Hyperbolic Geometry

Work on Non-Euclidean Geometries

Absolute Geometry

Foundation of Descriptive Geometry

Riemann Integral

Classification of various Geometries

Poincaré's Circular Disk Model and Poincaré's Half-Plane Model (Hyperbolic Geometry)

Formal Axiomatic Structure of the Geometry

Real Objects

"Proof" of the parallel postulate

Proclus' alternative to the parallel postulate

Omar Khayyam's "proof" of the parallel postulate

al-Tusi's "proof" of the parallel postulate

John Wallis's "proof" of the Euclidean postulate

Saccheri's attempt to prove Euclid's fifth postulate

Circle of Apollonios

Stewart’s theorem

Inscribed angle theorem and lune of Hippocrates

Theorem of Helly

Angle sum in the triangle and in regular polygons

The Central Angle Theorem

Cosine theorem

Gapless coverage of an area by polygons