Real Objects
Spatial Geometry: Volume of various Bodies
Plane Geometry: Area of Triangle, Quadrangle, Circle (256 divided by 81 as approximation for pi)
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)
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
Inscribed angle theorem and lune of Hippocrates
Exhaustion Method
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
Geometric mean Theorem and Leg rule
Axiomatization of Geometry
Angle sum in the triangle and in regular polygons
The Central Angle Theorem
Cosine theorem
Arbitrarily accurate Approximation of pi
Calculation of Areas and Volumes (Variation of Cavalieri's Principle)
The Apollonic Problem
Definition of Ellipse, Circle, Parbola, and Hyperbola by Conic Sections
Circle of Apollonios
The Heron Method
Heron's formula
Formula for the Surface Area of Pyramid, Cylinder, Sphere
Formula for the Volume of Pyramid, Cylinder, Sphere, Prism
Chord Function (Forerunner of the Sine)
Spherical Geometry
Theorem of Menelaus
Theorem of Ptolemy about Chordal Quadrangles
"Proof" of the parallel postulate
Triangulation and Chord Tables (Forerunner of Trigonometry)
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)
Proclus' alternative to the parallel postulate
Approximation of pi to 3.1416
A Basis for the Sine Function
Formula of Brahmagupta for calculating the Area of Chord Quadrangles
Theorem of Brahmagupta about Chord Quadrangles
pi as a Number
Volume of Sphere and Cylinder as a Number, not as a Ratio
Law of sines with Proof
Introduction of the Sine
Identity tangens = quotient of sine and cosine without the terms cosine and tangens
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
Omar Khayyam's "proof" of the parallel postulate
Summary and Translation of the Geometry of the Arabs
al-Tusi's "proof" of the parallel postulate
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
Gapless coverage of an area by polygons
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
John Wallis's "proof" of the Euclidean postulate
Theorem of Ceva
Exponential, Sine and Cosine Series
Saccheri's attempt to prove Euclid's fifth postulate
Stewart’s theorem
Founder of many symbols: pi for the Circle Constant, standard notation for the Triangle, i for the imaginary unit
Eulerian Polyhedron Theorem
Foundation of Descriptive Geometry
Simplification and Modernization of the Elements of Euclid
Construction Criterium for regular n-edged Polygons
First Ideas on Non-Euclidean Geometry
Gaussian Number Plane
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
Absolute Geometry
Work on Non-Euclidean Geometries
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
Theorem of Helly