Real Objects
Spatial Geometry: Volume of various Bodies
Plane Geometry: Area of Triangle, Quadrangle, Circle (\\(\\pi=\\frac{256}{81}\\))
Spatial Geometry: Surface Area of various Bodies
"Theorem of Pythagoras"
General rules for calculating Areas
General rules for calculating Volumes
Perimeter and Area of a Circle (\\(\\pi =3\\))
Area of an Isosceles Triangle (\\(A = \\frac{1}{2} \\cdot g \\cdot h\\))
Perimeter and Area of a Circle (\\(\\pi = \\frac{256}{81}\\))
Volume of a Cylinder (\\(\\pi = \\frac{256}{81}\\))
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)
Exhaustion Method
Volume of a Cone = \\(\\frac{1}{3}\\) of the Volume of the Cylinder
Volume of a Pyramid = \\(\\frac{1}{3}\\) of the Volume of the corresponding Prism
Axiomatization of Geometry
Geometric mean Theorem, Leg rule, Cosine Theorem
Arbitrarily accurate Approximation of \\(\\pi\\)
Calculation of Areas and Volumes (Varition of Cavalieri's Principle)
Definition of Ellipse, Circle, Parbola, and Hyperbola by Conic Sections
The Appolonic Problem
Heron's formula
Formula for the Surface Area of Pyramid, Cylinder, Sphere
Formula for the Volume of Pyramid, Cylinder, Sphere, Prism
The Heron Method
Spherical Geometry
Chord Function (Forerunner of the Sine)
Theorem of Menelaus
Triangulation and Chord Tables (Forerunner of Trigonometry)
Theorem of Ptolemy about Chordal Quadrangles
"Proof" of the parallel postulate
Theorem of Pappos
Area theorem of Pappos (Pythagoras for arbitrary Triangles using Parallelograms instead of Squares)
Formulas for the Surface Area and Volume of Bodies of Rotation
Proclus' alternative to the parallel postulate
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
Volume of Sphere and Cylinder as a Number, not as a Ratio
\\(\\pi\\) as a Number
Identity \\(\\tan = \\frac{\\sin}{\\cos}\\) without the terms \\(\\cos\\), \\(\\tan\\)
Introduction of the Sine
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
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\\)
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
Eulerian Polyhedron Theorem
Founder of many symbols: \\(\\pi\\) for the Circle Constant, standard notation for the Triangle, \\(i\\) for \\(\\sqrt{-1}\\)
Foundation of Descriptive Geometry
Simplification and Modernization of the Elements of Euclid
Construction Criterium for regular n-edged Polygons
Gaussian Number Plane
First Ideas on Non-Euclidean Geometry
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
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