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The Elements of Euclid

with highlights

An interactive rendering of the renowned 2300 year-old textbook on Geometry
--- with all 482 figures

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Proposition 43

For any parallelogram, the complements of the parallelograms about the diagonal are equal to one another.

Let ABCD be a parallelogram, and AC its diagonal. And let EH and FG be the parallelograms about AC,and BK and KD the so-called complements (about AC). I say that the complement BK is equal to the complement KD.

For since ABCD is a parallelogram, and AC its diagonal, triangle ABC is equal to triangle ACD [Prop. 1.34]. Again, since EH is a parallelogram, and AK is its diagonal, triangle AEK is equal to triangle AHK [Prop. 1.34]. So, for the same (reasons), triangle KFC is also equal to (triangle) KGC. Therefore, since triangle AEK is equal to triangle AHK, and KFC to KGC, triangle AEK plus KGC is equal to triangle AHK plus KFC. And the whole triangle ABC is also equal to the whole (triangle) ADC. Thus, the remaining complement BK is equal to the remaining complement KD.

Thus, for any parallelogramic figure, the complements of the parallelograms about the diagonal are equal to one another. (Which is) the very thing it was required to show.

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At the ages old difficulties of communication

"As for ancient geometrical analysis and modern algebra, even apart from the fact that they deal only in highly abstract matters that seem to have no practical application, the former is so closely tied to the consideration of figures that it is unable to exercise the intellect without greatly tiring the imagination, while in the latter case one is so much a slave to certain rules and symbols that it has been turned into a confused and obscure art that bewilders the mind instead of being a form of knowledge that cultivates it."

— R. Descartes, A Discourse on the Method, 1637

"... geometry is nothing other than that part of universal mechanics which reduces the art of measuring to exact propositions and demonstrations. But geometry is commonly used in reference to magnitude, and mechanics in reference to motion. In this sense rational mechanics will be the science, expressed in exact propositions and demonstrations, of the motions that result from any forces whatever and of the forces that are required for any motions whatever."

— I. Newton, Preface to Principia, 1687

Full Table of Contents

Book 1 - Fundamentals of Plane Geometry Involving Straight-Lines

Start with Definitions, Postulates, Common Notions or First proposition.
Interesting proofs: Complements in parallelograms, Pythagorean theorem

Book 2 - Fundamentals of Geometric Algebra

Start with Definitions or First proposition.
Interesting construction: Golden section

Book 3 - Fundamentals of Plane Geometry Involving Circles

Start with Definitions or First proposition.
Interesting propositions: Angles at center and circumference, Relations of lines from outside a circle

Book 4 - Construction of Rectilinear Figures In and Around Circles

Start with Definitions or First proposition.
Interesting construction: Regular Pentagon

Book 5 - Proportion

Start with Definitions or First proposition.
Interesting proof: Commutativity of multiplication

Book 6 - Similar Figures

Start with Definitions or First proposition.
Interesting propositions: Ratios of similar areas, Solution of a quadratic equation

Book 7 - Elementary Number Theory

Start with Definitions or First proposition.
Interesting propositions: Greatest common divisor algorithm

Book 8 - Continued Proportion

Start with First proposition.
Interesting proof: Prime factors in series

Book 9 - Applications of Number Theory

Start with First proposition.
Interesting proof: Infinity of primes

Book 10 - Incommensurable Magnitudes

Start with Definitions I, Definitions II or Definitions III.
Interesting proof: Method of exhaustion

Book 11 - Elementary Stereometry

Start with Definitions or First proposition.
Interesting construction: Solid angle

Book 12 - Proportional Stereometry

Start with First proposition.
Interesting proof: Pyramid as the third of a prism

Book 13 - The Platonic Solids

Start with First proposition.
Interesting propositions: Icosahedron, Dodecahedron, Finitude of regular polyhedra

Full Table of Contents

Based on this translation.

Care by ratherthanpaper as code to be read.

Euclid's Elements

Book 1 - Fundamentals of Plane Geometry Involving Straight-Lines

Start with Definitions, Postulates, Common Notions or First proposition.
Interesting proofs: Complements in parallelograms, Pythagorean theorem

Book 2 - Fundamentals of Geometric Algebra

Start with Definitions or First proposition.
Interesting construction: Golden section

Book 3 - Fundamentals of Plane Geometry Involving Circles

Start with Definitions or First proposition.
Interesting propositions: Angles at center and circumference, Relations of lines from outside a circle

Book 4 - Construction of Rectilinear Figures In and Around Circles

Start with Definitions or First proposition.
Interesting construction: Regular Pentagon

Book 5 - Proportion

Start with Definitions or First proposition.
Interesting proof: Commutativity of multiplication

Book 6 - Similar Figures

Start with Definitions or First proposition.
Interesting propositions: Ratios of similar areas, Solution of a quadratic equation

Book 7 - Elementary Number Theory

Start with Definitions or First proposition.
Interesting propositions: Greatest common divisor algorithm

Book 8 - Continued Proportion

Start with First proposition.
Interesting proof: Prime factors in series

Book 9 - Applications of Number Theory

Start with First proposition.
Interesting proof: Infinity of primes

Book 10 - Incommensurable Magnitudes

Start with Definitions I, Definitions II or Definitions III.
Interesting proof: Method of exhaustion

Book 11 - Elementary Stereometry

Start with Definitions or First proposition.

Book 12 - Proportional Stereometry

Start with First proposition.
Interesting proof: Pyramid as the third of a prism

Book 13 - The Platonic Solids

Start with First proposition.
Interesting propositions: Icosahedron, Dodecahedron, Finitude of regular polyhedra


Elements


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