"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
Start with Definitions, Postulates, Common Notions or First proposition.
Interesting proofs:
Complements in parallelograms,
Pythagorean theorem
Start with Definitions or First proposition.
Interesting construction:
Golden section
Start with Definitions or First proposition.
Interesting propositions:
Angles at center and circumference,
Relations of lines from outside a circle
Start with Definitions or First proposition.
Interesting construction: Regular Pentagon
Start with Definitions or First proposition.
Interesting proof:
Commutativity of multiplication
Start with Definitions or First proposition.
Interesting propositions:
Ratios of similar areas,
Solution of a quadratic equation
Start with Definitions or First proposition.
Interesting propositions: Greatest common divisor algorithm
Start with First proposition.
Interesting proof: Prime factors in series
Start with First proposition.
Interesting proof: Infinity of primes
Start with Definitions I, Definitions II or Definitions III.
Interesting proof: Method of exhaustion
Start with Definitions or First proposition.
Interesting construction: Solid angle
Start with First proposition.
Interesting proof: Pyramid as the third of a prism
Start with First proposition.
Interesting propositions: Icosahedron, Dodecahedron, Finitude of regular polyhedra
Based on this translation.
Care by ratherthanpaper as code to be read.
Start with Definitions, Postulates, Common Notions or First proposition.
Interesting proofs:
Complements in parallelograms,
Pythagorean theorem
Start with Definitions or First proposition.
Interesting construction:
Golden section
Start with Definitions or First proposition.
Interesting propositions:
Angles at center and circumference,
Relations of lines from outside a circle
Start with Definitions or First proposition.
Interesting construction: Regular Pentagon
Start with Definitions or First proposition.
Interesting proof:
Commutativity of multiplication
Start with Definitions or First proposition.
Interesting propositions:
Ratios of similar areas,
Solution of a quadratic equation
Start with Definitions or First proposition.
Interesting propositions: Greatest common divisor algorithm
Start with First proposition.
Interesting proof: Prime factors in series
Start with First proposition.
Interesting proof: Infinity of primes
Start with Definitions I, Definitions II or Definitions III.
Interesting proof: Method of exhaustion
Start with Definitions or First proposition.
Start with First proposition.
Interesting proof: Pyramid as the third of a prism
Start with First proposition.
Interesting propositions: Icosahedron, Dodecahedron, Finitude of regular polyhedra
A straight-line falling across parallel straight-lines makes the alternate angles equal to one another, the external (angle) equal to the internal and opposite (angle), and the (sum of the) internal (angles) on the same side equal to two right-angles.
For let the straight-line EF fall across the parallel straight-lines AB and CD. I say that it makes the alternate angles, AGH and GHD, equal, the external angle EGB equal to the internal and opposite (angle) GHD, and the (sum of the) internal (angles) on the same side, BGH and GHD, equal to two right-angles.
For if AGH is unequal to GHD then one of them is greater. Let AGH be greater. Let BGH have been added to both. Thus, (the sum of) AGH and BGH is greater than (the sum of) BGH and GHD. But, (the sum of) AGH and BGH is equal to two right-angles [Prop. 1.13]. Thus, (the sum of) BGH and GHD is [also] less than two right-angles. But (straight-lines) being produced to infinity from (internal angles whose sum is) less than two right-angles meet together [Post. 5]. Thus, AB and CD, being produced to infinity, will meet together. But they do not meet, on account of them (initially) being assumed parallel (to one another) [Def. 1.23]. Thus, AGH is not unequal to GHD. Thus, (it is) equal. But, AGH is equal to EGB [Prop. 1.15]. And EGB is thus also equal to GHD. Let BGH be added to both. Thus, (the sum of) EGB and BGH is equal to (the sum of) BGH and GHD.
But, (the sum of) EGB and BGH is equal to two right-angles [Prop. 1.13]. Thus, (the sum of) BGH and GHD is also equal to two right-angles.
Thus, a straight-line falling across parallel straight-lines makes the alternate angles equal to one another, the external (angle) equal to the internal and opposite (angle), and the (sum of the) internal (angles) on the same side equal to two right-angles. (Which is) the very thing it was required to show.
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