====== Euclid's Elements ======

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Euclid's Elements form one of the most beautiful and influential works of science in the history of humankind. Its beauty lies in its logical development of geometry and other branches of mathematics. It has influenced all branches of science but none so much as mathematics and the exact sciences. The Elements have been studied 24 centuries in many languages starting, of course, in the original Greek, then in Arabic, Latin, and many modern languages.

Euclid's Elements is by far the most famous mathematical work of classical antiquity, and also has the distinction of being the world's oldest continuously used mathematical textbook. Little is known about the author, beyond the fact that he lived in Alexandria around 300 BCE. The main subjects of the work are geometry, proportion, and number theory.

Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner, so as to demonstrate (admittedly, not always with the rigour demanded by modern mathematics) that they necessarily follow from five simple axioms. Euclid is also credited with devising a number of particularly ingenious proofs of previously discovered theorems: e.g., Theorem 48 in Book 1.

The geometrical constructions employed in the Elements are restricted to those which can be achieved using a straight-rule and a compass. Furthermore, empirical proofs by means of measurement are strictly forbidden: i.e., any comparison of two magnitudes is restricted to saying that the magnitudes are either equal, or that one is greater than the other.

The Elements consists of thirteen books.

  - [[Euclid's Elements Book 1|Book 1]] outlines the fundamental propositions of plane geometry, including the three cases in which triangles are congruent, various theorems involving parallel lines, the theorem regarding the sum of the angles in a triangle, and the Pythagorean theorem.
  - [[Euclid's Elements Book 2|Book 2]] is commonly said to deal with "geometric algebra", since most of the theorems contained within it have simple algebraic interpretations.
  - [[Euclid's Elements Book 3|Book 3]] investigates circles and their properties, and includes theorems on tangents and inscribed angles.
  - [[Euclid's Elements Book 4|Book 4]] is concerned with regular polygons inscribed in, and circumscribed around, circles.
  - [[Euclid's Elements Book 5|Book 5]] develops the arithmetic theory of proportion.
  - [[Euclid's Elements Book 6|Book 6]] applies the theory of proportion to plane geometry, and contains theorems on similar figures.
  - [[Euclid's Elements Book 7|Book 7]] deals with elementary number theory: e.g., prime numbers, greatest common denominators, etc.
  - [[Euclid's Elements Book 8|Book 8]] is concerned with geometric series.
  - [[Euclid's Elements Book 9|Book 9]] contains various applications of results in the previous two books, and includes theorems on the infinitude of prime numbers, as well as the sum of a geometric series.
  - [[Euclid's Elements Book 10|Book 10]] attempts to classify incommensurable (i.e., irrational) magnitudes using the so-called ``method of exhaustion'', an ancient precursor to integration.
  - [[Euclid's Elements Book 11|Book 11]] deals with the fundamental propositions of three-dimensional geometry.
  - [[Euclid's Elements Book 12|Book 12]] calculates the relative volumes of cones, pyramids, cylinders, and spheres using the method of exhaustion.
  - [[Euclid's Elements Book 13|Book 13]] investigates the five so-called Platonic solids.

  * [[http://www.ganino.com/files/thirteenbookseu03heibgoog.pdf|The Thirteen Books Of Euclid Elements Cambridge University Press 1908]]
  * [[http://www.ganino.com/files/ElementsHeiberg.pdf|Euclid's Elements Of Geometry J.L. Heiberg - Richard Fitzpatrick]]

====== Non-Euclid Geometry ======

  * [[http://www.ganino.com/files/TheElementsofnonEuclideanGeometrybyJulianLowellCoolidge.pdf|The Elements of non Euclidean Geometry by Julian Lowell Coolidge]]
  * [[http://www.ganino.com/files/NonEuclideanGeometrybyHenryManning.pdf|Non-Euclidean Geometry by Henry Manning]]