Book I : The fundamentals of geometry: theories of triangles, parallels, and area
Definitions (23)
Postulates (5)
Common Notions (5)
Propositions (48)
Book II : Geometric algebra
Definitions (2)
Propositions (14)
Book III : Theory of circles
Definitions (11)
Propositions (37)
Book IV : Constructions for inscribed and circumscribed figures
Definitions (7)
Propositions (16)
Book V : Theory of abstract proportions.
Definitions (18)
Propositions (25)
Book VI : Similar figures and proportions in geometry
Definitions (11)
Propositions (37)
Book VII : Fundamentals of number theory
Definitions (22)
Propositions (39)
Book VIII : Continued proportions in number theory
Propositions (27)
Book IX : Number theory
Propositions (36)
Book X : Classification of incommensurables
Definitions I (16)
Propositions (115)
Book XI : Solid geometry
Definitions (28)
Propositions (39)
Book XII : Measurement of figures
Propositions (18)
Book XIII : Regular solids
Propositions (18)

This is an Internet update of avjewe's writeup, and an augmentation of cbustapeck's wonderful scholarship in Euclid. The Internet's resources now give useful background information on Euclid's writings, and also modernize to a considerable extent his language and the impact of his mathematical findings. Nothing makes Euclid come alive quite so much as seeing a Java applet where you can drag around corners of a triangle and watch as the triangle sides and circles and lines of tangency move along with them.

Don Allen, a professor of mathematics at Texas A&M University, has written readable and understandable explanatory notes that would enhance the understanding of a close reading of the Elements. He includes a bulletized list of "basic facts" about Euclid's Elements:

  • Written about 2300 years ago
  • No copies extant
  • A few potsherds dating from 225 BC contain notes about some propositions
  • Many new editions were issued (e.g. Theon of Alexandria, 379 cent. AD)
  • Earliest copy dates from 888AD -- in Oxford
  • Style: no examples, no motivations, no calculation, no witty remarks, no introduction, no preamble --- nothing but theorems and their proofs.

From Richard Fitzpatrick's background on Euclid and his tour de force, the Elements:

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.

Fitzpatrick gives some background into his scholarship of Euclid, and what he's produced as a result:

I have prepared a new edition of Euclid's Elements which presents the definitive (and completely out-of-print) Greek text - i.e., that edited by J.L. Heiberg (1883-1885) - accompanied by a modern English translation, as well as a Greek-English lexicon. Neither the spurious books 14 and 15, nor the extensive scholia which have added to the Elements over the centuries, are included in the edition. The aim of the translation is to make the mathematical argument as clear and unambiguous as possible, whilst still adhering closely to the meaning of the original Greek. Text within square parenthesis (in both Greek and English) indicates material identified by Heiberg as being later interpolations to the original text (some particularly obvious or unhelpful interpolations have been omitted altogether). Text within round parenthesis (in English) indicates material which is implied, but not actually present, in the Greek text.

Aren't you tempted to buy a copy? I am.

References: Useful books and references on geometry

  1. H.S.M. Coxeter, Introduction to Geometry, 2nd Ed., (c) 1969
  2. Dan Pedoe, Geometry: A Comprehensive Course, Dover, (c)1970
  3. J.L. Heilbron, Geometry Civilized, ©2000
  4. Alfred S. Posamentier and Charles T. Salkind, Challenging Problems in Geometry, Dover, (c)1988
  5. Stanley Ogilvy, Excursions in Geometry, Dover, (c)1969
  6. Roland Deaux, Introduction to the Geometry of Complex Numbers, translated by Howard Eves, Dover, (c)1956?
  7. Nicholas D. Kazarinoff, Ruler and the Round: Classic Problems in Geometric Constructions, Dover, (c)1970

Internet References

  1. Richard Fitzpatrick , "Euclid's Elements of Geometry," U. Texas, Austin. Fitzpatrick is a professor of physics. He's modernized the Elements and made it available as a hardcover made-to-order book, a softcover made-to-order, a PDF file, and a LaTeX version. He also has an online version of Ptolemy's Almagest.
  2. Don Allen, "Euclid," Texas A&M University. Dr. Allen is a professor of mathematics at TAMU.
  3. Wikipedia, "Euclid's Elements"
  4. David E. Joyce, Homepage at Clark University. Joyce is a professor of Mathematics and Computer Science at Clark University, Worcester, MA. He has rendered Euclid's Elements for the computer and added numerous Java applets in the next reference. In my opinion, his site is one of the greatest reasons that geometers should use the Internet.
  5. David E. Joyce, Euclid's Elements. Euclid's Elements on steroids. The java applets make geometry and geometric proofs come alive. I defy you to read a few of his propositions, try out the applets, and then not be amazed at Euclid's genius.
  6. Paul Yiu, List of papers, Yiu is a professor in the Dept. of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL. He's written a series of elegant papers on classical geometry that provide an important grounding in the fundamental elements of geometry. His work is spare and elegant. He focuses on results, not on proofs. He provides figures to illustrate concepts, and they're always right to the point.
  7. Paul Yiu, "A Tour of Triangle Geometry." Elegant paper jam packed with amazing geometrical relationships.
  8. Encyclopedia of Triangle Centers. A major resource for geometers
  9. Wikipedia, "Encyclopedia of Triangle Centers
  10. Triangle Geometers. Mathematicians, both professional and laymen, who specialized in triangle geometry. Some famous names, and some not so famous: Euclid, Apollonius, Heron, Pythagoras, Menelaus, Pascal, Fermat, Ceva, Steiner, Feuerbach, Soddy, Coxeter, and even Napolean!

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