The
frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570
Euclid's Elements (Greek: Στοιχεῖα) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Egypt during the early 3rd century BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. Euclid's books are in the fields of Euclidean geometry, as well as the ancient Greek version of number theory. The Elements is one of the oldest extant axiomatic deductive treatments of geometry, and has proved instrumental in the development of logic and modern science.
It is considered one of the most successful textbooks ever written: the Elements was one of the very first books to go to press, and is second only to the Bible in number of editions published (well over 1000). For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century did it cease to be considered something all educated people had read. It is still (though rarely) used as a basic introduction to geometry today.
Contents
Although Elements is a geometric work, it also includes results that today would be classified as number theory. The contents of the work are as follows:
Books 1 through 4 deal with plane geometry:
- Book 1 contains the basic properties of geometry: the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area).
- Book 2 is commonly called the "book of geometrical algebra," because the material it contains may easily be interpreted as algebra.
- Book 3 deals with circles and their properties: inscribed angles, tangents, the power of a point.
- Book 4 is concerned with inscribing and circumscribing triangles and regular polygons.
Books 5 through 10 introduce ratios and proportions:
Books 11 through 13 deal with spatial geometry:
- Book 11 generalizes the results of Books 1–6 to space: perpendicularity, parallelism, volumes of parallelepipeds.
- Book 12 calculates areas and volumes by using the method of exhaustion: cones, pyramids, cylinders, and the sphere.
- Book 13 generalizes Book 4 to space: golden section, the five regular (or Platonic) solids inscribed in a sphere.
External links
- a bilinguial edition (typset in PDF format, with the original Greek and an English translation on facing pages; free in PDF form, available in print)
- in English (HTML, with the figures in the form of Java applets that the user can manipulate)
- Heath's translation (HTML, without the figures, public domain)
- in ancient Greek (typeset in PDF format, public domain)
- Oliver Byrne's 1847 edition - an unusual version using color rather than labels such as ABC (scanned page images, public domain)
- Reading Euclid - a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures)
Complete and fragmentary manuscripts of versions of Euclid's Elements :
|
