Euclid's Elements

Euclid's Elements (Greek Στοιχεία) is a mathematical and geometric treatise, consisting of 13 books, written by the Greek Mathematician Euclid around 300 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 the most successful textbook 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.

First principles

Euclid based his work in Book I on 23 definitions, such as Point, line and Surface, five postulates and five "common notions" (both of which are today called axioms).

Postulates in Book I:

  1. A straight line segment can be drawn by joining any two points.
  2. A straight line segment can be extended indefinitely in a straight line.
  3. Given a straight line segment, a circle can be drawn using the segment as radius and one endpoint as center.
  4. All right angles are congruent.
  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Common notions in Book I:

  1. Things which equal the same thing are equal to one another.
  2. If equals are added to equals, then the sums are equal.
  3. If equals are subtracted from equals, then the remainders are equal.
  4. Things which coincide with one another are equal to one another.
  5. The whole is greater than the part.

These basic principles reflect the constructive geometry Euclid, along with his contemporary Greeks, was interested in. The first three postulates basically describe the constructions one can carry out with a compass and an unmarked Straightedge or Ruler.

Success

The success of Elements is due primarily to its logical presentation of much of the mathematical knowledge available to Euclid. Most of the material is not original to him, although a few of the proofs are his. Its systematic development from a small set of axioms to deep results encouraged its use as a textbook for hundreds of years, and still influences modern geometry books.

Until the late 19th century, Elements was considered one of the best examples (if not the best) of a complete deductive structure: all of its components were thought to follow logically from previous components. However, the publication of David Hilbert's 'Grundlagen der Geometrie' (Foundations of Geometry) made evident many previously overlooked logical flaws in the Elements. Nevertheless, the Elements continues to be used as an adequate example of the application of Logic, and, historically, it has been enormously influential in many areas of Science. European scientists Nicolaus Copernicus, Johannes Kepler, Galileo Galilei and especially Sir Isaac Newton were all influenced by the Elements, and applied their knowledge of it to their work. Mathematicians (Bertrand Russell, Alfred North Whitehead) and philosophers (Baruch Spinoza) have also attempted to provide their own Elements; that is, axiomatized deductive structures of their own respective disciplines.

Of the five postulates Euclid used, the last, so-called "Parallel postulate" seemed less obvious than the others. Many geometers suspected that it may be provable from the other postulates but all attempts to do this failed. By the mid-19th century, it was shown that no such proof exists, because one can construct non-Euclidean geometries where the parallel postulate is false, while the other postulates remain true. Mathematicians say that the parallel postulate is independent of the other postulates. Two alternatives are possible: either an infinite number of parallel lines can be drawn through a point not on a straight line (Hyperbolic geometry, also called Lobachevskian geometry), or none can (Elliptic geometry, also called Riemannian geometry). That other geometries could be logically consistent was one of the most important discoveries in mathematics, with vast implications for science and philosophy. Indeed, Albert Einstein's theory of General relativity shows that the "real" space in which we live can be non-Euclidean (for example, around black holes and neutron stars). That Euclid recognized the independence of the parallel postulate long before other mathematicians accepted it is a testament to Euclid's dedication to a logical development from as few assumptions as possible.

History

Elements was written in approximately 300 BC by Euclid, an ancient Greek mathematician who probably studied under the pupils of Plato. Although most of the theorems had been developed earlier, Elements was so impressive and comprehensive that the Greeks had no use for the older books, and little is known about earlier geometers today.

It was translated later into Arabic after being gifted to the Arabs by Byzantium and from those secondary translations into Latin. The first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Copies of the Greek text also exist, e.g. in the Vatican Library and the Bodlean library in Oxford. However, the manuscripts available are of very variable quality and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been drawn about the contents of the original text (copies of which are no longer available). Texts which refer to the Elements itself and mathematical theories which were current at the time it was written are also important in this process. Such analyses are conducted by J.L. Heiberg and Sir Thomas L. Heath in their editions of the text.

Also of importance are the scholia, or footnotes to the text. These additions, which often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or elucidation. Some of these are useful and add to the text, but many are not.

Later axiomizations

Surprisingly, mathematicians in the nineteenth century discovered that Euclid's proofs require additional assumptions, ones not stated among either his postulates or his common notions. For example:
  • One of his theorems is to the effect that any line segment is part of a triangle, which he constructs in the usual way, by drawing circles around both endpoints and taking their intersection and them as three corners. However, his axioms do not guarantee that the circles actually do intersect.
  • It is proven by superposition that two triangles are congruent if two of their sides and the angle between them are respectively equal. The method relies on three assumptions: that the straight line making a given angle with another on one side is unique, that the straight line connecting two points is unique, and that the properties of a figure are independent of where it is situated.

David Hilbert gave a revised list containing no fewer than 23 separate axioms

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.

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