Euclid: Difference between revisions
imported>Paul Wormer (→Outline of the Elements: I put an extra link to the article "Euclid's elements"; to be honest I don't see why its contents must be repeated here.) |
imported>Peter Jackson |
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Book VI applies this theory to geometry. | Book VI applies this theory to geometry. | ||
Book VII deals with multiples, divisors, and ratio and proportion among numbers. | |||
Book VIII deals with geometric sequences of numbers, and with square and cube numbers. | |||
Book IX continues these topics, adding treatment of prime numbers. It ends with the proof that, if a power of 2 is 1 more than a prime, then the product of that prime and the previous power of 2 is a perfect number. | |||
Book X gives a detailed theory of (recursively) quadratic irrational numbers. | Book X gives a detailed theory of (recursively) quadratic irrational numbers. | ||
Revision as of 05:48, 6 November 2008
Euclid (Εύκλείδες, c. 300 BCE) was a Greek mathematician. He worked in Alexandria at the Museum founded by Ptolemy I. He systematized the geometric and arithmetic knowledge of his times in thirteen Books—Euclid's elements (Στοιχεία).
Nothing else is known of Euclid's life. That he lived and worked in Alexandria in the days of Ptolemy I is reported to us by the Greek philosopher Proclus (c. 410–485 CE) in his "summary" of famous Greek mathematicians. Probably Euclid is older than Archimedes (c. 290/280–212/211 BCE).
Outline of the Elements
Book I deals with triangles and parallelograms, concluding with Pythagoras' Theorem and its converse.
Book II covers elementary quadratic algebra in the form of theorems about the areas of rectangles etc.
Book III deals with circles.
Book IV gives constructions for polygons inscribed in and circumscribed round circles.
Book V covers the general theory of ratio and proportion, developed by Eudoxus.
Book VI applies this theory to geometry.
Book VII deals with multiples, divisors, and ratio and proportion among numbers.
Book VIII deals with geometric sequences of numbers, and with square and cube numbers.
Book IX continues these topics, adding treatment of prime numbers. It ends with the proof that, if a power of 2 is 1 more than a prime, then the product of that prime and the previous power of 2 is a perfect number.
Book X gives a detailed theory of (recursively) quadratic irrational numbers.
Book XI deals with basic solid geometry.
Book XII is concerned with the "method of exhaustions", a precursor of integral calculus. Circles, spheres, cones and cylinders are treated as limits of polygons and polyhedra.
Book XIII deals with the five regular solids: how to construct them; the ratios of the sides of different solids inscribed in the same sphere, classified by the methods of Book X; proof that there are only five.
Other works
Besides the Elements, Euclid wrote numerous other books. We have the texts of four others, Data, On Divisions of Figures, the Phaenomena, and the Optics. Data is a collection of 94 geometric propositions, On Division of Figures deals with problems of dividing a given figure by one or more straight lines into others with desired properties of shape and area. It was used by Leonardo of Pisa in his Practica geometriae of 1220. The Optics is the first Greek treatise on perspective, with geometrical propositions on the rectilinear propagation of light. The Phaenomena is an introduction to mathematical astronomy. Since it closely resembles the style of the earlier treatise "On the Moving Sphere" by Autolycus (c. 330 BCE), historians have concluded that Euclid did not necessarily invent the form of his works.
From Greek commentators one learns of several books that have been lost. They include Pseudaria (on fallacies); Porisms (on conditions determining curves); Conics (which was superseded by a similar work of Apollonius); Surface Loci (perhaps dealing with cones, spheres, and cylinders, or with curves on these surfaces); Elements of Music (possibly including the Pythagorean theory of harmony); and Catoptrics (on the properties of mirrors). A surviving Catoptrics bearing Euclid's name is in reality a later compilation—possibly by Theon of Alexandria (c. 350 CE)—but it probably is based upon an authentic Euclidean work of the same name and in the same form. Arabic writers also attribute to Euclid various treatises on mechanics including books on the balance and on specific gravity.
Bibliography
- Allman, George J. Greek Geometry from Thales to Euclid (1976)
- Artmann, Benno. Euclid - The Creation of Mathematics (2001) excerpt and text search
- Cajori, Florian. A History of Mathematics (1919) complete text online free
- Heath, Thomas L. A History of Greek Mathematics, (2 vol 1981).
- Heath, Thomas L. (trans.), The Thirteen Books of Euclid's Elements, Cambridge University Press, reprinted Dover Books, 3 volumes (heavily annotated)
- Mueller, Ian. Philosophy of Mathematics and Deductive Structure in Euclid's Elements (1981).