Philosophy of Mathematics and Deductive Structure in Euclids Elements (Dover Books on Mathematics)

Euclid's Elements
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Philosophy of Mathematics and Deductive Structure in Euclid's Elements

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The Elements is often considered as one of the documents, next to the Bible, that had the most impact on the Western culture. However, according to modern mathematical standards of rigor, the Elements show some shortcomings. These have been repaired as late as the s by the German mathematician Hilbert. Book I starts with definitions of the type: a point is that which has no part and: a line is a breadthless length.

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Most of the other books begin with their own collections of definitions. It then goes on to the formulation of the five postulates of Euclidean geometry:. After these come axioms or common notions, such as "Things that are equal to the same thing are equal to each other. The first 26 propositions theorems or constructions are based on the first four of the above postulates, and treat mainly congruence of triangles and of other geometric figures.

After the parallel postulate is first used, the subject largely shifts to parallelograms.

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Buy Philosophy of Mathematics and Deductive Structure in Euclid's Elements ( Dover Books on Mathematics) on oraszeuta.tk ✓ FREE SHIPPING on qualified. Philosophy of Mathematics and Deductive Structure in Euclid's Elements and Deductive Structure in Euclid's Elements is a Dover reprint of the classic. In Chapter 1, Mueller explores the first book of the Elements.

One of the most famous propositions is the sixteenth which states that the exterior angle of a triangle is greater than either remote interior angle. The sixteenth proposition, together with the fifth postulate, implies the existence of parallel lines, see the upper part of the figure. Euclid realized that this result could not be proven on basis of his first four postulates, and therefore he added his fifth most famous postulate.

For many centuries workers have tried to prove the fifth postulate from the other four, but all attempts failed. Indeed, a geometry—a non-Euclidean geometry—without this postulate is possible and logically consistent in contrast to, for instance, Immanuel Kant 's view, who gave a philosopher's proof of the necessity of Euclidean geometry. Another famous proposition in Book I is number the Pythagorean theorem , which is proved by a technique still used in high school texts.

This is followed by its converse, which ends Book I. Book II is on geometrical algebra. All quantities in Book II are handled geometrically.

Numbers are represented by line segments and products of numbers by areas. Most of division is postponed to Book V.

Book IV deals with figures inscribed in and circumscribed about circles. For example, triangles, squares, regular pentagons and hexagons. Book V is probably the most important of the books.

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It is based on Eudoxus ' work on proportions. It is about ratios, including incommensurable ratios, but avoids irrational numbers by an ingenious method. The approach covers proportions for all kinds of magnitudes. Book V proves twenty-five theorems about magnitudes and ratios of magnitudes. Book VI, about similar figures, uses the theory of Book V.

Euclid's Elements - Arithmetic and Geometry Math Foundations 19 - N J Wildberger

It gives the solution to the general quadratic equation with positive discriminant. Book VII deals with theory of numbers, i.

It covers divisors and multiples, and develops a theory of ratio and proportion of such numbers independently of Book V.