By Irving Adler

More than a hundred workouts with solutions and 2 hundred diagrams light up the textual content. academics, scholars (particularly these majoring in arithmetic education), and mathematically minded readers will delight in this amazing exploration of the position of geometry within the improvement of Western clinical thought.

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If s = m/n, then m2/n2 = s2 = 2, and m2 = 2n2. That is, m2 must be an even number. This implies that m is an even number, because if m were odd, m2 would also be odd. Consequently there exists a whole number k such that m = 2k. Then m2 = 4k2. But m2 = 2n2. Equating these two expressions for m2, we see that 2n2 = 4k2, or n2 = 2k2. That is, n2 must be even, and therefore n must be even. Notice that while m and n are not both even, we have been compelled to say that they are both even. We were led into this absurdity as a result of assuming that the length s may be expressed as a ratio of whole numbers.

Includes bibliographical references and index. Geometry. I. Title. com Contents Introduction to the Dover Edition 1 One Book and Three Metaphors 2 Geometry Before Euclid 3 Euclid’s 4 Geometry Geometry via Numbers 5 Geometry via Arrows 6 Geometry via Reflections 7 Geometry in Newtonian Physics 8 Non-Euclidean Geometry 9 The Calculus and Geometry 10 Projective Geometry 11 Geometry in Relativity Physics 12 Geometry Generalized 13 Geometry and Algebra Bibliography Answers to Exercises Index Introduction to the Dover Edition When it was first published in the USA by The John Day Company in 1966, this was, to my knowledge, the first book to provide an accessibly broad description of classical and modern geometries.

Thus introducing more points of division on the line. If m is any counting number, and n is any counting number except 0, we use the labels and respectively for the points to the right and left of 0 at a distance of units. The points labeled in this way constitute the rational number system. Each rational number is represented by many equivalent fractions. Rational numbers may be added and multiplied in accordance with the familiar rules Fields The operations addition and multiplication in the rational number system have the following properties: If x, y and z are any rational numbers, 1.

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Categories: Geometry Topology