Download Geometry by Its History (Undergraduate Texts in Mathematics) by Gerhard Wanner, Alexander Ostermann PDF
By Gerhard Wanner, Alexander Ostermann
During this textbook the authors current first-year geometry approximately within the order during which it used to be chanced on. the 1st 5 chapters convey how the traditional Greeks tested geometry, including its various useful functions, whereas newer findings on Euclidian geometry are mentioned besides. the subsequent 3 chapters clarify the revolution in geometry as a result of growth made within the box of algebra via Descartes, Euler and Gauss. Spatial geometry, vector algebra and matrices are taken care of in chapters nine and 10. The final bankruptcy deals an creation to projective geometry, which emerged within the nineteenth century.
Complemented by means of quite a few examples, workouts, figures and images, the e-book bargains either motivation and insightful reasons, and gives stimulating and stress-free interpreting for college kids and lecturers alike.
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Extra info for Geometry by Its History (Undergraduate Texts in Mathematics)
Sample text
Eucl. 8. If two triangles ABC and DEF have the same sides, they also have the same angles. The proof of Philo of Byzantium, which is given here, is more elegant than Euclid’s. We apply the triangle ABC (see Fig. 3 (b)) onto the triangle DEF in such a manner that the line BC is placed on EF and the point A which becomes G lies on the opposite side of EF to D (see Fig. 3 (c)). By hypothesis, DEG is isosceles and thus α = β (Eucl. 5). But DF G is also isosceles and hence γ = δ (Eucl. 5). Thus the angle at A (= β + δ) is equal to the angle at D (= α + γ).
Haben das Eigne, dass ... ihre Beweise ... ¨ ausserst versteckt liegen, und nur durch sehr tief eindringende Untersuchungen aufgesp¨ urt werden k¨ onnen. ” (Gauss, 1809; Werke, vol. 2, p. 152) 44 2 The Elements of Euclid Eucl. 2. Given two numbers not relatively prime, to find their greatest common measure. 9 Given a pair of distinct positive integers, say a, b with a > b, subtract the smaller from the larger. Then repeat this with the new pair a − b, b. Any common divisor of a and b also divides a − b and b, and conversely.
30. Diagonal in a particular quadrilateral (left); Archimedes’ Lemma (right; copied from Peyrard’s edition of Archimedes’ Opera, vol. 9 Exercises 25 15. Prove one of Archimedes’ Lemmata (see Fig. 30, right): The area of the moon-like region bounded by the semicircles AC, CD and DA is equal to the area of the circle with diameter BD. 16. A young couple, to celebrate their golden wedding, set up a tent whose base is a square of side length the golden ratio Φ (what else), held up by 5 tubes of length 1, see Fig.