By Alexander Ostermann, Gerhard Wanner (auth.)

In this textbook the authors current first-year geometry approximately within the order during which it was once chanced on. the 1st 5 chapters express how the traditional Greeks validated geometry, including its a number of sensible functions, whereas more moderen findings on Euclidian geometry are mentioned besides. the next 3 chapters clarify the revolution in geometry as a result of growth made within the box of algebra through Descartes, Euler and Gauss. Spatial geometry, vector algebra and matrices are taken care of in chapters nine and 10. The final bankruptcy bargains an advent to projective geometry, which emerged within the 19^{th}century.

Complemented by means of various examples, routines, figures and images, the publication bargains either motivation and insightful motives, and offers stimulating and relaxing examining for college kids and lecturers alike.

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3 (a), erected on the same base AB and on the same side of it. If a = a′ and b = b′ , then C = D. (a) b′ A D α β γ a′ b c (b) C A (c) D α γ δ a B C B F E β δ G Fig. 3. Triangles with equal sides Proof by Euclid . Suppose that C = D. Since DAC is isosceles by hypothesis, α + β = γ (Eucl. 5). Since DBC is isosceles, β = γ + δ (Eucl. 5). Thus we have on the one hand γ > β, and on the other hand γ < β, which is impossible. 1 Book I This is our ﬁrst indirect proof . J. Brouwer, 1881–1966). Eucl. 8. If two triangles ABC and DEF have the same sides, they also have the same angles.

Eucl. 14 (left) and its proof (right) Proof. e. by Eucl. 13, let γ + β = 2 . By hypothesis, α + β = 2 . These angles are equal by the fourth postulate, hence γ = α. Therefore, E and C lie on the same line. Eucl. 15. e. α = β in Fig. 7 (left). α β γ α β Fig. 7. Eucl. 15 (left) and its proof (right) Proof. By Eucl. 13, we have α + γ = 2 and also γ + β = 2 . By Post. 4, α + γ = γ + β. The result then follows from subtracting γ from each side. 1 Book I 35 Eucl. 16. If one side of a triangle is produced at C (see Fig.

3) and the midpoint of DE (Eucl. 10). The entrance of Postulate 4. “When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands”. (Def. 10 of Euclid’s ﬁrst book in the transl. of Heath, 1926). The fourth postulate expresses the homogeneity of the plane, the absence of any privileged direction, and allows one to compare, add and subtract the 34 2 The Elements of Euclid angles around a point.