Geometry by its history by Alexander Ostermann, Gerhard Wanner (auth.)

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 19thcentury.

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.

Show description

Read or Download Geometry by its history PDF

Similar geometry books

Geometry of Complex Numbers (Dover Books on Mathematics)

Illuminating, commonly praised publication on analytic geometry of circles, the Moebius transformation, and 2-dimensional non-Euclidean geometries. "This publication can be in each library, and each professional in classical functionality thought may be accustomed to this fabric. the writer has played a special carrier by means of making this fabric so with ease obtainable in one e-book.

Geometric Tomography (Encyclopedia of Mathematics and its Applications)

Geometric tomography offers with the retrieval of knowledge a few geometric item from info bearing on its projections (shadows) on planes or cross-sections via planes. it's a geometric relative of automated tomography, which reconstructs a picture from X-rays of a human sufferer. the topic overlaps with convex geometry and employs many instruments from that sector, together with a few formulation from crucial geometry.

First Steps in Differential Geometry: Riemannian, Contact, Symplectic (Undergraduate Texts in Mathematics)

Differential geometry arguably bargains the smoothest transition from the normal college arithmetic series of the 1st 4 semesters in calculus, linear algebra, and differential equations to the better degrees of abstraction and facts encountered on the higher department via arithmetic majors. at the present time it's attainable to explain differential geometry as "the examine of buildings at the tangent space," and this article develops this standpoint.

Additional info for Geometry by its history

Example text

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.

Download PDF sample

Rated 4.57 of 5 – based on 11 votes