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Über dieses Buch

The Russian edition of this book appeared in 1976 on the hundred-and-fiftieth anniversary of the historic day of February 23, 1826, when LobaeevskiI delivered his famous lecture on his discovery of non-Euclidean geometry. The importance of the discovery of non-Euclidean geometry goes far beyond the limits of geometry itself. It is safe to say that it was a turning point in the history of all mathematics. The scientific revolution of the seventeenth century marked the transition from "mathematics of constant magnitudes" to "mathematics of variable magnitudes. " During the seventies of the last century there occurred another scientific revolution. By that time mathematicians had become familiar with the ideas of non-Euclidean geometry and the algebraic ideas of group and field (all of which appeared at about the same time), and the (later) ideas of set theory. This gave rise to many geometries in addition to the Euclidean geometry previously regarded as the only conceivable possibility, to the arithmetics and algebras of many groups and fields in addition to the arith­ metic and algebra of real and complex numbers, and, finally, to new mathe­ matical systems, i. e. , sets furnished with various structures having no classical analogues. Thus in the 1870's there began a new mathematical era usually called, until the middle of the twentieth century, the era of modern mathe­ matics.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Spherical Geometry

Abstract
The first geometry other than Euclidean geometry was spherical geometry, or, as the ancients called it, Sphaerica. This geometry appeared after plane and solid Euclidean geometry. The main stimulus for the rise of plane and solid geometry was the need to measure the areas of fields and other plane figures and the capacities of vessels and storehouses of various shapes, that is, the volumes of different solids. The main stimulus for the rise of spherical geometry was the study of the starry heavens.
B. A. Rosenfeld

Chapter 2. The Theory of Parallels

Abstract
The first systematic account of the theory of parallels to come down to us is contained in Euclid’s Elements [173] dating from about 300 D.C. Euclid worked in Alexandria under Ptolemy I and was the head of the Museion, the most eminent scientific center of antiquity, which was founded at that time. Euclid’s Elements is a revised version of a number of Greek works from the fifth and fourth centuries D.C., namely the Elements attributed to Hippocrates of Chios (books I-IV and XI), the arithmetic works of the Pythagoreans (books VII-IX), Eudoxus ‘theory of similarity and ratios’ (books V-VI) and his method of exhaustion (book XII), and Theaetetus’ works on quadratic irrationalities (book X) and on regular polyhedra (book XIII). The Elements opens with a list of 23 definitions, many of which bear traces of ancient traditions. The last of these definitions deals with parallel lines:
Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction [173, vol. 1, p. 154].
B. A. Rosenfeld

Chapter 3. Geometric Transformations

Abstract
On a number of occasions we have come across the use of motions in geometry. In book I Euclid uses superposition in propositions 7 and 8 (theorems on the congruence of triangles) and later relies on these propositions. Although his definition of a circle (definition 15, book I) [173, vol. I. p. 153] does not involve the concept of motion, his definitions of a sphere, a circular cone, and a cylinder (definitions 14, 18, and 21 in book XI) do [173, vol. 3. pp. 261-262]. The motions involved are, respectively, the rotation of a semicircle about its diameter, of a right triangle about a leg, and of a rectangle about a side. The use of motions in these definitions seems to reflect an older tradition. In fact, we saw that in Theodosius’ later Sphaerica [578, p.1] a sphere is defined without the use of motions, in a manner analogous to Euclid’s definition of a circle.
B. A. Rosenfeld

Chapter 4. Geometric Algebra and the Prehistory of Multidimensional Geometry

Abstract
Our terms square and cube go back to the Pythagoreans, for whom quadratic numbers and cubic numbers were special cases of figurate numbers. These included plane numbers mƷn, solid numbers lƷmƷn, as well as the more complex triangular numbers n(n + 1)/2, pentagonal numbers n(3n – 1)/2, pyramidal numbers n(n + 1) (n + 2)/2Ʒ3, and so on.1 This terminology derives from the notion that points―which the Pythagoreans identified with units―are distributed in a discrete manner in figures in accordance with definite rules.
B. A. Rosenfeld

Chapter 5. Philosophy of Space

Abstract
The evolution of philosophical notions of space played an important role in the preparation of the discovery of non-Euclidean geometry and the subsequent generalizations of the idea of space. That is why we give a brief survey of this evolution.
B. A. Rosenfeld

Chapter 6. Lobačevskian Geometry

Abstract
Centuries of attempts to prove the parallel postulate led to the discovery of non-Euclidean geometry made at the beginning of the 19th century. This discovery was first published by the great Russian mathematician and professor at Kazan University Nikolaĭ Ivanovič Lobačevskiĭ in the paper On the principles of geometry (O načalah geometrii. Kazan, 1829) [333. vol. 1, pp. 185–261]. The first public announcement about this discovery was made during a meeting of the division of the physicomathematical sciences of Kazan University and took the form of a lecture entitled A brief exposition of the principles of geometry including a rigorous proof of the theorem on parallels (Exposition succincte des principles de la Géométrie avec une démonstration rigoureuse du théoremè des parallèles). Lobačevskiĭ notes that he drew on this lecture for the first part of the memoir “On the principles of geometry.” In the beginning of this part he writes:
Who would not agree that a Mathematical discipline must not start out with concepts as vague as those with which we, in imitation of Euclid, begin Geometry, and that nowhere in Mathematics should one tolerate the kind of insufficiency of rigor that one was forced to allow in the theory of parallel lines. … The initial concepts with which any discipline begins must be clear and reduced to the smallest possible number. It is only then that they can provide a firm and adequate foundation for the discipline. Such concepts must be learned by the senses — the inborn ones must not be trusted [333, vol. 1, pp. 185–186].
B. A. Rosenfeld

Chapter 7. Multidimensional Spaces

Abstract
Earlier we saw that the idea of a multidimensional space arose in connection with the geometric interpretation of, first, algebraic equations of degree higher than the third, and, later, of functions of three and more variables.
B. A. Rosenfeld

Chapter 8. The Curvature of Space

Abstract
By the curvature of a curve at a point we mean the limit of the ratio of the angle Δα between the tangents at the endpoints of an arc to the length Δα of that arc as the latter contracts to the point; that is, the limit
$$k = \frac{{d\alpha }} {{ds}} = \mathop {\lim }\limits_{\Delta s \to 0} \frac{{\Delta \alpha }}{{\Delta s}}. $$
(8.1)
k is also the reciprocal of the radius of curvature at the point in question, that is, the radius of the osculating circle at that point. (The osculating circle at a point P of a curve is dfined as the limit of circles determined by three points on the curve as they tend to P.) The concepts of the curvature of a curve and of the osculating (literally “kissing,” from the Latin osculans) circle were already known to Leibniz.1 Leibniz also suggested the possibility of characterizing the curvature of a surface by means of an osculating sphere.2
B. A. Rosenfeld

Chapter 9. Groups of Transformations

Abstract
The group concept was first defined for a certain class of concrete groups, namely groups of substitutions, which were studied in connection with attempts to obtain solutions in radicals of algebraic equations of degree n ≥ 5. Permutations of roots of algebraic equations were first studied by J. L. Lagrange in his Reflections on the solution of equations (Réflexions sur la resolution des éxquations. Berlin, 1771) [298, vol. 3, pp. 205–515]. Lagrange noticed that if x1, x 2, x 3 are the roots of a cubic equation, then each of the cubic radicals in the Cardano form can be written as 1/3(x 1 + ωx 2  + ω2 x 3), where ω is a cube root of 1. Since the function (x 1+ ωx 2+ ω 2 x 3)2 takes on two values under all possible permutations of the roots, it follows that this function is a root of a quadratic equation whose coefficients are rationally expressible in terms of the coefficients of the given equation. Lagrange also noticed that in the case of the fourth-degree equation the function x 1 x 2+x 3 x 4 of the four roots of this equation takes on only three values as a result of all permutations of the roots and is therefore a root of a cubic equation whose roots are rationally expressible in terms of the coefficients of the given equation. He called this pattern
the true principle, and, so to say, the metaphysics of the solution of an equation of third and fourth degree [298. vol. 3, p. 357].
B. A. Rosenfeld

Chapter 10. Application of Algebras

Abstract
The geometric interpretation of complex numbers as points of the plane appeared for the first time in the 18th century. After that there arose the natural idea of generalizing complex numbers in such a way that they could be interpreted as points of three-dimensional space. One of the earliest attempts of this kind was due to Caspar Wessel. It appeared in his previously mentioned Attempt to represent direction [624]. Having thought of the operation of multiplication of complex numbers in geometric terms, Wessel associated to a point in space with rectangular coordinates x, y, z the expression x + yε + , where ε and η are two different imaginary units, and interpreted by means of these numbers rotations about the Oy- and Oz-axes. Wessel used his “algebra” to solve problems involving spherical polygons.
B. A. Rosenfeld

Backmatter

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