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## Über dieses Buch

Based on the latest historical research, Worlds Out of Nothing is the first book to provide a course on the history of geometry in the 19th century. Topics covered in the first part of the book are projective geometry, especially the concept of duality, and non-Euclidean geometry. The book then moves on to the study of the singular points of algebraic curves (Plücker’s equations) and their role in resolving a paradox in the theory of duality; to Riemann’s work on differential geometry; and to Beltrami’s role in successfully establishing non-Euclidean geometry as a rigorous mathematical subject. The final part of the book considers how projective geometry rose to prominence, and looks at Poincaré’s ideas about non-Euclidean geometry and their physical and philosophical significance.

Three chapters are devoted to writing and assessing work in the history of mathematics, with examples of sample questions in the subject, advice on how to write essays, and comments on what instructors should be looking for.

## Inhaltsverzeichnis

### 1. Mathematics in the French Revolution

A brief account is given of the French revolution, with its implications for the organisation of advanced mathematical education in France. The Ecole Polytechnique was created, with Gaspard Monge as its Director. Monge’s descriptive geometry is introduced.

Jeremy Gray

### 2. Poncelet (and Pole and Polar)

Jean-Victor Poncelet, a former student of Monge, re-invented projective geometry while in prison in Russia in 1812–1813. For him, his patriotic feelings, his wish for simple, general methods in geometry, and his version of projective geometry were inextricably mixed. He described his approach to a non-metrical geometry at length in his

Traité des propriétés projectives des figures

in 1822. Some of his controversial ideas are introduced, notably the so-called method of continuity according to which non-intersecting lines and conic sections may still be said to meet.

The fundamental technique of pole and polar with respect to a conic is also described in an algebraically simple case.

Jeremy Gray

### 3. Theorems in Projective Geometry

This chapter is more mathematical. The theorems of Pappus, Desargues, and Pascal are introduced to show that there is a non-metrical geometry such as Poncelet had described. However, they are proved in a modern way, by reducing them to simple special cases and then using a mixture of elementary vector methods and theorems from elementary Euclidean geometry. The reductions use simple projective transformations, which leads to discussion of the idea of augmenting the familiar plane with a line at infinity. The concept of the cross-ratio of four points on a line is introduced and shown to be invariant under a projective transformation. Poncelet’s porism is described.

Jeremy Gray

### 4. Poncelet’s Traité

This returns to Poncelet’s

Traité

and looks in more detail at his theory of lines and conics ‘meeting’. It draws on Apollonius’s theory of conjugate diameters of a conic and of a conic auxiliary to a given one (with respect to a pair of conjugate diameters). An early version of Poncelet’s book met with a mixed reception: it was given a critical yet fair review by Cauchy, who objected precisely to Poncelet’s implausible ideas. Poncelet, however, rejected the algebraic remedy offered by Cauchy, and the final book retains those ideas. This shows the strength of Poncelet’s belief in the importance of simple general ideas in geometry which could match those of algebra.

Jeremy Gray

### 5. Duality and the Duality Controversy

The technique of pole and polar is applied to the theorems of projective geometry described in Chapter 3, and it shown that new theorems result. This leads to the idea of a duality in the new geometry, in which the words point and line are interchanged, and the adjectives collinear and concurrent are interchanged. This theory can be extended to yield a curve dual to a given one. Given a curve

C

one considers at each point

P

of

C

the tangent

t

P

to the curve at the point

P

. Each tangent is a line, and by duality it corresponds to a point. The set of these points is the dual curve to the curve

C

.

There was a controversy between Poncelet and another geometer, Joseph Diaz Gergonne, about duality. Gergonne believed it was a new fundamental process, Poncelet derived it from the theory of pole and polar. However, as Poncelet was able to show, Gergonne weakened his case by applying the theory of duality incorrectly when he attempted to describe the dual of a curve of degree greater than 2.

Jeremy Gray

### 6. Poncelet, Chasles, and the Early Years of Projective Geometry

Poncelet, faced with growing opposition to his ideas at the Ecole Polytechnique, and despite the enthusiasm with which some geometers adopted it elsewhere in France, eventually abandoned the subject for the mathematical analysis of machines. It was rescued by a younger mathematician, Michel Chasles, who dropped Poncelet’s strange ideas and founded the subject on the idea of the invariance of cross-ratio under projective transformations. This was presented in his historically rich

Aperçu historique

(1839) and at greater length in his

Théorie de géométrie supérieure

(1852), and established projective geometry as a legitimate, even rigorous discipline in France. More precisely, it established the theory of plane figures under projective transformations. It often proceeds, as we did in Chapter 3, by using a suitable transformation to reduce a figure to a special case and then arguing metrically. An important illustration of this is his projective definition of a conic section using only the property of cross-ratio, that was also given by Steiner independently.

A final section briefly describes the modern, formal treatment of real projective geometry, which will be taken up later, and notes that there are axiomatic formulations in which, for example, Desargues’ theorem is false.

Extracts are given from Chasles’s

Aperçu historique

to illustrate his thoughts about Monge, his descriptive geometry, and the school around him.

Jeremy Gray

### 7. Euclidean Geometry, the Parallel Postulate, and the Work of Lambert and Legendre

This chapter switches to the history of Euclidean geometry, and especially the issue of the parallel postulate. This states

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.

Alone among the postulates of Euclid’s

Elements

it lacks intuitive credibility, and many attempts were made to deduce it from the other postulates. We look at the ideas of Gerolamo Saccheri, who showed in 1733 that a geometry based on all of Euclid’s postulates except the parallel postulate must be one of at most three kinds, which are distinguished by the angle sums of triangles. Either every triangle has an angle sum greater that

π

, or equal to

π

, or less than

π

. He then showed that the first case cannot occur, but his attempts to show that the third case also leads to a contradiction, which would have left Euclidean geometry as the only possibility, failed.

Saccheri was followed by Johann Heinrich Lambert, who noted some more unusual features of a geometry based on the third of Saccheri’s hypotheses. Then came Adrien-Marie Legendre, who also made several unsuccessful attempts to refute the third hypothesis, one of which is considered in detail.

Extract: Lambert on the consequences of a non-Euclidean parallel postulate.

Jeremy Gray

### 8. Gauss (Schweikart and Taurinus) and Gauss’s Differential Geometry

In the 1820s the hitherto unthinkable was gradually thought. Friedrich Karl Schweikart, a law professor, wrote to Carl Friedrich Gauss with some further consequences of Saccheri’s and Lambert’s ideas, which Gauss accepted and improved. Schweikart’s nephew, Franz Adolf Taurinus, however, used a lengthy inverstigation as the basis for a fallacious refutation of the new geometry, and Gauss refused to be associated with his work. As for what Gauss knew, the question is complicated: he accepted the possibility of a new geometry but never gave a connected account of it, even when, as briefly discussed here, he had discovered the intrinsic nature of the curvature of a surface.

Jeremy Gray

### 9. János Bolyai

János Bolyai’s father, Farkas (Wolfgang) was a friend of Gauss and an author of numerous defences of Euclidean geometry. His son, János, however, fully accepted the idea of a new geometry and published it as a 24-page appendix to his father’s two-volume work on geometry in 1832. The content of this remarkable 24-page essay is described, noting the importance of the study of three-dimensional geometry (without which there can be no claim that the new geometry applies to physical space). János Bolyai also gave trigonometric formulae for his geometry, noting reassuringly that they reduced to the familiar Euclidean ones for small triangles.

Extract: Section 32 of Bolyai’s essay, which gives an approach that is close to that of intrinsic differential geometry.

Jeremy Gray

### 10. Lobachevskii

Nicolai Ivanovich Lobachevskii, the other discoverer of non-Euclidean geometry, lived and worked in Kasan in Russia, and when circumstances permitted started by 1830 to publish accounts of his new geometry. This early essay is looked at briefly: it derives the crucial trigonometric formulae from a consideration of the intrinsic differential geometry of the new geometry. This work is seldom described in recent histories of mathematics. Lobachevskii’s later, and better known, work in his

Geometrische Untersuchungen

(1840) is compared with the treatment given by János Bolyai.

Extract: a lengthy extract from the conclusion to Lobachevskii’s

Geometrische Untersuchungen

.

Jeremy Gray

### 11. Publication and Non-Reception up to 1855

In the 1830s Ferdinand Minding described a surface of revolution with constant negative curvature. He showed that figures can be moved around freely on it, but made no connection to non-Euclidean geometry. In the 1840s the Bolyais found out about Lobachevskii’s work, which they generally liked, but they did not get in touch with him. The most notable contact was Gauss’s reply to the Appendix written by János Bolyai, where Gauss famously claimed that to praise it would be to praise himself. This reply contributed to a painful separation between the Bolyais, and convinced János that there was no point in publishing again. Gauss’s praise for Lobachevskii went more smoothly. Gauss nominated him for membership of the Göttingen Academy of Sciences and praised him among his small circle of friends, but otherwise did nothing to promote his work. Only after Gauss died and his endorsement of non-Euclidean geometry could be traced in the papers he left behind did the reception of the new geometry change. By then both János Bolyai and Lobachevskii were dead.

Jeremy Gray

### 12. On Writing the History of Geometry – 1

This chapter is given over to assessment, and offers advice on writing essays in the history of mathematics. As an illustrative point, Gauss’s survey of the mountains of Hanover is discussed, which is often said to indicate his commitment to the non-Euclidean nature of physical space.

Jeremy Gray

### 13. Across the Rhine – Möbius’s Algebraic Version of Projective Geometry

In 1827 August Möbius published his

Der barycentrische Calcul

(

The barycentric calculus

). This assigns coordinates to points in the plane by regarding each point as the centre of gravity (or barycentre) of three weights attached at the vertices of a fixed but arbitrary triangle. Each choice of weights (other than all three zero) defines a pair of ratios that specifies a point in the plane except for a one-dimensional family of weights that defines a line “at infinity”. Möbius’s barycentric coordinates, which are very simply related to the usual system of homogeneous coordinates that were introduced later by other mathematicians, are thus well suited to describing projective transformations and therefore the projective geometry of conic sections. However, they are even better suited to handling duality, and provide a good algebraic approach to projective geometry. Möbius also described how to introduce a system of projective coordinates in the plane without recourse to Euclidean ideas or methods.

Möbius also extended his notion of duality to higher dimensional spaces, and found a new form of duality that is not a generalisation of pole and polar and applies only in spaces of odd dimension.

Jeremy Gray

### 14. Plücker, Hesse, Higher Plane Curves, and the Resolution of the Duality Paradox

The study of algebraic curves other than conics was revived by Julius Plücker in the 1830s. He succeeded in showing how the duality paradox could be resolved for such curves. The paradox is that a curve of degree

n

will seemingly have a dual of degree

n

(

n

−1) that will in its turn have a dual of degree

n

(

n

−1)(

n

(

n

−1)−1). But by duality the dual of the dual of a curve must be the original curve, which forces

n

(

n

−1)(

n

(

n

−1)−1)=

n

, an equation that is plainly false for

n

>2. Plücker observed that each double point on a curve lowers the degree of the dual by 2, and each cusp lowers the degree of the dual by 3. Moreover, the dual of a double point is a bitangent and of a cusp an inflection point. He showed that a non-singular curve of degree

n

has 3

n

(

n

−2) inflection points, so the degree of the double dual will be reduced by 9

n

(

n

−2). A simple calculation then shows that this degree would therefore be reduced to

n

, and the paradox resolved, if the original curve has

$\frac{1}{2}n(n-2)(n^{2}-9)$

bitangents. Plücker could only conjecture this result, which was proved in 1850 by Jacobi, but he did investigate the special case of the 28 bitangents to a curve of degree 4, and showed that they could all be real.

Plücker then turned to experimental physics and the study of cathode rays, and the subject he had opened up was developed by Otto Hesse, who made more systematic use of homogeneous coordinates and used the eponymous Hessian to locate inflection points on curves. The work of Plücker and Hesse successfully established the subject of algebraic projective geometry.

Jeremy Gray

### 15. The Plücker Formulae

The chapter looks at the use of Plücker’s theory of singular points to examine plane curves of degrees 3 and 4. It concludes with Plücker’s proof that the 28 bitangents to a plane quartic can all be real.

Jeremy Gray

### 16. The Mathematical Theory of Plane Curves

The chapter develops the theory of singular points on plane algebraic curves, using homogeneous coordinates to first and higher polars to a curve, inflections points, the Hessian of curve, and to give a method for finding tangents to a curve.

Jeremy Gray

### 17. Complex Curves

Bezout’s theorem, that curves of degrees

k

and

m

meet in

km

points, forces geometers to allow complex points on a curve, but despite this incentive the step to admit complex algebraic curves was resisted until Riemann’s work in the 1850s. An important stimulus was the study of elliptic integrals, which are necessarily complex.

Jeremy Gray

### 18. Riemann: Geometry and Physics

Bernhard Riemann turned to mathematics away from theology at Göttingen University and gradually developed a series of profound ideas in the theory of complex functions and in geometry. He argued that geometry can be studied in any setting where one may speak of lengths and angles. Typically this meant differential geometry but in a space of any arbitrary number of dimensions and with an arbitrary (positive definite) metric. Such a geometry was intrinsically defined, that is without any reference to an ambient Euclidean space. He supplied metrics for spaces of constant curvature in any dimension, which in two dimensions lead to spherical geometry (positive curvature), Euclidean geometry (zero curvature), and non-Euclidean geometry (negative curvature) although Riemann did not refer to it by that name.

Extract: a lengthy series of passages from Riemann’s Lecture of 1854 (published as his (1867)).

Jeremy Gray

### 19. Differential Geometry of Surfaces

The basic techniques in the differential geometry of surfaces are developed, including maps of surfaces onto a plane. Beltrami’s

Saggio

(1868) and

Teoria

(1870) are discussed.

Extract: a lengthy series of passages from Beltrami’s

Saggio

.

Jeremy Gray

### 20. Beltrami, Klein, and the Acceptance of Non-Euclidean Geometry

Eugenio Beltrami’s presentation of non-Euclidean geometry began the successful reception of non-Euclidean geometry. The influence of Gauss is another factor, but Kantian philosophy, contrary to what the historian Roberto Bonola suggested in 1906, was probably not a strong influence. In two major papers of 1871 and 1873, and also in his Erlangen program (1872), Felix Klein unified most of the existing geometries, including non-Euclidean geometry, by showing that they were special cases of projective geometry. Klein’s approach to non-Euclidean geometry is described, using his extension of the idea of Cayley metric. The influence of the Erlangen program seems to have been less than some mathematical historians have thought. The chapter ends with the Weierstrass-Killing hyperboloid model of non-Euclidean geometry, and a comparison of it with the Beltrami model.

Jeremy Gray

### 21. On Writing the History of Geometry – 2

This chapter discusses how to write essays in the history of mathematics, specifically to explain selections from Cremona’s

Elements of projective geometry

, Lobachevskii’s

Geometrical researches on the theory of parallels

, and Salmon’s

A treatise on the higher plane curves

, all available in the Digital Mathematics Library.

Jeremy Gray

### 22. Projective Geometry as the Fundamental Geometry

Projective geometry became regarded by the mid 19th century as the fundamental geometry. This was very much the view of the English mathematicians Arthur Cayley, James Joseph Sylvester, and Henry Smith, and of George Salmon in Ireland, as it was of the Italian mathematician Luigi Cremona, whose book

Elementi di geometria projettiva

may be the first to have given the subject its present name. We look briefly at this book, concentrating on Cremona’s treatment of duality. Then we look at the disquiet over the foundations of geometry that were addressed by Moritz Pasch in his

Vorlesungen über neuere Geometrie

(1882), who attempted to build elementary geometry axiomatically from ideas laid down empirically by Hermann von Helmholtz. Helmholtz, whose acceptance of non-Euclidean geometry was influential, based his approach on an analysis of what is required for a geometry to capture the idea of the free mobility of figures in space.

Jeremy Gray

### 23. Hilbert and his Grundlagen der Geometrie

After a brief biography of David Hilbert, we look at how he came to take up elementary geometry and to write his

Grundlagen der Geometrie

(1899). In this book he gave a careful exposition of axiom systems for elementary geometry that, in Hurwitz’s opinion, created the subject of axiomatics. Hilbert was able to show how different geometries may be studied and found to be inter-related, and how they may be established rigorously in terms of arithmetic. His book ran to 10 editions and opened the way to the axiomatic method in other branches of mathematics. The place of Desargues’ theorem in projective geometry is considered in some detail, using Forest Ray Moulton’s presentation (adopted by Hilbert in the 2nd and subsequent editions).

More extracts from Hilbert’s

Grundlagen der Geometrie

are given at the end of Chapter 28.

Jeremy Gray

### 24. The Foundations of Projective Geometry in Italy

Italian views on the foundations of projective geometry were divided between the school of Giuseppe Peano on the one hand, and that of Corrado Segre on the other. Peano, supported by Mario Pieri, adhered to a rigorous formal approach written in a dry artificial language and restricted for pedagogic reasons to 3 dimensions; Segre and Federigo Enriques were less precise but wished to promote the projective geometry of

n

dimensions, following the example of Giuseppe Veronese, who had earlier shown how to use the method of projection and section to resolve the singularities of some algebraic surfaces.

Jeremy Gray

### 25. Henri Poincaré and the Disc Model of non-Euclidean Geometry

After a brief biography of Henri Poincaré we look at his route to his discovery of non-Euclidean geometry and Fuchsian functions in 1880. His disc model of non-Euclidean geometry closely resembles Riemann’s; he himself explained how it is related to Beltrami’s. We look at the correspondence between Poincaré and Klein in 1881–1882, and a long series of exercises develops the basic properties of the Poincaré disc.

Jeremy Gray

### 26. Is the Geometry of Space Euclidean or Non-Euclidean?

Is space Euclidean or non-Euclidean? This question was much discussed around 1900, and we look at Poincaré’s surprising answer that it will be impossible to tell. This is derived from his philosophy of conventionalism, which is conveyed through extensive extracts from his popular essays.

Jeremy Gray

### 27. Summary: Geometry to 1900

A short summary of the state of geometry around 1900 in the light of the preceding chapters.

Jeremy Gray

### 28. What is Geometry? The Formal Side

Ernest Nagel argued in 1939 that the principle of duality, by putting points and lines on a par, was an important stimulus for abstract, non-intuitive geometry, because it is intuitive that a plane is made up of points but not that it is made up of lines. One can go further: projective geometry made up spaces whose elements (points) were curves somewhere else (the space of all conic sections, for example).

More extracts are given from Hilbert’s

Grundlagen der Geometrie

.

Jeremy Gray

### 29. What is Geometry? The Physical Side

This largely concentrates on Albert Einstein’s special theory of relativity and some of its famous paradoxes, but it also considers his work on the rotating disc (as a model of a certain kind of a gravitational field with a geometric model) and concludes with a lengthy extract on Riemannian differential geometry from his

Relativity: The special and general theory

(1916).

Jeremy Gray

### 30. What is Geometry? Is it True? Why is it Important?

We look briefly at truth versus proof in geometry, at the controversy between Gottlob Frege and Hilbert, and at the idea of a relative consistency proof.

Extract: Poincaré on the relative consistency of Euclidean and non-Euclidean geometry.

Jeremy Gray

### 31. On Writing the History of Geometry – 3

This is the third and last of the chapters on how to write the history of mathematics, and considers how one might the events in the history of geometry in the 19th century and their importance by around 1900.

Jeremy Gray

### Backmatter

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