# Geometry

## Modern Era

Significant Western interest in mathematics ebbed for a long time during and after the Roman Empire, before flowing at times in the Middle Ages. Only in the sixteenth century did a continual process of growth begin, aided by the rediscovery of Greek and Arabic texts and the publication of editions of Euclid's *Elements* and the works of Apollonius and Archimedes. At the same time, the discovery of the method of single-focused perspective transformed first architecture and then the practice of painting, where it produced a dramatically heightened realism. The technique proved eminently teachable, although few painters apart from Piero della Francesca (c. 1420–1492) also understood the mathematics involved.

*Analytic and projective geometry.*

Girard Desargues (1591–1661) brought together projective ideas from both architecture and painting to create the first fully unified theory of conic sections (all nondegenerate conic sections are projections of a circle). This theory naturally highlights those aspects that are projective (such as tangency questions) and it led directly or indirectly to a number of novel discoveries over the next century before it petered out. It was then rediscovered by Gaspard Monge (1746–1818) and Jean-Victor Poncelet (1788–1867) at the time of the French Revolution. In the form of simple horizontal and vertical projections it became the core technique of descriptive geometry or engineering drawing, a mainstay of French mathematical education throughout the nineteenth century, and, of course, it is still in use in the early twenty-first century.

Poncelet's breakthrough at the start of the nineteenth century was to see that, for many geometric properties a curve is equivalent to any of its "shadows" (its images under central projection). His own way of doing this was not found to be acceptable by later mathematicians, but Michel Chasles (1793–1880) in France and August Ferdinand Möbius (1790–1868) and Julius Plücker (1801–1868) in Germany all independently found more rigorous ways of making his insight work, and the resulting subject of projective geometry became the fundamental geometry of the nineteenth century. Although the details remained obscure for some time, the key point was that projective geometry discussed geometric properties of figures that do not involve the concept of distance. Any theorem in projective geometry is true in Euclidean geometry, but not vice versa, and so projective geometry is more basic than Euclidean geometry.

Desargues's contemporary, René Descartes (1596–1650) was much more successful with a work that was ruthlessly modern in its approach, and entirely eclipsed earlier attempts. Descartes took contemporary algebra, rewrote it in simpler notation, and proceeded to solve geometric problems by recasting them in algebraic terms and solving them by algebraic means, then reinterpreting the solution in geometric terms. Typically, the algebraic solution will be a single equation between two unknowns. Descartes interpreted this as defining a curve, and gave an elaborate discussion of how, given an equation, the corresponding curve can be drawn. He published his ideas as an appendix (entitled *La géométrie*) to a longer philosophical work in 1637.

Descartes did not explain the more elementary parts of his approach. This was done by a number of Dutch scholars who came after him, and the study of geometry by means of algebra (Cartesian, analytic, or coordinate geometry) was swiftly established. It took about a century for mathematicians to decide that the algebraic equation was an acceptable answer to a geometric problem, and to drop Descartes's search for geometric constructions, but the idea that geometric figures are naturally and fruitfully described by algebra has remained central to much of mathematics ever since.

*Differential geometry.*

Differential geometry, on the other hand, began as the study of curves and surfaces where the calculus is allowed. It is connected to such questions as: when a map of the earth's surface (assumed to be a sphere) is made on a plane, what geometric properties can be preserved? The decisive reformulation of the early nineteenth century came when Carl Friedrich Gauss (1777–1855) investigated the curvature of surfaces in space. The curvature of a surface at a point (and generally the curvature of a surface varies from point to point) is a measure of the best fitting sphere, plane, or saddle at that point (see figs. 2 and 3).

Gauss found that this quantity is intrinsic: it can be determined by measurements taken in the surface itself without reference to the ambient Euclidean space. This property was so unexpected Gauss called the result an exceptional theorem.

*Gaussian curvature and the emergence of non-Euclidean geometries.*

After Gauss's death it emerged that he, alone of the mathematicians of his time, had had some sympathy with efforts to show that Euclidean geometry was not the only possible geometry of space, and indeed his astronomer friends Friedrich Wilhelm Bessel (1784–1846) and Heinrich Wilhelm Matthäus Olbers (1758–1840) had also inclined that way. This leads back to the question of the parallel postulate.

Around 1830 János Bolyai (1802–1860) in present-day Hungary and Nicolai Ivanovich Lobachevsky (1792–1856) in remote Kazan in Russia, wrote down and published detailed accounts of what a geometry would look like in which the parallel postulate was false and the angle sum of a triangle is always less than two right angles (reprinted in English translation in Bonola, 1912). Although independent, their work is remarkably similar and can be described together. They studied geometry in two and three dimensions, and found formulas for triangles in the plane analogous to the formulas of spherical trigonometry for triangles on the sphere. These new formulas showed that small regions of the new geometry differed only slightly from small regions of the Euclidean plane, thus explaining why the new geometry had not been noticed hitherto, but they also showed what many a previous defence of the parallel postulate had hinted at—that the new geometry was different from Euclidean geometry in many respects.

Such was the novelty of Bolyai's and Lobachevsky's work that few read it and the published responses to it were extreme in their hostility. Most people instinctively found it incoherent; they "knew" it was wrong but were not willing to find out where. Gauss, however, wrote to Bolyai to say that he agreed with János's presentation but implying that he knew it all already (a claim for which there is little surviving evidence). János was so enraged he never published again. In 1840 Gauss nominated Lobachevsky to the Göttingen Academy of Sciences, but did nothing else to promote the new geometry. The result was that both men died without getting the acclaim their discoveries undoubtedly merited.

*Riemann's generalization for spaces of higher dimensions.*

The hegemony of Euclidean geometry came to an end not with the discoveries of Bolyai and Lobachevsky, but in stages, starting with Riemann's wholly novel approach to geometry that severely undercut it. Bernhard Riemann (1826–1866) was a student of mathematics at the University of Göttingen in the mid-1850s. In his postdoctoral thesis he set out the view that geometry was the study of any "space" of points upon which one could talk about lengths, and he indicated a variety of ways in which the techniques of the calculus could do such service. This is a rather natural and elementary idea, the problems come in spelling out the details in any useful way.

Riemann concentrated on intrinsic properties of the space, such as Gauss's idea of the curvature of a surface, and he noted that there would be different geometries on spaces with different intrinsic properties. That includes spaces of different dimensions, and also spaces of dimension two, say, but different curvatures. What it does not do is say that Euclidean space of some dimension is the source of geometric concepts, thus Euclidean geometry is overthrown.

Riemann's thesis was published posthumously in 1867, just in time to resolve the doubts of an Italian mathematician, Eugenio Beltrami (1835–1899), who had come to some of the same ideas. He immediately published his reworking of the geometry of Bolyai and Lobachevsky as the geometry of a surface of constant negative curvature, of which he had a description in a disc of unit (Euclidean) radius. Beltrami's map and Riemann's philosophy of geometry convinced mathematicians, but not all philosophers, of the validity of non-Euclidean geometry, as the Bolyai-Lobachevsky geometry became known.

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