A Survey of the Development of Geometry up to 1870∗ Eldar Straume Department of mathematical sciences Norwegian University of Science and Technology (NTNU) N-9471 Trondheim, Norway September 4, 2014 Abstract This is an expository treatise on the development of the classical ge- ometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a ”true” geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective. arXiv:1409.1140v1 [math.HO] 3 Sep 2014 Contents 1 Euclideangeometry,thesourceofallgeometries 3 1.1 Earlygeometryandtheroleoftherealnumbers . 4 1.1.1 Geometric algebra, constructivism, and the real numbers 7 1.1.2 Thedownfalloftheancientgeometry . 8 ∗This monograph was written up in 2008-2009, as a preparation to the further study of the early geometrical works of Sophus Lie and Felix Klein at the beginning of their career around 1870. The author apologizes for possible historiographic shortcomings, errors, and perhaps lack of updated information on certain topics from the history of mathematics. Comments or corrections from the reader are most welcome, and may contribute to an improved later edition. 1 1.1.3 The ancient geometry: Its failures and its final algebraiza- tion .............................. 9 1.2 The decline of pure geometry and rise of analytic geometry . 11 1.3 Theadventofthenewgeometries. 13 2 Classicaldifferentialgeometry 16 2.1 Curvesandtheirgeometricinvariants . 16 2.2 Surfacesandtheircurvatureinvariants . 19 2.2.1 TheGaussianapproach . 20 2.2.2 French and Italian response to the Gaussian approach . 25 2.2.3 Curves on a surface and the Gauss-Bonnet theorem . 26 2.3 Surfaces in classical analytic geometry . 28 2.3.1 Ruled and developable surfaces . 28 2.3.2 Minimalsurfaces ....................... 31 2.3.3 Dupin’s cyclides and some related topics . 33 2.3.4 Pseudosphericalsurfaces. 35 2.3.5 Coordinate geometry, triply orthogonal systems, and gen- eralizedcyclides ....................... 36 2.4 Riemann and the birth of modern differential geometry . 38 2.4.1 Riemann’s lecture and his metric approach to geometry . 39 2.4.2 The beginning of modern differential geometry . 42 3 Projective geometry 43 3.1 Theoriginsofprojectivegeometry . 43 3.1.1 Developmentsinthe 16thand17thcentury . 44 3.1.2 Euler’s affine geometry and Monge’s descriptive geometry 46 3.2 The riseofprojectivegeometryin the 19thcentury . 47 3.2.1 Poncelet and the creation of projective geometry . 48 3.2.2 The Principle of Continuity and the Principle of Duality . 50 3.2.3 Thecross-ratioandharmonicsets . 52 3.3 Theanalystandthesynthesist . 54 3.3.1 Poncelet,Gergonne,andChasles . 54 3.3.2 M¨obius,Pl¨ucker,andSteiner . 56 3.3.3 Steiner,vonStaudt,andCremona . 58 3.4 Somebasicanalyticdevelopments. 63 3.4.1 Algebraization and homogeneous coordinates . 64 3.4.2 M¨obius and his approach to duality . 68 3.4.3 Classical algebraic geometry and Pl¨ucker’s formulae for planecurves.......................... 70 3.4.4 “Projective geometry is all geometry” (Cayley 1858) . 74 3.5 Linegeometry............................. 77 3.5.1 Raysystemsandfocalsurfaces . 77 3.5.2 Basic ideas and definitions . 78 3.5.3 Pl¨ucker’snewappoach. 80 2 4 Non-Euclidean geometry 82 4.1 The discoveries of Bolyai, Lobachevsky, and Gauss . 83 4.2 Absolute geometry and the Euclidean parallel postulate . 84 4.3 Attemptedproofsoftheparallelpostulate . 85 4.4 Theemergenceofnon-Euclideangeometry . 88 4.4.1 The trigonometry of Lambert, Schweikart, and Taurinus . 88 4.4.2 OntheworkofBolyaiandLobachevsky . 90 4.4.3 Beltrami and his model of non-Euclidean geometry . 92 5 The classical geometries: Euclidean, spherical, and hyperbolic 94 5.1 A unified view of the three classical geometries . 94 5.2 The conception of higher-dimensional geometry . 96 5.3 Foundations of geometryand the geometryof Space . 98 5.4 Riemann–Helmholtz–Lie space problem . 99 1 Euclidean geometry, the source of all geome- tries At the end of the 18th century, the notion of geometry was largely synony- mous with Euclidean geometry, namely the classical Greek geometry which had prevailed for more than 2000 years. In the 17th century, Kepler, Galileo and Newton were leading figures in the Copernican revolution which had paved the way for the birth of modern science, and moreover, which finally abandoned the long lasting doctrines and the supreme authority of Aristotle among the scholars. Certainly, the 17th century was also a century of great advances in mathematics, such as the rise of calculus, but still the basic role of Euclidean geometry was not challenged. Indeed, in his famous work Principia Mathemat- ica (1687) Newton was careful to recast his demonstrations in geometric terms, and analytical calculations are almost completely missing. In particular, New- ton formulated his basic laws for the universe in the framework of Euclidean geometry. However, during the 18th century another dominating authority had estab- lished himself in the intellectual world, namely the German philosopher Im- manuel Kant (1724–1804). He maintained that Euclidean geometry was the only absolute geometry, known to be true a priori in our mind as an inevitable necessity of thought, and no other geometry was thinkable. Moreover, Kant regarded the Newtonian universe as the true model of the physical space, sup- ported by our endowed intuition about space and time, and independent of experience. In reality, the Kantian doctrine on the nature of space and geom- etry hampered the development of science, until the outburst of the inevitable geometric ”revolution” in the 19th century led to the discovery of non-Euclidean geometries and radically changed geometry as a mathematical science. 3 1.1 Early geometry and the role of the real numbers Geometry is in fact encountered in the first written records of mankind. But what are the origins of the Euclidean geometry? In his truly remarkable work, organized into 13 books usually referred to as Euclid’s Elements, Euclid (325– 265 BC) presented a large part of the geometric and algebraic knowledge of Babylonian, Egyptian and Greek scholars at his time. He did this in a deduc- tive style, which has become known as the axiomatic method of mathematics. Geometry as developed in the Elements is usually referred to as synthetic geom- etry. The basic undefined geometric objects are points, lines, planes, whereas notions such as line segments, angles and circles can be defined, and moreover, the undefined notion of congruence expresses ”equality” among them. Start- ing from five basic postulates and five common notions (or rules of logic), a logical chain consisting of 465 propositions were deduced and presented in the Elements. For the convenience of the reader, let us be more specific and state the postulates — somewhat modernized— as follows: E1 : There is a unique line passing through any two distinct points. E2 : Any segment on a line may be extended by any given segment. E3 : For every point O and every other point P , there is circle with center O and radius OP. E4 : All right angles are congruent to each other. E5 : If a line intersects two other lines so that the two interior angles on one side of it are together less than two right angles, then the two lines will meet at a point somewhere on that side. The ancient geometry flourished for about 1000 years, primarily in Greece and Alexandria in Egypt. Besides Euclid some of the prominent men from this epoch are Thales of Miletus (ca. 600 BC), Pythagoras (585–501 BC), Plato (429–348 BC), Eudoxus of Cnidus (408–355? BC), Aristotle (384–322 BC), Archimedes of Syracuse (287–212 BC), Appolonius of Perga (262–190 BC), and Pappus of Alexandria (290–350 AD). The Pythagorean school played a crucial role, developing number theory which they also linked to geometry, number mystics and music theory. In fact, Pythagoras himself is said to have coined the words ”philosophy” and ”mathematics”. The Platonic Academy in Athens became the mathematical center of the world and, for example, Aristotle and Eudoxus had been students at this academy. Many ot the ancient mathematical texts have been lost, including advanced works by Euclid and Appolonius. For example, Appolonius introduced the names ellipse, parabola and hyperbola in his famous treatise Conic Sections, but some of the volumes are irretrievably lost. Moreover, Euclid’s three-volume work called the Porisms, apparently on advanced geometry, is also lost. However, the Elements of Euclid have survived throught the centuries by manuscript copies which have transmitted the geometric truths to new generations, and with minor 4 changes. Through Euclids’s authority, the Elements teach geometry with a definite approach, but with no motivation even of the most sophisticated terms. Moreover, previous or alternative approaches, which could have filled some of the gaps, are ignored. The first printed edition was in 1482, and over thousand editions have appeared since then. In effect, the Elements have remained in use as a textbook practically unchanged for more than 2000 years, and up to modern times the first six books have served as the student’s usual introduction to geometry. It is somewhat surprising that Euclid did not use numbers in his geome- try. Namely, the geometric objects such as line segments, angles, areas etc.