Non-Euclidean geometry

two geometries based on axioms closely related to those specifying Euclidean geometry

non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. This article contains a variety of entries focusing on the history and development of the subject.

Lines with a common perpendicular in 3 types of geometry


  • There was a period when cosmology got started. There were some important works in the 30s—the Einstein-Infeld-Hoffman ideas [equations]. ...Unified Field theories were the bane of GR in those days. Einstein... was convinced that physics should be primarily geometry... about 10 years later, maybe 15, Steven Weinberg was convinced that geometry was irrelevant... the important stuff is just field theory. ...Weinberg, later... collaborated in proving that physics really is geometry. Except not the geometry of space-time... it's the geometry of the graph paper on which the properties of space-time are conceptually plotted... the idea of a curved connection. If you want to plot... any physical quantity... like a magnetic field, quarks, gluons, etc. you need to plot it on curved graph paper. But Einstein... didn't have that broad an idea of geometry...
    • Ignazio Ciufolini, Richard A. Matzner, General Relativity and John Archibald Wheeler (2010) p. 23.
  • [The] empirical origin of Euclid's geometrical axioms and postulates was lost sight of, indeed was never even realised. As a result, Euclidean geometry was thought to derive its validity from certain self-evident universal truths; it appeared as the only type of consistent geometry of which the mind could conceive. Gauss had certain misgivings on the matter, but... the honor of discovering non-Euclidean geometry fell to Lobatchewski and Bolyai. ...
    From the difference in geometric premises important variations followed. Thus, whereas in Euclidean geometry the sum of the angles of any triangles is always equal to two right angles, in non-Eudlidean geometry the value of this sum varies with the size of the triangles. It is always less than two right angles in Lobatchewski's, and always greater in Riemann's. Again, in Euclidean geometry, similar figures of various sizes can exist; in non-Euclidean geometry, this is impossible.
    It appeared then, that the universal truth formerly credited to Euclidean geometry would have to be shared by these two other geometrical doctrines. But truth, when divested of its absoluteness, loses much of its significance, so this co-presence of conflicting universal truths brought the realisation that a geometry was true only in relation to our more or less arbitrary choice of a system of geometrical postulates. ...The character of self-evidence which had been formerly credited to the Euclidean axioms was seen to be illusory.
    • A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927) pp. 35-36
  • The decisive steps toward a clear understanding of non-Euclidean geometry were taken by Riemann, Helmholtz, and Poincaré, who recognized the essential unity of geometry and physics. However, the understanding did not come into its own until Einstein showed that such a combination of geometry and physics was really necessary for the derivation of phenomena which had actually been observed.
    • Philipp Frank, Philosophy of Science: The Link Between Science and Philosophy (1957)
  • Selecting the  -axis as an axis of revolution, a point on the surface generated by rotating the curve   is defined by two coordinates...   and  . ...Now   where   is the displacement along the meridian and   the displacement along the the parallel of latitude. ...since   ...The [arbitrary] line element   is... defined by the relation
    The line element   is thus defined by the relation:
    This is the first of the generalized forms of equations in curved surface theory in which   and   are parameters. ...
    For a generalized curved surface with an arbitrarily selected orthoganal coordinate system defined by the coordinates   and  , eq. (1.1) assumes the generalized form
    ...the coefficients will now be functions of   and  . We may again write:
    Equations (1.1) and (1.3) are of great importance in the theory of curved surfaces and hence in comprehending shell theory. By means of these equations the geometry of the surface is described as a two-dimensional configuration similar to the method used to define a point on a flat surface, i.e. two normalized orthogonal coordinates. ...If a set of orthogonal coordinates can be selected such that   and   are independent of   and  , the geometry in the neighborhood of a point on the curved surface does not differ from that of a flat plate. Then the cartesian-coordinate relationship:
    is still valid.
    Developable surfaces include figures such as cones or this cylinder.
    This classification includes the developable surfaces such as the cone and the cylinder. ...the distance between two points on the surface does not change in the development.
    For that reason, when a curved surface defined by the generalized equation, eq. (1.3), can be reduced by using a suitable set of coordinates   and   to the form of eq. (1.4) with   and   constant, the so-called conditions of euclidean geometry will be satisfied. ...When it becomes impossible to select   and   coordinates for which   and   are constant, the geometry of the curved surface becomes different from from that of a flat surface... eq. (1.4), is no longer valid and a non-euclidean geometry must be applied. Such surfaces are not developable, i.e. they cannot be folded out into a flat surface under the condition that any line element   remains invariant. This class of surfaces includes the sphere, the ellipsoid, the paraboloid and the hyperboloid.
  • Let us then examine the extension of this universe to ascertain whether there exists there an infinitely great. The opinion that the world was infinite was a dominant idea for a long time. Up to Kant and even afterward, few expressed any doubt in the infinitude of the universe.
    Here too modern science, particularly astronomy, raised the issue anew and endeavored to decide it not by means of inadequate metaphysical speculations, but on grounds which rest on experience and on the application of the laws of nature. There arose weighty objections against the infinitude of the universe. It is Euclidean geometry which leads to infinite space as a necessity. ...Einstein showed that Euclidean geometry must be given up. He considered this cosmological question too from the standpoint of his gravitational theory and demonstrated the possibility of a finite world; and all the results discovered by the astronomers are consistent with this hypothesis of an elliptic universe.
  • He dwells only on broad impressions of vast angles and stone surfaces—surfaces too great to belong to any thing right or proper for this earth, and impious with horrible images and hieroglyphs. I mention his talk about angles because it suggests something Wilcox had told me of his awful dreams. He had said that the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours.
  • Although K. F. Gauss, one if the spiritual fathers of non-Euclidean geometry... proposed a possible test of the flatness of space by measuring the interior angles of a terrestrial triangle, it remained for... K. Schwarzschild to formulate the procedure and to attempt to evaluate [curvature]   on the basis of astronomical data... Schwarzschild's pioneer attempt is so inspiring in its conception and so beautiful in its expression...[!]
  • Howard P. Robertson, "Geometry as a Branch of Physics" (1949) from Albert Einstein: Philosopher-Scientist, ed. Paul Arthur Schilpp.
  • In the decades leading up to the period of relativity theory the architecture of space was revolutionized. Until then the mathematical imagination, and with it all of scientific thinking, had been dominated by a single book. ...Yet the mathematical framework the Elements espoused grants an unfounded privilege to one view, excluding the very idea of non-Euclidean geometries. The roots of a more flexible attitude to geometry reach back to the Renaissance creators of linear perspective, but the development... into the modern discipline... had to await the... great mathematicians such as Poncelet, Cayley and Klein. By the time of Einstein, non-Euclidean geometries and the even more comprehensive theory of projective geometry had broken the grip of Euclid on mathematical and spatial thinking, and a new imagination of space could be born.
    • Arthur Zajonc, Catching the Light: The Entwined History of Light and Mind (1993)
Florian Cajori, 2nd edition (1919).
  • In geometry the axioms have been searched to the bottom, and the conclusion has been reached that the space defined by Euclid's axioms is not the only possible non-contradictory space. Euclid proved (I, 27) that "if a straight line falling on two other straight lines make the alternate angles equal to one another, the two straight lines shall be parallel to one another." Being unable to prove that in every other case the two lines are not parallel, he assumed this to be true in what is now generally called the 5th "axiom," by some the 11th or the 12th "axiom."
    Simpler and more obvious axioms have been advanced as substitutes. As early as 1663, John Wallis of Oxford recommended: "To any triangle another triangle, as large as you please, can be drawn, which is similar to the given triangle." G. Saccheri assumed the existence of two similar, unequal triangles. Postulates similar to Wallis' have been proposed also by J. H. Lambert, L. Carnot, P. S. Laplace, J. Delboeuf. A. C. Clairaut assumes the existence of a rectangle; W. Bolyai postulated that a circle can be passed through any three points not in the same straight line, A. M. Legendre that there existed a finite triangle whose angle-sum is two right angles, J. F. Lorenz and Legendre that through every point within an angle a line can be drawn intersecting both sides, C. L. Dodgson that in any circle the inscribed equilateral quadrangle is greater than any one of the segments which lie outside it. But probably the simplest is the assumption made by Joseph Fenn in his edition of Euclid's Elements, Dublin, 1769, and again sixteen years later by William Ludlam... and adopted by John Playfair: "Two straight lines which cut one another can not both be parallel to the same straight line." It is noteworthy that this axiom is distinctly stated in Proclus's note to Euclid, I, 31.
  • The most numerous efforts to remove the supposed defect in Euclid were attempts to prove the parallel postulate. After centuries of desperate but fruitless endeavor, the bold idea dawned upon the minds of several mathematicians that a geometry might be built up without assuming the parallel-axiom. While A. M. Legendre still endeavored to establish the axiom by rigid proof, Lobachevski brought out a publication which assumed the contradictory of that axiom, and which was the first of a series of articles destined to clear up obscurities in the fundamental concepts, and greatly to extend the field of geometry.
  • Nicholaus Ivanovich Lobachevski['s]... views on the foundation of geometry were first set forth in a paper laid before the physico-mathematical department of the University of Kasan in February, 1826. This paper was never printed and was lost. His earliest publication was in the Kasan Messenger for 1829 and then in the Gelehrte Schriflen der Universtät Kasan, 1836-1838... "New Elements of Geometry, with a complete theory of Parallels." ...remained unknown to foreigners, but even at home it attracted no notice. In 1840 he published a brief statement of his researches in Berlin, under the title Geometrische Untersuchungen zur Theorie der Parallellinien. Lobachevski constructed an "imaginary geometry," as he called it, which has been described by W. K. Clifford as "quite simple, merely Euclid without the vicious assumption." A remarkable part of this geometry is this, that through a point an indefinite number of lines can be drawn in a plane, none of which cut a given line in the same plane. A similar system of geometry was deduced independently by the Bolyais in Hungary, who called it "absolute geometry."
  • Wolfgang Bolyai de Bolya... after studying at Jena... went to Göttingen, where he became intimate with K. F. Gauss, then nineteen years old. Gauss used to say that Bolyai was the only man who fully understood his views on the metaphysics of mathematics. Bolyai became professor at the Reformed College of Maros-Vásárhely, where for forty-seven years he had for his pupils most of the later professors of Transylvania. ...he was truly original in his private life as well as in his mode of thinking. ...No monument, said he, should stand over his grave, only an apple-tree, in memory of the three apples; the two of Eve and Paris, which made hell out of earth, and that of I. Newton, which elevated the earth again into the circle of heavenly bodies. His son, Johann Bolyai... once accepted the challenge of thirteen officers on condition that after each duel he might play a piece on his violin, and he vanquished them all.
  • The chief mathematical work of Wolfgang Bolyai appeared in two volumes, 1832-1833 entitled Tentamen juventutem studiosam in elementa matheseos puræ... introducendi. It is followed by an appendix composed by his son Johann. Its twenty-six pages make the name of Johann Bolyai immortal. He published nothing else but he left behind one thousand pages of manuscript.
  • While Lobachevski enjoys priority of publication, it may be that Bolyai developed his system somewhat earlier. Bolyai satisfied himself of the non-contradictory character of his new geometry on or before 1825; there is some doubt whether Lobachevski had reached this point in 1826. Johann Bolyai's father seems to have been the only person in Hungary who really appreciated the merits of his son's work. For thirty-five years this appendix, as also Lobachevski's researches, remained in almost entire oblivion. Finally Richard Baltzer of the University of Giessen, in 1867, called attention to the wonderful researches.
  • In 1866 J. Hoüel translated Lobachevski's Geometrische Unter suchungen into French. In 1867 appeared a French translation of Johann Bolyai's Appendix. In 1891 George Bruce Halsted, then of the University of Texas, rendered these treatises easily accessible to American readers by translations brought out under the titles of J. Bolyai's The Science Absolute of Space and N. Lobachevski's Geometrical Researches on the Theory of Parallels of 1840.
  • A copy of the Tentamen reached K. F. Gauss, the elder Bolyai's former roommate at Göottingen, and this Nestor of German mathematicians was surprised to discover in it worked out what he himself had begun long before, only to leave it after him in his papers. As early as 1792 he had started on researches of that character. His letters show that in 1799 he was trying to prove a priori the reality of Euclid's system; but some time within the next thirty years he arrived at the conclusion reached by Lobachevski and Bolyai. In 1829 he wrote to F. W. Bessel, stating that his "conviction that we cannot found geometry completely a priori has become, if possible, still firmer," and that "if number is merely a product of our mind, space has also a reality beyond our mind of which we cannot fully foreordain the laws a priori." The term non-Euclidean geometry is due to Gauss.
  • It is surprising that the first glimpses of non-Euclidean geometry were had in the eighteenth century. Geronimo Saccheri... a Jesuit father of Milan, in 1733 wrote Euclides ab omni naevo vindicatus (Euclid vindicated from every flaw). Starting with two equal lines AC and BD, drawn perpendicular to a line AB and on the same side of it, and joining C and D, he proves that the angles at C and D are equal. These angles must be either right, or obtuse, or acute. The hypothesis of an obtuse angle is demolished by showing that it leads to results in conflict with Euclid I, 17: Any two angles of a triangle are together less than two right angles. The hypothesis of the acute angle leads to a long procession of theorems, of which the one declaring that two lines which meet in a point at infinity can be perpendicular at that point to the same straight line, is considered contrary to the nature of the straight line; hence the hypothesis of the acute angle is destroyed. Though not altogether satisfied with his proof, he declared Euclid "vindicated."
    • Footnote: See English translation of Euclides ab omni naevo vindicatus by G. B. Halsted in Am. Math. Monthly, Vols. 1-5, 1894-1898.
  • J. H. Lambert... in 1766 wrote a paper "Zur Theorie der Parallellinien," published in the Leipziger Magazin für reine und angewandte Mathematik, 1786, in which: (1) The failure of the parallel-axiom in surface spherics gives a geometry with angle-sum > 2 right angles; (2) In order to make intuitive a geometry with angle-sum < 2 right angles we need the aid of an "imaginary sphere" (pseudo-sphere); (3) In a space with the angle-sum differing from 2 right angles, there is an absolute measure (Bolyai's natural unit for length). Lambert arrived at no definite conclusion on the validity of the hypotheses of the obtuse and acute angles.
  • Among the contemporaries and pupils of K. F. Gauss, three deserve mention as writers on the theory of parallels, Ferdinand Karl Schweikart... professor of law in Marburg, Franz Adolf Taurinus... a nephew of Schweikart, and Friedrich Ludwig Wachter... a pupil of Gauss in 1809 and professor at Dantzig. Schweikart sent Gauss in 1818 a manuscript on "Astral Geometry" which he never published, in which the angle-sum of a triangle is less than two right angles and there is an absolute unit of length. He induced Taurinus to study this subject. Taurinus published in 1825 his Theorie der Parallellinien in which he took the position of Saccheri and Lambert, and in 1826 his Geometriæ prima elementa, in an appendix of which he gives important trigonometrical formulæ for non-Euclidean geometry by using the formulæ of spherical geometry with an imaginary radius. His Elementa attracted no attention. In disgust he burned the remainder of his edition. Wachter's results are contained in a letter of 1816 to Gauss and in his Demonstratio axiomatis geometrici in Euclideis undecimi, 1817. He showed that the geometry on a sphere becomes identical with the geometry of Euclid when the radius is infinitely increased, though it is distinctly shown that the limiting surface is not a plane.
  • The researches of K. F. Gauss, N. I. Lobachevski and J. Bolyai have been considered by F. Klein as constituting the first period in the history of non-Euclidean geometry. It is a period in which the synthetic methods of elementary geometry were in vogue. The second period embraces the researches of G. F. B. Riemann, H. Helmholtz, S. Lie and E. Beltrami, and employs the methods of differential geometry.
  • It was in 1854 that Gauss heard from his pupil, Riemann, a marvellous dissertation which considered the foundations of geometry from a new point of view. Riemann was not familiar with Lobachevski and Bolyai. He developed the notion of n-ply extended magnitude, and the measure-relations of which a manifoldness of n dimensions is capable, on the assumption that every line may be measured by every other. Riemann applied his ideas to space. He taught us to distinguish between "unboundedness" and "infinite extent." According to him we have in our mind a more general notion of space, i.e. a notion of non-Euclidean space; but we learn by experience that our physical space is, if not exactly, at least to a high degree of approximation, Euclidean space. Riemann's profound dissertation was not published until 1867, when it appeared in the Göttingen Abhandlungen.
  • Before this, the idea of n dimensions had suggested itself under various aspects to Ptolemy, J. Wallis, D'Alembert, J. Lagrange, J. Plücker, and H. G. Grassmann. The idea of time as a fourth dimension had occurred to D'Alembert and Lagrange. About the same time with Riemann's paper, others were published from the pens of H. Helmholtz and E. Beltrami. This period marks the beginning of lively discussions upon this subject. Some writers—J. Bellavitis, for example—were able to see in non-Euclidean geometry and n-dimensional space nothing but huge caricatures, or diseased outgrowths of mathematics. H. Helmholtz's article was entitled Thatsachen, welche der Geometrie zu Grunde liegen, 1868, and contained many of the ideas of Riemann. Helmholtz popularized the subject in lectures, and in articles for various magazines. Starting with the idea of congruence, and assuming the free mobility of a rigid body and the return unchanged to its original position after rotation about an axis, he proves that the square of the line-element is a homogeneous function of the second degree in the differentials.
  • Helmholtz's investigations were carefully examined by S. Lie who reduced the Riemann-Helmholtz problem to the following form: To determine all the continuous groups in space which, in a bounded region, have the property of displacements. There arose three types of groups which characterize the three geometries of Euclid, of N. I. Lobachevski and J. Bolyai and of F. G. B. Riemann.
    • Footnote: Lie, Theorie der Transformationsgruppen, Bd. III, Leipzig, 1893, pp. 437-543; Roberto Bonola, Non-Euclidean Geometry trans. by H. S. Carslaw, Chicago, 1912, p. 154.
  • Beltrami wrote in 1868 a classical paper, Saggio di interpretazione della geometria non-euclidea (Giorn. di Matem., 6) which is analytical (and... should be mentioned elsewhere were we to adhere to a strict separation between synthesis and analysis). He reached the brilliant and surprising conclusion that in part the theorems of non-Euclidean geometry find their realization upon surfaces of constant negative curvature. He studied, also, surfaces of constant positive curvature, and ended with the interesting theorem that the space of constant positive curvature is contained in the space of constant negative curvature.
  • These researches of Beltrami, H. Helmholtz, and G. F. B. Riemann culminated in the conclusion that on surfaces of constant curvature we may have three geometries,—the non-Euclidean on a surface of constant negative curvature, the spherical on a surface of constant positive curvature, and the Euclidean geometry on a surface of zero curvature. The three geometries do not contradict each other, but are members of a system,—a geometrical trinity.
  • The ideas of hyper-space were brilliantly expounded and popularised in England by Clifford.
  • Beltrami's researches on non-Euclidean geometry were followed, in 1871, by important investigations of Felix Klein, resting upon Cayley's Sixth Memoir on Quantics, 1859. The development of geometry in the first half of the nineteenth century had led to the separation of this science into two parts: the geometry of position or descriptive geometry which dealt with properties that are unaffected by projection, and the geometry of measurement in which the fundamental notions of distance, angle, etc., are changed by projection. Cayley's Sixth Memoir brought these strictly segregated parts together again by his definition of distance between two points. The question whether it is not possible so to express the metrical properties of figures that they will not vary by projection (or linear transformation) had been solved for special projections by M. Chasles, J. V. Poncelet, and E. Laguerre, but it remained for A. Cayley to give a general solution by defining the distance between two points as an arbitrary constant multiplied by the logarithm of the anharmonic ratio in which the line joining the two points is divided by the fundamental quadric. These researches, applying the principles of pure projective geometry, mark the third period in the development of non-Euclidean geometry.
  • F. Klein showed the independence of projective geometry from the parallel-axiom, and by properly choosing the law of the measurement of distance deduced from projective geometry, the spherical, Euclidean, and pseudospherical geometries, named by him respectively, the elliptic, parabolic, and hyperbolic geometries. This suggestive investigation was followed up by numerous writers, particularly by G. Battaglini of Naples, E. d'Ovidio of Turin, R. de Paolis of Pisa, F. Aschieri, A. Cayley, F. Lindemann of Munich, E. Schering of Göttingen, W. Story of Clark University, H. Stahl of Tubingen, A. Voss of Munich, Homersham Cox, A. Buchheim.
    • Footnote: G. Loria, Die hauptsächlichsten Theorien der Geometrie 1888, p. 102.

History of Modern Mathematics (1896)

David Eugene Smith, source
  • The Non-Euclidean Geometry is a natural result of the futile attempts which had been made from the time of Proklos to the opening of the nineteenth century to prove the fifth postulate, (also called the twelfth axiom, and sometimes the eleventh or thirteenth) of Euclid. The first scientific investigation of this part of the foundation of geometry was made by Saccheri (1733), a work which was not looked upon as a precursor of Lobachevsky, however, until Beltrami (1889) called attention to the fact. Lambert was the next to question the validity of Euclid's postulate in his Theorie der Parallellinien (posthumous, 1786), the most important of many treatises on the subject between the publication of Saccheri's work and those of Lobachevsky and Bolyai. Legendre also worked in the field, but failed to bring himself to view the matter outside the Euclidean limitations.
    • pp. 565-566.
  • During the closing years of the eighteenth century Kant's doctrine of absolute space, and his assertion of the necessary postulates of geometry, were the object of much scrutiny and attack. At the same time Gauss was giving attention to the fifth postulate, though on the side of proving it. It was at one time surmised that Gauss was the real founder of the non-Euclidean geometry, his influence being exerted on Lobachevsky through his friend Bartels, and on Johann Bolyai through the father Wolfgang, who was a fellow student of Gauss's. But it is now certain that Gauss can lay no claim to priority of discovery, although the influence of himself and of Kant, in a general way, must have had its effect.
    • p. 566.
  • Bartels went to Kasan in 1807, and Lobachevsky was his pupil. The latter's lecture notes show that Bartels never mentioned the subject of the fifth postulate to him, so that his investigations, begun even before 1823, were made on his own motion and his results were wholly original. Early in 1826 he sent forth the principles of his famous doctrine of parallels, based on the assumption that through a given point more than one line can be drawn which shall never meet a given line coplanar with it. The theory was published in full in 1829-30, and he contributed to the subject... until his death.
    • p. 566.
  • Johann Bolyai received through his father, Wolfgang, some of the inspiration to original research which the latter had received from Gauss. When only twenty-one he discovered, at about the same time as Lobachevsky, the principles of non-Euclidean geometry, and refers to them in a letter of November, 1823. They were committed to writing in 1825 and published in 1832. Gauss asserts in his correspondence with Schumacher (1831-32) that he had brought out a theory along the same lines as Lobachevsky and Bolyai, but the publication of their works seems to have put an end to his investigations. Schweikart was also an independent discoverer of the non-Euclidean geometry, as his recently recovered letters show, but he never published anything on the subject, his work on the theory of parallels (1807), like that of his nephew Taurinus (1825), showing no trace of the Lobachevsky-Bolyai idea.
    • p. 567.
  • The hypothesis was slowly accepted by the mathematical world. Indeed, it was about forty years after its publication that it began to attract any considerable attention. ...
    Of all these contributions the most noteworthy from the scientific standpoint is that of Riemann. In his Habilitationsschrift (1854) he applied the methods of analytic geometry to the theory, and suggested a surface of negative curvature, which Beltrami calls "pseudo-spherical," thus leaving Euclid's geometry on a surface of zero curvature midway between his own and Lobachevsky's. He thus set forth three kinds of geometry, Bolyai having noted only two. These Klein (1871) has called the elliptic (Riemann's), parabolic (Euclid's), and hyperbolic (Lobachevsky's).
    • pp. 567-568.
  • There have contributed to the subject many of the leading mathematicians of the last quarter of a century, including... Cayley, Lie, Klein, Newcomb, Pasch, C. S. Peirce, Killing, Fiedler, Mansion, and McClintock. Cayley's contribution of his "metrical geometry" was not at once seen to be identical with that of Lobachevsky and Bolyai. It remained for Klein (1871) to show, this thus simplifying Cayley's treatment and adding one of the most important results of the entire theory. Cayley's metrical formulas are, when the Absolute is real, identical with those of the hyperbolic geometry; when it is imaginary, with the elliptic; the limiting case between the two gives the parabolic (Euclidean) geometry. The question raised by Cayley's memoir as to how far projective geometry can be defined in terms of space without the introduction of distance had already been discussed by von Staudt (1857) and has since been treated by Klein (1873) and by Lindemann (1876).

A Short History of Mathematics (1912)

by W. W. Rouse Ball
  • The question of the truth of the assumptions usually made in our geometry had been considered by J. Saccheri as long ago as 1773; and in more recent times had been discussed by N. I. Lobatschewsky of Kasan, in 1826 and again in 1840; by Gauss, perhaps as early as 1792, certainly in 1831 and in 1846; and by J. Bolyai in 1832 in the appendix to the first volume of his father's Tentamen; but Riemann's memoir of 1854 attracted general attention to the subject... and the theory has been since extended and simplified by various writers, notably A. Cayley... E. Beltrami... by H. L. F. von Helmholtz... by T. S. Tannery... by F. C. Klein... and by A. N. Whitehead... in his Universal Algebra. The subject is so technical that I confine myself to a bare sketch of the argument from which the idea is derived.
  • The Euclidean system of geometry, with which alone most people are acquainted, rests on a number of independent axioms and postulates. Those which are necessary for Euclid's geometry have, within recent years, been investigated and scheduled. They include not only those explicitly given by him, but some others which he unconsciously used. If these are varied, or other axioms are assumed, we get a different series of propositions, and any consistent body of such propositions constitutes a system of geometry. Hence there is no limit to the number of possible Non-Euclidean geometries that can be constructed.
  • Among Euclid's axioms and postulates is one on parallel lines, which is usually stated in the form that if a straight line meets two straight lines, so a to make the sum of the two interior angles on the same side of it taken together less than two right angles, then these straight lines being continually produced will at length meet upon that side on which are the angles which are less than two right angles. Expressed in this form the axiom is far from obvious, and from early times numerous attempts have been made to prove it. All such attempts failed, and it is now known that the axiom cannot be deduced from the other axioms assumed by Euclid.
    • Footnote) Some of the more interesting and plausible attempts have been collected by T. P. Thompson in his Geometry without Axioms, London, 1833, and later by J. Richard in his Philo: ie de mathématique, Paris, 1903.
  • The earliest conception of a body of Non-Euclidean geometry was due to the discovery, made independently by Saccheri, Lobatschewsky, and John Bolyai, that a consistent system of geometry of two dimensions can be produced on the assumption that the axiom on parallels is not true, and that through a point a number of straight (that is, geodetic) lines can be drawn parallel to a given straight line. The resulting geometry is called hyperbolic.
  • Riemann later distinguished between boundlessness space and its infinity, and showed that another consistent system of geometry of two dimensions can be constructed in which all straight lines are of finite length, so that a particle moving along a straight line will return to its original position. This leads to a geometry of two dimensions, called elliptic geometry, analogous to the hyperbolic geometry, but characterised by the fact that through a point no straight line can be drawn which, if produced far enough, will not meet any other given straight line. This can be compared with the geometry of figures drawn on the surface of a sphere.
    Thus according as no straight line, or only one straight line, or a pencil of straight lines can be drawn through a point parallel to a given straight line, we have three systems of geometry of two dimensions known respectively as elliptic, parabolic or homaloidal or Euclidean, and hyperbolic.
  • In the parabolic and hyperbolic systems straight lines are infinitely long. In the elliptic they are finite. In the hyperbolic system there are no similar figures of unequal size; the area of a triangle can be deduced from the sum of its angles, which is always less than two right angles; and there is a finite maximum to the area of a triangle. In the elliptic system all straight lines are of the same finite length; any two lines intersect; and the sum of the angles of a triangle is greater than two right angles.
  • In spite of these and other peculiarities of hyperbolic and elliptic geometries, it is impossible to prove by observation that one of them is not true for the space in which we live. For in measurements in each of these geometries we must have a unit of distance; and we live in a space whose properties are those of either of these geometries, and such that the greatest distances with which we are acquainted (ex. gr. the distances of the fixed stars) are immensely smaller than any unit, natural to the system, then it may be impossible for us by our observations to detect the discrepancies between the three geometries. It might indeed be possible by observations of the parallaxes of stars to prove that the parabolic system and either the hyperbolic or elliptic system were false, but never can it be proved by measurements that Euclidean geometry is true. Similar difficulties might arise in connection with excessively minute quantities. In short, though the results of Euclidean geometry are more exact than present experiments can verify for finite things, such as those with which we have to deal, yet for much larger things or much smaller things or for parts of space at present inaccessible to us they may not be true.
  • Other systems of Non-Euclidean geometry might be constructed by changing other axioms and assumptions made by Euclid. Some of these are interesting, but those mentioned above have a special importance from the somewhat sensational fact that they lead to no results inconsistent with the properties of the space in which we live.
  • In order that a space of two dimensions should have the geometrical properties with which we are familiar, it is necessary that it should be possible at any place to construct a figure congruent to a given figure; and this is so only if the product of the principle radii of curvature at every point of the space or surface be constant. The product is constant in the case (i) of spherical surfaces, where it is positive; (ii) of plane surfaces (which leads to Euclidean geometry), where it is zero; and (iii) of pseudo-spherical surfaces, where it is negative. A tractroid is an instance of a pseudo-spherical surface; it is saddle-shaped at every point. Hence on spheres, planes, and tractroids we can construct normal systems of geometry. These systems are respectively examples of elliptic, Euclidean, and hyperbolic geometries. Moreover, if any surface be bent without dilation or contraction, the measure of the curvature remains unaltered. Thus these three species of surfaces are types of three kinds on which congruent figures can be constructed. For instance a plane can be rolled into a cone, and the system of geometry on a conical surface is similar to that on a plane.
    • Note: In the preceding sketch of the foundations of Non-Euclidean geometry I have assumed tacitly that the measure of a distance remains the same everywhere.
  • The above refers only to hyper-space of two dimensions. Naturally there arises the question whether there are different kinds of hyper-space of three or more dimensions. Riemann showed that there are three kinds of hyper-space of three dimensions having properties analogous to the three kinds of hyper-space of two dimensions already discussed. These are differentiated by the test whether at every point no geodetical surfaces, or one geodetical surface, or a fasciculus of geodetical surfaces can be drawn parallel to a given surface; a geodetical surface being defined as such that every geodetic line joining two points on it lies wholly on the surface.

Elements of Non-Euclidean Geometry (1919)

Duncan M'Laren Young Sommerville, source
  • The common notions of Euclid are five in number, and deal exclusively with equalities and inequalities of magnitudes. The postulates are also five in number and are exclusively geometrical. The first three refer to the construction of straight lines and circles. The fourth asserts the equality of all right angles, and the fifth is the famous Parallel Postulate...
    It seems impossible to suppose that Euclid ever imagined this to be self-evident, yet the history of the theory of parallels is full of reproaches against the lack of self-evidence of this "axiom." Sir Henry Savile referred to it as one of the great blemishes in the beautiful body of geometry; D'Alembert called it "l'écueil et le scandale des élémens de Géométrie."
    Such considerations induced geometers (and others), even up to the present day, to attempt its demonstration. From the invention of printing onwards a host of parallel-postulate demonstrators existed, rivalled only by the "circle-squarers," the "flat-earthers," and the candidates for the Wolfskehl "Fermat" prize. ...Modern research has vindicated Euclid, and justified his decision in putting this great proposition among the independent assumptions which are necessary for the development of euclidean geometry as a logical system.
    All this labour has not been fruitless, for it has led in modern times to a rigorous examination of the principles not only of geometry, but of the whole of mathematics, and even logic itself, the basis of mathematics. It has had a marked effect upon philosophy, and has given us a freedom of thought which in former times would have received the award meted out to the most deadly heresies.
    • Chapter 1. Historical, pp. 2-4.
  • One of the commonest of the equivalents used for Euclid's axiom in school text-books is Playfair's axiom (really due to Ludlam).
    • Chapter 1. Historical, p. 4.
  • A... fallacy is contained in all proofs [of the Parallel Postulate] based upon the idea of direction. ...
    Another class of demonstrations is based upon considerations of infinite areas. [In] Bertrand's Proof... The fallacy... consists in applying the principle of superposition to infinite areas as if they were finite magnitudes.
    • Chapter 1. Historical, pp. 6-8.
  • Non-euclidean geometry has made it clear that the ideas of parallelism and equidistance are quite distinct. The term parallel (Greek... running alongside) originally connoted equidistance, but the term is used by Euclid rather in the sense "asymptotic" (Greek... non-intersecting), and this term has come to be used in the limiting case of curves which tend to coincidence, or the limiting case between intersection and non-intersection. In non-euclidean geometry parallel straight lines are asymptotic in this sense, and equidistant straight lines in a plane do not exist. This is just one instance of two distinct ideas which are confused in euclidean geometry, but are quite distinct in non-euclidean.
    • Chapter 1. Historical, pp. 10-11.
  • Among the early postulate demonstrators there stands a unique figure that of a Jesuit Gerolamo Saccheri, a contemporary and friend of Ceva. This man devised an entirely different mode of attacking the problem, in an attempt to institute a reductio ad absurdum. At that time the favourite starting-point was the conception of parallels as equidistant straight lines, but Saccheri, like some of his predecessors, saw that it would not do to assume this in the definition. ...Saccheri keeps an open mind, and proposes three hypotheses:
    (1) The Hypothesis of the Right Angle.
    (2) The Hypothesis of the Obtuse Angle.
    (3) The Hypothesis of the Acute Angle.
    The object of his work is to demolish the last two hypotheses and leave the first, the Euclidean hypothesis, supreme; but the task turns out to be more arduous than he expected. He establishes a number of theorems, of which the most important are the following:
    If one of the three hypotheses is true in any one case, the same hypothesis is true in every case.
    On the hypothesis of the right angle, the obtuse angle, or the acute angle, the sum of the angles of a triangle is equal to, greater than, or less than two right angles. ...
    Saccheri demolishes the hypothesis of the obtuse angle in his Theorem 14 by showing that it contradicts Euclid I. 17 (that the sum of any two angles of a triangle is less than two right angles); but he requires nearly twenty more theorems before he can demolish the hypothesis of the acute angle, which he does by showing that two lines which meet in a point at infinity can be perpendicular at that point to the same straight line. In spite of all his efforts, however, he does not seem to be quite satisfied with the validity of his proof, and he offers another proof in which he loses himself, like many another, in the quicksands of the infinitesimal.
    If Saccheri had had a little more imagination and been less bound down by tradition, and a firmly implanted belief that Euclid's hypothesis was the only true one, he would have anticipated by a century the discovery of the two non-euclidean geometries which follow from his hypotheses of the obtuse and the acute angle.
    • Chapter 1. Historical, pp. 11-13.
  • J. H. Lambert, fifty years after Saccheri, also fell just short... His starting point is very similar to Saccheri's, and he distinguishes the same three hypotheses; but he went further than Saccheri. He actually showed that on the hypothesis of the obtuse angle the area of a triangle is proportional to the excess of the sum of its angles over two right angles, which is the case for the geometry on the sphere, and he concluded that the hypothesis of the acute angle would be verified on a sphere of imaginary radius. ...
    He dismisses the hypothesis of the obtuse angle, since it requires that two straight lines should enclose a space, but his argument against the hypothesis of the acute angle, such as the non-existence of similar figures, he characterises as arguments ab amore et invidia ducta [guided by love and jealousy]. Thus he arrived at no definite conclusion, and his researches were only published some years after his death.
    • Chapter 1. Historical, pp. 13-14.
  • About... 1799 the genius of Gauss was being attracted to the question, and, although he published nothing on the subject except a few reviews, it is clear from his correspondence and fragments of his notes that he was deeply interested in it. He was a keen critic of the attempts made by his contemporaries to establish the theory of parallels; and while at first he inclined to the orthodox belief, encouraged by Kant, that Euclidean geometry was an example of a necessary truth, he gradually came to see that it was impossible to demonstrate it. He declares that he refrained from publishing anything because he feared the clamour of the Boeotians, or, as we should say, the Wise Men of Gotham; indeed at this time the problem of parallel lines was greatly discredited, and anyone who occupied himself with it was liable to be considered as a crank.
    • Chapter 1. Historical, p. 14.
  • Gauss was probably the first to obtain a clear idea of the possibility of a geometry other than that of Euclid, and we owe the very name Non-Euclidean Geometry to him. It is clear that about the year 1820 he was in possession of many theorems of non-euclidean geometry, and though he meditated publishing his researches when he had sufficient leisure to work them out in detail with his characteristic elegance, he was finally forestalled by receiving in 1832, from his friend W. Bolyai, a copy of the now famous Appendix by his son, John Bolyai.
    • Chapter 1. Historical, p. 14.
  • Among the contemporaries and pupils of Gauss... F. K. Schweikart, Professor of Law in Marburg, sent to Gauss in 1818 a page of MS. explaining a system of geometry which he calls "Astral Geometry," in which the sum of the angles of a triangle is always less than two right angles, and in which there is an absolute unit of length.
    He did not publish any account of his researches, but he induced his nephew, F.A. Taurinus, to take up the question. ...a few years later he attempted a treatment of the theory of parallels and having received some encouragement from Gauss he [Taurinus] published a small book, Theorie der Parallellinien, in 1825. After its publication he came across [J. W.] Camerer's new edition of Euclid in Greek and Latin, which in an Excursus to Euclid I. 29, contains a very valuable history of the theory of parallels, and there he found that his methods had been anticipated by Saccheri and Lambert. Next year, accordingly, he published another work, Oeometriae prima elementa and in the Appendix... works out some of the most important trigonometrical formulae for non-euclidean geometry by using the fundamental formulae of spherical geometry with an imaginary radius. Instead of the notation of hyperbolic functions, which was then scarcely in use, he expresses his results in terms of logarithms and exponentials, and calls his geometry the "Logarithmic Spherical Geometry."
    Though Taurinus must be regarded as an independent discoverer of non-euclidean trigonometry, he always retained the belief, unlike Gauss and Schweikart, that Euclidean geometry was necessarily the true one. Taurinus himself was aware, however, of the importance of his contribution... and it was a bitter disappointment to him when he found that his work attracted no attention. In disgust he burned the remainder of the edition of his Elementa, which is now one of the rarest of books.
    • Chapter 1. Historical, pp. 14-15.
  • The third... having arrived at the notion of a geometry in which Euclid's postulate is denied is F. L. Wachter, a student under Gauss. It is remarkable that he affirms that even if the postulate be denied, the geometry on a sphere becomes identical with the geometry of Euclid when the radius is indefinitely increased, though it is distinctly shown that the limiting surface is not a plane. This was one of the greatest discoveries of Lobachevsky and Bolyai. If Wachter had lived he might have been the discoverer of non-euclidean geometry, for his insight into the question was far beyond that of the ordinary parallel-postulate demonstrator.
    • Chapter 1. Historical, p. 15.
  • While Gauss, Schweikart, Taurinus and others were working in Germany,... just on the threshold of... discovery, in France and Britain... there was a considerable interest in the subject inspired chiefly by A. M. Legendre. Legendre's researches were published in the various editions of his Éléments, from 1794 to 1823. and collected in an extensive article in the Memoirs of the Paris Academy in 1833.
    Assuming all Euclid's definitions, axioms and postulates, except the parallel-postulate and all that follows from it, he proves some important theorems, two of which, Propositions A and B, are frequently referred to in later work as Legendre's First and Second Theorems.
    Prop. A. The sum of the three angles of a rectilinear triangle cannot be greater than two right angles (π). ...
    Prop. B. If there exists a single triangle in which the sum of the angles is equal to two right angles, then in every triangle the sum of the angles must likewise be equal to two right angles.
    This proposition was already proved by Saccheri, along with the corresponding theorem for the case in which the sum of the angles is less than two right angles... Legendre's proof... proceeds by constructing successively larger and larger triangles in each of which the sum of the angles = π. ...
    In this proof there is a latent assumption and also a fallacy. ...Legendre's other attempts make use of infinite areas. He makes reference to Bertrand's proof, and attempts to prove the necessity of Playfair's axiom...
    • Chapter 1. Historical, pp. 16-19.
  • Nikolai Ivanovich Lobachevsky, Professor of Mathematics at Kazan, was interested in the theory of parallels from at least 1815. Lecture notes of the period 1815-17 are extant, in which Lobachevsky attempts in various ways to establish the Euclidean theory. He proves Legendre's two propositions, and employs also the ideas of direction and infinite areas. In 1823 he prepared a treatise on geometry for use in the University, but it obtained so unfavourable a report that it was not printed. The MS. remained buried in the University Archives until it was discovered and printed in 1909. In this book he states that "a rigorous proof of the postulate of Euclid has not hitherto been discovered; those which have been given may be called explanations, and do not deserve to be considered as mathematical proofs in the full sense."
    Just three years afterwards, he read to the physical and mathematical section of the University of Kazan a paper entitled "Exposition succinte des principes de la géométrie avec une démonstration rigoureuse du théorème des parallèles." In this paper... Lobachevsky explains the principles of his "Imaginary Geometry," which is more general than Euclid's, and in which two parallels can be drawn to a given line through a given point, and in which the sum of the angles of a triangle is always less than two right angles.
    • Chapter 1. Historical, p. 21.
  • Bolyai János (John) was the son of Bolyai Farkas (Wolfgang), a fellow-student and friend of Gauss at Göttingen. The father was early interested in the theory of parallels, and without doubt discussed the subject with Gauss while at Göttingen. The professor of mathematics at that time, A. G. Kaestner, had himself attacked the problem and with his help G. S. Klügel, one of his pupils, compiled in 1763 the earliest history of the theory of parallels.
    • Chapter 1. Historical, pp. 21-22.
  • In 1804, Wolfgang Bolyai... sent to Gauss a "Theory of Parallels," the elaboration of his Göttingen studies. In this he gives a demonstration very similar to that of [Henry] Meikle and some of Perronet Thompson's, in which he tries to prove that a series of equal segments placed end to end at equal angles, like the sides of a regular polygon, must make a complete circuit. Though Gauss clearly revealed the fallacy, Bolyai persevered and sent Gauss, in 1808, a further elaboration of his proof. To this Gauss did not reply, and Bolyai, wearied with his ineffectual endeavours to solve the riddle of parallel lines, took refuge in poetry and composed dramas. During the next twenty years, amid various interruptions, he put together his system of mathematics, and at length in 1832-3, published in two volumes an elementary treatise on mathematical discipline which contains all his ideas with regard to the first principles of geometry.
    Meanwhile, John Bolyai... had been giving serious attention to the theory of parallels, in spite of his father's solemn adjuration to let the loathsome subject alone. At first, like his predecessors, he attempted to find a proof for the parallel-postulate, but gradually, as he focussed his attention more and more upon the results which would follow from a denial of the axiom, there developed in his mind the idea of a general or "Absolute Geometry" which would contain ordinary or euclidean geometry as a special or limiting case. Already, in 1823, he had worked out the main ideas of the non-euclidean geometry, and in a letter of 3rd November he announces to his father his intention of publishing a work on the theory of parallels, "for," he says, "I have made such wonderful discoveries that I am myself lost in astonishment, and it would be an irreparable loss if they remained unknown. When you read them, dear Father, you too will acknowledge it. I cannot say more now except that out of nothing I have created a new and another world. All that I have sent you hitherto is as a house of cards compared to a tower." Wolfgang advised his son, if his researches had really reached the desired goal, to get them published as soon as possible, for new ideas are apt to leak out, and further, it often happens that a new discovery springs up spontaneously in many places at once, "like the violets in springtime." Bolyai's presentment was truer than he suspected, for at this very moment Lobachevsky at Kazan, Gauss at Gottingen, Taurinus at Cologne, were all on the verge of this great discovery. It was not, however, till 1832 that... the work was published. It appeared in Vol. I of his father's Tentamen, under the title "Appendix, scientiam absolute veram exhibens."
    ...the son, although he continued to work at his theory of space, published nothing further. Lobachevsky's Geometrische Untersuchungen came to his knowledge in 1848, and this spurred him on to complete the great work on "Raumlehre," which he had already planned at the time of the publication of his "Appendix," but he left this in large part as a rudis indigestaque moles, and he never realised his hope of triumphing over his great Russian rival.
    • Chapter 1. Historical, pp. 22-23.
  • Lobachevsky never seems to have heard of Bolyai, though both were directly or indirectly in communication with Gauss. Much has been written on the relationship of these three discoverers, but it is now generally recognised that John Bolyai and Lobachevsky each arrived at their ideas independently of Gauss and of each other; and, since they possessed the convictions and the courage to publish them which Gauss lacked, to them alone is due the honour of the discovery.
    • Chapter 1. Historical, p. 24.
  • The ideas inaugurated by Lobachevsky and Bolyai did not for many years attain any wide recognition, and it was only after Baltzer had called attention to them in 1867, and at his request Hoüel had published French translations of the epoch making works, that the subject of non-euclidean geometry began to be seriously studied.
    It is remarkable that while Saccheri and Lambert both considered the two hypotheses, it never occurred to Lobachevsky or Bolyai or their predecessors, Gauss, [F. K.] Schweikart, [F. A.] Taurinus, and [F. L.] Wachter, to admit the hypothesis that the sum of the angles of a triangle may be greater than two right angles. This involves the conception of a straight line as being unbounded but yet of finite length. Somewhere "at the back of beyond" the two ends of the line meet and close it. We owe this conception first to Bernhard Riemann in his Dissertation of 1854 (published only in 1866 after the author's death), but in his Spherical Geometry two straight lines intersect twice like two great circles on a sphere. The conception of a geometry in which the straight line is finite, and is, without exception, uniquely determined by two distinct points, is due to Felix Klein. Klein attached the now usual nomenclature to the three geometries; the geometry of Lobachevsky he called Hyperbolic, that of Riemann Elliptic, and that of Euclid Parabolic.
    • Chapter 1. Historical, pp. 24-25.

A History of Geometrical Methods (1940)

by Julian Lowell Coolidge
  • In general the Greeks looked upon an axiom as something which was so self-evident that no reasonable person would object... while a postulate was a request that something be allowed. Now Euclid's fifth postulate... whatever else this postulate may be, self-evident it is not, and this was early perceived. ...
    The first line of attack was, naturally, the attempt to prove this postulate by the aid of others, and the axioms. Such, presumably, was Ptolemy's idea. But even if we grant that all of Euclid's axioms are self-evident, it does not... follow that he puts in his list all of the assumptions that he really uses.
  • The way that geometers... went about proving the fifth postulate was to smuggle in somewhere some unavowed assumption. A common practice was to assume that two straight lines could not approach one another assymptotically, that... they ultimately intersected. Or, again, it was assumed that a straight line was not a closed circuit... legitimate as long as avowed.
    A franker, and so more admirable way... was to change the definition of parallel lines into something else that seemed to avoid the trouble, or else to reword the axiom in a less objectionable form. A real step in advance... is known as Playfair's axiom, though it is casually mentioned in Proclus...There are... a great many alternatives. One of the most famous is to define two coplaner lines as parallel if they are everywhere the same distance apart... but how do we know there are such pairs... A still neater method consists in defining two lines as parallel if they have the same direction, or opposite directions. But here we introduce a totally new undefined concept, direction...
  • A writer who clearly saw the fallacy under the constant distance assumption was Girolamo Saccheri, S. J., whose 'Euclides ab omne naevo vindicatus' [Euclid Freed of Every Flaw]... in 1733, marked perhaps the most important single step in advance ever taken in the attempt to solve the parallel difficulty. This careful logician undertook to prove the correctness of Euclid's postulate by showing that when it is replaced by another, a contradiction is sure to arise.
  • Having disposed, as he thinks, of the obtuse-angled hypothesis, Saccheri turns boldly to the task of destroying the acute-angle one also. He shows that under this hypothesis there passes through each point without [outside of] a given line two parallels thereto... Most unfortunately he speaks of parallels as intersecting at infinity... and then speaks of ultra-infinite points beyond them. His proof... breaks down just there. ...In Segre we find an elaborate argument to the effect that subsequent writers who approached the parallel postulate problem through the means of elementary geometry were directly, or indirectly, influenced by him. The greatest, if the least communicative, of these was Gauss.
  • Gauss... wrote little on the subject beyond correcting the vagaries of his friend Schumacher, but it is certain that he reflected deeply, and arrived at conclusions subsequently supported by others. His revolutionary view, that Saccheri was wrong and that a consistent geometry can be developed... was carried through with complete success by Nicholai Ivanovitch Lobachevski.
  • Fourteen years before Beltrami published... a greater than he had studied the whole of the non-Euclidean problem from a more lofty and difficult point of view. This was Bernhard Riemann, who offered to Gauss three topics for his projected trial lecture as Privatsozent at Göttingen. Gauss chose the most difficult, wondering what so young a man could make of such an arduous subject; he learned. ...'Ueber die Hypothesen welche der Geometrie su Grunde liegen' ...was read in 1854, but never published till 1868.
    Riemann's approach is far different from anything that anyone had tried previously. ...The modern theory of relativity, on its mathematical side, is merely an elaboration of Riemann's analysis.
  • Riemann... made the important distinction, which had escaped previous writers, between the infinite and the unlimited. All of our experience tends to show that the universe is unlimited; a given segment may be extended indefinitely in either direction, but we know nothing as to whether it is infinite or not. If space have constant positive curvature, a geodesic surface is applicable to a Euclidean sphere where a geodesic is a circle, unlimited but not infinite. This possibility destroys the validity of Euclid's proof that an exterior angle of a triangle is greater than either opposite interior angle.
    Of all methods devised for attacking the problem of the bases of geometry Riemann's has proved by far to be the most fruitful. That is probably because it is the most flexible, and applicable to the greatest number of problems. In the twentieth century reverence for Euclid has been replaced by reverence for the differential equation
  • Beltrami's idea was to find in space a surface with the property that if you define distance thereon in terms of geodesic length, you have the geometry of Lobachevski. An analogous idea is to find a new definition for distance such that, starting from our familiar space, if we redefine distance in this way we may have the obtuse-angled geometry, elliptic geometry, or the acute-angled, hyperbolic geometry of Lobachevski. An illuminating example of this sort was worked out by Klein following a hint dropped by Cayley. The root of the matter goes back to Laguerre... in 1858...
  • A scruple... has troubled conscientious writers. We take Euclidean space as we know it, we take Cartesian geometry in that space, we set up certain point functions in that space and call them distances, certain transformations and call them motions, and find at last a set of objects which obey the presuppositions of non-Euclidean geometry. But is there not here, perhaps, a vicious circle around which the kitten is chasing its tail? The basis is a Euclidean space, and a Cartesian coordinate system in that space, which is based upon Euclidean measurements, and cross ratios which depend upon distances. How do we know that without all of these it would be possible to erect a consistent non-Euclidean geometry? ...
    We begin by setting up a system of axioms for a projective geometry in a space of as many dimensions as we please. The undefined elements are point, line as a system of points, and separation of pairs of collinear points. Other choices are possible... The idea of taking separation as fundamental was introduced by Vailati.
  • If we are to set up a system of axioms for a particular sort of geometry, two qualities are essential, and two desirable. The essential qualities are that:
    1) They should be consistent.
    2) They should contain all of the assumptions necessary for the purposes in hand.
    3) They should be independent of one another and include nothing unnecessary.
    4) The mathematical system built on them should be interesting rather than trivial.
    The first work where the problem of setting up geometrical axioms in this way was Pasch in 1882. The way opened by him was subsequently followed by a goodly number of others, among whom one might mention Peano, Pieri, Vahlen, HIlbert, E. H. Moore, R. L. Moore, Veblen, Huntington, and others or lesser note.
    • Ref note: Pasch, Vorlesungen über nuere Geometie [Lectures on newer Geometry], Leipzig, 1882.

An Introduction to the History of Mathematics (1953)

by Howard Eves
  • The attempts to derive the parallel postulate as a theorem from the remaining nine "axioms" and "postulates" occupied geometers for over two thousand years and culminated in some of the most far-reaching developments in modern mathematics. Many "proofs" of the postulate were offered, but each was sooner or later shown to rest upon a tacit assumption equivalent to the postulate itself.
    Not until 1733 was the first really scientific investigation... Gerolamo Saccheri received permission to print... Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw). ...Saccheri had become charmed with the powerful method of reductio ad absurdum and... easily showed... that if, in a quadrilateral... [base] angles... are right angles and [vertical] sides... are equal, then [ceiling] angles... are equal. Then there are three possibilities: [ceiling] angles are equal acute... equal right... or equal obtuse angles. The plan was to show that the assumption of either... the acute angle or... the obtuse angle would lead to a contradiction. ...Tacitly assuming the infinitude of the straight line, Saccheri readily eiliminated the hypothesis of the obtuse angle, but... After obtaining many of the now classical theorems of... non-Euclidean geometry, Saccheri lamely forced... an unconvincing contradiction.
  • Johann Heinrich Lambert... went considerably beyond Sacherri in deducing propositions under the hypotheses of the acute and obtuse angles. Thus, with Sacherri, he showed that in the three hypotheses the sum of the angles of a triangle is less than, equal to, or greater than two right angles, respectively, and... in addition, that the deficiency... in the hypothesis of the acute angle, or the excess, in the hypothesis of the obtuse angle, is proportional to the area of the triangle. He observed the resemblance of the geometry following the... obtuse angle to spherical geometry... and conjectured that the geometry following from... the acute angle could perhaps be verified on the sphere of imaginary radius.
  • It is no wonder that no contradiction was found under the hypothesis of the acute angle, for... the geometry developed from a collection of axioms comprising a basic set plus the acute angle hypothesis is as consistent as the Euclidean geometry developed from the same basic set plus the hypothesis of the right angle; that is, the parallel postulate is independent of the remaining postulates and therefore cannot be deduced from them.

Mathematics and the Physical World (1959)

by Morris Kline
  • In the field of non-Euclidean geometry, Riemann... began by calling attention to a distinction that seems obvious once it is pointed out: the distinction between an unbounded straight line and an infinite line. The distinction between unboundedness and infiniteness is readily illustrated. A circle is an unbounded figure in that it never comes to an end, and yet it is of finite length. On the other hand, the usual Euclidean concept of a straight line is also unbounded in that it never reaches an end but is of infinite length.
    ...he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded. He also proposed to adopt a new parallel axiom... In brief, there are no parallel lines. This ... had been tried... in conjunction with the infiniteness of the straight line and had led to contradictions. However... Riemann found that he could construct another consistent non-Euclidean geometry.
  • Non-Euclidean geometry was the most weighty intellectual creation of the nineteenth century, or, at worst, might have to share honors with the theory of evolution.
  • Unlike those of science, the conclusions of mathematics had always regarded as deduced from basic truths. ...the very reason that mathematicians persisted for so many centuries in attempting to find simple equivalents for Euclid's parallel axiom, instead of entertaining contradictory possibilities, is that they could not conceive of geometry being anything else than the true geometry of physical space.
  • The creation of non-Euclidean geometry showed... that mathematics could no longer be regarded as a body of unquestionable truths. ...Mathematics retained its deductive method of establishing its conclusions, but it was soon appreciated that mathematics offers only certainty of proof on the basis of uncertain axioms.
  • What was the effect of non-Euclidean geometry on the future progress of mathematics? ...Mathematics passed from serfdom to freedom. Up to [that] time... mathematicians were fettered to the physical world. ...Had not the history of non-Euclidean geometry shown that seemingly absurd ideas may prove to be not only illuminating but of actual use to science? ...Mathematicians found their house burned to the ground only to find gold under the floor boards.
  • Even the mathematicians of the late nineteenth century did not take non-Euclidean geometry seriously for physical applications, though they derived a great deal of pleasure from the new concepts and relating them to other domains of mathematics. The scientific world did not awaken to the reality on non-Euclidean geometry until the creation of the special theory of relativity in 1905.

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Pure math

Abstract algebraAlgebraAnalysisAlgebraic geometry (Sheaf theory) • Algebraic topologyArithmeticCalculusCategory theoryCombinatoricsCommutative algebraComplex analysisDifferential calculusDifferential geometryDifferential topologyErgodic theoryFoundations of mathematicsFunctional analysisGame theoryGeometryGlobal analysisGraph theoryGroup theoryHarmonic analysisHomological algebraInvariant theoryLogicNon-Euclidean geometryNonstandard analysisNumber theoryNumerical analysisOperations researchRepresentation theoryRing theorySet theorySheaf theoryStatisticsSymplectic geometryTopology

Applied math

Computational fluid dynamicsEconometricsFluid mechanicsMathematical physicsScience

History of math

Ancient Greek mathematicsEuclid's ElementsHistory of algebraHistory of calculusHistory of logarithmsIndian mathematicsPrincipia Mathematica


Mathematics and mysticismMathematics educationMathematics, from the points of view of the Mathematician and of the PhysicistPhilosophy of mathematicsUnification in science and mathematics