"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."

"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."

"Among the contemporaries and pupils of Gauss... F. K. Schweikart, Professor of Law in , 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."

"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."

"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..."

"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."

"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."

"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."

"Propositio XXXIII. Hypothesis anguli acuti est absolute falsa; quia repugnans naturae lineae rectae. [Proposition 33. The hypothesis of acute angle is absolutely false; because repugnant to the nature of the straight line.]"

"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 , s, s, etc. you need to plot it on curved graph paper. But Einstein... didn't have that broad an idea of geometry..."

"[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."

"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."

"Selecting the z-axis as an axis of revolution, a point on the surface generated by rotating the curve r = f(z) is defined by two coordinates... z and \theta. ...Now ds^2 = ds_1^2 + ds_2^2 where ds_1 is the displacement along the meridian and ds_2 the displacement along the parallel of latitude. ...since ds_1^2 = dz^2 + dr^2 ...The [arbitrary] line element ds is... defined by the relation {{center|1=ds_1 = dz\sqrt{1 + (\frac{dr}{dz})^2}}}and The line element ds is thus defined by the relation:{{center|1=ds^2 = dz^2[1 + (\frac{dr}{dz})^2] + r^2d\theta^2 = A^2dz^2 + B^2d\theta^2 \qquad (1.1)}}where{{center|1=A = \sqrt{1 + (\frac{dr}{dz})^2} \quad and \; B = r \qquad \qquad (1.2)}}This is the first of the generalized forms of equations in curved surface theory in which A and B are parameters. ... For a generalized curved surface with an arbitrarily selected orthoganal coordinate system defined by the coordinates \alpha and \beta, eq. (1.1) assumes the generalized form...the coefficients will now be functions of \alpha and \beta. We may again write:{{center|1=ds_1 = Ad\alpha \quad \text{for} \quad \beta = c_1 ds_2 = Bd\beta \quad \text{for} \quad \alpha = c_2}}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. ...by two normalized orthogonal coordinates. ...If a set of orthogonal coordinates can be selected such that A and B are independent of \alpha and \beta, 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. This classification includes the s 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 \alpha and \beta to the form of eq. (1.4) with A and B constant, the so-called conditions of euclidean geometry will be satisfied. ...When it becomes impossible to select \alpha and \beta coordinates for which A and B are constant, the geometry of the curved surface becomes different 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 ds remains invariant. This class of surfaces includes the , the , the 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] K on the basis of astronomical data... Schwarzschild's pioneer attempt is so inspiring in its conception and so beautiful in its expression...[!]"

", "Geometry as a Branch of Physics" (1949) from Albert Einstein: Philosopher-Scientist, ed. ."

"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 had broken the grip of Euclid on mathematical and spatial thinking, and a new imagination of space could be born."

"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.""

"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."

"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."

"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."

"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."

"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."

"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."

"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)."

"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)."

"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."