History Of Mathematics

"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."

"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."

"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."

"... 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."

"Nikolai Ivanovich Lobachevsky"

"Carl Friedrich Gauss"

"Howard P. Robertson"

"Bernhard Riemann"

"History of mathematics"

"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."

"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."

"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..."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"One of the commonest of the equivalents used for Euclid's axiom in school text-books is Playfair's axiom (really due to Ludlam)."

"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."

"The discussion on the Non-Euclidean geometry brought into prominence the logical foundations of the subject. The question of the principles of and underlying assumptions made in mathematics have been discussed as late by J. W. R. Dedekind... G. Cantor... G. Peano... the Hon. B. A. W. Russell, A. N. Whitehead, and E. W. Hobson..."

"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."

"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."

"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 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."

"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."

"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."

"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."

"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."

"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."

"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."

"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 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)."

"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."

"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."

"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."

"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."

"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."

"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."

"The ideas of hyper-space were brilliantly expounded and popularised in England by Clifford."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."