Geometry

"The classic example of an is that of plane geometry formulated by Euclid... It forms the model of all rigorous mathematical schemes. The axioms are the initial assumptions... From them, logical deductions can proceed under stipulated rules of reasoning... analogous to the scientists' laws of Nature, whilst the axioms play the role of s. We are not free to pick any axioms... They must be logically consistent... Euclid and most other pre-nineteenth-century mathematicians... were also strongly biased towards picking axioms which mirrored the way the world was observed to work... Later mathematicians did not feel so encumbered and have required only consistency from their lists of axioms."

"Each of five men—Lobachewsky, Bolyai, Plücker, Riemann, Lie—invented as part of his lifework as much (or more) new geometry as was created by all the Greek mathematicians in the two or three centuries of their greatest activity."

"The chemist smiles at the childish efforts of alchemists but the mathematician finds the geometry of the Greeks and the arithmetic of the Hindoos as useful and admirable as any research of today."

"When the value of mathematical training is called in question, quote the inscription over the entrance into the academy of Plato, the philosopher: "Let no one who is unacquainted with geometry enter here.""

"Students in analytical geometry should know something of Descartes, and, after taking up the differential and integral calculus, they should become familiar with the parts that Newton, Leibniz, and Lagrange played in creating that science."

"The Egyptians carried geometry no further than was absolutely necessary for their practical wants. The Greeks, on the other hand, had within them a strong speculative tendency. They felt a craving to discover the reasons for things. They found pleasure in the contemplation of ideal relations and loved science as science."

"The Eudemian Summary says that "Pythagoras changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom, and investigated its theorems in an immaterial and intellectual manner." His geometry was connected closely with his arithmetic. He was especially fond of those geometrical relations which admitted of arithmetical expression."

"Aristotle (384-322 B.C.), the systematiser of deductive logic, though not a professed mathematician, promoted the science of geometry by improving some of the most difficult definitions. His Physics contains passages with suggestive hints of the principle of virtual velocities. About this time there appeared a work called Mechanica, of which he is regarded by some as the author. Mechanics was totally neglected by the Platonic school."

"When Ptolemy once asked Euclid if geometry could not be mastered by an easier process than by studying the Elements, Euclid returned the answer, "There is no royal road to geometry.""

"It is a remarkable fact in the history of geometry, that the Elements of Euclid, written two thousand years ago, are still regarded by many as the best introduction to the mathematical sciences."

"I would myself say that the purely imaginary objects are the only realities, the ὂντως ὂντα, in regard to which the corresponding physical objects are as the shadows in the cave; and it is only by means of them that we are able to deny the existence of a corresponding physical object; and if there is no conception of straightness, then it is meaningless to deny the conception of a perfectly straight line."

"The discovery of rigid objects in nature is of fundamental importance. Without it, the concept of measurement would probably never have arisen and metrical geometry would have been impossible. ...As for the physical definition of straightness, it could have been arrived at in a number of ways, either by stretching a rope between two points or by appealing to the properties of these rigid bodies themselves. ...Equipped in this way, the first geometricians (those who built the pyramids, for instance) were able to execute measurements on the earth's surface and later to study the geometry of solids, or space-geometry. Thanks to their crude measurements, they were in all probability led to establish in an approximate empirical way a number of propositions whose correctness it was reserved for the Greek geometers to demonstrate with mathematical accuracy. Thus there is not the slightest doubt that geometry in its origin was essentially an empirical and physical science, since it reduced to a study of the possible dispositions of objects (recognised as rigid) with respect to one another and to parts of the earth. ... Now an empirical science is necessarily approximate, and geometry as we know it to-day is an exact science. It professes to teach us that the sum of the three angles of a Euclidean triangle is equal to 180°, and not a fraction more or a fraction less. Obviously no empirical determination could ever lay claim to such absolute certitude. Accordingly, geometry had to be subjected to a profound transformation, and this was accomplished by the Greek mathematicians Thales, Democritus, Pythagoras, and finally Euclid. ... But this 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."

"To-day, thanks to Einstein, we have definite reasons for believing that ultra-precise observation of nature has revealed our natural geometry arrived at with solids and light rays to be slightly non-Euclidean and to vary from place to place. So although the non-Euclidean geometers never suspected it (with the exception of Gauss, Riemann and Clifford), our real world happens to be one of the dream-worlds whose possible existence their mathematical genius forsaw."

"A more thorough study of Euclid's axioms and postulates proved them to be inadequate for the deduction of Euclid's geometry. ...Hilbert and others succeeded in filling the gap by stating explicitly a complete system of postulates for Euclidean and non-Euclidean geometries alike. Among the postulates missing in Euclid's list was the celebrated postulate of Archimedes, according to which, by placing an indefinite number of equal lengths end to end along a line, we should eventually pass any point arbitrarily selected on the line. Hilbert, by denying this postulate, just as Lobatchewski and Riemann had denied Euclid's parallel postulate, succeeded in constructing a new geometry known as non-Archimedean. It was perfectly consistent but much stranger than the classical non-Euclidean varieties. Likewise, it was proved possible to posit a system of postulates which would yield Euclidean or non-Euclidean geometries of any number of dimensions; hence, so far as rational requirements of the mind were concerned, there was no reason to limit geometry to three dimensions."

"[T]he ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds, and therefore no conclusion based upon such ratios can be accepted as rigorous and exact."

"Although there is no study which presents so simple a beginning as that of geometry, there is none in which difficulties grow more rapidly as we proceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear. This necessarily follows in a science which consists of reasoning from the very commencement, for it is evident that every student will feel a claim to have his objections answered, not by authority, but by argument, and that the intelligent student will perceive more readily than another the force of an objection and the obscurity arising from an unexplained difficulty, as the greater is the ordinary light the more will occasional darkness be felt. To remove some of these difficulties is the principal object of this Treatise."

"Since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art."

"[T]he system of concepts of axiomatic geometry alone cannot make any assertions as to the behavior of... practically-rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the coordination of real objects of experience with the empty conceptual schemata of axiomatic geometry. To accomplish this, we need only add the proposition: solid bodies are related, with respect to their proper dispositions, as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the behavior of practically-rigid bodies. Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience... not on logical inferences only. We will call this completed geometry "practical geometry," and shall distinguish it from "purely axiomatic geometry.""

"One is ordinarily accustomed to study geometry divorced from any relation between its concepts and experience. There are advantages in isolating that which is purely logical and independent of what is, in principle, incomplete empiricism. ... for our purpose it is necessary to associate the fundamental concepts of geometry with natural objects; without such an association geometry is worthless for the physicist. The physicist is concerned with the question as to whether the theorems of geometry are true or not."

"And the whole [is] greater than the part."

"There is no royal road to geometry."

"The contemporary decline in interest in geometry and its gradual disappearance from school curricula... should be deplored... Geometry is the most visual of the mathematical disciplines. It is not in principle divorced from numbers, and hence neither is it divorced from algebra. Many a pupil's understanding of algebraic proofs would be considerably reinforced by... visual geometrical proofs which were the hallmark of Greek mathematics and to some extent of Arab mathematics also. ...where a geometrical proof is clear and immediate, as... with... many algebraic identities such as (a \pm b)^2 = a^2 \pm 2ab + b^2\!, the geometry should not be forgotten. The Greeks were some of the greatest teachers of all time... [and] geometric algebra was in many ways [their] greatest achievement ..."

"The geometrical spirit is not so tied to geometry that it cannot be detached from it and transported to other branches of knowledge. A work of morals or politics or criticism, perhaps even of eloquence, would be better (other things being equal) if it were done in the style of a geometer. The order, clarity, precision and exactitude which have been apparent in good books for some time might well have their source in this geometric spirit. ...Sometimes one great man gives the tone to a whole century; Descartes], to whom one might legitimately be accorded the glory of having established a new art of reasoning, was an excellent geometer."

"The way I have taken seems not to lead to the goal, but much rather to make the truth of geometry doubtful."

"Geometry can in no way be viewed... as a branch of mathematics, instead, geometry relates to something already given in nature, namely, space. I... realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to those of geometry."

"I was informed by the priests at Thebes, that king Sesostris made a distribution of the territory of Egypt among all his subjects, assigning to each an equal portion of land in the form of a quadrangle, and that from these allotments he used to derive his revenue by exacting every year a certain tax. In cases however where a part of the land was washed away by the annual inundations of the Nile, the proprietor was permitted to present himself before the king, and signify what had happened. The king then used to send proper officers to examine and ascertain, by admeasurement, how much of the land had been washed away, in order that the amount of tax to be paid for the future, might be proportional to the land which remained. From this circumstance I am of opinion, that Geometry derived its origin; and from hence it was transmitted into Greece."

"The authors on may be divided into... theoretical and practical... [N]one... have combined the theory with the practice... to render the subject plain and intelligible... [T]he most valuable and scientifical are... abstruse, and the practical scarcely furnish... the rationale... The object of the ensuing treatise is to simplify the theory, yet to retain a methodical and accurate... investigation, and to exemplify this theory by... important... useful examples. ...[D]emonstrations are frequently founded on principles strictly Geometrical ...and sometimes ...by algebraical signs, particularly where the Geometrical ...would require a complicated figure, or a ...tedious process. ...[T]he algebraical mode of deduction tends greatly to simplify... yet... definitions and... elementary parts... must be acquired from Geometrical principles illustrated by diagrams; otherwise a student will never obtain a clear and satisfactory knowledge... Should any person attempt to teach the elementary principles of the science by... algebraic characters, and algebraic formulae alone, without the aid of Geometry, he would... deceive both himself and his pupils."

"Geometry has two great treasures: one is the Theorem of Phythagoras, the other the division of a line in extreme and mean ratio. The first we can compare to a mass of gold; the other we may call a precious jewel."

"Geometry enlightens the intellect and sets one's mind right."

"Historically, it was Euclidean geometry that, developed to a large extent as a votive offering to the God of Reason, opened men's eyes to the possibility of design and to the possibility of uncovering it by the pursuit of mathematics."

"The use of canon raised numerous questions concerning the paths of projectiles. ...One might determine... what type of curve a projectile follows and.... prove some geometrical facts about this curve, but geometry could never answer such questions as how high the projectile would go or how far from the starting point it would land. The seventeenth century sought the quantitative or numerical information needed for practical applications, and such information is provided by algebra."

"Descartes... complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof... What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish. ...historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is... paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do."

"Over and above the specific theorems created by men such as Desargues, Pascal and La Hire, several new ideas and outlooks were beginning to appear. The first is the idea of continuous change of a mathematical entity from one state to another... [i.e., of a] a geometrical figure. It was Kepler, in his Astronomiae Optica of 1604, who first seemed to grasp the fact that parabola, ellipse, hyperbola, circle, and the degenerate conic consisting of a pair of lines are continuously derivable from each other. ...The notion of a continuous change in a figure was also employed by Pascal. He allowed two consecutive vertices of his hexagon to approach each other so that the figure became a pentagon. In the same manner he passed from pentagons to quadrilaterals. The second idea to emerge from the work of the projective geometers is that of transformation and invariance. To project a figure from some point and then take a section of that projection is to transform the figure to a new one. The properties... of interest are those that remain invariant under transformation. Other geometers of the seventeenth century, for example, Gregory of St. Vincent... and Newton, introduced transformations other than projection and section."

"A geometrician has learned to perform the most difficult demonstrations and calculations, as a monkey has learned to take his little hat off and on... All has been accomplished through signs, every species has learned what it could understand, and in this way men have acquired symbolic knowledge..."

"He said that the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours."

"I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid — a term used in this work to denote all of standard geometry — Nature exhibits not simply a higher degree but an altogether different level of complexity … The existence of these patterns challenges us to study these forms that Euclid leaves aside as being "formless," to investigate the morphology of the "amorphous.""

"I conceived, developed and applied in many areas a new geometry of nature, which finds order in chaotic shapes and processes. It grew without a name until 1975, when I coined a new word to denote it, fractal geometry, from the Latin word for irregular and broken up, fractus. Today you might say that, until fractal geometry became organized, my life had followed a fractal orbit."

"If the Greeks had had a mind to reduce mathematics to one field... their only choice would have been to reduce arithmetic to geometry... it is hardly surprising that for nearly two millennia geometry took pride of place in mathematics. And it would have been obvious to any mathematician that a geometrical problem could not be stated or solved in the language of numbers, since the geometrical universe had more structure than the numerical universe. If one desired to translate geometrical problems into the language of numbers, one would have to invent (or discover) more numbers."

"Let us calculate the motion of bodies, but also consult the plans of the Intelligence that makes them move. It seems that the ancient philosophers made the first attempts at this sort of science, in looking for metaphysical relationships between numbers and material bodies. When they said that God occupies himself with geometry, they surely meant that He unites in that science the works of His power with the perspectives of His wisdom. From the all too few ancient geometers who undertook such studies, we have little that is intelligible or well-founded. The perfection which geometry has acquired since their time puts us in a better position to succeed, and may more than compensate for the advantages that those great minds had over us."

"O king, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all."

"The doctrine of Proportion, in the Fifth Book of Euclid's Elements, is obscure, and unintelligible to most readers. It is not taught either in foreign or American colleges, and is now become obsolete. It has therefore been omitted in this edition of Euclid's Elements, and a different method of treating Proportion has been substituted for it. This is the common algebraical method, which is concise, simple, and perspicuous; and is sufficient for all useful purposes in practical mathematics. The method is clear and intelligible to all persons who know the first principles of algebra. The rudiments of algebra ought to be taught before geometry, because algebra may be applied to geometry in certain cases, and facilitates the study of it."

"Geometry is almost the only subject as to which we find truths wherein all men agree; and one cause of this is, that geometers alone regard the true laws of demonstration."

"At a very early period the study of Geometry was regarded as a very important mental discipline, as may be shewn from the seventh book of the Republic of Plato. To his testimony may be added that of the celebrated Pascal (Œuvres, Tom. I. p. 66,) which Mr. Hallam has quoted in his History of the Literature of the Middle Ages. "Geometry," Pascal observes, "is almost the only subject as to which we find truths wherein all men agree; and one cause of this is, that geometers alone regard the true laws of demonstration." These are enumerated by him as eight in number. 1. To define nothing which cannot be expressed in clearer terms than those in which it is already expressed. 2. To leave no obscure or equivocal terms undefined. 3. To employ in the definition no terms not already known. 4. To omit nothing in the principles from which we argue, unless we are sure it is granted 5. To lay down no axiom which is not perfectly evident. 6. To demonstrate nothing which is as clear already as we can make it. 7. To prove every thing in the least doubtful, by means of self-evident axioms, or of propositions already demonstrated. 8. To substitute mentally the definition instead of the thing defined. Of these rules he says, "the first, fourth, and sixth are not absolutely necessary to avoid error, but the other five are indispensable; and though they may be found in books of logic, none but the geometers have paid any regard to them."

"Various relations being established in geometry between lines constituted under given conditions, as parts of geometrical figures, if we choose to adopt the idea of expressing these lines by numerical measures, we are then brought to the distinction of such lines being in some cases commensurable in their numerical values, in others not so. Their geometrical relations however are absolutely general, and do not refer to any such distinction."

"All those who have written histories [of geometry] bring to this point their account of the development of this science. Not long after these men [pupils of Plato] came Euclid… Not much younger than these [pupils of Plato] is Euclid, who put together the Elements ,…bringing to irrefragable demonstration the things which had been only loosely proved by his predecessors. This man [must have] lived in the time of the first Ptolemy; for Archimedes, who followed closely the first [Ptolemy? book?] makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorter way to study geometry…to which he replied that there was no royal road to geometry. He is therefore younger than Plato’s circle, but older than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says."

"It is remarkable that this generalization of plane geometry to surface geometry is identical with that generalization of geometry which originated from the analysis of the axiom of parallels. ...the construction of non-Euclidean geometries could have been equally well based upon the elimination of other axioms. It was perhaps due to an intuitive feeling for theoretical fruitfulness that the criticism always centered around the axiom of parallels. For in this way the axiomatic basis was created for that extension of geometry in which the metric appears as an independent variable. Once the significance of the metric as the characteristic feature of the plane has been recognized from the viewpoint of Gauss' plane theory, it is easy to point out, conversely, its connection with the axiom of parallels. The property of the straight line as being the shortest connection between two points can be transferred to curved surfaces, and leads to the concept of straightest line; on the surface of the sphere the great circles play the role of the shortest line of connection... analogous to that of the straight line on the plane. Yet while the great circles as "straight lines" share the most important property with those of the plane, they are distinct from the latter with respect to the axiom of the parallels: all great circles of the sphere intersect and therefore there are no parallels among these "straight lines". ...If this idea is carried through, and all axioms are formulated on the understanding that by "straight lines" are meant the great circles of the sphere and by "plane" is meant the surface of the sphere, it turns out that this system of elements satisfies the system of axioms within two dimensions which is nearly identical in all of it statements with the axiomatic system of Euclidean geometry; the only exception is the formulation of the axiom of the parallels. The geometry of the spherical surface can be viewed as the realization of a two-dimensional non-Euclidean geometry: the denial of the axiom of the parallels singles out that generalization of geometry which occurs in the transition from the plane to the curve surface."

"Visual forms are not perceived differently from colors or brightness. They are sense qualities, and the visual character of geometry consists in these sense qualities."

"The influence of geometry upon philosophy and scientific method has been profound. Geometry, as established by the Greeks, starts with axioms which are (or are deemed to be) self-evident, and proceeds, by deductive reasoning, to arrive at theorems which are very far from self-evident. The axioms and theorems are held to be true of actual space, which is something given in experience. It thus appeared to be possible to discover things about the actual world by first noticing what is self-evident and then using deduction. This view influenced Plato and Kant, and most of the intermediate philosophers... The eighteenth century doctrine of natural rights is a search for Euclidean axioms in politics. The form of Newton's Principia, in spite of its admittedly empirical material, is entirely dominated by Euclid. Theology, in its exact scholastic forms, takes its style from the same source."

"The Greeks... discovered mathematics and the art of deductive reasoning. Geometry, in particular, is a Greek invention, without which modern science would have been impossible."

"[…] nor did he [Thibaut] formulate the obvious conclusion, namely, that the Greeks were not the inventors of plane geometry, rather it was the Indians. At least this was the message that the Greek scholars saw in Thibaut’s paper. And they didn’t like it... If the Indians invented plane geometry, what was to become of Greek ‘genius’ or of the Greek ‘miracle’?"

".. general relativity is, of course, based on , which is one of the richest frameworks for our understanding of ordinary geometry. ... Many ambitious physicists and mathematicians have gone off into the in search of some fundamental generalization of geometry that would reconcile gravity with quantum mechanics but, generally speaking, they have come back empty-handed — if at all."

"Physics deals with a great many quantities that have both size and direction, and it needs a special mathematical language —the language of vectors —to describe those quantities. This language is also used in engineering, the other sciences, and even in common speech."

"While the move from dimension 2 to dimension 3 appears to be the obvious step there is a sense in which one should move from 2 to 4. This comes from the consideration of complex algebraic geometry. For complex dimension 1 this theory was started by Abel and continued by Riemann. For algebraic varieties of complex dimension n the real dimension is 2n, so the case n = 2 leads to 4-dimensional real manifolds. The key figures in the topology of higher-dimensional algebraic varieties were Lefschetz, Hodge, Cartan and Serre. While general algebraic geometry was one of the major developments of the second half of the 20th century, the topology of real 4-manifolds had a great surprise in store when Simon Donaldson made spectacular discoveries opening up an entirely new area."

"Should you just be an algebraist or a geometer? is like saying Would you rather be deaf or blind? If you are blind, you do not see space: if you are deaf, you do not hear, and hearing takes place in time. On the whole, we prefer to have both faculties."

"In the various forms of geometry (differential, metric, affine, algebraic), the central object is the variety, considered as a set of points."

"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."

"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 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."

"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."

"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 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."

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

"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."

"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."

"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."

"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."

"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 (&pi;). ... 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 = &pi;. ... 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."

"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."

"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."

"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{{center|1=ds^2 = \sum_{ij}^{} a_{ij} dx_i dx_j.}}"

"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."

"It is to the doubts about Euclid's parallel postulate, and efforts of such thinkers as Saccheri, Lobachevski, Bolyai, Beltrami, Riemann, and Pasch to settle these doubts, that we owe the whole modern abstract conception of mathematical science."

"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."

"... 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."

"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."

"Edwin Abbott Abbott"

"Euclid’s Elements"

"Carl Friedrich Gauss"

"Nikolai Ivanovich Lobachevsky"

"Mathematics"

"History of mathematics"

"Bernhard Riemann"

"Howard P. Robertson"

"Differential geometry originally sneaked into theoretical physics through Einstein's theory of general relativity."

"現代幾何学においても、特性類の重要性は減ずるどころかますます大きくなっている。しかも単なるコホモロジー類としてだけでではなく、それを表わす微分形式自 身の詳細な解析がなされるなど、より深い役割を果たすようになってきている。たとえば、de RhamコホモロジーとRiemann計量の関わりを記述する理論である調和積 分論を、ずっと大きな枠組みのなかで一般化するという、壮大な試みが進行中である。そしてこれらの新しい展開のなかで、微分形式は、たとえてみれば生物にとっ ての水や空気のような役割を果たしているといっても過言ではないだろう。"

"Symplectic geometry originates with Hamilton's formulation of optical laws."

"Group theory is the mathematical expression of symmetry, providing appropriate mathematical tools for the implementation of symmetry considerations in theoretical physics."

"The introduction of the digit 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps...-Alexander Grothendieck"

"... The way in which string theory addresses the cosmological constant problem can be summarized as follows: • Fundamentally, space is nine-dimensional. There are many distinct ways (perhaps 10500) of turning nine-dimensional space into three-dimensional space by compactifying six dimensions. ... • Distinct compactifications correspond to different three-dimensional metastable vacua with different amounts of vacuum energy. In a small fraction of vacua, the cosmological constant will be accidentally small. • All vacua are dynamically produced as large, widely separated regions in space-time. • Regions with Λ 1 contain at most a few bits of information and thus no complex structures of any kind. Therefore, observers find themselves in regions with Λ ≪ 1."

"The theoretical view of the actual universe, if it is in correspondence to our reasoning, is the following. The curvature of space is variable in time and place, according to the distribution of matter, but we may roughly approximate it by means of a spherical space. ...this view is logically consistent, and from the standpoint of the general theory of relativity lies nearest at hand [i.e. is most obvious]; whether, from the standpoint of present astronomical knowledge, it is tenable, will not be discussed here. In order to arrive at this consistent view, we admittedly had to introduce an extension of the field equations of gravitation, which is not justified by our actual knowledge of gravitation. It is to be emphasized, however, that a positive curvature of space is given by our results, even if the supplementary term [] is not introduced. The term is necessary only for the purpose of making possible a quasi-static distribution of matter, as required by the fact of the small velocity of the stars."

"Most constants are adjusted with a deviation of one percent, which means that if the value differs by one percent everything collapses. Physicists can certainly claim that this is a fluke, but it must be acknowledged that this cosmological constant is adjusted to an accuracy of 1/10120. No one thinks that this is solely a fluke. It is the most extreme example of hyperfine regulation... (Leonard Susskind)"

"Much later, when I was discussing cosmological problems with Einstein, he remarked that the introduction of the cosmological term was the biggest blunder he ever made in his life."

"After putting the finishing touches on general relativity in 1915, Einstein applied his new equations for gravity to a variety of problems. ... Despite the mounting successes of general relativity, for years after he first applied his theory to the most immense of all challenges—understanding the entire universe—Einstein absolutely refused to accept the answer that emerged from the mathematics. Before the work of Friedmann and Lemaître... Einstein, too, had realized that the equations of general relativity showed that the universe could not be static; the fabric of space could stretch or it could shrink, but it could not maintain a fixed size. This suggested that the universe might have had a definite beginning, when the fabric was maximally compressed, and might even have a definite end. Einstein stubbornly balked at this... because he and everyone else "knew" that the universe was eternal and, on the largest scales, fixed and unchanging. Thus, notwithstanding the beauty and successes of general relativity, Einstein reopened his notebook and sought a modification of the equations... It didn't take him long. In 1917 he achieved the goal by introducing a new term... the cosmological constant."

"The miracle of physics that I'm talking about here is something that was actually known since the time of Einstein's general relativity; that gravity is not always attractive. Gravity can act repulsively. Einstein introduced this in 1916... in the form of the cosmological constant, and the original motivation of modifying the equations of general relativity to allow this was because Einstein thought that the universe was static, and he realized that ordinary gravity would cause the universe to collapse if it was static. ...The fact that general relativity can support this gravitational repulsion, still being consistent with all the principles that general relativity incorporates, is the important thing which Einstein himself did discover.."

"In 1917 de Sitter showed that Einstein's field equations could be solved by a model that was completely empty apart from the cosmological constant—i.e. a model with no matter whatsoever, just dark energy. This was the first model of an expanding universe. although this was unclear at the time. The whole principle of general relativity was to write equations for physics that were valid for all observers, independently of the coordinates used. But this means that the same solution can be written in various different ways... Thus de Sitter viewed his solution as static, but with a tendency for the rate of ticking clocks to depend on position. This phenomenon was already familiar in the form of gravitational time dilation... so it is understandable that the de Sitter effect was viewed in the same way. It took a while before it was proved (by Weyl, in 1923) that the prediction was of a redshifting of spectral lines that increased linearly with distance (i.e. Hubble's law). ..."

"Even today, our picture of a world woven together by a gravitational force, and electromagnetic force, a strong force, and a weak force may be incomplete. Astronomers are gathering evidence that an additional fundamental interaction, a repulsive effect opposite to gravity, may be at work over vast distances and possibly changing with time."

"In Einstein's scheme there was no end, no outside. Shoot an arrow or a light beam infinitely far in any direction and it would come back and hit you in the butt. ...But there was a problem with the curved-back universe. Such a configuration was unstable, it would fly apart or collapse. Einstein didn't know about galaxies. He thought, and was reassured as much by the best astronomers of the time, that the universe was a static cloud of stars. To explain why his curved universe didn't collapse like a struck tent, therefore, he fudged his equations with a term he called the cosmological constant, which produced a long-range repulsive force to counteract cosmic gravity. It made the equations ugly and he never really liked it. That was in 1917, twelve years before Hubble showed that the universe was full of galaxies rushing away from each other."

"When the Higgs field froze and symmetry broke, Tye and Guth knew, energy had to be released... Under normal circumstance this energy went into beefing up the masses of particles like the weak force bosons that had been massless before. If the universe supercooled, however, all this energy would remain unreleased... according to Einstein, it was the density of matter and energy in the universe that determined the dynamics of space-time. ...The issue of vacuum energy had been a tricky problem for physics ever since Einstein. According to quantum theory, even the ordinary "true" vacuum should be boiling with energy—infinite energy... due to the the so-called s that produced the transient dense dance of s. This energy... could exert a repulsive force on the cosmos just like the infamous cosmological constant... quantum theories had reinvented it in the form of vacuum fluctuations. The orderly measured pace of the expansion of the universe suggested strongly that the cosmological constant was zero, yet quantum theory suggested it was infinite. Not even Hawking claimed to understand the cosmological constant problem... a trapdoor deep at the heart of physics."

"It's a term that Einstein recognized as allowed by his theory — he threw it in and then, in disgust, threw it out again ... It's back!"

"[Einstein's cosmological constant] is a name without any meaning. ...We have, in fact, not the slightest inkling of what it's real significance is. It is put in the equations in order to give the greatest possible degree of mathematical generality."

"There is no direct observational evidence for the curvature [of space], the only directly observed data being the mean density and the expansion, which latter proves that the actual universe corresponds to the non-statical case. It is therefore clear that from the direct data of observation we can derive neither the sign nor that value of the curvature, and the question arises whether it is possible to represent the observed facts without introducing the curvature at all. Historically the term containing the 'cosmological constant λ' was introduced into the field equations in order to enable us to account theoretically for the existence of a finite mean density in a static universe. It now appears that in the dynamical case this end can be reached without the introduction of λ."

"It was early 1932, when Einstein and I both were at the California Institute of Technology in Pasedena, and we just decided to look for a simple relativistic model that agreed reasonably well with the known observational data, namely, the Hubble recession rate and the mean density of matter in the universe. So we took the space curvature to be zero and also the cosmological constant and the pressure term to be zero, and then it follows straightforwardly that the density is proportional to the square of the Hubble constant. It gives a value for the density that is high, but not impossibly high. That's about all there was to it. It was not an important paper, although Einstein apparently thought that it was. He was pleased to have a simple model with no cosmological constant. That's it."

"String theory seems to be incompatible with a world in which a cosmological constant has a positive sign, which is what the observations indicate."

"The most far-reaching implication of general relativity... is that the universe is not static, as in the orthodox view, but is dynamic, either contracting or expanding. Einstein, as visionary as he was, balked at the idea... One reason... was that, if the universe is currently expanding, then... it must have started from a single point. All space and time would have to be bound up in that "point," an infinitely dense, infinitely small "singularity." ...this struck Einstein as absurd. He therefore tried to sidestep the logic of his equations, and modified them by adding... a "cosmological constant." The term represented a force, of unknown nature, that would counteract the gravitational attraction of the mass of the universe. That is, the two forces would cancel... it is the kind of rabbit-out-of-the-hat idea that most scientists would label ad-hoc. ...Ironically, Einstein's approach contained a foolishly simple mistake: His universe would not be stable... like a pencil balanced on its point."

"Our particular laws are not at all unique. ...they could change from place to place and from time to time. The Laws of Physics are much like the weather... controlled by invisible influences in space almost the same way as that temperature, humidity, air pressure, and wind velocity control how rain and snow and hail form. ...The Landscape... is the space of possibilities... all the possible environments permitted by the theory. ...[T]heoretical physicists ...have always believed that the laws of nature are the unique, inevitable consequence of some elegant mathematical principle. ...the empirical evidence points much more convincingly to the opposite conclusion. The universe has more in common with a Rube Goldberg machine than with a unique consequence of mathematical symmetry. ...Two key discoveries are driving the paradigm shift—the success of inflationary cosmology and the existence of a small cosmological constant."

"At about the time of Malcadena's discovery, physicists started to become convinced (by cosmologists) that we live in a world with a nonvanishing cosmological constant [footnote: 10-23 in Planck units...[t]he incredible smallness... had fooled almost all physicists into believing that it didn't exist.], smaller by far than any other physical constant... the main determinant of the future history of the universe... also known as ... a thorn in the side of physicists for almost a century. ...If \Lambda is positive, the cosomological term creates a repulsive force that increases with distance; if it is negative, the new force is attractive; if \Lambda is zero, there is no new force and we can ignore it."

"The cosmological constant['s]... most important consequence: the repulsive force, acting at cosmological distances, causes space to expand exponentially. There is nothing new about the universe expanding, but without a cosmological constant, the rate of expansion would gradually slow down. Indeed, it could even reverse itself and begin to contract, eventually imploding in a giant cosmic crunch. Instead, as a consequence of the cosmological constant, the universe appears to be doubling in size about every fifteen billion years, and all indications are that it will do so indefinitely."

"The models of Einstein and de Sitter are static solutions of Einstein's modified gravitational equations for a world-wide homogeneous system. They both involve a positive cosmological constant &lambda;, determining the curvature of space. If this constant is zero, we obtain a third model in classical infinite Euclidean space. This model is empty, the space-time being that of Special Relativity. It has been shown that these are the only possible static world models based on Einstein's theory. In 1922, Friedmann... broke new ground by investigating non-static solutions to Einstein's field equations, in which the radius of curvature of space varies with time. This Possibility had already been envisaged, in a general sense, by Clifford in the eighties."

"It is quite easy to include a weight for empty space in the equations of gravity. Einstein did so in 1917, introducing what came to be known as the cosmological constant into his equations. His motivation was to construct a static model of the universe. To achieve this, he had to introduce a negative mass density for empty space, which just canceled the average positive density due to matter. With zero total density, gravitational forces can be in static equilibrium. Hubble's subsequent discovery of the expansion of the universe, of course, made Einstein's static model universe obsolete. ...The fact is that to this day we do not understand in a deep way why the vacuum doesn't weigh, or (to say the same thing in another way) why the cosmological constant vanishes, or (to say it in yet another way) why Einstein's greatest blunder was a mistake."

"De Sitter proposed three types of nonstatic universes: the oscillating universes and the expanding universes of the first or second kiind. The main characteristic of the expanding "family" of the first kiind is that the radius is continually increasing from a definite initial time when it had the value zero. The universe becomes infinitely large after an infinite time. In the second kind... the radius possesses at the initial time a definite minimum value... in the Einstein model... the cosmological constant is supposed to be equal to the reciprocal of R2, whereas de Sitter computed for his interpretation the constant to be equal to 3/R2. Whitrow correctly points out the significant fact that in special relativity the cosmological constant is omitted..."

"The theory ... noncommutative geometry ... rests on two essential points:"

"The use of noncommutative geometry (NCG) as a tool for constructing particle physics models originated in the 1990s ... The main idea can be heuristically regarded as similar to the idea of "extra dimensions" in String Theory, except for the fact that the nature and scope of these extra dimensions is quite different. In the NGC model one considers an "almost commutative geometry", which is a product (or locally a product in a more refined and more recent version ...) of a four-dimensional spacetime manifold and a space of inner degrees of freedom, which is a "finite" noncommutative space, whose ring of functions is a sum of matrix algebras. According to the choice of this finite geometry, one obtains different possible particle contents for the resulting physics model. The physical content is expressed through an action functional, the spectral action ..., which is defined for more general noncommutative spaces, in terms of the spectrum of a Dirac operator."

"Gaitsgory gives an informal introduction to the geometric Langlands program. This is a new and very active area of research which grew out of the theory of automorphic forms and is closely related to it. Roughly speaking, in this theory we everywhere replace functions—like automorphic forms—by sheaves on algebraic varieties; this allows us to use powerful methods of algebraic geometry in order to construcut "automorphic sheaves.""

"Langlands outlined the first version of the programme in 1967, when he was a young mathematician visiting the IAS. His starting point was the theory of algebraic equations (such as the quadratic, or second-degree, equations that children learn in school). In the 1800s, French mathematician Évariste Galois discovered that, in general, equations of higher degree can be solved only partially. ... Inspired by subsequent developments in Galois’s theory, Langlands’ approach allowed researchers to translate algebra problems into the ‘language’ of harmonic analysis, the branch of mathematics that breaks complex waveforms down into simpler, sinusoidal building blocks. In the 1980s, Vladimir Drinfel’d, a Ukrainian-born mathematician now at the University of Chicago in Illinois, and others proposed a similar connection between geometry and harmonic analysis. Although this idea seemed to be only loosely inspired by the Langlands programme, mathematicians subsequently found stronger evidence that the two fields are connected. (Drinfel’d received a Fields Medal in 1990.)"

"Hecke transformations are one of the most important ingredients in geometric Langlands. What they mean in terms of physics had bothered me for a long time and eventually had been the last major stumbling block in interpreting geometric Langlands in terms of physics and gauge theory. Finally, while on an airplane flying home from Seattle, it struck me that a Hecke transformation in the context of geometric Langlands is simply an algebraic geometer’s way to describe the effects of a “’t Hooft operator” of quantum gauge theory. I had never worked with ’t Hooft operators, but they were familiar to me, as they had been introduced in the late 1970s as a tool in understanding quantum gauge theory. The basics of how to work with ’t Hooft operators and what happens to them under electric-magnetic duality were well known, so once I could reinterpret Hecke transformations in terms of ’t Hooft operators, many things were clearer to me."

"If our geometry is to resemble differential geometry we must adjoin some uniqueness properties. Now in those geometries the geodesics, and more generally the externals in the calculus of variations, are given by differential equations of the second order, and under the hypotheses usually made in those fields, there is just one solution through a given line element. Thus a geodesics has a unique prolongation, though the shortest geodesic are joining two points even on simple surfaces such as the sphere, need not be unique."

"A geodesic that is not a null geodesic has the property that ∫ds, taken along a section of the track with the end points P and Q, is stationary if one makes a small variation of the track keeping the end points fixed."

"String topology has been used to study closed geodesics on Riemannian manifolds through Morse theory on the energy functional ..."

"We begin by recalling that geodesics can be obtained as solutions of the Euler-Lagrange equation of a Lagrangian given by the kinetic energy. We define symplectic and contact manifolds and we set up the basic geometry of the tangent bundle; we introduce the connection map, horizontal and vertical subbundles, the Sasaki metric, the symplectic form and the contact form. We describe the main properties of these objects and we show that the geodesic flow is a Hamiltonian flow. Also, when we restrict the geodesic flow to the unit sphere bundle of the manifold, we obtain a contact flow. The contact form naturally induces a probability measure that is invariant under the geodesic flow and is called the Liouville measure."

"The idea of a parallel displacement along some given curve in a two-dimensional surface can be given an intuitive interpretation. Suppose the surface is developable. Then we can unroll it on to a plane and parallel-displace vectors in the plane. The surface is then rolled back and we have the required parallel-transported vector. If a given surface is not developable, we must first select a path for parallel transport, then erect a tangent plane at each point of the path. These tangent planes will envelope a developable surface. This new developable surface can then be unrolled and the operations of parallel transport and rerolling carried out. If the curve along which the parallel displacement is to be carried out happens to be a geodesic, it becomes a straight line when unrolled on to a plane. It is then clear that the angle between a geodesic and a vector remains unchanged in a parallel displacement."

"Like Moses, André Weil caught sight of the Promised Land, but unlike Moses, he was unable to cross the Red Sea on dry land, nor did he have an adequate vessel. For his own work, he had already reconstructed algebraic geometry on a purely algebraic basis, in which the notion of a “field” is predominant. To create the required arithmetic geometry, it is necessary to replace the algebraic notion of a field by that of a commutative ring, and above all to invent an adaptation of homological algebra able to tame the problems of arithmetic geometry. André Weil himself was not ignorant of these techniques nor of these problems, and his contributions are numerous and important (adeles, the so-called Tamagawa number, class field theory, deformation of discrete subgroups of symmetries). But André Weil was suspicious of “big machinery” and never learned to feel familiar with sheaves, homological algebra or categories, contrarily to Grothendieck, who embraced them wholeheartedly."