Geometry

"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."

"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."

"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."

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

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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

"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."

"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."

"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."

"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.""

"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."

"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..."

"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."

"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."