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’?"