Quotes about geometry (302 quotes, page 2)

“When I had the honour of his conversation, I endeavoured to learn his thoughts upon mathematical subjects, and something historical concerning his inventions, that I had not been before acquainted with. I found, he had read fewer of the modern mathematicians, than one could have expected; but his own prodigious invention readily supplied him with what he might have an occasion for in the pursuit of any subject he undertook. I have often heard him censure the handling geometrical subjects by algebraic calculations; and his book of Algebra he called by the name of Universal Arithmetic, in opposition to the injudicious title of Geometry, which Des Cartes had given to the treatise, wherein he shews, how the geometer may assist his invention by such kind of computations. He frequently praised Slusius, Barrow and Huygens for not being influenced by the false taste, which then began to prevail. He used to commend the laudable attempt of Hugo de Omerique to restore the ancient analysis, and very much esteemed Apollonius's book De sectione rationis for giving us a clearer notion of that analysis than we had before.”

Henry Pemberton (1694–1771) British doctor

Preface; The bold passage is subject of the 1809 article " Remarks on a Passage in Castillione's Life' of Sir Isaac Newton http://books.google.com/books?id=BS1WAAAAYAAJ&pg=PA519." By John Winthrop, in: The Philosophical Transactions of the Royal Society of London, from Their Commencement, in 1665, to the Year 1800: 1770-1776: 1770-1776. Charles Hutton et al. eds. (1809) p. 519.
Preface to View of Newton's Philosophy, (1728)

Books , Mathematics , Learning , Universe

“We can not… escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction. …Neither can this rule come to us from experience… This rule, inaccessible to analytic demonstration and to experience, is the veritable type of the synthetic a priori judgment. On the other hand, we can not think of seeing in it a convention, as in some of the postulates of geometry. …it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it.”

Henri Poincaré book Science and Hypothesis

Source: Science and Hypothesis (1901), Ch. I. (1905) Tr. George Bruce Halstead

Thinking

“More than any of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundations of the science, insisting upon accurate definitions, clear assumptions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic is… clear. …That Plato should hold the view… is not a cause for surprise. The world's thinkers have always held it. No man has ever created a mathematical theory for practical purposes alone. The applications of mathematics have generally been an afterthought.”

David Eugene Smith (1860–1944) American mathematician

Source: History of Mathematics (1923) Vol.1, p. 90

World , Mathematics , Science

“1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves.
2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra.
3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry.
Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics.”

Nicholas Murray Butler (1862–1947) American philosopher, diplomat, and educator

Editor's Introduction, The Teaching of Elementary Mathematics https://books.google.com/books?id=NKoAAAAAMAAJ (1906) by David Eugene Smith

Mathematics , Study , Universe , Time

“It has been said that to appreciate what virtue and morals mean, men must live virtuous and moral lives. It is equally true, that a knowledge of the objects of science is not to be attained by any scheme of definitions, however carefully contrived. He who would know what geometry is, must venture boldly into its depths and learn to think and feel as a geometer.”

James Joseph Sylvester (1814–1897) English mathematician

J. J. Sylvester. "A Probationary Lecture on Geometry", Collected Mathematical Papers, Vol. 2 (1908), p. 9 https://babel.hathitrust.org/cgi/pt?id=miun.aas8085.0002.001;view=1up;seq=25

Men , Feeling , Knowledge , Thinking

“Archytas of Tarentum found the two mean proportionals by a very striking construction in three dimensions, which shows that solid geometry, in the hands of Archytas at least, was already well advanced. The construction was usually called mechanical, which it no doubt was in form, though in reality it was in the highest degree theoretical. It consisted in determining a point in space as the intersection of three surfaces: (a) a cylinder, (b) a cone, (c) an "anchor-ring" with internal radius = 0.”

Thomas Little Heath (1861–1940) British civil servant and academic

Achimedes (1920)

Reality

“It is to the director of workshops and factories that it is suitable to make, by means of geometry and applied mechanics, a special study of all the ways to economize the efforts of workers… For a man to be a director of others, manual work has only a secondary importance; it is his intellectual ability (force intellectuelle) that must put him in the top position, and it is in instruction such as that of the Conservatory of the Arts and Professions, that he must develop it.”

Charles Dupin (1784–1873) French mathematician

Charles Dupin (1826), Geometrie et Mechanique des Arts et Metiers et des Beaux Arts Paris: Bachelier; Cited and translated by John Hoaglund, "Management Before Frederick Taylor," p. 30.; and cited in Wren & Bedeian (2005, 74)

Art , Way , Effort , Study

“This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles…. for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals.”

John Wallis (1616–1703) English mathematician

Arithmetica Infinitorum (1656)

“Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.”

William Kingdon Clifford On the Space-Theory of Matter

Abstract
On the Space-Theory of Matter (read Feb 21, 1870)

Help

“It is less than four years since cohomological methods (i. e. methods of Homological Algebra) were introduced into Algebraic Geometry in Serre's fundamental paper[11], and it seems certain that they are to overflow the part of mathematics in the coming years, from the foundations up to the most advanced parts. … [11] Serre, J. P. Faisceaux algébriques cohérents. Ann. Math. (2), 6, 197–278”

Alexander Grothendieck (1928–2014) French mathematician

1955
[1960, Cambridge University Press, The cohomology theory of abstract algebraic varieties, Proc. Internat. Congress Math.(Edinburgh, 1958), 103–118, https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/CohomologyVarieties.pdf] (p. 103)

Mathematics , Parting

“Sane judgment abhors nothing so much as a picture perpetrated with no technical knowledge, although with plenty of care and diligence. Now the sole reason why painters of this sort are not aware of their own error is that they have not learnt Geometry, without which no one can either be or become an absolute artist; but the blame for this should be laid upon their masters, who are themselves ignorant of this art.”

Albrecht Dürer (1471–1528) German painter, printmaker, mathematician, and theorist

The Art of Measurement (1525).

Art , Error , Knowledge

“The surfaces of three-dimensional space are distinguished from each other not only by their curvature but also by certain more general properties. A spherical surface, for instance, differs from a plane not only by its roundness but also by its finiteness. Finiteness is a holistic property. The sphere as a whole has a character different from that of a plane. A spherical surface made from rubber, such as a balloon, can be twisted so that its geometry changes…. but it cannot be distorted in such a way as that it will cover a plane. All surfaces obtained by distortion of the rubber sphere possess the same holistic properties; they are closed and finite. The plane as a whole has the property of being open; its straight lines are not closed. This feature is mathematically expressed as follows. Every surface can be mapped upon another one by the coordination of each point of one surface to a point of the other surface, as illustrated by the projection of a shadow picture by light rays. For surfaces with the same holistic properties it is possible to carry through this transformation uniquely and continuously in all points. Uniquely means: one and only one point of one surface corresponds to a given point of the other surface, and vice versa. Continuously means: neighborhood relations in infinitesimal domains are preserved; no tearing of the surface or shifting of relative positions of points occur at any place. For surfaces with different holistic properties, such a transformation can be carried through locally, but there is no single transformation for the whole surface.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

Way , Mathematics , Tear , Light

“First, the physicists in the persons of Faraday and Maxwell, proposed the "electromagnetic field" in contradistinction to matter, as a reality of a different category. Then, during the last century, the mathematicians, … secretly undermined belief in the evidence of Euclidean Geometry. And now, in our time, there has been unloosed a cataclysm which has swept away space, time, and matter hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope and entailing a deeper vision. This revolution was promoted essentially by the thought of one man, Albert Einstein.”

Hermann Weyl (1885–1955) German mathematician

Introduction
Space—Time—Matter (1952)

Nature , Reality , Science , Time

“Above all, do not allow yourself to be bewitched by the evil charms of geometry.”

François Fénelon (1651–1715) Catholic bishop

Sur-tout ne vous laissez point ensorceler par les attraits diaboliques de la géométrie.
Lettres Spirituelles, no. 59, cited from Correspondance de Fénelon, archevêque de Cambrai (Paris: Ferra Jeune, 1827) vol. 5, p. 514; translation from Georges Duby and Michelle Perrot (eds.) A History of Women in the West (Cambridge, Mass.: Belknap Press of Harvard University Press, 1994) vol. 3, p. 405.Œuvres complètes De François de Salignac De La Mothe Fénélon. TOME V Briand 1810 LETTRE CXLII (142) p.106.

Evil

“I thought fit to… explain in detail in the same book the peculiarity of a certain method, by which it will be possible… to investigate some of the problems in mathematics by means of mechanics. This procedure is… no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards… But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.”

Archimedes book The Method of Mechanical Theorems

The Method of Mechanical Theorems

Books , Mathematics , Question , Knowledge

“But if men either knew themselves thoroughly, or had the slightest knowledge of God, they would never claim as their own a divine and immortal nature; nor would they think themselves something great because they have made for themselves gridirons, basins, and bowls, because they have made under-shirts, outer-shirts, cloaks, plaids, robes of state, knives, cuirasses and swords, mattocks, hatchets, ploughs. Never, I say, carried away by pride and arrogance, would they believe themselves to be deities of the first rank, and fellows of the highest in his exaltation, because they had devised the arts of grammar, music, oratory, and geometry. For we do not see what is so wonderful in these arts, that because of their discovery the soul should be believed to be above the sun as well as all the stars, to surpass both in grandeur and essence the whole universe, of which these are parts.”

Arnobius (255–327) Christian apologist

Book II, § 19
Adversus Nationes

Men , God , Art , Music

“A proper school should teach nothing but bookkeeping, agriculture, geometry, dead languages made deader by leaving out all the amusing literature, and the Hebrew Bible as interpreted by men superbly trained to ignore contradictions, men technically called "Fundamentalists."”

Sinclair Lewis book Elmer Gantry

Elmer Gantry (1927)

Men , School , Literature , Training

“All this (the early excitement of Cybernetics) is now history, and in the decade which elapsed since these early baby steps of interdisciplinary communication, many more threads were picked up and interwoven into a remarkable tapestry of knowledge and endeavour: Bionics. It is good omen that at the right time the right name was found. For, bionics extends a great invitation to all who are willing not to stop at the investigation of a particular function or its realization, but to go on and to seek the universal signiﬁcance of these functions in living or artiﬁcial organisms.
The reader who goes through the following papers which constitute the transactions of the ﬁrst symposium held under the name Bionics will be surprised by the multitude of astonishing and unforeseen connections between concepts he believed to be familiar with. For instance, a couple of years ago, who would have thought to relate the reliability problem to multi-valued logics; or, who would have thought that integral or differential geometry would serve as an adequate tool in the theory of abstraction? It is hard to say in all these cases who was teaching whom: The life-sciences the engineering sciences, or vice versa? And rightly so, for it guarantees optimal information ﬂow, and everybody gains…”

Heinz von Foerster (1911–2002) Austrian American scientist and cybernetician

Von Foerster (1960) as cited in Peter M. Asaro (2007). "Heinz von Foerster and the Bio-Computing Movements of the 1960s," http://cybersophe.org/writing/Asaro%20HVF%26BCL.pdf
1960s

Life , History , Communication , Optimism

“As Gauss first pointed out, the problem of cyclotomy, or division of the circle into a number of equal parts, depends in a very remarkable way upon arithmetical considerations. We have here the earliest and simplest example of those relations of the theory of numbers to transcendental analysis, and even to pure geometry, which so often unexpectedly present themselves, and which, at first sight, are so mysterious.”

George Ballard Mathews (1861–1922) British mathematician

Part 1, sect. 167.
Theory of Numbers, 1892

Way , Problem , Parting , Dependency

“Paradoxical as it may seem, a Latin prose or a geometry problem, even though they are done wrong, may be of a great service one day, provided we devote the right kind of effort to them. Should the occasion arise, they can one day make us better able to give someone in affliction exactly the help required to save him, at the supreme moment of his need.”

Simone Weil (1909–1943) French philosopher, Christian mystic, and social activist

Waiting on God (1950), Reflections on the Right Use of School Studies with a View to the Love of God

Problem , Effort , Affliction , Help

“The arithmetization of mathematics… which began with Weierstrass… had for its object the separation of purely mathematical concepts, such as number and correspondence and aggregate, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics.
These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words, may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure thought.”

Tobias Dantzig (1884–1956) American mathematician

p, 125
Number: The Language of Science (1930)

Mathematics , Idea

“1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves.
2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra.
3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry.
Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics.”

David Eugene Smith (1860–1944) American mathematician

David Eugene Smith, "Editor's Introduction," in: The Teaching of Elementary Mathematics https://books.google.com/books?id=NKoAAAAAMAAJ (1906)

Mathematics , Study , Universe , Time

“As result a new kind of theory to be applied on a class of motions … these geometric motions are that which acquire different parts of a system of bodies, without neither perturb themselves nor the other and consequently these motions do not depend of the action or reaction among the bodies, but only upon the conditions of their connections, and thus being determined only by geometry and not dependent of the rules of dynamics.”

Lazare Carnot (1753–1823) French political, engineering and mathematical figure

On geomatric motion. A History of the Work Concept: From Physics to Economics, by Agamenon Oliveira, p. 154.

Parting , Dependency

“The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid did to geometry.”

Eric Temple Bell (1883–1960) mathematician and science fiction author born in Scotland who lived in the United States for most of his li…

The Search for Truth (1934), p. 191

Way , Thinking

“In order to see the difference which exists between… studies,—for instance, history and geometry, it will be useful to ask how we come by knowledge in each. Suppose, for example, we feel certain of a fact related in history… if we apply the notions of evidence which every-day experience justifies us in entertaining, we feel that the improbability of the contrary compels us to take refuge in the belief of the fact; and, if we allow that there is still a possibility of its falsehood, it is because this supposition does not involve absolute absurdity, but only extreme improbability.
In mathematics the case is wholly different… and the difference consists in this—that, instead of showing the contrary of the proposition asserted to be only improbable, it proves it at once to be absurd and impossible. This is done by showing that the contrary of the proposition which is asserted is in direct contradiction to some extremely evident fact, of the truth of which our eyes and hands convince us. In geometry, of the principles alluded to, those which are most commonly used are—
I. If a magnitude is divided into parts, the whole is greater than either of those parts.
II. Two straight lines cannot inclose a space.
III. Through one point only one straight line can be drawn, which never meets another straight line, or which is parallel to it.
It is on such principles as these that the whole of geometry is founded, and the demonstration of every proposition consists in proving the contrary of it to be inconsistent with one of these.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

Source: On the Study and Difficulties of Mathematics (1831), Ch. I.

Truth , Feeling , History , Mathematics

“Only geometry can hand us the thread [which will lead us through] the labyrinth of the continuum’s composition, the maximum and the minimum, the infinitesimal and the infinite; and no one will arrive at a truly solid metaphysic except he who has passed through this [labyrinth].”
Nam filum labyrintho de compositione continui deque maximo et minimo ac indesignabili at que infinito non nisi geometria praebere potest, ad metaphysicam vero solidam nemo veniet, nisi qui illac transiverit.

Gottfried Leibniz (1646–1716) German mathematician and philosopher

Dissertatio Exoterica De Statu Praesenti et Incrementis Novissimis Deque Usu Geometriae (Spring 1676)
Source: Leibniz, Leibnizens Mathematische Schriften, Herausgegeben Von C.I. Gerhardt. Bd. 1-7. 1850-1863. Halle. The quotation is found in vol. 7. on page 326 in ”Dissertatio Exoterica De Statu Praesenti et Incrementis Novissimis Deque Usu Geometriae”. Link https://archive.org/stream/leibnizensmathe12leibgoog
Source: Geometry and Monadology: Leibniz's Analysis Situs and Philosophy of Space by Vincenzo de Risi. Page 123. Link https://books.google.no/books?id=2ptGkzsKyOQC&lpg=PA123&ots=qz2aKxAYtp&dq=Dissertatio%20Exoterica%20De%20Statu%20Praesenti%20et%20Incrementis%20Novissimis%20Deque%20Usu%20Geometriae%E2%80%9D&hl=no&pg=PA123#v=onepage&q&f=false

“It is not true, out of geometry, that the mathematical sciences are, in all their parts those models of finished accuracy which many suppose. The extreme boundaries of analysis have always been as imperfectly understood as the tract beyond the boundaries was absolutely unknown. But the way to enlarge the settled country has not been by keeping within it, but by making voyages of discovery, and I am perfectly convinced that the student should be exercised in this manner; that is, that he should be taught how to examine the boundary, as well as how to cultivate the interior. …allowing all students whose capacity will let them read on the higher branches of applied mathematics, to have each his chance of being led to the cultivation of those parts of analysis on which rather depends its future progress than its present use in the sciences of matter.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

The Differential and Integral Calculus (1836)

Way , Exercises , Mathematics , Science

“The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858.”

Richard Dedekind (1831–1916) German mathematician

p, 125
Stetigkeit und irrationale Zahlen (1872)

Way , Time , Dependency

“To give here an elaborate account of Pappus would be to create a false impression. His work is only the last convulsive effort of Greek geometry which was now nearly dead and was never effectually revived. It is not so with Ptolemy or Diophantus. The trigonometry of the former is the foundation of a new study which was handed on to other nations indeed but which has thenceforth a continuous history of progress. Diophantus also represents the outbreak of a movement which probably was not Greek in its origin, and which the Greek genius long resisted, but which was especially adapted to the tastes of the people who, after the extinction of Greek schools, received their heritage and kept their memory green. But no Indian or Arab ever studied Pappus or cared in the least for his style or his matter. When geometry came once more up to his level, the invention of analytical methods gave it a sudden push which sent it far beyond him and he was out of date at the very moment when he seemed to be taking a new lease of life.”

James Gow (scholar) (1854–1923) scholar

p, 125
A Short History of Greek Mathematics (1884)

Life , People , School , History

“Arithmetic and geometry, according to Plato, are the two wings of the mathematician. The object indeed of all mathematical questions, is to determine the ratios of numbers, or of magnitudes; and it may even be said, to continue the comparison of the ancient philosopher, that arithmetic is the mathematician's right wing; for it is an incontestable truth, that geometrical determinations would, for the most part, present nothing satisfactory to the mind, if the ratios thus determined could not be reduced to numerical ratios. This justifies the common practice, which we shall here follow, of beginning with arithmetic.”

Jacques Ozanam (1640–1718) French mathematician

Source: Recreations in Mathematics and Natural Philosophy, (1803), p. 2

Truth , Mathematics , Question , Parting

“Suppose the reasoning centers of the brain can get their hands on the mechanisms that plop shapes into the array and that read their locations out of it. Those reasoning demons can exploit the geometry of the array as a surrogate for keeping certain logical constraints in mind. Wealth, like location on a line, is transitive: if A is richer than B, and B is richer than C, then A is richer than C. By using location in an image to symbolize wealth, the thinker takes advantage of the transitivity of location built into the array, and does not have to enter it into a chain of deductive steps. The problem becomes a matter of plop down and look up. It is a fine example of how the form of a mental representation determines what is easy or hard to think.”

Steven Pinker book How the Mind Works

Source: How the Mind Works (1997), p. 291

Problem , Thinking , Reading , Wealth

“In all these congruence geometries, except the Euclidean, there is at hand a natural unit of length R = \frac{1}{K^\frac{1}{2}}; this length we shall, without prejudice, call the "radius of curvature" of the space.”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

Nature

“Once a definition of congruence is given, the choice of geometry is no longer in our hands; rather, the geometry is now an empirical fact.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“The objects of abstract Geometry possess in absolute precision properties which are only approximately realized in the corresponding objects of physical Geometry.”

E. W. Hobson (1856–1933) British mathematician

Source: Squaring the Circle (1913), p. 5

“Take the case of a famous problem which plays a great part in the history of Greek geometry, the doubling of the cube, or its equivalent, the finding of two mean proportionals in continued proportion between two given straight lines. …if all the recorded solutions are collected together, it is much easier to see the relations, amounting in some cases to substantial identity, between them, and to get a comprehensive view of the history of the problem. I have therefore dealt with this problem in a separate section of the chapter devoted to 'Special Problems,' and I have followed the same course with the other famous problems of squaring the circle and trisecting any angle.”

Thomas Little Heath (1861–1940) British civil servant and academic

A History of Greek Mathematics (1921) Vol. 1. From Thales to Euclid

History , Problem , Parting

“The fact that arithmetic and geometry took such a notable step forward… was due in no small measure to the introduction of Egyptian papyrus into Greece. This event occurred about 650 B. C., and the invention of printing in the 15th century did not more surely effect a revolution in thought than did this introduction of writing material on the northern shores of the Mediterranean Sea just before the time of Thales.”

David Eugene Smith (1860–1944) American mathematician

History of Mathematics (1923) Vol.1

Time , Sea

“Let a thin, flat metal plate be heated… so that the temperature T is not uniform… clamp or otherwise constrain the plate to keep it from buckling… [and] remain [reasonably] flat… Make simple geometric measurements… with a short metal rule, which has a certain coefficient of expansion c… What is the geometry of the plate as revealed by the results of those measurements? …[T]he geometry will not turn out to be Euclidean, for the rule will expand more in the hotter regions… [T]he plate will seem to have a negative curvature K… the kind of structure exhibited… in the neighborhood of a "saddle point."”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“I hold in fact
(1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.
(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.
(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.
(4) That in the physical world nothing else takes place but this variation, subject possibly to the law of continuity.”

William Kingdon Clifford On the Space-Theory of Matter

Abstract
On the Space-Theory of Matter (read Feb 21, 1870)

World , Law , Nature

“"The last thing we expect of you, General, is a lesson in geometry!"”

Pierre-Simon Laplace (1749–1827) French mathematician and astronomer

"La dernière chose que nous attendions de vous, Général, est une leçon de géométrie !"
Laplace to Napoléon, after the latter had reported on some new elementary geometry results[citation needed]

“Some fundamental unity was surely to be discerned either in the matter or the structure of things. The Ionic philosophers chose the former field: Pythagoras took the latter. …The geometry which he had learnt in Egypt was merely practical. …It was natural to nascent philosophy to draw, by false analogies, and the use of a brief and deceptive vocabulary, enormous conclusions from a very few observed facts: and it is not surprising if Pythagoras, having learnt in Egypt that number was essential to the exact description of forms and of the relations of forms, concluded that number was the cause of form and so of every other quality. Number, he inferred, is quantity and quantity is form and form is quality.
Footnote2 Primitive men, on seeing a new thing, look out especially for some resemblance in it to a known thing, so that they may call both by the same name. This developes a habit of pressing small and partial analogies. It also causes many meanings to be at attached to the same word. Hasty and confused theories are the inevitable result.”

James Gow (scholar) (1854–1923) scholar

A Short History of Greek Mathematics (1884)

Men , Nature , Drawing , Falseness

“It [ non-Euclidean geometry ] would be ranked among the most famous achievements of the entire [nineteenth] century, but up to 1860 the interest was rather slight.”

Ivor Grattan-Guinness (1941–2014) Historian of mathematics and logic

Source: The Rainbow of Mathematics: A History of the Mathematical Sciences (2000), p. 400.

“As in geometry, the oblique must be known, as well as the right; and in arithmetic, the odd as well as the even; so in actions of life, who seeth not the filthiness of evil, wanteth a great foil to perceive the beauty of virtue.”

Philip Sidney (1554–1586) English diplomat

Aphorisms of Sir Philip Sidney; with remarks, by Miss Porter (1807), p. 23. London: Longman, Hurst, Rees and Orme https://babel.hathitrust.org/cgi/pt?id=uc1.aa0000617332;view=1up;seq=53

Life , Beauty , Evil

“I will add that I have recently received from Hungary a little paper on non-Euclidean geometry in which I rediscover all my own ideas and results worked out with great elegance… The writer is a very young Austrian officer, the son of one of my early friends, with whom I often discussed the subject in 1798, although my ideas were at that time far removed from the development and maturity which they have received through the original reflections of this young man. I consider the young geometer J. Bolyai a genius of the first rank.”

Carl Friedrich Gauss (1777–1855) German mathematician and physical scientist

Letter to Gerling (1832)

Idea , Time

“But if some mind very different from ours were to look upon some property of some curved line as we do on the evenness of a straight line, he would not recognize as such the evenness of a straight line; nor would he arrange the elements of his geometry according to that very different system, and would investigate quite other relationships as I have suggested in my notes.
We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line.”

Roger Joseph Boscovich (1711–1787) Croat-Italian physicist

"Boscovich's mathematics", an article by J. F. Scott, in the book Roger Joseph Boscovich (1961) edited by Lancelot Law Whyte.
"Transient pressure analysis in composite reservoirs" (1982) by Raymond W. K. Tang and William E. Brigham.
"Non-Newtonian Calculus" (1972) by Michael Grossman and Robert Katz.

Fashion , Nature

“Brook Taylor… in his Methodus Incrementorum Directa et Inversa (1715), sought to clarify the ideas of the calculus but limited himself to algebraic functions and algebraic differential equations. …Taylor's exposition, based on what we would call finite differences, failed to obtain many backers because it was arithmetical in nature when the British were trying to tie the calculus to geometry or to the physical notion of velocity.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 427

Idea , Nature

“I freely confess that I never had taste for study or research either in physics or geometry except in so far as they could serve as a means of arriving at some sort of knowledge of the proximate causes… for the good and convenience of life, in maintaining health, in the practice of some art,… having observed that a good part of the arts is based on geometry, among others that cutting of stone in architecture, that of sundials, that of perspective in particular.”

Girard Desargues (1591–1661) French mathematician and engineer

ca. 1640) as quoted by William Thompson Sedgwick, Harry Walter Tyler, A Short History of Science https://books.google.com/books?id=Wl8AAAAAMAAJ (1917

Life , Health , Art , Knowledge

“Euclidean geometry is only one of several congruence geometries… Each of these geometries is characterized by a real number K, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces… K may be interpreted as the curvature of the surface into the third dimension—whence it derives its name…”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“Mathematics and philosophy are cultivated by two different classes of men: some make them an object of pursuit, either in consequence of their situation, or through a desire to render themselves illustrious, by extending their limits; while others pursue them for mere amusement, or by a natural taste which inclines them to that branch of knowledge. It is for the latter class of mathematicians and philosophers that this work is chiefly intended j and yet, at the same time, we entertain a hope that some parts of it will prove interesting to the former. In a word, it may serve to stimulate the ardour of those who begin to study these sciences; and it is for this reason that in most elementary books the authors endeavour to simplify the questions designed for exercising beginners, by proposing them in a less abstract manner than is employed in the pure mathematics, and so as to interest and excite the reader's curiosity. Thus, for example, if it were proposed simply to divide a triangle into three, four, or five equal parts, by lines drawn from a determinate point within it, in this form the problem could be interesting to none but those really possessed of a taste for geometry. But if, instead of proposing it in this abstract manner, we should say: "A father on his death-bed bequeathed to his three sons a triangular field, to be equally divided among them: and as there is a well in the field, which must be common to the three co-heirs, and from which the lines of division must necessarily proceed, how is the field to be divided so as to fulfill the intention of the testator?"”

Jean-Étienne Montucla (1725–1799) French mathematician

This way of stating it will, no doubt, create a desire in most minds to discover the method of solving the problem; and however little taste people may possess for real science, they will be tempted to try iheir ingenuity in finding the answer to such a question at this.
Source: Preface to Recreations in Mathematics and Natural Philosophy. (1803), p. ii; As cited in: Tobias George Smollett. The Critical Review: Or, Annals of Literature http://books.google.com/books?id=T8APAAAAQAAJ&pg=PA410, Volume 38, (1803), p. 410

Death , Men , People , Books

“Vladimir Voevodsky and his collaborators have provided us with a very interesting candidate-category of motives: a category (of sheaves relative to an extraordinarily fine Grothendieck-style topology on the category of schemes) which in some intuitive sense “softens algebraic geometry” so as to allow for a good notion of homotopy in an algebro-geometric setup and is sufficiently directly connected to concrete algebraic geometry to have already yielded some extraordinary applications. The quest for a full theory of motives is a potent driving force in complex analysis, algebraic geometry, automorphic representation theory, the study of L functions, and arithmetic. It will continue to be so throughout the current century.”

Barry Mazur (1937) American mathematician

"WHAT IS...a Motive?" 2004

Motivation , Sense , Study

“Grégoire de Saint-Vincent… was a Jesuit, taught mathematics in Rome and Prag (1629-1631), and was afterwards called to Spain by Phillip IV as tutor to his son… He wrote two works on geometry [Principia Matheseos Univerales (1651); Exercitationum Mathematicarum Libri quinque (1657)], giving in one of them the quadrature of the hyperbola referred to its asymptotes, and showing that as the area increased in arithmetic series the abscissas increased in geometric series.”

David Eugene Smith (1860–1944) American mathematician

p, 125
History of Mathematics (1923) Vol.1

Mathematics

“There is no single rule that governs the use of geometry. I don't think that one exists.”

Benoît Mandelbrot (1924–2010) Polish-born, French and American mathematician

New Scientist interview (2004)

Thinking

“Tequila, scorpion honey, harsh dew of the doglands, essence of Aztec, crema de cacti; tequila, oily and thermal like the sun in solution; tequila, liquid geometry of passion; Tequila, the buzzard God who copulates in midair with the ascending souls of dying virgins; tequila, firebug in the house of good taste; O tequila, savage water of sorcery, what confusion and mischief your sly, rebellious drops do generate!”

Tom Robbins book Still Life with Woodpecker

Still Life with Woodpecker (1980)

God , Soul , Water , Passion

“Common to the two geometries is only the general property of one-to-one correspondence, and the rule that this correspondence determines straight lines as shortest lines as well as their relations of intersection.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“I propose to show that God, in creating the universe and arranging the spheres, had in view the five regular solids of geometry, and fixed by their dimensions the number, proportions and motions of the spheres. Take the sphere of the earth as a unit and circumscribe it with a regular dodecahedron. The sphere that contains this dodecahedron is the sphere of Mars.”

Johannes Kepler book Mysterium Cosmographicum

As Quoted in "The Discovery of Kepler's Laws," Scientific American: Supplement (Apr 29, 1911) Vol. 71, No. 1843, p. 278 https://books.google.com/books?id=ov4-AQAAMAAJ&pg=PA258.
Mysterium Cosmographicum (1596)

God , Universe

“The Pythagoreans discovered the existence of incommensurable lines, or of irrationals. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of √2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable… showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established…. The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes.”

Thomas Little Heath (1861–1940) British civil servant and academic

p, 125
Achimedes (1920)

Books , Universe

“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.”

Pierre Louis Maupertuis (1698–1759) French mathematician, philosopher and man of letters

Accord de différentes loix de la nature qui avoient jusqu’ici paru incompatibles (1744)

Wisdom , God , Intelligence , Study

“Two other sciences in the same way will accurately treat of 'size': geometry, the part that abides and is at rest, [and] astronomy, that which moves and revolves.”

Nicomachus (60–120) Ancient Greek mathematician

Book I, Chapter III, p.184
Nicomachus of Gerasa: Introduction to Arithmetic (1926)

Way , Science , Parting

“Although the Mathematicks according to its Etymology, signifies only Discipline, yet it merits the Name of Science better than any other, because its Principles are self-evident, and independent on any sensible Experience, and its Propositions demonstrated beyond all possible Doubt or Opposition. Youth were anciently instructed herein before Philosophy, on which Account Aristotle called it the Science of Children. This was taught them not only to raise and excite their Genius, but also as a fit preparative to the Study of Nature; and it was upon this Account that the Divine Plato inscribed on his School… that none wholly ignorant of Geometry should be admitted there.”

Jacques Ozanam (1640–1718) French mathematician

Source: A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms, 1702, p. 1, The Introduction; Lead paragraph

Children , Youth , School , Nature

“How many theorems in geometry which have seemed at first impracticable are in time successfully worked out!”

Archimedes book On Spirals

On Spirals (225 B.C.)

Time

“My dissertation for the Ph. D. degree at the University of Michigan was on applications of vectorial methods to metric geometry (in the sense of the Menger school), especially with a view to the merging of metric geometry in that sense with differential geometry. Professor S B Myers at the University of Michigan sponsored my dissertation, but I was particularly close to R L Wilder there.”

Leonard Jimmie Savage (1917–1971) American mathematician

Leonard Jimmie Savage, cited in: W.A. Wallis, "Leonard Jimmie Savage 1917-1971," in E Shils (ed.), Remembering the University of Chicago: teachers, scientists, and scholars. (University of Chicago Press, Chicago, 1991), 436-451; Quoted in: J J O'Connor and E F Robertson, " Leonard Jimmie Savage http://www-history.mcs.st-and.ac.uk/Biographies/Savage.html," at history.mcs.st-and.ac.uk, November 2010.

School , Sense , Universe

“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.”

Bernard Le Bovier de Fontenelle (1657–1757) French writer, satirist and philosopher of enlightenment

"The Utility of Mathematics," i.e. "Préface sur l'utitlité des mathématiques et de la physique et sur les travaux de le Académie des Sciences," Œuvres de Monsieur de Fontenelle (1753) Vol. 6, pp.37-50, as quoted by Herbert Butterfield, The Origins of Modern Science 1300-1800 (1949).

Books , Art , Knowledge , Time

“Without the aid of these, then, it is not possible to deal accurately with the forms of being nor to discover the truth in things, knowledge of which is wisdom, and evidently not even to philosophize properly, for "just as painting contributes to the menial arts toward correctness of theory, so in truth lines, numbers, harmonic intervals, and the revolutions of circles bear aid to the learning of the doctrines of wisdom," says the Pythagorean Androcydes. Likewise Archytas of Tarentum, at the beginning of… On Harmony, says… in about these words: "It seems to me that they do well to study mathematics, and it is not at all strange that they have correct knowledge about each thing, what it is. For if they knew rightly the nature of the whole, they were also likely to see well what is the nature of the parts. About geometry, indeed, and arithmetic and astronomy, they have handed down to us a clear understanding, and not least also about music. For these seem to be sister sciences; for they deal with sister subjects, the first two forms of being."”

Nicomachus (60–120) Ancient Greek mathematician

Nicomachus of Gerasa: Introduction to Arithmetic (1926)

Wisdom , Truth , Art , Music

“What is the true geometry of the plate? …Anyone examining the situation will prefer Poincaré's common-sense solution… to attribute it Euclidean geometry, and to consider the measured deviations… as due to the actions of a force (thermal stresses in the rule). …On employing a brass rule in place of one of steel we would find that the local curvature is trebled—and an ideal rule (c = 0) would… lead to Euclidean geometry.”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

Sense

“Development of Western Science is based on two great achievements, the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationships by systematic experiment (Renaissance). In my opinion one has not to be astonished that the Chinese sages have not made these steps. The astonishing thing is that these discoveries were made at all.”

Albert Einstein (1879–1955) German-born physicist and founder of the theory of relativity

Letter to J.S. Switzer (23 April 1953), quoted in The Scientific Revolution: a Hstoriographical Inquiry By H. Floris Cohen (1994), p. 234 http://books.google.com/books?id=wu8b2NAqnb0C&lpg=PP1&pg=PA234#v=onepage&q&f=false, and also partly quoted in The Ultimate Quotable Einstein edited by Alice Calaprice (2010), p. 405 http://books.google.com/books?id=G_iziBAPXtEC&lpg=PP1&pg=PA405#v=onepage&q&f=false
1950s

Science

“Cartan developed a general scheme of infinitesimal geometry in which Klein's notions were applied to the tangent plane and not to the n-dimensional manifold M itself.”

Hermann Weyl (1885–1955) German mathematician

On the foundations of general infinitesimal geometry. Bull. Amer. Math. Soc. 35 (1929) 716–725 [10.1090/S0002-9904-1929-04812-2] (quote on p. 716)

“Some philosophers have believed that a philosophical clarification of space also provided a solution of the problem of time. Kant presented space and time as analogous forms of visualization and treated them in a common chapter in his major epistemological work. Time therefore seems to be much less problematic since it has none of the difficulties resulting from multidimensionality. Time does not have the problem of mirror-image congruence, i. e., the problem of equal and similarly shaped figures that cannot be superimposed, a problem that has played some role in Kant's philosophy. Furthermore, time has no problem analogous to non-Euclidean geometry. In a one-dimensional schema it is impossible to distinguish between straightness and curvature. …A line may have external curvature but never an internal one, since this possibility exists only for a two-dimensional or higher continuum. Thus time lacks, because of its one-dimensionality, all those problems which have led to philosophical analysis of the problems of space.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

Problem , Time

“If heat were the affecting force, direct indications of its presence could be found which would not make use of geometry as an indirect method. …direct evidence for the presence of heat is based on the fact that it affects different materials in different ways. …The forces… which we have introduced… have two properties: (a) They affect all materials in the same way. (b) There are no insulating [or isolating] walls. …the definition of the insulating wall may be added here: it is a covering made of any kind of material which does not act upon the enclosed object with forces having property a. Let us call the forces which have the properties a and b universal forces; all other forces are called differential forces. Then it can be said that differential forces, but not universal forces, are directly demonstrable.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

Way , Universe

“Trained in a less severe school than that of geometry and physics, his reasonings are almost always loose and inconclusive. His generalizations seem to have been reached before he had obtained the materials upon which he rests them: His facts, though frequently new and interesting, are often little more than conjectures; and the grand phenomena of the world of life, and instinct, and reason, which other minds have woven into noble and elevating truths, have thus become in Mr. Darwin's hands the basis of a dangerous and degrading speculation.”

David Brewster (1781–1868) British astronomer and mathematician

Referring to Charles Darwin
The facts and fancies of Mr. Darwin (1862)

Life , Truth , World , School

“There never has been, and till we see it we never shall believe that there can be, a system of geometry worthy of the name, which has any material departures (we do not speak of corrections or extensions or developments) from the plan laid down by Euclid.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

"Short Supplementary Remarks on the First Six Books of Euclid's Elements" (Oct, 1848) Companion to the Almanac for 1849 as quoted by Sir Thomas Little Heath, The Thirteen Books of Euclid's Elements Vol.1 https://books.google.com/books?id=UhgPAAAAIAAJ, Introduction and Books I, II. Preface, p. v.

Departure

“Music is the arithmetic of sounds as optics is the geometry of light.”

Claude Debussy (1862–1918) French composer

As quoted in Greatness : Who Makes History and Why by Dean Keith Simonton, p. 110

Music , Light

“Phys. I know that it is often a help to represent pressure and volume as height and width on paper; and so geometry may have applications to the theory of gases. But is it not going rather far to say that geometry can deal directly with these things and is not necessarily concerned with lengths in space?
Math. No. Geometry is nowadays largely analytical, so that in form as well as in effect, it deals with variables of an unknown nature. …It is literally true that I do not want to know the significance of the variables x, y, z, t that I am discussing. …
Phys. Yours is a strange subject. You told us at the beginning that you are not concerned as to whether your propositions are true, and now you tell us you do not even care to know what you are talking about.
Math.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

That is an excellent description of Pure Mathematics, which has already been given by an eminent mathematician <nowiki>[</nowiki>Bertrand Russell<nowiki>]</nowiki>.
Space, Time and Gravitation (1920)

Mathematics , Nature , Help

“It seems that those, who have hitherto treated of this Subject, have been more conversant in the Practice of Designing, than in the Principles of Geometry… that might have enabled them to render the Principles of it more universal, and more convenient for Practice. In this Book I have endeavour'd to do this; and have done my utmost to render the Principles of the Art as general, and as universal as may be, and to devise such Constructions, as might be the most simple and useful in Practice.”

Brook Taylor (1685–1731) English mathematician

New Principles of Linear Perspective (1715, 1749)

Books , Art , Universe

“We see that experience plays an indispensable role in the genesis of geometry; but it would be an error thence to conclude that geometry is, even in part, an experimental science. If it were experimental it would be only approximative and provisional. And what rough approximation!
…The object of geometry is the study of a particular 'group'; but the general group concept pre-exists… in our minds. It is imposed on us, not as form of our sense, but as form of our understanding. Only, from among all the possible groups, that must be chosen… will be… the standard to which we shall refer natural phenomena.
Experience guides us in this choice without forcing it upon us; it tells us not which is the truest geometry, but which is the most convenient.
Notice that I have been able to describe the fantastic worlds… imagined without ceasing to employ the language of ordinary geometry.”

Henri Poincaré book Science and Hypothesis

Source: Science and Hypothesis (1901), Ch. IV: Space and Geometry, Conclusions (1905) Tr. https://books.google.com/books?id=5nQSAAAAYAAJ George Bruce Halstead

World , Error , Imagination , Understanding

“Arthropods and vertebrates share some broad features of general organization - elongated, bilaterally symmetrical bodies, with sensory organs up front, excretory structures in the back, and some form of segmentation along the major axis. But the geometry of major internal organs could hardly be more different… Arthropods concentrate their nervous system on their ventral (belly) side as two major cords running along the bottom surface of the animal. The mouth also opens on the ventral side, with the esophagus passing between the two nerve cords, and the stomach and remainder of the digestive tube running along the body above the nerve cords. In vertebrates, and with maximal contrast, the central nervous system runs along the dorsal (top) surface as a single tube culminating in a bulbous brain at the front end. The entire digestive system then runs along the body axis below the nerve cord.”

Stephen Jay Gould book Leonardo's Mountain of Clams and the Diet of Worms

"Brotherhood by Inversion", p. 321
Leonardo's Mountain of Clams and the Diet of Worms (1998)

Animals

“I… praise the newly opened halls of fossil mammals at the American Museum of Natural History. …teaching us about evolutionary trees by organizing the entire hall as a central trunk and set of branches… placing our brains in our feet and letting us learn by walking. …the chosen geometry of evolutionary organization… violates the traditional picture of life's history, thus illustrating… an important principle in the history of science: the central role of pictures, graphs, and other forms of visual representation in channeling and constraining our thought. …Words are an evolutionary afterthought. …My colleagues have actually done it. …They have ordered all the fossils into an unconventional iconographic tree that fractures the bias of progress. …so that we can preambulate along the tree of life and absorb the new scheme viscerally by walking… They have taken Colbert's radical idea and arranged all the fossils by their branching order, not their later "success" or "advancement."”

Stephen Jay Gould book Dinosaur in a Haystack

Groups that branch early appear early in the hall... Sea cows and elephants are at the end of the hall, horses in the middle, and primates near the beginning.
"Evolution by Walking", pp. 249-254.
Dinosaur in a Haystack (1995)

Life , History , Idea , Tree

“Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i. e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but "logical relations" or "artificial manifolds". They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

Mathematics , Reality , Sense , Study

“A finished or even a competent reasoner is not the work of nature alone… education develops faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to learn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something must be reasoned upon, it matters not much what it is, provided that it can be reasoned upon with certainty. The properties of mind or matter, or the study of languages, mathematics, or natural history may be chosen for this purpose. Now, of all these, it is desirable to choose the one… in which we can find out by other means, such as measurement and ocular demonstration of all sorts, whether the results are true or not.
.. Now the mathematics are peculiarly well adapted for this purpose, on the following grounds:—
1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing.
2. The first principles are self-evident, and, though derived from observation, do not require more of it than has been made by children in general.
3. The demonstration is strictly logical, taking nothing for granted except the self-evident first principles, resting nothing upon probability, and entirely independent of authority and opinion.
4. When the conclusion is attained by reasoning, its truth or falsehood can be ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if… reason is not to be the instructor, but the pupil.
5. There are no words whose meanings are so much alike that the ideas which they stand for may be confounded.
…These are the principal grounds on which… the utility of mathematical studies may be shewn to rest, as a discipline for the reasoning powers. But the habits of mind which these studies have a tendency to form are valuable in the highest degree. The most important of all is the power of concentrating the ideas which a successful study of them increases where it did exist, and creates where it did not. A difficult position or a new method of passing from one proposition to another, arrests all the attention, and forces the united faculties to use their utmost exertions. The habit of mind thus formed soon extends itself to other pursuits, and is beneficially felt in all the business of life.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

Source: On the Study and Difficulties of Mathematics (1831), Ch. I.

Life , Children , Truth , Education

“Geometry is grasping space. And since it is about the education of children, it is grasping that space in which the child lives, breathes and moves. The space that the child must learn to know, explore, conquer, in order to live, breathe and move better in it.”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: Mathematics as an Educational Task (1973), p. 403

Children , Education , Learning

“He took their facts for granted. He knew no more than a firefly about rays — or about race or sex — or ennui — or a bar of music — or a pang of love — or a grain of musk — or of phosphorus — or conscience — or duty — or the force of Euclidian geometry — or non-Euclidian — or heat — or light — or osmosis — or electrolysis — or the magnet — or ether — or vis inertiae — or gravitation — or cohesion — or elasticity — or surface tension — or capillary attraction — or Brownian motion — or of some scores, or thousands, or millions of chemical attractions, repulsions or indifferences which were busy within and without him; or, in brief, of Force itself, which, he was credibly informed, bore some dozen definitions in the textbooks, mostly contradictory, and all, as he was assured, beyond his intelligence.”

Henry Adams (1838–1918) journalist, historian, academic, novelist

The Education of Henry Adams (1907)

Love , Music , Sex , Intelligence

“So intimate is the union between Mathematics and Physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create for each successive class phenomena a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature. Sometimes the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armory of the mathematician the weapons which he needed ready made to his hand. But much oftener, the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma.”

Henry John Stephen Smith (1826–1883) mathematician

As quoted in The Century: A Popular Quarterly (1874) ed. Richard Watson Gilder, Vol. 7, pp. 508-509, https://books.google.com/books?id=ceYGAQAAIAAJ&pg=PA508 "Relations of Mathematics to Physics". Earlier quote without citation in Nature, Volume 8 (1873), page 450.
Also quoted partially in Michael Grossman and Robert Katz, Calculus http://babel.hathitrust.org/cgi/mb?a=listis;c=216746186|Non-Newtonian (1972) p. iv. ISBN 0912938013.

Mathematics , Question , Nature , Knowledge

“In fact it is a stupidity, Maurice Princet told me in the presence of Juan Gris, to claim to be able to bring together in a single system of relations, colour, which is a sensation that only needs to be received, and form which is an organisation that has to be understood (14); and, introducing us to the non-Euclidean geometries, he urged us to create a geometry for painters.”

Jean Metzinger (1883–1956) French painter

Cubism was born

Stupidity

“The student of the Differential Calculus may… be brought to think it possible that the terms and ideas which that science requires may exist in his own mind in the same rude form as that of a straight line in the conceptions of a beginner in geometry. …he must be prepared to stop his course until he can form exact notions, acquire precise ideas, both of resemblance between those things which have appeared most distinct, and of distinction between those which have appeared most alike. To do this… formal definitions would be useless; for he cannot be supposed to have one single notion in that precise form which would make it worth while to attach it to a word. One reason of the great difficulty which is found in treatises on this subject… the tacit assumption that nothing is necessary previously to actually embodying the terms and rules of the science, as if mere statement of definitions could give instantaneous power of using terms rightly. We shall here attempt… a wider degree of verbal explanation than is usual with the view of enabling the student to come to the definitions in some state of previous preparation.”

Augustus De Morgan (1806–1871) British mathematician, philosopher and university teacher (1806-1871)

The Differential and Integral Calculus (1836)

Idea , Thinking , Appearance , Science

“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.”

Benoît Mandelbrot (1924–2010) Polish-born, French and American mathematician

As quoted in Encyclopedia of World Biography (1997) edited by Thomson Gale

Life , Nature

“The geometry of space changes when things in the universe change their relationships to one another.”

Lee Smolin book Three Roads to Quantum Gravity

Three Roads to Quantum Gravity (2000)

Universe , Change

“We unfold out of the Idea of Space the propositions of geometry, which are plainly truths of the most rigorous necessity and universality. But if the idea of space were merely collected from observation of the external world, it could never enable or entitle us to assert such propositions: it could never authorize us to say that not merely some lines, but all lines, not only have, but must have, those properties which geometry teaches. Geometry in every proposition speaks a language which experience never dares to utter; and indeed of which she but half comprehends the meaning.”

William Whewell (1794–1866) English philosopher & historian of science

Part 1, Book 1, ch. 7, art. 1.
Philosophy of the Inductive Sciences (1840)

Truth , World , Idea , Universe

“By the time of Hippocrates of Chios the scope of Greek geometry was no longer even limited to the Elements; certain special problems were also attacked which were beyond the power of the geometry of the straight line and circle, and which were destined to play a great part in determining the direction taken by Greek geometry in its highest flights. The main problems in question were three: (1) the doubling of the cube, (2) the trisection of any angle, (3) the squaring of the circle; and from the time of Hippocrates onwards the investigation of these problems proceeded pari passu with the completion of the body of the Elements.”

Thomas Little Heath (1861–1940) British civil servant and academic

p, 125
Achimedes (1920)

Question , Problem , Time , Parting

“Now it is the practice of astronomers to assume that brightness falls off inversely with the square of the "distance" of an object—as it would do in Euclidean space, if there were no absorption… We must therefore examine the relation between this astronomer's "distance" d… and the distance r which appears as an element of the geometry.”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

Appearance

“I sought refuge in my schoolbooks, among the flowers of Greek poetry, or the magical figures of geometry.”

Jean Metzinger (1883–1956) French painter

Cubism was born

Flowers

“It is time to employ fractal geometry and its associated subjects of chaos and nonlinear dynamics to study systems engineering methodology (SEM). Systematic codification of the former is barely 15 years old, while codification of the latter began 45 years ago… Fractal geometry and chaos theory can convey a new level of understanding to systems engineering and make it more effective”

Arthur D. Hall (1925–2006) American electrical engineer

A.D. Hall III (1989) "The fractal architecture of the systems engineering method", in: Systems, Man and Cybernetics, Part C: Applications and Reviews, IEEE Transactions on Volume 28, Issue 4, Nov 1998 Page(s):565 - 572

Understanding , Study , Time

“If vision is a field, think about what we could do differently to create one. We would do our best to get it permeating through the entire organization so that we could take advantage of its formative properties. All employees, in any part of the company, who bumped up against the field, would be influenced by it. Their behavior could be shaped as a result of “field meetings”, where their energy would link with the fields form to create behavior congruent with the organizations goals. In the absence of that field, in areas of the organization that hadn’t been reached, we could hold no expectation of desired behaviors. If the field hadn’t extended into that space, there would be nothing there to help behaviors materialize, no invisible geometry working on our behalf”

Margaret J. Wheatley (1941) American writer

Source: Leadership and the New Science (1992), p. 54

Behavior , Thinking , Help , Parting

“The researches of the last thirty or forty years into the history of mathematics (I need only mention such names as those of [Carl Anton] Bretschneider, Hankel, Moritz Cantor, [Friedrich] Hultsch, Paul Tannery, Zeuthen, Loria, and Heiberg) have put the whole subject upon a different plane. I have endeavoured in this edition to take account of all the main results of these researches up to the present date. Thus, so far as the geometrical Books are concerned, my notes are intended to form a sort of dictionary of the history of elementary geometry, arranged according to subjects; while the notes on the arithmetical Books VII.-IX. and on Book X follow the same plan.”

Thomas Little Heath (1861–1940) British civil servant and academic

p, 125
The Thirteen Books of Euclid's Elements (1908)

Books , History , Mathematics

“The efforts of a multitude of writers have rather been directed towards producing alternatives for Euclid which shall be more suitable, that is to say, easier, for schoolboys. It is of course not surprising that, in these days of short cuts, there should have arisen a movement to get rid of Euclid and to substitute "a royal road to geometry"; the marvel is that a book which was not written for schoolboys but for grown men (as all internal evidence shows, and in particular the essentially theoretical character of the work and its aloofness from anything of the nature of "practical" geometry) should have held its own as a schoolbook for so long.”

Thomas Little Heath (1861–1940) British civil servant and academic

The Thirteen Books of Euclid's Elements (1908)

Men , Books , Nature , Effort

“What I have given in the second book on the nature and properties of curved lines, and the method of examining them, is, it seems to me, as far beyond the treatment in the ordinary geometry, as the rhetoric of Cicero is beyond the a, b, c of children.”

René Descartes (1596–1650) French philosopher, mathematician, and scientist

Letter to Marin Mersenne (1637) as quoted by D. E. Smith & M. L. Latham Tr. The Geometry of René Descartes (1925)

Children , Books , Nature

“The very spirits of the winds, when they were sent to carry the grateful harvest to the thirsting fields of Calabria, did not forget the geometry which they had studied in the caverns of Æolus and of which the geologist is daily discovering the diagrams.”

Benjamin Peirce (1809–1880) American mathematician

Ben Yamen's Song of Geometry (1853)

Study , Forgetting

“The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces… The intrinsic geometry of such a space of curvature K provides formulae for the surface area S and the volume V of a "small sphere" of radius r, whose leading terms are 3)S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + …),
V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + …).”

Howard P. Robertson (1903–1961) American mathematician and physicist

Geometry as a Branch of Physics (1949)

“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.”

Hans Reichenbach (1891–1953) American philosopher

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.
The Philosophy of Space and Time (1928, tr. 1957)

Way , Feeling , Idea , Understanding

“[Computer science] is not really about computers -- and it's not about computers in the same sense that physics is not really about particle accelerators, and biology is not about microscopes and Petri dishes…and geometry isn't really about using surveying instruments. Now the reason that we think computer science is about computers is pretty much the same reason that the Egyptians thought geometry was about surveying instruments: when some field is just getting started and you don't really understand it very well, it's very easy to confuse the essence of what you're doing with the tools that you use.”

Hal Abelson (1947) computer scientist

Source: Introductory lecture to Structure and Interpretation of Computer Programs http://www.youtube.com/watch?v=zQLUPjefuWA

Understanding , Thinking , Sense , Science

“Really, universally, relations stop nowhere, and the exquisite problem of the artist is eternally but to draw, by a geometry of his own, the circle within which they shall happily appear to do so.”

Henry James Roderick Hudson

Roderick Hudson.
Prefaces (1907-1909)

Problem , Universe , Appearance , Drawing

“The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.”

Nicholas Murray Butler (1862–1947) American philosopher, diplomat, and educator

The Meaning of Education and other Essays and Addresses https://books.google.com/books?id=H9cKAAAAIAAJ (1898) p. 45 as quoted by Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book https://books.google.com/books?id=G0wtAAAAYAAJ (1914)

History , Mathematics , Sense , Science

“If one would understand the Greek genius fully, it would be a good plan to begin with their geometry.”

Thomas Little Heath (1861–1940) British civil servant and academic

Preface p. vi
A History of Greek Mathematics (1921) Vol. 1. From Thales to Euclid

Understanding

Quotes about geometry page 2

Quotes about geometry
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