Quotes about geometry
page 3

“Globalisation is highly selective. It proceeds by linking up all that, according to dominant interests, has value anywhere in the planet, and discarding anything which has no value or becomes devalued, in a variable geometry of creative destruction and destructive creation of value.”

Manuel Castells (1942) Spanish sociologist (b.1942)

Materials for an exploratory theory of the network society (2000)

“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."”

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

As quoted in a review of The Fractal Geometry of Nature by J. W. Cannon in The American Mathematical Monthly, Vol. 91, No. 9 (November 1984), p. 594

Nature, Study

“Hodge cohomology, algebraic de Rham cohomology, crystalline cohomology, the étale ℓ-adic cohomology theories for each prime number ℓ … A strategy to encapsulate all the different cohomology theories in algebraic geometry was formulated initially by Alexandre Grothendieck, who is responsible for setting up much of this marvelous cohomological machinery in the first place. Grothendieck sought a single theory that is cohomological in nature that acts as a gateway between algebraic geometry and the assortment of special cohomological theories, such as the ones listed above—that acts as the motive behind all this cohomological apparatus. …”

Barry Mazur (1937) American mathematician

"WHAT IS...a Motive?" 2004

Motivation, Nature

“Geometry… is of much assistance in architecture, and in particular it teaches us the use of the rule and compasses, by which especially we acquire readiness in making plans for buildings in their grounds, and rightly apply the square, the level, and the plummet. By means of optics… the light in buildings can be drawn from fixed quarters of the sky. …Difficult questions involving symmetry are solved by means of geometrical theories and methods.”

Vitruvius book De architectura

Source: De architectura (The Ten Books On Architecture) (~ 15BC), Book I, Chapter I, Sec. 4

Question, Light

“More evidently still astronomy attains through arithmetic the investigations that pertain to it, not alone because it is later than geometry in origin—for motion naturally comes after rest—nor because the motions of the stars have a perfectly melodious harmony, but also because risings, settings, progressions, retrogressions, increases, and all sorts of phases are governed by numerical cycles and quantities. So then we have rightly undertaken first the systematic treatment of this, as the science naturally prior, more honorable, and more venerable, and, as it were, mother and nurse of the rest; and here we will take our start for the sake of clearness.”

Nicomachus (60–120) Ancient Greek mathematician

Book I, Chapter V
Nicomachus of Gerasa: Introduction to Arithmetic (1926)

Star, Nature, Science

“In geometry the following theorems are attributed to him [Thales]—and their character shows how the Greeks had to begin at the very beginning of the theory—(1) that a circle is bisected by any diameter (Eucl. I., Def. 17), (2) that the angles at the base of an isosceles triangle are equal (Eucl. I., 5), (3) that, if two straight lines cut one another, the vertically opposite angles are equal (Eucl. I., 15), (4) that, if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (Eucl. I., 26). He is said (5) to have been the first to inscribe a right-angled triangle in a circle: which must mean that he was the first to discover that the angle in a semicircle is a right angle. He also solved two problems in practical geometry: (1) he showed how to measure the distance from the land of a ship at sea (for this he is said to have used the proposition numbered (4) above), and (2) he measured the heights of pyramids by means of the shadow thrown on the ground”

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

this implies the use of similar triangles in the way that the Egyptians had used them in the construction of pyramids
Achimedes (1920)

Way, Problem, Sea

“In the application of the ordinary principles of geometry and trigonometry to such Astronomical measures… it may sometimes be expedient to resolve the process into several successive steps, and these steps may perhaps require different kinds of treatment. But… all are simple and within ordinary comprehension, and the only complexity arises from the circumstance that the student may find it necessary to have a clear view of several such steps at once…”

George Biddell Airy (1801–1892) English mathematician and astronomer

Introduction
Popular Astronomy: A Series of Lectures Delivered at Ipswich (1868)

Success

“The lecture concludes with a demonstration showing that a free body in hyperbolic non Euclidean space may be so fashioned as in real time to carry out the actual motions of the top. The form of such a body and the forces to actuate it are specified. Klein lays great stress on the beauty of this generalization. …The full geometry of this case is not carried out in these lectures, however, and Klein regrets that the development of the automorphic functions has recently fallen into abeyance.”

Carl Barus (1856–1935) U.S. physicist

"The Mathematical Theory of the Top" (April 8, 1898)

Beauty, Fashion, Time

“J. J. Sylvester was an enthusiastic supporter of reform [in the teaching of geometry]. The difference in attitude on this question between the two foremost British mathematicians, J. J. Sylvester, the algebraist, and Arthur Cayley, the algebraist and geometer, was grotesque. Sylvester wished to bury Euclid "deeper than e'er plummet sounded" out of the schoolboy's reach; Cayley, an ardent admirer of Euclid, desired the retention of Simson's Euclid. When reminded that this treatise was a mixture of Euclid and Simson, Cayley suggested striking out Simson's additions and keeping strictly to the original treatise.”

Florian Cajori book A History of Mathematics

Source: A History of Mathematics (1893), p. 285; Cited in: Moritz (1914, 148); Persons and anecdotes

Question, Support

“Three distinct geometries on S7 arise as solutions of the classical equations of motion in eleven dimensions. In addition to the conventional riemannian geometry, one can also obtain the two exceptional Cartan-Schouten compact flat geometries with torsion.”

François Englert (1932) Belgian theoretical physicist

[10.1016/0370-2693(82)90684-0, 1982, Spontaneous compactification of eleven-dimensional supergravity, Physics Letters B, 119, 4–6, 339–342]

“If we wished to relate the space of the [Cubist] painters to geometry, we should have to refer it to the non-Euclidean mathematicians; we should have to study, at some length, certain of Riemann's theorems.”

Jean Metzinger (1883–1956) French painter

Du Cubisme (1912)

Study

“You may find this work (if I judge rightly) quite new. For I see no reason why I should not proclaim it; nor do I believe that others will take it wrongly. …it teaches all by a new method, introduced by me for the first time into geometry, and with such clarity that in these more abstruse problems no-one (as far as I know) has used…”

John Wallis (1616–1703) English mathematician

Arithmetica Infinitorum (1656)

Problem, Time

“How did it happen that the Reverend Charles Dodgson, thirty years of age, lecturer on geometry at Christ Church, Oxford, hitherto remarkable chiefly for his precision, on a single July afternoon, while rowing up the Isis with a brother don and three little girls, parthenogenetically gave birth to one of the most famous stories of all time?”

Florence Becker Lennon

Page 23
The Life of Lewis Carroll (1962)

Age, Girl, Birth, Time

“I have endeavour'd to make every thing so plain, that a very little Skill in Geometry may be sufficient to enable one to read this Book by himself. And upon this occasion I would advise all my Readers, who desire to make themselves Masters of this Subject, not to be contented with the Schemes they find here; but upon every Occasion to draw new ones of their own, in all the Variety of Circumstances they can think of. This will take up a little more Time at first; but in a little while they will find the vast Benefit of it, by the extensive Notions it will give them of the Nature of these Principles.”

Brook Taylor (1685–1731) English mathematician

New Principles of Linear Perspective (1715, 1749)

Books, Nature, Thinking, Time

“Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras (jabbre and maqabeleh) are geometric facts which are proved by propositions five and six of Book two of Elements.”

Omar Khayyám (1048–1131) Persian poet, philosopher, mathematician, and astronomer

As quoted in "A Paper of Omar Khayyam" by A.R. Amir-Moez in Scripta Mathematica 26 (1963). This quotation has often been abridged in various ways, usually ending with "Algebras are geometric facts which are proved", thus altering the context significantly.

Books, Thinking, Appearance

“For 12 years I have had the honor of teaching geometry and mechanics applied to the arts, in favor of the industrial class… on the most important questions to the well-being, education, and morality of the workers, to the progress of national industry, to the development of all means of prosperity that work can produce for the splendor and happiness of our country.”

Charles Dupin (1784–1873) French mathematician

Charles Dupin (1831), Discours sur le Sort des Ouvriers [Discourse on the Condition of the Workers] Paris: Bachelier Librairie. p. 1. ; Translation Wren & Bedeian (2005, 73)

Happiness, Education, Art, Question

“Until Abstract Expressionism you had to have something to paint about, some kind of subject matter. Even though Kandinsky and w:Arthur Dove were improvising earlier, it didn't take. They had to have symbols, suggested natural images or geometry, which was something real structurally. That gave them something to paint about. What was new was the idea that something you looked at could be like something you heard.”

Kenneth Noland (1924–2010) American artist

Kenneth Noland, p. 8
Conversation with Karen Wilkin' (1986-1988)

Idea, Nature, Painting

“Newton… (after having remarked that geometry only requires two of the mechanical actions which it postulates, namely, to describe a straight line and a circle) says: geometry is proud of being able to achieve so much while taking so little from extraneous sources. One might say of metaphysics, on the other hand: it stands astonished, that with so much offered it by pure mathematics it can effect so little. In the meantime, this little is something which mathematics indispensably requires in its application to natural science, which, inasmuch as it must here necessarily borrow from metaphysics, need not be ashamed to allow itself to be seen in company with the latter.”

Immanuel Kant (1724–1804) German philosopher

Preface, Tr. Bax (1883) citing Isaac Newton's Principia
(1786)

Mathematics, Nature, Science

“We must… maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds.”

Hans Reichenbach (1891–1953) American philosopher

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

Mathematics, Understanding, Science

“Very late in life, when he was studying geometry, some one said to Lacydes, "Is it then a time for you to be learning now?"”

Diogenes Laërtius (180–240) biographer of ancient Greek philosophers

"If it is not," he replied, "when will it be?"
Lacydes, 5.
The Lives and Opinions of Eminent Philosophers (c. 200 A.D.), Book 4: The Academy

Life, Study, Learning, Time

“The Essential Parts of the Simple or Pure Mathematicks are Arithmetick and Geometry, which mutually assist one another, and are independent on any other Sciences, except perhaps on Artificial Logick: But doubtless Natural Logick may be sufficient to a Man of Sense. The other parts are chiefly Physical Subjects explained by the Principles of Arithmetics or Geometry.”

Jacques Ozanam (1640–1718) French mathematician

Source: A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms, 1702, p. 2, The introduction

Nature, Sense, Science, Parting

“We find in the history of ideas mutations which do not seem to correspond to any obvious need, and at first sight appear as mere playful whimsies — such as Apollonius' work on conic sections, or the non-Euclidean geometries, whose practical value became apparent only later.”

Arthur Koestler (1905–1983) Hungarian-British author and journalist

as quoted by Michael Grossman in the The First Nonlinear System of Differential and Integral Calculus (1979).
The Sleepwalkers: A History of Man's Changing Vision of the Universe (1959)

History, Idea, Appearance

Albert Gleizes (1881–1953) French painter

1910s, Du Cubisme (1912)

Study

“You start out with simple concepts,.. All of this beautiful edifice of geometry, for example, comes out of this configuration of lines and points… Think of the simple opening--dum dum dum dum!--of Beethoven's Fifth Symphony. In music you start out with a trivial motif and it turns into this beautiful, intricate composition,… Again, it all stems from this small, very simple idea.”

Noam Elkies (1966) American mathematician

Talking about "a stark, basic principle underpins even the most complex symphony or mathematical application."
Music + Math: A Common Equation?, 1988

Beauty, Music, Idea, Thinking

“The last thirty years [1925 to 1955] have seen an enormous improvement in the position of geometry as a branch of mathematics, or, rather, have seen the re-integration of geometry into the main fabric of mathematics. Indeed, one can go further and say that with the restoration of geometry to its rightful place in the mathematical scheme the process of fragmentation which had been doing so much harm to mathematics has been reversed, and we may look forward to the day in which there are no longer analysts, algebraists, geometers and so on, but simply mathematicians. Mathematical research has two aspects, motivation and technique, and when the latter gains control the result is apt to be excessive specialisation. The revolution of geometrical thought, and the reinstatement of geometry as one of the major mathematical disciplines, have helped to bring about a unification of mathematics which we may justly regard as one of the major contributions of the last quarter century to the subject.”

W. V. D. Hodge (1903–1975) British mathematician

W. V. D. Hodge, Changing Views of Geometry. Presidential Address to the Mathematical Association, 14th April, 1955, The Mathematical Gazette 39 (329) (1955), 177-183.

Mathematics, Motivation, Help

“In string theory one studies strings moving in a fixed classical spacetime. …what we call a background-dependent approach. …One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves—at the rate predicted by general relativity to the… accuracy of eleven decimal places—tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up.”

Lee Smolin (1955) American cosmologist

"Loop Quantum Gravity," The New Humanists: Science at the Edge (2003)

Study, Time, Dependency

“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.”

Florian Cajori book A History of Mathematics

Source: A History of Mathematics (1893), p. 30 Reported in Memorabilia mathematica or, The philomath's quotation-book by Robert Edouard Moritz. Published 1914.

History, Mathematics, Science

“The pure normative standpoint that Kant’s ethics tries to occupy, a standpoint in which we consider only the normatively relevant features of a possible world, abstracting strictly from the real world and the empirical accidents of concrete situations, is an expression of what Dewey called “the quest for certainty.” In an insecure world, weak humans struggle convulsively to reach some kind of stability; the a priori is an overcompensation in thought for experienced human weakness. This is one of the origins of Kant’s notorious rigidity, his authoritarian devotion to “principles,” and his tendency to promote local habits of thought to constituents of the absolute framework in which alone (purportedly) any coherent experience was possible; thus, Euclidean geometry is declared the a priori condition of human experience, and sadistic remnants of Puritanism become demands of pure practical reason. Classical liberalism rejected Kant’s practical philosophy, but perhaps this is not enough. Perhaps one should also reject the very idea of a pure normative standpoint.”

Raymond Geuss (1946) British philosopher

“Liberalism and its Discontents,” pp. 20-21.
Outside Ethics (2005)

World, Idea

“The absolute requisites for the study of this work… are a knowledge of algebra to the binomial at least, plane and solid geometry, plane trigonometry, and the most simple part of the usual applications of algebra to geometry.
…A. De Morgan. London July 1, 1836”

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

The Differential and Integral Calculus (1836)

Knowledge, Study, Parting

“Geometry is one and eternal shining in the mind of God. That share in it accorded to humans is one of the reasons that humanity is the image of God.”

Johannes Kepler book Harmonices Mundi

Book III, Ch. 1 as quoted in "Astrology in Kepler's Cosmology" by Judith V. Field, in Astrology, Science, and Society: Historical Essays (1987) edited by P. Curry, p. 154
Geometry, coeternal with God and shining in the divine Mind, gave God the pattern... by which he laid out the world so that it might be best and most beautiful and finally most like the Creator.
As quoted in Kepler's Geometrical Cosmology (1988), p. 123
Geometry is one and eternal shining in the mind of God. That share in it accorded to men is one of the reasons that Man is the image of God.
Unsourced variant
Harmonices Mundi (1618)

God

“The philosophy of the foundations of probability must be divorced from mathematics and statistics, exactly as the discussion of our intuitive space concept is now divorced from geometry.”

William Feller (1906–1970) Croatian-American mathematician

Introduction, The Nature of Probability Theory, p. 3.
An Introduction To Probability Theory And Its Applications (Third Edition)

Mathematics

“When mons. Descartes's philosophical Romance, by the Elegance of its Style and the plausible Accounts of natural Phænomena, had overthrown the Aristotelian Physics, the World received but little Advantage by the Change: For instead of a few Pedants, who, most of them, being conscious of their Ignorance, concealed it with hard Words and pompous Terms; a new Set of Philosophers started up, whose lazy Disposition easily fell in with a Philosophy, that required no Mathematicks to understand it, and who taking a few Principles for granted, without examining their Reality or Consistence with each other, fancied they could solve all Appearances mechanically by Matter and Motion; and, in their smattering Way, pretended to demonstrate such things, as perhaps Cartesius himself never believed; his Philosophy (if he bad been in earnest) being unable to stand the test of the Geometry which he was Master of.”

John Theophilus Desaguliers (1683–1744) French-born British natural philosopher and clergyman

Source: Course of Experimental Philosophy, 1745, p. vi-v: Preface

World, Way, Understanding, Nature

“Although the Special Theory of Relativity does not account for electromagnetic phenomena, it explains many of their properties. General Relativity, however, tells us nothing about electromagnetism. In Einstein's space-time continuum gravitational forces are absorbed in the geometry, but the electromagnetic forces are quite unaffected. Various attempts have been made to generate the geometry of space-time so as to produce a unified field theory incorporating both gravitational and electromagnetic forces.”

Gerald James Whitrow (1912–2000) British mathematician

p, 125
The Structure of the Universe: An Introduction to Cosmology (1949)

Time

“Let him [who would be an architect] be educated, skillful with the pencil, instructed in geometry, know much history, have followed the philosophers with attention, understand music, have some knowledge of medicine, know the opinions of the jurists, and be acquainted with astronomy and the theory of the heavens.”

Vitruvius book De architectura

Source: De architectura (The Ten Books On Architecture) (~ 15BC), Book I, Chapter I, Sec. 3

Education, Music, History, Understanding

“What would geometry be without Gauss, mathematical logic without Boole, algebra without Hamilton, analysis without Cauchy?”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Mathematics

“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.”

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

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

Way, Feeling, Intelligence, Study

“I am prepared to concede without further argument that all the theoretical constructions, including geometry, which are used in the various branches of physics are only imperfect instruments to enable the world of empirical fact to be reconstructed in our minds.”

Richard von Mises (1883–1953) Austrian physicist and mathematician

First Lecture, The Definition of Probability, p. 8
Probability, Statistics And Truth - Second Revised English Edition - (1957)

World

“A new theory by the author has been added, which draws the physical inferences consequent on the extension of the foundations of geometry beyond Reimann… and represents an attempt to derive from world-geometry not only gravitational but also electromagnetic phenomena. Even if this theory is still only in its infant stage, I feel convinced that it contains no less truth than Einstein's Theory of Gravitation—whether this amount of truth is unlimited or, what is more probable, is bounded by the Quantum Theory.”

Hermann Weyl (1885–1955) German mathematician

From the Author's Preface to Third Edition (1919)
Space—Time—Matter (1952)

Truth, World, Feeling, Drawing

“The farmer needs to learn different skills if he wants to cultivate, just as the doctor and hunter need to learn many things if they are to heal or hunt. And so must he who wishes to gain from the scripture and from Christ learn all rational and logical skills. He must go to geometry, music, grammar and philosophy itself and take from them what is useful, in order to defend his faith against those who plot to destroy it. If he does not do this, he is, as Plato says, like the athlete who turns up unprepared for the games.”

Eric Osborn (1922–2007)

Clement of Alexandria (Cambridge University Press: 2008), p. 63

Faith, Music, Game, Learning

“At the time the book of Marquis de l'Hôpital had appeared, and almost all mathematicians began to turn to the new geometry of the infinite [that is, the new infinitesimal calculus], until then little known. The surprising universality of the methods, the elegant brevity of the proofs, the neatness and speed of the most difficult solutions, a singular and unexpected novelty, all attracted the mind and there was in the mathematical world a well marked revolution [une révolution bien marquée.”

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

1792) as quoted by I. Bernard Cohen, Revolution in Science (1985

Books, World, Mathematics, Universe

“Sanskrit is constructed like geometry and follows a rigorous logic. It is theoretically possible to explain the meaning of the words according to the combined sense of the relative letters, syllables and roots. Sanskrit has no meanings by connotations and consequently does not age. Panini's language is in no way different from that of Hindu scholars conferring in Sanskrit today.”

Alain Daniélou (1907–1994) French historian, musicologist, Indologist and expert on Shaivite Hinduism

Alain Danielou in: Virtue, Success, Pleasure, and Liberation: The Four Aims of Life in the Tradition of Ancient India https://books.google.co.in/books?id=IMSngEmfdS0C&pg=PA17, Inner Traditions / Bear & Co, 1 August 1993 , p. 17.

Age, Way, Sense

“The concept of congruence in Euclidean geometry is not exactly the same as that in non-Euclidean geometry. …"Congruent" means in Euclidean geometry the same as "determining parallelism," a meaning which it does not have in non-Euclidean geometry.”

Hans Reichenbach (1891–1953) American philosopher

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

“I taught myself decimals, equations, the square, cube, and biquadrate roots. I got some knowledge of logarithms, and some of algebra. I readily got through a small schoolbook of geometry; and having an odd volume, the first, of Williamson's ' Euclid,' I attacked it vigorously and perseveringly. Williamson's is by no means the best book on the subject, yet I am still of opinion that it is the best book I could have had for the purpose of teaching myself.”

Francis Place (1771–1854) English social reformer

Source: The life of Francis Place, 1771-1854, 1898, p. 18

Books, Knowledge

“Music is geometry in time.”

Arthur Honegger (1892–1955) Swiss composer

I am a Composer (1951)

Music, Time

“The truth is that other systems of geometry are possible, yet after all, these other systems are not spaces but other methods of space measurements. There is one space only, though we may conceive of many different manifolds, which are contrivances or ideal constructions invented for the purpose of determining space.”

Paul Carus (1852–1919) American philosopher

Science, Vol. 18 (1903), p. 106, as reported in Memorabilia Mathematica; or, The Philomath's Quotation-Book https://archive.org/stream/memorabiliamathe00moriiala#page/81/mode/2up, (1914), by Robert Edouard Moritz, p. 352

Truth

“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.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 89

Men, God, Mathematics

“In Plato's Republic, when the interlocutor of Socrates appears to bring certain plausible reasons to bear upon the mathematical sciences, to show that they are useful to human life, arithmetic for reckoning, distributions, contributions, exchanges, and partnerships, geometry for sieges, the founding of cities and sanctuaries, and the partition of land, music for festivals, entertainment, and the worship of the gods, and the doctrine of the spheres, or astronomy, for farming, navigation and other undertakings, revealing beforehand the proper procedure and suitable season, Socrates, reproaching him says: "You amuse me, because you seem to fear that these are useless studies that I recommend; but that is very difficult, nay, impossible. For the eye of the soul, blinded and buried by other pursuits, is rekindled and aroused again by these and these alone, and it is better that this be saved than thousands of bodily eyes, for by it alone is the truth of the universe beheld."”

Nicomachus (60–120) Ancient Greek mathematician

Nicomachus of Gerasa: Introduction to Arithmetic (1926)

Life, God, Truth, Fear

“Pascal said that if geometry stirred us emotionally as much as politics we would not be able to expound it so well.”

André Maurois (1885–1967) French writer

Un Art de Vivre (The Art of Living) (1939), The Art of Friendship

“Gregorian chant, Romanesque architecture, the Iliad, the invention of geometry were not, for the people through whom they were brought into being and made available to us, occasions for the manifestation of personality.”

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

Source: Simone Weil : An Anthology (1986), Human Personality (1943), p. 55

People, Personality

“That the existence of these motions (the "axiom of free mobility") is a desideratum, if not… a necessity, for a geometry applicable to physical space, has been forcefully argued on a priori grounds by von Helmholtz, Whitehead, Russell and others; for only in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained.”

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

Geometry as a Branch of Physics (1949)

“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.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics for the Nonmathematician (1967), pp. 255-256.

Truth, Exercises, Imagination, Idea

“You well know … for which reason I began searching for a number of demonstrations proving a statement due to the ancient Greeks … and which passion I felt for the subject … so that you reproached me my preoccupation with these chapters of geometry, not knowing the true essence of these subjects, which consists precisely in going in each matter beyond what is necessary. … Whatever way he [the geometer] may go, through exercise will he be lifted from the physical to the divine teachings, which are little accessible because of the difficulty to understand their meaning … and because the circumstance that not everybody is able to have a conception of them, especially not the one who turns away from the art of demonstration.”

Al-Biruni (973–1048) Persian scholar and polymath

Book on the Finding of Chords.

Art, Way, Exercises, Search

“Our model of the cosmos must be as inexhaustible as the cosmos. A complexity that includes not only duration but creation, not only being but becoming, not only geometry but ethics. It is not the answer we are after, but only how to ask the question.”

Ursula K. Le Guin Hainish Cycle

Source: Hainish Cycle, (1974), Chapter 7 (p. 226)

Question

“A new point is determined in Euclidean Geometry exclusively in one of the three following ways:
Having given four points A, B, C, D, not all incident on the same straight line, then
(1) Whenever a point P exists which is incident both on (A, B) and on (C, D), that point is regarded as determinate.
(2) Whenever a point P exists which is incident both on the straight line (A, B) and on the circle C(D), that point is regarded as determinate.
(3) Whenever a point P exists which is incident on both the circles A(B), C(D), that point is regarded as determinate.
The cardinal points of any figure determined by a Euclidean construction are always found by means of a finite number of successive applications of some or all of these rules (1), (2) and (3). Whenever one of these rules is applied it must be shown that it does not fail to determine the point. Euclid's own treatment is sometimes defective as regards this requisite.
In order to make the practical constructions which correspond to these three Euclidean modes of determination, correponding to (1) the ruler is required, corresponding to (2) both ruler and compass, and corresponding to (3) the compass only.
…it is possible to develop Euclidean Geometry with a more restricted set of postulations. For example it can be shewn that all Euclidean constructions can be carried out by means of (3) alone…”

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

Source: Squaring the Circle (1913), pp. 7-8

Way, Success

“Suppose then I want to give myself a little training in the art of reasoning; suppose I want to get out of the region of conjecture and probability, free myself from the difficult task of weighing evidence, and putting instances together to arrive at general propositions, and simply desire to know how to deal with my general propositions when I get them, and how to deduce right inferences from them; it is clear that I shall obtain this sort of discipline best in those departments of thought in which the first principles are unquestionably true. For in all 59 our thinking, if we come to erroneous conclusions, we come to them either by accepting false premises to start with—in which case our reasoning, however good, will not save us from error; or by reasoning badly, in which case the data we start from may be perfectly sound, and yet our conclusions may be false. But in the mathematical or pure sciences,—geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves,—we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline. When Plato wrote over the portal of his school. “Let no one ignorant of geometry enter here,” he did not mean that questions relating to lines and surfaces would be discussed by his disciples. On the contrary, the topics to which he directed their attention were some of the deepest problems,—social, political, moral,—on which the mind could exercise itself. Plato and his followers tried to think out together conclusions respecting the being, the duty, and the destiny of man, and the relation in which he stood to the gods and to the unseen world. What had geometry to do with these things? Simply this: That a man whose mind has not undergone a rigorous training in systematic thinking, and in the art of drawing legitimate inferences from premises, was unfitted to enter on the discussion of these high topics; and that the sort of logical discipline which he needed was most likely to be obtained from geometry—the only mathematical science which in Plato’s time had been formulated and reduced to a system. And we in this country [England] have long acted on the same principle. Our future lawyers, clergy, and statesmen are expected at the University to learn a good deal about curves, and angles, and numbers and proportions; not because these subjects have the smallest relation to the needs of their lives, but because in the very act of learning them they are likely to acquire that habit of steadfast and accurate thinking, which is indispensable to success in all the pursuits of life.”

Joshua Girling Fitch (1824–1903) British educationalist

Source: Lectures on Teaching, (1906), pp. 291-292

Life, God, Truth, World

“The progression of a painter's work, as it travels in time from point to point, will be toward clarity: toward the elimination of all obstacles between the painter and the idea, and between the idea and the observer. As examples of such obstacles, I give (among others) memory, history or geometry, which are swamps of generalization from which one might pull out parodies of ideas (which are ghosts) but never an idea in itself. To achieve this clarity is, inevitably, to be understood.”

Mark Rothko (1903–1970) American painter

In Tiger’s Eye, Vol. 1, no 9, October 1949; as quoted in Abstract Expressionism Creators and Critics, ed. Clifford Ross, Abrams Publishers New York 1990, p. 170
1940's

Travel, History, Idea, Memory

“M. Desargues puts me under obligations on account of the pains that it has pleased him to have in me, in that he shows that he is sorry that I do not wish to study more in geometry, but I have resolved to quit only abstract geometry, that is to say, the consideration of questions which serve only to exercise the mind, and this, in order to study another kind of geometry, which has for its object the explanation of the phenomena of nature… You know that all my physics is nothing else than geometry.”

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

Letter to Marin Mersenne (July 27, 1638) as quoted by Florian Cajori, A History of Mathematics (1893) letter dated in The Philosophical Writings of Descartes Vol. 3, The Correspondence (1991) ed. John Cottingham, Robert Stoothoff, Dugald Murdoch

Pain, Exercises, Question, Nature

“It would be inconvenient to interrupt the account of Menaechmus's solution of the problem of the two mean proportionals in order to consider the way in which he may have discovered the conic sections and their fundamental properties. It seems to me much better to give the complete story of the origin and development of the geometry of the conic sections in one place, and this has been done in the chapter on conic sections associated with the name of Apollonius of Perga. Similarly a chapter has been devoted to algebra (in connexion with Diophantus) and another to trigonometry”

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

under Hipparchus, Menelaus and Ptolemy
A History of Greek Mathematics (1921) Vol. 1. From Thales to Euclid

Way, Problem

“String theory at its finest is, or should be, a new branch of geometry. …I, myself, believe rather strongly that the proper setting for string theory will prove to be a suitable elaboration of the geometrical ideas upon which Einstein based general relativity.”

Edward Witten (1951) American theoretical physicist

"Edward Witten" interview, Superstrings: A Theory of Everything? (1992) ed. P.C.W. Davies, Julian Brown

Idea

“Not much younger than these (sc. Hermotimus of Colophon and Philippus of Mende) is Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy), makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry. He is then younger than pupils of Plato but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says.”

Proclus (412–485) Greek philosopher

As quoted by Sir Thomas Little Heath, The Thirteen Books of Euclid's Elements (1908) Vol.1 https://books.google.com/books?id=UhgPAAAAIAAJ Introduction and Books I, II p.1, citing Proclus ed. Friedlein, p. 68, 6-20.

Way, Time, Perfection

“If geometry exists, arithmetic must also needs be implied”

Nicomachus (60–120) Ancient Greek mathematician

Nicomachus of Gerasa: Introduction to Arithmetic (1926)
Context: If geometry exists, arithmetic must also needs be implied... But on the contrary 3, 4, and the rest might be 5 without the figures existing to which they give names. Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry.<!--Book I, Chapter IV

“It may not be premature to propose that spatial imagination has evolved as a reflection of the physics and geometry of the external world.”

Roger Shepard (1929) American psychologist

L.A. Cooper and R.N. Shepard (1984). "Turning something over in the mind." Scientific American 251(6), 106-114; p. 114.
Context: In spite of some unresolved issues, the close match we have found between mental rotation and their counterparts in the physical world leads inevitably to speculations about the functions and origin of human spatial imagination. It may not be premature to propose that spatial imagination has evolved as a reflection of the physics and geometry of the external world. The rules that govern structures and motions in the physical world may, over evolutionary history, have been incorporated into human perceptual machinery, giving rise to demonstrable correspondences between mental imagery and its physical analogues.

World, Imagination

“I make no difference between the Plane of the Horizon, and any other Plane whatsoever; for since Planes, as Planes, are alike in Geometry, it is most proper to consider them as so”

Brook Taylor (1685–1731) English mathematician

New Principles of Linear Perspective (1715, 1749)
Context: I make no difference between the Plane of the Horizon, and any other Plane whatsoever; for since Planes, as Planes, are alike in Geometry, it is most proper to consider them as so, and to explain their Properties in general, leaving the Artist himself to apply them in particular Cases, as Occasion requires.

“Ascend with me above the dust, above the cloud, to the realms of the higher geometry, where the heavens are never clouded”

Benjamin Peirce (1809–1880) American mathematician

Ben Yamen's Song of Geometry (1853)
Context: Ascend with me above the dust, above the cloud, to the realms of the higher geometry, where the heavens are never clouded; where there is no impure vapour, and no delusive or imperfect observation, where the new truths are already arisen, while they are yet dimly dawning on the world below; where the earth is a little planet; where the sun has dwindled to a star; where all the stars are lost in the Milky Way to which they belong; where the Milky Way is seen floating through space like any other nebula; where the whole great girdle of nebulae has diminished to an atom and has become as readily and completely submissive to the pen of the geometer, and the slave of his formula, as the single drop, which falls from the clouds, instinct with all the forces of the material world.

“It is to Sir Isaac Newton's Application of Geometry to Philosophy, that we owe the routing of this Army of Goths and Vandals in the philosophical World;”

John Theophilus Desaguliers (1683–1744) French-born British natural philosopher and clergyman

Source: Course of Experimental Philosophy, 1745, p. vi: Preface
Context: It is to Sir Isaac Newton's Application of Geometry to Philosophy, that we owe the routing of this Army of Goths and Vandals in the philosophical World; which he has enriched with more and greater Discoveries, than all the Philosophers that went before him: And has laid such Foundations for future Acquisitions, that even after his Death, his Works still promote natural Knowledge. Before Sir Isaac, we had but wild Guesses at the Cause of the Motion of the Comets and Planets round the Sun', but now he has clearly deduced them from the universal Laws of Attraction (the Existence of which he has proved beyond Contradiction) and has shewn, that the seeming Irregularities of the Moon, which Astronomers were unable to express in Numbers, are but the just Consequences of the Actions of the Sun and Earth upon it, according to their different Positions. His Principles clear up all Difficulties of the various Phænomena of the Tides; and the true Figure of the Earth is now plainly shewn to be a flatted Spheroid higher at the Equator than the Poles, notwithstanding many Assertions and Conjectures to the contrary.

World

“There is proof enough furnished by every science, but by none more than geometry, that the world to which we have been allotted is peculiarly adapted to our minds, and admirably fitted to promote our intellectual progress.”

Benjamin Peirce (1809–1880) American mathematician

Ben Yamen's Song of Geometry (1853)
Context: There is proof enough furnished by every science, but by none more than geometry, that the world to which we have been allotted is peculiarly adapted to our minds, and admirably fitted to promote our intellectual progress. There can be no reasonable doubt that it was part of the Creator's plan. How easily might the whole order have been transposed! How readily might we have been assigned to some complicated system which our feeble and finite powers could not have unravelled!

World, Science

“Everything one invents is true, you may be perfectly sure of that. Poetry is as precise as geometry.”

Gustave Flaubert (1821–1880) French writer (1821–1880)

14 August 1853
Correspondence, Letters to Madame Louise Colet

“A transition, therefore, is not undeservedly made from sense to consideration, and from this to the nobler energies of intellect. Hence, as the certain knowledge of numbers received its origin among the Phœnicians, on account of merchandise and commerce, so geometry was found out among the Egyptians from the distribution of land. When Thales, therefore, first went into Egypt, he transferred this knowledge from thence into Greece: and he invented many things himself, and communicated to his successors the principles of many. Some of which were, indeed, more universal, but others extended to sensibles.”

Proclus (412–485) Greek philosopher

Chap. IV.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788)

Communication, Knowledge, Sense, Universe

“Let us now explain the origin of geometry, as existing in the present age of the world. For the demoniacal Aristotle observes, that the same opinions often subsist among men, according to certain orderly revolutions of the world: and that sciences did not receive their first constitution in our times, nor in those periods which are known to us from historical tradition, but have appeared and vanished again in other revolutions of the universe; nor is it possible to say how often this has happened in past ages, and will again take place in the future circulations of time. But, because the origin of arts and sciences is to be considered according to the present revolution of the universe, we must affirm, in conformity with the most general tradition, that geometry was first invented by the Egyptians, deriving its origin from the mensuration of their fields: since this, indeed, was necessary to them, on account of the inundation of the Nile washing away the boundaries of land belonging to each. Nor ought It to seem wonderful, that the invention of this as well as of other sciences, should receive its commencement from convenience and opportunity. Since whatever is carried in the circle of generation proceeds from the imperfect to the perfect.”

Proclus (412–485) Greek philosopher

Chap. IV. On the Origin of Geometry, and its Inventors, pp. 98-99. Footnote (Taylor's): Aristotle was called demoniacal by the Platonic philosophers, in consequence of the encomium bestowed on him by his master, Plato, "That he was the dæmon of nature." Indeed, his great knowledge in things subject to the dominion of nature, well deserved this encomium, and the epithet divine, has been universally ascribed to Plato, from his profound knowledge of the intelligible world.
The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788)

Men, Age, World, Art

“I presume that to the uninitiated the formulae will appear cold and cheerless; but let it be remembered that, like other mathematical formulae, they find their origin in the divine source of all geometry.”

Benjamin Peirce Linear Associative Algebra

Preface.
Linear Associative Algebra (1882)
Context: I presume that to the uninitiated the formulae will appear cold and cheerless; but let it be remembered that, like other mathematical formulae, they find their origin in the divine source of all geometry. Whether I shall have the satisfaction of taking part in their exposition, or whether that will remain for some more profound expositor, will be seen in the future.

Mathematics, Appearance

“The right way to ask the question is: How does the concept of a point fit into the logical structure of Euclid's geometry? …It cannot be answered by a definition.”

Freeman Dyson book Infinite in All Directions

Source: Infinite in All Directions (1988), Ch. 2 : Butterflies and Superstrings, p. 17
Context: Euclid... gave his famous definition of a point: "A point is that which has no parts, or which has no magnitude." …A point has no existence by itself. It exists only as a part of the pattern of relationships which constitute the geometry of Euclid. This is what one means when one says that a point is a mathematical abstraction. The question, What is a point? has no satisfactory answer. Euclid's definition certainly does not answer it. The right way to ask the question is: How does the concept of a point fit into the logical structure of Euclid's geometry?... It cannot be answered by a definition.

Way, Question

“I resumed my seat, again exclaiming, "The boy is a fool, I say; 3³ can have no meaning in Geometry." At once there came a distinctly audible reply, "The boy is not a fool; and 3³ has an obvious Geometrical meaning."”

Edwin Abbott Abbott book Flatland

Source: Flatland: A Romance of Many Dimensions (1884), PART II: OTHER WORLDS, Chapter 15. Concerning a Stranger from Spaceland
Context: There I sat by my Wife's side, endeavouring to form a retrospect of the year 1999 and of the possibilities of the year 2000, but not quite able to shake off the thoughts suggested by the prattle of my bright little Hexagon. Only a few sands now remained in the half-hour glass. Rousing myself from my reverie I turned the glass Northward for the last time in the old Millennium; and in the act, I exclaimed aloud, "The boy is a fool."Straightway I became conscious of a Presence in the room, and a chilling breath thrilled through my very being. "He is no such thing," cried my Wife, "and you are breaking the Commandments in thus dishonouring your own Grandson." But I took no notice of her. Looking round in every direction I could see nothing; yet still I FELT a Presence, and shivered as the cold whisper came again. I started up. "What is the matter?" said my Wife, "there is no draught; what are you looking for? There is nothing." There was nothing; and I resumed my seat, again exclaiming, "The boy is a fool, I say; 3³ can have no meaning in Geometry." At once there came a distinctly audible reply, "The boy is not a fool; and 3³ has an obvious Geometrical meaning."

Fools, Boy

“And thus far historians produce the perfection of this science. But Euclid was not much junior to these, who collected elements, and constructed many of those things which were invented by Eudoxus; and perfected many which were discovered by Theætetus. Besides, he reduced to invincible demonstrations, such things as were exhibited by others with a weaker arm. But he lived in the times of the first Ptolemy: for Archimedes mentions Euclid, in his first book, and also in others. Besides, they relate that Euclid was asked by Ptolomy, whether there was any shorter way to the attainment of geometry than by his elementary institution, and that he answered, there was no other royal path which led to geometry. Euclid, therefore, was junior to the familiars of Plato, but more ancient than Eratosthenes and Archimedes (for these lived at one and the same time, according to the tradition of Eratosthenes) but he was of the Platonic sect, and familiar with its philosophy: and from hence he appointed the constitution of those figures which are called Platonic, as the end of his elementary institutions.”

Proclus (412–485) Greek philosopher

Source: The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788), Ch. IV. On the Origin of Geometry, and its Inventors.

Books, Way, Science, Time

“I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof”

Pierre de Fermat (1601–1665) French mathematician and lawyer

Letter to Frénicle (1657) Oeuvres de Fermat Vol.II as quoted by Edward Everett Whitford, The Pell Equation http://books.google.com/books?id=L6QKAAAAYAAJ (1912) <br class="br">Context: There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square.

Question, Problem

“Again, Amyclas the Heracleotean, one of Plato's familiars, and Menæchmus, the disciple, indeed, of Eudoxus, but conversant with Plato, and his brother Dinostratus, rendered the whole of geometry as yet more perfect. But Theudius, the Magnian, appears to have excelled, as well in mathematical disciplines, as in the rest of philosophy. For he constructed elements egregiously, and rendered many particulars more universal. Besides, Cyzicinus the Athenian, flourished at the same period, and became illustrious in other mathematical disciplines, but especially in geometry. These, therefore, resorted by turns to the Academy, and employed themselves in proposing common questions.”

Proclus (412–485) Greek philosopher

Source: The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788), Ch. IV.

Mathematics, Question, Universe, Appearance

“Geometry, to which I have devoted my life, is honoured with the title of the Key of Sciences”

Benjamin Peirce (1809–1880) American mathematician

Ben Yamen's Song of Geometry (1853)
Context: Geometry, to which I have devoted my life, is honoured with the title of the Key of Sciences; but it is the Key of an ever open door which refuses to be shut, and through which the whole world is crowding, to make free, in unrestrained license, with the precious treasures within, thoughtless both of lock and key, of the door itself, and even of Science, to which it owes such boundless possessions, the New World included. The door is wide open and all may enter, but all do not enter with equal thoughtlessness. There are a few who wonder, as they approach, at the exhaustless wealth, as the sacred shepherd wondered at the burning bush of Horeb, which was ever burning and never consumed. Casting their shoes from off their feet and the world's iron-shod doubts from their understanding, these children of the faithful take their first step upon the holy ground with reverential awe, and advance almost with timidity, fearful, as the signs of Deity break upon them, lest they be brought face to face with the Almighty.

Life, Science

“It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry.”

Solomon Lefschetz (1884–1972) American mathematician

[Carl C. Gaither, Alma E. Cavazos-Gaither, Gaither's Dictionary of Scientific Quotations: A Collection of Approximately 27,000 Quotations Pertaining to Archaeology, Architecture, Astronomy, Biology, Botany, Chemistry, Cosmology, Darwinism, Engineering, Geology, Mathematics, Medicine, Nature, Nursing, Paleontology, Philosophy, Physics, Probability, Science, Statistics, Technology, Theory, Universe, and Zoology, https://books.google.com/books?id=zQaCSlEM-OEC&pg=PA29, 5 January 2012, Springer Science & Business Media, 978-1-4614-1114-7, 29]

“Time carries him as the river carries
A leaf in the downstream water.
No matter. The enchanted one insists
And shapes God with delicate geometry.”

Jorge Luis Borges (1899–1986) Argentine short-story writer, essayist, poet and translator, and a key figure in Spanish language literature

"Baruch Spinoza", as translated in Spinoza and Other Heretics: The Marrano of Reason (1989) by Yirmiyahu Yovel
Context: Time carries him as the river carries
A leaf in the downstream water.
No matter. The enchanted one insists
And shapes God with delicate geometry.
Since his illness, since his birth,
He goes on constructing God with the word.
The mightiest love was granted him
Love that does not expect to be loved.

God, Water, Time

“Newton saw the truth of many propositions of geometry without reading the demonstrations, but not, we may be sure, without their flashing through his mind. A truth, or supposed truth, which is really the result of a very rapid inference, may seem to be apprehended intuitively. It has long been agreed by thinkers of the most opposite schools, that this mistake is actually made in so familiar an instance as that of the eyesight. There is nothing of which we appear to ourselves to be more directly conscious, than the distance of an object from us. Yet it has long been ascertained, that what is perceived by the eye, is at most nothing more than a variously coloured surface; that when we fancy we see distance, all we really see is certain variations of apparent size, and degrees of faintness of colour; and that our estimate of the object's distance from us is the result of a comparison (made with so much rapidity that we are unconscious of making it) between the size and colour of the object as they appear at the time, and the size and colour of the same or of similar objects as they appeared when close at hand, or when their degree of remoteness was known by other evidence.”

John Stuart Mill book A System of Logic

Source: A System of Logic (1843), p. 4
Context: [W]e may fancy that we see or feel what we in reality infer. Newton saw the truth of many propositions of geometry without reading the demonstrations, but not, we may be sure, without their flashing through his mind. A truth, or supposed truth, which is really the result of a very rapid inference, may seem to be apprehended intuitively. It has long been agreed by thinkers of the most opposite schools, that this mistake is actually made in so familiar an instance as that of the eyesight. There is nothing of which we appear to ourselves to be more directly conscious, than the distance of an object from us. Yet it has long been ascertained, that what is perceived by the eye, is at most nothing more than a variously coloured surface; that when we fancy we see distance, all we really see is certain variations of apparent size, and degrees of faintness of colour; and that our estimate of the object's distance from us is the result of a comparison (made with so much rapidity that we are unconscious of making it) between the size and colour of the object as they appear at the time, and the size and colour of the same or of similar objects as they appeared when close at hand, or when their degree of remoteness was known by other evidence. The perception of distance by the eye, which seems so like intuition, is thus, in reality, an inference grounded on experience; an inference, too, which we learn to make; and which we make with more and more correctness as our experience increases; though in familiar cases it takes place, so rapidly as to appear exactly on a par with those perceptions of sight which are really intuitive, our perceptions of colour.

Truth, School, Appearance, Time

“The Art of Sight Recognition, being no longer needed, was no longer practised; and the studies of Geometry, Statics, Kinetics, and other kindred subjects, came soon to be considered superfluous, and fell into disrespect and neglect even at our University. The inferior Art of Feeling speedily experienced the same fate at our Elementary Schools….”

Edwin Abbott Abbott book Flatland

Source: Flatland: A Romance of Many Dimensions (1884), PART I: THIS WORLD, Chapter 9. Of the Universal Colour Bill
Context: The Art of Sight Recognition, being no longer needed, was no longer practised; and the studies of Geometry, Statics, Kinetics, and other kindred subjects, came soon to be considered superfluous, and fell into disrespect and neglect even at our University. The inferior Art of Feeling speedily experienced the same fate at our Elementary Schools.... Year by year the Soldiers and Artisans began more vehemently to assert — and with increasing truth — that there was no great difference between them and the very highest class of Polygons, now that they were raised to an equality with the latter, and enabled to grapple with all the difficulties and solve all the problems of life, whether Statical or Kinetical, by the simple process of Colour Recognition. Not content with the natural neglect into which Sight Recognition was falling, they began boldly to demand the legal prohibition of all "monopolizing and aristocratic Arts" and the consequent abolition of all endowments for the studies of Sight Recognition, Mathematics, and Feeling. Soon, they began to insist that inasmuch as Colour, which was a second Nature, had destroyed the need of aristocratic distinctions, the Law should follow in the same path, and that henceforth all individuals and all classes should be recognized as absolutely equal and entitled to equal rights.

Fate, School, Art, Feeling

“After Pythagoras, Anaxagoras the Clazomenian succeeded, who undertook many things pertaining to geometry. And Oenopides the Chian, was somewhat junior to Anaxagoras, and whom Plato mentions in his Rivals, as one who obtained mathematical glory. To these succeeded Hippocrates, the Chian, who invented the quadrature of the lunula, and Theodorus the Cyrenean, both of them eminent in geometrical knowledge. For the first of these, Hippocrates composed geometrical elements: but Plato, who was posterior to these, caused as well geometry itself, as the other mathematical disciplines, to receive a remarkable addition, on account of the great study he bestowed in their investigation. This he himself manifests, and his books, replete with mathematical discourses, evince: to which we may add, that he every where excites whatever in them is wonderful, and extends to philosophy. But in his time also lived Leodamas the Thasian, Architas the Tarentine, and Theætetus the Athenian; by whom theorems were increased, and advanced to a more skilful constitution. But Neoclides was junior to Leodamas, and his disciple was Leon; who added many things to those thought of by former geometricians. So that Leon also constructed elements more accurate, both on account of their multitude, and on account of the use which they exhibit: and besides this, he discovered a method of determining when a problem, whose investigation is sought for, is possible, and when it is impossible.”

Proclus (412–485) Greek philosopher

Source: The Philosophical and Mathematical Commentaries of Proclus on the First Book of Euclid's Elements Vol. 1 (1788), Ch. IV.

Books, Mathematics, Knowledge, Problem

“She was no scholar in geometry or aught else, but she felt intuitively that the bend and slant of the way she went were somehow outside any other angles or bends she had ever known.”

C. L. Moore (1911–1987) American author

Black God's Kiss (1934)
Context: It was a long way down. Before she had gone very far the curious dizziness she had known before came over her again, a dizziness not entirely induced by the spirals she whirled around, but a deeper, atomic unsteadiness as if not only she but also the substances around her were shifting. There was something queer about the angles of those curves. She was no scholar in geometry or aught else, but she felt intuitively that the bend and slant of the way she went were somehow outside any other angles or bends she had ever known. They led into the unknown and the dark, but it seemed to her obscurely that they led into deeper darkness and mystery than the merely physical, as if, though she could not put it clearly even into thoughts, the peculiar and exact lines of the tunnel had been carefully angled to lead through poly-dimensional space as well as through the underground — perhaps through time, too.

Way

“A point has no existence by itself. It exists only as a part of the pattern of relationships which constitute the geometry of Euclid.”

Freeman Dyson book Infinite in All Directions

Parting

“He fell well short of mastering the art of demonstration; he had only a mediocre knowledge of analysis and geometry; what he knew best was to make lenses for microscopes.”

Baruch Spinoza (1632–1677) Dutch philosopher

Gottfried Wilhelm Leibniz, letter to Count Ernst von Hessen-Rheinfels (Aug. 14, 1683) in Gottfried Wilhelm Leibniz: Sämtliche Schriften und Briefe (1923-) II.ii. p. 535, as translated by Matthew Stewart, The Courtier and the Heretic (2006) pp. 228-229.
Context: Regarding Spinoza, whom M. Arnauld has called the most impious and most dangerous man of this century, he was truly an Atheist, [i. e., ] he allowed absolutely no Providence dispensing rewards and punishments according to justice.... The God he puts on parade is not like ours; he has no intellect or will.... He fell well short of mastering the art of demonstration; he had only a mediocre knowledge of analysis and geometry; what he knew best was to make lenses for microscopes.

Art, Knowledge

“It was a long way down. Before she had gone very far the curious dizziness she had known before came over her again, a dizziness not entirely induced by the spirals she whirled around, but a deeper, atomic unsteadiness as if not only she but also the substances around her were shifting. There was something queer about the angles of those curves. She was no scholar in geometry or aught else, but she felt intuitively that the bend and slant of the way she went were somehow outside any other angles or bends she had ever known. They led into the unknown and the dark, but it seemed to her obscurely that they led into deeper darkness and mystery than the merely physical, as if, though she could not put it clearly even into thoughts, the peculiar and exact lines of the tunnel had been carefully angled to lead through poly-dimensional space as well as through the underground—perhaps through time, too.”

C. L. Moore (1911–1987) American author

Black God's Kiss (1934); pp. 10-11
Short fiction, Jirel of Joiry (1969)

Way, Time

“Well aware that the opinions and belief of men depend not on their own will, but follow involuntarily the evidence proposed to their minds; that Almighty God hath created the mind free, and manifested his supreme will that free it shall remain by making it altogether insusceptible of restraint; that all attempts to influence it by temporal punishments, or burthens, or by civil incapacitations, tend only to beget habits of hypocrisy and meanness, and are a departure from the plan of the holy author of our religion, who being lord both of body and mind, yet choose not to propagate it by coercions on either, as was in his Almighty power to do, but to exalt it by its influence on reason alone; that the impious presumption of legislature and ruler, civil as well as ecclesiastical, who, being themselves but fallible and uninspired men, have assumed dominion over the faith of others, setting up their own opinions and modes of thinking as the only true and infallible, and as such endeavoring to impose them on others, hath established and maintained false religions over the greatest part of the world and through all time: That to compel a man to furnish contributions of money for the propagation of opinions which he disbelieves and abhors, is sinful and tyrannical; … that our civil rights have no dependence on our religious opinions, any more than our opinions in physics or geometry; and therefore the proscribing any citizen as unworthy the public confidence by laying upon him an incapacity of being called to offices of trust or emolument, unless he profess or renounce this or that religions opinion, is depriving him injudiciously of those privileges and advantages to which, in common with his fellow-citizens, he has a natural right; that it tends also to corrupt the principles of that very religion it is meant to encourage, by bribing with a monopoly of worldly honours and emolumerits, those who will externally profess and conform to it; that though indeed these are criminals who do not withstand such temptation, yet neither are those innocent who lay the bait in their way; that the opinions of men are not the object of civil government, nor under its jurisdiction; that to suffer the civil magistrate to intrude his powers into the field of opinion and to restrain the profession or propagation of principles on supposition of their ill tendency is a dangerous fallacy, which at once destroys all religious liberty, … and finally, that truth is great and will prevail if left to herself; that she is the proper and sufficient antagonist to error, and has nothing to fear from the conflict unless by human interposition disarmed of her natural weapons, free argument and debate; errors ceasing to be dangerous when it is permitted freely to contradict them.”

Thomas Jefferson (1743–1826) 3rd President of the United States of America

A Bill for Establishing Religious Freedom, Chapter 82 (1779). Published in The Works of Thomas Jefferson in Twelve Volumes http://oll.libertyfund.org/ToC/0054.php, Federal Edition, Paul Leicester Ford, ed., New York: G. P. Putnam's Sons, 1904, Vol. 1 http://oll.libertyfund.org/Texts/Jefferson0136/Works/0054-01_Bk.pdf, pp. 438–441. Comparison of Jefferson's proposed draft and the bill enacted http://web.archive.org/web/19990128135214/http://www.geocities.com/Athens/7842/bill-act.htm <br class="br">1770s

Men, Money, God, Truth

“In the famous dispute with the Nominalists, the Realists had a good deal of reason. General ideas are essences. They are our gods: they round and ennoble the most partial and sordid way of living. Our proclivity to details cannot quite degrade our life, and divest it of poetry. The day-laborer is reckoned as standing at the foot of the social scale, yet he is saturated with the laws of the world. His measures are the hours; morning and night, solstice and equinox, geometry, astronomy, and all the lovely accidents of nature play through his mind. Money, which represents the prose of life, and which is hardly spoken of in parlors without an apology, is, in its effects and laws, as beautiful as roses. Property keeps the accounts of the world, and is always moral. The property will be found where the labor, the wisdom, and the virtue have been in nations, in classes, and (the whole life-time considered, with the compensations) in the individual also. How wise the world appears, when the laws and usages of nations are largely detailed, and the completeness of the municipal system is considered! Nothing is left out.”

Ralph Waldo Emerson (1803–1882) American philosopher, essayist, and poet

1840s, Essays: Second Series (1844), Nominalist and Realist

Love, Life, Wisdom, Money

Immanuel Kant book Metaphysical Foundations of Natural Science

In the meantime, this little is something which mathematics indispensably requires in its application to natural science, which, inasmuch as it must here necessarily borrow from metaphysics, need not be ashamed to allow itself to be seen in company with the latter.
Preface, Tr. Bax (1883) citing Isaac Newton's Principia
Metaphysical Foundations of Natural Science (1786)

Mathematics

“Nature, therefore, is subject with absolute precision to all the precepts of geometry as to all the properties of space there demonstrated, this being the subjective condition, not hypothetically but intuitively given, of every phenomenon in which nature can ever be revealed to the senses.”

Immanuel Kant (1724–1804) German philosopher

Kant's Inaugural Dissertation (1770), Section III On The Principles Of The Form Of The Sensible World

Nature, Sense

“Phenomena of the external sense are examined and set forth in physics; those of the internal sense in empirical psychology. Pure mathematics considers space in geometry and time in pure mechanics. To these is to be added a certain concept, intellectual to be sure in itself, but whose becoming actual in the concrete requires the auxiliary notions of time and space in the successive addition and simultaneous juxtaposition of separate units, which is the concept of number treated in arithmetic.”

Immanuel Kant (1724–1804) German philosopher

Kant's Inaugural Dissertation (1770), Section II On The Distinction Between The Sensible And The Intelligible Generally

Mathematics, Sense, Time, Success

“The sensuous may be exceedingly distinct, while intellectual concepts are extremely confused. The former we observe in the prototype of sensuous knowledge geometry; the latter, in the organon of all intellectual concepts, metaphysics. It is evident how much toil the latter is expending to dispel the fogs of confusion darkening the common intellect, though not always with the happy success of the former science.”

Immanuel Kant (1724–1804) German philosopher

Kant's Inaugural Dissertation (1770), Section II On The Distinction Between The Sensible And The Intelligible Generally

Happiness, Knowledge, Science, Success

“Civil freedom, gentlemen, is not, as many have endeavoured to persuade you, a thing that lies hid in the depth of abstruse science. It is a blessing and a benefit, not an abstract speculation; and all the just reasoning that can bo upon it, is of so coarse a texture, as perfectly to suit the ordinary capacities of those who are to enjoy, and of those who are to defend it. Far from any resemblance to those propositions in geometry and metaphysics, which admit no medium, but must be true or false in all their latitude; social and civil freedom, like all other things in common life, are variously mixed and modified, enjoyed in very different degrees, and shaped into an infinite diversity of forms, according to the temper and circumstances of every community. The extreme of liberty (which is its abstract perfection, but its real fault) obtains no where, nor ought to obtain any where. Because extremes, as we all know, in every point which relates either to our duties or satisfactions in life, are destructive both to virtue and enjoyment. Liberty too must be limited in order to be possessed. The degree of restraint it is impossible in any case to settle precisely. But it ought to be the constant aim of every wise public counsel, to find out by cautious experiments, and rational, cool endeavours, with how little, not how much of this restraint, the community can subsist. For liberty is a good to be improved, and not an evil to be lessened. It is not only a private blessing of the first order, but the vital spring and energy of the state itself, which has just so much life and vigour as there is liberty in it. But whether liberty be advantageous or not, (for I know it is a fashion to decry the very principle,) none will dispute that peace is a blessing; and peace must in the course of human affairs be frequently bought by some indulgence and toleration at least to liberty. For as the sabbath (though of divine institution) was made for man, not man for the sabbath, government, which can claim no higher origin or authority, in its exercise at least, ought to conform to the exigencies of the time, and the temper and character of the people, with whom it is concerned; and not always to attempt violently to bend the people to their theories of subjection. The bulk of mankind on their part are not excessively curious concerning any theories, whilst they are really happy; and one sure symptom of an ill-conducted state, is the propensity of the people to resort to them.”

Edmund Burke (1729–1797) Anglo-Irish statesman

Letter to the Sheriffs of Bristol (1777)

Life, Happiness, People, Peace

“Swati Tirunal, now thirteen… took up a book of mathematics and selecting the forty seventh proposition of Euclid sketched the figure on a country slate but what astonished me most was his telling us in English that Geometry was derived from the Sanskrit, which as Jaw metor (Jyamiti) to measure the earth and that many of our mathematical terms were also derived from the same source such as hexagon, heptagon, octagon… This promising boy is now, I conclude, sovereign of the finest country in India for he was to succeed to the Musnud (throne) the moment he had attained his 16th year.”

Swathi Thirunal Rama Varma (1813–1846) Maharajah of Travencore

Colonel Welsh, in "The Monarch musician"
About Swathi Thirunal

Books, Boy, Mathematics, Country

“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.”

Ptolemy (100–170) Greco-Egyptian writer and astronomer of Alexandria

James Gow, A Short History of Greek Mathematics https://books.google.com/books?id=9d8DAAAAMAAJ (1884) p.308.

History, Effort, Study, Falseness

“I, therefore, did learn a lesson: The female mind is capable of understanding analytic geometry. Those people who have for years been insisting (in the face of all obvious evidence to the contrary) that the male and female are equally capable of rational thought may have something. The difficulty may just be that we have never yet discovered a way to communicate with the female mind. If it is done in the right way, you may be able to get something out of it.”

Richard Feynman (1918–1988) American theoretical physicist

address " What is Science? http://www.fotuva.org/feynman/what_is_science.html", presented at the fifteenth annual meeting of the National Science Teachers Association, in New York City (1966), published in The Physics Teacher, volume 7, issue 6 (1969), p. 313-320

People, Way, Communication, Understanding

“[O]ur purpose is to give a presentation of geometry... in its visual, intuitive aspects. With the aid of visual imagination we can illuminate the manifold facts and problems... beyond this, it is possible... to depict the geometric outline of the methods of investigation and proof, without... entering into the details... In this manner, geometry being as many-faceted as it is and being related to the most diverse branches of mathematics, we may even obtain a summarizing survey of mathematics as a whole, and a valid idea of the variety of problems and the wealth of ideas it contains. Thus a presentation of geometry in large brushstrokes... and based on the approach through visual intuition, should contribute to a more just appreciation of mathematics by a wider range of people than just the specialists.”

David Hilbert Geometry and the Imagination

Preface
Geometry and the Imagination (1952)

People, Imagination, Mathematics, Idea

“In mathematics, as in any scientific research, we find two tendencies... [T]he tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material in a systematic and orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects... a live rapport with them... which stresses the concrete meaning of their relations. ...[I]ntuitive understanding plays a major role in geometry. ...[S]uch concrete intuition is of great value not only for the research worker, but... for anyone who wishes to study and appreciate the results of research in geometry.”

David Hilbert Geometry and the Imagination

Preface
Geometry and the Imagination (1952)

Mathematics, Understanding, Study, Relation ships

“I think a lot of people may take some heart from this: a lot of the math, the heavy math in projective geometry, [...] took me a long time. There were many many years, a decade, when I was considered this graphics guru genius, when I really couldn't do from scratch [...] the mathematics that underpins a lot of that. But slowly, eventually, with a couple decades of experience, most of it did eventually sink in on me.”

John D. Carmack (1970) American computer programmer, engineer, and businessman

Speaking about mathematics in engineering, Quoted in https://www.youtube.com/watch?v=dSCBCk4xVa0&t=1271s

People, Mathematics, Thinking, Time

“The various branches of geometry are all interrelated closely and quite often unexpectedly. This shows up in many places in the book. Even so... it was necessary to make each chapter...self-contained... We hope that... we have rendered each chapter taken by itself... understandable and interesting. We want to take the reader on a leisurely walk... in the big garden that is geometry, so that each may pick for himself a bouquet to his liking.”

David Hilbert Geometry and the Imagination

Preface
Geometry and the Imagination (1952)

Books, Making love, Hope, Understanding

“A philosopher who has no connection to geometry is only half a philosopher, and a mathematician who has no philosophical vein is only half a mathematician.”

Gottlob Frege (1848–1925) mathematician, logician, philosopher

Original: (de) Ein Philosoph, der keine Beziehung zur Geometrie hat, ist nur ein halber Philosoph, und ein Mathematiker, der keine philosophische Ader hat, ist nur ein halber Mathematiker.

Gottlob Frege: Erkenntnisquellen der Mathematik und der mathematischen Naturwissenschaften, 1924/1925, submitted to Wissenschaftliche Grundlagen; posthumously published in: Frege, Gottlob: Nachgelassene Schriften und Wissenschaftlicher Briefwechsel. Felix Meiner Verlag, 1990, p. 293

Quotes about geometry page 3

Quotes about geometry
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