Quotes about geometry (302 quotes) | Quotes of famous people

“There is no royal road to geometry.”
Non est regia ad Geometriam via.

Euclid (-323–-285 BC) Greek mathematician, inventor of axiomatic geometry

μὴ εἶναι βασιλικὴν ἀτραπὸν ἐπί γεωμετρίαν, Non est regia [inquit Euclides] ad Geometriam via
Reply given when the ruler Ptolemy I Soter asked Euclid if there was a shorter road to learning geometry than through Euclid's Elements.The "Royal Road" was the road built across Anatolia and Persia by Darius I which allowed rapid communication and troop movement, but use of ἀτραπός (rather than ὁδός) conveys the connotation of "short cut".
The Greek is from Proclus (412–485 AD) in Commentary on the First Book of Euclid's Elements, the Latin translation is by Francesco Barozzi, 1560) the English translation follows Glenn R. Morrow (1970), p. 57 http://books.google.com/books?id=JZEHj2fEmqAC&q=royal#v=snippet&q=royal&f=false.
Attributed

“In geometry his greatest achievement was an accurate value of π.”

Aryabhata (476–550) Indian mathematician-astronomer

His rule is stated as: dn^2+(2a-d)n=2s, which implies the approximation 3.1416 which is correct to the last decimal place.
In, p. 245.
Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures

“The science of Chaos teaches us that everything is interconnected, but the contemporary developments in neuroscience, getting started with the brain neurons and their multiple connections, reveal the topology of the brain, a miniature of the universal geometry of everything.”

Alexis Karpouzos (1967)

Source: https://www.goodreads.com/author/quotes/14108295.alexis_karpouzos?page=2

Universe , Science

“The problem, of course, was that turning into a monster was the brighter of my two choices. Choice Number 1: I turn into a vampyre, which equals a monster in just about any human’s mind. Choice Number 2: My body rejects the Change and I die. Forever.

So the good news is that I wouldn’t have to take the geometry test tomorrow.”

Kristin Cast book Marked

Source: Marked

Problem , Change

“I cannot say whether I will still be doing geometry ten years from now. It also seems to me that the mine has maybe already become too deep and unless one finds new veins it might have to be abandoned. Physics and chemistry now offer a much more glowing richness and much easier exploitation. Also, the general taste has turned entirely in this direction, and it is not impossible that the place of Geometry in the Academies will someday become what the role of the Chairs of Arabic at the universities is now.”

Joseph Louis Lagrange (1736–1813) Italian mathematician and mathematical physicist

Letter to d'Alembert (1781) cited in R. Laubenbacher, D. Pengelly: Mathematical Expeditions: Chronicles by the Explorers (1999) Springer, pp. 233–234.

Universe

“Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians' observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real. …There may be three dimensions in this room and five next door. As a professional mathematician, I have no idea; I can only ask some competent physicist to instruct me in the facts.
The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. …Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers.”

G. H. Hardy (1877–1947) British mathematician

"The Theory of Numbers," Nature (Sep 16, 1922) Vol. 110 https://books.google.com/books?id=1bMzAQAAMAAJ p. 381

Beauty , Mountain , Mathematics , Idea

“An ancient writer said that arithmetic and geometry are the wings of mathematics; I believe one can say without speaking metaphorically that these two sciences are the foundation and essence of all the sciences which deal with quantity. Not only are they the foundation, they are also, as it were, the capstones; for, whenever a result has been arrived at, in order to use that result, it is necessary to translate it into numbers or into lines; to translate it into numbers requires the aid of arithmetic, to translate it into lines necessitates the use of geometry.”

Joseph Louis Lagrange (1736–1813) Italian mathematician and mathematical physicist

Dans Les Leçons Élémentaires sur les Mathématiques (1795) Leçon cinquiéme, Tr. McCormack, cited in Moritz, Memorabilia mathematica or, The philomath's quotation-book (1914) Ch. 15 Arithmetic, p. 261. https://archive.org/stream/memorabiliamathe00moriiala#page/260/mode/2up

Mathematics , Science

“The ancients considered mechanics in a twofold respect; as rational, which proceeds accurately by demonstration, and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accurate is called geometrical; what is less so is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic: and if any could work with perfect accuracy, he would be the most perfect mechanic of all; for the description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn; for it requires that the learner should first be taught to describe these accurately, before he enters upon geometry; then it shows how by these operations problems may be solved.”

Isaac Newton book Philosophiae Naturalis Principia Mathematica

Preface (8 May 1686)
Philosophiæ Naturalis Principia Mathematica (1687)

Drawing , Art , Perfection , Problem

“It is necessary to learn geometry only so far as might enable a man to measure land for the purposes of buying and selling.”

Socrates (-470–-399 BC) classical Greek Athenian philosopher

Diogenes Laertius

Learning

“No story is a straight line. The geometry of a human life is too imperfect and complex, too distorted by the laughter of time and the bewildering intricacies of fate to admit the straight line into its system of laws.”

Pat Conroy book Beach Music

Source: Beach Music

Life , Fate , Law , Laughter

“If I died, would it get me out of my geometry test tomorrow? One could only hope.”

Kristin Cast book Marked

Source: Marked

Hope

“If I died, would it get me out of my geometry test tomorrow? One could only hope." - Zoey Mongomery (Redbird)”

P. C. Cast book Marked

Source: Marked

Hope

“Some forms of Kantianism put great weight on “what we can imagine” holding that this can be a source of insight into necessary connections. Thus various of Kant’s arguments about space and time depend on the purported fact that it is impossible for us to imagine certain things: we can know, Kant claims, a priori that space has only three dimensions because we cannot imagine it as having more than three dimensions. History in the form of non-Euclidean geometry and modern physics has put paid to that particular line of argument, but in general we should beware of depending too much on “What we can imagine?,” especially in politics. As Nietzsche puts it somewhere, sometimes the fact that you can’t imagine a situation in which things are very different from the way they are now is not an especially good argument for the claim that they must be as they now are, but, rather, represents a failure of your powers about which you should feel mildly apologetic.”

Raymond Geuss book Philosophy and Real Politics

Source: Philosophy and Real Politics (2008), pp. 48-49.

Way , Feeling , Imagination , History

“Integral geometry, started by the English geometer M. W. Crofton, has received recently important developments through the works of W. Blaschke, L. A. Santaló, and others. Generally speaking, its principal aim is to study the relations between the measures which can be attached to a given variety.”

Shiing-Shen Chern (1911–2004) mathematician (1911–2004), born in China and later acquiring U.S. citizenship; made fundamental contributio…

[Differential geometry and integral geometry, Proc. Int. Congr. Math. Edinburgh, 1958, 411–449, http://www.mathunion.org/ICM/ICM1958/Main/icm1958.0441.0453.ocr.pdf]

Study

“The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves.”

Shiing-Shen Chern (1911–2004) mathematician (1911–2004), born in China and later acquiring U.S. citizenship; made fundamental contributio…

[Differential geometry, its past and its future, Actes, Congrès inter. math, 1970, 41–53, http://www.math.harvard.edu/~hirolee/pdfs/2014-fall-230a-icm1970-chern-differential-geometry.pdf]

Study

“I wanted to have a water jet in my garden: Euler calculated the force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sans Souci. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces from the reservoir. Vanity of vanities! Vanity of geometry!”

Frederick II of Prussia (1712–1786) king of Prussia

Je voulus faire un jet d’eau dans mon jardin; Euler calcula l’effort des roues pour faire monter l’eau dans un bassin, d’où elle devait retomber par des canaux, afin de jaillir à Sans-Souci. Mon moulin a été exécuté géométriquement, et il n’a pu élever une goutte d’eau à cinquante pas du bassin. Vanité des vanités! vanité de la géométrie!
Letter H 7434 from Frederick to Voltaire (1778-01-25)

Water

“Not all the geometrical structures are "equal". It would seem that the riemannian and complex structures, with their contacts with other fields of mathematics and with their richness in results, should occupy a central position in differential geometry. A unifying idea is the notion of a G-structure, which is the modern version of an equivalence problem first emphasized and exploited in its various special cases by Elie Cartan.”

Shiing-Shen Chern (1911–2004) mathematician (1911–2004), born in China and later acquiring U.S. citizenship; made fundamental contributio…

Mathematics , Idea , Problem

“Equations are just the boring part of mathematics. I attempt to see things in terms of geometry.”

Stephen Hawking (1942–2018) British theoretical physicist, cosmologist, and author

As quoted in Stephen Hawking: A Biography (2005) by Kristine Larsen, p. 43

Mathematics , Parting

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

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

Forward, as quoted by Mario Livio, Is God a Mathematician? (2009)
Ausdehnungslehre (1844)

Law , Way , Mathematics , Nature

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

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

The Art of Measurement (1525).

Art , Painting

“My research work was based in pictures. I needed to visualise things and projective geometry was often most useful e. g. in figuring out how a particular quantity transforms under Lorentz transf[ormation]. When I came to publish the results I suppressed the projective geometry as the results could be expressed more concisely in analytic form.”

Paul Dirac (1902–1984) theoretical physicist

"Recollections of an Exciting Era," three lectures given at Varenna, 5 August 1972, quoted in Peter Galison, "The Suppressed Drawing: Paul Dirac's Hidden Geometry", Representations, No. 72 (Autumn, 2000)

“I feel entitled to hope that I have found in this new analysis the only natural method according to which mathematics should be applied to nature, and according to which geometry may also be treated, whenever it leads to general and to fruitful results.”

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

Ausdehnungslehre (1844)

Hope , Feeling , Mathematics , Nature

“all the standard equations of mathematical physics can be separated and solved in Kerr geometry.”

Subrahmanyan Chandrasekhar (1910–1995) physicist

From Chandrasekhar's Nobel lecture, in his summary of his work on black holes; Republished in: D. G. Caldi, ‎George D. Mostow (1989) Proceedings of the Gibbs Symposium: Yale University, May 15-17, 1989 p. 230

Mathematics

“After the same Manner in Geometry, if a Line drawn any certain Way be reckon'd for Affirmative, then a Line drawn the contrary Way may be taken for Negative: As if AB be drawn to the right, and BC to the left; and AB be reckon'd Affirmative, then BC will be Negative; because in the drawing it diminishes AB…”

Isaac Newton book Arithmetica Universalis

p, 125
Arithmetica Universalis (1707)

Way , Drawing

“Either one or the other [ analysis or synthesis ] may be direct or indirect. The direct procedure is when the point of departure is known-direct synthesis in the elements of geometry. By combining at random simple truths with each other, more complicated ones are deduced from them. This is the method of discovery, the special method of inventions, contrary to popular opinion.”

André-Marie Ampère (1775–1836) French physicist and mathematician

André-Marie Ampè, in André-Marie Ampère: Enlightenment and Electrodynamics http://books.google.co.in/books?id=QWZKQWB-sbQC&pg=PA159, Cambridge University Press, 1996, p. 159

Truth , Departure

“As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited. But when these sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace toward perfection.”

Joseph Louis Lagrange (1736–1813) Italian mathematician and mathematical physicist

Dans Les Leçons Élémentaires sur les Mathématiques (1795) Leçon cinquiéme,Tr. McCormack, cited in Robert Edouard Moritz, Memorabilia mathematica or, The philomath's quotation-book (1914) Ch. V The teaching of mathematics, p. 81. https://archive.org/stream/memorabiliamathe00moriiala#page/80/mode/2up

Science , Perfection

“The first impulse came from the consideration of negatives in geometry; I was accustomed to viewing the distances AB and BA as opposite magnitudes. Arising from this idea was the conclusion that if A, B, and C are points of a straight line, then in all cases AB + BC = AC, this being true whether AB and BC are directed in the same direction or in opposite directions (where C lies between A and B). In the latter case AB and BC were not viewed as merely lengths, but simultaneously their considered since they were oppositely directed, Thus dawned the distinction between the sum of lengths and the sum of distances which were fixed in direction. From this resulted the requirement for establishing this latter concept of sum, not simply for the case where the distances were directed in the same or opposite directions, but also for any other case. This could be done in the most simple manner, since the law that AB + BC = AC remains valid when A, B, and C do not lie on a straight line.
This then was the first step which led to a new branch of mathematics… I did not however realize how fruitful and how rich was the field that I had opened up; rather that result seemed scarcely worthy of note until it was combined with a related idea.”

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

Ausdehnungslehre (1844)

Lie , Law , Mathematics , Idea

“In the beginning of the year 1665 I found the method of approximating Series and the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of tangents of Gregory and Slusius, and in November had the direct method of Fluxions, and the next year in January had the Theory of Colours, and in May following I had entrance into the inverse method of Fluxions. And the same year I began to think of gravity extending to the orb of the Moon, and having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere, from Kepler's Rule of the periodical times of the Planets being in a sesquialterate proportion of their distances from the centers of their orbs I deduced that the forces which keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the earth, and found them answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and minded Mathematicks and Philosophy more than at any time since. What Mr Hugens has published since about centrifugal forces I suppose he had before me. At length in the winter between the years 1676 and 1677 I found the Proposition that by a centrifugal force reciprocally as the square of the distance a Planet must revolve in an Ellipsis about the center of the force placed in the lower umbilicus of the Ellipsis and with a radius drawn to that center describe areas proportional to the times. And in the winter between the years 1683 and 1684 this Proposition with the Demonstration was entered in the Register book of the R. Society. And this is the first instance upon record of any Proposition in the higher Geometry found out by the method in dispute. In the year 1689 Mr Leibnitz, endeavouring to rival me, published a Demonstration of the same Proposition upon another supposition, but his Demonstration proved erroneous for want of skill in the method.”

Isaac Newton (1643–1727) British physicist and mathematician and founder of modern classical physics

(ca. 1716) A Catalogue of the Portsmouth Collection of Books and Papers Written by Or Belonging to Sir Isaac Newton https://books.google.com/books?id=3wcjAAAAMAAJ&pg=PR18 (1888) Preface
Also partially quoted in Sir Sidney Lee (ed.), The Dictionary of National Biography Vol.40 http://books.google.com/books?id=NycJAAAAIAAJ (1894)

Books , Age , Thinking , Winter

“Though a state of affairs that would contravene the laws of physics can be represented by us spatially, one that would contravene the laws of geometry cannot.”

Ludwig Wittgenstein (1889–1951) Austrian-British philosopher

3.0321
Original German: Wohl können wir einen Sachverhalt räumlich darstellen, welcher den Gesetzen der Physik, aber keinen, der den Gesetzen der Geometrie zuwiderliefe.
1920s, Tractatus Logico-Philosophicus (1922)

Law

“In my Judgment no Lines ought to be admitted into plain Geometry besides the right Line and the Circle.”

Isaac Newton book Arithmetica Universalis

p, 125
Arithmetica Universalis (1707)

“It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretical reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no analogue for orthogonal groups in n (>4) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifols of four dimensions, because their tangent spaces possess a geometry of this kind.”

Shiing-Shen Chern (1911–2004) mathematician (1911–2004), born in China and later acquiring U.S. citizenship; made fundamental contributio…

[On Riemannian manifolds of four dimensions, Bulletin of the American Mathematical Society, 51, 12, 1945, 964–971, http://www.ams.org/journals/bull/1945-51-12/S0002-9904-1945-08483-3/S0002-9904-1945-08483-3.pdf]

Knowledge , Study , Universe

“Of such are the mathematical sciences alone; that is, geometry and arithmetic, in which the Divine intellect indeed knows infinitely more propositions, since it knows all. But with regard to those few which the human intellect does understand, I believe its knowledge equals the Divine in objective certainty, for here it succeeds in understanding necessity, beyond which there can be no greater sureness.”

Galileo Galilei book Dialogue Concerning the Two Chief World Systems

In the 1661 translation by Thomas Salusbury: … such are the pure Mathematical sciences, to wit, Geometry and Arithmetick: in which Divine Wisdom knows infinite more propositions, because it knows them all; but I believe that the knowledge of those few comprehended by humane understanding, equalleth the divine, as to the certainty objectivè, for that it arriveth to comprehend the necessity thereof, than which there can be no greater certainty." p. 92 (from the Archimedes Project http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.cgi?page=92;dir=galil_syste_065_en_1661;step=textonly)
In the original Italian: … tali sono le scienze matematiche pure, cioè la geometria e l’aritmetica, delle quali l’intelletto divino ne sa bene infinite proposizioni di piú, perché le sa tutte, ma di quelle poche intese dall’intelletto umano credo che la cognizione agguagli la divina nella certezza obiettiva, poiché arriva a comprenderne la necessità, sopra la quale non par che possa esser sicurezza maggiore." (from the copy at the Italian Wikisource).
Dialogue Concerning the Two Chief World Systems (1632)

Mathematics , Understanding , Knowledge , Science

“Geometry is to the plastic arts what grammar is to the art of the writer.”

Guillaume Apollinaire (1880–1918) French poet

La géométrie est aux arts plastiques ce que la grammaire est à l'art de l'écrivain.
Les peintres cubistes (1913), reprinted in Oeuvres en prose complètes (Paris: Gallimard, 1991) vol. 2, p. 11; translation from Lionel Abel (trans.) The Cubist Painters (New York: Wittenborn, 1949) p. 13.

Art

“I think the first thing that led me toward philosophy (though at that time the word 'philosophy' was still unknown to me) occurred at the age of eleven. My childhood was mainly solitary as my only brother was seven years older than I was. No doubt as a result of much solitude I became rather solemn, with a great deal of time for thinking but not much knowledge for my thoughtfulness to exercise itself upon. I had, though I was not yet aware of it, the pleasure in demonstrations which is typical of the mathematical mind. After I grew up I found others who felt as I did on this matter. My friend G. H. Hardy, who was professor of pure mathematics, enjoyed this pleasure in a very high degree. He told me once that if he could find a proof that I was going to die in five minutes he would of course be sorry to lose me, but this sorrow would be quite outweighed by pleasure in the proof. I entirely sympathized with him and was not at all offended. Before I began the study of geometry somebody had told me that it proved things and this caused me to feel delight when my brother said he would teach it to me. Geometry in those days was still 'Euclid.' My brother began at the beginning with the definitions. These I accepted readily enough. But he came next to the axioms. 'These,' he said, 'can't be proved, but they have to be assumed before the rest can be proved.' At these words my hopes crumbled. I had thought it would be wonderful to find something that one could prove, and then it turned out that this could only be done by means of assumptions of which there was no proof. I looked at my brother with a sort of indignation and said: 'But why should I admit these things if they can't be proved?”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

He replied, 'Well, if you won't, we can't go on.'
Source: 1950s, Portraits from Memory and Other Essays (1956), p. 19

Pleasure , Age , Hope , Feeling

“It has been the error of schools to teach astronomy, and all the other sciences and subjects of natural philosophy, as accomplishments only; whereas they should be taught theologically, or with reference to the Being who is the Author of them: for all the principles of science are of divine origin. Man cannot make, or invent, or contrive principles; he can only discover them, and he ought to look through the discovery to the Author.
When we examine an extraordinary piece of machinery, an astonishing pile of architecture, a well-executed statue, or a highly-finished painting where life and action are imitated, and habit only prevents our mistaking a surface of light and shade for cubical solidity, our ideas are naturally led to think of the extensive genius and talent of the artist. When we study the elements of geometry, we think of Euclid. When we speak of gravitation, we think of Newton. How, then, is it that when we study the works of God in creation, we stop short and do not think of God? It is from the error of the schools in having taught those subjects as accomplishments only and thereby separated the study of them from the Being who is the Author of them.”

Thomas Paine (1737–1809) English and American political activist

1790s, Discourse to the Theophilanthropists (1798)

Life , God , School , Error

“Algebraic geometry has developed in waves, each with its own language and point of view. The late nineteenth century saw the function-theoretic approach of Riemann, the more geometric approach of Brill and Noether, and the purely algebraic approach of Kronecker, Dedekind, and Weber. The Italian school followed with Castelnuovo, Enriques, and Severi, culminating in the classification of algebraic surfaces. Then came the twentieth-century "American" school of Chow, Weil, and Zariski, which gave firm algebraic foundations to the Italian intuition. Most recently, Serre and Grothendieck initiated the French school, which has rewritten the foundations of algebraic geometry in terms of schemes and cohomology, and which has an impressive record of solving old problems with new techniques. Each of these schools has introduced new concepts and methods.”

Robin Hartshorne book Algebraic Geometry

Algebraic Geometry, Springer, (1977), p. xiii
Algebraic Geometry (1977)

School , Problem

“Geometry does not teach us to draw these lines, but requires them to be drawn; for it requires that the learner should first be taught to describe these accurately, before he enters upon geometry; then it shows how by these operations problems may be solved.”

Isaac Newton book Philosophiae Naturalis Principia Mathematica

Preface (8 May 1686)
Philosophiae Naturalis Principia Mathematica (1687)
Context: The ancients considered mechanics in a twofold respect; as rational, which proceeds accurately by demonstration, and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accurate is called geometrical; what is less so is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic: and if any could work with perfect accuracy, he would be the most perfect mechanic of all; for the description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn; for it requires that the learner should first be taught to describe these accurately, before he enters upon geometry; then it shows how by these operations problems may be solved.

Problem , Drawing

“But the Moderns advancing yet much farther, have receiv'd into Geometry all Lines that can be express'd by Æquations, and have distinguish'd, according to the Dimensions of the Æquations, those Lines into Kinds; and have made it a Law, that you are not to construct a Problem by a Line of a superior Kind, that may be constructed by one of an inferior one.”

Isaac Newton book Arithmetica Universalis

Arithmetica Universalis (1707)
Context: The Antients, as we learn from Pappus, in vain endeavour'd at the Trisection of an Angle, and the finding out of two mean Proportionals by a right line and a Circle. Afterwards they began to consider the Properties of several other Lines. as the Conchoid, the Cissoid, and the Conick Sections, and by some of these to solve these Problems. At length, having more throughly examin'd the Matter, and the Conick Sections being receiv'd into Geometry, they distinguish'd Problems into three Kinds: viz. (1.) Into Plane ones, which deriving their Original from Lines on a Plane, may be solv'd by a right Line and a Circle; (2.) Into Solid ones, which were solved by Lines deriving their Original from the Consideration of a Solid, that is, of a Cone; (3.) And Linear ones, to the Solution of which were requir'd Lines more compounded. And according to this Distinction, we are not to solve solid Problems by other Lines than the Conick Sections; especially if no other Lines but right ones, a Circle, and the Conick Sections, must be receiv'd into Geometry. But the Moderns advancing yet much farther, have receiv'd into Geometry all Lines that can be express'd by Æquations, and have distinguish'd, according to the Dimensions of the Æquations, those Lines into Kinds; and have made it a Law, that you are not to construct a Problem by a Line of a superior Kind, that may be constructed by one of an inferior one. In the Contemplation of Lines, and finding out their Properties, I like their Distinction of them into Kinds, according to the Dimensions thy Æquations by which they are defin'd. But it is not the Æquation, but the Description that makes the Curve to be a Geometrical one.<!--pp.227-228

Law , Problem

“Geometry was invented that we might expeditiously avoid, by drawing Lines, the Tediousness of Computation.”

Isaac Newton book Arithmetica Universalis

Arithmetica Universalis (1707)
Context: Geometry was invented that we might expeditiously avoid, by drawing Lines, the Tediousness of Computation. Therefore these two Sciences ought not to be confounded. The Antients did so industriously distinguish them from one another, that they never introduc'd Arithmetical Terms into Geometry. And the Moderns, by confounding both, have lost the Simplicity in which all the Elegancy of Geometry consists. Wherefore that is Arithmetically more simple which is determin'd by the more simple Æquations, but that is Geometrically more simple which is determin'd by the more simple drawing of Lines; and in Geometry, that ought to be reckon'd best which is Geometrically most simple. Wherefore, I ought not to be blamed, if with that Prince of Mathematicians, Archimedes and other Antients, I make use of the Conchoid for the Construction of solid Problems.<!--p.230

Drawing

“Useful Things, though Mechanical, are justly preferable to useless Speculations in Geometry”

Isaac Newton book Arithmetica Universalis

Arithmetica Universalis (1707), p.248

“But after these, Pythagoras changed that philosophy, which is conversant about geometry itself, into the form of a liberal doctrine, considering its principles in a more exalted manner; and investigating its theorems immaterially and intellectually; who likewise invented a treatise of such things as cannot be explained in geometry, and discovered the constitution of the mundane figures.”

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)

Change

“Logic has borrowed, perhaps, the rules of geometry, without comprehending their force”

Blaise Pascal (1623–1662) French mathematician, physicist, inventor, writer, and Christian philosopher

The Art of Persuasion
Context: Logic has borrowed, perhaps, the rules of geometry, without comprehending their force... it does not thence follow that they have entered into the spirit of geometry, and I should be greatly averse... to placing them on a level with that science that teaches the true method of directing reason.

“I affirm, in words as true and literal as any that belong to geometry, that the man who withholds knowledge from a child not only works diabolical miracles for the destruction of good, but for the creation of evil also.”

Horace Mann (1796–1859) American politician

Congressional speech (1849)
Context: I affirm, in words as true and literal as any that belong to geometry, that the man who withholds knowledge from a child not only works diabolical miracles for the destruction of good, but for the creation of evil also. He who shuts out truth, by the same act opens the door to all the error that supplies its place. Ignorance breeds monsters to fill up all the vacuities of the soul that are unoccupied by the verities of knowledge. He who dethrones the idea of law, bids chaos welcome in its stead. Superstition is the mathematical complement of religious truth; and just so much less as the life of a human being is reclaimed to good, just so much more is it delivered over to evil. The man or the institution, therefore, that withholds knowledge from a child, or from a race of children, exercises the awful power of changing the world in which they are to live, just as much as though he should annihilate all that is most lovely and grand in this planet of ours, or transport the victim of his cruelty to some dark and frigid zone of the universe, where the sweets of knowledge are unknown, and the terrors of ignorance hold their undisputed and remorseless reign.

Knowledge , Evil

“Be not, O Greeks, so very hostilely disposed towards the Barbarians. To the Babylonians you owe astronomy; to the Persians, magic; to the Egyptians, geometry; to the Phoeicians, instruction by alphabetic writing. Cease, then, to miscall these imitations inventions of your own. Orpheus, again, taught you poetry and song; from him, too, you learned the mysteries.”

Tatian (120–180) Syrian writer

Ante-Nicene Christian library: v. 3 p. 5
Address to the Greeks

Learning , Hostility

“Our civil rights have no dependence upon our religious opinions more than our opinions in physics or geometry.”

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
1770s
Variant: Our civil rights have no dependence on our religious opinions any more than our opinions in physics or geometry...
Source: The Statute Of Virginia For Religious Freedom
Context: 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.

Dependency

“Rachel bit her lip. "I hope you're right. I'm a little worried. What if someone asks what's on the next math test and I start spouting a prophecy in the middle of geometry class? Gods, that would be embarrassing.”

Rick Riordan book The Last Olympian

Source: The Last Olympian

God , Hope

“There is still a difference between something and nothing, but it is purely geometrical and there is nothing behind the geometry.”

Martin Gardner (1914–2010) recreational mathematician and philosopher

The Mathematical Magic Show (1978)

“There is geometry in the humming of the strings. There is music in the spacings of the spheres.”

Pythagoras (-585–-495 BC) ancient Greek mathematician and philosopher

As quoted in the preface of the book entitled Music of the Spheres by Guy Murchie (1961)
The Golden Verses

Powerful , Music

“Everything you invent is true: you can be sure of that. Poetry is a subject as precise as geometry.”

Julian Barnes book Flaubert's Parrot

Source: Flaubert's Parrot

“A firm rule must be imposed upon our nation before it destroys itself. The United States needs some theology and geometry, some taste and decency. I suspect that we are teetering on the edge of the abyss.”

John Kennedy Toole A Confederacy of Dunces

Source: A Confederacy of Dunces

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

Johannes Kepler book Mysterium Cosmographicum

As quoted by Karl Fink, Geschichte der Elementar-Mathematik (1890) translated as A Brief History of Mathematics https://books.google.com/books?id=3hkPAAAAIAAJ (1900, 1903) by Wooster Woodruff Beman, David Eugene Smith. Also see Carl Benjamin Boyer, A History of Mathematics (1968).
Mysterium Cosmographicum (1596)

“Cézanne showed us forms living in the reality of light; Picasso gives us a material report of their real life in the mind. He establishes a free, mobile perspective, in such a way that the shrewd mathematician Maurice Princet has deduced an entire geometry.”

Jean Metzinger (1883–1956) French painter

Note sur la Peinture (1910)

Life , Way , Reality , Light

“There are infinitely many variations of the initial situation and therefore no doubt indefinitely many theorems of moral geometry.”

John Rawls book A Theory of Justice

Source: A Theory of Justice (1971; 1975; 1999), Chapter III, Section 21, pg. 126

“Andre Agassi, in his account of becoming an embittered prodigy, seems never to have liked tennis much, except as a vehicle for achievement. The kids who do like life inside the lines can find the flow within that green-and-white geometry.”

Adam Gopnik (1956) American journalist

How to Raise a Prodigy, The New Yorker (2018)

Life

“Some other Course therefore must be taken to promote the Search of Knowledge. Some other kind of Art for Inquiry than what hath been hitherto made use of, must be discovered; the Intellect is not to he suffer'd to act without its Helps, but is continually to be assisted by some Method or Engine, which shall be as a Guide to regulate its Actions, so as that it shall not be able to act amiss: Of this Engine, no Man except the incomparable Verulam hath had any Thoughts, and he indeed hath promoted it to a very good pitch; but there is yet somewhat more to be added, which he seem'd to want time to compleat. By this, as by that Art of Algebra in Geometry, 'twill be very easy to proceed in any Natural Inquiry, regularly and certainly: And indeed it may not improperly be call'd a Philosophical Algebra, or an Art of directing the Mind in the search after Philosophical Truths, for as 'tis very hard for the most acute Wit to find out any difficult Problem in Geometry. without the help of Algebra to direct and regulate the Acts of the Reason in the Process from the question to the quœsitum, and altogether as easy for the meanest Capacity acting by that Method to compleat and perfect it, so will it be in the inquiry after Natural Knowledge.”

Robert Hooke (1635–1703) English natural philosopher, architect and polymath

"The Present State of Natural Philosophy, and wherein it is deficient," The Posthumous Works of Robert Hooke https://books.google.com/books?id=6xVTAAAAcAAJ (1705) ed., Richard Waller, pp. 6-7.

Truth , Art , Search , Question

“The geometry of Tlön comprises two somewhat different disciplines: the visual and the tactile. The latter corresponds to our own geometry and is subordinated to the first.”

Jorge Luis Borges book Tlön, Uqbar, Orbis Tertius

Tlön, Uqbar, Orbis Tertius (1940)

“…the differential element of non-Euclidean spaces is Euclidean. This fact, however, is analogous to the relations between a straight line and a curve, and cannot lead to an epistemological priority of Euclidean geometry, in contrast to the views of certain authors.”

Hans Reichenbach (1891–1953) American philosopher

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

“Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.”

Michael Atiyah (1929–2019) British mathematician

[Michael Atiyah, Collected works. Vol. 6, The Clarendon Press Oxford University Press, Oxford Science Publications, http://www.math.tamu.edu/~rojas/atiyah20thcentury.pdf, 978-0-19-853099-2, 2160826, 2004]

Soul , Question

“The strongest use of the symbol is to be found in its magical power of doubling the actual universe, and placing by its side an ideal universe, its exact counterpart, with which it can be compared and contrasted, and, by means of curiously connecting fibres, form with it an organic whole, from which modern analysis has developed her surpassing geometry.”

Benjamin Peirce (1809–1880) American mathematician

On the Uses and Transformations of Linear Algebra (1875)

Universe

“Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra, speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, “that it reaches out and indissolubly connects itself each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic 31 geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics” (he might, if speaking at the present moment, go on to add the theory of elasticity and the development of the integral calculus) “have all felt its influence.”

James Joseph Sylvester (1814–1897) English mathematician

James Joseph Sylvester. "A Plea for the Mathematician, Nature," Vol. 1, p. 238; Collected Mathematical Papers, Vol. 2 (1908), pp. 655, 656.

Law , Mathematics , Idea , Effort

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

Hans Reichenbach (1891–1953) American philosopher

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

Color , Sense

“For to pass by those Ancients, the wonderful Pythagoras, the sagacious Democritus, the divine Plato, the most subtle and very learned Aristotle, Men whom every Age has hitherto acknowledged as deservedly honored, as the greatest Philosophers, the Ring-leaders of Arts; in whose Judgments how much these Studies [mathematics] were esteemed, is abundantly proclaimed in History and confirmed by their famous Monuments, which are everywhere interspersed and bespangled with Mathematical Reasonings and Examples, as with so many Stars; and consequently anyone not in some Degree conversant in these Studies will in vain expect to understand, or unlock their hidden Meanings, without the Help of a Mathematical Key: For who can play well on Aristotle’s Instrument but with a Mathematical Quill; or not be altogether deaf to the Lessons of natural Philosophy, while ignorant of Geometry? Who void of (Geometry shall I say, or) Arithmetic can comprehend Plato’s 218 Socrates lisping with Children concerning Square Numbers; or can conceive Plato himself treating not only of the Universe, but the Polity of Commonwealths regulated by the Laws of Geometry, and formed according to a Mathematical Plan?”

Isaac Barrow (1630–1677) English Christian theologian, and mathematician

Source: Mathematical Lectures (1734), pp. 26-27

Men , Children , Age , Law

“Experience has convinced me that the proper way of teaching is to bring together that which is simple from all quarters, and, if I may use such a phrase, to draw upon the surface of the subject a proper mean between the line of closest connexion and the line of easiest deduction. This was the method followed by Euclid, who, fortunately for us, never dreamed of a geometry of triangles, as distinguished from a geometry of circles, or a separate application of the arithmetics of addition and subtraction; but made one help out the other as he best could.”

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

Advertisement, p.4
The Differential and Integral Calculus (1836)

Dreams , Way , Help , Drawing

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

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 148

Question

“The more I looked at his face [of Saint Sebastian] the more curious it seemed. That said, I seemed to have always known it and the aseptic morning light revealed its smallest details which such clarity, such purity, that I was impossible moved... In the upper part of the heliometer was St. Sebastian's magnifying glass... I put my eye to the magnifying glass, product of a slow distillation, at once numerical and intuitive. Each drop of water a number. Each drop of blood a geometry.”

Salvador Dalí (1904–1989) Spanish artist

Quote from his poem 'Sant Sebastia', Salvador Dali 1927 - dedicated to the Spanish poet Lorca; as quoted in Dali and Me, Catherine Millet, - translation Trista Selous -, Scheidegger & Spiess AG, 8001 Zurich Switzerland, p. 46
Quotes of Salvador Dali, 1920 - 1930

Light , Water , Parting

“Around 1650 I came across the mathematical writings of Torricelli, where among other things, he expounds the geometry of indivisibles of Cavalieri. …His method, as taught by Torricelli… was indeed all the more welcome to me because I do not know that anything of that kind was observed in the thinking of almost any mathematician I had previously met.”

John Wallis (1616–1703) English mathematician

Arithmetica Infinitorum (1656)

Mathematics , Thinking

“Fractal geometry is not just a chapter of mathematics, but one that helps Everyman to see the same world differently.”

Benoît Mandelbrot book The Fractal Geometry of Nature

The Fractal Geometry of Nature (1982)

World , Mathematics , Help

“I now saw, that a science is either deductive or experimental, according as, in the province it deals with, the effects of causes when conjoined, are or are not the sums of the effects which the same causes produce when separate. It followed that politics must be a deductive science. It thus appeared, that both Macaulay and my father were wrong; the one in assimilating the method of philosophising in politics to the purely experimental method of chemistry; while the other, though right in adopting a deductive method, had made a wrong selection of one, having taken as the type of deduction, not the appropriate process, that of the deductive branches of natural philosophy, but the inappropriate one of pure geometry, which, not being a science of causation at all, does not require or admit of any summing-up of effects.”

John Stuart Mill book Autobiography

Source: Autobiography (1873)
Source: https://archive.org/details/autobiography01mill/page/160/mode/1up pp. 160-161

Nature , Appearance , Science

“I expect that mathematicians have classified such fuzzy logics. Certainly they have been much used by physicists. But is there not something to be said for the approach of Euclid? Even now that we know that Euclidean geometry is (in some sense) not quite true? Is it not good to know what follows from what, even if it is not necessarily FAPP? Suppose for example that quantum mechanics were found to resist precise formulation. Suppose that when formulation beyond FAPP was attempted, we find an unmovable finger obstinately pointing outside the subject, to the mind of the observer, to the Hindu scriptures, to God, or even only Gravitation? Would that not be very, very interesting?”

John S. Bell (1928–1990) Northern Irish physicist

Against 'measurement' (1990)

God , Sense

“Since geometry is co-eternal with the divine mind before the birth of things, God himself served as his own model in creating the world (for what is there in God which is not God?), and he with his own image reached down to humanity.”

Johannes Kepler book Harmonices Mundi

Book IV, Ch. 1, as quoted in "Kepler's Astrology"in Kepler, Four Hundred Years (1975) edited by Arthur and Peter Beer.
Harmonices Mundi (1618)

God , World , Birth

“[T]he application of algebra to geometry… far more than any of his metaphysical speculations, has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences.”

John Stuart Mill (1806–1873) British philosopher and political economist

An Examination of Sir William Hamilton's Philosophy (1865) as quoted in 5th ed. (1878) p. 617. https://books.google.com/books?id=ojQNAQAAMAAJ&pg=PA617

Science

“There is only one other survey, Datta and Singh’s 1938 History of Hindu Mathematics, recently reprinted but very hard to obtain in the West (I found a copy in a small specialized bookstore in Chennai). They describe in some detail the Indian work in arithmetic and algebra and, supplemented by the equally hard to find Geometry in Ancient and Medieval India by Sarasvati Amma (1979), one can get an overview of most topics.”

David Mumford (1937) American mathematician

[David Mumford, Book Review, Notices of the AMS, March 2010, 57, 3, http://www.ams.org/notices/201003/rtx100300385p.pdf]

History , Mathematics

“From Pythagoras to Boethius, when pure mathematics consisted of arithmetic and geometry while applied mathematics consisted of music and astronomy, mathematics could be characterized as the deductive study of 'such abstractions as quantities and their consequences, namely figures and so forth' (Acquinas ca. 1260). But since the emergence of abstract algebra it has become increasingly difficult to formulate a definition to cover the whole of the rich, complex and expanding domain of mathematics.”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

Music , Mathematics , Study

“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.”

Pāṇini ancient Sanskrit grammarian

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

“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.”

Baruch Spinoza (1632–1677) Dutch philosopher

Jorge Luis Borges, "Baruch Spinoza", as translated in Spinoza and Other Heretics, Vol. 1: The Marrano of Reason (1989) by Yirmiyahu Yovel
A - F

Love , God , Birth , Water

“The word science of administration has been used. There are many who object to the term. Now if by science is meant a conceptual scheme of things in which every particularity coveted may be assigned a mathematical value, then administration is not a science. In this sense only astro-physics may be called a science and it is well to remember that mechanical laws of the heavens tell us nothing about the color and composition of the stars and as yet cannot account for some of the disturbances and explosions which seem accidental. If, on the other hand, we may rightly use the term science in connection with a body of exact knowledge derived from experience and observation, and a body of rules or axioms which experience has demonstrated to be applicable in concrete practice, and to work out in practice approximately as forecast, then we may, if we please, appropriately and for convenience, speak of a science of administration. Once, when the great French mathematician, Poincaré, was asked whether Euclidean geometry is true, he replied that the question had no sense but that Euclidean geometry is and still remains the most convenient. The Oxford English Dictionary tells us that a science is, among other things, a particular branch of knowledge or study; a recognized department of learning.”

Charles A. Beard (1874–1948) American historian

Source: Philosophy, Science and Art of Public Administration (1939), p. 660-1

Law , Color , Star , Mathematics

“Nevertheless, it remains conceivable that the measure relations of space in the infinitely small are not in accordance with the assumptions of our geometry [Euclidean geometry], and, in fact, we should have to assume that they are not if, by doing so, we should ever be enabled to explain phenomena in a more simple way.”

Bernhard Riemann (1826–1866) German mathematician

Memoir (1854) Tr. William Kingdon Clifford, as quoted by A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein https://archive.org/details/TheEvolutionOfScientificThought (1927) p. 55.

Way

“One finds in this subject a kind of demonstration which does not carry with it so high a degree of certainty as that employed in geometry; and which differs distinctly from the method employed by geometers in that they prove their propositions by well-established and incontrovertible principles, while here principles are tested by inferences which are derivable from them. The nature of the subject permits of no other treatment. It is possible, however, in this way to establish a probability which is little short of certainty. This is the case when the consequences of the assumed principles are in perfect accord with the observed phenomena, and especially when these verifications are numerous; but above all when one employs the hypothesis to predict new phenomena and finds his expectations realized.”

Christiaan Huygens book Treatise on Light

Treatise on Light (1690) - preface, Translated by Michael R. Matthews, Scientific Background to Modern Philosophy. 1989. p. 126

Way , Nature , Perfection

“My specific… object has been to contain, within the prescribed limits, the whole of the student's course, from the confines of elementary algebra and trigonometry, to the entrance of the highest works on mathematical physics. A learner who has a good knowledge of the subjects just named, and who can master the present treatise, taking up elementary works on conic sections, application of algebra to geometry, and the theory of equations, as he wants them, will, I am perfectly sure, find himself able to conquer the difficulties of anything he may meet with; and need not close any book of Laplace, Lagrange, Legendre, Poisson, Fourier, Cauchy, Gauss, Abel, Hindenburgh and his followers. or of any one of our English mathematicians, under the idea that it is too hard for him.”

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

Preface, p. iii
The Differential and Integral Calculus (1836)

Books , Mathematics , Idea , Knowledge

“What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze.”

Paul Klee (1879–1940) German Swiss painter

1921 - 1930
Source: 'Bauhaus prospectus 1929'; as quoted in Artists on Art, from the 14th – 20th centuries, ed. by Robert Goldwater and Marco Treves; Pantheon Books, 1972, London, p. 444

Art , Music , Mathematics , Study

“You see, some mathematicians have clear and far-ranging. "programs". For instance, Grothendieck had such a program for algebraic geometry; now Langlands has one for representation theory, in relation to modular forms and arithmetic. I never had such a program, not even a small size one.”

Jean-Pierre Serre (1926) French mathematician

An Interview with Jean-Pierre Serre - Singapore Mathematical Society https://sms.math.nus.edu.sg/smsmedley/Vol-13-1/An%20interview%20with%20Jean-Pierre%20Serre(CT%20Chong%20&%20YK%20Leong).pdf

“"Some fool has put the head of this nail on the wrong end." "You idiot, it's for the opposite wall!" To be sure, if the space of physical objects allowed motions of translation only, and not also rotations. That space has such geometry is a fantasy; experience shows otherwise. It is only experience that makes this complaint and the rejoinder a dialogue of madmen.”

Abraham Kaplan (1918–1993) American philosopher

Source: "The Conduct of Inquiry", p. 35.

Fools

“Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers… were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians… The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences… Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency… it did trouble deeply the European mathematicians.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p.144

History , Mathematics , Problem , Reading

“"All things excellent are as difficult as they are rare." These words which bring to a close Spinoza's masterpiece Ethics, after the manner of Geometry, sum up the experience of a life as rare as it was difficult.”

Edgar A. Singer, Jr. (1873–1954) American philosopher

Source: Modern thinkers and present problems, (1923), p. 37: Chapter 2. Benedict de Spinoza, 1632-1677

Life

“Mathematicks therefore is a Science which teaches or contemplates whatever is capable of Measure or Number as such. When it relates to Number, it is called Arithmetick; but when to measure, as Length, Breadth, Depth, Degrees of Velocity in Motion, Intenseness or Remissness of Sounds, Augmentation or Diminution of Quality, 6tc. it is called Geometry.”

Jacques Ozanam (1640–1718) French mathematician

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

Science

“A psychological space is established for any set of stimuli by determining metric distances between the stimuli such that the probability that a response learned to any stimulus will generalize to any other is an invariant monotonic function of the distance between them. To a good approximation, this probability of generalization (i) decays exponentially with this distance, and (ii) does so in accordance with one of two metrics, depending on the relation between the dimensions along which the stimuli vary. These empirical regularities are mathematically derivable from universal principles of natural kinds and probabilistic geometry that may, through evolutionary internalization, tend to govern the behaviors of all sentient organisms.”

Roger Shepard (1929) American psychologist

Source: "Toward a universal law of generalization for psychological science," 1987, p. 1317

Behavior , Mathematics , Nature , Learning

“The geometry of innocence flesh on the bone
Causes Galileo’s math book to get thrown
At Delilah who sits worthlessly alone
But the tears on her cheeks are from laughter”

Bob Dylan (1941) American singer-songwriter, musician, author, and artist

Song lyrics, Highway 61 Revisited (1965), Tombstone Blues

Books , Tear , Laughter

“Once the first principles are disposed of, the body of doctrine contained in the recent textbooks of elementary geometry does not, and from the nature of the case cannot, show any substantial differences from that set forth in the Elements.”

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

Introduction, p. v
The Thirteen Books of Euclid's Elements (1908)

Nature

“There is perhaps no question that occupies, comparatively, a larger space in the history of Greek geometry than the problem of the Doubling of the Cube. The tradition concerning its origin is given in a letter from Eratosthenes of Cyrene to King Ptolemy Euergetes quoted by Eutocius…
"Eratosthenes to King Ptolemy greeting.
"There is a story that one of the old tragedians represented Minos as wishing to erect a tomb for Glaucus and as saying, when he heard that it was a hundred feet every way,Too small thy plan to bound a royal tomb.
Let it be double; yet of its fair form
Fail not, but haste to double every side.But he was clearly in error; for when the aides are doubled, the area becomes four times as great, and the solid content eight times as great. Geometers also continued to investigate the question in what manner one might double a given solid while it remained in the same form.”

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

Apollonius of Perga (1896)

Way , Error , Quotes , History

“I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect... Geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.”

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

As quoted in Solid Shape (1990) by Jan J. Koenderink

“Thus, all unknown quantities can be expressed in terms of a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth.
But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by anyone at all familiar with ordinary geometry and with algebra, who will consider carefully all that is set forth in this treatise.”

René Descartes book La Géométrie

First Book
La Géométrie (1637)

Pleasure , Problem , Science , Training

“The main objection to the theory of pure visualization is our thesis that the non-Euclidean axioms can be visualized just as rigorously if we adjust the concept of congruence. This thesis is based on the discovery that the normative function of visualization is not of visual but of logical origin and that the intuitive acceptance of certain axioms is based on conditions from which they follow logically, and which have previously been smuggled into the images. The axiom that the straight line is the shortest distance is highly intuitive only because we have adapted the concept of straightness to the system of Eucidean concepts. It is therefore necessary merely to change these conditions to gain a correspondingly intuitive and clear insight into different sets of axioms; this recognition strikes at the root of the intuitive priority of Euclidean geometry. Our solution of the problem is a denial of pure visualization, inasmuch as it denies to visualization a special extralogical compulsion and points out the purely logical and nonintuitive origin of the normative function. Since it asserts, however, the possibility of a visual representation of all geometries, it could be understood as an extension of pure visualization to all geometries. In that case the predicate "pure" is but an empty addition, since it denotes only the difference between experienced and imagined pictures, and we shall therefore discard the term "pure visualization."”

Hans Reichenbach (1891–1953) American philosopher

Instead we shall speak of the normative function of the thinking process, which can guide the pictorial elements of thinking into any logically permissible structure.
The Philosophy of Space and Time (1928, tr. 1957)

Imagination , Problem , Thinking , Change

“The confidence placed in physical theory owes much to its possessing the same kind of excellence from which pure geometry and pure mathematics in general derive their interest, and for the sake of which they are cultivated. … We cannot truly account for our acceptance of such theories without endorsing our acknowledgement of a beauty that exhilarates and a profundity that entrances us.”

Michael Polanyi (1891–1976) Hungarian-British polymath

Source: Personal Knowledge (1958), p. 15

Beauty , Confidence , Mathematics

“One of the most curious of these cases [geometrical paradoxers] was that of a student, I am not sure but a graduate, of the University of Virginia, who claimed that geometers were in error in assuming that a line had no thickness. He published a school geometry based on his views, which received the endorsement of a well-known New York school official and, on the basis of this, was actually endorsed, or came very near being endorsed, as a text-book hi the public schools of New York.”

Simon Newcomb (1835–1909) American astronomer

Simon Newcomb, The Reminiscences of an Astronomer, (Boston and New York, 1903), p. 388. Reported in Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book https://archive.org/stream/memorabiliamathe00moriiala#page/81/mode/2up, (1914), p. 368

Books , School , Error , Universe

“vagueness is a 'romantic' value.... an emphasis on geometry is an emphasis on the 'known', on order and knowledge.”

Ad Reinhardt (1913–1967) American painter

Quote of Ad Reinhardt in: Abstract Expressionist Painting in America, W.C, Seitz, Cambridge Massachusetts, 1983, p. 107
after 1967 - posthumous

Knowledge

“But to return to the Newtonian Philosophy: Tho' its Truth is supported by Mathematicks, yet its Physical Discoveries may be communicated without. The great Mr. Locke was the first who became a Newtonian Philosopher without the help of Geometry; for having asked Mr. Huygens, whether all the mathematical Propositions in Sir Isaac's Principia were true, and being told he might depend upon their Certainty; he took them for granted, and carefully examined the Reasonings and Corollaries drawn from them, became Master of all the Physics, and was fully convinc'd of the great Discoveries contained in that Book.”

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

Source: Course of Experimental Philosophy, 1745, p. viii: Preface; Cited in Joseph Schwartz (1992), The creative moment: how science made itself alien to modern culture, p. 20

Truth , Books , Communication , Mathematics