Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence.

**Ibn Khaldun**

geometry-enlightens-intellect-sets-ones-mind-right-all-its-proofs-are-clear-orderly-it-is-hardly-possible-for-errors-to-enter-into-geometrical-ibn-khaldun

In fact, Gentlemen, no geometry without arithmetic, no mechanics without geometry... you cannot count upon success, if your mind is not sufficiently exercised on the forms and demonstrations of geometry, on the theories and calculations of arithmetic ... In a word, the theory of proportions is for industrial teaching, what algebra is for the most elevated mathematical teaching.

**Jean-Victor Poncelet**

in-fact-gentlemen-no-geometry-without-arithmetic-no-mechanics-without-geometry-you-cannot-count-upon-success-if-your-mind-is-not-sufficiently-jeanvictor-poncelet

I conceived, developed and applied in many areas a new geometry of nature, which finds order in chaotic shapes and processes. It grew without a name until 1975, when I coined a new word to denote it, fractal geometry, from the Latin word for irregular and broken up, fractus. Today you might say that, until fractal geometry became organized, my life had followed a fractal orbit.

**Benoit Mandelbrot**

i-conceived-developed-applied-in-many-areas-new-geometry-nature-which-finds-order-in-chaotic-shapes-processes-it-grew-without-name-until-1975-benoit-mandelbrot

i-like-my-things-hurried-haunted-night-tea-darktime-sacred-geometry-secret-geometry-petalflame-whisper-i-am-here-you-arent-virginia-petrucci

Abstraction didn't have to be limited to a kind of rectilinear geometry or even a simple curve geometry. It could have a geometry that had a narrative impact. In other words, you could tell a story with the shapes. It wouldn't be a literal story, but the shapes and the interaction of the shapes and colors would give you a narrative sense. You could have a sense of an abstract piece flowing along and being part of an action or activity. That sort of turned me on.

**Frank Stella**

abstraction-didnt-have-to-be-limited-to-kind-rectilinear-geometry-even-simple-curve-geometry-it-could-have-geometry-that-had-narrative-impact-in-frank-stella

As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer...

**Oliver Heaviside**

as-to-need-improvement-there-can-be-no-question-whilst-reign-euclid-continues-my-own-idea-useful-course-is-to-begin-with-arithmetic-then-not-euclid-but-algebra-next-not-euclid-bu

The best that Gauss has given us was likewise an exclusive production. If he had not created his geometry of surfaces, which served Riemann as a basis, it is scarcely conceivable that anyone else would have discovered it. I do not hesitate to confess that to a certain extent a similar pleasure may be found by absorbing ourselves in questions of pure geometry.

**Albert Einstein**

the-best-that-gauss-has-given-us-was-likewise-exclusive-production-if-he-had-not-created-his-geometry-surfaces-which-served-riemann-as-basis-it-is-scarcely-conceivable-that-anyon

The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry [as did Clifford], for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.

**Eric Temple Bell**

the-full-impact-lobachevskian-method-challenging-axioms-has-probably-yet-to-be-felt-it-is-no-exaggeration-to-call-lobachevsky-copernicus-geometry-as-did-clifford-for-geometry-is-

Pi is not merely the ubiquitous factor in high school geometry problems; it is stitched across the whole tapestry of mathematics, not just geometry's little corner of it. Pi occupies a key place in trigonometry too. It is intimately related to e, and to imaginary numbers. Pi even shows up in the mathematics of probability

**Robert Kanigel**

pi-is-not-merely-ubiquitous-factor-in-high-school-geometry-problems-it-is-stitched-across-whole-tapestry-mathematics-not-just-geometrys-little-robert-kanigel

one-geometry-cannot-be-more-true-than-another-it-can-only-be-more-convenient-geometry-is-not-true-it-is-advantageous

It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.

**Bernhard Riemann**

it-is-well-known-that-geometry-presupposes-not-only-concept-space-but-also-first-fundamental-notions-for-constructions-in-space-as-given-in-advance-it-only-gives-nominal-definiti

Kepler's discovery would not have been possible without the doctrine of conics. Now contemporaries of Kepler-such penetrating minds as Descartes and Pascal-were abandoning the study of geometry... because they said it was so UTTERLY USELESS. There was the future of the human race almost trembling in the balance; for had not the geometry of conic sections already been worked out in large measure, and had their opinion that only sciences apparently useful ought to be pursued, the nineteenth century would have had none of those characters which distinguish it from the ancien regime.

**Charles Sanders Peirce**

keplers-discovery-would-not-have-been-possible-without-doctrine-conics-now-contemporaries-keplersuch-penetrating-minds-as-descartes-pascalwere-abandoning-study-geometry-because-t

... geometry became a symbol for human relations, except that it was better, because in geometry things never go bad. If certain things occur, if certain lines meet, an angle is born. You cannot fail. It's not going to fail; it is eternal. I found in rules of mathematics a peace and a trust that I could not place in human beings. This sublimation was total and remained total. Thus, I'm able to avoid or manipulate or process pain.

**Louise Bourgeois**

geometry-became-symbol-for-human-relations-except-that-it-was-better-because-in-geometry-things-never-go-bad-if-certain-things-occur-if-certain-louise-bourgeois

About Thomas Hobbes: He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read the proposition "By God", sayd he, "this is impossible:" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that truth. This made him in love with geometry.

**John Aubrey**

about-thomas-hobbes-he-was-40-years-old-before-he-looked-on-geometry-which-happened-accidentally-being-in-gentlemans-library-euclids-elements-lay-john-aubrey

The Greeks made Space the subject-matter of a science of supreme simplicity and certainty. Out of it grew, in the mind of classical antiquity, the idea of pure science. Geometry became one of the most powerful expressions of that sovereignty of the intellect that inspired the thought of those times. At a later epoch, when the intellectual despotism of the Church, which had been maintained through the Middle Ages, had crumbled, and a wave of scepticism threatened to sweep away all that had seemed most fixed, those who believed in Truth clung to Geometry as to a rock, and it was the highest ideal of every scientist to carry on his science 'more geometrico.

**Hermann Weyl**

the-greeks-made-space-subjectmatter-science-supreme-simplicity-certainty-out-it-grew-in-mind-classical-antiquity-idea-pure-science-geometry-became-one-most-powerful-expressions-t

And so will I here state just plainly and briefly that I accept God. But I must point out one thing: if God does exist and really created the world, as we well know, he created it according to the principles of Euclidean geometry and made the human brain capable of grasping only three dimensions of space. Yet there have been and still are mathematicians and philosophers-among them some of the most outstanding-who doubt that the whole universe or, to put it more generally, all existence was created to fit Euclidean geometry; they even dare to conceive that two parallel lines that, according to Euclid, never do meet on earth do, in fact, meet somewhere in infinity. And so my dear boy, I've decided that I am incapable of understanding of even that much, I cannot possibly understand about God.

**Fyodor Dostoyevsky**

and-will-i-here-state-just-plainly-briefly-that-i-accept-god-but-i-must-point-out-one-thing-if-god-does-exist-really-created-world-as-we-well-know-he-created-it-according-to-prin

inspiration-is-needed-in-geometry-just-as-much-as-in-poetry-alexander-pushkin

there-is-still-difference-between-something-nothing-but-it-is-purely-geometrical-there-is-nothing-behind-geometry-martin-gardner

it-has-adopted-geometry-most-advantageous-to-species-in-other-words-most-convenient

no-employment-can-be-managed-without-arithmetic-no-mechanical-invention-without-geometry-benjamin-franklin

everything-you-invent-is-true-you-can-be-sure-that-poetry-is-subject-as-precise-as-geometry-julian-barnes

i-think-i-still-like-science-art-better-but-geometry-is-big-improvement-over-algebra

the-detailed-geometry-coenzyme-molecule-as-whole-is-fascinating-in-its-complexity-dorothy-hodgkin

all-ones-inventions-are-true-you-can-be-sure-of-that-poetry-is-as-exact-a-science-as-geometry

everything-one-invents-is-true-you-may-be-perfectly-sure-of-that-poetry-is-as-precise-as-geometry

there-is-geometry-in-the-humming-of-the-strings-there-is-music-in-the-spacing-of-the-spheres

Some persons fancy that bias and counter-bias are favorable to the extraction of truth-that hot and partisan debate is the way to investigate. This is the theory of our atrocious legal procedure. But Logic puts its heel upon this suggestion. It irrefragably demonstrates that knowledge can only be furthered by the real desire for it, and that the methods of obstinacy, of authority and every mode of trying to reach a foregone conclusion, are absolutely of no value. These things are proved. The reader is at liberty to think so or not as long as the proof is not set forth, or as long as he refrains from examining it. Just so, he can preserve, if he likes, his freedom of opinion in regard to the propositions of geometry; only, in that case, if he takes a fancy to read Euclid, he will do well to skip whatever he finds with A, B, C, etc., for, if he reads attentively that disagreeable matter, the freedom of his opinion about geometry may unhappily be lost forever.

**Charles Sanders Peirce**

some-persons-fancy-that-bias-counterbias-are-favorable-to-extraction-truththat-hot-partisan-debate-is-way-to-investigate-this-is-theory-our-atrocious-legal-procedure-but-logic-pu

in-creating-world-god-used-arithmetic-geometry-likewise-astronomy-nicholas-cusa

let-noone-ignorant-geometry-enter-said-to-have-been-inscribed-above-door-platos-academy-plato

one-great-challenges-modern-cosmology-is-to-discover-what-geometry-universe-really-is

preliminary-accounting-banking-surveying-known-as-arithmetic-algebra-geometry-alan-watts

one-great-sources-employment-for-people-with-phds-in-geometry-is-animation-industry-margaret-wertheim

plane-geometry-is-sort-key-course-where-you-learn-about-proving-things-abstraction

life-is-merely-fracas-on-unmapped-terrain-universe-geometry-stricken-with-epilepsy-emile-m-cioran

arithmetic-algebra-geometry-grandiose-trinity-luminous-triangle-whoever-has-not-known-you-is-without-sense-isidore-ducasse-lautreamont

the-composer-opens-cage-door-for-arithmetic-draftsman-gives-geometry-its-freedom-jean-cocteau

who-thinks-all-science-as-all-virtue-vain-who-counts-geometry-numbers-toys-john-dryden

people-were-going-to-geometry-class-i-was-swimming-through-vats-chili-on-even-stevens-it-was-like-dream-shia-labeouf

experience-proves-that-anyone-who-has-studied-geometry-is-infinitely-quicker-to-grasp-difficult-subjects-than-one-who-has-not-plato

there-are-infinitely-many-variations-initial-situation-therefore-no-doubt-indefinitely-many-theorems-moral-geometry-john-rawls

i-find-purer-philosophy-in-poem-than-in-conclusion-geometry-chemical-analysis-physical-law

i-think-universe-is-pure-geometry-basically-beautiful-shape-twisting-around-dancing-over-spacetime-antony-garrett-lisi

a-work-morality-politics-criticism-will-be-more-elegant-other-things-being-equal-if-it-is-shaped-by-hand-geometry-bernard-le-bovier-de-fontenelle

geometry-in-every-proposition-speaks-language-which-experience-never-dares-to-utter-indeed-which-she-but-halfway-comprehends-meaning-william-whewell

only-fraction-cameras-possibilities-interests-me-marvelous-mixture-emotion-geometry-together-in-single-instant-henri-cartierbresson

and-since-geometry-is-right-foundation-all-painting-i-have-decided-to-teach-its-rudiments-principles-to-all-youngsters-eager-for-art

eternity-is-depth-which-no-geometry-can-measure-no-arithmetic-calculate-no-imagination-conceive-no-rhetoric-describe-hannah-more

mathematicians-didnt-invent-infinity-until-1877-so-they-thought-it-was-impossible-that-africans-could-be-using-fractal-geometry

i-am-present-at-sowing-seed-world-with-geometry-sunbeams-soul-lays-foundations-nature-ralph-waldo-emerson

while-fractal-geometry-is-often-used-in-high-tech-science-its-patterns-are-surprisingly-common-in-traditional-african-designs

the-pits-tangles-are-more-than-blemishes-distorting-classic-shapes-euclidian-geometry-they-are-often-keys-to-essence-thing-james-gleick

the-creator-fountain-all-wisdom-approver-perpetual-order-eternal-superessential-spring-geometry-harmonics-johannes-kepler

it-is-hard-to-convince-highschool-student-that-he-will-encounter-lot-problems-more-difficult-than-those-algebra-geometry-e-w-howe

i-have-infamously-low-capacity-for-visualizing-relationships-which-made-study-geometry-all-subjects-derived-from-it-impossible-for-me-sigmund-freud

celestial-mechanics-is-origin-dynamical-systems-linear-algebra-topology-variational-calculus-symplectic-geometry-vladimir-arnold

the-two-elements-the-traveler-first-captures-in-the-big-city-are-extra-human-architecture-and-furious-rhythm-geometry-and-anguish

a-tree-nowhere-offers-straight-line-regular-curve-but-who-doubts-that-root-trunk-boughs-leaves-embody-geometry-george-iles