Euclidean Quotes & Sayings

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

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser. — Euclid

Of the two alternatives - a curved manifold in a Euclidean space of ten dimensions or a manifold with non-Euclidean geometry and no extra dimensions - which is right? I would rather not attempt a direct answer, because I fear I should get lost in a fog of metaphysics. But I may say at once that I do not take the ten dimensions seriously; whereas I take the non-Euclidean geometry of the world very seriously, and I do not regard it as a thing which needs explaining away. — Arthur Stanley Eddington

The once-surprising existence of non-Euclidean models of Euclid's first four axioms can be seen as a sort of mathematical joke. — John Allen Paulos

I walked downhill to the rental place, my backpack ten pounds heavier than it was this morning because of three huge textbooks: one on government from world history class; one from English class called Catastrophes of New England: 1650 to 1875; and a much-used book from my last class of the day, Non-Euclidean Geometry. The class was taught by Mr. Gint, a pale, balding man who barely looked at us. The entire class period he sat at his desk with a protractor and pencil, drawing pictures and muttering to himself. — Daryl Gregory

Free will is an idealization of human beings that makes the ethics game playable. Euclidean geometry requires idealizations like infinite straight lines and perfect circles, and its deductions are sound and useful even though the world does not really have infinite straight lines or perfect circles. The world is close enough to the idealization that the theorems can usefully be applied. Similarly, ethical theory requires idealizations like free, sentient, rational, equivalent agents whose behavior is uncaused, and its conclusions can be sound and useful even though the world, as seen by science, does not really have uncaused events. As long as there is no outright coercion or gross malfunction of reasoning, the world is close enough to the idealization of free will that moral theory can meaningfully be applied to it. — Steven Pinker

The classical theorists resemble Euclidean geometers in a non-Euclidean world who, discovering that in experience straight lines apparently parallel often meet, rebuke the lines for not keeping straight as the only remedy for the unfortunate collisions which are occurring. Yet, in truth, there is no remedy except to throw over the axiom of parallels and to work out a non-Euclidean geometry. — John Maynard Keynes

DOCTOR: Oh. My, God! Oh, it's bigger!
RIVER: Well, yes.
DOCTOR: On the inside,
RIVER: We need to concentrate.
DOCTOR: Than it is
RIVER: I know where you're going with this, but I need you to calm down.
DOCTOR: On the outside!
RIVER: You've certainly grasped the essentials.
DOCTOR: My entire understanding of physical space has been transformed! Three-dimensional Euclidean geometry has been torn up, thrown in the air and snogged to death! My grasp of the universal constants of physical reality has been changed forever. — The Doctor

For authors, the shortest distance between two points is a straight line only if you are writing the letter I. — Michael A. Arnzen

There have been and still are geometricians and philosophers, and even some of the most distinguished, who doubt whether the whole universe, or to speak more widely the whole of existence, was only created in Euclid's geometry; they even dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity. I have come to the conclusion that, since I can't understand even that, I can't expect to understand about God. I acknowledge humbly that I have no faculty for settling such questions, I have a Euclidean earthly mind, and how could I solve problems that are not of this world? And I advise you never to think about it either, my dear Alyosha, especially about God, whether He exists or not. All such questions are utterly inappropriate for a mind created with an idea of only three dimensions. — Fyodor Dostoyevsky

In Fyodor Dostoyevsky's great novel The Brothers Karamazov, there is a scene in which two people are talking about suffering. Ivan Karamazov is talking about there being any possibility that we can make sense of suffering, and here's what he says: "I believe like a child that suffering will be healed and made up for, that all the humiliating absurdity of human contradictions will vanish like a pitiful mirage, like the despicable fabrication of the impotent and infinitely small Euclidean mind of man, that in the world's finale, at the moment of eternal harmony, something so precious will come to pass that it will suffice for all hearts, for the comforting of all resentments, for the atonement of all the crimes of humanity, of all the blood that they've shed; that it will make it not only possible to forgive but to justify all that has happened."11 — Timothy Keller

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

The victory over Euclidean space was not achieved by isolated individuals, but by a field of young rebels opposed to all absolutes. — Marshall McLuhan

Even today, I am in total awe of the following wondrous chain of ideas and interconnections. Guided throughout by principles of symmetry, Einstein first showed that acceleration and gravity are really two sides of the same coin. He then expanded the concept to demonstrate that gravity merely reflects the geometry of spacetime. The instruments he used to develop the theory were Riemann's non-Euclidean geometries-precisely the same geometries used by Felix Klein to show that geometry is in fact a manifestation of group theory (because every geometry is defined by its symmetries-the objects it leaves unchanged). Isn't this amazing? — Mario Livio

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

But I still read Shaw on a regular basis. What I love is the nakedness of the polemic and the irresistible good humour. For me, 'Major Barbara' is the greatest of all the plays in that it starts from the rational and proceeds to the ecstatic in a spectacular way, and leaves you very confused if you cling to Euclidean logic. — Tony Kushner

Because one is using Euclidean space-times, in which the time direction is on the same footing as directions in space, it is possible for space-time to be finite in extent and yet to have no singularities that formed a boundary or edge. Space-time would be like the surface of the earth, only with two more dimensions. The surface of the earth is finite in extent but it doesn't have a boundary or edge: if you sail off into the sunset, you don't fall off the edge or run into a singularity. (I know, because I have been round the world!) — Stephen Hawking

I see now the virtue in madness, for this country knows no law nor any boundary. I pity the poor shades confined to the Euclidean prison that is sanity — Grant Morrison

I believe like a child that suffering will be healed and made up for, that all the humiliating absurdity of human contradictions will vanish like a pitiful mirage, like the despicable fabrication of the impotent and infinitely small Euclidean mind of man, that in the world's finale, at the moment of eternal harmony, something so precious will come to pass that it will suffice for all hearts, for the comforting of all resentments, for the atonement of all the crimes of humanity, for all the blood that they've shed; that it will make it not only possible to forgive but to justify all that has happened. — Fyodor Dostoyevsky

The world of shapes, lines, curves, and solids is as varied as the world of numbers, and it is only our long-satisfied possession of Euclidean geometry that offers us the impression, or the illusion, that it has, that world, already been encompassed in a manageable intellectual structure. The lineaments of that structure are well known: as in the rest of life, something is given and something is gotten; but the logic behind those lineaments is apt to pass unnoticed, and it is the logic that controls the system. — David Berlinski

Non- Euclidean calculus and quantum physics are enough to stretch any brain; and when one mixes them with folklore, and tries to trace a strange background of multi-dimensional reality behind the ghoulish hints of Gothic tales and the wild whispers of the chimney-corner, one can hardly expect to be wholly free from mental tension.
(Dreams In The Witch-House) — H.P. Lovecraft

I have a childlike conviction that the sufferings will be healed and smoothed over, that the whole offensive comedy of human contradiction will disappear like a pitiful mirage, a vile concoction of man's Euclidean mind, feeble and puny as an atom, and that ultimately, at the world's finale, in that moment of eternal harmony, there will occur and be revealed something so precious that it will suffice for all hearts, to allay all indignation, to redeem all human villainy, all bloodshed; it will suffice not only to make forgiveness possible, but also to justify everything that has happened with men
let this, let off of this, come true and be revealed. — Fyodor Dostoyevsky

We must here follow the first course so as to be able to pass on later to generalisations which extend beyond the limits of Euclidean geometry. — Hermann Weyl

No very good sense can be given to the idea that the elements of Euclidean geometry may be found in nature because either everything is found in nature or nothing is. Euclidean geometry is a theory, and the elements of a theory may be interpreted only in terms demanded by the theory itself. Euclid's axioms are satisfied in the Euclidean plane. Nature has nothing to do with it. — David Berlinski

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

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

Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself. — George Polya

You might ask why we cannot teach physics by just giving the basic laws on page one and then showing how they work in all possible circumstances, as we do in Euclidean geometry, where we state the axioms and then make all sorts of deductions. (So, — Richard Feynman

She wanted to run her hands over him as he whispered the impassioned corollaries of non-Euclidean geometry. — Sherry Thomas

[The Euclidean algorithm is] the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day. — Donald Knuth

I left Caen, where I was living, to go on a geological excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go to some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for convenience sake, I verified the result at my leisure. — Henri Poincare

At that stage of my youth, death remained as abstract a concept as non-Euclidean geometry or marriage. I didn't yet appreciate its terrible finality or the havoc it could wreak on those who'd entrusted the deceased with their hearts. I was stirred by the dark mystery of mortality. I couldn't resist stealing up to the edge of doom and peering over the brink. The hint of what was concealed in those shadows terrified me, but I caught sight of something in the glimpse, some forbidden and elemental riddle that was no less compelling than the sweet, hidden petals of a woman's sex.
In my case - and, I believe, in the case of Chris McCandless - that was a very different thing from wanting to die. — Jon Krakauer

I entered an omnibus to go to some place or other. At that moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with non-Euclidean geometry. — Henri Poincare

One can say that Javert is our conscience. The ever lurking presence of the law and our own condemnation. The tension between who we were and who we are and who we can be. Javert represents that inescapable, shameful past that forever haunts and persues one's conscience. Javert is the man of the law, and ... There are no surprises with the law. The principle of retribution is simple and monotonous, like Euclidean logic. It's closed to all alternatives and shut up against divine or human intervention ... Indeed, Javert represents the merciless application of the law, the blind Justice that in the end is befuddled by hope and the possibility of redemption without punishment. — Cristiane Serruya

But then, Cap'n Crunch in a flake form would be suicidal madness; it would last about as long, when immersed in milk, as snowflakes sifting down into a deep fryer. No, the cereal engineers at General Mills had to find a shape that would minimize surface area, and, as some sort of compromise between the sphere that is dictated by Euclidean geometry and whatever sunken treasure related shapes that the cereal aestheticians were probably clamoring for, they came up with this hard -to-pin-down striated pillow formation. — Neal Stephenson

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

Proofs of the Euclidean [parallel] postulate can be developed to such an extent that apparently a mere trifle remains. But a careful analysis shows that in this seeming trifle lies the crux of the matter; usually it contains either the proposition that is being proved or a postulate equivalent to it. — Johann Heinrich Lambert

Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience. — Arthur Cayley