Quotes about Euclid

75 Sourced Quotes

The Elements of Euclid is as small a part of mathematics as the Iliad is of literature; or as the sculpture of Phidias is of the world's total art.

Cassius Jackson Keyser

He now understood navigation. He had started with Gower's Treatise on the Theory and Practice of Navigation....The book itself was the ocean; when he closed it he could cover up deep water. He had read Moore's Practical Navigator and had tried Euclid. He found arithmetic easy, because nobody pushed him.

Four circles to the kissing come, The smaller are the benter. The bend is just the inverse of The distance from the centre. Though their intrigue left Euclid dumb There's now no need for rule of thumb. Since zero bend's a dead straight line And concave bends have minus sign, The sum of squares of all four bends Is half the square of their sum.

Frederick Soddy

Old Euclid drew a circle On a sand-beach long ago, He bound it and enclosed it With angles thus and so.

Some people have been found to regard all mathematics, after the 47 proposition of Euclid, as a sort of morbid secretion, to be compared only with the pearl said to be generated in the diseased oyster.

James Joseph Sylvester

The attempt to avoid a direct affirmation about infinite parallel straight lines caused Euclid to phrase the parallel axiom in a rather complicated way. He realized that, so worded, this axiom lacked the self-sufficiency of the other nine axioms, and there is good reason to believe that he avoided using it until he had to. Many Greeks tried to find substitute axioms for the parallel axiom or to prove it on the basis of the other nine....Simplicius cites others who worked on the problem and says further that people "in ancient times" objected to the use of the parallel postulate.

The early study of Euclid made me a hater of geometry.

James Joseph Sylvester

The pages of Euclid are like the opening bars of the music of the Grand Opera of Nature's great drama. So to say, they lift the veil and show to our vision a glimpse of a vast world of natural knowledge awaiting study.

Mountains are not pyramids and trees are not cones. God must love gunnery and architecture if Euclid is his only Geometry.

The outstanding personalities of Euclid and Archimedes demand chapters to themselves. Euclid, the author of the incomparable Elements, wrote on almost all the other branches of mathematics known in his day. Archimedes's work, all original and set forth in treatises which are models of scientific exposition, perfect in form and style, was even wider in its range of subjects. The imperishable and unique monuments of the genius of these two men must be detached from their surroundings and seen as a whole if we would appreciate to the full the pre-eminent place which they occupy, and will hold for all time, in the history of science.

Thomas Little Heath

Let's suppose there are two kinds of geometry — we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y2, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment.

All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors, and learned in what had been before them. There is not one exception. I do not say that every man has made direct acquaintance with the whole of his mental ancestry... But... it is remarkable how many of the greatest names in all departments of knowledge have been real antiquaries in their several subjects.
I may cite among those... in science, Aristotle, Plato, Ptolemy, Euclid, Archimedes, Roger Bacon, Copernicus, Francis Bacon, Ramus, Tycho Brahe, Galileo, Napier, Descartes, Leibnitz, Newton, Locke.

Augustus De Morgan

Every night as I gazed up at the window I said to myself softly the word paralysis. It had always sounded strangely in my ears, like the word gnomon in the Euclid and the word simony in the Catechism. But now it sounded to me like the name of some maleficent and sinful being. It filled me with fear, and yet I longed to be nearer to it and to look upon its deadly work.

In other branches of science, where quick publication seems to be so much desired, there may possibly be some excuse for giving to the world slovenly or ill-digested work, but there is no such excuse in mathematics. The form ought to be as 102 perfect as the substance, and the demonstrations as rigorous as those of Euclid. The mathematician has to deal with the most exact facts of Nature, and he should spare no effort to render his interpretation worthy of his subject, and to give to his work its highest degree of perfection. Pauca sed matura was Gauss's motto.

James Whitbread Lee Glaisher

The volume of the cone was discovered by Democritus... He did not prove it, he guessed it... not a blind guess, rather it was reasoned conjecture. As Archimedes has remarked, great credit is due to Democritus for his conjecture since this made proof much easier. Eudoxes... a pupil of Plato, subsequently gave a rigorous proof. Surely the labor or writing limited his manuscript to a few copies; none has survived. In those days editions did not run to thousands or hundreds of thousands of copies as modern books—especially, bad books—do. However, the substance of what he wrote is nevertheless available to us.... Euclid's great achievement was the systematization of the works of his predecessors. The Elements preserve several of Eudoxes' proofs.

Science and art both project the world along certain coordinates. Differences in form are due only to differences in the coordinates. All realistic forms are projections along the fixed, plane coordinates of Euclid's world. These coordinates do not exist in nature. Nor does the finite, fixed world; this world is a convention, an abstraction, an unreality. And therefore Realism — be it "socialist" or "bourgeois" — is unreal. Far closer to reality is projection along speeding, curved surfaces — as in the new mathematics and the new art. Realism that is not primitive, not realia but realiora, consists in displacement, distortion, curvature, non-objectivity. Only the camera lens is objective.

Yevgeny Zamyatin

Lunacy!
But it had not been lunacy to Emma and Scott. They thought differently. They used x logic. Those notes Emma had made on the page — she'd translated Carroll's words into symbols both she and Scott could understand.
The random factor had made sense to the children. They had fulfilled the conditions of the time-space equation. And the mome raths outgrabe —
Paradine made a rather ghastly little sound, deep in his throat. He looked at the crazy pattern on the carpet. If he could follow it, as the kids had done — but he couldn't. The pattern was senseless. The random factor defeated him. He was conditioned to Euclid.
Even if he went insane, he still couldn't do it. It would be the wrong kind of lunacy.

Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

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 connerion 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

From the imputation of confounding axioms with assumed concepts Euclid himself, however, is free. Euclid incorporated the former among his postulates while he separated the latter as common concepts—a proceeding which even on the part of his commentators was no longer understood, and likewise with modern mathematicians, unfortunately for science, has met with little imitation. As a matter of fact, the abstract methods of mathematical science know no axioms at all.

Hermann Grassmann

Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions. Then, in the alleged proof, be alert for inexplicit assumptions. Euclid's notorious oversights drove this lesson home.

Eric Temple Bell

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

Thomas Little Heath

I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid — a term used in this work to denote all of standard geometry — Nature exhibits not simply a higher degree but an altogether different level of complexity … The existence of these patterns challenges us to study these forms that Euclid leaves aside as being "formless," to investigate the morphology of the "amorphous."

Benoît Mandelbrot

"Your mind has been conditioned to Euclid," Holloway said. "So this — thing — bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believe what he sees."
"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded.
"Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child especially a baby might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas."
"Hardening of the thought-arteries," Jane interjected.

Euclid's work will live long after all the text books of the present day are superseded and forgotten. It is one of the noblest monuments of antiquity; no mathematician worthy of the name can afford not to know Euclid, the real Euclid as distinct from any revised or rewritten versions which will serve for schoolboys or engineers. And, to know Euclid, it is necessary to know his language, and, so far as it can be traced, the history of the "elements" which he collected in his immortal work.

Thomas Little Heath

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

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

The golden age of mathematics - that was not the age of Euclid, it is ours.

Cassius Jackson Keyser

It is a remarkable fact in the history of geometry, that the Elements of Euclid, written two thousand years ago, are still regarded by many as the best introduction to the mathematical sciences.

Euclid defines the angle as an inclination of lines…he meant halflines, because otherwise he would not be able to distinguish adjacent angles from each other… Euclid does not know zero angles, nor straight and bigger than straight angles…Euclid takes the liberty of adding angles beyond two and even four right angles; the result cannot be angles according to the original definitions…Nevertheless one feels that Euclid's angle concept is consistent.

Hans Freudenthal