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There is no royal road to Geometry.

Euclid (Greek: Εὐκλείδης), also known as Euclid of Alexandria, was a Greek mathematician who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323–283 BC). Neither the year nor place of his birth have been established, nor the circumstances of his death. He is famous for writing one of the earliest comprehensive mathematics textbooks, the Elements.


  • Ὅπερ ἔδει δεῖξαι.
    • Which was to be proved.
    • Elements, Book I, proposition 5.
    • Latin translation: Quod erat demonstrandum (often abbreviated QED).
  • And the whole [is] greater than the part.
    • Elements, Book I, Common Notions 5
  • A prime number is one (which is) measured by a unit alone.
    • Elements, Book 7, Definitions 11

"The laws of nature are but the mathematical thoughts of God."

Quotes about Euclid[edit]

  • Upon these accounts it appeared necessary and I hope will prove acceptable to all lovers of accurate reasoning and of mathematical learning to remove such blemishes and restore the the principal Books of the Elements to their original accuracy, as far as I was able; especially since these Elements are the foundation of a science by which the investigation and discovery of very useful truths, at least in mathematical learning, is promoted as far as the limited powers of the mind allow; and which likewise is of the greatest use in the arts both of peace and war, to many of which geometry is absolutely necessary. This I have endeavoured to do, by taking away the inaccurate and false reasonings which unskilful editors have put into the place of some of the genuine Demonstrations of Euclid, who has ever been justly celebrated as the most accurate of geometers, and by restoring to him those things which Theon or others have suppressed, and which have these many ages been buried in oblivion.
  • 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.
  • Comparatively few of the propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him.
    • Florian Cajori, A History of Mathematics (1893)
  • The term 'axiom' was used by Proclus, but not by Euclid. He speaks, instead, of 'common notions'—common either to all men or to all sciences.
    • Florian Cajori, A History of Mathematics (1893)
  • The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals; but when examined in the light of strict mathematical logic, it has been pronounced by C.S. Peirce to be "riddled with fallacies." The results are correct only because the writer's experience keeps him on his guard.
  • The Greeks elaborated several theories of vision. According to the Pythagoreans, Democritus, and others vision is caused by the projection of particles from the object seen, into the pupil of the eye. On the other hand Empedocles, the Platonists, and Euclid held the strange doctrine of ocular beams, according to which the eye itself sends out something which causes sight as soon as it meets something else emanated by the object.
  • Professor Klein then speaks of "that artistic finish that we admire in Euclid's Elements," and mentions Allman's important historical work. I heartily concur in this estimate of Euclid, and desire to contrast it with the error of Charles S. Peirce, in the Nation, where he speaks of "Euclid's proof (Elements Bk. I., props. 16 and 17)" as "really quite fallacious, because it uses no premises not as true in the case of spherics." Our bright American seems to have forgotten Euclid's Postulate 6 (Axiom 12 in Gregory, Axiom 9 in Heiberg), "Two straight lines cannot enclose a space;" that is, two straights having crossed never recur.


  • There is no royal road to geometry. (μή εἶναι βασιλικήν ἀτραπόν 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.
    • Proclus (412–485 AD) in Commentary on the First Book of Euclid's Elements as translated by Glenn R. Morrow (1970), p. 57. ἀτραπός "road, trail, track" here takes the more specific sense of "short cut". The Latin translation is by Francesco Barozzi, 1560)
  • Give him threepence, since he must make gain out of what he learns. (Δός αὐτῷ τριώβολον, ἐπειδὴ δεῖ αὐτῷ ἐξ ὧν μανθάνει κερδαίνειν)
    • Said to be a remark made to his servant when a student asked what he would get out of studying geometry.
    • 'threepence' renders τριώβολον "three-obol-piece". This amount increases the sarcasm of Euclid's reply, as it was the standard fee of a Dikastes for attending a court case (μίσθος δικαστικός), thus inversing the role of teacher and pupil to that of accused and juror.
    • The English translation is by The History of Greek Mathematics by Thomas Little Heath (1921), p. 357. The quote is recorded by Stobaeus' Florilegium iv, 114 (ed. Teubner 1856, p. 205; see also here). Stobaeus attributes the anecdote to Serenus.


  • The laws of nature are but the mathematical thoughts of God.
    • The earliest published source found on google books that attributes this to Euclid is A Mathematical Journey by Stanley Gudder (1994), p. xv. However, many earlier works attribute it to Johannes Kepler, the earliest located being in the piece "The Mathematics of Elementary Chemistry" by Principal J. McIntosh of Fowler Union High School in California, which appeared in School Science and Mathematics, Volume VII (1907), p. 383. Neither this nor any other source located gives a source in Kepler's writings, however, and in an earlier source, the 1888 Notes and Queries, Vol V., it is attributed on p. 165 to Plato. It could possibly be a paraphrase of either or both of the following to comments in Kepler's 1618 book Harmonices Mundi (The Harmony of the World)': "Geometry is one and eternal shining in the mind of God" and "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".


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