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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.

See also Euclid’s Elements


Euclid’s Elements[edit]

  • Ὅπερ ἔδει δεῖξαι.
    • Which was to be proved.
    • Elements, Book I, Proposition 4.
    • Latin translation: Quod erat demonstrandum (often abbreviated Q.E.D.).
  • ὅπερ ἔδει ποιῆσαι.
    • Which was to be done.
    • Elements, Book I, Proposition 1.
    • Latin translation: Quod erat faciendum (often abbreviated Q.E.F.).
  • Καὶ τὸ ὅλον τοῦ μέρους μεῖζον [ἐστιν].
    • And the whole [is] greater than the part.
    • Elements, Book I, Common Notion 5
    • Cf. Aristotle, Metaphysics, Book Η 1045a 8–10: "... the totality is not, as it were, a mere heap, but the whole is something besides the parts ... [πάντων γὰρ ὅσα πλείω μέρη ἔχει καὶ μὴ ἔστιν οἷον σωρὸς τὸ πᾶν]"
  • Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος.
    • A prime number is one (which is) measured by a unit alone.
    • Elements, Book 7, Definition 11 (12 in certain editions)


  • 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".

Quotes about Euclid[edit]

  • 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 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.

See also[edit]

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