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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.


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 8 (5 in certain editions)
    • 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]

  • With the completion of Euclid's Elements... For the first time in history masses of isolated discoveries were unified and correlated by a single guided principle, that of rigid deduction from explicitly stated assumptions. ...If the Pythagorean dream of a mathematized science is to be realized, all of the sciences must eventually submit to the discipline that geometry accepted from Euclid.
  • 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.
  • There is irrefutable evidence that a substantial portion of the material recorded in the Elements was known before Euclid, and there is nothing either in the style or in the plan of the treatise to suggest that it was intended as a collection of original contributions. Thus, on the whole... the chief objective... was to put system and rigour into the work of his predecessors.
  • There never has been, and till we see it we never shall believe that there can be, a system of geometry worthy of the name, which has any material departures (we do not speak of corrections or extensions or developments) from the plan laid down by Euclid.
    • Augustus De Morgan, "Short Supplementary Remarks on the First Six Books of Euclid's Elements" (Oct, 1848) Companion to the Almanac for 1849 as quoted by Sir Thomas Little Heath, The Thirteen Books of Euclid's Elements (1908) Vol.1, Introduction and Books I, II. Preface, p. v
  • 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.
    • Freeman Dyson, in Infinite in All Directions (1988), "Butterflies and Superstrings"
  • Euclid is said to have written the Elements of Music. Two treatises are attributed to Euclid in... the Musici, the... Sectio canonis (the theory of the intervals) and the... (introduction to harmony). The first, resting on the Pythagorean theory of music, is mathematical and clearly and well written, the style and the form of the propositions agreeing well with what we find in the Elements. Its genuineness is confirmed not only by internal evidence... Euclid is twice mentioned by name, in the commentary on Ptolemy's Harmonica published by Wallis... The second treatise is not Euclid's...
    • Sir Thomas Little Heath, The Thirteen Books of Euclid's Elements (1908) Vol.1 Introduction and Books I, II p.17, citing Proclus p. 69 3; Musici Scriptores Graeci, ed. Jan (Teubner, 1895) pp. 113-166; Jan, Musici Scriptores Graeci, p. 116
  • Those who have written the history of geometry have thus far carried the development of this science. Not much later than these is Euclid, who wrote the 'Elements,' arranged much of Eudoxus' work, completed much of Theaetetus's and brought to irrefragable proof propositions which had been less strictly proved by his predecessors.
  • Not much younger than these (sc. Hermotimus of Colophon and Philippus of Mende) is Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy), makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry. He is then younger than pupils of Plato but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says.
    • Proclus (ca. 335 BC) as quoted by Sir Thomas Little Heath, The Thirteen Books of Euclid's Elements (1908) Vol.1 Introduction and Books I, II p.1, citing Proclus ed. Friedlein, p. 68, 6-20
  • Inasmuch as many things, while appearing to rest on truth and to follow from scientific principles, really tend to lead one astray from the principles and deceive the more superficial minds, he has handed down methods for the discriminative understanding of these things as well, by the use of which methods we shall be able to give beginners in this study practice in the discovery of paralogisms and to avoid being misled. This treatise, by which he puts this machinery in our hands, he entitled (the book) of Pseudaria, enumerating in order their various kinds, exercising our intelligence in each case by theorems of all sorts, setting the true side by side with the false, and combining the refutation of error with practical illustration. This book then is by way of cathartic and exercise, while the Elements contain the irrefragable and complete guide to the actual scientific investigation of the subjects of geometry.
    • Proclus (ca. 335 BC) as quoted by Sir Thomas Little Heath, The Thirteen Books of Euclid's Elements (1908) Vol.1 Introduction and Books I, II p.7, citing Proclus, p. 70 1-18

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