Euclid

"Ὅπερ ἔδει δεῖξαι."

"ὅπερ ἔδει ποιῆσαι."

"Καὶ τὸ ὅλον τοῦ μέρους μεῖζον [ἐστιν]."

"Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος."

"μὴ εἶναι βασιλικὴν ἀτραπὸν ἐπί γεωμετρίαν."

"Non est regia [inquit Euclides] ad Geometriam via."

"Δός αὐτῷ τριώβολον, ἐπειδὴ δεῖ αὐτῷ ἐξ ὧν μανθάνει κερδαίνειν."

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

"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 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."

"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."

"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 history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several preceding centuries."

"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..."

"Euclid alone has looked on Beauty bare."

"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."

"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."