The doctrine of ratios and proportion is introduced by **Euclid** as a part of his system of geometry ; and the student seldom fails to remark, that in the treatises on algebra, the same subject is presented under a considerably different form; though he is usually quite unable to determine wherein the essential difference consists; and would probably find but few teachers who could precisely point out the distinction to him.

— Rev. Baden Powell,

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."

— As quoted in a review of

**Euclid** … manages to obtain a rigorous proof without ever dealing with infinity, by reducing the problem [of the infinitude of primes] to the study of finite numbers. This is exactly what contemporary mathematical analysis does.

— 2.4, "Discrete Mathematics and the Notion of Infinity", p. 45

How little inventiveness there is in man, Grave copier of copies, I give thanks For a new relish, careless to inquire My pleasure's pedigree, if so it please, Nobly, I mean, nor renegade to art. The Grecian gluts me with its perfectness, Unanswerable as **Euclid** , self-contained, The one thing finished in this hasty world, Forever finished, though the barbarous pit, Fanatical on hearsay, stamp and shout As if a miracle could be encored.

—

**Euclid** 's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument.

— p. 70 (How to Solve It (1945))

To avoid any assertion about the infinitude of the straight line, **Euclid** says a line segment (he uses the word "line" in this sense) can be extended as far as necessary. Unwillingness to involve the infinitely large is seen also in **Euclid**'s statement of the parallel axiom. Instead of considering two lines that extend to infinity and giving a direct condition or assumption under which parallel lines might exist, his parallel axiom gives a condition under which two lines will meet at some finite point.

— Morris Kline,

Closely related to the problem of the parallel postulate is the problem of whether physical space is infinite. **Euclid** assumes in Postulate 2 that a straight-line segment can be extended as far as necessary; he uses this fact, but only to find a larger finite length for example in Book I, Propositions 11, 16, and 20. For these proofs Heron gave new proofs that avoided extending the lines, in order to meet the objection of anyone who would deny that the space was available for the extension.

— Morris Kline,

Jaisingh collected and studied all the available astronomical works...Several European works were translated into Sanskrit under his orders, particularly **Euclid** ’s elements, with a treatise on plane and spherical trigonometry ; and on the construction and use of logarithms ...and also a treatise on conical sections...maps and globes of the Ferenghis were obtained from Surat .

— G.R. Kaye, on the efforts made by Jai Singh to set up the Astronomical Observatory in Jaipur, in p.213

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.

— Florian Cajori,

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.

— Florian Cajori,

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.

— Florian Cajori,

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,