I am coming more and more to the conviction that the necessity of our **geometry** cannot be demonstrated, at least neither by, nor for, the human intellect. . . **geometry** should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.

— As quoted in

It [ non-Euclidean **geometry** ] would be ranked among the most famous achievements of the entire [nineteenth] century, but up to 1860 the interest was rather slight.

— p. 400 (The Rainbow of Mathematics: A History of the Mathematical Sciences (2000))

There is still a difference between something and nothing, but it is purely geometrical and there is nothing behind the **geometry**.

—

There is **geometry** in the humming of the strings, there is music in the spacing of the spheres.

— As quoted in

There is **geometry** in the humming of the strings. There is music in the spacings of the spheres.

— As quoted in the preface of the book entitled

There are infinitely many variations of the initial situation and therefore no doubt indefinitely many theorems of moral **geometry**.

— Chapter III, Section 21, pg. 126

In **geometry** his greatest achievement was an accurate value of ?. His rule is stated as: dn^2+(2a-d)n=2s, which implies the approximation 3.1416 which is correct to the last decimal place.

— In, p.245 (Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures)

There is no royal road to **geometry**.

—

—

— Reply given when the ruler Ptolemy I Soter asked Euclid if there was a shorter road to learning geometry than through Euclid's

— Attributed to Euclid by Proclus (412–485 AD) in

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."

— Benoît Mandelbrot As quoted in a review of

The Pythagoreans were... familiar with the construction of a polygon equal in area to a given polygon and similar to another given polygon. This problem depends upon several important and somewhat advanced theorems, and testifies to the fact that the Pythagoreans made no mean progress in **geometry**.

—

Neither natural ability without instruction nor instruction without natural ability can make the perfect artist. Let him be educated, skilful with the pencil, instructed in **geometry**, know much history, have followed the philosophers with attention, understand music, have some knowledge of medicine, know the opinions of the jurists, and be acquainted with astronomy and the theory of the heavens.

— Chapter I, Sec. 3

— Variant translation (by Frank Granger): "For neither talent without instruction nor instruction without talent can produce the perfect craftsman."

Anaximander relies on the accuracy of **geometry** in matters beyond the range of any kind of verification in its application to cosmic proportions and also in contradiction to appearance, which suggests that the sun is about as large in diameter as the width of a human foot. The concept of geometrical similarity is also the precondition for Anaximander's attempt to construct a map of the world.

— Walter Burkert,

Once a definition of congruence is given, the choice of **geometry** is no longer in our hands; rather, the **geometry** is now an empirical fact.

— Hans Reichenbach, in 'The Philosophy of Space and Time (1928), as translated by Maria Reichenbach (1957), § 27

Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of **geometry**!

— Frederick the Great,

There is **geometry** in the humming of the strings, there is music in the spacing of the spheres.

— Pythagoras, as quoted in

**geometry**... is of much assistance in architecture, and in particular it teaches us the use of the rule and compasses, by which especially we acquire readiness in making plans for buildings in their grounds, and rightly apply the square, the level, and the plummet. By means of optics... the light in buildings can be drawn from fixed quarters of the sky. ...Difficult questions involving symmetry are solved by means of geometrical theories and methods.

— Chapter I, Sec. 4

The Greeks... discovered mathematics and the art of deductive reasoning. **geometry**, in particular, is a Greek invention, without which modern science would have been impossible.

— Bertrand Russell (1945)