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

— Chapter III, Section 21, pg. 126

The progression of a painter’s work, as it travels in time from point to point, will be toward clarity: toward the elimination of all obstacles between the painter and the idea, and between the idea and the observer. As examples of such obstacles, I give (among others) memory, history or **geometry**, which are swamps of generalization from which one might pull out parodies of ideas (which are ghosts) but never an idea in itself. To achieve this clarity is, inevitably, to be understood.

—

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)

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

I conceived, developed and applied in many areas a new [[**geometry** of nature, which finds order in chaotic shapes and processes. It grew without a name until 1975, when I coined a new word to denote it, fractal **geometry**, from the Latin word for irregular and broken up, fractus . Today you might say that, until fractal **geometry** became organized, my life had followed a fractal orbit.

— Benoît Mandelbrot As quoted in

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**.

—

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,

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,

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

— Pythagoras, as quoted in

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)