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,

Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and **geometry** are different in appearance. Algebras (jabbre and maqabeleh) are geometric facts which are proved by propositions five and six of Book two of Elements.

— As quoted in "A Paper of Omar Khayyam" by A.R. Amir-Moez in

I need **geometry** to set the grammar of the image expressive language . The structural skeleton, the composition and the geometric layout provide a perspective from which one can read the image; otherwise we would do what Dadaists did when they put words in a little bag and then took them out at random in order to compose a poem.

— As quoted in: interview by Flavia Squarcio for

Music is the arithmetic of sounds as optics is the **geometry** of light.

— As quoted 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."

— As quoted in a review of

Visual forms are not perceived differently from colors or brightness. They are sense qualities, and the visual character of **geometry** consists in these sense qualities.

—

There is (gentle reader) nothing (the works of God only set apart) which so much beautifies and adorns the soul and mind of man as does knowledge of the good arts and sciences . Many arts there are which beautify the mind of man; but of all none do more garnish and beautify it than those arts which are called mathematical , unto the knowledge of which no man can attain, without perfect knowledge and instruction of the principles, grounds, and Elements of **geometry**.

— ?"The Mathematical Preface" to Henry Billingsley's English translation of Euclid's

It is noteworthy that modern Platonists , almost without exception, are ignorant of mathematics, in spite of the immense importance that Plato attached to arithmetic and **geometry**, and the immense influence that they [these studies] had on his philosophy. This is an example of the evils of specialization: a man must not write on Plato unless he has spent so much of his youth on Greek as to have no time for the things that Plato thought important.

— Book One, Part II, Chapter XV, The Theory of Ideas, p. 132

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

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 spacings of the spheres.

— As quoted in the preface of the book entitled

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."

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)