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

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."

all the standard equations of mathematical physics can be separated and solved in Kerr **geometry**.

— From Chandrasekhar's Nobel lecture, in his summary of his work on black holes; Republished in: D. G. Caldi, ?George D. Mostow (1989)

Trained in a less severe school than that of **geometry** and physics, his reasonings are almost always loose and inconclusive. His generalizations seem to have been reached before he had obtained the materials upon which he rests them: His facts, though frequently new and interesting, are often little more than conjectures; and the grand phenomena of the world of life, and instinct, and reason, which other minds have woven into noble and elevating truths, have thus become in Mr. Darwin's hands the basis of a dangerous and degrading speculation.

— Referring to Charles Darwin

Grosseteste was the first medieval writer to break with the ancient tradition of discounting refraction in the rainbow. In another way, however, Grosseteste is firmly bound to his predecessors' rainbows an experimentalist in principle, in practice he strays very little from the Aristotelian tradition of selective observation constrained by **geometry**. Furthermore, Grosseteste's rainbow theory often appeals to authority, leading historian Bruce Eastwood to remark: "When sources fail, he invents, and the invention is never contradictory to literary sources." (In fairness, Grosseteste made several observations about the rainbow that gainsaid ancient authority.)

— Raymond L. Lee & Alistair B. Fraser,

In matters connected with **geometry**, nothing is to be taken upon trust: mere opinion, unsupported by reasonings which elevate it into proof, must be regarded, in this subject , as of but little worth.

— John Radford Young, "Proportion. A Treatise intended as a Substitute for Euclid's Book V," in

I have no fault to find with those who teach **geometry**. That science is the only one which has not produced sects; it is founded on analysis and on synthesis and on the calculus; it does not occupy itself with the probable truth; moreover it has the same method in every country.

— Frederick the Great in: G.E. Martin

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

Certainly it is permitted to anyone to put forward whatever hypotheses he wishes, and to develop the logical consequences contained in those hypotheses. But in order that this work merit the name of **geometry**, it is necessary that these hypotheses or postulates express the result of the more simple and elementary observations of physical figures.

— "Sui fondamenti della geometria" (1894), p. 141, as quoted in "The Mathematical Philosophy of Giuseppe Peano" by Hubert C. Kennedy, in