Euclid (325 BC to 265 BC)

There is no royal road to geometry.

A youth who had begun to read geometry with Euclid, when he had learnt the first proposition, inquired, "What do I get by learning these things?" So Euclid called a slave and said "Give him threepence, since he must make a gain out of what he learns."
Stobaeus, Extracts

Archimedes (287 BC to 212 BC) [A quotation by Plutarch]

... being perpetually charmed by his familiar siren, that is, by his geometry, he neglected to eat and drink and took no care of his person; that he was often carried by force to the baths, and when there he would trace geometrical figures in the ashes of the fire, and with his finger draws lines upon his body when it was anointed with oil, being in a state of great ecstasy and divinely possessed by his science.

Albrecht Dürer (1471 to 1528)

Welcher aber ... durch die Geometria sein Ding beweist und die grüdliche Wahrheit anzeigt, dem soll alle Welt glauben. Denn da ist man gefangen.
Whoever ... proves his point and demonstrates the prime truth geometrically should be believed by all the world, for there we are captured.
Von menschlicher Proportion

John Dee (1527 to 1608)

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

Thomas Hobbes (1588 to 1629) [A quotation by John Aubrey]

He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras's Theorem]. He read the proposition . "By God", sayd he, "this is impossible:' So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that trueth. This made him in love with geometry.
Quoted in O L Dick, Brief Lives

Geometry (which is the only science that it hath pleased God to bestow on mankind) ...
Leviathan 1651

Sir Isaac Newton (1643 to 1727)

The description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn.
Principia Mathematica

Bernard de Fontenelle (1657 to 1757)

A work of morality, politics, criticism will be more elegant, other things being equal, if it is shaped by the hand of geometry.
Preface sur l'Utilité des Mathématiques et de la Physique, 1729.

Joseph-Louis Lagrange (1736 to 1813)

As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection.

Sir Christopher Wren (1632 to1723)

In things to be seen at once, much variety makes confusion, another vice of beauty. In things that are not seen at once, and have no respect one to another, great variety is commendable, provided this variety transgress not the rules of optics and geometry.

Farkas Bolyai (1775 to 1856)

For God's sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.
[A letter to his son János urging him to give up work on non-Euclidean geometry.]

János Bolyai (1802 to 1860)

Out of nothing I have created a strange new universe.
[A reference to the creation of a non-euclidean geometry.]

Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard.

Carl Friedrich Gauss (1777 to 1855)

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.

Augustus De Morgan (1806 to 1871)

As to writing another book on geometry, the middle ages would have as soon thought of composing another New Testament.

Joseph J Sylvester (1814 to 1897)

The early study of Euclid made me a hater of geometry.

Arthur Cayley (1821 to 1895)

Projective geometry is all geometry.

Felix Klein (1849 to 1925)

Projective geometry has opened up for us with the greatest facility new territories in our science, and has rightly been called the royal road to our particular field of knowledge.

Henri Poincaré (1854 to 1912)

...by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous.
Science and Method

Eric T Bell (1883 to 1960)

The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid did to geometry.
The Search For Truth

John E Littlewood (1885 to 1977)

I constantly meet people who are doubtful, generally without due reason, about their potential capacity [as mathematicians]. The first test is whether you got anything out of geometry. To have disliked or failed to get on with other [mathematical] subjects need mean nothing; much drill and drudgery is unavoidable before they can get started, and bad teaching can make them unintelligible even to a born mathematician.
A Mathematician's Miscellany