I presume that to the uninitiated the formulae will appear cold and cheerless; but let it be remembered that, like other mathematical formulae, they find their origin in the divine source of all geometry. Whether I shall have the satisfaction of taking part in their exposition, or whether that will remain for some more profound expositor, will be seen in the future.
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.carl friedrich gauss
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.ivor grattan-guinness
There is still a difference between something and nothing, but it is purely geometrical and there is nothing behind the geometry.martin gardner
There is geometry in the humming of the strings, there is music in the spacing of the spheres.pythagoras
There is geometry in the humming of the strings. There is music in the spacings of the spheres.pythagoras
There are infinitely many variations of the initial situation and therefore no doubt indefinitely many theorems of moral geometry.john rawls
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.
There is no royal road to geometry.
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."
The Eudemian Summary says that "Pythagoras changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom, and investigated its theorems in an immaterial and intellectual manner." His geometry was connected closely with his arithmetic. He was especially fond of those geometrical relations which admitted of arithmetical expression.
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.Florian Cajori
The regular solids were studied so extensively by the Platonists that they received the name of "Platonic figures." The statement of Proclus that the whole aim of Euclid in writing the Elements was to arrive at the construction of the regular solids, is obviously wrong. The fourteenth and fifteenth books, treating of solid geometry, are apocryphal .
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.
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.
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.
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!
There is geometry in the humming of the strings, there is music in the spacing of the spheres.
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.vitruvius
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.