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.
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.
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.omar khayyám
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.augusto de luca
Music is the arithmetic of sounds as optics is the geometry of light.claude debussy
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
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.Hans reichenbach
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.john dee
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.
The Eudemian Summary ascribes to Thales the invention of the theorems on the equality of vertical angles, the equality of the angles at the base of an isosceles triangle, the bisection of a circle by any diameter, and the congruence of two triangles having a side and the two adjacent angles equal respectively. The last theorem he applied to the measurement of the distances of ships from the shore. Thus Thales was the first to apply theoretical geometry to practical uses.
Plato gave a healthful stimulus to the study of stereometry [solid geometry], which until his time had been entirely neglected. The sphere and the regular solids had been studied to some extent, but the prism, pyramid, cylinder, and cone were hardly known to exist. All these solids became the subjects of investigation by the Platonic school.
We have seen the birth of geometry in Egypt, its transference to the Ionian Islands, thence to Lower Italy and to Athens. We have witnessed its growth in Greece from feeble childhood to vigorous manhood, and now we shall see it return to the land of its birth and there derive new vigour.
" Dioptra ," says Venturi , were instruments which had great resemblance to our modern theodolites . The book Dioptra is a treatise on geodesy containing solutions, with aid of these instruments, of a large number of questions in geometry , such as to find the distance between two points, of which one only is accessible, or between two points, which are visible but both inaccessible; from a given point to draw a perpendicular to a line which cannot be approached; to find the difference of level between two points; to measure the area of a field without entering it.
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.
all the standard equations of mathematical physics can be separated and solved in Kerr geometry.subrahmanyan chandrasekhar
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.david brewster
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.)
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.
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.
A pretty little story is related by Stobæus: "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 his slave and said, 'Give him threepence, since he must make gain out of what he learns.'"
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.
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.
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.giuseppe peano