...to characterize the import of pure geometry, we might use the standard form of a movie-disclaimer: No portrayal of the characteristics of geometrical figures or of the spatial properties of relationships of actual bodies is intended, and any similarities between the primitive concepts and their customary geometrical connotations are purely coincidental.
Carl Gustav Hempel
It is impossible to anticipate all of the misdeeds engendered by the universal conflict of human passions. They multiply at a compound rate with the growth in population and the interlacing of particular interests that cannot be directed with geometrical precision towards the public utility.
Nobody since Newton has been able to use geometrical methods to the same extent for the like purposes; and as we read the Principia we feel as when we are in an ancient armoury where the weapons are of gigantic size; and as we look at them we marvel what manner of man he was who could use as a weapon what we can scarcely lift as a burden.
Each organic being is striving to increase in a geometrical ratio . . . each at some period of its life, during some season of the year, during each generation or at intervals, has to struggle for life and to suffer great destruction . . . The vigorous, the healthy, and the happy survive and multiply.
If I should throw down a thousand beans at random upon a table, I could doubtless, by eliminating a sufficient number of them, leave the rest in almost any geometrical pattern you might propose to me, and you might then say that that pattern was the thing prefigured beforehand, and that the other beans were mere irrelevance and packing material. Our dealings with Nature are just like this.
Surprise quality can be achieved in many ways. It may be produced by a certain stimulating geometrical relationship between elements in the picture or through the human interest of the situation photographed or by calling our attention to some commonplace but fascinating thing we have never noticed before or it can be achieved by looking at an everyday thing in a new interesting way.
To pray is to pay attention to something or someone other than oneself. Whenever a man so concentrates his attention - on a landscape, a poem, a geometrical problem, an idol, or the True God - that he completely forgets his own ego and desires, he is praying. The primary task of the schoolteacher is to teach children, in a secular context, the technique of prayer.
W. H. Auden
... 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.
Remembering... that Eratosthenes of Cyrene, employing mathematical theories and geometrical methods, discovered from the course of the sun, the shadows cast by an equinoctial gnomon, and the inclination of the heaven that the circumference of the earth is two hundred and fifty-two thousand stadia, that is, thirty-one million five hundred thousand paces.
Marcus Vitruvius Pollio
In this science the illustrations and examples are not confined in their effect merely to the practice they afford in the analytical art, but [...] they also store the mind with independent geometrical and physical knowledge. Besides, it should be considered, that the only effectual method of impressing abstract formulae and rules upon the memory, and, indeed, of making them fully and clearly apprehended by the understanding, is by examples of their practical application.
It is to geometry that we owe in some sort the source of this discovery [of beryllium]; it is that [science] that furnished the first idea of it, and we may say that without it the knowledge of this new earth would not have been acquired for a long time, since according to the analysis of the emerald by M. Klaproth and that of the beryl by M. Bindheim one would not have thought it possible to recommence this work without the strong analogies or even almost perfect identity that Citizen Hae¼y found for the geometrical properties between these two stony fossils.
... legislators cannot invent too many devices for subdividing property... Another means of silently lessening the inequality of property is to exempt all from taxation below a certain point, and to tax the higher portions or property in geometrical progression as they rise. Whenever there are in any country uncultivated lands and unemployed poor, it is clear that the laws of property have been so far extended as to violate natural right.
Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence.
In the candle's flickering light, the library's thousands of books emerged from the shadows, and for a moment Nicholas could not help admiring them again. During free time he had almost never looked up from the pages he was reading, but now he saw the books anew, from without rather than from within, and was reminded of how beautiful they were simply as objects. The geometrical wonder of them all, each book on its own and all the books together, row upon row, the infinite patterns and possibilities they presented. They were truly lovely.
Trenton Lee Stewart
In geometry I find certain imperfections which I hold to be the reason why this science, apart from transition into analytics, can as yet make no advance from that state in which it came to us from Euclid.As belonging to these imperfections, I consider the obscurity in the fundamental concepts of the geometrical magnitudes and in the manner and method of representing the measuring of these magnitudes, and finally the momentous gap in the theory of parallels, to fill which all efforts of mathematicians have so far been in vain.
I review all I know, but can synthesize no meaning. When I doze, the Fact, the certain accomplished calamity, wakes me roughly like a brutal nurse. I see it crouching inflexibly in a corner of the ceiling. It comes down in geometrical diagonal like lightning.It says, I remain, I AM, I shall never cease to be: your memory will grow a deathly glaze: you will forget, you will fade out, but I cannot be undone.Thus every quarter hour it puts the taste of death in my mouth, and shows me, but not gently, how I go whoring after oblivion.
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.
Marcus Vitruvius Pollio
The personality and style of a photographer usually limits the type of subject with which he deals best. For example Cartier-Bresson is very interested in people and in travel; these things plus his precise feeling for geometrical relationships determine the type of pictures he takes best. What is of value is that a particular photographer sees the subject differently. A good picture must be a completely individual expression which intrigues the viewer and forces him to think.
We may, I believe, anticipate that the chemist of the future who is interested in the structure of proteins, nucleic acids, polysaccharides, and other complex substances with high molecular weight will come to rely upon a new structural chemistry, involving precise geometrical relationships among the atoms in the molecules and the rigorous application of the new structural principles, and that great progress will be made, through this technique, in the attack, by chemical methods, on the problems of biology and medicine.
Philosophy [nature] is written in that great book which ever is before our eyes -- I mean the universe -- but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.
Philosophy [nature] is written in that great book which ever is before our eyes - I mean the universe - but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.
The universe will finally become a ball of radiation, becoming more and more rarified and passing into longer and longer wave-lengths. The longest waves of radiation are Hertzian waves of the kind used in broadcasting. About every 1500 million years this ball of radio waves will double in diameter; and it will go on expanding in geometrical progression for ever. Perhaps then I may describe the end of the physical world as-one stupendous broadcast.
Large department stores, with their luxuriant abundance of canned goods, foods, and clothing, are like the primary landscape and the geometrical locus of affluence. Streets with overcrowded and glittering store windowsthe displays of delicacies, and all the scenes of alimentary and vestimentary festivity, stimulate a magical salivation. Accumulation is more than the sum of its products: the conspicuousness of surplus, the final and magical negation of scarcitymimic a new-found nature of prodigious fecundity.
Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings.
The concept of an independent system is a pure creation of the imagination. For no material system is or can ever be perfectly isolated from the rest of the world. Nevertheless it completes the mathematician's "blank form of a universe" without which his investigations are impossible. It enables him to introduce into his geometrical space, not only masses and configurations, but also physical structure and chemical composition. Just as Newton first conclusively showed that this is a world of masses, so Willard Gibbs first revealed it as a world of systems.
Lawrence Joseph Henderson
The mathematical thermology created by Fourier may tempt us to hope that, as he has estimated the temperature of the space in which we move, me may in time ascertain the mean temperature of the heavenly bodies: but I regard this order of facts as for ever excluded from our recognition. We can never learn their internal constitution, nor, in regard to some of them, how heat is absorbed by their atmosphere. We may therefore define Astronomy as the science by which we discover the laws of the geometrical and mechanical phenomena presented by the heavenly bodies.
I think the Greeks first took astrology to India and took from the Hindus the science of astronomy and carried it back with them from Europe. Because in India you will find old altars made according to a certain geometrical plan, and certain things had to be done when the stars were in certain positions, therefore I think the Greeks gave the Hindus astrology, and the Hindus gave them astronomy.
The differences between religions are reflected very clearly in the different forms of sacred art: compared with Gothic art, above all in its 'flamboyant' style, Islamic art is contemplative rather than volitive: it is 'intellectual' and not 'dramatic', and it opposes the cold beauty of geometrical design to the mystical heroism of cathedrals. Islam is the perspective of 'omnipresence' ('God is everywhere'), which coincides with that of 'simultaneity' ('Truth has always been'); it aims at avoiding any 'particularization' or 'condensation', any 'unique fact' in time and space, although as a religion it necessarily includes an aspect of 'unique fact', without which it would be ineffective or even absurd. In other words Islam aims at what is 'everywhere center', and this is why, symbolically speaking, it replaces the cross with the cube or the woven fabric: it 'decentralizes' and 'universalizes' to the greatest possible extent, in the realm of art as in that of doctrine; it is opposed to any individualist mode and hence to any 'personalist' mysticism. To express ourselves in geometrical terms, we could say that a point which seeks to be unique, and which thus becomes an absolute center, appears to Islam-in art as in theology-as a usurpation of the divine absoluteness and therefore as an 'association' (shirk); there is only one single center, God, whence the prohibition against 'centralizing' images, especially statues; even the Prophet, the human center of the tradition, has no right to a 'Christic uniqueness' and is 'decentralized' by the series of other Prophets; the same is true of Islam-or the Koran-which is similarly integrated in a universal 'fabric' and a cosmic 'rhythm', having been preceded by other religions-or other 'Books'-which it merely restores. The Kaaba, center of the Muslim world, becomes space as soon as one is inside the building: the ritual direction of prayer is then projected toward the four cardinal points. If Christianity is like a central fire, Islam on the contrary resembles a blanket of snow, at once unifying and leveling and having its center everywhere.
Homesickness is a great teacher. It taught me, during an endless rainy fall, that I came from the arid lands, and like where I came from. I was used to dry clarity and sharpness in the air. I was used to horizons that either lifted into jagged ranges or rimmed the geometrical circle of the flat world. I was used to seeing a long way. I was used to earth colors--tan, rusty red, toned white--and the endless green of Iowa offended me. I was used to a sun that came up over mountains and went down behind other mountains. I missed the color and smell of sagebrush, and the sight of bare ground.
His brown eyes would roam around the various sentimental and artistic bric-a-brac present, and his own banal toiles (the conventionally primitive eyes, sliced guitars, blue nipples and geometrical designs of the day), and with a vague gesture toward a painted wooden bowl or veined vase, he would say "Prenez donc une des ces poires. La bonne dame d'en face m'en offre plus que je n'en peux savourer." Or: "Mississe Taille Lore vient de me donner ces dahlias, belles fleurs que j'exe¨cre.
Then she understood that what she needed was the motion to a purpose, no matter how small or in what form, the sense of an activity going step by step to some chosen end across a span of time. The work of cooking a meal was like a closed circle, completed and gone, leading nowhere. But the work of building a path was a living sum, so that no day was left to die behind her, but each day contained all those that preceded it, each day acquired its immortality on every succeeding tomorrow. A circle, she thought, is the movement proper to physical nature, they say that there's nothing but circular motion in the inanimate universe around us, but the straight line is the badge of man, the straight line of a geometrical abstraction that makes roads, rails and bridges, the straight line that cuts the curving aimlessness of nature by a purposeful motion from a start to an end. The cooking of meals, she thought, is like the feeding of coal to an engine for the sake of a great run, but what would be the imbecile torture of coaling an engine that had no run to make? It is not proper for man's life to be a circle, she thought, or a string of circles dropping off like zeros behind him-man's life must be a straight line of motion from goal to farther goal, each leading to the next and to a single growing sum, like a journey down the track of a railroad, from station to station to-oh, stop it!