Geometry Quotes - page 9
Sanskrit is constructed like geometry and follows a rigorous logic. It is theoretically possible to explain the meaning of the words according to the combined sense of the relative letters, syllables and roots. Sanskrit has no meanings by connotations and consequently does not age. Panini's language is in no way different from that of Hindu scholars conferring in Sanskrit today.
Pāṇini
It is to the director of workshops and factories that it is suitable to make, by means of geometry and applied mechanics, a special study of all the ways to economize the efforts of workers... For a man to be a director of others, manual work has only a secondary importance; it is his intellectual ability (force intellectuelle) that must put him in the top position, and it is in instruction such as that of the Conservatory of the Arts and Professions, that he must develop it.
Charles Dupin
For 12 years I have had the honor of teaching geometry and mechanics applied to the arts, in favor of the industrial class... on the most important questions to the well-being, education, and morality of the workers, to the progress of national industry, to the development of all means of prosperity that work can produce for the splendor and happiness of our country.
Charles Dupin
Nevertheless, it remains conceivable that the measure relations of space in the infinitely small are not in accordance with the assumptions of our geometry [Euclidean geometry], and, in fact, we should have to assume that they are not if, by doing so, we should ever be enabled to explain phenomena in a more simple way.
Bernhard Riemann
You see, some mathematicians have clear and far-ranging. "programs". For instance, Grothendieck had such a program for algebraic geometry; now Langlands has one for representation theory, in relation to modular forms and arithmetic. I never had such a program, not even a small size one.
Jean-Pierre Serre
In string theory one studies strings moving in a fixed classical spacetime. ...what we call a background-dependent approach. ...One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves-at the rate predicted by general relativity to the... accuracy of eleven decimal places-tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up.
Lee Smolin
A psychological space is established for any set of stimuli by determining metric distances between the stimuli such that the probability that a response learned to any stimulus will generalize to any other is an invariant monotonic function of the distance between them. To a good approximation, this probability of generalization (i) decays exponentially with this distance, and (ii) does so in accordance with one of two metrics, depending on the relation between the dimensions along which the stimuli vary. These empirical regularities are mathematically derivable from universal principles of natural kinds and probabilistic geometry that may, through evolutionary internalization, tend to govern the behaviors of all sentient organisms.
Roger Shepard
Be not, O Greeks, so very hostilely disposed towards the Barbarians. To the Babylonians you owe astronomy; to the Persians, magic; to the Egyptians, geometry; to the Phoeicians, instruction by alphabetic writing. Cease, then, to miscall these imitations inventions of your own. Orpheus, again, taught you poetry and song; from him, too, you learned the mysteries.
Tatian
But to return to the Newtonian Philosophy: Tho' its Truth is supported by Mathematicks, yet its Physical Discoveries may be communicated without. The great Mr. Locke was the first who became a Newtonian Philosopher without the help of Geometry; for having asked Mr. Huygens, whether all the mathematical Propositions in Sir Isaac's Principia were true, and being told he might depend upon their Certainty; he took them for granted, and carefully examined the Reasonings and Corollaries drawn from them, became Master of all the Physics, and was fully convinc'd of the great Discoveries contained in that Book.
John Theophilus Desaguliers
When mons. Descartes's philosophical Romance, by the Elegance of its Style and the plausible Accounts of natural Phænomena, had overthrown the Aristotelian Physics, the World received but little Advantage by the Change: For instead of a few Pedants, who, most of them, being conscious of their Ignorance, concealed it with hard Words and pompous Terms; a new Set of Philosophers started up, whose lazy Disposition easily fell in with a Philosophy, that required no Mathematicks to understand it, and who taking a few Principles for granted, without examining their Reality or Consistence with each other, fancied they could solve all Appearances mechanically by Matter and Motion; and, in their smattering Way, pretended to demonstrate such things, as perhaps Cartesius himself never believed; his Philosophy (if he bad been in earnest) being unable to stand the test of the Geometry which he was Master of.
John Theophilus Desaguliers
Because certain elements of geometry have for a long time been included in the general course of education, every educated man is able to distinguish between the practical task of the land surveyor and the theoretical investigation of the geometer. The corresponding distinction between the theory of probability and statistics has yet to be recognized.
Richard von Mises
The researches of the last thirty or forty years into the history of mathematics (I need only mention such names as those of [Carl Anton] Bretschneider, Hankel, Moritz Cantor, [Friedrich] Hultsch, Paul Tannery, Zeuthen, Loria, and Heiberg) have put the whole subject upon a different plane. I have endeavoured in this edition to take account of all the main results of these researches up to the present date. Thus, so far as the geometrical Books are concerned, my notes are intended to form a sort of dictionary of the history of elementary geometry, arranged according to subjects; while the notes on the arithmetical Books VII.-IX. and on Book X follow the same plan.
Thomas Little Heath
Algebraic geometry has developed in waves, each with its own language and point of view. The late nineteenth century saw the function-theoretic approach of Riemann, the more geometric approach of Brill and Noether, and the purely algebraic approach of Kronecker, Dedekind, and Weber. The Italian school followed with Castelnuovo, Enriques, and Severi, culminating in the classification of algebraic surfaces. Then came the twentieth-century "American" school of Chow, Weil, and Zariski, which gave firm algebraic foundations to the Italian intuition. Most recently, Serre and Grothendieck initiated the French school, which has rewritten the foundations of algebraic geometry in terms of schemes and cohomology, and which has an impressive record of solving old problems with new techniques. Each of these schools has introduced new concepts and methods.
Robin Hartshorne
The Pythagoreans discovered the existence of incommensurable lines, or of irrationals. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of √2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable... showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established.... The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes.
Thomas Little Heath
The efforts of a multitude of writers have rather been directed towards producing alternatives for Euclid which shall be more suitable, that is to say, easier, for schoolboys. It is of course not surprising that, in these days of short cuts, there should have arisen a movement to get rid of Euclid and to substitute "a royal road to geometry"; the marvel is that a book which was not written for schoolboys but for grown men (as all internal evidence shows, and in particular the essentially theoretical character of the work and its aloofness from anything of the nature of "practical" geometry) should have held its own as a schoolbook for so long.
Thomas Little Heath
