It is not known so generally that Kepler was... a geometrician and algebraist of considerable power, and that he, Desargues, and perhaps Galileo, may be considered as forming a connecting link between the mathematicians of the renaissance and those of modern times. Kepler's work in geometry consists rather in certain general principles enunciated, and illustrated by a few cases, than in any systematic exposition of the subject. In a short chapter on conics inserted in his Paralipomena, published in 1604, he lays down what has been called the principle of continuity, and gives as an example the statement that a parabola is at once the limiting case of an ellipse and of a hyperbola; he illustrates the same doctrine by reference to the foci of conics (the word focus was introduced by him); and he also explains that parallel lines should be regarded as meeting at infinity. He introduced the use of the eccentric angle in discussing properties of the ellipse.
Johannes KeplerRelated topics
angle case certain continuity doctrine eccentric example exposition few forming general geometrician geometry gives illustrated infinity known limiting link once parabola parallel perhaps power renaissance should short times use word work Kepler galileoRelated quotes
It is true that God is necessity, or, as we may also put it, that he is the absolute Thing: he is however no less the absolute Person. That he is the absolute Person however is a point which the philosophy of Spinoza never reached: and on that side it falls short of the true notion of God which forms the content of religious consciousness in Christianity. Spinoza was by descent a Jew; and it is upon the whole the Oriental way of seeing things, according to which the nature of the finite world seems frail and transient, that has found its intellectual expression in his system. This Oriental view of the unity of substance certainly gives the basis for all real further development. Still it is not the final idea. It is marked by the absence of the principle of the Western world, the principle of individuality, which first appeared under a philosophic shape, contemporaneously with Spinoza, in the Monadology of Leibnitz.
Baruch Spinoza
Had it merely called to our attention the existence and exact nature of certain fundamental gaps in economic theory, the Theory of Economic Behavior by von Neumann and Morgenstern would have been a book of outstanding importance. But it does more than that. It is essentially constructive: where existing theory is considered to be inadequate, the authors put in its place a highly novel analytical apparatus designed to cope with the problem.
It would be doing the authors an injustice to say that theirs is a contribution to economics only. The scope of the book is much broader. The techniques applied by the authors in tackling economic problems are of sufficient generality to be valid in political science, sociology, or even military strategy. The applicability to games proper (chess and poker) is obvious from the title. Moreover, the book is of considerable interest from a purely mathematical point of view.
Leonid Hurwicz
The ancients considered mechanics in a twofold respect; as rational, which proceeds accurately by demonstration, and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accurate is called geometrical; what is less so is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic: and if any could work with perfect accuracy, he would be the most perfect mechanic of all; for 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; for it requires that the learner should first be taught to describe these accurately, before he enters upon geometry; then it shows how by these operations problems may be solved.
Isaac Newton
We must grant the artist his subject, his idea, what the French call his donnée; our criticism is applied only to what he makes of it. Naturally I do not mean that we are bound to like it or find it interesting: in case we do not our course is perfectly simple - to let it alone. We may believe that of a certain idea even the most sincere novelist can make nothing at all, and the event may perfectly justify our belief; but the failure will have been a failure to execute, and it is in the execution that the fatal weakness is recorded. If we pretend to respect the artist at all we must allow him his freedom of choice, in the face, in particular cases, of innumerable presumptions that the choice will not fructify. Art derives a considerable part of its beneficial exercise from flying in the face of presumptions, and some of the most interesting experiments of which it is capable are hidden in the bosom of common things.
Henry James
Art' is a Renaissance invention which has been carried to a state of extreme refinement in the present day. This is the so-called 'abstract art'. The production of good works of art was achieved only at the cost of an enormous concentration upon certain matters. This concentration could be achieved only through neglecting 'life', through the very loss of life- just as religion had experienced before. 'Today this situation is no longer tolerable'. Today life is paramount. Modern life in general rejects all tendencies towards isolation and ivory tower-like exclusiveness. It is absolute un-modern to concentrate upon just one thing (as did the middle Ages!) Modern life is based upon the construction, which is to say, upon a system of tensions or the neutralization of the system of carry and support. In agreement with this concept we too must distribute our vitality over the whole range of life taken in the broadest possible sense. All other attitudes towards life produce tragedy.
Theo van Doesburg