The Collected Mathematical Papers of Arthur Cayley, Volume 11

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University Press, 1896 - Mathematics - 664 pages
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Page 459 - Yet I doubt not through the ages one increasing purpose runs, And the thoughts of men are widened with the process of the suns.
Page 431 - Nihil in intellectu quod non prius in sensu (There is nothing in the Understanding not derived from the Senses, or — There is nothing conceived that was not previously perceived ;) he replied — prater intellectum ipsum (except the Understanding itself).
Page 561 - ... line is a conic section, which is an ellipse, a parabola or a hyperbola according as the given ratio is less than, equal to, or greater than unity*.
Page 435 - ... geometry. My own view is that Euclid's twelfth axiom in Playfair's form of it does not need demonstration, but is part of our notion of space, of the physical space of our experience — the space, that is, which we become acquainted with by experience, but which is the representation lying at the foundation of all external experience.
Page 435 - A more extended experience and more accurate measurements would teach them that the axioms were each of them false; and that any two lines if produced far enough each way, would meet in two points: they would in fact arrive at a spherical geometry, accurately representing the properties of the two-dimensional space of their experience.
Page 468 - ... from it by the theory of reciprocal polars (or that of geometrical duality), viz. we do not demonstrate the first theorem and deduce from it the other, but we do at one and the same time demonstrate the two theorems; our (a:, y, e) instead of meaning point-coordinates may mean line-coordinates, and the demonstration is in every step thereof a demonstration of the correlative theorem.
Page 445 - ... with the older physical sciences, Astronomy and Mechanics: the mathematical theory is in the first instance suggested by some question of common life or of physical science, is pursued and studied quite independently thereof, and perhaps after a long interval comes in contact with it, or with quite a different question.
Page 441 - ... wish to regard any four or more magnitudes as the coordinates of a point in space of a corresponding number of dimensions. Starting with the hypothesis of such a space, and of points therein each determined by means of its coordinates, it is found possible to establish a system of...
Page 449 - It is difficult to give an idea of the vast extent of modern mathematics. This word " extent " is not the right one : I mean extent crowded with beautiful detail — not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood, and flower.
Page 491 - ... b", c" +a" b , c b, c , b',c' , d' , d" , c'", d'" — a b", c", d" +a" b'", c'", d'" b , c , d b'", c'", d'" -a'" b , c , d b , c , d b', c', d' b' , c' , d' b", c", d

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