## The collected mathematical papers of Arthur Cayley: Vol. XI |

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### Saturs

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### Bieži izmantoti vārdi un frāzes

absolutely convergent acnodal algebraical arbitrary assumed axes axis centre circle coefficients condition cone congruence conic considered contains coordinates corresponding covariants crunodal cubic cubic curve cubic equation cusp cuspidal curve denote determined differential equation distance dodecahedron elliptic functions equal expression fact factor figure finite foregoing formula geometry give given line group of 12 hence homographic homographic transformations icosahedron imaginary infinite infinity instance integral function intersections linear Mathematics memoir Messenger of Mathematics multiple nodal curve obtain prime number quartic question quintic equation rational and integral rational function regard relation respectively right angles root of unity roots rotation satisfied sextic side similarly singularities solution squares substitutions surface symmetrical functions taking tangent plane thence theorem theory theta-functions torse triangle values variable writing

### Populāri fragmenti

459. lappuse - Yet I doubt not through the ages one increasing purpose runs, And the thoughts of men are widened with the process of the suns.

431. lappuse - 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).

561. lappuse - ... 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*.

435. lappuse - ... 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.

435. lappuse - 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.

468. lappuse - ... 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.

445. lappuse - ... 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.

441. lappuse - ... 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...

449. lappuse - 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.

491. lappuse - ... 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