The Collected Mathematical Papers of Arthur Cayley, 11. sējums
University Press, 1896 - 664 lappuses
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algebraical angles appears arbitrary assumed axis becomes called centre circle coefficients complete condition conic considered constants contains coordinates corresponding course covariants cubic curve denote depends determined differential distance double equal equation expression fact factor figure follows foregoing formula four function further geometry give given hence imaginary included infinite instance integral intersection letters linear manner Mathematics means meets memoir multiple negative notion observe obtained once original pair particular pass plane positive powers present prime quadric quantities question rational and integral referred regard relation remaining remarked represented respectively result roots satisfied sides signs similarly singularities solution squares substitutions suppose surface taken taking tangent thence theorem theory thing third transformation values variable whole writing written
457. 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 - English school of mathematicians said a few years ago: *"I would myself say that the purely imaginary objects are the only realities, the Zv-cio-; fara in regard to which the corresponding physical objects are as the shadows in the cave; and it is only by means of them that we are able to deny the existence of a corresponding physical object; and if there is no conception of straightness, then it is meaningless to deny the conception of a perfectly straight line.
433. 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.
357. lappuse - On a sheet of paper ruled in squares, and which is read as a continuous column from the bottom of one column to the top of the next...
431. lappuse - A line, as defined by geometers, is wholly inconceivable. We can reason about a line as if it had no breadth; because we have a power, which is the foundation of all the control we can exercise over the operations of our minds; the power, when a perception is present to our senses, or a conception to our intellects, of attending to a part only of that perception or conception, instead of the whole.
430. lappuse - To get rid of this difficulty, and at the same time to save the credit of the supposed system of necessary truth, it is customary to say that the points, lines, circles, and squares which are the subject of geometry, exist in our conceptions merely, and are part of our minds; which minds, by working on their own materials, construct an a priori science, the evidence of which is purely mental, and has nothing whatever to do with outward experience.
501. lappuse - We have in fact a double algebra as the instrument for the complete treatment of all higher analysis, except that in which one of higher multiplicity is used. The field of Quantics has been brilliantly cultivated by Cayley, Sylvester and others.
430. lappuse - It remains to inquire what is the ground of our belief in axioms what is the evidence on which they rest? I answer, they are experimental truths, generalizations from observation. The proposition, "Two straight lines cannot...
433. 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.