The Collected Mathematical Papers of Arthur Cayley, Volume 11

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The University Press, 1896 - Mathematics - 643 pages
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Page 457 - 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 429 - 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 431 - 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.
Page 433 - ... 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 357 - 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...
Page 431 - 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.
Page 430 - 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.
Page 501 - 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.
Page 430 - 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...
Page 433 - 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.

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