## An Essay on the Foundations of Geometry |

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absolute abstract actual analytical anharmonic ratio apodeictic apriority argument assumed axiom of Free Chap Chapter circle comparison Congruence construction contradictions corresponding curve deduced defined definition discuss distance distinction distinguish diversity element elliptic elliptic Geometry elliptic space empirical empty space Epistemology Erdmann Euclid Euclidean Geometry Euclidean space experience finite follows form of externality Free Mobility fundamental geodesies given Hence homogeneity of space imaginary important impossible independent infinitely divisible infinitesimal intuition involves judgment Kant Kant's knowledge logical Lotze Lotze's manifold mathematical matter means measure of curvature Metageometry method metrical Geometry motion necessary non-Euclidean Geometry non-Euclidean spaces object obtain philosophical plane possible presupposes principle priori projective Geometry projective transformation properties proposition prove purely quantity question reference regarded relativity of position rigid bodies sensation sense sense-perception space-constant spatial figures spatial magnitudes spatial order spatial relations Sphereland spherical Geometry straight line supposed surface things wholly

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Page 43 - For all the fruitful uses of imaginaries, in Geometry, are those which begin and end with real quantities, and use imaginaries only for the intermediate steps. Now in all such cases, we have a real spatial interpretation at the beginning and end of our argument, where alone the spatial interpretation is important : in the intermediate links, we are dealing in a purely algebraical manner with purely algebraical quantities, and may perform any operations which are algebraically permissible.

Page 181 - Reality is given for me in present sensuous perception, and in the immediate feeling of my own sentient existence that goes with it. The real world, as a definite organized system, is for me an extension of this present sensation and self feeling by means of judgment, and it is the essence of judgment to effect and sustain such an extension... The subject in every judgment of Perception is some given spot or point in sensuous contact with the percipient self.

Page 198 - that projective geometry, which has no reference to quantity, is necessarily true of any form of externality." "In metrical geometry is an empirical element, arising out of the alternatives of Euclidean and non-Euclidean space.

Page 40 - ... about it; but at present, and, considering the prominent position which the notion occupies — say even that the conclusion were that the notion belongs to mere technical mathematics, or has reference to nonentities in regard to which no science is possible, still it seems to me that (as a subject of philosophical discussion) the notion ought not to be thus ignored ; it should at least be shown that there is a right to ignore it.

Page 169 - The line is the relation The relation of position between the top and bottom points of a vertical line is that line, and nothing else." If I had been willing to use this doctrine at the beginning, I might have avoided all discussion. A unique relation between two points must in this case, involve a unique line between them. But it seemed better to avoid a doctrine not universally accepted, the more so as I was approaching the question from the logical, not the psychological, side. After disposing...

Page 12 - ... were mathematically possible in an extended manifold. Their philosophy, which seems to me not always irreproachable, will be discussed in Chapter II. ; here, while it is important to remember the philosophical motive of Riemann and Helmholtz, we shall confine our attention to the mathematical side of their work. In so doing, while we shall, I fear, somewhat maim the system of their thoughts, we shall secure a closer unity of subject, and a more compact account of the purely mathematical development....

Page 130 - ... infinite division, the zero of extension, is called a point \ III. Any two points determine a unique figure, called a straight line, any three in general determine a unique figure, the plane. Any four determine a corresponding figure of three dimensions, and for aught that appears to the contrary, the same may be true of any number of points. But this process comes to an end, sooner or later, with some number of points which determine the whole of space. For if this were not the case, no number...

Page 155 - A particle preserves in its state of rest, or of rectilinear motion with velocity varying as/(£), except in so far as it is compelled to alter that state by the action of external forces. Such a hypothesis is mathematically possible, but, like the similar one for space, it is excluded by the fact that it involves absolute time, as a determining agent in change, whereas time can never, philosophically, be anything but a passive holder of events, abstracted from change. I have introduced this parallel...

Page 47 - It becoines obvious, not only that exceptions within a certain region, but also that limitation to a certain region, of the axiom of Free Mobility, are philosophically quite impossible and inconceivable. How can a certain line, or a certain surface, form an impassable barrier in space, or have any mobility different in kind from that of all other lines or surfaces ? The notion cannot, in philosophy, be permitted for a moment, since it destroys that most fundamental of all the axioms, the homogeneity...

Page 148 - ... superposed, if it can ever occur, must occur always, whatever path be pursued in bringing it about. Hence, if mere motion could alter shapes, our criterion of equality would break down. It follows that the application of the conception of magnitude to figures in space involves the following axiom2: Spatial magnitudes can be moved from place to place without distortion; or, as it may be put, Shapes do not in any way depend upon absolute position in space.