## An Essay on the Foundations of Geometry |

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à priori actual appear applied argument assumed axiom becomes bodies called Chapter circle comparison completely conception Congruence consider construction contradictions coordinates corresponding criticism curve deals deduced defined definition depends determine dimensions discuss distance distinction distinguish diversity element empirical empty space equal Euclid Euclidean existence experience fact figure follows form of externality four Free Mobility function fundamental give given ground Helmholtz Hence homogeneity important impossible independent infinite intuition involves judgment knowledge logical magnitude manifold mathematical matter means measurement mere Metageometry method metrical metrical Geometry motion nature necessary non-Euclidean object obtain philosophical plane points position possible present principle priori projective Geometry properties prove purely qualitative quantity question reference regarded relation relativity remains Riemann rigid says seems seen sense similar space spatial straight line supposed surface theory things third unique whole wholly

### Popular passages

Page 73 - Definition dessen aufgestellt, was wir als anschaulich vorstellbar anerkennen müssten, die dahin lautet, dass dazu erforderlich sei die vollständige Vorstellbarkeit derjenigen Sinneseindrücke, welche das betreffende Object in uns nach den bekannten Gesetzen unserer Sinnesorgane unter allen denkbaren Bedingungen der Beobachtung erregen, und wodurch es sich von anderen ähnlichen Objecten unterscheiden würde.

Page 171 - But just as, in the field of quantity, the relation between two numbers is another number, so in the field of space the relations are facts of the same order with the facts they relate.

Page 45 - If the quantities with which we end are capable of spatial interpretation, then, and only then, our results may be regarded as geometrical. To use geometrical language, in any other case, is only a convenient help to the imagination. To speak, for example, of projective properties which refer to the circular points, is a mere memoria technica for purely algebraical properties; the circular points are not to be found in space, but only in the auxiliary quantities by which geometrical equations are...

Page 200 - 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 171 - ... completeness, with nothing further to be •done. Just so the relation of direction between two lines is identical with the peculiar sensation of shape of the space enclosed between them. This is commonly called an angular relation. If these relations are sensations, no less so are the relations of position. The relation of position between the top and bottom points of a vertical line is that line, and nothing else.

Page 183 - Reality is given for me in present sensuous perception, and in the immediate feeling of my own sentient existence that goes with it.

Page 42 - ... 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 31 - In that theory (the Non-Euclidian Geometry) it seems äs if we try to replace our ordinary notion of distance between two points by the logarithm of a certain anharmonic ratio. But this ratio itself involves the notion of distance measured in the ordinary way. How then can we supersede the old notion of distance by the Non-Euclidian notion, inasmuch äs the very definition of the latter involves the former"?

Page 6 - Everything in physical science, from the law of gravitation to the building of bridges, from the spectroscope to the art of navigation, would be profoundly modified by any considerable inaccuracy in the hypothesis that our actual space is Euclidean. The observed truth of physical science, therefore, constitutes overwhelming empirical evidence that this hypothesis is very approximately correct, even if not rigidly true.

Page 40 - ... 2. I assume that this space is affected with such curvature that a right line shall always return into itself at the end of a finite and real distance...