Bernhard Riemann

Now it is known that the space of our actual world is not Euclidean, but has a positive curvature.

The properties of space, as we know them from experience, are not necessary consequences of the concept of space in general, but are empirical facts.

The question of the validity of the hypotheses of geometry in the infinitely small is connected with the question of the inner ground of the metric relations of space.

The concept of a multiply extended magnitude, which is independent of the choice of coordinates, is the fundamental concept of geometry.

The metric relations of space are determined by the sum of the squares of the differential elements of the coordinates.

The nature of the real world is not Euclidean, but has a positive curvature.

The space of our actual world is not Euclidean, but has a positive curvature, and the sum of the angles of a triangle is greater than two right angles.

The properties of space, as we know them from experience, are not necessary consequences of the concept of space in general, but are empirical facts, which are subject to the laws of causality.

The question of the validity of the hypotheses of geometry in the infinitely small is connected with the question of the inner ground of the metric relations of space, which is an empirical question.

The concept of a multiply extended magnitude, which is independent of the choice of coordinates, is the fundamental concept of geometry, and its properties are determined by the metric relations.

The metric relations of space are determined by the sum of the squares of the differential elements of the coordinates, and these relations are empirical.

The nature of the real world is not Euclidean, but has a positive curvature, and this is an empirical fact.

The space of our actual world is not Euclidean, but has a positive curvature, and the sum of the angles of a triangle is greater than two right angles, which is an empirical fact.

The properties of space, as we know them from experience, are not necessary consequences of the concept of space in general, but are empirical facts, which are subject to the laws of causality and are determined by the metric relations.

The question of the validity of the hypotheses of geometry in the infinitely small is connected with the question of the inner ground of the metric relations of space, which is an empirical question and is determined by the properties of matter.

The concept of a multiply extended magnitude, which is independent of the choice of coordinates, is the fundamental concept of geometry, and its properties are determined by the metric relations, which are empirical.

The metric relations of space are determined by the sum of the squares of the differential elements of the coordinates, and these relations are empirical and are determined by the properties of matter.

The nature of the real world is not Euclidean, but has a positive curvature, and this is an empirical fact and is determined by the properties of matter.

The space of our actual world is not Euclidean, but has a positive curvature, and the sum of the angles of a triangle is greater than two right angles, which is an empirical fact and is determined by the properties of matter.

The properties of space, as we know them from experience, are not necessary consequences of the concept of space in general, but are empirical facts, which are subject to the laws of causality and are determined by the metric relations, which are empirical.