[2029]. Think of the image of the world in a convex mirror.... A well-made convex mirror of moderate aperture represents the objects in front of it as apparently solid and in fixed positions behind its surface. But the images of the distant horizon and of the sun in the sky lie behind the mirror at a limited distance, equal to its focal length. Between these and the surface of the mirror are found the images of all the other objects before it, but the images are diminished and flattened in proportion to the distance of their objects from the mirror.... Yet every straight line or plane in the outer world is represented by a straight [?] line or plane in the image. The image of a man measuring with a rule a straight line from the mirror, would contract more and more the farther he went, but with his shrunken rule the man in the image would count out exactly the same number of centimeters as the real man. And, in general, all geometrical measurements of lines and angles made with regularly varying images of real instruments would yield exactly the same results as in the outer world, all lines of sight in the mirror would be represented by straight lines of sight in the mirror. In short, I do not see how men in the mirror are to discover that their bodies are not rigid solids and their experiences good examples of the correctness of Euclidean axioms. But if they could look out upon our world as we look into theirs without overstepping the boundary, they must declare it to be a picture in a spherical mirror, and would speak of us just as we speak of them; and if two inhabitants of the different worlds could communicate with one another, neither, as far as I can see, would be able to convince the other that he had the true, the other the distorted, relation. Indeed I cannot see that such a question would have any meaning at all, so long as mechanical considerations are not mixed up with it.—Helmholtz, H.
On the Origin and Significance of Geometrical Axioms; Popular Scientific Lectures, second series (New York, 1881), pp. 57-59.
[2030]. That space conceived of as a locus of points has but three dimensions needs no argument from the mathematical point of view; but just as little can we from this point of view prevent the assertion that space has really four or an infinite number of dimensions though we perceive only three. The theory of multiply-extended manifolds, which enters more and more into the foreground of mathematical research, is from its very nature perfectly independent of such an assertion. But the form of expression, which this theory employs, has indeed grown out of this conception. Instead of referring to the individuals of a manifold, we speak of the points of a higher space, etc. In itself this form of expression has many advantages, in that it facilitates comprehension by calling up geometrical intuition. But it has this disadvantage, that in extended circles, investigations concerning manifolds of any number of dimensions are considered singular alongside the above-mentioned conception of space. This view is without the least foundation. The investigations in question would indeed find immediate geometric applications if the conception were valid but its value and purpose, being independent of this conception, rests upon its essential mathematical content.—Klein, F.
Mathematische Annalen, Bd. 43 (1893), p. 95.
[2031]. We are led naturally to extend the language of geometry to the case of any number of variables, still using the word point to designate any system of values of n variables (the coördinates of the point), the word space (of n dimensions) to designate the totality of all these points or systems of values, curves or surface to designate the spread composed of points whose coördinates are given functions (with the proper restrictions) of one or two parameters (the straight line or plane, when they are linear fractional functions with the same denominator), etc. Such an extension has come to be a necessity in a large number of investigations, in order as well to give them the greatest generality as to preserve in them the intuitive character of geometry. But it has been noted that in such use of geometric language we are no longer constructing truly a geometry, for the forms that we have been considering are essentially analytic, and that, for example, the general projective geometry constructed in this way is in substance nothing more than the algebra of linear transformations.—Segre, Corradi.
Rivista di Matematica, Vol. I (1891), p. 59. [J. W. Young.]
[2032]. Those who can, in common algebra, find a square root of −1, will be at no loss to find a fourth dimension in space in which ABC may become ABCD: or, if they cannot find it, they have but to imagine it, and call it an impossible dimension, subject to all the laws of the three we find possible. And just as √−1 in common algebra, gives all its significant combinations true, so would it be with any number of dimensions of space which the speculator might choose to call into impossible existence—De Morgan, A.
Trigonometry and Double Algebra (London, 1849), Part 2, chap. 3.
[2033]. The doctrine of non-Euclidean spaces and of hyperspaces in general possesses the highest intellectual interest, and it requires a far-sighted man to foretell that it can never have any practical importance.—Smith, W. B.
Introductory Modern Geometry (New York, 1893), p. 274.