should be like ours in every other way, but should have another or fourth dimension; and this led to the question, "May not our space have a fourth dimension?"
Now, our only way of reasoning about the matter is to take analogous cases in the picturable spaces of two dimensions, and carry the analogies up from two to three, from three to four dimensions.
Let us take the easiest illustration. Suppose beings not living on the surface of a sphere, but in the surface of a sphere, and so having no conception of the third dimension of space.
Now, if they were so small as only to perceive a small portion of their surface, they might easily think it a plane, as the ancients thought our earth, and so their geometry would be the same as Euclid's.
But, if they were originally created so large in proportion to their spherical surface as to be immediately affected by its positive curvature, then they would never gather any experience of parallel lines, or of geometrical similarity between figures of different sizes. A straight line being the shortest distance between two points, then all their straight lines or geodesic lines would return upon themselves; and as also any two straightest lines on a sphere must meet somewhere, our imaginary surface men could never learn our theory of parallels and geometry, unless, as has been suggested, they should produce mathematicians sufficiently powerful first to imagine and investigate a surface in which two straight lines might be drawn so as to remain at the same distance apart to infinity; that is, if they could in any way be supposed to have the idea of infinity. Then, as Land says: "Reasoning on this, and a few more suppositions, they might discover the analytical geometry of the plane. Combining this with their original spherical theorems, some genius among them might conceive the bold hypothesis of a third dimension in space, and demonstrate that actual observations are perfectly explained by it. Henceforth there would be a double set of geometrical axioms; one the same as ours, belonging to science, and another resulting from experience in a spherical surface only, belonging to daily life."
In reference to our own science of today, the two analogous questions are:
1. May we not be drawing wrong conclusions about space from our limited experience of space, just as the Greeks concluded that the earth was flat?
2. If our conclusions so far are true, yet may there not be, in addition to the three dimensions we know, still another or fourth dimension in space?
The idea of space of four dimensions has been successfully used by Salmon, Clifford, and Sylvester, in their researches, and Cayley has published "Chapters in the Analytical Geometry of n Dimensions." Spaces of two and three dimensions with constant curvature have been carefully investigated by Beltrami, Helmholtz, and now Frankland. I mention these as among the most important and easily procurable writings on the subject. In reality there have been about a hundred books and memoirs treating of new or non-Euclidean space.
Of that kind of non-Euclidean surface now being discussed in Nature, a very pretty idea may be obtained by likening it to a hemisphere on which, when any moving point has reached the edge, it is supposed, without any jump or any further motion, to have reached the corresponding point on the opposite edge, so that the meridians, great circles, shortest lines, instead of intersecting twice, as they do on the earth, only intersect once and yet return upon themselves. This, like the sphere, is called a surface of positive curvature, in reference to the plane, which has no curvature.
Now, just as to the plane corresponds an uncurved or homaloidal space, so to a surface of positive curvature corresponds a space of positive curvature; and if the space in which we live can be proved to have the slightest positive curvature, it instantly follows that the universe is only finite in extent, and that every physical straight line, for example, every ray of light, if sufficiently produced, returns into itself.
| Yours very truly, | |
| George Bruce Halsted. | |
| Johns Hopkins University, Baltimore, May 20, 1877. |
|
"THE EARLY MAN OF NORTH AMERICA."
To the Editor of the Popular Science Monthly.
In an article in your March number, upon "The Early Man of North America," the writer says of the Esquimaux, "They are from their speech a branch of the Turanian family, and allied to the Hungarian, Turkish, Lapp, and Basque races." Whitney, in his work on "Life and Growth of Language," says of the Basque: "It stands entirely alone; no kindred having been found for it in any part of the world." He further says: "Attempts have been made to connect them" (the American languages) "with some dialect or family in the Old World, but with obviously unavoidable ill success. . . . There appears to be no tolerable prospect that, even supposing the American languages derived from the Old World, they can ever be proved so, or traced to their parentage." In the same article there is the statement that the Esquimaux "extend in scattered companies for nearly five hundred miles on the coast of Asia beyond Behring's Straits," while other writers assert that the Esqui-
