CORRESP ONDENCE. 



365 



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 dimen- 

 sion of space. 



Now, if they were so small as only to per- 

 ceive 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 sur- 

 face 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 paral- 

 lels and geometry, unless, as has been sug- 

 gested, they should produce mathematicians 

 sufficiently powerful first to imagine and in- 

 vestigate 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: " Rea- 

 soning 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 hy- 

 pothesis 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 experi- 

 ence in a spherical surface only, belonging 

 to daily life." 



In reference to our own science of to- 

 day, the two analogous questions are : 



1. May we not be drawing wrong con- 

 clusions about space from our limited ex- 

 perience of space, just as the Greeks con- 

 cluded 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, Clif- 

 ford, and Sylvester, in their researches, and 



Cayley has published " Chapters in the Ana- 

 lytical 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 impor- 

 tant 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 ojiposite edge, so that the meridians, 

 great circles, shortest lines, instead of inter- 

 secting twice, as they do on the earth, only 

 intersect once and yet return upon them- 

 selves. 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 sur- 

 face 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 fol- 

 lows that the universe is only finite in 

 extent, and that every physical straight 

 line, for example, every ray of light, if suf- 

 ficiently produced, returns into itself. 



Yours very truly, 



George Bruce Halsted. 



Johns Hopkins University, ) 

 Baltimore, 3Iay 20, 18T7. j 



"THE EARLY MAN OF NOETH AMERICA." 



To the Editor of the Popular Science Mordhly. 



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 dia- 

 lect 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 de- 

 rived from the Old World, they can ever 

 be proved so, or traced to their parentage." 

 In the same article thei"e is the statement 

 that the Esquimaux " extend in scattered 

 companies for nearly five hundred miles on 

 the coast of Asia beyond Beliring's Straits," 

 while other writers assert that the Esqui- 



