272 



SCIENCE. 



[Vol. XIX. No. 484 



SCIENCE; 



A WEEKLY NEWSPAPER OF ALL THE ARTS AND SCIENCES. 



PUBLISHED BY 



N. D. C. HODGES, 



874 Broadway, New York. 



StJBSCRiPTiONs. — United States and Canada g3,50 a year. 



Great Britain and Europe 4.50 a year. 



Communications will be welcomed from any quarter. Abstracts of scientific 

 papers are solicited, and one hundred copies of tbe issue containing such will 

 be mailed the author on request in advance. Rejected manuscripts will be 

 returned to the authors only when the requisite amount of postage accom- 

 panies the manuscript. Whatever is intended for insertion must be authenti- 

 oated by the name and address of the writer; not necessarily for publication, 

 but as a guaranty of good faith. We do not hold ourselves responsible for 

 any view or opinions expressed in the communications of our correspondents. 



Attention is called to the "Wants " column. It is invaluable to those who 

 use it in soliciting information or seeking new positions. The name and 

 address of applicants should be given in full, so that answers will go direct to 

 them. The "Exchange " column is likewise open. 



For Advertising Rates apply to Henry F. Taylor, 47 Lafayette Place, New 

 fork. 



THE POSSIBILITY OF A REALIZATION OF FOUR- 

 FOLD SPACE.' 



Any magnitude that is a functioQ of a single variable may 

 be represented geometrically by a straight line. Functions 

 of two variables are represented by curved lines or by plane 

 areas; and functions of three variables by either twisted 

 curves, curved surfaces, or volumes. The conceptions of 

 length, area, and volume when used in this way are evidently 

 independent of any of the properties of matter except exten- 

 sion. The question now before us is this, Can we develop 

 and use in a similar way a space-concept which can fully 

 represent a function of four independent variables ? 



Perhaps most of us can remember times in the course of 

 our education when new conceptions of quantity entered 

 into our conscious life, conceptions which correspond in a 

 general way with those of length, area, and volume, in that 

 they enable us to find at once such relationships as are most 

 frequently required for practical purposes by a general, 

 synthetic, instinctive' method. A medical student, instead 

 of memorizing the exact amount of each dose under all possi- 

 ble conditions of the patient, fixes in his mind as in a frame 

 work the medicinal outline of each drug. The student of 

 chemistry does something similar with the elements; the 

 architect has such a concept of structural beauty; the hunter, 

 of the most likely place for game. The sense of propriety, 

 the sense of honor, and numberless other " inbred "or " in- 

 stinctive " concepts are examples of this mental tendency. 

 There is therefore nothing inherently absurd or improbable 

 in the supposition that any of us may attain to a conception 

 of four-fold space, ' ' as clear as the designer and the draughts- 

 man have of three-fold space." " Such a conception would 

 be of great value to all classes of scientists. The biologist 



' Digest of a paper read before the Canadltin Club of Clark University by 

 T. Proctor Hall, PluD 



» " A New E a of Tbought," by C. H. Hlnton, M.A. 



could set in this four-fold framework a complete picture of 

 genetic or race relationships ; the theologian could use it for 

 the world of spirits; the physicist for forces, etc. By this 

 means (ordinary men may become able to see and to develop 

 easily new truths, such as are now revealed only to men of 

 genius and inspiration. 



It may be objected that our conception of three-fold space 

 is derived directly from sensations in three fold space, and 

 that the conception of four-fold space cannot be derived in a 

 similar way, nor yet from sensations in three-fold space. 

 But It is evident that from any sen^e, from sight, for in- 

 stance, we get at most a two-dimentional sensation, and it is 

 only by the kind of changes that occur in the sensation that 

 we can infer that a given retinal picture represents extension 

 in two or in three dimensions. In other words, granting, 

 for the sake of the argument, that in sight we perceive di- 

 rectly the existence of two dimensions, it is clear that the 

 existence of a third dimension is solely a matter of inference. 

 It is the simplest hypothesis we can get to explain our sen- 

 sations. It is conceivable that the hypothesis of a fourth 

 dimension, if it could be made as real to us, might be found 

 of nearly equal value in the simplification of ordinary phe- 

 nomena. This would be the case if ordinary phenomena 

 involve motion in four independent directions, or if some of 

 the relations of things in the universe, relations not in space, 

 are capable of complete representation in four-fold space. 

 But before we can decide whether or not space and objects 

 of four dimensions exist we must have our ideas of four- 

 fold space developed sufficiently to know what sensations, 

 what visible and tangible phenomena, would be obtained 

 from objects of four dimensions. Up to this time discussions 

 on the reality of four-fold space have been (necessarily) 

 characterized by the absence of evidence for or against. 



To develop a clear conception of four-fold space only one 

 course seems to be open, namely, the synthetical study of 

 four-fold geometric figures in the same way that we now 

 study geometric solids. Having given the number and form 

 of the boundaries of a solid we can, by the process of visuali- 

 zation, find more or less easily its appearance (plane projec- 

 tion) in various positions, tlie possible plane sections, the 

 distance between any two of its points, and so on. In the 

 study of a tessaract (four- fold figure) we should deal simi- 

 larly with its solid boundaries, finding the possible solid sec- 

 tions, solid projections, and so forth, studying the tessaract 

 by means of conceptions already familiar (length, area, vol- 

 ume), but in new relations. It this way may be developed 

 gradually such a knowledge of the properties of tessaracts as 

 will enable us to "see" them clearly, and to comprehend 

 quickly a new shape. Models of the solid projections and 

 sections are indispensible to rapid progress. Difficulties may, 

 in general, be overcome by considering the analogous diffi- 

 culties an imaginary plane being, that is to say, a being- 

 who has no conception of volume, would have in trying to 

 understand a geometric solid. 



The First Lesson. 



A point moving in one direction traces a straight line. A. 

 line moving perpendicular to itself, in one plane, traces a 

 square ; and a square moving similarly traces a cube. How 

 could a plane being learn the number and relations of the- 

 faces of a cube ? He could readily understand that as the 

 square moves in a direction perpendicular to all of its sides 

 each side traces a new square, and that the moving square 

 in its first and last positions forms the remaining pair of 

 opposite faces. In this way he could count up the six faces, 

 twelve edges, and eight corners of the cube, and might pro- 



