AugtLst 15, 1878] 



NATURE 



411 



.of feelings and aflfections, of the secret springs of character and 

 motives of conduct, it frequently happens that the writer is 

 driven to imagery, to an analogy, or even to a paradox, in order 

 to give utterance to that of which there is no direct counterpart 

 in recognised speech. And yet which of us cannot find a mean- 

 ing for these literary figures, an inward response, to imaginative 

 poetry, to social fiction, or even to those tales of giant and fairy- 

 land written, it is supposed, only for the nursery or schoolroom ? 

 But in order thus to reanimate these things with a meaning 

 beyond that of the mere words, have we not to reconsider our 

 iirst position, to enlarge the ideas with which we started ; have 

 we not to cast about for something which is common to the idea 

 ■conveyed and to the subject actually described, and to seek for 

 the sympathetic spring which underlies both ; have we not, like 

 the mathematician, to go back as it were to some first principles, 

 or as it is pleasanter to describe it, to become again as a little 

 child ? 



Manifold Space. 



Passing to the second of the three methods, viz., that of mani" 

 fold space, it may first be remarked that our whole experience 

 of space is in three dimensions, viz., of that which has length, 

 breadth, and thickness ; and if for certain purposes we restrict 

 ■our ideas to two dimensions as in plane geometry, or to one 

 dimension as in the division of a straight line, we do this only 

 by consciously and of deliberate purpose setting aside, but not 

 annihilating, the remaining one or two dimensions. Negation, 

 as Hegel has justly remarked, implies that which is negatived, 

 or, as he expresses it, affirms the opposite. It is by abstraction 

 from previous experience, by a limitation of its results, and not 

 by any independent process, that we arrive at the idea of space 

 whose dimensions are less than three. 



It is doubtless on this account that problems in plane geometry 

 which, although capable of solution on their own account, become 

 much more intelligible, more easy of extension, if viewed in 

 •connection with solid space, and as special cases of correspond- 

 ing problems in solid geometry. So eminently is this the case 

 that the very language of the more general method often leads 

 us almost intuitively to conclusions which from the more restricted 

 ^oint of view require long and laborious proof. Such a change 

 in the base of operations has, in fact, been successfully made in 

 •geometry of two dimensions, and although we have not the same 

 •experimental data for the further steps, yet neither the modes of 

 reasoning, nor the validity of its conclusions, are in any way 

 affected by applying an analogous mental process to geometry 

 of three dimensions, and by regarding figures in space of three 

 ■dimensions as sections of figures in space of four in the same 

 way that figures in piano are sometimes considered as sections of 

 figures in solid space. The addition of a fourth dimension to 

 space not only extends the actual properties of geometrical 

 fiigures, but it also adds new properties which are often useful 

 for the purposes of transformation or of proof. Thus it has 

 recently been shown that in four dimensions a closed material 

 shell could be turned inside out^ by simple flexure, without 

 either stretching or tearing, 2 and that in such a space it is 

 impossible to tie a knot. 



Again, the solution of problems in geometry is often effected 

 by means of algebra : and as three measurements, or co-ordi- 

 ■nates as they are called, determine the position of a point in 

 ^pace, so do three letters or measurable quantities serve for the 

 same purpose in the language of algebra. Now many alge- 

 braical problems involving three unknown or variable quantities 

 admit of being generalised so as to give problems involving many 

 such quantities. And as, on the one hand, to every algebraical 

 problem involving unknown quantities or variables by ones, or 

 by twos, or by threes, there corresponds a problem in geometry 

 of one or of two or of three dimensions, so on the other it may 

 be said that to every algebraical problem involving many vari- 

 ables there corresponds a problem in geometry of many 

 dimensions. 



There is, however, another aspect under which even ordinary 

 space presents to us a four-fold, or indeed a manifold character. 

 >In modern physics space is not regarded as a vacuum in which 

 bodies are placed and forces have play, but rather as a plenum 

 with which matter is co-extensive. And from a physical point 

 of view the properties of space are the properties of matter, or 

 of the medium which fills it. Similarly from a mathematical 

 point of view, space may be regarded as a locus in quo, as a 



I S. Newcomb " On Certain Transformations of Surfaces," American 

 Journal 0/ Mathematics, \o\. i. p. i. 



^ T^t "On Knots," Transactions of the Royal Society fof Edinburgh, 

 vol. xxvui. p. 145. Klein, Mathematische Annalen, ix. p. 478. 



plenum, filled with those elements of geometrical magnitude 

 which we take as fundamental. These elements need not always 

 be the same. For different purposes different elements may be 

 chosen ; and upon the degree of complexity of the subject of our 

 choice will depend the internal structure or manifoldness of 

 space. 



Thus, beginning with the simplest case, a point may have any 

 singly infinite multitude of positions in a line, which gives a one- 

 fold system of points in a line. The line may revolve in a plane 

 about any one of its points, giving a two-fold system of points 

 in a plane ; and the plane may revolve about any one of the 

 lines, giving a three-fold system of points in space. 



Suppose, however, that we take a straight line as our element, 

 and conceive space as filled with such lines. This will be the 

 case if we take two planes, e.g., two parallel planes, and join 

 every point in one with every point in the other. Now the 

 points in a plane form a two-fold system, and it therefore follows 

 that the system of lines is four-fold ; in other words, space 

 regarded as a plenum of lines is four-fold. The same result 

 follows from the consideration that the lines in a plane, and the 

 planes through a point, are each two-fold. 



Again, if we take a sphere as our element we can through any 

 point as a centre draw a singly infinite number of spheres, but 

 the number of such centres is triply infinite ; hence space as a 

 plenum of spheres is four-fold. And generally, space as a 

 plenum of surfaces has a manifoldness equal to the number of 

 constants required to determine the surface. Although it would 

 be beyond our present purpose to attempt to pursue the subject 

 further, it should not pass unnoticed that the identity in the 

 four-fold character of space, as derived on the one hand from a 

 system of straight lines, and on the other from a system of 

 spheres, is intimately connected with the principles estjfblished by 

 Sophus Lie in his researches on the correlation of these figures. 



If we take a circle as oiu: element we can around any point in 

 a plane as a centre draw a singly infinite system of circles ; but 

 the number of such centres in a plane is doubly infinite ; hence 

 the circles in a plane form a three-fold system, and as the planes 

 in space form a three-fold system, it follows that space as a 

 plenum of circles is six-fold. 



Again, if we take a circle as our element, we may regard it as 

 a section either of a sphere, or of a right cone (given except in 

 position) by a plane perpendicular to the axis. In the former 

 case the position of the centre is three-fold ; the directions of the 

 plane, lik e that of a pencil of lines perpendicular thereto, two- 

 fold ; and the radius of the sphere one-fold ; six-fold in all. In 

 the latter case, the position of the vertex is three-fold ; the direc- 

 tion of the axis two-fold ; and the distance of the plane of 

 section one-fold ; six-fold in all, as before. Henc; space as a 

 plenum of circles is six-fold. 



Similarly, if we take a conic as our element we may regard it 

 as a section of a right cone (given except in position) by a plane. 

 If the natiure of the conic be defined, the plane of section will 

 be inclined at a fixed iangle to the axis ; otherwise it will be free 

 to take any inclination whatever. This being so, the position of 

 the vertex will be three-fold, the direction of the axis two-fold, 

 the distance of the plane of section from the vertex one-fold, 

 and the direction of that plane one-fold if the conic be defined, 

 two-fold if it be not defined. Hence, space as a plenum of 

 definite conies will be seven-fold, as a plenum of conies in 

 general, eight-fold. And so on for curves of higher degrees. 



This is, in fact, the whole story and mystery of manifold 

 space. If not seriously regarded as a reality in the same sense as 

 ordinary space, it is a mode of representation, or a method 

 which, having served its piu-pose, vanishes from the scene. 

 Like a rainbow, if we try to grasp it, it eludes our very touch ; 

 but, like a rainbow, it arises out of real conditions of known and 

 tangible quantities, and if rightly apprehended it is a true and 

 valuable expression of natural laws, and serves a definite 

 purpose in the science of which it forms part. 



Illustrations. 

 Again, if we seek a counterpart of this in common life, I 

 might remind you that perspective in drawing is itself a method 

 not altogether dissimilar to that of which I have been speaking ; 

 and that the third dimension of space, as represented in a pictiu-e, 

 has its origin in the painter's mind, and is due to his skiU, but 

 has no real existence upon the canvas which is the groundwork 

 of his art. Or again, turning to literature, when in legendary 

 tales, or in works of fiction, things past and future are pictured 

 as present, has not the poetic fancy brought time into correlation 

 with the three dimensions of space, and brought all alike to a 



