328 



NA TURE 



[February i, 1894 



this idea directed language, that we stillextend the useof the word 

 " equal " to all cases of this particular kind of linear equality : 

 we say that a circle is equal to the rectangle contained by its 

 radius and half its circumference. We have therefore been 

 obliged to invent the word "congruent " for absolute equality 

 in all points, or to qualify the adjective "equal " by " identical," 

 aying "identically equal." There is of course no objection 

 to the comparison of magnitudes in this way by reference to one 

 dimensional measures, or by presupposing that the change which 

 one magnitude must undergo in order to be transformed into 

 the other is to be expressed by a single parameter, but changes 

 involving two or any number of parameters, are just as important 

 as those involving one, and the attempt to express all metric 

 relations by referring them to a single parameter has placed such 

 restrictions on thought that it seems to me appropriate to apply 

 the term emancipation to our act in freeing ourselves from 

 them. With us mathematics is no longer the science of quan- 

 tity. But even if we consider that the ultimate object of mathe- 

 matics is relations between quantities, we have reaped a rich 

 reward by the emancipation, for we are enabled by the use of 

 our broader ideas to reach new conclusions as to metric relations. 

 The idea of groups of operations, as I have tried to develop 

 it, has in recent years been so extended as to cover a large part 

 of the fields of algebra and geometry. Among the leaders in 

 this extension has been Sophus Lie. Considered from the alge- 

 braic point of view, his idea in its simplest form may be ex- 

 pressed thus : We have a certain quantity, say x. We have also 

 an operation of any sort which we may perform upon this 

 quantity. Let this operation depend on a certain quantity, a, 

 which necessarily enters into it. As one of the simplest possible 

 examples, we may consider the operation to be that of adding 

 a to ,r. As the quantity a may take an infinity of values, it 

 follows that there will be an infinity of operations all belonging 

 to one class, which operations will be distinguished by the par- 

 ticular value of a in each case. We thus operate on ,v with one 

 of these operators, and get a certain result, say v'. We operate 

 on ,1' with a second operator, of the i-ame class, and get a 

 second result, say .\". If whatever operators we choose from the 

 class, the result i" could have been obtained from the original 

 quantity r by some operation of the class, then these operations 

 are sucti that the product of any two is equivalent to the per- 

 formance of some one of them. Thus, by repealing them for 

 ever, we could get no results except such as could be obtained 

 by some one operator. To illustrate by one simple example : 

 if our operation consists in the addition of an arbitrary quantity 

 to V, then we change .\ into v' by adding a certain quantity a 

 and ,v' into x" by adding a second quantity /'. The result of 

 these two additions is the same as if we had added in the first 

 place the quantity a -f h. It need hardly be said that the mul- 

 tiplication by v of any quantity would be another example of 

 the same kind. The perlormance of any number of successive 

 multiplications on a quantity is always equal to a single multi- 

 plication by the product of all the factors of the separate 

 multiplications. 



These operations are not confined to single quantities. We 

 may consider the operation to be performed upon a system of 

 quantities, which are thus transformed into an equal number of 

 different quantities, each of these new quantities corresponding 

 to one of the first system. If a repetition of the operation upon 

 the second system of quantities gives rise to a third system, 

 which could have been formed from the first system by an 

 operation of the same class, then all these possible operations 

 form a group. 



The idea of such systems of operations is by no means 

 new. It has always been obvious, since the general theory 

 of algebraic operations has been studied, that any com- 

 bination of the operations of addition, multiplication, and 

 division could always be reduced to a system in which there 

 would be only a single operation of division necessary — just as 

 in arithmetic a complex fraction, no matter what the order of 

 complexity of its terms, can always be reduced to a single simple 

 fraction, that is, to a ratio of two integers, but cannot, in general, 

 be reduced to an integer. Abel made use of this theorem in 

 his celebrated Memoir on the impossibility of solving the general 

 equation of the fifth degree. 



Another field of mathematical thought, quite distinct from 

 that at which we have just glanced, may be called the fairy- 

 land of geometry. To make a mathematician, we must have 

 a higher development of his special power than falls to the 

 lot of other men. When he enters fairyland he must, to do 



himself justice, take wings which will carry him far above the 

 flights, and even above the sight, of ordinary mortals. To 

 the most imaginative of the latter, a being enclosed in a 

 sphere, the surface of which was absolutely impenetrable, would 

 be so securely imprisoned that not even a spirit could escape ex- 

 cept by being so ethereal that it could pass through the substance 

 of the sphere. But the mathematical spirit, in four-dimensional 

 space, could step out without even touching any part of the 

 globe. Taking his stand at a short distance from the earth, he 

 could with his telescope scan every particle of it, from centre to 

 surface, without any necessity that the light should pass through 

 any part of the substance of the earth. If a practised gymnast, 

 he could turn a somersault and come down right side left, just as 

 he looks to our eyes when seen by reflection in a mirror, and that 

 without suffering any distortion or injury whatever. A straight 

 line, or a line which to all our examination would appear 

 straight, if followed far enough, might return into itself. Sp.ace 

 itself may have a boundary, or, rather, there may be only a 

 certain quantity of it ; go on for ever, and we would find our- 

 selves always coming back to the starting-point. All these re- 

 sult^, too, are reached not merely by facetious forms, but by 

 rigorous geometrical demonstration. 



The considerations which lead to the study of these forms of 

 space are so simple that they can be traced without difficulty. 

 When the youth begins the study of plane geometry his attention 

 is devoted entirely to figures lying in a plane. For him space 

 has only two dimensions. To a given point on a straight line 

 only one perpendicular can be drawn. By moving a line of any 

 sort in the plane he can describe a surface, but a solid is wholly 

 without his field. He cannot draw a line from the outside to the 

 inside of a circle without intersecting it. On a given base only 

 two triangles with given sides can be erected, one being on one 

 side of the base, the other on the other. When he reaches solid 

 geometry his conceptions are greatly extended. He can draw 

 any number of perpendiculars to the same point of a straight 

 line. If he has two straight lines perpendicular to each other, 

 he can draw a third straight line which shall be perpendicular to 

 both. A plane surface is not confined to its own plane, but can 

 be moved up and down in such a way as to describe a solid. 

 The characteristic of this motion is that it constantly carries 

 every part of the plane to a position which no part occupied 

 before. 



Now, it is a fundamental principle of pure science that the 

 liberty of making hypotheses is unlimited. It is not necessary 

 that we shall prove the hypothesis to be a reality before we are 

 allowed to make it. It is legitimate to anticipate all the possi- 

 bilities. It is, therefore, a perfectly legitimate exercise of 

 thought to imagine what would result if we should not stop at 

 three dimensions in geometry, but construct one for space 

 having four. As the boy, at a certain stage in his studies, 

 passes from two to three dimensions, so may the mathematician 

 pass from three to four dimensions with equal facility. He 

 does indeed meet with the obstacle that he cannot draw figures 

 in four dimensions, and his faculties are so limited that he can- 

 not construct in his own mind an image of things as they 

 would look in space of four dimensions. But this need not 

 prevent his reasoning on the subject, and one of the most ob- 

 vious conclusions he would reach is this : As in space of two 

 dimensions one line can be drawn perpendicular to another at 

 a given point, and by adding another dimension to space a 

 third line can be drawn perpendicular to these two ; so in a 

 fourth dimension we can draw a line which shall be perpen- 

 dicular to all three. True, we cannot imagine how the line 

 would look, or where it would be placed, but this is merely 

 because of the limitations of our faculties. As a surface de- 

 scribes a solid by continually leaving the space in which it lies 

 at the moment, so a four-dimensional solid will be generated by 

 a three-dimensional one by a continuous motion which shall 

 constantly be directed outside of this three-dimensional space in 

 which our universe appears to exist. As the man confined in a 

 circle can evade it by stepping over it, so the mathematician, if 

 placed inside a sphere in four-dimensional space, would simply 

 step over it as easily as we should over a circle drawn on the 

 floor. Add a fourth dimension to space, and there is room for 

 an indefinite number of universes, all alongside of each other, 

 as there is for an indefinite number of sheets of paper when we 

 pile them upon each other. 



From this point of view of physical science, the question 

 whether the actuality of a fourth dimension can be considered 

 admissible is a very interesting one. All we can say is that, 



NO. 1266, VOL 49] 



