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NATURE 



[Sept. 20, 1883 



in this point of view they are Klein's elliptic, parabolic, and 

 hyperbolic geometries respectively. 



Next as regards solid geometry : we can by a modification of 

 the notion of distance (such as has just been explained in regard 

 to Lobatschewsky's system) pass from our present system to a 

 non-Euclidian system ; for the other mode of passing to a non- 

 Euclidian system it would be necessary to regard our space as a 

 flat three-ciiinensional space existing in a space of four dimen- 

 sions (i.e. as the analogue of a plane existing in ordinary space) ; 

 and to substitute for such fiat three-dimensional space a curved 

 three-dimensional space, say of constant positive or negative 

 curvature. In regarding the physical space of our experience as 

 possibly non-Euclidian, Kiemann's idea seems to be that of 

 modifying the notion of distance, not that of treating it as a locus 

 in four-dimensional space. 



I have just come to speak of four-dimensional space. What 

 meaning do w e attach to it ? or can we attach to it any mean- 

 ing? It may be at once admitted that we cannot conceive of a 

 fourth dimension of space ; that space as we conceive of it, and 

 the physical space of our experience, are alike three-dimensional ; 

 but we can, I think, conceive of space as being two- or even 

 one-dimensional ; we can imagine rational beings living in a 

 one-dimensional space (a line) or in a two-dimensional space (a 

 surface), and conceiving of space accordingly, and to whom, 

 therefore, a two-dimensional space, or (as the case may be) a 

 three-dimensional space, would be as inconceivable as a four- 

 dimensional space is to us. And very curious speculative ques- 

 tions arise. Suppo-e the one-dimensional space a right line, and 

 that it afterwards becomes a curved line : would there be any 

 indication of the change ? Or, if originally a curved line, would 

 there be anything to suggest to them that it was not a right line ? 

 Probably not, for a one-dimensional geometry hardly exists. 

 But let the space be two-dimensional, and imagine it originally a 

 plane, and afterwards bent (converted, that is, into some form of 

 developable surface) or converted into a curved sur ace ; or imagine 

 it originally a developable or curved surface. In the former 

 case there should bean indication of the change, for the geometry 

 originally applicable to the space of their experience (our own 

 Euclidian geometry) would cease to be applicable ; but the 

 change could not be apprehended by them as a bending or 

 deformation of the plane, for this would imply the notion of a 

 three-dimensional space in which this bending or deformation 

 could take place. In the latter case their geometry would be 

 that appropriate to the developable or curved surface which is 

 their space : viz. this would be their Euclidian geometry : would 

 they ever have arrived at our own more simple system? But 

 take the case where the two-dimensional space is a plane, and 

 imagine the beings of such a space familiar with our own 

 Euclidian plane geometry ; if, a third dimension being still 

 inconceivable by them, they were by their geometry or otherwise 

 led to the notion of it, there would be nothing to prevent them 

 from forming a science such as our own science of three-dimen- 

 sional geometry. 



Evidently all the foregoing questions present themselves in 

 regard to ourselves, and to three-dimensional space as we con- 

 ceive of it, and as the physical space of our experience. And 1 

 need hardly say that the first step is the difficulty, and that 

 granting a fourth dimension we may assume as many more 

 dimensions as we please. But whatever answer be given to 

 them, we have, as a branch of mathematics, potentially, if not 

 actually, an analytical geometry of «-dimensional space. I shall 

 have to speak again upon this. 



Coming now to the fundamental notion already referred to, 

 that of imaginary magnitude in analysis and imaginary space in 

 geometry : I connect this with two great discoveries in n athe- 

 tnatics made in the first half of the seventeenth century, Harriot's 

 representation of an equation in the form f(x) = o, and the 

 consequent notion of the roots of an equaiion as derived from 

 the linear factors of /(.<) (Harriot 1500-1621 : his " Algebra," 

 published after his death, has the date 1631), and Descartes' 

 method of coordinate-, as given in the "Geometrie," forming 

 a short supplement to his " Traite de la Methode, &c." 

 (Leyden, 1637). 



1 show how by these we are led analytically to the notion of 

 imaginary points in geometry ; for instance, we arrive at the 

 theorem that a straight line and circle in the same plane intersect 

 always in two points, real or imaginary. The conclusion as to 

 the two points of intersection cannot be contradictea by expe- 

 rience : take a sheet of paper and draw on it the straight line 

 and circle, and try. But you might say, or at least be strongly 



tempted to say, that it is meaningless. The question of course 

 arises, What is the meaning of an imaginary point ? and, further, 

 In what manner can the notion be arrived at geometrically ? 



There is a well known construction in perspective for drawing 

 lines through the intersection of two lines which are so nearly 

 parallel as not to meet within the limits of the sheet of paper. 

 You have two given lines which do not meet, and you draw a 

 third line, which, when the lines are all of them produced, is 

 found to pass through the intersection of the given lines. If 

 instead of lines we have two circular arcs not meeting each 

 other, then we can, by means of these arcs, construct a line ; 

 and if on completing the circles it is found that the circles inter- 

 sect each other in two real points, then it will be found that the 

 line passes through these two points : if the circles appear not 

 to intersect, then the line will appear not to intersect either of 

 the circles. But the geometrical construction being in each case 

 the same, we say that in the second case also the line passes 

 through the two intersections of the circles. 



Of course it may be said in reply that the conclusion is a very 

 1 atural one, provided we assume the existence of imaginary 

 points ; and that, this assumption not being made, then, if the 

 circles do not intersect, it is meaningless to assert that the line 

 passes through their points of intersection. The difficulty is not 

 got over by the analytical method before referred to, for this 

 introduces difficulties of its own : is there in a plane a point the 

 coordinates of which have given imaginary values ? As a matter 

 of fact, we do consider in plane geometry imaginary points intro- 

 duced into the theory analytically or geometrically as above. 



The like considerations apply to solid geometry, and we thus 

 arrive at the notion of imaginary space as a locus in quo of 

 imaginary points and figures. 



1 have used the word imaginary rather than complex, and I 

 repeat that the word has been used as including real. But, this 

 once understood, the word becomes in many cases superfluous, 

 and the use of it would even be misleading. Tims, "a problem 

 has so many solutions : " this means so many imaginary (includ- 

 ing real) solutions. But if it were said that the problem had 

 " so many imaginary solutions," the word " imaginary " would 

 here be understood to be used in opposition to real. I give this 

 explanation the better to point out how wide the application of 

 the notion of the imaginary is, viz. (unless expressly or by im- 

 plication excluded) it is a notion implied and presupposed in all 

 the conclusions of modern analysis and geometry. It is, as I 

 have said, the fundamental notion underlying and pervading the 

 v hole of these branches of mathematical science. 



I consider the question of the geometrical representation of 

 an imaginary variable. We represent the imaginary variable 

 x + iy by means of a point in a plane, the coordinates of which 

 are (x, y). This idea, due to Gauss, dates from about the year 

 1831. We thus picture to our-elves the succession of values of 

 the imaginary variable .r + iy by means of the motion of the 

 representative point : for instance, the succession of values corre- 

 sponding to the motion of the point along a closed curve to its 

 original position. The value X + A* of the function can of 

 course be represented by means of a point (taken for greater 

 convenience in a different plane), the coordinates of which are 



We may consider in general two points, moving each in its 

 own plane, so that the position of one of them determines the 

 position of the other, and consequently the motion of the one 

 determines the motion of the other : for instance, the two points 

 may be the tracing-point and the pencil of a pentagraph. You 

 may with the first point draw any figure you please, there will 

 be a corresponding figure draw n by the second point : for a gocd 

 pentagraph a copy on a different scale (it may be); for a badly- 

 adjusted pentagraph, a distorted Copy; but the one figure will 

 always be a sort of copy of the first, so that to each point of the 

 one figure there will correspond a point in the other figure. 



In the case above referred to, where one point represents the 

 value x + iy of the imaginary variable and the other the value 

 X + iY of some function <f> [x + iy) of that variable, there is a 

 remarkable relation between the two figures : this is the relation 

 of orthomorphic projection, the same which presents itself 

 I etween a ] ortion of the earth's surface and the representation 

 thereof by a map on the stereographic projection or on Mer- 

 talor's projection — viz., any indefinitely small area of the one 

 figure is represented in the other figure by an indefinitely small 

 area of the same shape. There will possibly be for different 

 parts of the figure great variations of scale, but the shape will 

 be unalteied ; if for the one area the boundary is a circle, then 



