482 



SCIENCE. 



[Vol. II., No, 35. 



geometry originally applicable to'tlie 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 defor- 

 mation could take place. In the latter case their 

 geometry would be that appropriate to the develop- 

 able 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 tal<e 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-dimensional geometry. 



Evidently, all the foregoing questions present them- 

 selves in regard to ourselves, and to three-dimension- 

 al space as we conceive of it, and as the physical space 

 of our experience. And I need hardly say that the 

 first step is the difficulty, and that, granting a fourth 

 dimension, we may assume as many rnore dimensions 

 as we please. But, whatever answer be given to 

 them, we have, as a branch of mathematics, poten- 

 tially if not actually, an analytical geometry of n- 

 dimensional space. I shall have to speak again upon 

 this. 



Coming now to the fundamental notion already re- 

 ferred to, — that of imaginary magnitude in analysis, 

 and imaginary space in geometry ; 1 connect this 

 with two great discoveries in mathematics, made in 

 the first half of the seventeenth century, — Harriot's 

 representation of an equation in the form/(x)= 0, 

 and the consequent notion of the roots of an equa- 

 tion as derived from the linear factors of /(s) (Har- 

 riot, 1560-1621: his 'Algebra,' published after his 

 death, has the date 1631}; and Descartes' method of 

 co-ordinates, as given in the ' Geometric ' forming a 

 short supplement to his ' Traits de la m^thode,' etc. 

 (Leyden, 1637). 



I show liow by these we are led analytically to the 

 notion of imaginary points in geometry. For in- 

 stance : we arrive at tlie 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 contradicted by 

 experience. 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, furtlier, In 

 what manner can the notion be arrived at geometri- 

 cally ? 



There is a ■well-known construction in perspective 

 for drawing lines through the intersection of two 

 lines which are so nearly parallel as Jiot 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 intersect each other 

 in two real points, then it will be found tb.at the line 

 passes through these two points: if the circles appear 

 not to intersect, then the line will appear not to inter- 

 sect either of tlie circles. But the geometrical con- 

 struction 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 conclu- 

 sion is a very natural one, provided we assume the 

 existence of imaginary points; and that, this assump- 

 tion not being made, then, if the circles do not inter- 

 sect, it is meaningless to assert that the line passes 

 through tlieir points of intersection. The difficulty 

 is not got over by the analytical method before 

 referred to, for tliis introduces difficulties of its own. 

 Is there, in a plane, a point the co-ordinates of 

 which have given imaginary values? As a matter of 

 fact, we do consider, in plane geometry, imaginary 

 points introduced 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. 



I have used the word ' imaginary ' rather than ' com- 

 plex,' and I repeat that the word has been used as in- 

 cluding real. But, this once understood, the word 

 becomes in many eases superfluous, and the use of it 

 would even be misleading. Thus: 'a problem has 

 so many solutions.' This means so many imaginary 

 (including 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 implication excluded), it is a notion implied 

 and presupposed in all the conclusions of modern 

 analysis and geometry. It is, as I have said, the fun- 

 damental notion underlying and pervading the whole 

 of these branches of mathematical science. . 



I consider the question of the geometrical repre- 

 sentation of an Imaginary variable. We represent 

 the imaginary variable x + iy by means of a point in 

 a plane, the co-ordinates of whicli are (x, y). Tliis 

 idea, due to Gauss, dates from about the year 1831. 

 We thus picture to ourselves the succession of values 

 of the imaginary variable x -\- iy by means of the 

 motion of the representative point: for instance, the 

 succession of values corresponding to the motion of 

 the point along a closed curve to its original position. 

 The value X+ iY of the function can, of course, be 

 represented by means of a point (taken for greater 

 convenience in a different plane), the co-ordinates of 

 which are X, Y. 



We may consider, in general, two points, moving 

 each in its own plane; so that the position of one of 

 tliem determines the position of the otlier, and con- 

 sequently the motion of the one determines the mo- 

 tion of the other. For instance : the two points may 



