Nov. 2, 1883.] 



♦ KNOWLEDGE ♦ 



281 



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 1 and further. In what manner can the 

 notion be arrived at geometrically 1 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 arc 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, con- 

 struct 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 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 con- 

 clusion is a very natural 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 got over by the 

 analytical method before referred to, for this introduces 

 difficulties of its own ; is there in a plane a point the co- 

 ordinates of which have given imaginary values 1 As a 

 matter of fact, we do not 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 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 mis- 

 leading. 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 be here under- 

 stood to bo used in opposition to real. I give this 

 explanation the bettor to point out how wide the applica- 

 tion of the notion of the imaginary is — viz. (unless ex- 

 pressly 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 fundamental 

 notion underlying and.pervading the whole of these branches 

 of mathematical science. 



In geometry it is the curve, whether defined by means 

 of its equation, or in any other manner, which is the sub- 

 ject for contemplation and study. Hut we also use the 

 curve as a representation of its equation — that is of tlie 

 relation existing between two magnitudes x, y, which are 

 taken as the co-ordinates of a point on the curve. Such 

 employment of a curve for all sorts of purposes — the fluc- 

 tuations of till' buronieter, the Camliridgc boat races, or 

 the Funds — is familiar to most of you. 



It is in like manner convenient in analysis for exhibiting 

 tlu! relations between any three magnitudes, .r, y, -, to 

 regard tliem as the co-ordinates of a point in space ; and, 

 on the like groimd, we should at least wisli to regard any 

 four or more magnitudes as the co-ordinates of a point in 

 space of a corresponding number of dimensions. Starting 

 with the hypothesis of such a space, and of points tliercin 

 each determined by means of its co-ordinates, it is found 



possible to establish a system of ?i-dimensional geometry 

 analogous in every respect to our two- and three- dimen- 

 sional geometries, and to a very considerable extent ser%-ing 

 to exhibit the relations of the variables. It is to be borne 

 in mind that the space, whatever its dimensionality may be, 

 must always be regarded as an imaginary or complex space 

 such as the two or three- dimensional space of ordinary 

 geometry ; the advantages of the representation would 

 otherwise altogether fail to be obtained. 



©ur CI) ess Column. 



Bv MKfHISTO. 



PKOBLEM No. 103. 



E E. N. Fbaxk:esstein. 



Black. 



Whits. 



Wliite to play and self-mate in 5 moves. 



(A Tery ingenious conception. — Ed.) 



Solution-. — Pkoblem Xo. 102, p. 250. 



1. Kt to Kt sq Kt to BO (cli), or 1. Kt to B8 



2. B takes Kt Kt to ]vt3 (ch) 2. B to B7 P to K3 



3. K to K6 K to K5 3. B to Q8 any move 



4. K to K2 mate mates accordingly 



Members of the fairer sc.\ have a more sensitive and complex 

 nervous system, rendering them more susceptible to mental in- 

 fluence than their less <rifted masculine fellow-creatures. Uence, in 

 accordance with the theory advanced in our recent article on the 

 subject of mind influence, the Chess play of ladies would be more 

 incliued to vary through this influence than that of men. The idea 

 is certainly supported by the fact that we have not many strong- 

 lady Chess players. We will not, however, now examine whether 

 this disadvantage only forms the minor obstacle to success, or 

 whether the want of perseverance and powers of concentration of 

 thought, are the real causes of failure in feminine Chess practice. 

 Wc will chivalrously assume the former to be the major and not 

 the minor obstacle, especially as it may bo possible that a too 

 extensive practice of Chess playing amongst ladies would result to 

 the <iisadvantage of us men. 



Notwithstanding, a little more attention might be given by ladies 

 to our noble game ; for only in this wise can the earnest ideal of 

 Mr. Lamb, the President of the North London Chess Club, be 

 realised. At the recent very successful annual dinner of the North 

 London Chess Club, the worthy President expressed it as his 

 sincere conviction that Chess ought to bo taught to the young. 

 " Chess strengthens the intellect," he remarked, "induces cautious 

 habits and steadiness, and is likely to keep the superabundant spirit 

 of yonth within bounds, liesides initiating the child in a recreation 

 whieh if indulged in moderately would bo a constant source of 

 pleasure in after-life." Who is there, we ask, moro«iit to teach 

 Cliess to the young than ladies ? 



To assist such of our fair readers as wish to gain a good insight 

 into the subtleties of Chess, wo will this week enter more fully than 



