262 POPULAR SCIENCE MONTHLY. 



acquire a thorough familiarity with the country through which we pass. 

 The analytical method, however, affords abundant opportunity for men- 

 tal activity, although of a different kind from that required in the 

 other. First, the most advantageous analytic expression for the given 

 geometric conditions must be sought; then the proper line of analytic 

 transformation must be determined upon; and finally the result must 

 be interpreted geometrically. This last step requires keen insight in 

 order to ensure the full value of the result, for it is here that we often 

 find far more than we anticipated, or than a casual glance will reveal. 



The obligation thus incurred by geometry to analysis has been 

 largely repaid by the aid which analysis has derived from geometry. 

 The study of pure analysis is unquestionably the most abstruse branch 

 of mathematics, but it is now advancing with rapid strides and demands 

 less and less the aid of geometry. The results of the analytic method 

 in geometry, however, are too fruitful for it to be either desirable or 

 possible for us to go back to a condition of complete separation of these 

 two methods. 



Amongst the distinctly modern developments of geometry is what 

 is known as hyper-geometry, the geometry of space of more than three 

 dimensions. The fact that the product of two linear dimensions is 

 representable by an area, and the product of three linear dimensions by 

 a volume, naturally leads us to ask what is the geometric representa- 

 tive of the product of four or more linear dimensions. The answer to 

 this question leads to the ideal conception of space of four or more 

 dimensions. Just as in space of three dimensions, the space of our 

 every-day experience, we can draw three concurrent straight lines such 

 that each one is perpendicular to each of the other two, so in space of 

 four dimensions it must be possible to draw four concurrent straight 

 lines such that each one is perpendicular to each of the other three. 

 It is needless to say it transcends the power of the human mind to 

 form such a conception, nevertheless it is possible to study the geome- 

 try of such a space, and much has been done in this way both analyti- 

 cally and by the methods of pure geometry. If our space has a fourth 

 dimension (not to speak of any higher dimension) as some mathemati- 

 cians seem disposed seriously to maintain, a body moved from any posi- 

 tion in the direction of the fourth dimension will disappear from view. 

 In fact, it will be annihilated so far as we are concerned. Again, a 

 body placed in an inclosed space can be removed therefrom while the 

 walls of the envelope remain intact; or the envelope itself can be turned 

 inside out without rupturing the walls. For example, it would be 

 possible to extract the meat from an egg and leave the shell unbroken. 

 For most persons, however, the geometry of four-dimensional space is 

 likely to remain a mathematical curiosity, serving no useful purpose 

 except to furnish an opportunity for acute logical reasoning, for in- 



