acquire a thorough familiarity with the country through which we pass. The analytical method, however, affords abundant opportunity for mental 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 representative 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 geometry of such a space, and much has been done in this way both analytically and by the methods of pure geometry. If our space has a fourth dimension (not to speak of any higher dimension) as some mathematicians seem disposed seriously to maintain, a body moved from any position 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
