Page:Popular Science Monthly Volume 18.djvu/706

688 form of the earth, but the elephant and the tortoise would be more than ever necessary to support it. There would be no science of astronomy, no knowledge of the law of gravitation, none of physics, so far as it was dependent upon astronomy. The sciences so act and react on one another that it is difficult to say just how far the absence of one would affect the others, but we know it would greatly. Such suppositions as we have made may be more or less fully realized in other worlds, and we can thus see that their science may differ very widely from ours and still be no less correct. Science is everywhere relative to the facts with which it has to deal. The difficulty of conceiving physical and spatial relations different from those we are familiar with does not prove them non-existent; nor does the ease of such conceptions prove their existence.

Helmholtz has very ably shown that our geometrical axioms have a truth relative only to the space they are applied to. He supposes beings, of the same mental capacity as ourselves, but of two dimensions only, to inhabit a plane surface. They would possess a plane geometry like ours, but would have no solid geometry whatsoever. Thickness would be as inconceivable to them as a fourth dimension in space is to us. Transplant these beings to the surface of a sphere, and their planimetry would change to a spherical geometry. Defining a straight line as the shortest distance between two points, all their straight lines would be arcs of great circles, and every straight line, when sufficiently extended, would return to itself. Between two points, half the circumference of a great circle apart, an infinite number of straight lines, of equal length, could be drawn; and, as two points would always cut the great circle on which they were situated into two arcs of unequal length, there would always be, besides the shortest straight line connecting the points, a longer straight line (i. e., a line made up of shorter lines, each of which is the shortest distance between its extremities) also connecting them. There could be no parallel straight lines and no similar triangles. The sum of the angles of a triangle would always be more than two right angles, and the amount of the excess would depend upon the length of the sides.

Helmholtz again supposes these beings of two dimensions to be placed upon (what has been called by Beltrami) a pseudo-spherical surface—a surface shaped somewhat like the sides of an hour-glass. Here our axiom of parallels does not hold good. Through a given point, a whole pencil of straight lines may be drawn, none of which shall cut a given line, though infinitely produced, and limited, at the two extremes, by lines that cut the given line at infinite distances in the opposite directions. Helmholtz makes other suppositions which it is unnecessary to follow, as we have already gone far enough to show that even the fundamental axioms of geometry are quite as dependent upon the conditions under which they are used as upon any intuitive necessity we may think belongs to them.