Page:Popular Science Monthly Volume 58.djvu/271

Rh studying the geometry of such space we have only our reasoning powers to guide us and cannot fall back upon experience, as we so often do more or less unconsciously, perhaps, in ordinary geometry.

Geometry of three-dimensional space is often studied by projecting the solid in question upon two or more planes and working with these plane projections instead of with the solid itself. This is done exclusively in descriptive geometry, the geometry of the engineer and builder with their plan and elevation, so called. The geometry of four-dimensional figures has been studied in an analogous way. A four-dimensional figure, it should be remarked, is a figure whose bounding parts are three dimensional figures, just as a three-dimensional figure is one whose bounding parts are surfaces or two-dimensional figures. A four-dimensional figure can be projected on a three-dimensional space and its properties studied from such projections made from different points of view, corresponding to the plan and elevation of ordinary geometry. The mathematical department of the University of Pennsylvania has in its possession wire models of solid projections of all the possible regular four-dimensional hyper-solids, the number of which is limited in the same way as is the number of regular three-dimensional solids. These models were constructed, after a careful study of the question, by Dr. Paul E. Heyl, a recent student and graduate of the University.

Amongst the subjects of most profound interest to mathematicians of recent years has been an investigation into the foundations of geometry and analysis. It was found, as the growth of the science proceeded, that much of fundamental importance, which hitherto had been accepted without question, would not bear searching scrutiny, and it began to be feared that the foundation might collapse in places altogether. We are concerned here with this only so far as it relates to geometry. Whatever may be said of geometry as a science which proceeds by pure reason from certain axioms, postulates and definitions, it is undoubtedly true that for at least the most fundamental conceptions we are thrown back upon experience; and that in the matter of axioms or postulates there is some latitude as to what we shall accept. Amongst the axioms or postulates given by Euclid is one known as the parallel-postulate, which states that if two coplanar straight lines are intersected by a third straight line (transversal) and if the interior angles on one side of the transversal are together less than two right angles, the two straight lines, if produced far enough, will meet on the same side of the transversal on which the sum of the interior angles is less than two right angles. This is, in fact, a theorem, and it is hardly possible to suppose that Euclid did not adopt it as a postulate only after finding that he could neither prove it nor do without it. It belongs to a set of theorems which are so connected that if the truth of