GEOMETRY: ANCIENT AND MODERN. 263 



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 exclu- 

 sively in descriptive geometry, the geometry of the engineer and builder 

 with their plan and elevation, so called. The geometry of four-dimen- 

 sional figures has been studied in an analogous way. A four-dimen- 

 sional 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 Pennsyl- 

 vania 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 mathe- 

 maticians 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 pro- 

 ceeds by pure reason from certain axioms, postulates and definitions, 

 it is undoubtedly true that for at least the most fundamental concep- 

 tions 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 be- 

 longs to a set of theorems which are so connected that if the truth of 



