588 



SCIENCE 



[N. S. Vol. XLHI. No. 1113 



rise of a geometry freed from Euclid's pos- 

 tulate of parallels. The chains forged by 

 habit and by authority were broken, and 

 with ease when once they had dared the 

 attempt, men found that the existence of 

 one parallel to a line through a given point 

 was not the only workable hypothesis: all 

 other axioms and postulates of Euclid 

 might stand, while instead of this one they 

 substituted either no parallel, or more than 

 one. Lobachewski, Bolyai, Saccheri, and 

 probably the great Gauss, were leaders in 

 this memorable emancipation. It was left 

 for the later decades to reflect upon the 

 reasons and to furnish illustrations of the 

 various possible kinds of systems of points, 

 lines and surfaces. Along with parallels, 

 of course right angles, the measures of all 

 angles and the measurement of distances 

 were subjected to revision; and late in the 

 century Cayley and Klein invented the 

 theory of projective measurement of linear 

 segments and of angles, to re-combine the 

 divergent kinds of systems into one har- 

 monious theory. It is easy to misunder- 

 stand. I do not mean to say that non- 

 Euclidean geometries require substantia- 

 tion or sanction from the older system of 

 Euclid; or that the kind of space which 

 they describe presupposes a Euclidean 

 space within which it may exist. The ques- 

 tion of the true nature of the space we live 

 in is equally foreign to all pure geometries. 

 But our common experience accords suffi- 

 ciently with the description given by Eu- 

 clid, and men will always, no doubt, find 

 his axioms preferable. Hence it was and 

 always will be advantageous for us to have 

 as illustrations of non-Euclidean geometries 

 pictures of definite portions of Euclidean 

 space and of objects therein which fit the 

 described relations of other systems in other 

 kinds of space. 



Fortunately for my present theme, and 

 its secular limitation, the end of the cen- 



tury brought a full and satisfactory dis- 

 cussion of the fundamental postulates of 

 geometry, by Hilbert of Gottingen. This 

 gave us a model for the examination of not 

 only the traditional Euclidean and the two 

 traditional divergent non-Euclidean geom- 

 etries, but also for the testing of any other 

 proposed system of fundamental postulates. 

 For the first time, the consistency and inde- 

 pendence of sets of axioms were tried and 

 proven. And this was a boon equally to 

 teachers of all grades; for the redundancy 

 of text-book definitions and axioms in geom- 

 etry had become an intolerable incubus to 

 teachers of critical classes, who yet had not 

 the patience nor the time for finding the 

 solution of their own difficulties. Kant 

 lived and philosophized too early. Axioms 

 must now be judged by their utility for the 

 purpose intended. But whatever they have 

 lost in sacrosanctity and authority, far 

 more is gained in freedom and in power. 



Chronologically it is false, but in the in- 

 evitable logic of events it is true, that pro- 

 jective geometry developed simultaneously 

 with non-Euclidean. The latter clung to 

 measures but looked at parallels differently, 

 the former viewed distance as changeable 

 and considered parallels as intersecting. 

 Descriptive geometry, the body of rules and 

 relations collected in orthogonal projection, 

 parallel projection, and central projection, 

 acted as a stimulus or challenge. Here 

 were a set of observed phenomena, partly 

 reasoned, ready for precise definition and 

 logical arrangement. On the other hand 

 were visible the beginnings of algebraic 

 geometry, presenting general methods and 

 highly general theorems, threatening to en- 

 gulf and obliterate all pure geometry except 

 the most elementary. Let any student of 

 analytic geometry reflect on how few the- 

 orems from elementary geometry the whole 

 analytic superstructure rests! No wonder 

 that those who preferred things rather than 



