406 



SCIENCE. 



[N. S. Vol. XIX. No. 480. 



9. DARWINISM AND GEOMETRY. 



The doctrine of evolution as commonly 

 exponnded postulates a world independent 

 of man, and teaches the production of man 

 from lower forms of life by wholly natural 

 and unconscious causes. In this statement 

 of the world of evolution there is need of 

 some rudimentary approximative practical 

 geometry. 



The mighty examiner is death. The 

 puppy, though born blind, must still be 

 able to superimpose his mouth upon the 

 source of his nourishment. The little chick 

 must be able, responding to the stimulus 

 of a small bright object, to bring his beak 

 into contact with the object so as to grasp 

 and then swallow it. The springing goat, 

 that too greatly misjudges an abyss, does 

 not survive and thus is not the fittest. 



So, too^ with man. We are taught that 

 his ideas must in some way and to some 

 degree of approximation correspond to this 

 independent world, or death passes upon 

 him an adverse judgment. 



But it is of the very essence of the doc- 

 trine of evolution that man's knowledge of 

 this independent world, having come by 

 gradual betterment, trial, experiment, ad- 

 aptation, and through imperfect instru- 

 ments, for example the eye, can not be 

 metrically exact. 



If two natural hard objects, susceptible 

 of high polish, be ground together, their 

 surfaces in contact may be so smoothed as 

 to fit closely together and slide one on the 

 other, without separating. If now a third 

 surface be ground alternately against each 

 of these two smooth surfaces until it ac- 

 curately fits both, then we say that each 

 of the three surfaces is approximately 

 plane. If one such plane surface cut 

 through another, we say the common 

 boundary or line where they cross is ap- 

 proximately a straight line. If three 

 approximately plane surfaces on objects 



cut through a fourth, in general they make 

 a figure we may call an approximate tri- 

 angle. Such triangles vary greatly in 

 shape. But no matter what the shape, if 

 we cut off the six ends of any two such 

 and place them side by side on a plane 

 with their vertices at the same point, the 

 six are found, with a high degree of ap- 

 proximation, just to fill up the plane about 

 the point. Thus the six angles of any two 

 approximate triangles are found to be to- 

 gether approximately four right angles. 



Now, does the exactness of this approxi- 

 mation depend only on the straightness of 

 the sides of the original two triangles, or 

 also upon the size of these triangles? 



If we know with absolute certitude, as 

 the Yale professors imagine, that the size 

 of the triangles has nothing to do with it, 

 then we know something that we have no 

 right to know, according to the doctrine 

 of evolution; something impossible for us 

 ever to have learned evolutionally. 



10. THE NEW EPOCH. 



Yet before the epoch-making ideas of 

 Lobachevski and John Bolyai every one 

 made this mistake, every one supposed we 

 were perfectly sure that the angle-sum of 

 an actual approximate triangle approached 

 two right angles with an exactness depend- 

 ent only on the straightness of the sides, 

 and not at all on the size of the triangle. 



11. THE SLIPS OF PHILOSOPHY. 



The Scotch philosophy accounted for this 

 absolute metrically exact knowledge by 

 teaching that there are in the human mind 

 certain synthetic theorems, called intui- 

 tions, supernaturally inserted there. Dr. 

 McCosh elaborated this doctrine in a big: 

 book entitled 'The Intuitions of the Mind 

 Inductively Investigated.' One of these 

 supernatural intuitions was Euclid's par- 

 allel-postulate ! Voild! 



'Yet,' to quote a sentence from Wenley's 



