MEETINGS OF SOCIETIES. 23 I 



was asked how it was possible to possess one of these "self-evident" 

 truths except by inheritance without breaking the chain of cause and 

 effect. Such a statement as that " Things which are equal to the same 

 thing are equal to each other " was not a " self-evident " truth ; it 

 required reasoning from experience before the mind could place faith in 

 it. The purely mental conception of a line as having " length without 

 breadth" could not be called useless (although it could not be practically 

 represented) because arithmetical figures used in trigonometry proved 

 that boundaries of geometrical figures really had position but not 

 magnitude (of breadth). So that this is almost a necessary truth, and 

 although abstract truths were little more than hypotheses, still if they 

 were " working hypotheses " they were of enormous value. He might 

 instance the value of the 47th problem of the 1st book of Euclid ; the 

 discoverer of the principle in this problem offered up a hetacornb of 

 oxen to the gods for so great a truth being found, and it had proved of 

 inestimable value to the world in astronomy, navigation, engineering, 

 &c. He could understand the schoolboys' delight if allowed to prove 

 the truth of the fifth problem of the first book of Euclid by turning the 

 ti'iangle on its back, but he hardly thought such a simplification would 

 be allowed, although many of the propositions in Euclid might be swept 

 away as being evident at sight, and not made clearer by the attempted 

 proof. As to Mr. Maskell's assertion that the Bosjeman or any savage 

 had as much intellectual power as the civilised European, there would 

 be difficulty in measuring the amount of latent power in any individual, 

 but it was certain that the expression of that power was immensely 

 unequal. It would be almost imj)ossible to assert with gravity that the 

 mind of an African who, with great difficulty, could be taught the use 

 of numbers beyond 2 or 3, was equal to any one of the minds of Bacon, 

 Newton, or Herschel, although a potentiality of mind equal to great 

 intellectual effort might lie unrecognised in the brain of the savage. 



Mr. Carlile, in reply, expressed his gratification at the appreciative 

 criticism his paper had received. The Bresident had already explained 

 some of the matter to which exception had been taken. He had not 

 meant to suggest that the simplification of the proof of the V. 

 proposition which he suggested in any way detracted from its validity 

 and importance. There were several of the propositions at the 

 becrinnino- of the first book which were rather obscured than illustrated 

 by the proof furnished of them. The XIII., for instance. If we 

 regarded a point in a straight line as an angle of 180°, it was certain 

 that drawing any number of lines through this point could have no 

 tendency to alter the size of this angle, yet this was what was 

 elaborately proved. He thought a desideratum among the definitions 

 was a definition of what was meant by the size of an angle. We 

 proceeded to speak of the size of angles without furnishing any 

 criterion for their measurement. If this were furnished, it would 

 necessarily carry with it the proof of the IV., V., and VIII., and a 

 host of other propositions. The size of an angle and the length of the 

 subtending side in any triangle were, it seemed to him, two names for 

 the same thing. There was no need of propositions to prove the fact of 

 their concomitant variations. 



(2) " On the shifting of the Sand Dunes," by H. C. Field. The 

 paper gave results of forty years' observations on the coast from Paika^ 



