690 Proceedings. 



arithmetical figures used in trigonometry proved that the 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 Forty- 

 seventh Proposition of the First Book of Euclid : the discoverer of the 

 principle in this problem offered up a hecatomb 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 schoolboy's delight if allowed to prove the truth of the Fifth Propo- 

 sition of the First Book of Euclid by turning the triangle on its back, but 

 he hardly thought such a simplification would be allowed, although many 

 of the propositions might be swept away as being evident at sight, and 

 not made clearer by the attempted j)roof. As to Mr. Maskell's assertion 

 that the Bosjesman, 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 impossible to 

 assert with gravity that the mind of an African who with great difficulty 

 could be taught the use of numbers beyond two or three was equal 

 to any one of the miiids of Bacon, Newton, or Herschel, although a 

 potentiality of mind equal to great intellectual effort might lie unrecog- 

 nised in the brain of the savage. 



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

 criticism his paper had received. The President had already explained 

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

 meant to suggest that the simplification of the proof of the Fifth Pro- 

 position which he suggested in any way detracted from its validity or 

 importance. There were several of the propositions at the beginning of 

 the First Book which were rather obscured than illustrated by the proof 

 furnished of them : the Thirteenth, for instance. If we regard 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. It 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 Fourth, 

 Fifth, and Eighth, 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 thmg. There was no use of propositions to 

 prove the fact of their concomitant variations. 



2. "On the Shifting of Sand-dunes," by H. C. Field. 

 (Transactions, p. 561.) 



Sir James Hector said he thought the subject a most important one. 

 In a new country they should be very careful as to how they interfered 

 with the natural changes of the coast-line. He was of opinion tliat Mr. 

 Field had done good service in bringing this matter before the Society. 

 They in New Zealand would have to guard against selling lands situated 

 in dangerous positions on the coasts. They should also prevent mis- 

 chievous people from interfering with mouths of rivers, and thus pre- 

 venting natural changes. Mr. Field's paper had opened up a subject of 

 extreme practical importance to the colony. 



Mr. Boetham thought this a valuable paper. It would encourage 

 those who had the opportunity to note carefully such changes as had 

 been spoken of. There was no doubt great alterations had taken place on 

 our coasts and in our rivers owing to the causes mentioned by Mr. Field. 



