302 



DISCOVERY 



mode in which animals and plants had come into 

 existence, but he held that his special theory of 

 natural selection was now disproved and worthless. 

 For some reason the plant has advantages which 

 enable it to spread. His theory explained the 

 present distribution of plants and animals on a 

 mechanical principle so well that it could be applied 

 successfully to predicting unknown cases. His reason 

 for it was that he found that the area occupied by 

 species varied with their age, new species occupying 

 very small areas, and older species having a wider 

 distribution. Moreover, older genera had a larger 

 number of species than younger genera. The theory 

 explained evolution and the spread of animals and 

 plants as strictly mechanical processes depending in 

 no way on natural selection. 



This new theory was opposed by several important 

 speakers, who pointed out that the actual facts of the 

 distribution of plants did not agree with those pre- 

 dicted by the theory. The general opinion was that 

 no problem in biology could be determined by statis- 

 tics, for these were merely " smoothed abstractions of 

 actual facts." Observation and experiment were the 

 avenues to understand living things, and the more 

 patiently these were pursued in the spirit of Darwin, 

 the more they appeared to confirm the main lines of 

 his work. 



CRITICISM OF M. CODE'S METHOD 



A second piece of work that was adversely criticised 

 was the Coue method, which has recently created 

 great stir in the land. Dr. W. Brown dealt with it 

 incidentally in a paper to the Ps^xhology Section on 

 auto-suggestion. M. Coue has an extraordinarily clear 

 and penetrating insight into the facts of suggestion, 

 transparent smcerity, and untiring zeal. But he is 

 not a doctor, so that in a sense he is an amateur. 

 And medical men specialising in neurology and psycho- 

 therapy have employed similar methods of treatment 

 on suitable patients with success in no way inferior 

 to that claimed for his work. Their more profound 

 knowledge of the facts of physical and mental disease 

 has allowed them to make progress in psysho-therapy 

 which leaves the amateur far behind. Auto-sugges- 

 tion, or the patient's appeal to his own subconscious 

 mind, must alwaj's be supplemented — and some- 

 times so extensively as to be replaced — by knowledge 

 of many of the chief motive-forces actuating that 

 subconscious. Dr. Brown said it was quite unneces- 

 sary to repeat a parrot-phrase twenty times. (A later 

 speaker said that such repetition in curing a headache 

 had brought on sore throat !) He also gave an 

 account of his own methods of treatment by suggestion 

 and auto-suggestion (which are described in his recent 

 book Suggestion and Mental Analysis), and told how 



children, who could not concentrate their minds, 

 responded to it. For example, a little girl of thirteen, 

 who was very bad at writing essays, was treated by 

 him, and after eight hours' treatment was able to do 

 her essay-writing properh' and well. 



PROBLEMS IN THE THEORY OF NUMBER 



The presidential address to Section A (Mathematics 

 and Physics) was given this year by Mr. G. H. Hardy, 

 the Professor of Mathematics at Oxford. He pro- 

 pounded five problems in the theory of numbers 

 which are yet to be solved, three of which are of 

 general interest. The first is : When is a number the 

 sum of two cubes, and in how many ways may it be so 

 expressed ? The numbers 2 and 9 are sums of two 

 cubes, the former being equal to i' + i', the latter 

 to 2^ -f I' ; 3 and 4 are not. It is exceptional for a 

 number to be the sum of two cubes, and such numbers 

 become rarer and rarer as investigation extends to 

 larger and larger numbers. No simple test has yet 

 been discovered by which such numbers can be 

 distinguished. Again, 2 and 9 are sums of two cubes 

 and can be expressed in one way only, but there are 

 numbers so expressible in a variety of ways. 1729 

 may be expressed in two ways, 12' -f- i^ or 10^ -\- cj', 

 and is the smallest number to be expressed in this way. 

 This has been worked out mentally, but, when a 

 number is sought which is the sum of two cubes 

 expressible in three ways, the computer requires 

 paper. The smallest number indeed is 175,959,000, 

 which is 560' + 70^ or 552' -f 198' or 525^ -f 3153. 

 There is one number known which may be expressible 

 in four ways, 912,646,702,079,469,000, a number 

 so huge that when we are told that no one has dis- 

 covered a number which may be represented in more 

 than four ways we do not feel altogether surprised. 

 Theory, however, has run ahead of computation and 

 has shown that numbers exist which may be expressed 

 in five, six, or in any number of ways. 



The second problem is also a difficult one : Is every 

 large number the sum of five cubes only? It is known 

 that everv number without exception is the sum of 

 nine cubes and two numbers, 23 (which is 2^ + 2' 

 -1- i3 -f i3 + i3 + i3 4- i3 _^ i3 _^ i3) and 239 (53 4- 

 3^ + 3^ + 3' + 2^ 4- 2' -f 2^ 4- 2' 4- i3), cannot be ex- 

 pressed as a sum of less than nine. There are only 

 fifteen, the largest being 454, which need eight, and 

 one hundred and twenty-one, the largest being 8,042, 

 which need seven. Numbers expressible as the sum 

 of six cubes are probably aU less than 1,000,000. 

 Beyond this aU numbers may probably be expressed 

 as the sum of five, or, with very large numbers, of 

 four, but the numbers are so large that computation 

 is helpless. Numbers expressed as the sum of four 



