PROBLEMS OF MODERN SCIENCE 



the odds against the happening of some event. I 

 now direct your attention to a very similar type of 

 subject the Theory of Partitions which, especi- 

 ally through the work of Major Macmahon, is also 

 one of the most rapidly developing, and at the 

 same time most fundamental, branches of Pure 

 Mathematics properly so called. Let us take one 

 of the simplest possible instances the partitions of 

 the small number 5, or the various ways in which 

 we can make up 5 by the addition of smaller 

 numbers. They are as follows : 



I, I, I, I, I I, 2, 2, O, O , 



I, I, I, 2, O I, 4, O, O, O 



i, i, 3, o, o 5, o, o, o, o 



3, 2, o, o, o 



where the order is not regarded as relevant, 

 and the total is 7. In a case like this it is 

 easy to enumerate the partitions by writing 

 them down and counting, but it is the work 

 of a lifetime to find the number of partitions 

 of, let us say, the number 20,000 by any such 

 procedure. We need a theory of partitions, and 

 Major Macmahon has supplied us with a very 

 comprehensive one. But my readers may well, 

 at this point, ask me what is the practical value, 

 and what can ever be the practical value, of such an 

 investigation into the number of partitions of a 

 very large integer number. It might be compared 

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