24 SECTIONAL ADDRESSES. 



of the number of prime-pairs, p, p + 2, or p, p + i, or p, p + 6, both of 

 whose members are less than a hirge number ,r, is, it appeiu^s, 



X 



(log xr 



The order of magnitude of the corresponding number of triplets, of any 

 possible type, is 



X 



and so on generally. Further, we can assign the relative frequencies 

 of pairs or triplets of different types ; there are, for example, about twice 

 as many pairs whose difference is 6 as pairs whose diffei'ence is 2. All 

 these results have been tested by actual enumeration from the factor 

 tables of the first million numbers ; and a physicist would probably 

 regard them as proved, though we of course know very well that they 

 are not. 



There is a great deal of mathematics the purport of which is quite 

 impossible for any amateur to grasp, and which, however beautiful and 

 important it may be, must always remain the possession of a narrow 

 circle of experts. It is the peculiarity of the theory of numbers that 

 much of it could be published broadcast, and would win new readers for 

 the Daily Mail. The positive integers do not lie, like the logical foun- 

 dations of mathematics, in the hardly visible distance, nor in the un- 

 comfortably tangled foreground, like the immediate data of the physical 

 world, but at a decent middle distance, where the outlines are clear 

 and yet some element of mystery remains. There is no one so blind 

 that he does not see them, and no one so sharp-sighted that his vision 

 does not fail; they stand there a continual and inevitable challenge to 

 the curiosity of every healthy mind. I have merely directed your 

 attention for a moment to a few of the less immediately conspicuous 

 features of the landscape, in the hope that I might sharpen your 

 curiosity a little, and that some of you perhaps might feel tempted to 

 walk a little nearer and take a rather closer view. 



