726 SCIENCE AND HYPOTHESIS 



very small. I may give to < any value I choose, with one restriction : 

 this function must be continuous; and, in fact, from the point of 

 view of subjective probability, the choice of a discontinuous function 

 would have been unreasonable. What reason could I have, for in- 

 stance, for supposing that the initial longitude might be exactly o, 

 but that it could not lie between o and 1 ? 



The difficulty reappears if we look at it from the point of view of 

 objective probability; if we pass from our imaginary distribution in 

 which the supposititious matter was assumed to be continuous, to the 

 real distribution in which our representative points are formed as 

 discrete atoms. The mean value of sin (at + b) will be represented 

 quite simply by 



J2 "n (at+b), 



n being the number of minor planets. Instead of a double integral 

 referring to a continuous function, we shall have a sum of discrete 

 terms. However, no one will seriously doubt that this mean value is 

 practically very small. Our representative points being very close 

 together, our discrete sum will in general differ very little from an 

 integral. An integral is the limit towards which a sum of 

 terms tends when the number of these terms is indefinitely in- 

 creased. If the terms are very numerous, the sum will differ 

 very little from its limit that is to say, from the integral, 

 and what I said of the latter will still be true of the sum 

 itself. But there are exceptions. If, for instance, for all the minor 

 planets b = ?- at, the longitude of all the planets at the time t 

 would be , and the mean value in question would be evidently unity. 

 For this to be the case at the time o, the minor planets must have all 

 been lying on a kind of spiral of peculiar form, with its spires very 

 close together. All will admit that such an initial distribution is 

 extremely improbable (and even if it were realized, the distribution 

 would not be uniform at the present time for example, on the 1st 

 January 1900; but it would become so a few years later). Why, 

 then, do we think this initial distribution improbable ? This must be 

 explained, for if we are wrong in rejecting as improbable this absurd 

 hypothesis, our inquiry breaks down, and we can no longer affirm any- 

 thing on the subject of the probability of this or that present distri- 

 bution. Once more we shall invoke the principle of sufficient reason, 

 to which we must always recur. We might admit that at the begin- 

 ning the planets were distributed almost in a straight line. We 

 might admit that they were irregularly distributed. But it seems 

 to us that there is no sufficient reason for the unknown cause that 

 gave them birth to have acted along a curve so regular and yet so 

 complicated, which would appear to have been expressly chosen so 

 that the distribution at the present day would not be uniform. 



