NATUEE 731 



suit is so much. This is absolutely invalid: it would be true 

 only if we were sure that all the systematic errors were eliminated, 

 and of that we know absolutely nothing. We have two series of ob- 

 servations; by applying the law of least squares we find that the 

 probable error in the first series is twice as small as in the second. 

 The second series may, however, be more accurate than the first, 

 because the first is perhaps affected by a large systematic error. All 

 that we can say is, that the first series is probably better than the 

 second because its accidental error is smaller, and that we have no 

 reason for affirming that the systematic error is greater for one of 

 the series than for the other, our ignorance on this point being ab- 

 solute. 



VII. Conclusion. In the preceding lines I have set several pro- 

 blems, and have given no solution. I do not regret this, for perhaps 

 they will invite the reader to reflect on these delicate questions. 



However that may be, there are certain points which seem to be 

 well established. To undertake the calculation of any probability, and 

 even for that calculation to have any meaning at all, we must admit, 

 as a point of departure, an hypothesis or convention which has always 

 something arbitrary about it. In the choice of this convention we 

 can be guided only by the principle of sufficient reason. Unfortu- 

 nately, this principle is very vague and very elastic, and in the cur- 

 sory examination we have just made we have seen it assume different 

 forms. The form under which we meet it most often is the belief in 

 continuity, a belief which it would be difficult to justify by apodeictic 

 reasoning, but without which all science would be impossible. Finally, 

 the problems to which the calculus of probabilities may be applied 

 with profit are those in which the result is independent of the hypo- 

 thesis made at the outset, provided only that this hypothesis satisfies 

 the condition of continuity. 



Optics and Electricity l 



Fresnel's Theory. The best example that can be chosen is the 

 theory of light and its relations to the theory of electricity. It is 

 owing to Fresnel that the science of optics is more advanced than 

 any other branch of physics. The theory called the theory of undu- 

 lations forms a complete whole, which is satisfying to the mind; but 

 we must not ask from it what it cannot give us. The object of math- 

 ematical theories is not to reveal to us the real nature of things; that 

 would be an unreasonable claim. Their only object is to co-ordinate 

 the physical laws with which physical experiment makes us acquainted, 



i This section is mainly taken from the prefaces of two of my books 

 Thdorie Hathtmatique de la lumicre (Paris: Naud, 1889). and Electricity et 

 Optique (Paris: Naud, 1901). 



