NA TURE 



\_Ncv. 7, 1889 



potential between the metals ; it can only give us the 

 value of the temperature coefficient, which is equal to the 

 Peltier effect divided by the absolute temperature. Then, 

 again, the pyro-electricity of tourmaline is explained by 

 the unilateral conductivity of a tourmaline crystal whose 

 temperature is changing, discovered by the author and 

 Prof. Silvanus Thompson. If this unilateral conduc- 

 tivity is regarded as proving the existence of an electro- 

 motive force in a crystal which is increasing or decreasing 

 in temperature, the explanation is valid, but in the text 

 nothing is said about an electromotive force, and the 

 student might be led to infer that a mere difference in 

 resistance could explain pyro-electricity. The way in 

 Avhich a current flows past an insulating obstacle, the lines 

 of flow closing in on the obstacle, and leaving nothing 

 corresponding to " dead water " behind it, is given as a 

 proof that the electric current has no mechanical mo- 

 mentum ; but unless the corners of the obstacle were 

 infinitely sharp, a slowly-moving fluid might flow in the 

 same way as electricity, even though it possessed inertia, 

 so that the proof is not conclusive. It is also stated that 

 the effects on light produced by a magnetized body, dis- 

 covered by Dr. Kerr, of Glasgow, have been deduced by 

 Prof. Fitzgerald from Maxwell's theory of light. As a 

 matter of fact, however, the results deduced from this 

 theory by Fitzgerald do not coincide with those observed 

 by Dr. Kerr and Prof. Kundt. The production in an 

 unequally-heated conductor of an electromotive force is 

 explained by supposing the atoms in such a body to be 

 moving faster in one direction than the opposite, and 

 therefore, since they are supposed to drag the ether with 

 them, producing a flow of ether in the direction in which 

 they are moving fastest ; but, on the dualistic theory of 

 electricity adopted in this book, this ether stream would 

 consist of equal quantities of positive and negative elec- 

 tricity moving in the sa?ne direction, and this would not 

 produce any electrical effect. 



At the end of the book are three popular lectures de. 

 livered by Prof. Lodge, the first on the relation between 

 electricity and light, the second on the ether and its 

 functions, and the third his admirable one at the Royal 

 Institution, on the discharge of a Leyden jar, which is a 

 model of what such a lecture ought to be. 



Taken as a whole, we think that the book is one which 

 ought to be read by all advanced students of electricity ; 

 they will get from it many of the views which are guiding 

 those who are endeavouring to advance that science, and 

 it is so stimulating that no one can read it without being 

 inspired with a desire to work at the subject to which it 

 is devoted. 



THE CALCULUS OF PROBABILITIES. 



Calcul des Probabilith. Par J. Bertrand. (Paris : 

 Gauthier-Villars, 1 889.) 



" T^ VERYBODY makes errors in Probabilities at times^ 

 -L^ and big ones," writes De Morgan to Sir William 

 Hamilton. M. Bertrand appears to form an exception 

 to this dictum, or at least to its severer clause. He 

 avoids those slips in the philosophical part of the subject 

 into which the greatest of his mathematical predecessors 

 have fallen. Thus he points out that, in investigating the 



" causes " of an observed event, or the ways in which it 

 might have happened, by means of the calculus of prob- 

 abilities, it is usual to make certain unwarranted assump- 

 tions concerning the so-called a /rz'w/ probability of those 

 causes. Suppose that a number of black and white balls 

 have been drawn at random from an urn, and from this 

 datum let us seek to determine the proportion of black 

 and white balls in the urn. It is usual to assume, without 

 sufficient grounds, that a priori one proportion of balls, 

 one constitution of the urn, is as likely as another. Or 

 suppose a coin has been tossed up a number of times, and 

 from the observed proportion of heads and tails let it be 

 required to determine whether and in what degree the 

 coin is loaded. Some assumption must be made as to 

 the probability which, prior to, or abstracting from, our 

 observations, attaches to different degrees of loading. The 

 assumptions which are usually made have a fallacious 

 character of precision. 



Again, M. Bertrand points out that the analogy of urns 

 and dice has been employed somewhat recklessly by- 

 Laplace and Poisson. It is true that the ratio of male to- 

 female births has a constancy such as the statistics of 

 games of chance present. But, before we compare boys 

 and girls to black and white balls taken out at random 

 from an urn, we must attend not only to the average pro- 

 portion of m.ale to female births, but also to the deviations 

 from that average which from time to time or from place 

 to place may be observed. The analogy of urns and balls 

 is more decidedly inappropriate when it is applied to 

 determine the probable correctness of judicial decisions. 

 The independence of the judges or jurymen which the 

 theory supposes does not exist. 



" Quand un juge se trompe il y a pour cela des raisons r 

 il n'a pas reellement mis la main dans une urne ou le 

 hazard I'a mal servi. II a ajoute foi a une faux te- 

 moignage, le concours fortuit de plusieurs circonstances a 

 eveille a tort sa defiance, un avocat trop habile I'a emu, 

 de hautes influences peutetre I'ont ebranle. Ses collegues 

 ont entendu les memes temoins, on les a instruits des- 

 memes circonstances, le meme avocat a plaide devant 

 eux, on a tentc sur eux la meme pression." 



With equal force does M. Bertrand expose the futility 

 of the received reasoning by which it is pretended to deter- 

 mine the probability that the sun will rise to-morrow from 

 the fact that it has risen so many days in the past. 



These reflections are just and important ; but their 

 value is somewhat diminished by the fact that they have 

 been, for the most part, made by previous writers with 

 whom our author seems unacquainted. Thus Prof. 

 Lexis has more carefully considered the extent of the 

 error committed by Laplace and Poisson in applying to 

 male and female births and other statistics rules derived 

 from games of chance. The fundamental principles of 

 Probabilities have been more fully explored by Dr. Venn. 

 M. Bertrand, like Laplace, starts by defining the prob- 

 ability of an event as the ratio of the number of favour- 

 able cases to the number of possible cases. He does not 

 explain what constitutes a "favourable case " — that, when 

 a die is thrown, the probability of obtaining the 3 or 

 4 is one- sixth, because as a matter of fact each side in 

 the long run turns up once out of six times. Accordingly,, 

 when he argues that in a great number of trials each 

 event is most likely to occur with a frequency correspond- 



