Nov. 7. 1889] 



NATURE 



ing to its probability, he lays himself open to the charge 

 of circularity which Dr. Venn has brought against Ber- 

 nouilli's theorem. Without pronouncing on this delicate 

 ■question, we may safely say, with respect to the first 

 principles of the subject, that no point which has been 

 left obscure by Dr. Venn has been cleared up by M. 

 Bertrand. 



It is with respect to the purely mathematical portion of 

 the calculus, or that part of its metaphysics which is 

 inextricably mixed with mathematics, that we expected 

 and have found most assistance from M. Bertrand. 

 Hitherto the study of Probabilities has been barred by 

 the dilemma which M. Bertrand thus states: — 



" On ne peut bien connaitre le calcul des probabilit^s 

 sans avoir lu le livre de Laplace ; on ne peut lire le livre 

 de Laplace sans s'y preparer par les Etudes mathdmatiques 

 les plus profondes." 



Much of Laplace's analysis which must have affected 

 many eager students like stickjaw has been simplified 

 by M. Bertrand. He is in general more readable than 

 Poisson. Several of the theorems which he gives seem 

 to be new. His methods of determining from a given set 

 of observations the characteristic, or modulus, appertain- 

 ing to the source of error are specially interesting. 



M. Bertrand's mathematical power enables him to carry 

 the torch of common-sense to those perplexed parts of the 

 subject where less qualified critics, awed by the imposing 

 mass of symbols, have hesitated to differ from Laplace or 

 Poisson, Of this kind is the simultaneous determination 

 of several quantities from a great number of equations. 

 When Laplace computes that the odds are a million to 

 one against the occurrence of an error of assigned magni- 

 tude in the determination of Jupiter's mass, M. Bertrand 

 shows reasons for suspecting the accuracy of such com- 

 putations. In fact, he carries out Poinsot's witty direction ; 



" Apr^s avoir calcule la probability d'une erreur il 

 faudrait calculer la probabilitd d'une erreur dans le 

 calcul." 



The true import and proper application of the theory of 

 errors of observation are thus well expressed : — 



" On peut accepter sans crainte le rdsultat, mais il est 

 temdraire d'dvaluer en chiffres la confiance qu'il doit 

 inspirer." 



M. Bertrand teaches with authority — and not like those 

 who have not followed the higher mathematical reason- 

 ings of the calculus — in what spirit its conclusions should 

 be accepted. 



Still, even with regard to those parts of the subject 

 where a first-rate mathematician has so great an advant- 

 age, we venture to think that the work would have been 

 much more valuable if the writer had taken the trouble to 

 acquaint himself more fully with what his predecessors 

 had done. For example, in discussing the reasons for taking 

 the arithmetic mean of a set of observations (presumed to 

 be equally good) relating to a single quantity, M. Bertrand 

 does not dwell on the argument that the probability-curve 

 — with which the arithmetic mean is specially corre- 

 lated — is apt to represent the grouping of errors for this 

 reason, that an error may be regarded as a function of a 

 great number of elements each obeying some definite law 

 of facility, and that the values of such a function conform 

 to the probability-curve. It is true that Laplace, from 



whom this argument may be derived, has not himself 

 used it very directly. But in a writer on the method of 

 least squares we may expect some conversance with more 

 recent works, in particular with Mr. Glaisher's classical 

 paper in the Memoirs of the Astronomical Society 

 (London). Moreover, Laplace does employ the mathe- 

 matical theorem which we have indicated, not indeed to 

 prove that the law of facility for errors of observation in 

 general is the probability-curve, but that, whatever that 

 law of facility be, the most advantageous combination is a 

 certain linear function, A treatise in which this celebrated 

 argument is not discussed cannot be regarded as exhaust- 

 ive. But it is remarkable that with respect to the com- 

 bination of observations, M. Bertrand seems to defer 

 more to Gauss than to his own eminent countryman. 



M. I^ertrand has indeed slipped in a doctrine for which 

 the authority of Laplace may be quoted, that in choosing 

 the best combination of a set of observations "there is 

 an essential difference between the most probable 

 value of a quantity and the value which it is best to 

 adopt" (Bertrand, Art. 138) ; the latter being the mean 

 (first power) of the observations (Art. 155)— which M. 

 Bertrand rather awkwardly terms " la valeur probable.'' 

 M. Bertrand does not seem to realize the gravity of the 

 assumption which is contained in the latter clause. Later 

 on he employs Gauss's criterion of erroneousness — namely, 

 the mean square of error. But the ground, nature, and 

 relation of these two principles are not very clearly 

 explained by the writer. With respect to the philosophical 

 foundation of the method of least squares he has not 

 superseded the necessity of studying Laplace. 



With these reservations, M. Bertrand's work may be 

 regarded as one of the most complete treatises on the 

 subject. Nowhere else are the two elements so pecu- 

 liarly combined in the science of Probabilities — common- 

 sense and mathematical reasoning — to be found existing 

 together in such abundance. F. Y. E. 



ARGENTINE ORNITHOLOGY. 



Argentine Ornithology. By P. L. Sclater, Ph.D., F.R.S.^ 

 and W. H. Hudson, C.M.Z.S. Vol.11. (London : W, 

 H. Porter, 1889.) 



THE completion of this important work is an event 

 of considerable importance to every lover of neo- 

 tropical zoology, and the authors have both performed 

 their parts well, while the ten plates by Mr. Keulemans 

 are beautifully drawn and admirably coloured. Among 

 the increasing number of Englishmen who settle in the 

 Argentine Republic, there are sure to be many who will 

 pursue natural history studies, and to all such a well-exe- 

 cuted book like the present will be invaluable. The joint 

 authors of the work are happy in their association, for 

 while Dr. Sclater brings to the work a vast experience, 

 and a sound scientific knowledge of his subject, it is 

 certain that never was there a better describer of the 

 habits of birds than Mr. Hudson. Although of English 

 parentage, he is a native-born Argentine, and he has 

 grown up among the birds whose life and history he so 

 well knows how to portray. In turning over the pages 

 of this volume, we have found many interesting extra:ts 

 which we should have liked to present to our readers* 



