22 



MATHEMATICAL AND PHYSICAL SCIENCE. 



[Diss. VI. 



Probable 

 error. 



(86.) 



Probability 

 of Testi- 

 mony and 

 of Design. 



Laplace's 

 doctrines 

 questioned. 



method of interpolation will serve the same purpose, 

 as well as save a very tedious calculation. Perhaps too 

 much importance has been assigned to the " probable 

 mean error" of a single observation deduced from 

 manv individual errors by the same theory. If a 

 man* has but one shot at a target, it is perfectly un- 

 certain (by hypothesis) whether that shot be one of 

 his best, or one of his worst, or one of his middling 

 ones. But as there are more middling hits than 

 very bad or very good ones, there is a certain dis- 

 tance which it will be safe to bet (i. e. for which the 

 probability is ), that he will not exceed ; though it 

 would be a strange occurrence indeed, if he exactly 

 struck the ring in question. This is all that is meant 

 by the phrase " probable error ;" it is an entirely 

 artificial number, which serves to give a sort of nume- 

 rical value of the skill of the performer, but is other- 

 wise of no importance. Its application has sometimes 

 been strangely mistaken. Even the " rule of least 

 squares" is often misapplied, and empirical laws al- 

 together false have been deduced from it ; for it is 

 rare in practice that the chances of error of observa- 

 tion of a varying quantity are the same throughout 

 the limits of observation. 



But those applications of the doctrine of proba- 

 bility which pretend to give us a measure of our belief 

 in the constancy of natural laws, in the confidence due 

 to testimony and to the teachings of history, in the 

 proofs of design in cosmical arrangements, are 

 inquiries which, from their connection with meta- 

 physics, religion, and morals, have had a higher interest 

 for mankind at large than ordinary problems about 

 cards and dice. To these Laplace paid peculiar 

 attention ; and the reputation of his name tended to 

 create in others a belief that the analysis he so power- 

 fully wielded could communicate a portion of its 

 certainty to the data subjected to it, and gave a 

 currency to many of his conclusions to which we 

 believe them by no means entitled. Professor Play- 

 fair, one of the most ardent of Laplace's admirers, has 

 recorded (in a criticism in the Edinburgh Review, of 

 the works we are considering) his total dissent from 

 Laplace's doctrine that the transmission from age to 

 age of the historic record of a fact diminishes its credi- 

 bility in a geometrical ratio. But a Cambridge 

 mathematician and speculative philosopher of singu- 

 lar penetration, Mr Leslie Ellis, has most formally 

 assailed 1 the principle of nearly all Laplace's esti- 

 mates of our expectation of events arising from causes 

 unknown or assumed to be so, such, for instance, as that 

 a common cause determined the revolution of all the 

 planets in one direction. The subject of the meta- 

 physics of probability evidently requires a complete 

 reconsideration ; and, owing to the singular subtlety of 





the matter, it is one which few persons are competent 

 to handle. The state of health of Mr Ellis leaves us 

 little hope of his resuming the inquiry ; but two 

 eminent mathematicians, Mr De Morgan and Mr 

 Boole, have published considerable works chiefly 

 bearing upon it. 



As an implement bearing upon discovery in (87.) 

 science, the Calculus of Probabilities has as yet been P 

 of little service. Whilst Laplace tries to indicate how ^ 

 it guided his researches connected with planetary irre- covery. 

 gularities, every one sees at a glance that, with the 

 data before him, common sense must have outstripped 

 analysis. Laplace has called the doctrines of pro- 

 bability " good sense reduced to calculation." What 

 is to be feared is, that the calculation should outstrip 

 the good sense. 



V. Fifthly, We are to consider Laplace's charac- ( 88 

 ter as a general physicist and as a writer. 



In the former respect he stands in a higher posi- as a p hy- 

 tion than that usually attained by eminent analysts, sicist, and 

 In fact, if we compare him with any of his own as an au ~ 

 generation, we find him not only better acquainted 

 with physical principles, and mora scrupulous in 

 taking account of them in his mathematical discus- 

 sions, but even possessing skill and interest in ex- 

 periments. The Calorimeter for measuring the capa- 

 cities of bodies for heat was the joint invention of 

 Lavoisier and himself; at least their memoir does 

 not assign to either a predominant share in it, 2 and 

 their determinations of the expansions of the metals 

 by heat seem also to have been made in common. 

 His happy discovery of the chief or sole cause of the Memoirs on 

 discrepancy between the theoretical and observed Heat in 

 velocity of sound (due to the heat developed by com- ^ h mon 

 pression) would alone have given him a just reputa- Lavoisier. 

 tion, so anxiously had the matter been debated, and 

 so much was it involved in a purely mathematical 

 intricacy. Even in his great work on physical as- 

 tronomy he takes a peculiar pleasure in embracing 

 topics of terrestrial physics. We there find discussed 

 the theory of barometrical measurements, the ques- On Baro- 

 tion of atmospheric tides, the laws of capillary attrac- metrical 

 tion, and the constitution of the gases. As to the first Measure * 

 of these topics he made a practical improvement on capillary 

 the formulae of his predecessors, so that his rules are Attraction, 

 in fact still in use. As regards capillary attraction, & Astrono- 

 he was materially anticipated by Young, who evi- 

 dently considered his principles to have been pirated ; 

 yet his theory, though obscured by a display of re- 

 dundant mathematics, was a real improvement. His 

 theory of tides, and that of atmospheric refractions, 

 though closely connected with physical astronomy, 

 were in fact not less so with the doctrines of hydro- 



1 Cambridge Transactions, Vol. VIII. The writer of these pages has also given his reasons for dissenting from the argu- 

 ments of Michell (which have been sanctioned by the authority of Laplace) on an astronomical question as discussed by the 

 Theory of Probabilities. See Philosophical Magazine for December 1850. 



2 Dr Black, the discoverer of latent heat (who was probably well-informed), states in a letter to Watt that he believes 

 Laplace to have been the inventor. See Correspondence of Watt on the Decomposition of Water, p. 66. 



