CHAP. II., 2.] 



PHYSICAL ASTRONOMY LAPLACE. 



21 



ties. 



called attention. 1 It is very satisfactory to find that, 

 by their independent and very different modes of 

 analysis, Mr Airy and Dr Young have arrived at 

 results generally coincident. It is in the essay 

 of the former that most readers will now seek for an 

 acquaintance with Laplace's abstruse investigations, 

 whilst they will find in it the bearing of experiments 

 more recent than the time of Young on the propa- 

 gation of waves in canals, the theory of Mr Airy, 

 beginning as it were at the opposite end from that of 

 Laplace, and offering far more points of contact with 

 actual observation, particularly in the Tides of 

 Rivers and Estuaries. The theory of Young will 

 naturally be best studied in his own article TIDES, 

 in this Encyclopaedia. 



(83.) IV. In the fourth place, we connect the name of 



Pbabili- "- ja P^ ace ^th the progress, during the period we are 

 considering, of the curious doctrine of probability, 

 or the laws of chance and expectation. These he 

 discussed in two works, the Th4orie Analytique and 

 the Essai Philosophique sur les Probabilite"s the first 

 the most mathematically profound, the last the most 

 popular and elegant, account of the subject which 

 has yet been given. Nearly all mathematicians are 

 agreed on two points first, in considering this the 

 " most subtle" and " difficult to handle" of all the 

 applications of their science, involving a perpetual 

 recurrence to contingencies, and to elements of the 

 argument easily left out of account, and in which, 

 more than in any other, it is dangerous to let sleep 

 for a moment the severely reasoning faculty, or per- 

 mit it to be lulled to security amidst the maze of 

 symbolic transformation. In truth, from experience, 

 I am disposed to receive with doubt the solution of 

 even a tolerably simple problem of chances, unless 

 two competent persons at least have concurred in 

 verifying it. Secondly, Mathematicians are agreed 

 in considering Laplace's The"orie nearly, if not quite, 

 the ablest specimen of mathematical writing of his 

 age, notwithstanding a degree of obscurity and repe- 

 tition in addition to the inherent abstruseness of the 

 subject, which render it, in the opinion of one of the 

 most learned and extensively read of our pure ma- 

 thematicians, 2 " by very much the most difficult ma- 

 thematical work he ever met with." 



A single paragraph has been devoted to the subject 

 of probabilities in Sir John Leslie's Dissertation, 

 relating to its earlier history ; and the subject was so 

 popular during the last century, that there was 

 scarcely an eminent mathematician who did not add 

 something to its practical development ; so that La- 

 place may be considered rather to have enlarged 

 widely its applications by means of his almost unex- 

 ampled power in handling the calculus, than to have 

 improved or established its first principles, or even 



(84.) 



Improve- 

 ments in its 

 investiga- 

 tion by 

 Laplace. 



applied it to classes of problems altogether new. 

 We find that most of the principles of the Calculus 

 were established by James Bernouilli, in the earliest 

 part of the eighteenth century, who gave the first History of 

 application of the Binomial Theorem to determine theDoctrin( 

 the probability of a particular combination of a given 

 number of things occurring, in preference to all the 

 other equally possible combinations. Stirling dis- 

 covered a curious theorem for approximating to the 

 continued product of the arithmetical series of num- 

 bers carried to any extent, which perpetually occurs 

 in such calculations. Demoivre carried out Halley's 

 application of it to the laws of mortality. Condorcet 

 applied it to moral questions ; Mitchell to natural 

 phenomena, considered as the results of accident or 

 design ; Lagrange to errors of observation. The 

 chief applications of the Theory of Probability are Itschief ap. 

 the following : 1. To chances known a priori, as plications, 

 that of throwing two given numbers with dice, the 

 whole range of possibilities being known with preci- 

 sion. 2. The calculation of the expectation of future 

 events on a great or average scale, deduced from the 

 past course of events observed also on a great or 

 average scale. Of this description are the calcula- 

 tions of life assurance, first tabulated by Halley. 3. 

 To find the most probable result of a number of in- 

 dependent observations and problems of a like kind. 

 4. To the proof of causation as opposed to accident 

 or " random," derived from existing combinations of 

 facts. 5. To the probability of testimony, and the 

 confidence due to legal decisions. None of these 

 inquiries are peculiar to Laplace, or originated with 

 him. We select, however, for a brief notice (which 

 must be confined to a few sentences) the third and 

 fourth of these applications. 



The chances are enormous against the most expe- (85.) 

 rienced marksman's hitting the bull's eye of a target. To find th 

 But if he make many shots in succession, the balls a le ^" 

 will be lodged round about the spot at which he suit of a 

 aimed, and they will be fewer in each successive ring number of 

 of equal area drawn round the mark. The degree ^ n e t P o^ s " er . 

 of their scattering will depend upon the skill of the vations. 

 marksman ; but in all cases the most probable result 

 will be, that the point aimed at is the centre of gravity 

 of the shots. This may be shown to be equivalent to 

 saying that the most probable result of any number of 

 equally reliable observations is that which will make 

 the sum of the squares of the outstanding errors a 

 minimum. This rule was conjecturally proposed by Legendre's 

 Legendre in 1806. A demonstration of its truth was method of 

 first published by Laplace. It is of great practical 

 use in deducing the results of complex observations, 

 such as those of Astronomy, and generally in com- 

 bining " equations of condition" more numerous than 

 the quantities whose value is sought to be extracted 

 from them. In very many cases, however, a graphical 



1 See note to p. 262 of the Second Volume of " Young's Miscellaneous Works," by Peacock. 



2 Professor De Morgan in Encyclopaedia Metropolitana. 



