A 
Cnapr. II., § 2.] 
called attention.t 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 Tinzs, 
in this Encyclopedia. 
(88.) 
Doctrine of 
Probabili- 
ties. 
IV. In the fourth place, we connect the name of 
Laplace with 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 Théorie Analytique and 
the Essai Philosophique sur les Probabilités—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—irst, 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 Théorie 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,? “ by very much the most difficult ma- 
thematical work he ever met with.” 
(84) A single paragraph has been devoted to the subject 
Improve- of probabilities in Sir John Leslie’s Dissertation, 
— relating to its earlier history; and the subject was so 
investi 
_ tion le popular during the last century, that there was 
Laplace. 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 
PHYSICAL ASTRONOMY—LAPLACE. 
819 
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 thePoctrine 
the probability of a particular combination of a given aoe 
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 Itschiefap- 
the following :—1. To chances known @ 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, T° find the 
But if he make many shots in succession, the balls veered ah 
will be lodged round about the spot at which he sult of a 
aimed, and they will be fewer in each successive ring nares of 
of equal area drawn round the mark, The degree qo cer. 
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 anal 
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 Encyclopadia Metropolitana. 
