ai? On the Tides. 123 
show, therefore, the relation of the partial tides, and consequently 
__ the ratio of the actions of the heavenly bodies on the ocean; and 
__ by comparing it to the ratio of their masses, the increase pro- 
duced on it by the difference of their motion will be determined. 
This increase is almost insensible for the sun, on account of 
the slowness of its motion; but it is very evident for the moon, 
whose motion is thirteen times more rapid, and whose action 
on the sea is nearly three times greater. 
By comparing in the fourth book of the Mécanique Céleste, 
the observations of equinoctial and solstitial tides in the syzygies 
and the quadratures, I was led by this method to an increase of 
at least a tenth in the ratio of the action of the moon to that of 
the sun; but I remarked that an element so delicate ought to 
be determined by a greater number of observations. T ie col- 
lection of modern observations has procured me this advantage. 
These observations, employed in double number, confirm the in- 
crease indicated by the ancient observations, and they make it 
more than one-eighth. Another method founded on the com- 
parison of the tides towards the apogeum and perigeum of the 
moon, and applied to the ancient as well as modern observations, 
leads us also to a similar result.—'Thus the incr as of the action 
: of the heavenly bodies on the tides in the port of Brest ought 
5 not to leave any doubt. 
The results of observations being always susceptible of errors, 
it is necessary to know the probability that those errors are con- 
tained within given limits. It is conceived, and with truth, that 
the probability remaining the same, those limits are the more 
diminished as the observations are more numerous, and agree 
better with each other. But this general view of the subject 
is not sufficient to warrant the exactness of the results of obser- 
vations and the existence of regular causes which they seem to 
point out. Sometimes, indeed, it has induced us to seek for the 
cause of phenomena which were only the accidents of chance. 
The caleulation of probabilities can alone enable us to appre- 
ciate these objects, which renders its use of the highest import- 
ance in physical and moral sciences. The preceding researches 
atforded me an opportunity too favourable to be neglected, of 
wpplying the new formule which I have obtained in my Théorie 
analylique des Probabilités, to one of the grandest phenomena 
of nature. I there explain at full length the application I have 
_ made of it to the laws of the tides. My object has been, not 
_ only to confirm the truth of those laws, but to trace the wa 
which must be pursued in applications of this kind. Among 
these laws, the most delicate are those of the increase and de- 
crease of the tides towards their maximum and their minimum, 
and the influence which the declinations of the heavenly bodies 
é and 
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