1809.] 
ciples, and rigorously demonstrated every 
part of the seience. The law of areas 
Jed him to the consideration of 1a 
plane moving in a direction parallel to 
itself with the centre of the system, the 
position of which may be calculated ‘for 
any particular instant. ‘To a plane of 
this kind he refers the motions of the 
satellites of Jupiter, and thus he has 
been enabled to overcome the inextrica- 
ble difficulties of this particular system, 
which is ona small scale a representa- 
tion of the grand system of the universe, 
and which presents this advantage, that 
all-the changes, all the revolutions, are 
completed in it in periods infinitely 
shorter,and consequently more favorable, 
to the present enquiries. He has de- 
duced from observation the laws of 
Kepler, which serve him to prove the 
law of universal pravity. 
The geometricians of the last century, 
by framing for themselves methods of ap- 
proximation, have been enabled to re- 
duce to calcuiation the effects of at- 
traction. M. Lagrange had given new 
canons susceptible of still further illus- 
trations. M. Laplace has: made this 
problem the particular subject of his 
meditations ;. he discovered means for 
. Obtaining the secular equations, and for 
calculating separately the terms of all the 
orders to which it may be foreseen that 
integration may give a sensible value; 
means which have led him to the discovery 
of the equations for long periods, and to 
that of the secular equation of the moon, 
We shall dweil no longer on the ex- 
tract of the Celestial Mechanics; it 
will suffice to say, that in this work, 
every page of which displays the genius 
of analysis, the most fruitful of all in 
interesting applications, we everywhere 
meet with theories entirely belonging: to 
the author, or which he has appropriated 
to himself by the new forms which ‘he 
has given them. 
The author bas given a kind of trans- 
lation of it into common language, under 
the title of Exposition of the System 
of the World; in which, without using 
any calculations, he unfolds, to a reader 
who has some knowledge of geometry, 
the spirit of the methods, and the pro- . 
gress of the inventor. 
~ From these great problems of celestial 
physics, the author re-descends with 
equal success to phenomena, less im- 
posing, but not less difficult: thus he ex- 
plains the effects of capillary attraction, 
by two methods totally independent of 
each other, and which lead him to the 
same equations, MM. Legendre was the 
Progress of the Sctences. 199 
first who. demonstrated that the elliptic 
form alone could suit the equilibrium of 
a fluid mass impelled by a rotatory mo- 
tion, afl the molecules of which attract 
each other in the inverse ratio of the 
squares of the distances. By an equa-. 
tion, for which he was indebted to M.. 
Laplace, he proved that the same figure _ 
also suits spheroids covered with fluid 
plates or layers, and of densities varying 
according to any law whatever. He has, 
in short, extended his researches as.far 
as to heterogeneous spheroids which 
perform no revolution. 
The same equation has led M. Biot, 
by a very simple process, to several 
thearems of great generality, which he 
afterwards particularly applies to ellipti¢ 
spheroids. 3 
Lastly, the same equation, in the 
hands of M. Leyrange, has given the 
successive terms of the development of 
the perturbations, and this great geometri- 
cian has applied this method for the secu- 
lar equations to that of the moon, the ex. 
istence and greatness of which was first 
analytically ascertained by M. Laplace. 
We have hitherto alluded only to ra- 
tional mechanics: practical mechanics 
have however been honoured by useful 
inventions, which have revived our ma- 
nufactures so as to be in future nearly 
independent on foreign industry.. These 
valuable discoveries have not been de- 
scribed in any printed work within our 
knowledge, and we should have been 
fearful of disfiguring them by imperfect 
descriptions; but in our general report 
we have carefully collected all the in- 
formation which we could. procure: we 
can speak with more confidence of the 
watches for determining the longitude, 
which have obtained for Lewis Berthoud 
the prize of the institute, and Are raiaty of 
navigators; and mention the hydraulic 
ram of Montgolfier as a very ingenious 
‘Invention, the success of which appears 
certain, when a very large body of water 
is not required. Finally, amongst. the 
ideas approved of by the class of sciences, 
we shall name the pyriolophorus of MM, 
Lenieps, a new principle of motion 
which appears calevlated to produce the 
greatest effects; and the looms for trans« 
parent net-work, by M. Bellemere, wha, 
by rendering the movements of the Eng- 
lish loom much lighter, has found the 
means of forming a machine less ex- 
pensive by one half; the advantages of 
it have been already ascertained by two 
years’ experience, 
[To be continued in our next. | 
oT REM 
