(119.) 
Great work 
826 MATHEMATICAL AND PHYSICAL SCIENCE. [Diss. VI. 
give a more impressive idea of the condition at which 
Physical Astronomy has now arrived, than a glance at 
this mountain of intellectual labour, — especially 
as to the intricacy, inexpressible by words, of the 
motions of our own satellite. Here we have awork,not 
much smaller in appearance than the whole Mécanique 
Céleste, devoted to this one object. It is not super- 
fluous to add, that this is no chimerical undertaking 
—no curious puzzle—no learned trifling; the Lunar 
Theory is the grand basis of the Art of Navigation. 
The real and main use to mankind of our companion 
planet is a discovery of these latter ages: her cheer- 
ful and beneficial light, which all appreciate, and all 
enjoy, may almost be termed a secondary boon. 
The merit of M. Plana is not so much that of 
of the for- 20 original geometer, outstripping the theories of La- 
mer. 
place and Lagrange, as of a most intrepid and skilful 
calculator, who has contrived to place in complete 
order the whole mathematical and many of the arith- 
metical steps of the solution of one vast problem, ex- 
tending to some 2000 quarto pages. The calcula- 
tions he has made unaided! The details are ar- 
ranged in so lucid a manner, as to court enquiry 
from those interested in verifying them ; and though 
the readers of so abstruse and indeed repulsive a 
work must be few indeed, it has already proved 
of essential service in the way which was intended 
—the improvement of the lunar tables. The ap- 
proximations by series are in all cases carried to 
the fifth powers of small quantities, and, in some in- 
stances, to the seventh powers. Sir John Herschel 
has expressed in forcible and picturesque language 
the nature of what is wanting to the completeness 
of Laplace’s investigations and which M. Plana has 
supplied, as regards the theory of the moon’s motions: 
—‘“ In the Mécanique Céleste, we admire the elegance 
displayed in the alternate interlinking and develop- 
ment of formule, and exult in the power of the ana- 
lytical methods used; but when we come to the 
statement of numerical results, we quail before the 
vast task of filling in those distant steps ; and while 
cloud rolls after cloud in majesty and darkness, we 
feel our dependence on the conclusions attained 
to partake of superstitious trust, or of amicable 
confidence, rather than of clear and demonstrative 
conviction.”' The courage of M. Plana did not 
“quail” before the serried ranks of symbolic legions. 
He attacked them first, and finally became their com- 
mander. But more than this; on the same high 
authority, “his analysis is always graceful, his com- 
binations well considered, and his conceptions of the 
ultimate results to be expected from them perfectly 
just, and justified by the results when obtained.” 
I may here mention that M. Plana, in conjunc- 
tion with M. Carlini of Milan, undertook the calcula- 
tion of the moon’s motions from theory alone (that is, Lunar 
with only the fundamental constants required to be a 
given by observation), in compliance with a pro- Theory 
gramme issued by the French Academy of Sciences alone. 
on the proposition of Laplace; and that a French Peeler 
geometer, M. Damoiseau, on the same occasion, pro- easter 
duced an independent investigation with very elabo- 
rate and valuable tables founded thereon. 
M. Plana is the author of many other more (121.) 
circumscribed researches on important points in phy- 
sical astronomy, in geodesy, and in mathematical 
physics. 
M. Hansen, a German astronomer and analyst (122.) 
of great merit, has made the most recent considerable M. Hansen. 
improvement in the theory of the moon. We will 
first, however, allude to an invention which is ap- 
plicable to the whole theory of perturbation. 
We have seen, in the first section of this chap- _ (123.) 
ter, that the effects of perturbation may be considered bare 
either as applicable directly to the three co-ordinates parkas 
of the perturbed body, or to the variation of the ele- tions. 
ments of the orbit considered as instantaneously va- 
rying ; that the former method has the advantage of 
being in most cases, and especially in first approxi- 
mations, most direct ; the latter is most applicable 
to secular inequalities, and has a great recommenda- 
tion in the exact physical conceptions on which it is 
founded. M. Hansen has proposed a third method, 
a refinement, in fact, on that of Lagrange, in which, 
by an assumption purely mathematical and arbitrary, 
he throws the effect of perturbation entirely on the 
element of time, so that with the time so altered, and 
invariable elliptic elements, the conditions of pertur- 
bation may be satisfied, and the true place of the 
body may result from the calculation. Such is the 
general nature of the conception, which, to be carried 
out, requires the introduction of subsidiary terms, 
which serve to correct the latitude and radius vector. 
The advantages which are understood to follow from 
this highly artificial mode of proceeding are stated to 
be (1.) that the serieses expressing the perturbation of 
the co-ordinates are more convergent than in the 
other methods, and likewise the coefficients of the 
terms to be retained are more easily calculated; (2.) 
M. Hansen considers that his method enables him 
to ascertain with certainty those terms which, when 
fully calculated, will affect the result in a sensible 
manner. The inventor has applied his methods both 
to the Lunar and to the Planetary theory. 
We have said that physical astronomy is indebted (124.) 
to M. Hansen for a notable improvementin the theory ee ey 
of the Moon,—the discovery, in fact, of two inequali- oqualtiiins 
ties of long period, the existence of which had been 
more than suspected from observation, but which were 
1 Astronomical Society's Monthly Notices, vol. v., p. 37. The last expressions may astonish some persons, but experienced ana- 
lytical calculators agree in the same view. Mr Airy (probably the most competent authority in Britain) states the same thing 
in many passages of his writings, to the effect that the evidence for conclusions so obtained is rather that of moral than of ma- 
thematical certainty. 
