812 
ments (the sixth being the longitude of the planet on 
its orbit at a given time) may conveniently be con- 
sidered thus: 1st, the major axis, which, for one and 
the same system, involves the periodic time or mean 
motion; 2d, the excentricity and position of the line of 
apsides; 3d, the inclination and position of the line of 
nodes, Of these, the stability primarily depends upon 
the first. Ifthe major axis and mean period increase 
or diminish without limit, the planets will diverge 
into infinite space, or rush after myriads of ages to 
utter annihilation in the burning embrace of the sun. 
The latter alternative was the popular belief about 
the middle of last century, and was maintained by 
the grave authority of Euler; whilst Darwin, in his 
florid but picturesque language, described the order 
and beauty of the planetary system as but a little 
more permanent than the glowing ornaments of the 
gay parterre.!_ The principal reason for this conelu- 
sion, and its refutation, will be mentioned in the next 
section. 
(51.) The first person who perceived the probable sta- 
arene, bility of the major axes and mean motions was not 
discovery. Lagrange but Laplace, who, in a paper published in 
1773, gave a demonstration, the sufficiency of which 
has not been doubted, that the major axes are in- 
variable, so far as the influence of the principal 
terms of the disturbances are concerned, that is as 
far as terms containing the cubes of the excentrici- 
ties inclusive, and the first powers of the perturbing 
masses. Nor does Laplace appear to have doubted 
that the mutual distinction of the terms, including 
secular changes, was not accidental, but would ex- 
tend also to the farther approximations. Lagrange, 
however, in a celebrated though short memoir of 
1776, demonstrated the truth of the conclusion for 
the higher powers of quantities contained in the per- 
turbations of the first order, and that by methods 
peculiarly comprehensive and elegant, which he far- 
ther extended in 1781 to the other five orbital ele- 
ments, showing the periodicity within certain narrow 
limits of the excentricity and inclination, the only 
elements, except the major axis, whose variations 
menace the stability of the system. Yet it is quite 
impossible to separate completely the names of La- 
grange and Laplace in the effectual demonstration of 
this important truth, the former as frequently in- 
MATHEMATICAL AND PHYSICAL SCIENCE. 
[Diss. VI. 
dicating the means of overcoming the more purely 
mathematical difficulties, as the latter was suggestive 
and far-sighted in anticipating their application to 
the peculiarities of our system. 
Laplace discovered (1784) two remarkable theo- _ (62.) 
rems which limit the whole amount of the excentri- peace 
cities and inclinations of the orbits of the planetary eraae por 
system, showing that if once small, they must ever clinations 
remain so; and, in particular, that the most massive °f plane- 
planets of the system (Jupiter and Saturn) must also “*Y oTPits: 
undergo the most trifling variation in these respects. 
In the case of the small pianets between Mars and 
Jupiter, a wider range may occur (as indeed we prac- 
tically find to be the case), without endangering the 
permanency of the whole. Italso follows that these 
variations, though “secular,” are practically “ perio- 
dic ;” that is, that the excentricities and inclinations 
oscillate about certain mean values and within ex- 
tremely narrow limits, the periods of these oscillations 
being also of vast duration. Concerning such changes, 
theory is our only guide, The whole duration of astro- 
nomical records can barely reveal the existence of two 
or three of them, and tells us absolutely nothing of 
their remoter consequences. Lagrange calculated 
the superior limits of the excentricities of the larger 
planets, and M. Leverrier has recently, by more ac- 
curate methods, obtained results nearly coincident. 
According to him, the maximum excentricity of the 
Earth’s orbit is 0:07775, the minimum 0°003314, so 
that it can never be quite a circle. It is now di- 
minishing, and will continue (according to the same 
geometer) to do so for 24,000 years, when it will 
begin to increase. The inclinations of the Earth’s 
orbit to its equator, and also to a fixed plane, are 
confined within definite limits which are not perhaps 
very perfectly known. 
The motions of the apsides and nodes of the orbits _(53.) 
which gradually complete the entire circumference era o 
have manifestly no tendency to affect the stability of ae = 
the system. he grand cycle of the Harth’s perihe- 
lion will only be completed in 110,000 years. It 
coincided with the vernal equinox 4089 years before 
Christ, a period (as Laplace remarks) nearly coinci- 
dent with that assigned by chronologers tothe creation. 
These results may be considered as among the 
most astonishing with which science brings us ac- 
(54.) 
1 Roll on ye stars! exult in youthful prime, 
Mark with bright curves the printless steps of Time, 
Near and more near your beamy cars approach, 
And lessening orbs on lessening orbs encroach ; 
Flowers of the sky! ye too to age must yield, 
Frail as your silken sisters of the field! 
Star after star from heaven’s high arch shall rush, 
Suns sink on suns, and systems systems crush, 
Headlong, extinct, to one dark centre fall, 
And Death and Night and Chaos mingle all! 
*Till o’er the wreck, emerging from the storm, 
Immortal Nature lifts her changeful form, 
Mounts from her funeral pyre on wings of flame, 
And soars and shines, another and the same.” 
Darwin’s Botanic Garden, Canto iv., line 367. 
