14 



MATHEMATICAL AND PHYSICAL SCIENCE. 



[Diss. VI. 



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. If the 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. 1 The principal reason for this conclu- 

 sion, and its refutation, will be mentioned in the next 

 section. 

 (51.) The first person who perceived the probable sta- 



Laplace's ^lity O f the major axes and mean motions was not 

 share mthe T * , T > . , , .. . , . 



discovery. Lagrange but Laplace, who, in a paper published m 



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- 



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- (52.) 

 rems which limit the whole amount of the excentri- limits f 

 cities and inclinations of the orbits of the planetary tieTan" in- 

 system, showing that if once small, they must ever clinations 

 remain so ; and, in particular, that the most massive of plane- 

 planets of the system (Jupiter and Saturn) must also ary 01 

 undergo the most trifling variation in these respects. 

 In the case of the small planets 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. It also 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 excentricilies 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 

 have manifestly no tendency to affect the stability of 

 the system. The grand cycle of the Earth'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 chronologerstothe creation. 



These results may be considered as among the (54.) 

 most astonishing with which science brings us ac- 



' 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. 



