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 aw"ork,not 

 much smaller in appearance than the whole Mecanique 

 Celeste, 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- an 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 Mecanique Celeste, we admire the elegance 

 displayed in the alternate interlinking and develop- 

 ment of formulae, 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." 1 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 ^ able 

 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 Carlmi 

 L -\/r T\ J.T Damoiseat 



geometer, M. Damoiseau, on the same occasion, pro- 

 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 - Hansei 

 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 Gen < 5r * 1 



,i T 11 T ,1 ,T i method of 



either as applicable directly to the three co-ordinates pe rturba- 



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 improvement in the theory !? wo ne . w 

 of the Moon, the discovery, in fact, of two inequali- equalities", 

 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. 



