CHAP. II. , 2.] 



PHYSICAL ASTRONOMY LAPLACE. 



19 



Laplace's 



Dynamical 



Theory. 



(73.) 

 diffi- 



(75.) 

 Three 

 classes of 

 Tides. 



dynamics than of astronomy, and to estimate all the 

 causes of movement of the particles of a heavy fluid, 

 surrounding a spheroidal rotating nucleus exposed to 

 the attractions of the sun and moon. This he did in 

 a series of memoirs, more systematically condensed in 

 the Traite" de Mecanique Celeste, and it may safely 

 be affirmed that no other mathematician of his day 

 was equal to the labours and disappointments of an 

 investigation attended with every species of difficulty, 

 in which each result must be attained by a combina- 

 tion of general sagacity with mathematical rigour, and 

 for the verification of which observations were yet in 

 a great measure wanting. The Theory of the Tides 

 was, upon the whole, the most arduous and compli- 

 cated problem which could well be conceived, in a 

 branch of science (hydrodynamics) hitherto remark- 

 ably little successful in predicting the results of the 

 most simple and arbitrarily selected experiments. 



That Laplace has been in a measure successful 

 in such an undertaking must be considered the highest 

 test of his genius, especially in reducing his mathe- 

 matics to practical application ; but the result has 

 been a treatise so profound and obscure (I mean as 

 regards the tide theory), that very few persons have 

 attempted to master its difficulties. Mr Airy, the 

 present astronomer royal, has done a great service 

 to men of science, and to that far wider community 

 whom the laws of the tides nearly interest, by giving 

 a connected and tolerably elementary view of La- 

 place's investigation, which he states confidently to 

 be " the most obscure of the Mtcanique Celeste" 



In this theory the figure of the ocean at any 

 moment is considered as a dynamical problem ; and 

 that figure as a momentary state arising from the in- 

 ternal movements of the fluid itself, as well as from 

 the variation of the external forces. The resulting 

 differential equations, expressing the attractions of 

 the sun, moon, and earth, the rotatory movement of 

 the earth, and the pressure of the water itself in mo- 

 tion, are abundantly complex, and the solutions only 

 partial and imperfect. The inferences from these 

 solutions, too, partake not only of their imperfection, 

 but, since they take no cognisance of the irregular 

 distribution of land and water, present cases almost 

 impossible to verify by observation. Some of the 

 results are indeed so paradoxical, that without bet- 

 ter evidence of their truth we do not further allude 

 to them. 



The tidal effects are divided by Laplace into three 

 classes ; the distinction of which, however, cannot be 

 called a discovery of his. The first class are inde- 

 pendent of the earth's rotation, and are practically 

 insignificant. The second class includes the diurnal 

 tide occurring once in about 24 hours. Concerning 



it, Laplace draws this conclusion, that its rise and fall 

 (not, however, its horizontal motion) are insensible if 

 the depth of the ocean is uniform ; x and being practi- 

 cally insensible in moat latitudes, we have thence an 

 argument of more or less weight for a general 

 tendency to uniformity in the depth of the sea. The 

 third class of tides are the ordinary semi-diurnal 

 tides. They afford, as Newton acutely perceived, the 

 most direct and attainable measure of the relative at- 

 tractions of the sun and moon. We have the sum of 

 these attractions at the conjunctions and oppositions 

 of the luminaries, and the difference when they are 

 90 apart ; and a higher maximum when both bodies 

 are without latitude. From the observation of the 

 tides on the Severn near Bristol, Newton computed 

 the relative action of the moon and sun to be as 

 4*48 to 1 ; but this value is much too great, and 

 gave far too large a relative mass to the moon. The 

 result in the harbour of Brest, from observations made 

 under Laplace's direction, is about 2'90 to 1 ; and the 

 moon's mass ^th that of the earth, agreeing almost 

 identically with that deduced from the nutation of 

 the earth's axis caused by her attraction. Observa- 

 tions at London and Liverpool reduced by Sir John 

 Lubbock and Dr Whewell give about 2'66 to I. 2 



From the general theory of Laplace, the follow- (76.) 

 ing results have been deduced with confidence : (1.) Laplace's 

 That the stability of the ocean is secure, whilst the results - 

 density of the ocean is inferior to that of the earth 

 generally ; which it is about five or six times. (2.) 

 That the phenomenon of precession is not modified 

 by the fluid covering of the globe. 



In the application of his theory to special cases, (77.) 

 Laplace is compelled to have recourse to an assump- 

 tion entirely arbitrary namely, that the periodic 

 fluctuations, however otherwise modified by circum- 

 stances, recur in the same periods as the causes to 

 which they are due. In this manner he conciliates the 

 results of observation with his theory, which the latter 

 would have been altogether incompetent to predict. 



The general merits of Laplace's theory we will sum (78.) 

 up in the words of Mr Airy, who, of all his succes- Character 

 sors, has probably most attentively studied it : " If, [ a ^ in _ 

 putting from our thoughts the details of the investi- vestigation. 

 gation, we consider its general plan and objects, we 

 must allow it to be one of the most splendid works 

 of the greatest mathematician of the past age. To 

 appreciate this, the reader must consider first, the 

 boldness of the writer who, having a clear under- 

 standing of the gross imperfection of the methods of 

 his predecessors, had also the courage deliberately 

 to take up the problem on grounds fundamentally 

 correct (however it might be limited by suppositions 

 afterwards introduced) ; secondly, the general diffi- 



1 Dr Young asserts that the conclusion will not hold unless the depth be also evanescent. Laplace has shown in his Fifth 

 Book that the disappearance of diurnal tides will take place only when the nucleus is completely covered. 



2 These ratios, however, being found to depend upon the configuration of the coast or estuary, cannot be used directly to de- 

 termine the relative action of the sun and moon. See Phil. Tram., 1845, p. 42. 



