PHYSICAL SCIENCE. 



XXX} 



the results vary not less than those obtained by 

 trigonometrical measurement. Mr Ivory, from a 

 critical examination of all the data, has concluded 

 the ellipticity to be -J-. In this decision, from 

 the consummate mathematical skill of the author 

 of it, we are disposed to coincide. 



3. Application of mathematics to calculate all 

 the disturbances introduced by gravitation into the 

 solar system. 



The first person who improved the Newtonian 

 theory of the moon was Calandrini, professor of 

 mathematics in Geneva, who superintended the 

 printing of the Jesuits' copy of the Principia in 

 1739 and 1742. He investigated, by a direct 

 method, the principal lunar equations, and like- 

 wise the smaller inequalities which Newton had 

 left undemonstrated. He revised the investiga- 

 tion of the motion of the apsides ; but his calcula- 

 tions only gave half the quantity derived from 

 observation. Dr Matthew Stewart, professor of 

 mathematics in Edinburgh, discovered the true 

 motion of the line of the apsides by a simple 

 geometrical procedure. And in 1749, Walmesley 

 produced a correct analytical investigation of the 

 motion of the lunar apogee, which he extended 

 and completed in 1758. 



Clairaut began his investigations of the lunar 

 theory in 1743. At first he was satisfied with 

 merely studying the Newtonian procedure, and 

 converting it into analytical expressions; but 

 he gradually pushed his investigations farther, 

 and in 1747 comprised all the subordinate motions 

 of the moon under the famous general problem 

 of the three bodies. But after prodigious la- 

 bour, his solution assigned for the variation of 

 the lunar apogee only half the measure estab- 

 lished by observation. Euler and D'Alembert 

 arrived at a similar conclusion about the same 

 time. Clairaut resumed the subject, and, after 

 incredible labour, obtained a result which ac- 

 corded perfectly with observation, and thus con- 

 firmed the simple law of gravity, as laid down 

 by Newton. The knowledge of this result in- 

 duced Euler to resume his investigations, and by 

 quite a different procedure he also obtained the 

 true variation of the lunar apogee. D'Alembert 

 pushed his calculations still farther, and ap- 

 proached still nearer the result of observation. 

 Thus the law of attraction was for ever estab- 

 lished on the secure basis of mathematical de- 

 monstration. 



This great point being settled, mathematicians 

 set themselves with eagerness to improve the 

 lunar tables, which were obviously of such im- 

 portance for finding the longitude at sea. Clair- 

 aut bestowed intense application on the subject, 

 and produced a set of lunar tables, distinguished 

 by their superior accuracy. Euler devoted to the 

 same task the whole of his unrivaled analytical 



skill. But Mayer was the astronomer who dis- 

 tinguished himself most in this important inves- 

 tigation. He had been appointed director of the 

 observatory of Gottingen in 1751, and laboured 

 with so much intensity that he shortened his days. 

 He derived the elements of his lunar tables from 

 a discussion of numerous eclipses and occulta- 

 tions. He borrowed little from theory, though 

 he preferred the arrangement of the elements 

 adopted by Euler. He was the -first person that 

 employed conditional equations to find the true 

 value of the co-efficients. His tables were inserted 

 in the Gottingen transactions; and after the most 

 careful corrections, he sent them in 1755 to Lon- 

 don for the patronage of the Board of Longitude. 

 At his death in 1762, he left two copies, greatly 

 improved, one of which his widow transmitted to 

 that scientific body. After protracted delibera- 

 tions, the sum of Jt'3000 was at last awarded to 

 his family, with a present of 300 to Euler for 

 his excellent formulas. But another more com- 

 plete copy having been afterwards presented, the 

 Board of Longitude bestowed an additional re- 

 ward of 2000 at the instance of Dr Maskelyne, 

 who zealously undertook the editing of those 

 tables in the year 1770. 



The next point to which mathematicians di- 

 rected their attention, was the investigation of 

 the disturbing influence or mutual perturbations 

 of the larger and nearer planets. Euler, in 1747, 

 sent to the Academy of Sciences a most inge- 

 nious memoir on the derangement of Saturn's 

 motion, occasioned by the superior attraction of 

 Jupiter. It was now that he discovered that 

 there exist really no secular equations, but that 

 all deviations from the regular course are strictly 

 periodical, and return always in the same order, 

 though separated at vast intervals. His first in- 

 vestigation was rather imperfect ; but four years 

 after he produced another dissertation, which 

 gained the double prize of the academy. He 

 found that the mean motions of Jupiter and Sa- 

 turn are equally subject to a very slow increase 

 or diminution, which alternates, however, in the 

 lapse of 15,000 years. He gained the prizes for 

 1754 and 1756, by his theory of the inequalities 

 in the earth's motion caused by the planets. He 

 discovered four small anomalies to result from 

 their combined attractions, though it was scarcely 

 possible, for want of proper data, to assign the 

 precise measures of these aberrations. He esti- 

 mated the mean progression of the aphelia at 

 12'' annually, and the diminution of the ob- 

 liquity of the ecliptic at 49" in a century. He 

 found that the eccentricities of the aphelia of 

 Jupiter and Saturn are periodical, and complete 

 their cycle in the space of 30,000 years. 



The same subject was discussed by Clairaut in 

 1757. By comparing his formula with the ob- 



