SECULAR VARIATIONS OF THE PLANETARY ORBITS. 2G7 



the difference between the present value of precession and the maximum 

 and miiiiuuim values of tlie same quautit}', we shall tind that the tropi- 

 cal year may be shorter than at present by 59.13 seconds, and longer 

 than at present l>y 49.27 seconds. \Ye also find that the tropical year 

 is now shorter than in the time of llipparchus, by 11.30 seconds. 



The obliquity of the equator to the ai>parent ecliptic, and also to the 

 fixed ecliptic of 1850, has also been determined. The variations of this 

 element follow a law similar to that Avhich governs the variation of i)re- 

 cession, although the maximum values of the inequalities are consider- 

 ably smaller than those which affect this latter quantity. The mean 

 value of the obliquity of both the apparent and fixed ecliptics to the 

 equator is 23° 17' 17". The limits of the obliquity of the apparent ecliptic 

 to the equator are 21° 35' oS" and 21° 58' 3G"; w^hence it follows that the 

 greatest and least declinations of the sun at the solstices can never diHer 

 from each other to any gxeater extent than 2° 37' 22". And here we may 

 mention a few, among the many happy consequences, which residt from 

 the spheroidal form of the earth. Were the earth a perfect sphere there 

 would be no precession or change of obliquity arising" from the attraction 

 ofthesunand moon; the equinoctial circle would form an invariable 

 plane in the heavens, about which the solar orbit would revolve with an 

 inclination varying to the extent of twelve degrees, and a motion equal 

 to the planetary precession of the equinoctial points. The sun, when at 

 llie solstices, would, at some periods of time, attain the declination of 

 29^17', for many thousands of years; and again, at other periods, only 

 to 17^ 17'. The seasons would be subject to vicissitudes depending on 

 the distance of the tropics from the equator, and the distribution of solar 

 light and heat on the surface of the earth would be so modilied as essen- 

 tially to change the character of its vegetation, and the distribution of 

 its animal life. But the spheroidal form of the earth so modifies the 

 secular changes in the relative positions of the equator and ecliptic that 

 the inequalities of precession and obliquity" are reduced to less than one- 

 quarter part of what they would otherwise be. The periods of the secular 

 changes, which, in the case of a spherical earth, would require nearly 

 two millions of years to ])ass through a complete cycle of values, are now 

 reduced to periods which vary between 20,000 and 53,000 years. The 

 secular motions which would take place in the case of a spherical earth 

 are so modified by the actual condition of the terrestrial globe that 

 changes in the position of the equinox and equator are now produced in 

 a few centuries, which would otherwise require a period of many thou- 

 sands of years. This consideration is of much importance in the investi- 

 gation of the reputed antiquity and chronology of those ancient nations 

 wliicli attained proficiency in the science of astronomy, and the records 

 of whose astronomical labors are the only remaining- monument of a 

 highly intellectual people, of whose existence every other trace has long" 

 since passed away. For it is evident that, if these changes were much 

 slower than they are, a much longer time would be rcfiuired in order to 

 produce changes of sufficient magnitude to be detected by observation, 



