INTRODUCTION. xiii 



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 eclip- 

 tic 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 pass 

 through a complete cycle of values, are now reduced to periods which vary between 

 26,000 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 thousands of years. 

 This consideration is of much importance in the investigation of the reputed anti- 

 quity and chronology of those ancient nations which 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 required in order to pro- 

 duce changes of sufficient magnitude to be detected by observation, and we should 

 be unable to estimate the interval between the epochs of elements which differed 

 by only a few thousand years, since they would manifestly be so nearly identical 

 with our own that the value of legitimate conclusions would be greatly impaired 

 by the unavoidable errors of the observations on which they were based. 



The duration of the different seasons is also greatly modified by the eccentricity 

 of the earth's orbit. At present the sun is north of the equator scarcely 1863 days, 

 and south of the same circle about 178f days; thus making a difference of 7f days 

 between the length of the summer and winter at present. But when the eccen- 

 tricity of the orbit is nearly at its maximum, and its transverse axis also passes 

 through the solstices, both of which conditions have, in past ages, been fulfilled, 

 the summer, in one hemisphere, will have a period of 198| days, and a winter of 

 only 166 1 days, while, in the other hemispheres, these conditions will be reversed; 

 the winter having a period of 198f days, and a summer of only 166^ days. The 

 variations of the sun's distance from the earth in the course of a year, at such 

 times, are also enormous, amounting to almost one-seventh part of its mean distance 

 a quantity scarcely less than 13,000,000 of miles! 



Passing now to the consideration of the elements of the planet Mars, we find that 

 the eccentricity of his orbit always oscillates within the limits 0.018475 and 

 0.139655; and the mean motion of his perihelion is 17".784456. The maximum 

 inclination of his orbit to the fixed ecliptic of 1850, and to the invariable plane of 

 the planetary system, is 7 28' and 5 56' respectively. The minimum inclination 

 to both planes being nothing, the mean motion of the node is indeterminate. 



The secular variations of the orbits of Jupiter, Saturn, Uranus, and Neptune 

 present some curious and interesting relations. These four planets compose a sys- 

 tem by themselves, which is practically independent of the other planets of the 

 system. 



The maximum and minimum limits of the eccentricity of the orbits o f these four 

 planets are as follows: 



