1866.] Adams’ Recent Astronomical Discovery. 515 
about 2rds of a mile. These changes will progress (but more and 
more slowly after attaining a certain maximum rate) for about 
23,950 years, when the sun’s distance from the centre of the orbit 
will be reduced to 303,200 miles, and the semi-minor axis will 
only differ from the semi-major axis by about 500 miles.* 
In such minute variations, thus slowly propagated, Laplace 
found the origin of the secular acceleration of the moon’s mean 
motion. The average effect of the sun’s action on the moon is to 
diminish her gravitation to the earth by one-179th part. The diminu- 
tion of the earth’s influence over the moon during the month varies 
inversely as the cube of the earth’s distance from the sun. Now in 
the development of this inverse cube in a series proceeding accord- 
ing to sines and cosines of the earth’s mean motion and its 
multiples, there is a term 3 ¢” (e' being the eccentricity of the 
earth’s orbit). The dimmution of the moon’s angular velocity 
contains the product zs x e¢”, and this product would be confounded 
with the mean angular velocity of the moon if e’ were constant. 
But e’ is continually decreasing; and the decrease of e' causes a 
diminisbed diminution—that is, an acceleration—of the moon’s 
mean angular velocity. 
Laplace estimated the acceleration at 10'-1816213 ¢# (¢ as 
before), a result which agreed well with ancient eclipses, though not 
quite perfectly. When Lagrange applied with the requisite substi- 
tutions the formule he had already obtaimed, he arrived at a result 
almost exactly coincident with that of Laplace. Damoiseau, Plana, 
and Carlini obtamed similar results; the largest estimate of the 
co-efficient being that of Damoiseau, who made it 10-72. Hansen 
obtained the value 11-93 (in 1842), corrected in 1847 to 11'°47, 
in 1857 to 12'"18, and recently to 12'°557. 
But it was not merely the combination of six of the greatest 
names in the history of mathematics which seemed to point to this 
question as settled beyond dispute. A comparison of the results 
of calculation (when the larger values 11” 93, 12-18, or 12'°56 
were used) with the records of ancient eclipses exhibited a corres- 
pondence so complete and exact, that no doubt could (or can) exist 
that the actual acceleration lies between 11'°5 and 13’°0. Six re- 
markable total or nearly total eclipses of the sun, and nineteen lunar 
eclipses recorded in the Almagest, are represented so closely by an 
* Although the minuteness of the ellipticity of the terrestrial orbit is patent to 
calculation of the simplest kind, we repeatedly see the variation of this ellipticity 
cited as a possible cause of the prevalence at long past epochs of tropical or arctic 
climates in the temperate zones of our earth’s surface. It may be readily shown 
that the greatest possible decrease of mean annual temperature due to this cause is 
less than one-7,000th part of the present mean annual temperature, and the greatest 
possible increase is also quite insignificant. Compare with the above-named 
difference of one-7,000th the difference of one-15th (460 times larger) between 
the total amount of heat received’in a summer’s day, in the northern and southern 
hemispheres. 
