NOTES. 483 



16'0697 feet are to the space a body would fall through at the distance of the 

 sun by the earth's attraction, as the square of the distance of the sun from the 

 earth to the square of the distance of the centre of the earth from its surface; 

 that is, as the square of 95,000,000 miles to the square of 4000 miles. And thus, 

 by a simple question in the rule of three, the space which the sun would fall 

 through in a second by the attraction of the earth may be found in parts of a 

 mile. The space the earth would fall through in a second, by the attraction 

 of the sun, must now be found in miles also. Suppose mn, fig. 4, to be the 

 arc which the earth describes round the sun in C, in a second of time, by the 

 joint action of the sun and the centrifugal force. By the centrifugal force 

 alone the earth would move from m to T in a second, and by the sun's attrac- 

 tion alone it would fall through T M in the same time. Hence the length of 

 T n, in miles, is the space the earth would fall through in a second by the sun's 

 attraction. Now, as the earth's orbit is very nearly a circle, if 360 degrees be 

 divided by the number of seconds in a sidereal year of 365J days, it will give 

 mn, the arc which the earth moves through in a second, and then the tables 

 will give the length of the line C T in numbers corresponding to that angle ; 

 but, as the radius C n is assumed to be unity in the tables, if 1 be subtracted 

 from the number representing C T, the length of Tn will be obtained; and, 

 when multiplied by 95,000,000, to reduce it to miles, the space which the earth 

 falls through, by the sun's attraction, will be obtained in miles. By this simple 

 process it is found that, if the sun were placed in one scale of a balance, it 

 would require 354,936 earths to form a counterpoise. 



NOTE 135, p. 67. The sum of the greatest and least distances, S P, SA, 

 fig. 12, is equal to PA, the major axis; and their difference is equal to twice 

 the excentricity C S. The longitude <Y^ S P of the planet, when in the point 

 P, at its least distance from the sun, is the longitude of the perihelion. The 

 greatest height of the planet above the plane of the ecliptic E N e n, is equal to 

 the inclination of the orbit P N A n to that plane. The longitude of the planet, 

 when in the plane of the ecliptic, can only be the longitude of one of the points 

 N or n ; and, when one of these points is known, the other is given, being 180 

 distant from it. Lastly, the time included between two consecutive passages 

 of the planet through the same node N or n, is its periodic time, allowance 

 being made for the recess of the node in the interval. 



NOTE 136, p. 68. Suppose that it were required to find the position of a 

 point in space, as of a planet, and that one observation places it in n, fig. 34, 

 another observation places it in n', another 



in n", and so on ; all the points n, n ', n", n'", 

 &c. being very near to one another. The 

 true place of the planet P will not differ 

 much from any of these positions. It is evi- 

 dent, from this view of the subject; that 

 P M, P n', P n", &c., are the errors of observ- 

 ation. The true position of the Planet P is 

 found by this property, that the squares of 

 the numbers representing the lines Pn,Pn ', 

 &c., when added together, is the least pos- 

 sible. Each line P n, Pn', &c., being the whole error in the place of the 

 planet, is made up of the errors of all the elements ; and, when compared with 



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