NOTES. 413 



of the distance, it is clear that 16-0697 feet are to the space a body would 

 fall through :it the distance of the sun by the earth's attraction, as the 

 square nf tlie distance of the sun from the earth to the square of the 

 distance of the center of the earth from its surface ; that is, as the square 

 <>t'i.i.(KM),OOU 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 attrac- 

 tioa of the sun must now be found in miles also. Suppose m x, 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 attraction alone it would fall through T n 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 365$ 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 TC in numbers corresponding to that angle; but as the radius 

 C it is assumed to be unity in the tables, if 1 be subtracted from the 

 number representing CT, the length of Tre wHl 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. 



XOTE 135, p. 58. The sum of the greatest and least distances, S P, S A, 

 fis. 1-2, is equal to PA, the major axis; and their difference is equal to 

 twice the eccentricity CS. The longitude T S P of the planet, when in 

 the point P, at its least distance from the sun, is the longitude of the peri- 

 helion. 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. 59. 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', Fig. 34. 



another hi n", and so on ; all the points 

 n, ;t', 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 evident, from this view of 

 the subject, that P n, P ', P n", &c. are 

 the errors of observation. The true posi- 

 tion of the planet P is found by this prop- 

 erty, that the squares of the numbers 

 representing the lines P n, P n', &.C., when, v ., 

 added together, are the least possible. 

 Each line P n, P n', &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 the 

 errors obtained from theory, it affords the means of finding each. The 

 principle of least squares is of very general application ; its demonstration 

 cannot find a place here ; but the reader is referred to Biot's Astronomy, 

 vol. ii. p. 203. 



NOTE 137, p. 61. An axis that, Sre. Fig. 20 represents the earth 



M :i a 



