NOTES. 453 



primaiy planets are greater in proportion as the cubes of the numbers 

 representing the mean distances of their satellites are gi'eater, and also in 

 proportion as the squares of their periodic times are less. 



NOTE 136, p. 55. As hardly anything appears more impossible than 

 that man should have been able to weigh the sun as it were in scales and 

 the earth in a balance, the method of doing so may have some interest. 

 The attraction of the sun is to the attraction of the earth as the quantity 

 of matter in the sun to the quantity of matter in the earth ; and, as the 

 force of this reciprocal attraction is measured by its effects, the space the 

 earth would fall through in a second by the sun's attraction is to 

 the space which the sun would fall through by the earth's attraction 

 as the mass of the sun to the mass of the earth. Hence, as many times 

 as the fall of the earth to the sun in a second exceeds the fall of the sun 

 to the earth in the same time, so many times does the mass of the sun 

 exceed the mass of the earth. Thus the weight of the sun will be known 

 if the length of these two spaces can be found in miles or parts of a mile. 

 Nothing can be easier. A heavy body falls through 16-0697 feet in a 

 second at the surface of the earth by the earth's attraction ; and, as the 

 force of gravity is inversely as the square of the distance, it is clear that 

 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 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 365J days, it will give m n, 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 T n 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 eai'ths to form a counterpoise. 



NOTE 137, p. 59. The sum of the greatest and least distances S P, S A, 

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

 twice the excentricity C S. The longitude cyD 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, 



