i\OTES. 



403 



revolves and takes the successive positions Cn, Co, C'6, &.C., the sines* 

 n p, aq, br, &LC. of the arcs 7/171, ma, mh, &c. increase, while the corres 

 ponding cosines ( ' /<. C q, C r, &c. decrease, and when the revolving radius 

 takes the position (.'</, ut right angles to the diameter g i, the sine be- 

 comes equal to the radius Cd, and the cosine is zero. After passing the 

 point (/. the contrary happens; for the sines eK, IV, &c. diminish, and 

 the cosines CK, C V, &.c. go on increasing, till at g the sine is zero, and 

 the cosine is equal to the radius C g. The same alternation takes place 

 through the remaining parts g A, A?/, of the circle, so that a sine or cosine 

 never can exceed the radius. As the rotation of the earth is invariable, 

 each point of its surface passes through a complete circle, or 360 degrees, 

 in twenty-four hours, at a rate of 15 degrees in an hour. Time, there- 

 fore, becomes a measure of angular motion, and vice versd, the arcs of a 

 circle a measure of time, since these two quantities vary simultaneously 

 and equably, and as the sines and cosines of the arcs are expressed in 

 terms of the time, they vary with it. Therefore, however long the time 

 may be, and how often soever the radius may revolve round the circle, 

 the sines and cosines never can exceed the radius ; and us the radius is as- 

 sumed to be equal to unity, their values oscillate between unity and zero. 



NOTE 77, p. 21. The small eccentricities and inclinations of the plan- 

 etary orbits, and the revolutions of all the bodies in the sarae direction, 

 were proved by Euler, La Grange, and La Place, to be conditions neces- 

 sary for the stability of the solar system. Recently, however, the peri- 

 odicity of the terms of the series expressing the perturbations was sup- 

 posed to be sufficient alone, but M. Poisson has shown that to be a mistake ; 

 that these three conditions are requisite for the necessary convergence 

 of the series, and that therefore the stability of the system depends on 

 them conjointly with the periodicity of the sines and cosines of each 

 term. The author is aware that this note can only be intelligible to the 

 analyst, but she is desirous of correcting an error, and the more so as the 

 conditions of stability afford one of the most striking instances of design 

 in the original construction of our system, and of the foresight and su- 

 preme wisdom of the Divine Architect. 



NOTE 78, p. 21. Resisting medium. A fluid which resists the motions 

 of bodies such as atmospheric air, or the highly elastic fluid called ether, 

 with which it is presumed that space is filled. 



NOTK 79, p. 22. Obliquity of the ecliptic. The angle e T q, fig. 11, be- 

 tween the plane of the terrestrial equator q T Q, and the plane of the eclip 

 tic E T e. The obliquity is variable. 



A'OTK 80, p. 2-2. Invariable p'ane. In the earth the equator is the ia- 



Fig. 20 



