NOTES. 471 



compared with the motion of the node, as evidently appears from fig. 19, where 

 the angles npnf, n'p'n", Sac. are much smaller than the corresponding angles 

 nSn', Sn", &c. 



NOTK 76, p. 23. Sines and cosines. Figure 4 is a circle; np is the sine, and 

 Cp is the cosine of an arc mn. Suppose the radius Cm to begin to revolve at 

 m, in the direction mna; then at the point m the sine is zero, and the cosine 

 is equal to the radius Cm. As the line C m revolves and takes the successive 

 positions Cn, Ca, Cb, &c., the sines np, aq,br, &c. of the arcs mn, ma, mh, 

 &c. increase, while the corresponding cosines Cp, Cq, Cr, dec. decrease, and 

 when the revolving radius takes the position C d, at right angles to the diameter 

 gm, the sine becomes equal to the radius C d, and the cosine is zero. After passing 

 the point d, the contrary happens ; for the sines eK., I V, &c. diminish, and the 

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

 equal to the radius Cg. The same alternation takes place through the remain- 

 ing parts gh, hm, 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, therefore, becomes a measure of angular motion, 

 and vice versa, 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, as the radius is assumed 

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



NOTE 77, p. 24. The small excentricities and inclinations of the planetary 

 orbits, and the revolutions of all the bodies in the same direction, were proved 

 by Euler, La Grange, and La Place, to be conditions necessary for the stability 

 of the solar system. Recently, however, the periodicity of the terms of the 

 series expressing the perturbations was supposed to be sufficient alone, but M. 

 Poisson has shown that to be mistake; that these three conditions are re- 

 quisite 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 intelli- 

 gible 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 

 supreme wisdom of the Divine Architect. 



NOTE 78, p. 25. 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. 



NOTE 79, p. 26. Obliquity of the ecliptic. The angle e^Tq, fig. 11, between 

 the plane of the terrestrial equator q^T Q> and the plane of the ecliptic E<Y* e. 

 The obliquity is variable. 



NOTE 80, p. 27. Invariable plane. In the earth the equator is the invariable 

 plane which nearly maintains a parallel position with regard to itself while 

 revolving about the sun, as in fig. 20, where E Q represents it. The two hemi- 



