NOTES. 395 



ting a cone. A cone is a solid figure, like a sugar-loaf, fig. 5, of which A 

 is the apex, AD the axis, and the plane BECF the base. The axis 

 may or may not be perpendicular to the base, and the base may be a 

 circle, or any other curved line. When the axis is perpendicular to the 

 base, the solid is a right cone. If a right cone with a circular base be cut 

 at ri-jlit a'ngles to the base by a plane passing through the apex, the sec- 

 tion will be a triangle. If the cone be cut through both sides by a plane 

 parallel to the base, the section will be a circle. If the cone be cut slanting 

 quite through both sides, the section will be an ellipse, fig. 6. If the cone 

 be cut parallel to one of the sloping sides, as A B, the section will be a 

 parabola, fig. 7. And if the plane cut only one side of the cone, and be not 

 parallel to the other, the section will be a hyperbola, fig. 8. Thus there 

 are five conic sections. 



NOTE 23, p. 5. Inverse square of distance. The attraction of one body 

 for another at the distance of two miles is four times less than at the 

 distance of one mile ; at three miles, it is nine times less than at one ; at 

 four miles, it is sixteen times less, and so on. That is, the gravitating 

 force decreases in intensity as the squares of the distance increase. 



NOTE 24, p. 5. Ellipse. One of the conic sections, fig. 6. An ellipse 

 may be drawn by fixing the ends of a string to two points, S and F, in a 

 sheet of paper, and then carrying the point of a pencil round in the loop 

 of the string kept stretched, the length of the strkig being greater than 

 the distance between the two points. The points S and F are called the 

 foci, C the center, SC or CF the eccentricity, A P the major axis, QD 

 the minor axis, and P S the focal distance. It is evident that the less the 

 eccentricity CS, the nearer does the ellipse approach to a circle; and 

 from the construction it is clear that the length of the string SmF.is 

 equal to the major axis PA. If T t be a tangent to the ellipse at TO, then 

 the angle TmS is equal to the angle t mF; and as this is true for every 

 point in the ellipse, it follows, that in an elliptical reflecting surface, rays 

 of light or sound coming from one focus S will be reflected by the surface 

 to the other focus F, since the angle of incidence is equal to" the angle of 

 reflection by the theories of light and sound. 



NOTE 25, p. 5. Periodic time. The time in which a planet or comet 

 performs a revolution round the sun, or a satellite about its planet. 



NOTE 26, p. 5. Kepler discovered three laws in the planetary motions 

 by which the principle of gravitation is established : 1st law, That the 

 radii vectores of the planets and comets describe areas proportional to the 

 time. Let fig. 9 be the orbit of a planet ; Fig. 9. 



then supposing the spaces or areas PSp, 

 p S a, aSb, &c. equal to one another, the 

 radius vector S P, which is the line joining 

 the centers of the sun and planet, passes 

 over these equal spaces in equal times, 

 that Is, if the line S P passes to Sp in one p 

 day, it wHl come to So in two days, to S b 

 in three days, and so on. 2d law, That the 

 orbits or paths of the planets and comets 

 are conic sections, having the sun in one of 

 their foci. The orbits of the planets and 

 satellites are curves like fig. 6 or 9, called 

 ellipses, having the sun in the focus 8. Three comets are known to 

 move in ellipses, but the greater part seem to move in parabolas, fig. 7, 

 having the sun in S, though it is probable that they really move in very 

 long flat ellipses; others appear to move in hyperbolas, like fig. 8. The 

 third law is, that the squares of the periodic times of the planets are pro- 

 portional to the cubes of their mean distances from the sun. The square 

 of a number is that number multiplied by itself, and the cube of a mnu 





