462 



NOTES. 



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. 6. 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. 6. 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 string being greater than the distance between 

 the two points. The points S and F are called the foci, C the centre, SC or 

 C F the excentricity, AP the major axis, Q D the minor axis, and PS the focal 

 distance. It is evident that, the less the excentricity 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< be a tangent 

 to the ellipse at m, then the angle TmS is equal to the angle tmF; and, as 

 this is true for every point in the ellipse, it follows that, in an elliptical reflect- 

 ing 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. 6. Periodic time. The time in which a planet or comet per- 

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



NOTE 26, p. 7- Kepler discovered three laws in the planetary motions by 

 which the principle of gravitation is established: 1st law, That the radii vec- 

 tores of the planets and comets describe areas proportional to the time. Let 

 fig. 9 be the orbit of a planet; then, supposing Fig. 9. 



the spaces or areas PSp,p So, aSb, &c., equal 

 to one another, the radius vector S P, which is 

 the line joining the centres of the sun and 

 planet, passes over these equal spaces in equal 

 times, that is, if the lines S P passes to Sp in 

 one day, it will come to S a in two days, to S 6 

 in three days, and so on. 2nd 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 S. Three 

 comets are known to mo vein ellipses; but the greater part seem to move in para- 

 bolas, 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 proportional 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 number is that number twice 

 multiplied by itself. For example, the squares of the numbers 2, 3, 4, &c., are 

 4, 9, 16, &c., but their cubes are 8, 27, 64, &c. Then the squares of the numbers 

 representing the periodic times of two planets are to one another as the cubes 



