urging it to rise from the center; the two on an average balan- CHAP. I. cing each other, retains the moon in an orbit about the earth. Now what and where is this force? Is it around the earth, or within the earth? Is it electrical or magnetic? or . is it that same force (call it what we may) that makes a body fall toward the earth's center when unsupported on a resting base? Isaac New A trifling incident, the fall of an apple from a tree, seems Contempla. to have led the mind of Newton to the contemplation of this tions of Sir force which compels and causes bodies to fall, and he at once conceived this force to extend to the moon and to cause it to deviate from the tangent of its orbit. The next consideration was, whether if this were the force, it was the same at the distance of the moon, as on the surface of the earth; or if it extended with a diminished amount, what was the law of diminution? ton. theory of Newton now resorted to computation, and for a test he Incipient conceived the force in question to extend to the moon, undi- steps to the minished by the distance; and corresponding thereto he de- gravity. cided that the moon must then make a revolution in its orbit in 10 h. 55 m. But the actual time is 27 d. 7h. 43 m., which shows that if the force is the same which pervades a falling body on the surface of the earth, it must be greatly diminished. tions. Now by making a reverse computation, taking the actual Important time of revolution, and finding how far the moon did really computafall from the tangent of its orbit in one second of time, it was found to be about a part of 16 feet-the distance a body falls the first second of time. But the distance to the moon is about 60 times the radius of the earth, and the inverse square of this is 6', which corresponds to the actual fall of the moon in one second. (151.) It is a well-established fact in philosophy, and A principle geometrically demonstrated, that any force or influence exist- in philosophy ing at a point, must diminish as it spreads over a larger space, and in proportion to the increase of space. But space increases as the square of linear distance, as we see by Fig. 28. CHAP. 1. A double distance spreads the influence over four times the space, whatever that influence may be; a triple distance, nine times the space, etc., the space increasing as the square of Fig. 28. The theory of universal gravity. This theory well lished. the distance. Therefore, any influence spreading in all directions from its central point must be enfeebled as the square of the distance. From observations and considerations like these, Newton established the all-important and now universally admitted theory of gravity. This theory may be summarily stated in the following words: Every body of matter in the universe attracts every other body, in direct proportion to its mass, and in the inverse proportion to the square of the distance. Some attempts have been made, from time to time, to call estab- the truth of this theory in question, and substitute in its place the influence of light, caloric, and electricity; but any thing like a close application shows how feebly all such substitutes stand the test. Attraction lar body. The theory of gravity so exactly accounts for all the physical phenomena of the solar system, that it is impossible it should be false; and although we cannot determine its nature or its essence, it is as unreasonable to doubt its existence, as to doubt the existence of animate beings, because we know nothing of the principle of life. (152.) According to the theory of gravity, every particle of an irregu- composing a body has its influence, and a very irregular body may be divided in imagination into many smaller bodies, and the center of gravity of each taken as the point of attraction, and all the forces resolved into one will be the attraction of the whole body. In a sphere composed of homogeneous particles, the aggre- CHAP. L gate attraction of all of them will be the same as if all were Attraction of compressed at the center; but this will be true of no other a sphere. body. The earth is not a perfect sphere, and two lines of attraction from distant points on its surface may not, yea, will not, cross each other at the earth's center of gravity. (See Fig. 10.) Attraction spherical shell. (153.) A particle anywhere inside of a spherical shell of equal thickness and density, is attracted every way alike, and inside of a of course would show no indication of being attracted at all. Hence a body below the surface of the earth, as in a deep pit or well, will be less attracted than on the surface, as it will be attracted only by the diminished sphere below it. At the center of the earth a body would be attracted by the earth every way alike, and there would be no unbalanced force, a sphere. and of course no perceptible or sensible attraction.* Attraction at the center of (154.) The attractive power on the surface of any perfect Expression and homogeneous sphere may be expressed by the mass of the for the sphere divided by the square of the radius. traction on the surface of Consider the earth a sphere (as it is very nearly), and a sphere. put E to represent its mass, and r its mean radius, then and it is sufficient to cause bodies to fall 16 feet during the first second of time. If the earth had contained more matter, bodies would have fallen more than 16 feet the first second; if less, a less distance. With the same matter, but more compact, so that r2 would The definite E p2 attraction of be less with E the same, would be greater, and the attrac- the earth. tive power at the surface greater, and bodies would then fall more than 16 feet the first second of their fall. Now we say this 16 feet is the measure of the earth's attraction at its surface, and it is made the unit and standard measure, directly or indirectly, for all astronomical forces. * See Robinson's Natural Philosophy, page 16. CHAP. I. To find the For this reason, we call the undivided attention to this force, the known-the noted the all-important 16 feet. (155.) By the theory of gravity, we can readily obtain an attraction of analytical expression for the attraction of a sphere at any disany distance. tance from the center, after knowing the attraction at the a sphere at An expres mutual traction of surface. For example. Find the value of the attraction of the earth, at the distance of D from its center; r being the radius of the earth, and g the gravity at the surface; put z to represent the attraction sought. Then by the theory, 1 r2 g:x:: : 1 D2 ; Or, x=9(1) (5) As g and r are constant quantities, the variations to x will correspond entirely to the variations of D2. We shall often refer to this equation. (156.) As every particle of matter in the universe atsion for the tracts every other particle, therefore the moon attracts the at- earth as well as the earth attracts the moon; and the extent two bodies. by which they will draw together, depends on their mutual attraction. If m represents the mass of the moon, and R the radius of the lunar orbit; then, The earth will attract the moon by the force The moon will attract the earth by the force E R2 m R2 The two bodies will draw together by the force E+m If we substitute the value of g, as found in (154), in equa tion (5), and making R=D, then we have the expression E The spirit of these expressions will be more apparent when we make some practical applications of them, as we intend soon to do. CHAPTER II. KEPLER'S LAWS-DEMONSTRATION OF THE SECOND AND THIRD- (157.) In this chapter we design to make some examina- Chap. II. tion of Kepler's laws, recapitulating them in order. Examina ler's laws. The orbits of the planets are ellipses, having the sun at tions of Kepone of their foci. This law is but a concise statement of an observed fact, which never could have been drawn from any other source than observation; but the second law, namely, That the radius vector of any planet (conceived to be in motion) sweeps over equal areas in equal times is susceptible of a rigid mathematical demonstration, under the following general theorem. Any body, being in motion, and constantly urged toward any fixed point, not in a line with its motion, must describe equal areas in equal times round that point. A general theorem. Let a moving body be at A, having a veloci Fig. 29. But during this second must obey an impulse or tance, in the next second of time. Its demonstration. |