473 



GRAVITATION. 



GRAVITATION. 



474 



draws the sun through as many inches, or parts of an inch, in on 

 second of time as it draws the earth in the same time. 



(7.) The second rule is this: "Attraction is proportional to the 

 mass of the body which attracts, if the distances of different attracting 

 bodies be the same." 



(8.) Thus, suppose that the sun and Jupiter are at equal distances 

 from Saturn ; the sun is about a thousand times as big as Jupiter ; 

 then whatever be the number of inches through which Jupiter draws 

 Saturn in one second of time, the sun draws Saturn in the same time 

 through a thousand tunes that number of inches. 



(9.) The third rule is this : " If the same attracting body act upon 

 several bodies at different distances, the attractions are inversely 

 proportional to the square of the distances from the attracting body. 



(10.) Thus the earth attracts the sun, and the earth also attracts the 

 moon ; but the sun is four hundred times as far off as the moon, and 

 therefore, the earth's attraction on the sun is only Ts^sjth part of its 

 attraction on the moon ; or, as the earth's attraction draws the moon 

 through about ^th of an inch in one second of time, the earth's 

 attraction draws the sun through jjsfeooth of an inch in one second of 

 time. In like manner, supposing Saturn ten times as far from the sun 

 as the earth is, the sun's attraction upon Saturn is only one hundredth 

 part of his attraction on the earth. 



(11.) The same rule holds in comparing the attractions which one 

 body exerts upon another, when, from moving in different paths, and 

 with different degrees of swiftness, their distance is altered. Thus 

 Mars, in the spring of 1833, was twice as far from the earth as in the 

 autumn of 1832 ; therefore, in the spring of 1833, the earth's attrac- 

 tion on Mars was only one-fourth of its attraction on- Mars in the 

 autumn of 1832. Jupiter is three times as near to Saturn, when-they 

 are on the same side of the sun as when they are on opposite sides; 

 therefore, Jupiter's attraction on Saturn, and Saturn's attraction on 

 Jupiter, are nine times greater when they are on the same side of the 

 sun than when they are on opposite sides. 



(12.) The reader may ask, How is all this known to be true ? The 

 best answer is, perhaps, the following : We find that the force which 

 the earth exerts upon the moon bears the same proportion to gravity 

 on the earth's surface which it ought to bear in conformity with the 

 rule just given. For the motions of the planets, calculations are made, 

 which are founded upon these laws, and which will enable us to predict 

 their places with considerable accuracy, if the laws are true, but 

 which would be much in error if the laws were false. The accuracy 

 of astronomical observations is carried to a degree that can scarcely be 

 imagined ; and by means of these we can every day compare the 

 observed place of a planet with the place which was calculated before- 

 hand, according to the law of gravitation. It is found that they agree 

 so nearly as to leave no doubt of the truth of the law. The motion of 

 Jupiter, for instance, is so perfectly calculated, that astronomers have 

 computed ten years beforehand the time at which it will pass the 

 meridian of different places, and we find the predicted time correct 

 within half a second of time. 



SECTION II. On the Effect of Attraction upon a Body which u in motion, 



and on 'the Orbital Revolutions of Planets and Satellite!. 

 (13.) We have spoken of the simplest effects of attraction, namely, 

 the production of pressure, if the matter on which the attraction acts 

 is supported (as when a stone is held in the hand), and the production 

 of motion if the matter is set at liberty (as when a stone is dropped 

 from the hand). And it will easily be understood, that when a body 

 is projected, or thrown, in the same direction in which the force draws 



the motion of the stone may be calculated with the utmost accuracy 

 from the following rule, called the second law of motion (the accuracy 

 of which has been established by many simple experiments, and many 

 inferences from complicated motion). If A, fig. 1, is the point from 

 which the stone was thrown, arid A B the direction in which it was 



Fig. 1. 



thrown ; and if we wish to know where the stone will be at the end of 

 any particular time (suppose, for instance, three seconds), and if the 

 velocity with which it is thrown would, in three seconds, have carried 

 it to B, supposing gravity not to have acted on it ; and if gravity would 

 have made it fall from A to c, supposing it to have been merely dropped 

 from the hand ; then, at the end of three seconds, the stone really will 

 be at the point D, which is determined by drawing B D parallel and 

 equal to A c ; and it will have reached it by a curved path A D, of which 

 different points can be determined in the same way for different in- 

 stants of time. 



(16.) The calculation of the stone's course is easy, because, during 

 the whole motion of the stone, gravity is acting upon it with the same 

 force and in the same direction. The circumstances of the motion of 

 a body attracted by a planet, or by the sun (where the force, as we 

 have before mentioned, is inversely proportional to the square of the 

 distance, and therefore varies as the distance alters, and is not the 

 same, either in its amount or in its direction, at the point D as it is at 

 the point c), cannot be computed by the same simple method. But 

 the same method will apply, provided we restrict the intervals for 

 which we make the calculations to times so short, that the alterations 

 in the amount of the force, and in its direction, during each of those 

 times, will be very small. Thus, in the motion of the earth, as affected 

 by the attraction of the sun, if we used the process that we have 

 described, to find where the earth will be at the end of a month from 

 the present time, the place that we should find would be very far 

 wrong ; if we calculated for the end of a week, since the direction of 

 the force (always directed to the sun) and its magnitude (always pro- 

 portional inversely to the square of the distance from the sun) would 

 have been less altered, the circumstances would have been more similar 

 to those of the motion of the stone, and the error in the place that we 

 should find would be much less than before; if we calculated by this 

 rule for the end of a day, the error would be so small as to be per- 

 ceptible only in the nicest observations ; and if we calculated for the 

 end of a minute, the error would be perfectly insensible. 



(17.) Now a method of calculation has been invented, which amounts 

 to the same as making this computation for every successive small 

 portion of time, with the correct value of the attractive force and the 

 correct direction of force at every particular portion of time, and finding 

 thus the place where the body will be at the end of any time that we 

 may please to fix on, without the smallest error. The rules to which 

 this leads are simple : but the demonstration of the rules requires the 

 artifices of advanced science. We cannot here attempt to give any 

 steps of this demonstration ; but our plan requires us to give the 



it (as when a stone is thrown downwards), it will move with a greater 

 velocity than either of these causes separately would have given it ; 

 and if thrown in the direction opposite to that in which the force 

 draws it (as when a stone is thrown upwards), its motion will become 

 slower and slower, and will, at last, be turned into a motion in the 

 opposite direction. We have yet to consider a case much more im- 

 portant for astronomy than either of these : Suppose that a body is 

 projected in a direction tramvene to, or craning, the direction in which 

 the force draws it, how will it move ? 



(14.) The simplest instance of this motion that we can imagine is 

 the motion of a stone when it is thrown from the hand in a horizontal 

 direction, or in a direction nearly horizontal. We all know that the 

 stone soon falls to the ground ; and if we observe its motion with the 

 least attention, we see that it does not move in a straight line. It 

 begins to move in the direction in which it is thrown ; but this direc- 

 tion is speedily changed ; it continues to change gradually and con- 

 stantly, and the stone strikes the ground, moving at that time in a 

 direction much inclined to the original direction. The most powerful 

 effort that we can make, even when we use artificial means (as in pro- 

 ducing the motion of a bomb or a cannon-ball), is not sufficient to 

 prevent the body from falling at last. This experiment therefore will 

 not enable us immediately to judge what will become of a body (as a 

 planet) which is put in motion at a great distance from another body, 

 which attracts it (as the sun) ; but it will assist us much in judging 

 generally what is the nature of motion when a body is projected in a 

 direction transverse to the direction in which the force acts on it. 



(15.) It appears then that the general nature of the motion is this : 

 the body describes a curved path, of which the first part has the same 

 direction aa the lino in which it is projected. The circumstances of 



results. 



(18.) It is demonstrated that if a body (a planet for instance) is by 

 some force projected from A., fig. 1, in the direction AB, and if the 

 attraction of the sun, situated at s, begins immediately to act on it, 

 and continues to act on it according to the law that we have mentioned 



Fig. 2. 



(that is, being inversely proportional to the square of its distance 

 from B, and always directed to s) ; and if no other force whatever but 

 this attraction acts upon the body; then the body will move in 

 one of the following curves a circle, an ellipse, a parabola, or a 

 hyperbola. 



In every case the curve will, at the point A, have the same direction 

 as the line AB : or (to use the language of mathematicians), AB will be 

 a tangent to the curve at A. 



The curve cannot be a circle unless the line A B is perpendicular to 

 s A, and, moreover, unless the velocity with which the planet is pro- 

 jected is neither greater nor less than one particular velocity deter- 

 mined by the length of s A and the mass of the body s. If it differs 

 little from this particular velocity (either greater or less), the body will 



