42 



NATURE 



[Nov. 13, 1879 



cessive chords are so many inclined planes, and the move- 

 ment of the weight down the entire series, is identical 

 with the swing of the bob in the arc. More is necessary 

 to establish this completely than Galileo was able to 

 supply. In passing from plane to plane the particle must 

 be supposed to make a slight rebound at each, a rebound 

 which is less for each, according as the change of slope 

 from one to the other becomes less and less, but the 

 number of the planes, and therefore of the rebounds, in- 

 creases in the same proportion as the slope of each to 

 each diminishes. To reduce the swing of the bob in its 

 arc to th? fall of the mass down the planes it is necessary 

 to show that the effect of this great number of small re- 

 bounds is negligible, and Galileo had not advanced far 

 enough in the Fluxional Calculus to show it. 



The principle that the speed at any point of the down- 

 ward slope depends only on the vertical drop between the 

 two positions of the particle, is true independent of 

 ■friction which lowers the speed attained in a constant 

 proportion. But it would have been difficult to establish 

 the truth stated in this way by ordinary experiment. 

 What is the speed attained, and how are we to recognise 

 it ? As the body goes downwards it is increasing in speed 

 from moment to moment. It is easy to time a railway train 

 running at a uniform rate. When the first quarter milestone 

 he notices flies past him, a passenger sees, let us suppose, 

 that the second hand of his watch is at 5 seconds, while 

 at the next quarter milestone it is at 20, at the third 35, 

 at the fourth 50. Every one of these equal intervals is swept 

 over by the train in 15 seconds, or a quarter of a minute. 

 The train is going at the regular rate of a quarter mile 

 per quarter minute, or a mile a minute, or sixty miles an 

 hour. Had the intervals of time noted been different, 

 the problem would obviously have been mu:h more com- 

 plicated. Let us suppose that the two first 5 and 20, are 

 as before, that the next is 40, and that at the fourth the 

 second hand of the watch has again come round to 5 

 seconds past the minute. In that case the first quarter 

 mile interval is done in 15 seconds, the next in 20, the 

 third in 25. If the rates were uniform for each interval 

 these figures would give us sixty miles an hour for the 

 first quarter mile, forty-five miles per hour for the next, 

 thirty-six miles an hour for the third. The train is 

 slackening speed, and these are the average rates during 

 the time spent in covering each of these quarter miles. 

 But the train does not drop suddenly from one to the 

 other, and nothing in nature does so. Point by point it 

 lias a different rate, and the question, What is the rate 

 at any point ? is not easily answered. How, then, are we 

 to measure the rate of speed at a point when that rate is 

 ■constantly changing? We must seek some necessary 

 consequence of any law of change which we suppose, 

 and we must transform the question, the answer of which 

 it is difficult to verify, into one which it will be easy to 

 subject to an experimental test. Galileo appealed to 

 mathematics, and showed that if his theory, that the 

 velocity depends on the vertical drop, be true, the amount 

 of the vertical drop must be four times as great for two 

 seconds, and nine times as great for three seconds, as for 

 one second, and he set himself to compare the real with 

 the theoretic result. 



Let us consider what seems a simple thing, a fall in 

 space, where there is no inclined plane at all. What is 

 the amount of fall for so many seconds ? The difficulty in 

 answering accurately is that for even a short time the fall 

 is very large. It is of no use distinguishing between a 

 fall of 16 feet, for instance, and one of 20^ feet, if the 

 times of description, which are 1 second and i£ second, 

 are too nearly the same to be distinguished by our 

 measurement of time. In Galileo's day the measure- 

 ments of time were only beginning to be a little delicate, 

 chiefly through his own discoveries, and an error of £ of 

 a second in measurement is obviously easy to make, when 

 one of 4 feet is not easy. In the simpler case of free 



fall, therefore, Galileo could not compare spaces and 

 times conveniently, because his measures of space were 

 so much more accurate than those of time. The experi- 

 mental test can be more readily applied to the inclined 

 plane because the fall is slower and there is no other 

 vital alteration in the conditions of the problem. 



It is necessary to form some hypothesis about the law 

 which the falling body obeys, to deduce the mathematical 

 consequences of that law, to select one of them which 

 admits of an immediate and satisfactory experimental 

 verification. This was what Galileo did. He believed 

 that the force on the falling body was probably due to the 

 mass of the earth, and that it was at least likely that it 

 would be the same all through the motion, as the particle 

 all through it is practically equally far from the centre of 

 that mass. A constant force must be measured by its 

 constantly producing the same effect in the same time, 

 and the first obvious effect of any force on a falling body 

 is, like the effect of getting up steam on a locomotive, the 

 change of speed which it produces from moment to 

 moment. If this be uniform — so much extra speed put 

 on every second — there must be some way of connecting 

 mathematically the easily measurable spaces and times 

 instead of the less practicable but more direct speeds and 

 times, and the question whether the result and the theory 

 at the back of it agree can be tested over and over again 

 by experiment. The two answers do agree, and they 

 agree in every case. The theory, therefore, is right, un- 

 less some other theory about the effect of forces can be 

 found to lead to the same result. The hypothesis about 

 the earth force, that when a body falls from rest its speed 

 will be increased by the same amount in every equal time 

 interval, and that the speed of any body will be increased 

 just as much as that of any other, is a true hypothesis. 

 A 10 lb. weight falls neither faster nor slower than a 1 lb. 

 one. If the earth alone be acting on both, a feather falls 

 as fast as a guinea. It is so in vacuum, though in 

 ordinary air, of course, it is different. A force always 

 the same, producing, that is to say, always the same 

 amount of change of speed in the same time, is acting on 

 every equal particle of matter at the earth's surface. To 

 test this theory we can appeal practically to the inclined 

 plane, rough or smooth. The force on a body falling 

 along it at any moment bears a fixed proportion to that 

 in a free fall; a very small proportion, if the plane has 

 only a very slight slope. Obviously the length of the line 

 along such a plane, down which a body runs in a second, 

 is a very small proportion of that of the free fall in the 

 same tim2. In the latter case, what to Galileo's power 

 of measuring time was an almost imperceptible difference 

 involved a very marked difference in the spaces gone 

 through, so that it was difficult to verify the law. In the 

 former the spaces needed to be measured for experiments 

 lasting even a few seconds become reasonable. In three 

 seconds a body falling freely from the top of a steeple 

 144 feet high would fall to the bottom, and it would only 

 take five seconds to fall down Tennant's stalk, but it is 

 easy to make a plane such that a body will only fall down 

 14 feet along it in three seconds. 



It was in conneciion with his investigations of motion 

 on a plane that Galileo laid down the principle that 

 perhaps serves best as the basis of the theory of balancing 

 forces, the principle of what is called Virtual Velocities. 

 Every one is familiar with it, in the ordinary maxim, that 

 what is gained in speed is lost in power. In the board 

 laid across a fallen tree, on which children see-saw, the 

 lighter child is put at the extremity of the longer arm. 

 With a plank, 12 feet long, a child 50 lbs. weight will 

 be balanced against one 70 lbs. weight when the plank 

 rests on the tree 7 feet from the light child's end, and 5 

 feet from the heavy one's. When they swing, the amount 

 of swing is proportional to the distances from the fixed 

 point. If the plank moves, so that the child at the 7 feet 

 end rises through seven inches, the other goes down 



