NATURE AND LAW. 373 



results were found capable of being expressed by a very simple 

 formula that the total fall in any number of seconds is the pro 

 duct of the square of that number multiplied into the fall in the 

 first second. But there was no adequate ground for asserting, or 

 even for expecting, that this formula would hold good in regard to 

 a body let fall from a height of ten or a hundred times 256 feet. 

 The &quot;law&quot; was, in that stage, the simple generalized expression 

 of facts within the range of actual knowledge. No one had a 

 right to say how far above the general surface of the earth its 

 attractive force extends ; nor could it be affirmed with any cer 

 tainty, that the fall of bodies from great mountain heights would 

 follow the same &quot; law &quot; as their fall from the top of a tower. 



But a great advance was made, when Galileo applied to this 

 case the general doctrine of the action of &quot;accelerating forces,&quot; 

 to which his study of mechanics had led him. For he saw that 

 when the falling body is let go, it starts from a state of rest, its 

 velocity being o ; and that since it is receiving afresh, at every 

 instant of its fall, the same &quot; pull &quot; from the earth as that which 

 first puts it in motion, its rate of movement must undergo a con 

 tinual regular acceleration. On the basis of this conception, a 

 very simple computation showed that during the first second it 

 will have thus acquired a velocity, which, if there were no fresh 

 &quot;pull,&quot; would carry it through 32 feet in the next second, but 

 which, with the fresh &quot; pull,&quot; would cause it to descend 48 feet, 

 making 64 feet in the two seconds and so on. The simply 

 empirical law, then, which at first had no higher value than it 

 derived from its accordance with a very limited experience, and 

 which might, or might not, be found to hold good beyond the 

 range of that experience, acquired a rational value, as the ex 

 pression of what may be fairly anticipated to be the continually 

 accelerating rate of motion of falling bodies, due to the con- 

 stantly acting attraction of the earth upon all bodies within its 

 range. And thus it was reasonable to expect, that within the 

 range of the earth s attraction whatever that range might be 

 the rate of descent of bodies falling towards its surface would still 

 be found to conform to it. But no one could then form any 

 definite idea as to the extent of that range. It was, as we shall 

 presently see, the bold &quot;scientific imagination&quot; of Newton, which 



