AIMS, ETC., OF SCIENTIFIC THOUGHT. 9g 



increases very slowly as the body falls. We know also that this varia- 

 tion of the law from the truth is too small to be perceived by direct 

 observation on the change of velocity. But suppose we have invented 

 means for observing this, and have verified that the increase of velocity 

 is inversely as the squared distance from the earth's centre. Still the 

 law is not accurate ; for the earth does not attract accurately toward 

 her centre, and the direction of attraction is continually varying with 

 the motion of the sea; the body will not even fall in a straight line. 

 The sun and the planets, too, especially the moon, will produce devia- 

 tions ; yet the sum of all these errors will escape our new process of 

 observation, by being a great deal smaller than the necessary errors 

 of that observation. But when these again have been allowed for, 

 there is still the influence of the stars. In this case, however, we only 

 give up one exact law for another. It may still be held that if the 

 effect of every particle of matter in the universe on the falling body 

 were calculated according to the law of gravitation, the body would 

 move exactly as this calculation required. And if it were objected 

 that the body must be slightly magnetic or diamagnetic, while there 

 are magnets not an infinite way off; that a very minute repulsion, 

 even at sensible distances, accompanies the attraction ; it might be 

 replied that these phenomena are themselves subject to exact laws, 

 and that, when all the laws have been taken into account, the actual 

 motion will exactly correspond with the calculated motion. 



I suppose there is hardly a physical student (unless he has specially 

 considered the matter) who would not at once assent to the statement 

 I have just made; that, if we knew all about it, Nature would be 

 found universally subject to exact numerical laws. But let us just 

 consider for another moment what this means. 



The word " exact " has a practical and a theoretical meaning. 

 When a grocer weighs you out a certain quantity of sugar very care- 

 fully, and says it is exactly a pound, he means that the difference be- 

 tween the mass of the sugar and that of the pound-weight he enqrioys 

 is too small to be detected by his scales. If a chemist had made a 

 special investigation, wishing to be as accurate as he could, and told 

 you this was exactly a pound of sugar, he would mean that the mass 

 of the sugar differed from that of a certain standard piece of platinum 

 by a quantity too small to be detected by his means of weighing, 

 which are a thousandfold more accurate than the grocer's. But what 

 would a mathematician mean, if he made the same statement? He 

 would mean this. Suppose the mass of the standard pound to be 

 represented by a length, say a foot, measured on a certain line ; so 

 that half a pound would be represented by six inches, and so on. And 

 let the difference between the mass of the sugar and that of the stand- 

 ard pound be drawn upon the same line to the same scale. Then, if 

 that difference were magnified an infinite number of times, it would 

 still be invisible. This is the theoretical meaning of exactness ; the 



