148 WHAT IS SCIENCE? 



which is the idea especially associated by the form 

 of the deduction). The assumptions thus suggested by 

 mathematical deduction are almost invariably found to 

 be actually true. It is this fact which gives to mathe- 

 matical deduction its special significance for science. 



THE NEWTONIAN ASSUMPTION 



Again an example is necessary and we will take one 

 which brings us close to the actual use of mathematics 

 in science. Let us return to Table II which gives the 

 relation between the time for which a body has fallen 

 and the distance through which it has fallen. The falling 

 body, like all moving bodies, has a " velocity." By the 

 velocity of a body we mean the distance that it moves in 

 a given time, and we measure the velocity by dividing 

 that distance by that time (as we measure density by 

 dividing the weight by the volume). But this way of 

 measuring velocity gives a definite result only when the 

 velocity is constant, that is to say, when the distance 

 travelled is proportional to the time and the distance 

 travelled in any given time is always the same (compare 

 what was said about density on p. 130). This condition 

 is not fulfilled in our example ; the distance fallen in the 

 first second is i, in the next 3, in the third 5, in the next 7 

 and so on. We usually express that fact by saying 

 that the velocity increases as the body falls ; but we 

 ought really to ask ourselves whether there is such a 

 thing as velocity in this case and whether, therefore, the 

 statement can mean anything. For what is the velocity 

 of the body at the end of the 3rd second i.e. at the time 

 called 3. We might say that it is to be found by taking 

 the distance travelled in the second before 3, which is 5, 

 or in the second after 3, which is 7, or in the second of 

 which the instant " 3 " is the middle (from 2^ to 3^), which 

 turns out to be 6. Or again we might say it is to be found 



