28 ELEMENTS OF SCIENCE 



+ a 



- b + c and then add ; 

 when we have for result + a b + c, which is correct. 



Any quantity preceded by the sign - is a negative 

 quantity. 



On the rational principle of our language, that " two 

 negatives make an affirmative," to take away a negative 

 quantity from any other quantity is really to add to that 

 second quantity. Thus if 5 has been taken from 12, 

 so that 7 remains, and then that operation be negatived, 

 that amounts to adding 5 again to the 7 and so restor- 

 ing the original number 12. 



Similarly, if both 2 and 3 have to be taken from 10, 

 we may write it 10 - (2 + 3) = 5. But if no bracket be 

 used, we must of course change the sign in order to show 

 that both 2 and 3 are taken from 10, and write it, 

 10-2-3 = 5; 10-2 being 8, and 8-3 being 5. 



Suppose we have to subtract + by from + ^ax, the 

 difference is obviously $ax-by\ and thus the sign before 

 by is changed; but if instead of the positive quantity 

 + by we have to take the negative quantity - by from 

 + sax, the result then must be ytx + by. 



This may seem at first paradoxical to some readers, 

 but to take away a negative (i.e., to subtract a dimi- 

 nution) is evidently, in fact, to make an addition. To 

 cause a man to cease to have no hat is, of course, to 

 cause him to have one. 



The above statement may be made more plain by the 

 fifth axiom, for if we both add and subtract the same 

 quantity to and from $ax, then, of course, $ax will 

 remain unchanged and as it was. Now if we accord- 

 ingly add to and take from it cy, we shall have 

 ^ax + cy cy, which is simply the same quantity as $ax. 

 Let us then take - cy from' both, and the result must 



