30 ELEMENTS OF SCIENCE 



positive sign ( + ) prefixed to it. If they are unlike, 

 then the result must have the negative sign ( ) before 

 it. Thus + a x + b = + ab ; -ax + b = - ab; 

 + a x b = - ab, and ax -b = +ab. 



The reason of this is very simple, + a x + b = + ab, 

 because a has to be taken positively b times; and 



a x + b= ab, because the sum a has to be 

 taken b times, as is expressed by the result. But 



+ a x b also = ab, because multiplication being, as 

 before said, essentially the same as addition, multiplying 

 a by b is the same thing as adding a to itself b times. 

 Now, in this case, a has to be added to itself - b times, 

 which is of course less than once, or, in other words is 

 really subtraction. Thus, in this negative case, a has 

 to be subtracted from itself b times, and (as we have 

 seen in subtraction) the sign must be changed and 

 so a subtracted from itself b times is, and must be 



ab. Lastly a x - b = + ab, because here, on 

 the same principles as in the last case, -a has to be 

 subtracted from itself b times. Therefore it is -ab 

 which has to be subtracted ; but, as we have seen, to 

 subtract a negative quantity is the same thing as 

 adding a positive one, and therefore subtracting -ab 

 is the same thing as adding + ab, and therefore 



-ax - b = + a b. 



It may be useful to note the three following examples 

 of multiplication : 



a + b 

 multiplied by a + b 



a 2 + ab 

 + ab + b z 



a + 2ab + 6 2 



