Introduction. 9 



above all, that the notion should have outlived the belief in 

 judicial astrology, and the existence of witches, either of which 

 is ten thousand times more possible." 



This strong language is directed against the use of the words 

 *'less than" in the popular sense, but let the student keep in 

 mind that the words are used in a technical sense and there will be 

 no objection to such an inequality as — 2<o. 



Illustrations. — If we speak of temperature as indicated by a 

 thermometer scale, then ''greater t/ia7i'' means higher and ''less 

 tha7i ' ' means lower. If we speak of distance east and west and 

 agree that distances measured east are positive, then "greater 

 thafi ' ' means ' ' east of ' , and ' ' less tha7i ' ' means * ' ivest o/'\ If we 

 agree that distances measured north are positive and those meas- 

 ured south are negative, then "greater than'' means "north of'\ 

 and ' ' less tha?i ' ' means ' ' south of'\ etc. 



THE RULE OF SIGNS IN MULTIPLICATION AND DIVISION IN 



ALGEBRA. 



19. If we take a and b any two positive quantities, it is easy to 

 see that the notion of multiplication we get from arithmetic will 

 enable us to deal with any case of multiplication where the vinl- 

 tiplier is a positive quantity, for, evidently, a can be repeated b 

 times, and so can —a be repeated b times, but a caymot be repeated 

 — b times, e. g. 3, and also —3, can be repeated 5 times, but 3 

 cannot be repeated —5 times. 



Thus, when the mnltiplier is negative, multiplicatioii has no mean- 

 ing according to the arithmetical notio7i of multiplication, and so we 

 are obliged to broaden our ideas of multiplication in some way or 

 else exclude the operation when the multiplier is negative. 



20. The primary definition of multiplication is repeated addi- 

 tion, yet, even in arithmetic, the word outgrows its original 

 meaning, for, by no stretch of language, can the operation of mul- 

 tiplying I by |- be brought under the original definition. 



According to the original definition, multiplication, in arithme- 

 tic, is intelligible so long as the multiplier is a whole number. 



3 can be repeated 4 times, and so can -J- be repeated 4 times 

 but 4 ca7i7iot be repeated \ a time. 



A— 2- 



