REMARKS ON BARROW'S EUCLID. 379 



Definition 9. 

 " But a number evenly odd is that which an even number Definition 9. 

 measureth by an odd number." 



There appears to be something erroneous in these defini- Remarks 

 tions, since the same number may be found to apply to 

 both of them; for instance, the number 40 is evenly even, 

 because the even number 4 measures it by the even number 

 10; it is also evenly odd, because the even number 8 

 measures it by the odd number 5. These definitions would 

 perhaps be less exceptionable, if expressed thus : Definition 8. 

 A number evenly even is that which may be divided into two 

 equal parts, having each part an even number. Definition 9. 

 But a number evenly odd is that which may be divided into 

 two equal parts, having each part an odd number. 

 Definition 15. 

 <c One number is said to multiply another, when the Definition 15. 

 number multiplied is so often added to itself, as there are 

 units in the number multiplying, and another number is 

 produced. 



This definition appears to be improperly expressed: for Remarks. 

 if, for instance, it were required to multiply the number 

 3 by the number 2, it is necessary, to obtain the product 

 according to the definition, to add the number 3 to itself 

 so often as there are units in the number 2, that is to say 

 twice to itself; now the number 3 added once to itself gives 

 6, and added tziice to itself gives 9; thus 9 would be ob- 

 tained for the product of 3 multiplied by 2, which is evi- 

 dently erroneous. Perhaps this definition would be better 

 thus : one number is said to be multiplied by another, when 

 it is taken or repeated as many times as there are. units in 

 that other. To those, however, who may be disposed to 

 contend, that the words " taken or repeated" do not suf- 

 ficiently define the operation intended; and who may farther 

 insist, that multiplication is only a continued addition, 

 Euclid's definition may perhaps be preferred, if the words 

 less one be inserted after the word mult/plying. 

 Definition 23. 

 u One number is said to measure another by a third num. Definition S3, 

 ber, which, when it either multiplies, or is multiplied by 

 the measuring number, produces the number measured." 



Tins ' 



