OF QUANTITY AND NUMBER. 5 



Or a : JL=:«c : 6 ; for a : — zza : (b. — )=:(a : b) : — =(« : b) .c 

 c c ^ c * c 



(21), =ac : 6. 



Scholium. A beginner may perhaps render these demonstrations 

 more intelligible, by substituting any numbers at pleasure for the cha- 

 racters. For example, the demonstration of the first theorem may 

 be written thus. Twelve fourths, ':f, are equal to 12 divided by 4 ; 

 for, by the definition of a multiple fraction, ^^zz.\2.\f and multiplying 

 these equals by 4, 4.'|z:4.12.^ ; but by the definition of a simple frac- 

 tion 4.|zzl, therefore 4.12.|izl2, whence 4.',2zil2, and by the defini- 

 tion of division, 12 : AiZZ.^^. But, in fact, the proposition is too evi- 

 dent to admit much demonstrative confirmation. 



24. Theorem. A positive number or quantity being 

 multiplied by a positive one, the product is positive. 



For the repeated addition of a positive quantity must make the 

 result actually greater. What is true of numbers may practically 

 be affirmed of quantities in general (10). 



25. Theorem. A negative number or quantity being 

 multiplied by a positive one, the product is negative. 



For since adding a negative quantity is equivalent to subtracting 

 a positive one, the more of such quantities that are added, the 

 greater will the whole diminution be, and the sum of the whole, or 

 the product, must be negative. 



26. Theorem. A negative number or quantity being 

 multiplied by a negative one, the product is positive. 



Or — a.— 6=:a&. For «.— &=— a5(25): that is, when the positive 

 quantity a is multiplied by the negative b, the product indicates that 

 a must be subtracted as often as there are units in b : but when a is 

 negative, its subtraction is equivalent to the addition of an equal 

 positive number ; therefore in this case an equal positive number 

 must be added as often as there are units in b. 



27. Definition. If the quotients of two pairs of 

 numbers are equal, the numbers are proportional, and the 

 first is to the second, as the third to the fourth ; and any 

 quantities, expressed by such numbers, are also propor- 

 tional. 



