4 £8 On Geometrical Proportion. 



thus, we say one thing is twice, thrice, &c, as large as 

 another, or one thing will cost twice, thrice, &c, as much 

 as another : but, as our ideas expand, we wibh to compare 

 all kinds of magnitudes as exactly as possible ; and then it 

 is that a more ample view of the subject becomes necessary. 

 Arithmetic instructs us how to compare any two quantities 

 with each other, so as to determine fhe relation which sub- 

 sists between them : this is the first notion which we ac- 

 quire of proportion, and it is the foundation upon which 

 we must raise our future reasonings : this comparison of any 

 two quantities may be called a ratio, and hence we have the 

 following 



Definition I. — The word ratio signifies the relation 

 which subsists between two quantities with respect to their 

 magnitudes. One of the quantities thus compared is called 

 the antecedent, the other the consequent of the ratio, and they 

 are sometimes expressed by placing two points between them, 

 or more frequently by writing them in the form of a frac- 



3 

 tion : thus, 3 : 4, or-, is the manner in which we generally 



designate the ratio of 3 to 4, and a : b, or j, denotes the 



ratio of a. to b. 



6 3 1 



The ratio of 6 to 12, or — , is the same as - , or as ^ ; 



1 & O It 



hence it is plain that the terms of a ratio may vary, and the 

 ratio still continue the same : if, therefore, the terms of a 

 ratio be either multiplied or divided by the same quantity, the 



a 



ratio will not be altered; for : 7 = 7 , and ~ — T = the 



m 



same ratio. 



It will now be very easy to define the word proportion. 



Definition II. — Four quantities are proportional when the 

 ratio between the first and second is the same as the ratio 

 between the third and fourth ; and, in general, any number 

 of quantities are in the same proportion when they are com- 

 posed of equal ratios. 



Thus four quantities, a, b i c, d> are in the same pro- 

 portion 



