I06 ARITHMETIC. 



2. Consider whether the answer ought to be greater or 

 less than this number : if greater, write the greater of the 

 two remaining numbers on the right of it for the third, 



and 



2. Though numbers of the same kind, being either of the same 

 or of different denominations, have a real ratio, yet this ratio is 

 the same as that of the two numbers taken abstiactly, only when 

 they are of the same denomination. Thus the ratio of il. to 2I, 

 is the same as that of i to 2 =ri- ; is. has a real ratio to 2I. but 

 it is not the ratio of i to 2 ; it is the ratio of is. to 40s. that is, 

 of i to 40 ^=:fo' Therefore, as the first and third numbers have 

 the ratio, that is required between the second and answer, they 

 must, if not of the same denomination, be reduced to it ; and 

 then their ratio is that of the abstract numbers. 



3. The product of the extremes of four geometrical propor- 

 tionals is equal to the product of the means ; hence, if the pro- 

 duct of two numbers be equal to the- product of two other num- 

 bers, the four numbers are proportionals j. and if the product of 

 two numbers be divided by a third, the quotient will be a fourth, 

 proportional to those three numbers. Now as the question is re- 

 solvable into this, viz. to find a number of the same kind as the 

 second in tlie statement, and having the same ratio to it, that die 

 greater of the otJier two has to the less, or the less has to the great* 

 er ; and as these two, being of the same denomination, may be con* 

 stdered as abstract numbers ; it plainly follows, that the fourth 

 number or answer is truly found by multiplying the second by one 

 of the other two, and dividing the product by that, which remains. 



4. It is very evident, that, if the answer must be greater than 

 the second number, the greater of the other two numbers must 

 be the multiplier, and may occupy the third place ; but, if less, 

 the less number must be the multiplier. 



5. The reductibn of the second number Is only performed for 

 convenience in the subsequent multiplication and division, and not 

 to produce an abstract number. The reason of the reduction of 



the 



