194 PHILOSOPHICAL TRANSACTIONS. [aNNO 1787. 



known quantity exterminated has only one dimension; then the latter equation 

 can be predicated of the former; for in this case both equations have only one 

 and the same value of the unknown quantity exterminated. 



3. 1. If the quantity exterminated has more dimensions than one in the equa- 

 tions, then the proposition may not generally be true; for the equations may 

 have some roots the same, but not all. These observations may be applied to 

 more equations. 



4. From n given equations <2 = 0, Z» = 0, c = 0, &c. can easily be deduced 

 others dependent on them, by finding any direct algebraical functions of the 

 above-mentioned equations, that is, (p (a, b, c, &c.), which will always = 0; and 

 in like manner, from the relation between any lines being given, can be deduced 

 innumerable relations between the above-mentioned lines, and other lines de- 

 pendent on them. 



Part III. — 1. Ratios, which are supposed greater or less than others, can 

 easily be transformed into equations, which contain affirmative and negative 

 quantities. For example, let the ratio a : /; be greater than the ratio c : d, then 



will- = - — k; if it be less, then will - = - 4- ^, where h denotes an affirma- 

 a a ' a d ' ^ 



b c 



tive quantity; and, vice versd, if - = -r — k, then will the ratio of a : ^ be 



greater than the ratio of c : d, &c. 



2. If one quantity a is affirmed to be greater than another b, for a in the 

 given equations substitute its value b -{• k; if less, for a write b — k, where k 

 denotes an affirmative quantity. 



3. Reduce the equations, so as to take away their denominators, and the de- 

 monstration of the proposition will often very easily follow. 



4. Let k = - and k' == —,; and if p and q be affirmative, let p' and a' be 



affirmative; and, vice versd, if negative, negative; then, if k be affirmative, 

 will k' also be affirmative; the same also may be affirmed, if p and a have both 

 contrary signs to p' and q'; but if one has the same, and the other contrary, 

 then will k and k' have contrary signs. 



5. Let some affirmative quantities be less than others, then any direct affirma- 

 tive function of the former, viz. function in which no negative or impossible 

 quantities or indexes are contained, will be less than the same function of the 

 latter. The contrary happens when the indexes are all negative, and the quan- 

 tities affirmative as before: for example, let 2 quantities be less than 2 others, 

 then the product of the former 2 will be less than the product of the latter. 



Corol. Hence some quantities may often be known to be greater or less than 

 others, from their direct function being greater or less than the same functions 

 of the others : for example, let d^ — b"- be an affirmative quantity, then will a be 

 greater than b. 



