XiV. On the Resolution of Indeterminate Pro- 

 blems. By John Leslie, A..M. 



[Read by Mr Platfjir, Dec. i. 1788.] 



IT is a fundamental principle in Algebra, that a problem ad- 

 mits of folution, when the number of independent equa- 

 tions is equal to that of the unknown quantities. If fimple ex- 

 preflions only occur, the anfwers will always be found in num- 

 bers, either whole or fradlional. But if the higher funftions 

 be concerned, the values of the unknown quantities will com- 

 monly be involved in furds, which it is impoflible to exhibit on 

 any arithmetical fcale, and to which we can only make a re- 

 peated approximation. Hence the origin of that branch of 

 analyfis which is employed in the inveftigation of thofe pro- 

 blems, where the number of unknown quantities exceeds that of 

 the propofed equations, but where the values are required in 

 whole or fradlional numbers. The fubjedl is not merely an ob- 

 jedl of curiofity ; it can be applied with advantage to the 

 higher calculus. Yet the dodlrine of indeterminate equations 

 has been feldom treated in a form equally fyftematic with the 

 other parts of algebra. The folutions commonly given are de- 

 void t)f uniformity, and often require a variety of aflumptions.. 

 The objedl of this paper is to refolve the complicated expreflions 

 which we obtain in the folution of indeterminate problems, into 

 fimple equations, and to do fo, without framing a number of 

 alTumptions, by help of a fingle principle, which, though ex- 

 tremely fimple, admits of a very extenfive application. 

 Vol. II. b b Let 



