VOL. LXXVII.] PHILOSOPHICAL TRANSACTIONS. 1q7 



quantities are contained between given limits, and the demonstration of it is re- 

 quired ; from the given equations and the given functions find limits of the un- 

 known quantities respectively, and if the latter limits are contained between the 

 former, the proposition is generally true, otherwise not. 



7. From the above-mentioned principles can be found the cases in which an 

 unknown quantity x admits of one or more affirmative values. 



8. It appears from the principles before delivered, that the finding the number 

 of affirmative and negative roots of a given equation, necessarily includes the 

 finding the number of its impossible roots ; and therefore it may not be improper 

 to subjoin somewhat on what has been done on this subject. 



1. Descartes gave a method of finding the number of affirmative and negative 

 roots of a given equation, when all its roots are possible ; but all the roots in 

 equations of superior dimensions are very seldom possible, unless when the equa- 

 tion is purposely made. 



2. It has been demonstrated by others and myself, that the equation will at 

 least have so many changes of signs from + to — , and — to -|-, as there are 

 affirmative roots, and so many continued progresses, from -|- to + and — to — , 

 as there are negative roots. 



3. A rule for finding in general the number of affirmative or negative roots 

 in a biquadratic, and in the equation x" -j- kx"' -f- b = O, was first published in 

 the Medit. Algebr. 



4. Harriot demonstrated a method of finding the number of impossible roots 

 contained in a cubic equation. In the year 1757 I sent to the Royal Society a 

 method of finding the number of impossible roots contained in a biquadratic 

 and quadrato-cubic equations, and in the equation x" + kx"" ± b = O. 



5. Schooten gave a method of finding the number of impossible roots which 

 can be concluded from the deficient terms of an equation Newton gave a rule 

 which often discovers the number of impossible roots contained in a given equa- 

 tion. Campbell discovered a new rule on the same subject. Mr. Maclaurin 

 has added somewhat more general on these subjects: these rules may be rendered 

 more general by a principle first given in the Miscell. Analyt. viz. multiplying 

 the given equation into a quantity x — a or (^ — a) X i^x — ^), &c. and finding 

 from the rule the number of impossible roots contained in the given equation. 

 Similar and more general rules and principles have been added in the Medit. 

 Algebr. These rules, in equations of superior dimensions, seldom discover the 

 true number of impossible roots. I believe also, that I first gave a rule in the 

 Miscell. Analyt. for finding the number of impossible roots from finding an equa- 

 tion, whose roots are the squares &c. of the roots of a given equation, which 

 rule in equations of superior dimensions sometimes finds impossible roots, when 

 Newton's, Campbell's, &c. rules fail, and fails when they find them ; and also a 



