on converging Seriefes. g t 
» 
1. Descartes gave a method of finding the number of affir- 
mative and negative roots of a givqn equation, when all its 
roots are pofiible ; but all the roots are very feldom in equa- 
tions of fuperior dimenfions pofiible, unlefs when the equation 
is purpofedly made. 
2. It has been demonfirated by others and myfelf, that the 
equation will at leaf! have fo many changes of figns from + to 
— , and — to -f , as there are affirmative roots, and fo many con- 
tinued progrefies from + to + and — to - , as there are ne- 
gative roots. 
3. A rule for finding in general the number of affirmative or- 
negative roots in a biquadratic, and in the equation x” -f hx m 
+ B = o, was firft publiffied in the Medit. Algebr. 
4. Harriot demonftrated a method of finding the num- 
ber of impoffible roots contained in a cubic equation. In 
the year 1757 I fent to the Royal Society a method of finding 
the number of impoffible roots contained in a biquadratic and 
quadrato-cubic equations, and in the equation a,”c±;Aa* , ”±B = o. 
5. ScHooTENgave a method of finding the number of impoffi- 
ble roots which can be concluded from the deficient terms of an 
equation. Newton gave a rule which often difcovers the num- 
ber of impoffible roots contained in a given equation. Campbell 
difcovered a new rule on the lame lubjebt. Mr. Maclaurin 
has added fomewhat more general on thefe fubjects : thefe rules 
may be rendered more general by a principle firlt given in the 
Mifcell. Analyt, viz. multiplying the given equation into 
a quantity x - a or {x — d) x ( x-b ), &c. and finding from the 
rule the number of impoffible roots contained in the given 
equation. Similar and more general rules and principles have 
been added in the Medit. Algebr. Thefe rules, in equations of 
fuperior dimenfions, feldom difcover the true number of im- 
Vo l. LXXVIL M pofiible. 
