So Dr. Wa ring’s Obfervations 
from the above-mentioned quantities or equations exprefling 
the limits, And others denoting their limits, which do not con- 
tain more than ( n — 2) above-mentioned quantities, and fo on. 
4. Often from the fubflitution of the limits of given quan- 
tities can be acquired the limits of the remaining one (*). 
Find all the greatefb values of the quantity (a:) contained be- 
tween the above-mentioned limits, and thence can be deduced 
the limits fought. 
5. If there are given (m) equations involvin + 1) or 
more unknown quantities ; then fometimes with, and fome- 
times without, reducing them to others involving more few 
unknown quantities can be found by the preceding method 
limits ; and from comparing the limits fo acquired can fome- 
times be deduced the limits fought. 
6. If a given fun&ion of the unknown quantities (v, y, z, 
&c.) is aflerted to be contained between given limits, when 
other fundlions of the above-mentioned quantities are contained 
between given limits, and the demonftration of it is required ; 
from the given equations and the given functions find limits 
of the unknown quantities reflectively, and if the latter limits 
fire contained between the former, the propofition is generally 
true, otherwife not. 
7. From the above-mentioned principles can be found the , 
cafes in which an unknown quantity (a) 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 necefiarily includes the finding the number of 
its impoffible roots ; and therefore it may not be improper to 
fubjoin fomewhat on what has been done on this fubjedt. 
1. Descartes 
