[ 2 45 ] 
QP-Pj — qr — rs , is equal to 
a + b -{- c -f- d — 4 — d x 
n 
c-\-dx — cx — dx x b-\- cx + d x x — b x — cxx — d xxx 
tt n * 
t . a + bx-{-cxx-\-dxxx r , , r\ 
that is, to j confequently, when Qj 
is nothing, that is, when the curve defcribed by s , 
thing, and therefore equal alfo to a-\-b x-^cxx^dxxx j 
this laft being alfo, from the equation propofed, e- 
qual to nothing ; Qj therefore in thofe circumftances 
will be equal to a-\-bx -\-cxx-\- dxxx j and confc- 
quently whatever value of x or O (^renders a-\-bx 
-\-cxx-\-dxxx equal to nothing, will render Qr equal 
to nothing : but every value of x that renders a-^bx 
-\-cxx~\-dxxx equal to nothing, is a root of the pro- 
pofed equation a-\-bx-\-cxx-[-dxx: v=o j conlequent- 
ly the curve will crofs the bafe ZZ at every real root 
of that equation, whether negative or affirmative, and 
therefore, as every one acquainted with curve lines 
knows, will attempt to do fo, but not quite reach it, 
at every impojjible one. Q, E. D. 
This demonftration is adapted only to an equation 
of three dimenfions } but it is eafy to fee, it may be 
extended to any other. 
Note. To obtain the negative roots, the rulers 
muft be extended to the left of the line SS, as re- 
prefented fig. 2. where they are denoted by the 
fame letters as in the other figure i viz. the ruler 
C c mull be extended from c to q ; the ruler 
from b to r, and a A from a to s, and onwards 
towards the left, the two laft turning upon the 
centers A, B, in the fixed line SS. 
cuts the bafe, 
a-\-bx-\-cxx-\-dx* 
will be equal to no- 
n 
It 
