IV. A Letter from Mr, Colin Mac I aiirin, Profef- 
for of Mathematicks at Ediifburgh, and F.ii.S. 
' to Martin Folkes, Efq\ Pr. l<* S, concern 
ning Equations v^ith ift^pofjible Roots. 
•V 
SIR, 
} Wrote to you laO: Winter, that I had thought of a 
very ealle and fimple way oF demonftrating Sir 
IJaac Newton's Rule, by which it may be often difco- 
verd when an ^Fquation has impofTible Roots. This 
Method requiring nothing but tiie common y^lgebra, 
and being Founded on I'ome obvious Properties of 
Quantities demonflrated in the following Lemmata, 
without having recourfe to the Confideration of any 
Curve whatfoever, which does not feem (b proper a 
Method in a Matter purely Algebraical, I hope it will 
not be unacceptable. 
Lemma i. The Sum of the Squares of two real 
QuantitiesTs always greater than twice their Produdf. 
Thus is greater than z ab; becaufe the Excels 
4*4 ^^— 2 a h Is eqaal to and therefore is Pofirive; 
fince the Square ot any real Quantity, Negative or Po- 
I'ltive, is always Pofitive. 
Lemma 2 . The Sum of the Squares of three real 
Quantities is always greater than the Sum of the Pro. 
dudfs^that can be made by multiplying any tw'o of them. 
'Thus ^7 4^Qc^ is always greater than ah-Vac-^h z 
for ’t is plain, that the Excels a^'\rh'-\-c^--a b—ac—l c 
