( '05 ) 
5 fi — •>. a c — ‘2 J 2 <* ^-|«^^-|-.'J^ —2 a c-\-c^-\-h^ 
' 2 2 
— 2 that is, half the Sum of the 
“*^ -T- -■ « 
Sqjares of the Differences of the Q^iantities a^h,c\ 
But fince thefe Squares are Polltlve, it follows, that 
the Exccfs of a above ah-\-:i c-\-h c is Pofitive, 
. and that the Sum ot the Squares oT three Quantities 
muft be greater than the Sum of the Products made, 
by multiplying any two of them. • 
Lemma 3. The triple Sum of the vSquares of four 
Quantities is greater than the double Sum of the Pro- 
duv 5 ts,that can be made by muhiplying any two of them ; 
for 3 3 ^-4 3 c’4 3 d‘ — 7 - a h — 2 ac — 2. a d — ihc 
. — xh d — 2 c d z= a' — x a h‘- + — 2 a c -{• c a'^ 
— X a d ~\r d‘ -{■ — x h cl -V d^ -f — 2 i r -f c* -p 
X C d d^ =a — d a—c a^d “h b—c b~d "h c^-d^, 
the Sum of the Squares of the Differences of the tour 
Quantities a, c, d Therefore 3 -f 3 +3 
is greater than x a h x ac -^r x a d x h c x h d 
~V X c dj the Excels being always Pofitive. 
Lemma 4* Let the Number of the Quantities a hy 
Cy dy €y <Stc. be r/iy the Sum of their Squares Ay and the 
Sum of the ProduCf-s made by multiplying any two of 
them B. Then ‘dull m ~ \ x a be always greater than B, 
For by adding together the Squares of the Differen- 
ces a-hya-Cy a-d h-Cy h-dy c-d,&:.c. you add as 
often to it felf as there are Quantities more than a ; 
the fame is true of Ly d\ e\ &c. But the Redan- 
gles —2 a hy —2 aCy ~xa dy —x b c, —2 h dy &c. arife 
but once each. Therefore the Sum of ail the Squares 
n — b, a~c i b—C’ b—d,&i.C. =ni— iX n~\~)n-^iy.b^ m— 1 XcS 
&c. — 2 a h — 2 a c — xlcy^z, ~m~\y.d — iB. Bur n — V 
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