131

the same manner it may be shown that

hl=ce(-^) 1+2w

Hence since GK=CE (S 1 FI1+ ”

'CE»

GIG

(FX)2 + 2«

CE 2 ™

AlsoHL=CE(g|) 1 + a =

(FI)l + 2«CE 2 ™

; consequently

GK (FI)2+2«FIL ~CE 2 ™

(FI) 1 + _ FP,th FT~CE 2 « "F.H + 2 « —

GK*=FIxHL ;

that is, GK is a mean proportional betweenFI and HL.

Moreover, from the first part of the de-

CE 2 CE 2

monstration we have/i —grg-

and hi

CE 2

HL

: whence fi, g k, and h l are to

, , T CE\ CE*, •] CE*, i reach other as and or, by di-

viding each term by CE 2 , as FI, GK, andHL.

But it has been shown that FI, GK, andHL are to each other in geometrical pro-gression ; consequently fi, gk, and hi mustalso be to each other in geometrical pro-gression; and glc is therefore a mean propor-tional between fi and hi. q.e.d,

* k 2