Eawson’s Geometrical Theorems. 429 
(TT’ — Tg), consequently zg ..'TT’ + 
Tg? =Tg.TT’ and Tg? = TT. Tz. 19 
Which is the nineteenth Proposition. 
If the line Fo be taken of any length, 
so that the rectangle under Fo and a givem 
line P may be equal to F’T* and ot cut 
TT in Zz, then FE? = FO. P = (by the 
18th) Fg . Ft, hence Fo: Ft: : Fg: P:: 
wg: gt; therefore P . vg = Fg. gt = 
Teg. gT’. Whichis the twentieth Pro- 20 
position. 
Let E® perpendicular to nE meet GT 
produced in x, then because the 2s 4En, 
xGn are right, 4, E,n, Gare in a circle 
whose diameter is an, therefore the angle 
CD perpendicolar to the same meeting the circumference 
in Cand D, «and let E be the centre, and from C and D 
let CF, DF be inflected to any point F in the circumfer- 
ence meeting the diameter AB in G and H; I say the 
rectangle GEH is equal to the square of the radius AE. 
_ Prop. XXVI. In ABthe diameter of a circle let two 
points C and D be taken such that AC: CB: : AD: DB, 
and let D be without the circle, and DE perpendicular to 
BD, through the point C let any line be drawn meeting 
the circumference in F and G, and from the points F and 
G let FH and GH be inflected to any point H in the cir- 
cumference. meeting DE in K and L; J say the rectangle 
KDL is equal to the rectangle ADB. 
Prop. XXVII. In AB the diameter of a circle let be 
taken the point C, and CD be perpendicular to AB, meet- 
