Lawson's Geometrical Theorems. 42\ 
points F and G be assumed, such that 
FD: FH:: DG: GH, and from Gan 
indefinite perp. be erected, and through 
F any line be drawn to cut the same in g, 
and the circle in h and HE, then Fh: FE: : 
hg: gE. Which is the second Proposition. 2 
Also EF: Fh’:: EG : Gy = Gh. Which 
is the fourth Proposition. A 
Also if in the diameter of a circle, any 
point F be taken, and FhE be drawn 
meeting the circle in h and E, and hq be 
perpendicular to DH meeting it in q, and 
the circle again in h’ and Eh’ be joined | 
meeting DH inG;then FD: FH:: D@ 7 
:GH. Which is the seventh Proposition. 
Prop. VIII. If in the diameter of a circle AB two 
points C and H be taken such that AC: CB: : AH: HB, 
and from the points C and H be inflected to any point of 
the circumference E two lines CE, HE meeting the same 
again in D and G; I say that EC:CD:: EH: HG, 
Prop. IX. If in AB the diameter of a circle be taken 
any point C, and CD be drawn meeting the circumference 
in D and E, and from the point D be drawn DF perpen- 
dicular to CD, which meets the diameter AB in F and the 
circumference in G, then I say that DC: CE: ; DF : FG. 
Prop. X. If in AB the diameter of acircle two points 
Cand D be taken such that AC: CB:: AD: DB, and 
through the centre E a perpendicular to AB be drawn, and 
from C a line be drawn to meet the same in F, and if 
through D any line DG be drawn to meet the circle in G 
