Lawson’s Geometrical Theorems. 445 
Againif AB: AD:: AD: DB, mul- 
tiplying by AB, AB* : AB .AD:: AB. 
AD : AB’. DB. Which is Proposi- 51 
tion 51. 
Also AB? : AD? :: AB.: DB, and by 
composition AB* + AD*: AD* = AB. 
DB:: AB + DB: DB, and multiplying 
the consequents AB. DB and DB by AD, 
and dividing them by DB, we have AB* 
4+ AD: AB. AD:: AB + DB: AD. 52 
Which is Proposition 52. For it is evident 
that this holds whether AB be equal to 
AD + DB or not. 
Diacram VIII. 
The rectangle under the difference of 
two lines or quantities, and the difference 
of two other lines or quantities is easily 
shewn to be — the sum of the rectangles 
Ee nnn aaa 
Prop. L. If in a semicircle whose diameter is AB the 
chord of 60°. equal to the radius be inscribed and from the 
center E a perpendicular drawn thereto and produced to 
meet the circumference in F; then I say that AF, EF, BF 
are continual proportionals. 
Prov. LL If a line be cut in extreme and mean pro- 
portion; then I say that the square of the whole, the rect- 
angle under the whole and the greater segment, and the 
