GEOMETRICAL P1IQBLEMS. 353 
and O; then, instead of taking CD equal to the dif 
ference of the seini-di.a meters CF, OF, let C d betaken 
equal to their sum, and the rest of the process will be 
exactly the same. 
PROBLEM XL I. 
To draw a right-line AD through tivp circles GAEF, 
HCSR, given in magnitude and position, so as to cut off 
segments thereo f, A KB>71, CTD«, equal respectively to 
given segments JiQEti,. SER6. 
CONSTRUCTION. 
Upon the subtenses EF, SR frpm the centres G and 
H, let fall the perpendiculars GQ and IIP ; and from 
m 
the same centres, at the distances GQ, HP, let two 
circles GQK, HPT be described; then draw a right- 
line AD to touch both these circles, by the last propo 
sition, and the thing is done; for the lines FE, AB 
being at the same distance from the centre G, the seg 
ments cut off by them must consequently be equal: 
and, in like manner, the segments SPRi, CTD«, are 
also equal. 
PROBLEM XLII, 
To describe the circumference of a circle through a 
given point P, to touch two right-lines AB, AC, given 
by position. 
CONSTRUCTION. 
Join A, P, and bisect the angle B A C, with the right 
line AK, and, from any point Q in that line, draw Q I 
A A