[ 5i3 ] 
rectangle under CE D is given, and A in the red- 
angle under D A C to be found by applying to the 
given line C D a redangle exceeding by a fquare, 
which fhall bear a given ratio to the redangle under 
C E D now given ; whence by the ratio of A B to 
B E given the point B will be found in both cafes. 
The 2 2d propolition either way applied refers to 
the 3d cafe only, the 30th relates both to the 4th 
and 5th, and the 36th propolition to the remain- 
ing 6th. 
The 45th, and other following propolitions, are 
accommodated to the lolution of Apollonius’s prob- 
lem, when four points are given, and a fifth requir- 
ed, which with the given points fhall form four feg- 
rnents fuch, that the redangle under two fhall bear 
a given proportion to the redangle under the other 
two. The various cafes of this problem appear to 
have been the fubjed of the fecond book of the 
mentioned treatife of Apollonius 5 and, according to 
the charader given by Pappus of thofe proportions, 
thefe lemmas ferve to reduce them to problems in the 
firfl book, not thofe above mentioned, but thofe, 
where three points being given, the redangle under 
the fegments included by two, and a fourth point 
fhall bear a given proportion to the redangle under the 
fegment formed by the third point and a given line. 
For inflance the 46th propofition is this ; in the 
line A B four points , , , 
A,C,E,Bbeinggiven; ac D If B 
and the point F album- | j 
ed between E and B; G 
alfo D taken, according to the 41ft propofition, that the 
redangle under ADC be equal to that under BDE; 
if G be equal to the fum of A E, C B, the redan- 
