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which will be a fecond cafe. If it be required to fall 
between this point and the other two, this will be a 
third cafe. A fourth cafe will be, when the point 
fought fhall be required to fall between the other two 
points. Alfo when the given extreme of the feg- 
ment to conftitute the fquare lies between the other 
two given points, the point fought may be required to 
fall, either there alfo, or without, compofing the 5th, 
and 6th cafes. 
The propofitions in Pappus referring to thefe cafes, 
though but four in number, fuffice for them all, each 
proportion being applicable to the problem two ways. 
For inftance the thirty-fifth proposition, as exprefted 
by Pappus, is this, being the firft above cited. Three 
points C, D, E being taken in the line A B, fo that 
the redangle under ABE be equal . 
to that under C BD, AB is to B E q ^ £ jj 
as the rectangle under D A C to 
that under C E D. Now A B is to B E, both as 
the fquare of ABto the redangle under ABE, and 
as the redangle under ABE to the fquare of B E. 
Therefore if the ratio of A B to BE be given, 
the ratio of the fquare of AB to the rectangle under 
C B D will be given, which is the firft of the cafes 
above defcribed, and alfo the ratio of the rectangle 
under C B D to the fquare of B E given, which is 
the fecond cafe. In both cafes the redangle under 
DAC will be to that under CED in the given 
ratio of A B to BE. But in the firft the red- 
angle under DAC will be given, and the point E 
in the redangle under CED to be found by ap- 
plying a redangle, which fhall bear a given ratio to 
the given redangle under D A C to the given line C D 
exceeding by a fquare ; and in the fecond cafe the 
redangle 
