THE ELEMENTS. BOOK VI 
395 
the whole treatment of conic sections by Apollonius. The 
problems themselves are enunciated in 28, 29: ‘To a given 
straight line to apply a parallelogram equal to a given recti 
lineal figure and deficient (or exceeding) by a parallelogrammic 
figure similar to a given parallelogram’; and 27 supplies the 
Siopur/xos, or determination of the condition of possibility, 
which is necessary in the case of deficiency (28): ‘ The given 
rectilineal figure must (in that case) not be greater than the 
parallelogram described on the half of the straight line and 
similar to the defect.’ We will take the problem of 28 for 
examination. 
We are already familiar with the notion of applying a 
parallelogram to a straight line AB so that it falls short or 
exceeds by a certain other parallelogram. Suppose that D is 
the given parallelogram to which the defect in this case has to 
be similar. Bisect AB in E, and on the half EB describe the 
parallelogram GEBF similar and similarly situated to TJ. 
Draw the diagonal GB and complete the parallelogram 
HABF. Now, if we draw through any point T on HA a 
straight line TR parallel to AB meeting the diagonal GB in 
Q, and then draw PQS parallel to TA, the parallelogram TASQ 
is a parallelogram applied to AB but falling short by a 
parallelogram similar and similarly situated to IJ, since the 
deficient parallelogram is QSBR which is similar to EF (24). 
(In the same way, if T had been on HA r jyroduced and TR had 
met GB produced in R, we should have had a parallelogram 
applied to AB but exceeding by a parallelogram similar and 
similarly situated to D.) 
Now consider the parallelogram AQ falling short by SR 
similar and similarly situated to D. Since {AO) = (ER), and 
{OS) = (QF), it follows that the parallelogram AQ is equal to 
the gnomon U11' V, and the problem is therefore that of 
constructing the gnomon UWV such that its area is equal to 
that of the given rectilineal figure C. The gnomon obviously 
cannot be greater than the parallelogram EF, and hence the 
given rectilineal figure C must not be greater than that 
parallelogram. This is the Siopurpo^ proved in 27. 
Since the gnomon is equal to C, it follows that the parallelo 
gram GOQP which with it makes up the parallelogram EF is 
equal to the difference between {EF) and C. Therefore, in