APPLICATION OF AREAS 
151 
beyond it, it is said to fall short. Euclid, too, in the sixth 
book speaks in this way both ot‘ exceeding and falling short; 
«but in this place (I. 44) he needed the application simply, as 
he sought to apply to a given straight line an area equal 
to a given triangle, in order that we might have in our 
power, not only the construction (a-va-Tacns) of a parallelogram 
equal to a given triangle, but also the application of it to 
a limited straight line.’ 1 
The general form of the problem involving application 
with exceeding or falling short is the following: 
‘To apply to a given straight line a rectangle (or, more 
generally, a parallelogram) equal to a given rectilineal figure, 
and (1) exceeding or (2) falling short by a square figure (or, 
in the more general case, by a parallelogram similar to a given 
parallelogram).’- 
The most general form, shown by the words in brackets, 
is found in End. VI. 28, 29, which are equivalent to the 
geometrical solution of the quadratic equations 
G 
ax+ - x 2 = 
~ c 
t and YI. 27 gives the condition of possibility of a solution 
when the sign is negative and the parallelogram falls short. 
This general case of course requires the use of proportions; 
but the simpler case where the area applied is a rectangle, 
and the form of the portion which overlaps or falls short 
is a square, can be solved by means of Book II only. The 
proposition II. 11 is the geometrical solution of the particular 
quadratic equation a ^ ^ _ x z 
or x 2 + ax = a 2 . 
The propositions II. 5 and 6 are in the form of theorems. 
Taking, e. g., the figure of the former proposition, and sup 
posing AB — a, BD = x, we have 
ax — x 2 = rectangle AH 
= gnomon NOP. 
If, then, the area of the gnomon is given (= h 2 , say, for any 
area can be transformed into the equivalent square by means 
of the problems of Eucl. I. 45 and II. 14), the solution of the 
equation 
ax — x 2 = h 2 
1 Proclus on Enel. I, pp. 419. 15-420. 12.