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.