- EUCLID 297
Β A CD AFIGURE 17. Expression of a rectangle as the difference of two squares.by Euclid, we give just a summary description of the contents, since we have seen
them coming together in the work of the Pythagoreans, Plato, and Aristotle.
The contents of the first book of the Elements are covered in the standard
geometry courses given in high schools. This material involves the elementary
geometric constructions of copying angles and line segments, drawing squares, and
the like and the basic properties of parallelograms, culminating in the Pythagorean
theorem (Proposition 47). In addition, these properties are applied to the problem
of transformation of areas, leading to the construction of a parallelogram with a
given base angle, and equal in area to any given polygon (Proposition 45). There
the matter rests until the end of Book 2, where it is shown (Proposition 14) how
to construct a square equal to any given polygon.
Book 2 contains geometric constructions needed to solve problems that may
involve quadratic incommensurables without resorting to the Eudoxan theory of
proportion. For example, a fundamental result is Proposition 5: // á straight line is
cut into equal and unequal segments, the rectangle contained by the unequal segments
of the whole together with the square on the straight line between the points of the
section is equal to the square on the half. This proposition is easily seen using Fig.
17, in terms of which it asserts that [A + B) + D = 2A + C + D; that is, Β = A+C.
This proposition, in arithmetic form, appeared as a fundamental tool in the
cuneiform tablets. For if the unequal segments of the line are regarded as two
unknown quantities, then half of the segment is precisely their average, and the
straight line between the points (that is, the segment between the midpoint of the
whole segment and the point dividing the whole segment into unequal parts) is
precisely what we called earlier the semidifference. Thus, this proposition says that
the square of the average equals the product plus the square of the semidifference;
and that result was fundamental for solving the important problems of finding two
numbers given their sum and product or their difference and product. However,
those geometric constructions do not appear until Book 6. These application prob-
lems could have been solved in Book 2 in the case when the excess or defect is a
square. Instead, these special cases were passed over and the more general results,
which depend on the theory of proportion, were included in Book 6.
Book 2 also contains the construction of what came to be known as the Section,
that is, the division of a line in mean and extreme ratio so that the whole is to one
part as that part is to the other. But Euclid is not ready to prove that version
yet, since he doesn't have the theory of proportion. Instead, he gives what must
have been the original form of this proposition (Proposition 11): to cut a line so
that the rectangle on the whole and one of the parts equals the square on the other