Theorem of Pythagoras: Greece
The Pythagorean theorem is Proposition 47
of Book I of EE (Euclid's Elements),
and its converse is Proposition 48,
the last proposition of that Book.
The converse is the proof
that is needed to construct a temple,
that is, that the 3-4-5 rope trick
actually constructs a right angle.
Dependencies of EE-I.47
The official
list
of all elements
(Definitions, Axioms, Common Notions, and Propositions)
of EE (Euclid's Elements)
together with all of their dependencies, and so on
(that is, the transitively closed list),
as found in the Euclid Project of the
Visual Math Institute, includes more than
half of Book I of EE.
However, a close study of these
logical prerequisites reveals that
only a four are conceptual prerequisites:
- I.34: The diagonal of a parallelogram bisects its area.
- I.35: Parallelograms on the same base and in the
same parallels have the same area.
- I.37: Triangles on the same base and in the
same parallels have the same area.
- I.41: Given a parallelogram and a triangle on the same base and in the
same parallels, the area of the parallelogram is double the
area of the triangle.
The proof of I.47, occupies about two pages
in Heath's translation of EE,
or about 8 pages
in my expanded version, VEE
(The Visual Elements of Euclid.)
See VEE I.47
Dependencies of EE-I.48
The official list of all dependencies of Prop. I.48
is: at depth 1, only I.8 and I.47, of which
only I.47 is conceptually essential.
The proof of I.48, occupies about one page
in Heath's EE, or about 5 pages in VEE.
It proceeds by a very simple reductio ad absurdum.
Thus, if you believe I.47, you believe in
rope stretching.
Ralph H. Abraham, 28 April 1996.