EQUATIONS AND VARIETIES 51
2. If a circular bath has diameter 10 cubits, then the
circumference is 30 cubits.
3. A triangle whose three sides have lengths 3, 4, and 5 is a
The ﬁrst equation is a proposition from Euclid. The assertion is
claimed to hold for all isosceles triangles. So it is a general theorem.
The second equation is from the Bible, describing a large basin in
the Temple in Jerusalem. It is not exactly true, but no doubt it was
meant to be only approximate. In this book, we will be concerned
only with exact equations.
The third equation is an example of a solution to what is called
a Diophantine equation because it has unknowns (the three side-
lengths) which are constrained to be integers (whole numbers). The
equation we are thinking of here is x
, which expresses the
Pythagorean Theorem about the side lengths x, y ,andz of a right
triangle. As a mere equation about any old right triangles, we can
suppose x, y,andz are any positive (real) numbers. What makes
it a Diophantine equation is our self-imposed desire to restrict the
solutions we are interested in to whole numbers.
Diophantus was a Greek mathematician from postclassical
times whose writings inﬂuenced the founders of modern European
number theory. Because Diophantus lived before the invention of
symbolic algebra, his work can be hard to read. It is amazing what
ancient mathematicians accomplished without the aid of the slick
symbolic language we use now.
When symbolic algebra was ﬁrst set down in Europe in a treatise
by Franc¸ois Vi
ete in the late 1500s, he called it “the art of ﬁnding
the solutions to all problems.”
In short order, Ren
e Descartes added
the connection between algebra and geometry, and he also thought
that now all geometrical problems could be ﬁrst quantiﬁed and
then solved. Because he also thought that the physical universe
was governed entirely by geometrical laws, he felt that “analytic
geometry” or “algebraic geometry” could be used to discover all of
You can ﬁnd an English translation of this treatise at the end of (Klein, 1992).