**Some Introductory Material on Infinitary Arguments in Ancient Greek Mathematics**©

by Henry Mendell (Cal. State U., L.A.)

Return to Vignettes of Ancient Mathematics

It is commony supposed that Greek mathematicians avoided the infinite. This is not true. It is only true of the most commonly read texts, and certainly must have been used as a heuristic underlying the most commonly read texts. Underneath them lies much evidence for other types of arguments which involve infinitary analyses of figures. Is it correct to describe these as merely heuristics or should one concede that the philosophical objections to infinitary analyses establishes no more than a norm of rigor for the presentation of finitary arguments, some version of the Method of Exhaustion?

It is not my purpose here to argue for the use of infinitary arguments. Hence, I shall include reconstructions and fictional examples, as well as real examples.

I here distinguish five different kinds of infinitary arguments:

3. Archimedes' Mechanical Method, using convergence

4. Cavalieri's Method of Indivisibles

5. Archimedes' Mechanical Method, using indivisibles

We may contrast these with the conception of these methods in the 17th century. Here is a dramatic example from Isaac Barrow.

Constant convergence arguments have two forms corresponding to two types of arguments using the Method of Exhaustion.

- Approximation Arguments

This is the simplest type of infiniary arguments. Suppose that we
want to show that two figures A and B have some property, F(A, B).
We then create approximations of A_{1}, , A_{n}, ... and
B_{1}, , B_{n}, .., such that each series gets closer and
closer to A and B and, furthermore, it is always the case that F(A_{i},
B_{i}). Then it is reasonable to infer that at the limit, F(A,
B). Note that to make that inference, we need some implicit assumption,
such as that F is an appropriate property for this type of argument.
"x and y are equal in area" may be appropriate, 'x and y are both rectilinear"
is not. Our evidence for this type of argument in the 5th-3rd cent.
B.C.E. is indirect.

Constant convergence
arguments: a standard illustration based on Euclid, __Elements__
xii 2

- Compression Arguments

These are like Approximation Arguments. However, one approximates
the figure from the outside and from the insided. One shows that each
approximation has the requisite constant convergence. The figure in
between the two, then, must have the same property. Such arguments
were probably characteristic of Archimedes, but only survive in later authors.

These types of arguments are very sophisticated (if common from the 17
cent. on). I doubt if they were used before Archimedes. Suppose
that we want to show that two figures A and B have some property, F(A, B).
We then create approximations of A_{1}, , A_{n}, ... and
B_{1}, , B_{n}, .., such that each series gets closer and
closer to A and B and, furthermore,there is a relation F_{i}(A_{i},
B_{i}) such that the relation F_{i}(A_{i}, B_{i})
approaches as A approaches A_{i} and B_{i} approaches B.
Then it is reasonable to infer that at the limit of F_{1}(A_{1},
B_{1}), ..., F_{n}(A_{n}, B_{n}) is F(A,
B). Note that to make that inference, we need some implicit assumption,
such as that F is an appropriate property for this type of argument.

Archimedes tells us that he used a mechanical method to discover the area
of a parabola. While his presentation in the __Quadrature of the Parabola__
involves the method of exaustion, it is also clear that the method of exhaustion
would not have led to a discovery. We can assume that his method combined
an infinitary convergence argument with the mechanical method. The
mechanical method uses the principle of the balance.

This is named for the 17th century mathematician. Suppose that we
want to show that two figures A and B have some property, F(A, B).
Let A and B n-dimensional (i.e., where n = 2 or 3). We then take slivers
of n-1 dimensions, a's of A and b's of B and show that if the a's and b's
are ** paired up** in some appropriate way, for each pair, a, b,
it happens that G(a,b). We then show that it follows from G(a,b) that
for all pairs F(a+...+a+...,b+...+...). Since we conceive of A as composed
of the a's and B as composed of the b's, it follows that F(A,B).

We then create approximations of A

Cavalieri's
method of indivisibles: a sophisticated example from Archimedes, Method

Archimedes' Mechanical Method, using Indivisibles

In his book the __Method__, Archmedes also combines the mechanical
method (using the principle of the balance) with the treatment of an n-dimensional
figure as composed of an infinity of n-1-dimensional figures.

Archimedes mechanical method with indivisibles: a very trivial illustration

Archimedes, The Method, prop. 1, the area of a parabolic segment

Archimedes, The Method, prop. 2, comparing the volumes of a sphere, cone, and cylinder