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:

1.    constant convergence

2.    progressive convergence

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

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

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 A1, , An, ... and B1, , Bn, .., such that each series gets closer and closer to A and B and, furthermore, it is always the case that F(Ai, Bi).  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

An actual example from Pappus, Collectio iv 22, proving that an Archimedean spiral area of one rotation is 1/3 the circle with the radius of the generating line.

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 A1, , An, ... and B1, , Bn, .., such that each series gets closer and closer to A and B and, furthermore,there is a relation Fi(Ai, Bi) such that the relation Fi(Ai, Bi) approaches as A approaches Ai and Bi approaches B.  Then it is reasonable to infer that at the limit of F1(A1, B1), ..., Fn(An, Bn) 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 A1, , An, ... and B1, , Bn, .., such that each series gets closer and closer to A and B and, furthermore, it is always the case that F(Ai, Bi).  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.

Cavalieri's method of indivisibles: a simple but garbled example from Theon of Alexandria, Commentary on Ptolemy's Almagest

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