Is it really suitable to say Archimedes invented calculus when he didn't
have any notion of differentiation, and the relationship between
integration and differentiation, and some version of the so-called
Fundamental Theorem of Calculus? It seems like Archimedes ought to be
credited with a method of finding volumes which is similar to Riemann
integration, but not with inventing "the calculus"???
>> Can you imagine having to do all the problems in this class using
>> Archimedes's Method of Exhaustion?
>
> Yes, since Archimedes' method of exhaustion really IS just the
> epsilon-delta method for the integral calculus, except that since he
> uses inequalities he has to repeat the argument for every particular
> case.
It appears to me that what Archimedes had was a method for finding areas
and volumes by successive approximations with figures of known areas or
volumes, using a kind of limiting technique similar to that used by Euclid,
and based on the material in Book V of Euclid's *Elements*. This technique
bears some resemblance to the various varieties of Riemann integration
which also use approximations by figures of known length, area, volumes,
hypervolumes, usually rectilinear figures (unlike Archimedes, who used more
general polygons and polyhedra), using a kind of limiting technique. In
the latter case, the technique commonly used is an epsilon-delta technique
of the sort fully developed by Weierstrass. It is questionable if even
Cauchy used an epsilon-delta technique of the sort to which we are
accustomed today. A number of people, including myself, have written
articles on how Cauchy's limit theory differs from the later epsilon-delta
theory.
> How much further do you think Archimedes could have gone with our modern
> notation for polynomials and exponents? I can't help thinking that how
> close Archimedes was, with the exponential numbers he invented in _The
> Sand Reckoner_, to inventing logarithms.
>
It's hard to see, sometimes, how great mathematicians could have missed
certain discoveries in view of what they did discover. Of course, we see
that they missed these discoveries by using hindsight.
Gordon Fisher gfisher@shentel.net