>>> Which were, in your opinion, the notations that have permitted the
>>> greatest advances in mathematics?
>>
>> One candidate is the use of exponents. It is said that if Archimedes had
>> had a decimal notation, he would have invented calculus. This I doubt.
>
> But Archimedes DID invent the calculus - or at least, the integral
> calculus!
Yes, if you want to identify one single person as the inventor of calculus,
that one person would be Archimedes, who (as far as I know) was the first
person to do any significant work on finding integrals and using the Method
of Exhaustion to find limits.
"It is said that if Archimedes had had a decimal notation, he would have
invented calculus"---I'm pretty sure the loing-forgotten source I was quoting
meant that Archimedes would have developed calculus techniques to about the
level that Newton and Leibnitz did.
>> 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.
Yes, epsilon-delta is a fast and easy way of applying the Method of
Exhaustion (with the odd quirk that both the greater than and less than
cases have the same epsilons and deltas). The idea was that the
Archimedean method was so much more, uh, exhausting than epsilon-delta...
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.
James A Landau