>
> But Archimedes DID invent the calculus - or at least, the integral
> calculus!
>
>
> ... 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.
>
In a chapter on Leibniz in a popular book I'm writing, this is how I explain
in what sense Leibniz can be said to be (along with Newton) an inventor of
the calculus:
One kind of problem that could be solved using limit
processes, [is] that of finding areas of figures with curved
boundaries. Another kind of problem susceptible to attack using
limits was finding exact rates of change, such as the varying
speed of a moving body. During the last months of 1675, towards
the end of his stay in Paris, Leibniz made a number of conceptual
and computational breakthroughs in the use of limit processes
that, taken together, are called his "invention of the calculus":
1. Leibniz saw that the problems of finding areas and calculating
rates of change were paradigmatic, in the sense that many
different kinds of problems were reducible to one or the other of
these two types. [Thus, finding volumes and centers of gravity are problems
of the first kind, and computing accelerations and (in economic theory)
marginal elasticity are problems of the second type.]
2. He also perceived that the mathematical operations required
in calculating the solutions to problems of these two types were
in fact *inverse* to each other in much the same sense that
the operations of addition and subtraction (or multiplication and
division) are inverse to one another. Nowadays these operations
are called *integration* and *differentiation*,
respectively, and the fact that they are inverse is called, in
the textbooks, the "fundamental theorem of the calculus."
3. Leibniz developed an appropriate symbolism (the very
notation still in use today) for these operations. ...
Finally he found the mathematical rules needed for actually
carrying out the integrations and differentiations that occurred
in practice.
Taken together these discoveries transformed the use of limit
processes, from being an exotic method accessible only to a handful
of specialists, into a straightforward technique that could be taught
in textbooks to many thousands of people.
Martin Davis
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin@eipye.com
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http://www.eipye.com