- Archimedes' Method - using levers to find areas
- Archimedes and the method of exhaustion
- Leibniz - hints toward the fundamental theorem of calculus
The discussion page can be found here, for observations and remarks.
We walked through how Archimedes discovered the quadrature of the parabola using his method of moments.
Then we reviewed how he used the method of exhaustion to prove his result with an argument based theorems in Euclid (thus acceptable to the Greeks). As Shira insisted, those arguments were really about limits in the contemporary $\epsilon - \delta$ sense, without using the word "limit".
That observation is half the story. What's missing is the existence of limits - the completeness of the real line.
The wikipedia page on nonstandard analysis has a (reasonably) comprehensible account of the history and the definition of the hyperreal numbers, where infinitesimals make perfect logical sense.
- Our discussion followed pages 108-116 of the text