Informal pedagogical synopsis of the Analysis chapter
Analysis: Calculating Areas and Volumes

As with each chapter, the Introduction tells the story from beginning to end, and the subsequent sections each feature original sources which are high points of the story.

For the original source sections, we begin with two dramatically different texts by Archimedes, along with his introductory letters for them, each on the quadrature of the parabola. His result uses lovely properties of the parabola, mostly forgotten today in our curriculum, to prove that the area of any segment of a parabola (i.e., that cut off by a chord)  is four-thirds the area of the largest inscribed triangle on the base of the segment.  This delightful result seems so fresh to us and students today precisely because it is not any of the standard results learned in high school, like the area of a circle. Our first text is his proof by the classic method of exhaustion, while the second text shows how he himself says he discovered this and his other results, since it is the first proposition in The Method, in which he revealed his mechanical method of balancing indivisibles with his law of the lever, and summing them to obtain area.  His argument for calculating the area this way is clear genius.  Of course our standard knowledge today of the "area under part of a parabola" can be deduced from his result (Exercise!).  His "Method" remained lost for almost two-thousand years, forcing the reinvention of infinitesimals in the Renaissance, and making people think he hid his discovery methods.  The only copy we have is a palimpsest rediscovered in 1899, and it has remained in unknown private hands for almost 100 years.  It is being auctioned at Christie's on Park Avenue in New York on October 29, 1998 for about one million dollars [it actually sold for two million!].  It is one of the most important Greek mathematical documents in the world.  You can go and view and touch it in New York for several days prior to the auction.

Next is Cavalieri's text from Exercitationes Geometricae Sex, in which he computes areas and volumes for higher parabolas (what we would write as y=xn) by the method of indivisibles, using verbal algebra to create an inductive procedure on n.  To Cavalieri this is an n+1 - dimensional object; on the other hand, he doesn't really know what to make of something more than 3 - dimensional.  This was the first quadrature of a higher parabola after Archimedes.  The original text studies his argument for n=3 in detail (with photos of his original figures from the early 16th century), and then sees how he also reinterprets his results as plane areas involving the curves y=xn (of course without this notation).

The next section features Leibniz's 1693 paper proving the Fundamental Theorem of Calculus as a geometric procedure for finding areas, based on infinitesimals, "that the general problem of quadratures can be reduced to the finding of a line [curve] that has a given law of tangency".  Once one has this, one is equipped to find areas by producing antiderivatives (exercises!).  The section ends with salient excerpts from Bishop Berkeley's searing attack on the infinitesimals of Newton and Leibniz, assailing science as being no more rigorous than he claimed the scientists said his religion was.

Excerpts from Cauchy's Lecons on the differential and integral calculus are next.  Here we see him attempt to define limits and continuity using variables, in a way which avoids "infinitely small quantities", but still wanting to retain the word "infinitesimal" for a certain type of variable. The difficulties inherent here will still remain unresolved until later. But he does present much of our modern view of calculus, especially by removing the circularity in the relation between derivative and integral. He defines the definite integral as a limit of (Riemann) sums, and then actually proves the fundamental theorem from this definition.

Our final original source is from Abraham Robinson's resuscitation of infinitesimals in the year 1960, with the formal introduction of Non-Standard Analysis!  We hear Robinson discuss what he has done in historical context (he was interested "in getting into Leibniz's mind"), and we hear Kurt Godel's perspective on this discovery within mathematics as a whole, and its future.

The story thus goes around and around, from indivisibles to exhaustion, back to indivisibles, their banishment by Cauchy's successors, and resurrection by Robinson.  There are lots of spots where connections to philosophy and metaphysics abound.



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Teaching with Original Historical Sources in Mathematics