Ramble, Part 25: The Coming of Calculus

[stoutfellow]

Calculus did not spring full-grown from Newton's brow; the century or so preceding the publication of Principia mathematica brought forth many of the elements of calculus, at least in rudimentary form. Problems of tangency, of optimization, of approximation, and of the computation of areas were all being discussed, and the crucial connection between tangents and areas had been recognized before Newton's work. The details are under the cut.

1. I've already mentioned Viète's technique for finding approximate zeros of polynomials, which was a special case of what we now call Newton's Method.


2. Likewise, I've spoken of Fermat's discovery of a version of the First Derivative Test.


3. Johannes Kepler used a simple form of integration in developing his laws of motion, and Bonaventura Cavalieri systematized his insight in the "Method of Indivisibles".


4. Evangelista Torricelli and Gilles de Roberval independently came up with the "method of composition of motions", making it possible to construct tangents to a variety of new curves


5. John Wallis, extending earlier work by Galileo, computed the areas under the graphs of a number of functions, showing in particular that the integral from 0 to 1 of x^m is 1/(m+1) even if m is 0, negative, or fractional. (It was he who first systematically used exponents of this broader type.)


6. What we now call the Fundamental Theorem of Calculus was discovered by Isaac Barrow, who was Newton's mentor at Cambridge.

Let me expand on some of these items.

Kepler's Second Law - The radius vector joining a planet to the sun sweeps over equal areas in equal intervals of time - required him to consider the problem of computing areas of elliptic arcs. His method for doing this involved decomposing the arcs into infinitesimal triangles. Bonaventura Cavalieri systematized Kepler's idea, regarding areas and volumes as being composed of "indivisibles" - parallel cross-sections - and noting that if two areas (for example) had cross-sections in some fixed ratio, then their areas were in the same ratio. Cavalieri was, to some extent, reproducing Archimedes' long-lost work, but with the difference that in the less rigorous atmosphere of the time he felt free to publicize it, so that the idea could be further refined at the hands of, among others, Wallis and Newton.

The method of composition of motions was applicable to curves whose generation could be mechanically described, as Descartes had suggested. For example, consider an ellipse. The sum of the distances from a point P on the ellipse to the foci F, F' is fixed. Thus, an infinitesimal motion of P - i.e., motion along the tangent line - could be decomposed into, say, an infinitesimal motion away from F and another towards F'; and these infinitesimal vectors must be equal in length, since the sum of the distances must remain constant. The vectors yield an infinitesimal parallelogram, and the tangent line lies along its diagonal; since the vectors are equal in length, the parallelogram is a trapezoid rhombus, and the diagonal bisects the angle between the adjacent sides. Thus, the tangent line at P bisects the angle between the lines PF and PF'. Similar methods can be applied to the other conic sections, and to other curves including the cycloid. (In this case, the two motions are the forward motion of the circle as a whole, rolling along the line, and the circular motion of a point on the circle around its center.)

Barrow was the first Lucasian Professor of Mathematics at Cambridge; Newton was the second. (Stephen Hawking is the current incumbent; if "Star Trek: TNG" is to be believed, the android Data will one day hold the post!) Barrow's version of the Fundamental Theorem is rather opaquely presented, and applies only to functions which are strictly increasing or strictly decreasing. The proof follows traditional lines ("if this tangent falls short of the given point, then... :contradiction:; if the tangent surpasses the given point, then... :contradiction:; therefore the tangent passes through the given point"). Nonetheless, upon being unraveled, it clearly does illustrate the inverse nature of differentiation and integration.

None of the above should be taken as disparaging Newton's achievement, but only to indicate that its nature was a bit different than is commonly believed. More on that in the next post.

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Comments

[countrycousin.livejournal.com]

I knew about Kepler, but not about the others. And Newton didn't publish any of this, and then gets annoyed when someone else starts to come out with it? Well, amplified or not, he still had amazing accomplishments. I think there is a tendency in popularizers to amplify the difference between a select few and the rest. There are in fact large differences. But they are more widely shared than popular history would have us believe.

Thanks for the wider spotlight.