A History of Mathematics From Mesopotamia to Modernity

TheCalculus 183

du

5

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r

Fig. 7A cardioid,r= 2 ( 1 +cosθ); and an element of area.

which (you can either believe this or work it out) is a relatively simple integral, leading to the answer

3 πa^2 /2.

2. At university, calculus was taught rigorously (see above), and one had to forget all one thought
   one had learned about differentiation. However, when one came to study differential geometry—
   the geometry of curves and surfaces—it was another matter altogether. The textbooks (from the
   1950s) which were used still picturesquely define the ‘line element’dson a surface as the distance
   between two neighbouring points, as though a point had a neighbour; and similarly we learned
   about infinitely small areas and the shape of an infinitely small ellipse near a point. (This language
   was indeed what Einstein used to formulate his general theory of relativity (1919), which was
   differential geometry par excellence.) Such formulations disappeared from universities, at least in
   England, in the 1960s with the arrival of a serious modernization drive, which introduced the
   ideas of Elie Cartan and Georges de Rham. The termdxwas still allowed (and much the same things
   could be done with it), but it meant something finite, more abstract, sounder in logic but harder to
   grasp.


3. And today? Infinitesimals are still certainly used by working mathematicians in areas which
   have not been modernized; and physics, being more results-driven, is full of them. But to discuss
   the methods which physicists allow themselves would require much more space—see Chapter 10
   for some thoughts on the subject.

Exercise 8.Sketch some points on the curve r=a( 1 +cosθ), with a suitable choice of a, and verify

that it looks as I have drawn it.

Appendix A. Newton

(FromOn the method of fluxions and infinite series(in Newton 1967–81, 3 , pp. 121–7).

PROBLEM 4

TO DRAW TANGENTS TO CURVES

MODE 1.

Tangents are drawn to curves according to the various relationships of curves to straight lines

[i.e. according to the coordinate system]. And in the first place let the straight lineBDbe ordinate to

another straight lineABas base and terminate at the curveED. Let this ordinate move through an

indefinitely small space to the positionbdso that it increases by the momentcdwhileABincreases