A History of Mathematics From Mesopotamia to Modernity

(nextflipdebug2) #1

TheCalculus 183


du

5

–5


  • 55


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.


  1. 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.

  2. 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
Free download pdf