- BRANCHES AND ROOTS OF THE CALCULUS 475
were hidden for some time, and for a blissful century mathematicians and physicists
operated formally on power series as if they were finite polynomials. They did so
even though it had been known since the time of Oresme that the partial sums of
the harmonic series 1 + \ + \ + · • • grow arbitrarily large.3. Branches and roots of the calculus
The calculus grew organically, sending forth branches while simultaneously putting
down firm roots. The roots were the subject of philosophical speculation that even-
tually led to new mathematics as well, but the branches were natural outgrowths
of pure mathematics that appeared very early in the history of the subject. We
begin this section with the branches and will end it with the roots.3.1. Ordinary differential equations. Ordinary differential equations arose al-
most as soon as there was a language (differential calculus) in which they could
be expressed.^3 These equations were used to formulate problems from geometry
and physics in the late seventeenth century, and the natural approach to solving
them was to apply the integral calculus, that is, to reduce a given equation to
quadratures. Leibniz, in particular, developed the technique now known as sepa-
ration of variables as early as 1690 (Grosholz, 1987). In the simplest case, that of
an ordinary differential equation of first order and first degree, one is seeking an
equation /(x, y) = c, which may be interpreted as a conservation law if ÷ and y are
functions of time having physical significance. The conservation law is expressed
as the differential equation
df df
-dx + -dy = 0.The resulting equation is known as an exact differential equation. To solve this
equation, one has only to integrate the first differential with respect to x, adding
an arbitrary function g(y) to the solution, then differentiate with respect to y and
compare the result with ^ in order to get an equation for g'{y), which can then
be integrated.
If all equations were this simple, differential equations would be a very trivial
subject. Unfortunately, it seems that nature tries to confuse us, multiplying these
equations by arbitrary functions ì(÷, y). That is, when an equation is written down
as a particular case of a physical law, it often looks like
M(x, y) dx + N(x, y) dy = 0,
where M(x,y) = ì(÷^)|^ and N(x,y) = ì(÷,2/)|£, and no one can tell from
looking at Ì just which factors in it constitute ì and which constitute ||. To take
the simplest possible example, the mass y of a radioactive substance that remains
undecayed in a sample after time t satisfies the equation
dy - ky dx = 0,
where k is a constant. The mathematician's job is to get rid of ì(÷, y) by looking for
an "integrating factor" that will make the equation exact. One integrating factor
for this equation is 1/y; another is e~kx.(^3) This subsection is a summary of an unpublished paper that can be found in full at the following
website: http: //www.emba.uvm.edu/~cooke/ckthm.pdf