374 12. MODERN GEOMETRIESa much more complicated problem. It was, however, once again a cycloid. This
evolute made it possible to limit the oscillations of a cycloidal pendulum by putting
a complete cycloidal frame in place to stop the pendulum when the thread was
completely wound around the evolute.3.3. Leibniz. Leibniz' contributions to differential geometry began in 1684, when
he gave the rules for handling what we now call differentials. His notation is essen-
tially the one we use today. He regarded ÷ and ÷ + dx as infinitely near values of
÷ and í and dv as the corresponding infinitely near values of í on a curve defined
by an equation relating ÷ and v. At a maximum or minimum point he noted that
dv = 0, so that the equation defining the curve had a double root (v and í + dv) at
that point. He noted that the two cases could be distinguished by the concavity of
the curve, defining the curve to be concave if the difference of the increments ddv
(which we would now write as (d^2 v/dx^2 ) dx^2 ) was positive, so that the increments
dv themselves increased with increasing v. He defined a point where the increments
changed from decreasing to increasing to be a point of opposite turning (punctum
flexus contrarii), and remarked that at such a point (if it was a point where dv = 0
also), the equation had a triple root. What he said is easily translated into the
language of today, by looking at the equation 0 — f(x + h) - f(x). Obviously,
h = 0 is a root. At a maximum or minimum, it is a double root. If the point ÷
yields dv — 0 (that is, f'(x) = 0) but is not a maximum or minimum, then h = 0
is a triple root.
In 1686 he was the first to use the phrase osculating circle. He explained the
matter thus: