The History of Mathematics: A Brief Course

3. BRANCHES AND ROOTS OF THE CALCULUS 479

Q

FIGURE 6. Left: Fermat's principle. Choosing the point Ï so

that the time of travel from Ñ to Q through Ï is a minimum.

Right: the principle applied layer by layer when the speed increases

proportionally to the square root of the distance of descent.

As with the ordinary calculus, the development of calculus of variations pro-
ceeded from particular problems solved by special devices to general techniques and
algorithms based on theoretical analysis and rigorous proof. In the seventeenth cen-
tury there were three such special problems that had important consequences. The
first was the brachystochrone (shortest-time) problem for an object crossing an in-
terface between two media while moving from one point to another. In the simplest
case (Fig. 6), the interface is a straight line, and the point Ï is to be chosen so that
the time required to travel from Ñ to Ï at speed v, then from Ï to Q at speed
w, is minimized. If the two speeds are not the same, it is clear that the path of
minimum time will not be a straight line, since time can be saved by traveling a
slightly longer distance in the medium in which the speed is greater.
The second problem, that of finding the cross-sectional shape of the optimally
streamlined body moving through a resisting medium, is discussed in the scholium
to Proposition 34 (Theorem 28) of Book 2 of Newton's Principia.
Fermat's principle, which asserts that the path of a light ray is the one that re-
quires least time, came into play in a challenge problem stated by Johann Bernoulli
in 1696: To find the curve down which a frictionless particle will slide from point
Ñ to point Q under the influence of gravity in minimal time. Since the speed of
a falling body is proportional to the square root of the distance fallen, Bernoulli
reasoned that the sine of the angle between the tangent and the vertical would be
proportional to the square root of the vertical coordinate (assuming the vertical
axis directed downward). (Recall that ibn Sahl, al-Haytham, Harriot, Snell, and
Descartes had all derived the law of refraction which asserts that the ratio of the
sines of the angles of incidence and refraction at an interface are proportional to
the velocities in the two media.) In that way he arrived at a differential equation
for the curve:

dy = j y

dx y a + y
(We have taken y as the vertical coordinate. Bernoulli apparently took x.) He
recognized this equation as the differential equation of a cycloid and thus came to
the fascinating conclusion that this curve, which Huygens had studied because it
enabled a clock to keep theoretically perfect time (the tautochrone property), also
had the brachystochrone property. The challenge problem was solved by Bernoulli