The History of Mathematics: A Brief Course

3. DIFFERENTIAL GEOMETRY^375

In the infinitely small parts of a curve it is possible to consider not

only the direction or inclination or declination, as has been done

up to now, but also the change in direction or curvature (flexura),

and as the measures of the direction of curves are the simplest lines

of geometry having the same direction at the same point, that is,

the tangent lines, likewise the measure of curvature is the simplest

curve having at the same point not only the same direction but

also the same curvature, that is a circle not only tangent to the

given curve but, what is more, osculating.^22

Leibniz recognized the problem of finding the evolute as that of constructing

"not merely an arbitrary tangent to a single curve at an arbitrary point, but a

unique common tangent^23 of infinitely many curves belonging to the same order."

That meant differentiating with respect to the parameter and eliminating it between

the equation of the family and the differentiated equation. In short, Leibniz was

the first to discuss what is now called the envelope of a family of curves defined by

an equation containing a parameter.

3.4. The eighteenth century. Compared to calculus, differential equations, and

analysis in general, differential geometry was not the subject of a large number of

papers in the eighteenth century. Nevertheless, there were important advances.

Euler. According to Coolidge (1940, p. 325), Euler's most important contribution

to differential geometry came in a 1760 paper on the curvature of surfaces. In that

paper he observed that different planes cutting a surface at a point would generally

intersect it in curves having different curvatures, but that the two planes for which

this curvature was maximal or minimal would be at right angles to each other. For

any other plane, making angle á with one of these planes, the radius of curvature

would be

2/g

/ + 3+(g-/)cos2a'

where / and g are the minimum and maximum radii of curvature at the point.

Nowadays, because of an 1813 treatise of Monge's student Pierre Dupin (1784-

1873), this formula is written in terms of the curvature 1/r as

1 cos^2 a sin^2 a
r g f
where a is the angle between the given cutting plane and the plane in which the
curvature is minimal (l/g). The equation obviously implies that in a plane per-
pendicular to the given plane the curvature would be the same expression with the
cosine and sine reversed, or, what is the same, with / and g reversed.
Another fundamental innovation due to Euler was the introduction of the now-
familiar idea of a parameterized surface, in a 1770 paper on surfaces that can be
mapped into a plane. The canvas on which an artist paints and the paper on
which an engineer or architect draws plans are not only two-dimensional but also
flat, having curvature zero. Parameters allow the mathematician or engineer to
represent, information about any curved surface, as Euler remarked, in the form of

Literally, kissing.

The tangent was not necessarily to be a straight line.