- DIFFERENTIAL GEOMETRY 371
2.11. Arthur Cayley. Homogeneous coordinates provided important invariants
and covariants^19 in projective geometry. One such invariant under orthogonal
transformations (those that leave the sphere fixed) is the angle between two planes
Ax + By + Cz = D and A'x + B'y + C'z = D^1 , given byIn his "Sixth memoir on quantics," published in the Transactions of the London
Philosophical Society in 1858, Cayley fixed a "quantic" (quadratic form) ]Ð
whose zero set was a quadric surface that he called the absolute, and defined angles
by analogy with Eq. (2) and other metric concepts by a similar analogy. In this
way he obtained the general projective metric, commonly called the Cayley metric.
It allowed metric geometry to be included in descriptive-projective geometry. As
Cayley said, "Metrical geometry is thus a part of descriptive geometry and de-
scriptive geometry is all geometry." By suitable choices of the absolute, one could
obtain the geometry of all kinds of quadric curves and surfaces, including the non-
Euclidean geometries studied by Gauss, Lobachevskii, Bolyai, and Riemann. Klein
(1926, p. 150) remarked that Cayley's models were the most convincing proof that
these geometries were consistent.Differential geometry is the study of curves and surfaces (from 1852 onwards, mani-
folds) using the methods of differential calculus, such as derivatives and local series
expansions. This history falls into natural periods defined by the primary sub-
ject matter: first, the tangents and curvatures of plane curves; second, the same
properties for curves in three-dimensional space; third, the analogous properties
for surfaces, geodesies on surfaces, and minimal surfaces; fourth, the application
(conformal mapping) of surfaces on one another; fifth, very broad expansions of all
these topics, to embrace ç-dimensional manifolds and global properties instead of
local.
3.1. Huygens. Struik (1933) and Coolidge (1940, p. 319) agree that credit for
the first exploration of secondary curves generated by a plane curve —the involute
and evolute—occurred in Christiaan Huygens' work Horologium oscillatorium (Of
Oscillating Clocks) in 1673, even though calculus had not yet been developed. The
involute of a curve is the path followed by the endpoint of a taut string being wound
onto the curve or unwound from it. Huygens did not give it a name; he simply called
it the "line [curve] described by evolution." There are as many involutes as there
are points on the curve to begin or end the winding process.
Huygens was seeking a truly synchronous pendulum clock, and he needed a
pendulum that would have the same period of oscillation no matter how great
the amplitude of the oscillation was.^20 Huygens found the mathematically ideal
solution of the problem in two properties of the cycloid. First, a frictionless particle(^19) According to Klein (1926, p. 148), the distinction between an invariant and a covariant is
not essential. Any algebraic expression that remains unchanged under a family of changes of
coordinates is a covariant if it contains variables, and is an invariant if it contains only constants.
(^20) Despite the legend that Galileo observed a chandelier swinging and noticed that all its swings,
whether wide or short, required the same amount of time to complete, for circular arcs that obser-
vation is only true approximately for small amplitudes, as anyone who has done the experiment
in high-school physics will have learned.
(2)
AA' + BB' + CC