to translate the extraordinary
compounds in which K.C.F. Krause expounds his theory of the
categories. Notices of the changes introduced by Antonio
Rosmini-Serbati, and of Vincenzo Gioberti's remarkable theory, will
be found in Ragnisco's work referred to below.

[10] _System der Metaphysik_ (1844).

[11] _Logische Untersuchungen_, i. 376-377.

[12] _Essais de critique generale_ (2nd ed.), _La Logique_, i. pp.
184, 190, 207-225.

[13] _Discussions_, p. 577.

[14] _Logic_, i. 83; cf. Bain, _Ded. Log._, App. C.

CATENARY (from Lat. _catena_, a chain), in mathematics, the curve
assumed by a uniform chain or string hanging freely between two
supports. It was investigated by Galileo, who erroneously determined it
to be a parabola; Jungius detected Galileo's error, but the true form
was not discovered until 1691, when James Bernoulli published it as a
problem in the _Acta Eruditorum_. Bernoulli also considered the cases
when (1) the chain was of variable density, (2) extensible, (3) acted
upon at each point by a force directed to a fixed centre. These curves
attracted much attention and were discussed by John Bernoulli, Leibnitz,
Huygens, David Gregory and others.

The mechanical properties of the curves are treated in the article
MECHANICS, where various forms are illustrated. The simple catenary is
shown in the figure. The cartesian equation referred to the axis and
directrix is y = c cosh (x/c) or y = 1/2c[e^(x/c) + e^(-x/c)]; other
forms are s = c sinh (x/c) and y^2 = c^2 + s^2, s being the arc
measured from the vertex; the intrinsic equation is s = c tan [psi].
The radius of curvature and normal are each equal to c sec^2 [psi].

[Illustration]

The surface formed by revolving the catenary about its directrix is
named the _alysseide_. It is a minimal surface, i.e. the catenary
solves the problem: to find a curve joining two given points, which
when revolved about a line co-planar with the points traces a surface
of minimum area (see VARIATIONS, CALCULUS OF).

The involute of the catenary is called the _tractory_, _tractrix_ or
_antifriction_ curve; it has a cusp at the vertex of the catenary, and
is asymptotic to the directrix. The cartesian equation is
_ _
| c - [root](c^2 - y^2) |
x = [root](c^2 - y^2) + 1/2c log |-----------------------|