he construction of such apparatus affords a study which to
some is fascinating, and even in the abstract is not devoid of utility.
In each case a definite object is presented, and usually a choice of
methods of attaining it; success requires a thorough knowledge of the
properties of the curve in hand, while ingenuity is stimulated, and
familiarity with expedients is cultivated, by the effort to select the
most available of those properties, and to arrange parts whose motions
shall be in accordance with them. Such exercise of the inventive
faculties, then, is good training for the mechanician. And it must not be
forgotten that a mechanical movement thus devised for one purpose very
frequently is either itself applicable to a different one, or proves to
be the germ from which are developed new movements which can be made so;
the solution of one problem sometimes furnishing a hint or clew of great
value in dealing with another.

[Illustration: FIG. 2.]

We proceed, then, to describe a few instruments of this kind, which we
believe to be new, in the hope that in the manner just pointed out they
may render a greater service than that for which they are directly
intended.

The first of these, shown in Fig. 1, is for the purpose of describing the
hyperbola. The properties of the curve, upon which the action of the
instrument depends, are illustrated in Fig. 2, where MM, NN, are the two
branches of an hyperbola; C the center; AB the major axis; F and F' the
foci. If now a tangent TT be drawn at any point as P of either branch,
and a perpendicular let fall upon it from the nearer focus F be produced
to cut at G a line drawn from P to the farther focus F', then this
perpendicular will cut the tangent at a point I upon the circumference of
a circle described about C upon AB as a diameter, and also the distance
F'G will be equal to AB.

In Fig. 1, then, we have a crank CI, whose radius is equal to CB, half
the major axis, turning about a fixed center C. Upon the crank-pin I is
hung, so as to turn freely, a rigid cross composed of a long slotted
piece TT, in which slides a block, and two cylindrical arms at right
angles to it and in line with each other, the axis EE passing through I.
The arm on the right slides through a socket pivoted at the focus F; the
one on the left slides through a similar socket, which is pivoted at G to
a third socket longer than the others, which again is pivoted at the
focus F'; the distance F'G being