We are free to express the opinion at the outset, that for various
reasons the draughtsman is likely to gain very little advantage by the
use of mechanical devices for describing mathematical curves by
continuous motion. Such instruments are as a rule not only complicated
and expensive, but cumbersome and difficult of adjustment. It may be
suggested, _per contra_, that these objections do not apply to the
familiar combination of two pins and a string, for tracing the
"gardener's ellipse." But we question the propriety of classing a string
among strictly mechanical devices; it has its uses, to be sure, but in
respect to perfect flexibility and inextensibility it cannot be relied on
when rigid accuracy is required in drawing any of the conic sections.
[Illustration: FIG. 1.]
Nevertheless, the 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 equal to AB. Through this long socket
slides a rod KP, the end P being formed into an eye, by which this rod is
pivoted to the block which slides in the long slot, and thus controls the
motion of the block; and the pivot at P is centrally drilled to carry the
pencil. It is thus apparent that the center line of the slot TT must in
all positions be tangent to the hyperbola PBR, which will be traced by
the pencil, whose motions are so restricted as always to satisfy the
conditions explained in connection with Fig. 2.
The apparatus as thus represented does not at first sight appear unduly
complicated. But in order to render it adjustable, so that hyperbolas of
varying eccentricities and on different scales may be drawn with it,
several parts not here shown must be added. A frame must be provided, in
which to arrange supports for the pivots at F and F', and these supports
connected by a right and left handed screw, or equivalent means of
altering the distance between the foci; the crank CI and the socket F'G
must be of variable length, and these in each case would require to be
carefully adjusted. So that, as we stated in the beginning, it is
questionable whether a draughtsman of ordinary skill could draw the curve
any more readily by the aid of such a piece of mechanism than he could
without it; but it may claim a passing notice as a novel device, and the
first one, we believe, for describing the hyperbola by a combination of
rigid parts.
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