By Prof. C.W. MacCord, Sc.D.

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.

Instruments For Drawing Curves I The Hyperbola 530 11a

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.

Instruments For Drawing Curves I The Hyperbola 530 11b

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.