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.