PLATE 14, FIGURE 2. SECOND METHOD.-By Continuous Motion. Let BC be a ruler whose edge coincides with the directrix of the parabola, and let DEG be a square. Take a thread equal in length to EG, and attach one extremity of it at G, and the other at the focus F Then slide the side of the square DE along the ruler BC, and at the same time keep the thread continually stretched by means of the point of a pencil, A, in contact with the square; the pencil will describe one part of the required parabola. For, in every position of the square, AF÷AGAE + AG, and hence, AF AE; that is, the point A is always equally distant from the focus Fand the directrix BC. If the square be turned over and moved on the other side of the point F the other part of the same parabola may be described. PLATE 15, FIGURE 1. The Major Axis and Foci of an Ellipse being given, to Describe the Curve. FIRST METHOD.-By Points. Let AA' be the major axis and FF" the foci of an ellipse. Take E any point between the foci, and from F and F" as centers, with distances AE, A'E as radii, describe two circles cutting each other in the point D; D will be a point on the ellipse. For, join FD, F'D; then DF + DF' = EA + EA'AA'; and, at whatever point between the foci E is taken, the sum of DF and DF" will be equal to AA'. Hence, by Def. 1, D is a point on the curve; and in the same manner any number of points in the ellipse may be determined. Cor. The same circles determine two points of the curve D and D', one above and one below the major axis. It is also evident that these two points are equally distant from the axis; that is, the ellipse is symmetrical with respect to its major axis, and is bisected by it. PLATE 15, FIGURE 2. SECOND METHOD.-By Continuous Motion. Take a thread equal in length to the major axis of the ellipse, and fasten one of its extremities at F, the other at F'. Then let a pencil be made to glide along the thread, so as to keep it always stretched; the curve described by the point of the pencil will be an ellipse. For, in every position of the pencil, the sum of the distances DF, DF" will be the same, viz.: equal to.the entire length of the string. Scholium. The ellipse is evidently a continuous and closed curve. PLATE 16, FIGURE 1. The Transverse Axis and Foci of an Hyperbola being given, to Describe the Curve. FIRST METHOD.-By Points. Let AA' be the transverse axis, and FF" the foci of an hyperbola. In the transverse axis AA' produced, take point E, and from Fand F" as centers, with the distances AE, A'E as radii, describe two circles cutting each other in the point D; D will be a point in the hyperbola. For, join FD, F"D; then DF' — DF — EA' ЕÀ — AA' AA'; and at whatever point of the transverse axis produced E is taken, the difference between DF" and DF will be equal to AA'. Hence, by Def. 1, D is a point on the curve; and, in the same manner, any number of points in the hyperbola may be determined. In a similar manner the opposite branch may be constructed. Cor. The same circle determine two points of the curve D and D', one above and one below the transverse axis. It is also evident that these two points are equally distant from the axis; that is, the hyperbola is symmetrical with respect to its transverse axis. PLATE 16, Figure 2. SECOND METHOD.-By Continuous Motion. Take a ruler longer than the distance FF", and fasten one of its extremities at the point F". Take a thread shorter than the ruler, and fasten one end of it at F and the other to the end H of the ruler. Then move the ruler HDF' about the point F', while the thread is kept constantly stretched by a pencil pressed against the ruler; the curve described by the point of the pencil will be a portion of an hyperbola. For, in every position of the ruler, the difference of the lines DFDF' will be the same, viz.: the difference between the length of the ruler and the length of the string. If the ruler be turned and move on the other side of the point F, the other part of the same branch may be described. Also, if one end of the ruler be fixed in F, and that of the thread in F', the opposite branch may be described. It is evident that each portion of each branch will extend to an indefinitely great distance from the foci and center. |