An ellipse can generally be described to satisfy any five conditions. If five points be given, Pascal's theorem affords a solution; if five tangents, Brianchon's theorem is employed. The principle of involution solves such constructions as: given four tangents and one point, three tangents and two points, &c. If a tangent and its point of contact be given, it is only necessary to remember that a double point on the curve is given. A focus or directrix is equal to two conditions; hence such problems as: given a focus and three points; a focus, two points and one tangent; and a focus, one point and two tangents are soluble (very conveniently by employing the principle of reciprocation). Of practical importance are the following constructions:--(1) Given the axes; (2) given the major axis and the foci; (3) given the focus, eccentricity and directrix; (4) to construct an ellipse (approximately) by means of circular arcs. (1) If the axes be given, we may avail ourselves of several constructions, (a) Let AA', BB' be the axes intersecting at right angles in a point C. Take a strip of paper or rule and mark off from a point P, distances Pa and Pb equal respectively to CA and CB. If now the strip be moved so that the point a is always on the minor axis, and the point b on the major axis, the point P describes the ellipse. This is known as the _trammel_ construction. (b) Let AA', BB' be the axes as before; describe on each as diameter a circle. Draw any number of radii of the two circles, and from the points of intersection with the major circle draw lines parallel to the minor axis, and from the points of intersection with the minor circle draw lines parallel to the major axis. The intersections of the lines drawn from corresponding points are points on the ellipse. (2) If the major axis and foci be given, there is a convenient mechanical construction based on the property that the sum of the focal distances of any point is constant and equal to the major axis. Let AA' be the axis and S, S' the foci. Take a piece of thread of length AA', and fix it at its extremities by means of pins at the foci. The thread is now stretched taut by a pencil, and the pencil moved; the curve traced out is the desired ellipse.