Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/166



In this lemma, the name of conic ection is to be undertood in a large ene, comprehending as well the rectilinear ection thro' the vertex of the cone, as the circular one parallel to the bae. For if the point p happens to be in a right line, by which the points A and D or C and B are joined, the conic ection will be changed into two right lines, one of which is that right line upon which the point p falls, and the other is a right line that joins other two of the four points. If the two oppoite angles of the trapezium taken together are equal to two right angles, and if the four lines PQ, PR, PS, PT are drawn to the ides thereof at right angles, or any other equal angles, and the rectangle PQ x PR under two of the lines drawn PQ and PR, is equal to the rectangle PS x PT under the other two PS and PT the conic ection will become a circle. And the ame thing will happen, if the four lines are drawn in any angles, and the rectangle PQ x PR under one pair of the lines drawn, is to the rectangle PS x PT under the other pair, as the rectangle under the ines of the angles S, T in which the two lat lines PS, PT are drawn, to the rectangle under the ines of the angles Q, R, in which the two firt PQ, PR are drawn. In all other caes the locus of the point P will be one of the three figures, which pas commonly by the name of the conic ections. But in room of the trapezium ABCD, we may ubtitute a quadrilateral figure whoe two oppoite ides cros one another like diagonals. And one or two of the four points A, B, C, D may be uppoed to be removed