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

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Scholium.

In this lemma, the name of conic ſection is to be underſtood in a large ſenſe, comprehending as well the rectilinear ſection thro' the vertex of the cone, as the circular one parallel to the baſe. 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 ſection 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 oppoſite 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 ſides 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 ſection will become a circle. And the ſame 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 ſines of the angles S, T in which the two laſt lines PS, PT are drawn, to the rectangle under the ſines of the angles Q, R, in which the two firſt PQ, PR are drawn. In all other caſes the locus of the point P will be one of the three figures, which paſs commonly by the name of the conic ſections. But in room of the trapezium ABCD, we may ſubſtitute a quadrilateral figure whoſe two oppoſite ſides croſs one another like diagonals. And one or two of the four points A, B, C, D may be ſuppoſed to be removed