TRANSVERSAL (ML. transversalis, from Lat. transvcrsus, traverses, transverse, p. p. of transverterc, to cross, transverse, from trans, across, through + verterc, to turn). In geome try, a term commonly applied to a line cutting a pencil of parallels. In modern geometry the term is extended to mean any straight line cutting the other lines of a figure. Thus any line intersect ing the three lines forming a triangle ABC in P, Q, R, is a transversal of the triangle. The theory of transversal is one of the most impor tant in modern geometry. It has its origin in a theorem attributed to Ptolemy (q.v.), but which is found in the 'Spherics of Menelaus (q.v.), and which has been thought to go back to Hippa rebus (q.v.). This states that a straight line drawn arbitrarily in the plane of a triangle determines on the lines of its sides six segments such that the product of three not having a common ex tremity equals the product of the other three. Pappus (q.v.) in his Collections approaches the theory from another standpoint and shows that if a pencil of four lines be cut by a transversal in the points A, B, C, D, the ratio x AC i) = BC is BD constant for any position of the transversal. Pappus also showed that if a transversal cuts the sides and diagonals of a complete quadri lateral, the six segments determined on this transversal are such that the product of three not having a common extremity will equal the product of the other three; that each diagonal is cut harmonically by the other two; and that when a hexagon has three of its vertices collinear, and the other three also collinear, the intersee tions of the opposite sides are collinear also— a special case of Pascal's theorem on a hexagon inscribed in a conic. Desargues (q.v.) in his
Essai pour les coniques generalized the theorem of Pappus with respect to the quadrilateral. He showed that if a transversal cuts a conic and a quadrilateral inscribed therein, the product of the segments between either point of the conic and two opposite sides of the quadrilateral will have to the product of the segments between this point and the other two opposite sides, the same ratio as between the corresponding products when the other point of the conic is taken. The the ory was extended by Pascal. who was a friend of Desargues, and later by Newton, Cotes, and Maclaurin. In more recent times Carnot and Poncelet have been among the foremost to elabo rate the theory. To Carnot is due the introduc tion of negative lines in the theory of transver sals, and the treatment of the subject as related to modern geometry. See CONCURRENCE AND COLLINEARITY, Ceva's and Menelaus's theorems being important examples of the theory of trans versals.