SEC. V.J OF NATURAL PHILOSOPHY. 137 equal to the given angle CNM. Wherefore since (by supposition) the an gles MBD, NBP are equal, as also the angles MOD, NCP, take away the angles NBD and NOD that are common, and there will remain the angles NBM and PBT, NCM and PCR equal; and therefore the triangles NBM, PBT are similar, as also the triangles NCM, PCR. Wherefore PT is to NM as PB to NB ; and PR to NM as PC to NC. But the points, B, C, N, P are immovable: wheiefore PT and PR have a given ratio to NM, and consequently a given ratio between themselves; and therefore, (by Lemma XX) the point D wherein the moveable right lines BT and CR perpetually concur, will be placed in a conic section passing through the points B. C, P. Q.E.D. And, vice versa, if the moveable point D lies in a conic section passing through the given points B, C, A ; and the angle DBM is always equal to the given an gle ABC, and the angle DCM always equal to the given angle ACB, and when the point D falls successively on any two immovable points p, P, of the conic section, the moveable point M falls suc cessively on two immovable points /?, N. Through these points ??, N, draw the right line nN : this line nN will be the perpetual locus of that moveable point M. For, if possible, let the point M be placed in any curve line. Therefore the point D will be placed in a conic section passing through the five points B, C, A, p, P, when the point M is perpetually placed in a curve line. But from what was de monstrated before, the point D will be also placed in a conic section pass ing through the same five points B, C, A, p, P, when the point M is per petually placed in a right line. Wherefore the two conic sections will both pass through the same five points, against Corol. 3, Lem. XX. It is therefore absurd to suppose that the point M is placed in a curve line. QE.D. PROPOSITION XXII. PROBLEM XIV. To describe a trajectory that shall pass through Jive given points. Let the five given points be A, B, C, P, D. c From any one of them, as A, to any other sv two as B, C, which may be called the poles, draw the right lines AB, AC, and parallel to those the lines TPS, PRO, through the fourth point P. Then from the two poles B, C, draw through the fifth point D two indefinite lines BDT, CRD, meeting with the last drawn lines TPS, PRQ (the