Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/195

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To decribe a trajectory that may touch five right lines given by poition. Pl. 11. Fig. 5.

uppoing ABG, BCF, GCD, FDE, EA to be the tangents given by poition. Biect in M and N, AF, BE the diagonals of the quadrilateral figure ABFE contained under any four of them; and (by cor. 3. lem. 25) the right line MN drawn through the points of biection will pas through the centre of the trajectory. Again. biect in P and Q the diagonals (if I may o call them) BD, GF of the quadrilateral figure BGDF contained under any other four tangents, and the right line PQ drawn through the points of biection will pas through the centre of the trajectory. And therefore the centre will be given in the concoure of the biecting lines. Suppoe it to be O. Parallel to any tangent BC draw KL, at uch ditance that the centre O may be placed in the middle between the parallels; this KL will touch the trajectory to be decribed. Let this cut any other two tangents GCD, FDE, in L and K. Through the points C and K, F and L, where the tangents not parallel CL, FK meet the parallel tangents CF, KL, draw CK, FL meeting in R; and the right line OR drawn and produced, will cut the parallel tangents CF, KL, in the points of contact. This appears from cor. 3. lem. 24. And by the ame method the other points of contact may be found, and then the trajectory may be decribed by prob. 14. Q. E. F. Rh