Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/175

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Sect. V.
of Natural Philosophy.
115

will be alſo placed in a conic ſection, paſſing through the ſame five points B, C, A, p, P. when the point M is perpetually placed in a right line. Wherefore the two conic ſections will both paſs through the ſame given points, againſt corol. 3. lem. 20. It is therefore abſurd to ſuppoſe that the point M is placed in a curve line. Q. E. D.


Proposition XXII. Problem XIV.

To deſcribe a trajectory that ſhall paſs through five given given points. Pl. 9 Fig. 5.

Plate 9, Figure 5
Plate 9, Figure 5

Let the five given points be A, B, C, P, D. From any one of them as A, to any other two as B, C, which may be called the poles, draw the right lines AB, AO, and parallel to thoſe the lines TPS, PRQ through the fourth point P. Then from the two poles B, C, draw through the fifth point D two indefinite lines BDE, CRD, meeting with the laſt drawn lines TPS, PRQ (the former with the former, and the latter with the latter) in T and R. Then drawing the right line tr parallel to TR, cutting off from the right lines PT, PR, any ſegments Pt, Pr, proportional to PT, PR; and if through their extremities t, r, and the poles B, C, the right lines Bt, Cr are drawn, meeting in d, that point d will be placed in the trajectory required. For (by lem. 20.) that point d is placed in a conic ſection paſſing through the four points A, B, C, P; and the lines Rr, Tt vaniſhing, the point d comes to coincide with the point D. Wherefore the conic ſection paſſes through the five points A, B, C, P, D. Q. E. D.

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