Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/196

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132
Mathematical Principles
Book 1.


Scholium.

Under the preceding propoſitions are comprehended thoſe problems wherein either the centres or aſymptotes of the trajectories are given. For when points and tangents and the centre are given, as many other points and as many other tangents are given at an equal diſtance on the other ſide of the centre. And an aſymptote is to be conſidered as a tangent, and its infinitely remote extremity (if we may ſay ſo) is a point of contact. Conceive the point of contact of any tangent removed in infinitum, and the tangent will degenerate into an aſymptote, and the conſtructions of the preceding problem will be changed into the conſtructions of thoſe problems wherein the aſymptote is given.

Plate 12, Figure 1
Plate 12, Figure 1

After the trajectory is deſcribed, we may find its axes and foci in this manner. In the conſtruction and figure of lem. 21. (Pl. 12. Fig. 1.) let thoſe legs BP, CP, of the moveable angles PBN, PCN, by the concourſe of which the trajectory was deſcribed, be made parallel one to the other; and retaining that poſition, let them revolve about their poles in that figure. In the mean while let the other legs CN, BN of thoſe angles, by their concourſe K or k, deſcribe the circle BKGC. Let O be the centre of this circle; and from this centre upon the ruler MN, wherein thoſe legs CN, BN did concur while the trajectory was deſcribed, let fall the perpendicular OH meeting the circle in K and L. And when thoſe other legs CK, BK meet in the point K that is neareſt to the ruler, the firſt legs CP, BP will be parallel to the greater axis

and