Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/249

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Sect. VIII.
of Natural Philoſophy.

area's VDba, VIC are always equal; and the naſcent particles DcxE. XCY of the area's VDca, VCX are always equal; therefore the generated area VDba will be equal to the generated area VIC, and therefore proportional to the time; and the generated area VDca is equal to the generated ſector VCX. If therefore any time be given during which the body has been moving from V, there will be alſo given the area proportional to it VDba; and thence will be given the altitude of the body CD or CI; and the area VDca, and the ſector VCX equal thereto, together with its angle VCI. But the angle VCI, and the altitude CI being given, there is alſo given the place in which the body will be found at the end of that time. Q. E. I.

Cor. 1. Hence the greateſt and leaſt altitudes of the bodies, that is the apſides of the trajectories, may be found very readily. For the apſides are thoſe points in which a right line IC drawn thro' the centre falls perpendicularly upon the trajectory VIK; which comes to paſs when the right lines IK and NK become equal; that is, when the area ABFD is equal to ZZ.

Cor. 2. So alſo, the angle KIN in which the trajectory at any place cuts the line IC, may be readily found by the given altitude IC of the body: to wit, by making the ſine of that angle to radius as KN to IK; that is as Z to the ſquare root of the area ABFD.

Plate 17, Figure 5

Cor. 5. If to the centre C (Pl. 17. Fig. 5.) and the principal vertex V there be deſcribed a conic ſection VRS; and from any point thereof as R, there be drawn the tangent RT meeting the axe CV indefinitely produced, in the point