*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*.

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

*T;*