THE MATHEMATICAL PRINCIPLES [BOOK 1 trill be as the velocity, and therefore as the right line whose square is equal to the area ABFD, and the triangle ICK proportional to the time will be given, and therefore KN will be reciprocally as the altitude 1C : that is (if there be given any quantity Q, and the altitude 1C be called A), as -T-. This quantity — call Z, and suppose the magnitude of Q, to oe such that in some case v/ABFD may be to Z as IK to KN, and then in all cases V ABFD will be to Z as IK to KN, and ABFD to ZZ as IK2 to KN2, and by division ABFD — ZZ to ZZ as IN2 to KN2, and therefore V ABFD — ZZ to Z, or — as IN to KN; and therefore A x KN Q. x IN \vill be equal to — — — . Therefore since YX X XC is to A X KN — ZZ Q. X IN x CX2 as CX2, to AA, the rectangle XY X XC will be equal toAAv/ABFD — ZZ. Therefore in the perpendicular DF let there be taken continually I)//, IV Q ax ex2 equal to — , =. — respectively, and 2 v/ ABFD — ZZ 2AA V ABFD — ZZ let the curve lines ab, ac, the foci of the points b and c, be described : and from the point V let the perpendicular Va be erected to the line AC, cut ting off the curvilinear areas VD&a, VDra, and let the ordi nates Es:? E#, be erected also. Then because the rectangle D& X IN or DbzR is equal to half the rectangle A X KN, or to the triangle ICK ; and the rectangle DC X IN or Dc.rE is equal to half the rectangle YX X XC, or to the triangle XCY; that is, because the nascent particles I)6d3, ICK of the areas VD/>#, VIC are always equal; and the nascent particles Dc^-E, XCY of the areas VDca, VCX are always equal : therefore the generated area VD6a will be equal to the generated area VIC, and there fore proportional to the time; and the generated area VDco- 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 pro portional to it VD/>«; and thence will be given the altitude of the body CD or CI ; and the area VDca, and the sector VCX equal there' o, together with its angle VCL But the angb VCI, and the altitude CI being given, there is also given the place I, 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 1C drawn through the centre falls perpendicularly upon the trajectory VTK; which comes to pass when the right lines IK and NK become equal; that is, when the area ABFD ig C'nl to ZZ.