Page:Newton's Principia (1846).djvu/176

will 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 IC; that is (if there be given any quantity Q, and the altitude IC be called A), as $\scriptstyle {\frac {Q}{A}}$. This quantity $\scriptstyle {\frac {Q}{A}}$ call Z, and suppose the magnitude of Q to be such that in some case $\scriptstyle {\sqrt {ABFD}}$ may be to Z as IK to KN, and then in all cases $\scriptstyle {\sqrt {ABFD}}$ 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 $\scriptstyle {\sqrt {ABFD-ZZ}}$ to Z; or $\scriptstyle {\frac {Q}{A}}$ as IN to KN; and therefore A $\scriptstyle \times$ KN will be equal to $\scriptstyle {\frac {Q\times IN}{\sqrt {ABFD-ZZ}}}$. Therefore since YX $\scriptstyle \times$ XC is to A $\scriptstyle \times$ KN as CX², to AA, the rectangle XY $\scriptstyle \times$ XC will be equal to $\scriptstyle {\frac {Q\times IN\times CX^{2}}{AA{\sqrt {ABFD-ZZ}}}}$. Therefore in the perpendicular DF let there be taken continually Db, Dc equal to $\scriptstyle {\frac {Q}{2{\sqrt {ABFD-ZZ}}}}$, $\scriptstyle {\frac {Q\times CX^{2}}{2AA{\sqrt {ABFD-ZZ}}}}$ respectively, and 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, cutting off the curvilinear areas VDba, VDca, and let the ordinates Ez, Ex, be erected also. Then because the rectangle Db $\scriptstyle \times$ IN or DbzE is equal to half the rectangle A $\scriptstyle \times$ KN, or to the triangle ICK; and the rectangle Dc $\scriptstyle \times$ IN or DcxE is equal to half the rectangle YX $\scriptstyle \times$ XC, or to the triangle XCY; that is, because the nascent particles DbzE, ICK of the areas VDba, VIC are always equal; and the nascent particles DcxE, XCY of the areas 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 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 IC drawn through 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.