to DB, and through the vertex F describe the hyperbola FTVE, whose conjugate semi-diameters are DB and DF, and which cuts DA in E, and DP, DQ in T and V; and the time of the whole ascent will be as the hyperbolic sector TDE.
For the decrement PQ of the velocity, produced in a given particle of time, is as the sum of the resistance AP² + 2BAP and of the gravity AB² - BD², that is, as BP² - BD². But the area DTV is to the area DPQ as DT² to DP²; and, therefore, if GT be drawn perpendicular to DF, as GT² or GD² - DF² to BD², and as GD² to BP², and, by division, as DF² to BP² - BD². Therefore since the area DPQ is as PQ, that is, as BP² - BD², the area DTV will be as the given quantity DF². Therefore the area EDT decreases uniformly in each of the equal particles of time, by the subduction of so many given particles DTV, and therefore is proportional to the time. Q.E.D.
Case 3. Let AP be the velocity in the descent of the body, and AP² + 2BAP the force of resistance, and BD² - AB² the force of gravity, the angle DBA being a rirht one. And if with the centre D, and the principal vertex B, there be described a rectangular hyperbola BETV cutting DA, DP, and DQ produced in E, T, and V; the sector DET of this hyperbola will be as the whole time of descent.
For the increment PQ of the velocity, and the area DPQ proportional to it, is as the excess of the gravity above the resistance, that is, as BD² - AB² - 2BAP - AP² or BD² - BP². And the area DTV is to the area DPQ as DT² to DP²; and therefore as GT² or GD² - BD² to BP², and as GD² to BD², and, by division, as BD² to BD² - BP². Therefore since the area DPQ is as BD² - BP², the area DTV will be as the given quantity BD². Therefore the area EDT increases uniformly in the several equal particles of time by the addition of as many given particles DTV, and therefore is proportional to the time of the descent. Q.E.D.
Cor. If with the centre D and the semi-diameter DA there be drawn through the vertex A an arc At similar to the arc ET, and similarly subtending the angle ADT, the velocity AP will be to the velocity which the body in the time EDT, in a non-resisting space, can lose in its ascent, or acquire in its descent, as the area of the triangle DAP to the area of the sector DAt; and therefore is given from the time given. For the velocity in a non-resisting medium is proportional to the time, and therefore to this sector; in a resisting medium, it is as the triangle; and in both mediums, where it is least, it approaches to the ratio of equality, as the sector and triangle do.
