and there will remain the density of the medium in P, as . Let the spiral be given, and, because of the given ratio of OS to OP, the density of the medium in P will be as . Therefore in a medium whose[Pg 289] density is reciprocally as SP the distance from the centre, a body will revolve in this spiral. Q.E.D. COR. 1. The velocity in any place P, is always the same wherewith a body in a non-resisting medium with the same centripetal force would revolve in a circle, at the same distance SP from the centre. COR. 2. The density of the medium, if the distance SP be given, is as , but if that distance is not given, as . And thence a spiral may be fitted to any density of the medium. COR. 3. The force of the resistance in any place P is to the centripetal force in the same place as to OP. For those forces are to each other as and TQ, or as and , that is, as and PQ, or and OP. The spiral therefore being given, there is given the proportion of the resistance to the centripetal force; and, vice versa, from that proportion given the spiral is given. COR. 4. Therefore the body cannot revolve in this spiral, except where the force of resistance is less than half the centripetal force. Let the resistance be made equal to half the centripetal force, and the spiral will coincide with the right line PS, and in that right line the body will descend to the centre with a velocity that is to the velocity, with which it was proved before, in the case of the parabola (Theor. X, Book I), the descent would be made in a non-resisting medium, in the subduplicate ratio of unity to the number two. And the times of the descent will be here reciprocally as the velocities, and therefore given. COR. 5. And because at equal distances from the centre the velocity is the same in the spiral PQR as it is in the right line SP, and the length of the spiral is to the length of the right line PS in a given ratio, namely, in the ratio of OP to OS; the time of the descent in the spiral will be to the time of the descent in the right line SP in the same given ratio, and therefore given. COR. 6. If from the centre S, with any two given intervals, two circles are described; and these circles remaining, the angle which the spiral makes with the radius PS be any how changed; the number of revolutions which the body can complete in the space between the circumferences of those circles, going round in the spiral from one circumference to another, will be as , or as the tangent of the angle which the spiral makes with the radius PS; and[Pg 290] the time of the same revolutions will be as , that is, as the secant of the same angle, or reciprocally as the density of the medium.