COR. 1. Therefore Rr is equal to ; and therefore if RT be produced to X so that RX may be equal to , that is, if the parallelogram ACPY be completed, and DY cutting CP in Z be drawn, and RT be produced till it meets DY in X; Xr will be equal to , and therefore proportional to the time. COR. 2. Whence if innumerable lines CR, or, which is the same, innumerable lines ZX, be taken in a geometrical progression, there will be as many lines Xr in an arithmetical progression. And hence the curve DraF is easily delineated by the table of logarithms. COR. 3. If a parabola be constructed to the vertex D, and the diameter DG produced downwards, and its latus rectum is to 2 DP as the whole resistance at the beginning of the notion to the gravitating force, the velocity with which the body ought to go from the place D, in the direction of the right line DP, so as in an uniform resisting medium to describe the curve DraF, will be the same as that with which it ought to go from the same place D in the direction of the same right line DP, so as to describe[Pg 256] a parabola in a non-resisting medium. For the latus rectum of this parabola, at the very beginning of the motion, is ; and Vr is or . But a right line, which, if drawn, would touch the hyperbola GTS in G, is parallel to DK, and therefore Tt is , and N is . And therefore Vr is equal to , that is (because DR and DC, DV and DP are proportionals), to ; and the latus rectum comes out , that is (because QB and CK, DA and AC are proportional), , and therefore is to 2DP as DP × DA to CP × AC; that is, as the resistance to the gravity. Q.E.D. COR. 4. Hence if a body be projected from any place D with a given velocity, in the direction of a right line DP given by position, and the resistance of the medium, at the beginning of the motion, be given, the curve DraF, which that body will describe, may be found. For the velocity being given, the latus rectum of the parabola is given, as is well known. And taking 2DP to that latus rectum, as the force of gravity to the resisting force, DP is also given. Then cutting DC in A, so that GP × AC may be to DP × DA in the same ratio of the gravity to the resistance, the point A will be given. And hence the curve DraF is also given. COR. 5. And, on the contrary, if the curve DraF be given, there will be given both the velocity of the body and the resistance of the medium in each of the places r. For the ratio of CP × AC to DP × DA being given, there is given both the resistance of the medium at the beginning of the motion, and the latus rectum of the parabola; and thence the velocity at the beginning of the motion is given also. Then from the length of the tangent[Pg 257] L there is given both the velocity proportional to it, and the resistance proportional to the velocity in any place r.