854 THE MATHEMATICAL PRINCIPLES [BOOK IL COROL. 2. But the time being augmented in an arithmetical progression, the sum of that greatest velocity and the velocity in the ascent, and also their difference in the descent, decreases in a geometrical progression. COROL. 3. Also the differences of the spaces, which are described in equal differences of the times, decrease in the same geometrical progression. COROL. 4. The space described by the body is the difference of two spaces, whereof one is as the time taken from the beginning of the descent, and the other as the velocity; which [spaces] also at the beginning of the descent are equal among themselves. PROPOSITION IV. PROBLEM II. Supposing the force of gravity in any similar medium to be uniform, and to tend perpendicularly to the plane of the horizon ; to define the motion of a projectile therein, which suffers resistance proportional to its velocity. Let the projectile go from any place D in the direction of any right line DP, and let its velocity at the beginning of the motion be expounded by the length DP. From the point P let fall the perpendicular PC on the horizontal line DC, and cut DC in A, so that DA may be to AC as the resistance of the medium arising from the motion up wards at the beginning to the force of grav ity; or (which comes to the same) so that t ie rectangle under DA and DP may be to that under AC and CP as the whole resist ance at the beginning of the motion to the force of gravity. With the asymptotes DC, CP describe any hyperbola GTBS cut ting the perpendiculars DG, AB in G and B ; complete the parallelogram DGKC, and let its side GK cut AB in Q,. Take N in the same ratio to QB as DC is in to CP ; and from any point R of the right line DC erect RT perpendicular to it, meeting the hy] erbola'in T, and the right lines EH, GK, DP in I, t, and V ; in that perpendicular take Vr equal to — ~- , or which is the same thing, take Rr equal to (""""PIT? — ^T— ; and the projectile in the time DRTG will arrive at the point r describing the curve line DraF, the locus of the point r ; thence it will come to its greatest height a in the perpendicular AB j and afterwards