SEC. VLJ OF NATURAL PHILOSOPHY. 311 therefore, these velocities be expressed by those perpendiculars DE, de ; arid let DF be the velocity which it acquires in D by falling from B in the resisting medium. And if from the centre C with the interval OF we describe the circle F/M meeting the right lines de and AB in / and M, then M will be the place to which it would thenceforward, without farther resistance, ascend, and (//"the velocity it would acquire in d. Whence, also, if FO- represent the moment of the velocity which the body D, in de scribing the least space DC/, loses by the resistance of the medium ; and CN be taken equal to Cg ; then will N be the place to which the body, if it met no farther resistance, would thenceforward ascend, and MN will be the decrement of the ascent arising from the loss of that velocity. Draw F/n perpendicular to dft and the decrement Fg of the velocity DF gener ated by the resistance DK will be to the increment fm of the same velo city, generated by the force CD, as the generating force DK to the gener ating force CD. But because of the similar triangles F////, Fhg, FDC, fm is to Fm or Dd as CD to DF ; and, ex ceqno, Fg to Dd as DK to DF. Also Fh is to Fg as DF to CF ; and, ex ax/uo perturbate, Fh or MN to Do1 as DK to CF or CM ; and therefore the sum of all the MN X CM will be equal to the sum of all the Dd X DK. At the moveable point M suppose always a rectangular ordinate erected equal to the inde terminate CM, which by a continual motion is drawn into the whole length Aa ; and the trapezium described by that motion, or its equal, the rectangle Aa X