com A Hi E SEC. VII.] OF NATURAL PHILOSOPHY- 341 ratio compounded of the duplicate ratio of the velocities and the duplicate ratio of the diameters, and the ratio of the density of the mediums. COR. 2. If the breadth of the canal be not infinitely increased but the cylinder go forward in the direction of its length through an included quiescent medium, its axis all the while coinciding with the axis of the canal, its resistance will be to the force by which its whole motion, in the time in which it describes four times its length, K ............. I... ........ L may be generated or destroyed, in a ratio pounded of the ratio of EF2 to EF2 — i once, and the ratio of EF2 to EF2 — PQ,2 twice, and the ratio of the density of the medium to the density of the cylinder. COR. 3. The same thing supposed, and that a length L is to the quadruple of the length of the cylinder in a ratio compounded of the ratio EF2'-- iPQ2 to EF2 once, and the ratio of EF2 — PQ,2 to EF2 twice; the resistance of the cylinder will be to the force by which its whole motion, in the time during which it describes the length L, may be destroyed or generated, as the density of the medium to the density of the cylinder. SCHOLIUM. In this proposition we have investigated that resistance alone which arises from the magnitude of the transverse section of the cylinder, neg lecting that part of the same which may arise from the obliquity of the motions. For as, in Case 1, of Prop. XXXVL, the obliquity of the mo tions with which the parts of the water in the vessel converged on every side to the hole EF hindered the efflux of the water through the hole, so, in this Proposition, the obliquity of the motions, with which the parts of the water, pressed by the antecedent extremity of the cylinder, yield to the pressure, and diverge on all sides, retards their passage through the places that lie round that antecedent extremity, toward the hinder parts of the cylinder, and causes the fluid to be moved to a greater distance; which in creases the resistance, and that in the same ratio almost in which it dimin ished the efflux of the water out of the vessel, that is, in the duplicate ratio of 25 to 21, nearly. And as, in Case 1, of that Proposition, we made the parts of the water pass through the hole EF perpendicularly and in the greatest plenty, by supposing all the water in the vessel lying round the cataract to be frozen, and that part of the water whose motion was oblique, and useless to remain without motion, so in this Proposition, that the obliquity of the motions may be taken away, and the parts of the water may give the freest passage to the cylinder, by yielding to it witli the most direct and quick motion possible, so that only so much resistance may re