SEC. VII.] OF NATURAL PHILOSOPHY 343 LEMMA VI. The same supposition remaining, the fore- mentioned bodies are equally acted OIL by the water Jlowin g through the canal. This appears by Lein. V and the third Law. For tht water and the bodies act upon each other mutually and equally. LEMMA VIL If the water be at rest in the canal, and these bodies move with equil ve locity and the contrary way through the canal, their resistances will be equal among themselves. This appears from the last Lemma, for the relative motions remain the same among themselves. SCHOLIUM. The case is the same of all convex and round bodies, whose axes coincide with the axis of the canal. Some difference may arise from a greater or less friction; but in these Lemmata we suppose the bodies to be perfectly smooth, and the medium to be void of all tenacity and friction ; and that those parts of the fluid which by their oblique and superfluous motions may disturb, hinder, and retard the flux of the water through the canal, are at rest amorg themselves ; being fixed like water by frost, and adhering to the fore and hinder parts of the bodies in the manner explained in the Scholium of the last Proposition : for in what follows we consider the very least resistance that round bodies described with the greatest given trans verse sections can possibly meet with. Bodies swimming upon fluids, when they move straight forward, cause the fluid to ascend at their fore parts and subside at their hinder parts, especially if they are of an obtuse figure ; and thence they meet with a little more resistance than if they were acu*-e at the head and tail. And bodies moving in elastic fluids, if they are obtuse behind and before, con dense the fluid a little more at their fore parts, and relax the same at theii hinder parts ; and therefore meet also with a little more resistance than ii they were acute at the head and tail. But in these Lemmas and Proposi tions we are not treating of elastic but non-elastic fluids; not of bodies floating on the surface of the fluid, but deeply immersed therein. And when the resistance of bodies in non-elastic fluids is once known, we may then augment this resistance a little in elastic fluids, as our air; and in the surfaces of stagnating fluids, as lakes and seas. PROPOSITION XXXVIII. THEOREM XXX. If a globe move uniformly forward in a compressed, infinite, and no?t' elastic fluid, its resistance is to the force by which its whole