thing, or, if it so please thee, put its truth to the test of experience. First of all the following must be premised: 1. If there are two fluids of different density, the weights of which respectively are in the ratio G to L; the altitudes of cylinders of equal weight, and having the same base, will be in the ratio L to G. 2. Therefore, if the altitude A C of one fluid contained in the vessel A D to be the altitude E F of the other fluid contained in an open tube, as L to D; the fluids so placed will remain at rest. [Illustration] 3. Therefore, if A C to E F be in a greater ratio than L to G, the fluid in the tube will ascend; or if the tube be not sufficiently long, the fluid will escape by the orifice E. (These are proved by Hydrostatics.) 4. It is possible to have two fluids of different gravity, which are capable of being mixed one with the other. 5. It is possible to have a filter, strainer or other separator, by means of which the lighter fluid may be separated from the heavier. _Construction_ These being pre-supposed, I construct Perpetual Motion in the following manner: Let two fluids of different gravity and capable of mixing together (which is possible by Hyp. 4) be taken in any quantities, in equal quantities, if desired; let the ratios of their gravities be first determined, which suppose as G to L, the heavier to the lighter; and being mixed, let a vessel, A D, be filled to A. This having been done, let a tube be taken, open at both ends E F; and of such a length that A C : E F > 2 L : G + L; and the orifice F stopped, or rather filled with a filter or some substance separating the lighter fluid from the heavier (as is possible also by Hyp. 5); when the tube filled in this manner with fluid is immersed to the bottom of the vessel C D; I say that the fluid will continually ascend by the orifice of the tube F, and by the orifice E will fall into the fluid below. _Demonstration_ Because the orifice of the tube F is occupied by a filter (by Constr.) which separates the lighter fluid from the heavier; it follows, that if the tube be immersed to the bottom of the vessel, the fluid lighter by itself, which is mixed with the heavier fluid, must ascend in the tube, and as it will ascend above the surface of the surrounding fluid as A C : E F = 2 L : G + L : which is (by Const.) A C : E F > 2 L : G + L, it necessarily follows (by Hyp. 3) that the lighter liquid, through the orifice E, will fall in the vessel below; there it again mixes with the heavier (by Hyp. 4); and then, penetrating the filter, ascends again into the tube, and escapes by the upper orifice. So, therefore, the flow is continued perpetually.--Q. E. D. _Corollary_