309. Corol. 3. The pressure of the fluid on any horizontal surface or plane, is equal to the weight of a column of the fluid, whose base is equal to that plane, and altitude is its depth below the upper surface of the fluid. PROPOSITION LXI. 310. When a Fluid is Pressed by its own Weight, or by any other Force; at any Point it Presses Equally, in all Directions what ever. THIS arises from the nature of fluidity, by which it yields to any force in any direction. If it cannot recede from any force applied, it will press against other parts of the fluid in the direction of that force. And the pressure in all direc tions will be the same: for if it were less in any part, the fluid would move that way, till the pressure be equal every way. 311. Corol. 1. In a vessel containing a fluid; the pressure is the same against the bottom, as against the sides, or even upwards at the same depth. 312. Corol. 2. Hence, and from the last proposition, if ABCD be a vessel of water, and there be taken, in the base produced, DE, to represent the pressure at the bottom; joining AE, and drawing any parallels to the base, as FG, HI; then shall FG represent the pressure at A F H E D the depth AG, and HI the pressure at the depth AI, and sờ on; because the parallels FG, HI, ED, by sim. triangles, are as the depths AG, AI, AD: which are as the pressures, by the proposition. And hence the sum of all the FG, HI, &c, or area of the triangle ADE, is as the pressure against all the points G, I, &c, that is, against the line AD. But as every point in the line CD is pressed with a force as DE, and that thence the ́pressure on the whole line CD is as the rectangle ED. DC; while that against the side is as the triangle ADE or AD. DE; therefore the pressure on the horizontal line DC, is to the pressure against the vertical line DA, as DC to DA. And hence, if the vessel be an upright rectangular one, the pressure on the bottom, or whole weight of the fluid, is to the psessure against one side, as the base is to half that side. Therefore the weight of the fluid is to the pressure against 1 all all the four upright sides, as the base is to half the upright surface. And the same holds true also in any upright vessel, whatever the sides be, or in a cylindrical vessel. Or, in the cylinder, the weight of the fluid, is to the pressure against the upright surface, as the radius of the base is to double the altitude. Also, when the rectangular prism becomes a cube, it appears that the weight of the fluid on the base, is double the pressure against one of the upright sides, or half the pressure against the whole upright surface. 313. Corol. 3. The pressure of a fluid against any upright surface, as the gate of a sluice or canal, is equal to half the weight of a column of the fluid whose base is equal to the surface pressed, and its altitude' the same as the altitude of that surface. For the pressure on a horizontal base equal to the upright surface, is equal to that column; and the pressure on the upright surface, is but half that on the base, of the same area. So that, if b denote the breadth, and d the depth of such a gate or upright surface; then the pressure against it, is equal to the weight of the fluid whose magnitude is bď2 = LAB. AD2. Hence, if the fluid be water, a cubic foot of which weighs 1000 ounces, or 624 pounds; and if the depth AD be 12 feet, the breadth AB 20 feet; then the content, or AB. AD, is 1440 feet; and the pressure is 1440000 ounces, or 90000 pounds, or 40 tons weight nearly. PROPOSITION LXII. 314. The pressure of a Fluid on a Surface any how immersed in it, either Perpendicular, or Horizontal, or Oblique; is Equal to the Weight of a Column of the Fluid, whose Base is equal to the Surface pressed, and its Altitude equal to the Depth of the Centre of Gravity of the Surface pressed below the Top or Surface of the Fluid. FOR, Conceive the surface pressed to be divided into innumerable sections parallel to the horizon; and let s denote any one of those horizontal sections, also d its distance or depth below the top surface of the fluid. Then, by art. 309, the pressure of the fluid on the section is equal to the weight of ds; consequently the total pressure on the whole surface is equal to all the weights ds. But, if b denote the whole surface pressed, and g the depth of its centre of gravity below the top of the fluid; then, by art. 256 or 259, bg is equal to to the sum of all the ds. Consequently the whole pressure of the fluid on the body or surface b, is equal to the weight of the bulk bg of the fluid, that is, of the column whose base is the given surface b, and its height is g the depth of the centre of gravity in the fluid. PROPOSITION LXIII. 315. The Pressure of a Fluid, on the Base of the Vessel in which it is contained, is as the Base and Perpendicular Altitude; whatever be the Figure of the Vessel that contains it. If the sides of the base be upright, so that it be a prism of a uniform width throughout; then the case is evident; for then the base supports the whole fluid, and the pressure is just equal to the weight of the fluid. But if the vessel be wider at top than bottom; then the bottom sustains, or is pressed by, only the part contained within the upright lines ac, bD; because the parts Aca, BDb are supported by the sides AC, BD; and those parts have no other effect on the part abDc than keeping it in its position, by the lateral pressure against ac and bD, which Aa bB does not alter its perpendicular pressure downwards. And thus the pressure on the bottom is less than the weight of the contained fluid. And if the vessel be widest at bottom; then the bottom is still pressed with a weight which is equal to that of the whole upright column abDc. For, as the parts of the fluid are in equilibrio, all the parts have an equal pressure at the same depth; so that the parts within cc and do press equally as those in cd, and therefore equally the same as if the sides of the vessel had gone upright to a and b, the defect of fluid in the parts Aca and BDb being exactly compensated by the downward pressure or resistance of the sides AC and BD against the contiguous fluid. And thus the pressure on the base may be made to exceed the weight of the contained fluid, in any proportion whatever. So that, in general, be the vessels of any figure whatever, regular or irregular, upright or sloping, or variously wide and narrow in different parts, if the bases and perpendicular altitudes be but equal, the bases always sustain the same pressure. And as that pressure, in the regular upright VOL. II, Q vessel, vessel, is the whole column of the fluid, which is as the base and altitude; therefore the pressure in all figures is in that same ratio. 316. Corol. 1. Hence, 'when the heights are equal, the pressures are as the bases. And when the bases are equal, the pressure is as the height. But when both the heights and bases are equal, the pressures are equal in all, though their contents be ever so different. 317. Corol. 2. The pressure on the base of any vessel, is the same as on that of a cylinder, of an equal base and height. 318. Carol. 3. If there be an inverted syphon, or bent tube, ABC, containing two different fluids CD, ABD, that balance each other, or rest in equilibrio; then their heights in the two legs, AE, CD, above the point of meeting, will be reciprocally as their densities. For if they do not meet at the bottom, the part BD balances the part BE, and therefore the part CD balances the part AE; that is, the weight of CD is equal to the weight of AE. And as the surface at D is the same, where they act against each other, therefore AE CD density of CD: density of AE. A So, if CD be water, and AE quicksilver, which is near 14 times heavier; then CD will be 14AE; that is, if AE be 1 inch, CD will be 14 inches; if AE be 2 inches, GD will be 28 inches; and so on. PROPOSITION LXIV. 319. If a Body be Immersed in a Fluid of the Same Density or Specific Gravity; it will Rest in any Place where it is put. But a Body of Greater Density will Sink; and one of a Less Density will Rise to the Top, and Float. THE body, being of the same density, or of the same weight with the like bulk of the fluid, will press the fluid under it, just as much as if its space was filled with the fluid itself. The pressure then all around it will be the same as if the fluid were in its place; consequently there is no force, neither upward nor downward, to put the body out of its place. And therefore it will remain wherever it is put. But But if the body be lighter; its pressure downward will be less than before, and less than the water upward at the samę depth; therefore the great force will overcome the less, and push the body upward to A. And if the body be heavier than the fluid, the pressure downward will be greater than the fluid at the same depth; therefore the greater force will prevail, and carry the body down to the bottom at c. 320. Corol. 1. A body immersed in a fluid, loses as much weight, as an equal bulk of the fluid weighs. And the fluid gains the same weight. Thus, if the body be of equal density with the fluid, it loses all its weight, and so requires no force but the fluid to sustain it. If it be heavier, its weight in the water will be only the difference between its own weight and the weight of the same bulk of water; and it requires a force to sustain it just equal to that difference. But if it be lighter, it requires a force equal to the same difference of weights to keep it from rising up in the fluid: 321. Corol. 2. The weights lost, by immerging the same body in different fluids, are as the specific gravities of the fluids. And bodies of equal weight, but different bulk, lose, in the same fluid, weights which are reciprocally as the spe cific gravities of the bodies, or directly as their bulks. 4 322. Corol. 3. The whole weight of a body which will float in a fluid, is equal to as much of the fluid, as the immersed part of the body takes up, when it floats For the pressure under the floating body, is just the same as so much of the fluid as is equal to the immersed part; and therefore the weights are the same. 323. Corol. 4. Hence the magnitude of the whole body, is to the magnitude of the part immersed, as the specific gravity of the fluid, is to that of the body. For, in bodies of equal weight, the densities, or specific gravities, are reciprocally as their magnitudes. 324. Corol. 5. And because, when the weight of a body taken in a fluid, is subtracted from its weight out of the fluid, the difference is the weight of an equal bulk of the fluid, this therefore is to its weight in the air, as the specific gravity of the fluid, is to that of body. Therefore, if w be the weight of a body in air, w its weight in water, or any fluid, Q.2 then |