2. We all along fuppofe, and particularly in the laft experiment, that STATICS. the fmalleft glafs tubes employ'd are of a moderate fize, and not exceeding narrow; or, in cafe they are, that allowance be made for the property of water's rifing in them to a greater height, than can be attributed to the balance either of water or oil, that keeps it there fufpended. The direct preffure fuftain'd by a body, placed any where under water, with its upper furface parallel to the horizon, is that of a column of water, whose base is the horizontal fuperficies of the body, and height the perpendicular depth of the water; and if the water preffing upon a body, be contained in open tubes, its preffure is to be estimated by a column of the fame, the bafe whereof equals the lower orifice of the pipe, and height a perpendicular from thence to the top of the water and this, tho' the pipes ftand obliquely, be irregularly shaped, or wider in fome parts than the faid orifice. The former part of our propofition we might, with Stevinus, make more general, and thus demonftrate. Suppofe ABCD, a folid rectangular fi- Fig. 10. gure of water, whofe bafe DC, is parallel to the horizon, and height GE, a perpendicular let fall from the upper to the lower furface of the water; if that part of the bafe EF, be charged with more water than GEFH, the overplus must come from the adjacent water; fuppofe it then, if poffible, to come from AGED, and HBCF: then, for the fame reafon, the bafe DE, will have more water incumbent thereon than AGED; and the like may be faid of FC, whence the whole bafe DC will fuftain a greater weight than that of the whole water ABCD, which is abfurd. By the like reafoning, we prove that the bafe DC, fuftains no lefs a weight than that of the water GHFE; fince, then, it fuftains neither a greater weight nor a lefs, than the column of water GHFE, it must, exactly fuftain the fame. To this demonftration, we fhall annex fenfible experiments, to confirm and illuftrate each part of our propofition. Firft then, fuck oil of turpen- Fig. 11tine into a flender, glafs fyphon, till it fill the fhorter leg, and rife two or three inches high in the longer; then nimbly stop the upper orifice with your finger, and plunge the lower part of the fyphon into a deep glass of water, till the furface of the oil in the longer leg rifes but little higher than that of the external water, and upon removing your finger, the furface of the oil will either intirely, or almoft, preferve its former station; when if the fyphon be immerfed deeper, the oil in the fhorter leg will be deprefs'd; but upon gently raifing the fyphon towards the top of the water, it will exceed its former ftation, gradually flow out in drops, and emerge to the furface. Now, fince the water, at firft, kept the oil in the longer leg, fufpended no higher, than it would have been by a cylinder of water, equal in bafis to the orifice of the fhorter leg, and reaching, in length, from thence to the top of the water, as may be easily feen in a fyphon with legs fufficiently long; and fince, when by raifing the fyphon,, the: STATICS. the height of the incumbent water was diminish'd, the oil over-balanced the water, and run out; it may be fairly concluded, that tho' much of the water in the veffel, lay higher than the immerfed orifice of the fyphon, yet no more gravitated thereon, than what stood directly over it; that is, fuch a column as our paradox requires. Fig. 12, Fig. 13. COROLLAR Y. It follows from hence, that water may prefs as regularly upon an immerfed body, when not reftrained by pipes, as when included in them; we may, therefore, properly conceive a particular column of water, tho' it has no actual bounds, in an unlimited quantity of the fame fluid. The firft part of our paradox holds equally true, whether the incumbent water be free or confined in vefels of any irregular fhape. Thus, tho' the fhorter leg of the fyphon be fashion'd into a funnel, and filled with water, the oil in the other leg will refift the preffure thereof, fo that the furface of the oil in the longer leg will rife but little above that of the water in the funnel. For farther confirmation hereof, we try'd the experiment in a fyphon; in one leg whereof, a glafs fphere was made to communicate with the upper parts of the fame. Into the uniform leg of this fyphon, we poured a proper quantity of oil of turpentine, and fill'd the lower and globular part of the other, with water; which proved infufficient to fuftain the oil at a greater height, than if this leg alfo had been uniform: all the water in the spherical cavity, which fell not directly over the lower orifice thereof, being fupported by the fides of the fame. And when the irregular leg was fill'd with oil, and the other with water, the former fluid would not fuftain the latter to an equal height, notwithstanding its excess of quantity contain'd in the concave fphere. To clear up this matter ftill farther, we poured mercury into the fyphon ABCD, till it almost reach'd the bottom of the globular part, in the fhorter leg, and to an equal height in the longer; then letting a proper quantity of water run into the longer leg, it impell'd the mercury in the fhorter leg, till it more than half filled the cavity of the spherical part E, the tube AB, not proving fufficiently long to allow of its being quite fill'd; when, we obferv'd the furface of the quick-filver HG, to lie as high as the different fpecific gravities of the two fluids required: fo that notwithstanding the great weight of the mercury contain'd in the concave fphere E, no more preffed upon the fubjacent flender part of the leg EC, than food directly over its lower orifice: whence the water in the leg AB, appear'd prefs'd only as much as if the leg CD, had been uniform, and without the ipheri-. cal cavity E. And thus, if the fphere had been made larger, afmall quantity of water in the leg AB, would have fomewhat raifed a much greater weight of mercury than its own. The fecond part of our prefent propofition is proved thus. We took three open, glafs pipes, of the irregular figures in the fcheme, and plunged them in a glafs vefiel of water, the two crooked ones ftanding very oblique to the upper furface thereof, and the higher ends of them all coming thro' thro' particular ftrait holes made in a cork fitted to the veffel's mouth, STATICS. whilft the water afcended to a certain height, thro' the lower orifice of every pipe; then pouring a confiderable quantity of oil of turpentine thro' a funnel, into the veffel, the water was, thereby, impelled to the same height, of two or three inches, into the feveral tubes: whence 'tis apparent, that notwithstanding the irregularity of fhape in the pipes, and the different wideness in each of their bores, the preffure of the contain'd water upon their lower orifices did not exceed that of five cylinders of the fame, with each of thofe orifices for bafes, and the perpendicular depth of the water and oil therefrom for their height. For had each of the pipes contain'd only fuch a cylinder, its upper furface would have ftood at the fame. height; and if it did not deprefs the fubjacent fluid as much as the external force tended to raise it, that excefs of preffure muft have impelled it higher. Since then the water rofe to the fame height in the feveral pipes, tho' two of them, being greatly inclined, contain'd much more water than if they had ftood erect; by a like way of reafoning, we may conclude, that the imaginary plain cutting their immerfed ends, fuftain'd no greater preffure than that of a fhorter, erect cylinder of water. For in all cafes where the pipe is either inclined, or wider in one part than in another, the weight of the liquor it contains, is not wholly fupported by the body contiguous to the lower orifice, but in fome meafure by the fides of the pipe. Thus, when oil of turpentine, in a flender tube, balances the prefTure of the external water, if the tube be barely inclined to the fides of the containing glass, a confiderable quantity of water will get up into the pipe; because the oil no longer refting wholly upon the water, but partly upon the tube, its preffure on the water is diminished, while that, continuing its force, impels up the oil, and intrudes it into the pipe; which being reftored to a perpendicular pofition, the oil will again deprefs, and drive the water out of its cavity. We farther caufed three pipes to be Fig. 14. blown differently from the former, and filled their winding and irregular parts, as alfo their uniform ftems, to a proper height, with oil of turpentine ; we then plung'd them to a due depth under water, and there unftopping their upper ends, the furface of the oil they contain'd appear'd at the fame height above the water, it would have done had the tubes been strait; as we found by a cylindrical one fill'd and immerfed as the reft; tho' the quantity of oil in one of thefe pipes were, perhaps, thrice as great as that in the ftrait one. Hence we may fairly conclude, that the preffure of fluids contain❜d in pipes, must be computed by their perpendicular height, how wide, long, or irregular foever they are. SCHOLIUM. The learned Stevinus has a corollary from the preceding propofition, to this purpose. If thro' the upper furface of a concave cylinder, an open, cylindrical tube be erected perpendicularly, and the whole united cavity of both fill'd with water, the bottom circle will fuftain a preffure equal to that of a cylinder of water, whose basis is that of the veffel, and height that of VOL. II. Rr the Մ Fig. 15. STATICS. the tube therefrom. The truth of this paradox we attempted to try in the following manner. Having provided a latton vefiel ABEHGF, furnish'd with a falfe wooden bottom CD, which was cover'd with a fine bladder, and oiled about the lower edges, to facilitate its rifing from the rim of wood HG, that lay every where contiguous to the infide of the metal, and kept the water from paffing between; to the middle of this loose bottom we alfo faften'd a ftrong ftring, which came out at the orifice A. Thro' this orifice we then pour'd in water, which preffing the false bottom CD, help'd to tighten the veffel, and keep its parts clofe. When the whole cavity was fill'd, we faften'd the upper end of the ftring A, to the beam of a balance, and gradually placed weights in the oppofite fcale, till they elevated the falfe bottom CD, from the rim HG, and confequently lifted the incumbent water, which foon ran down between them. Now, we had before-hand found what weight fufficed to raife the bottom CD, alone; deducting that, therefore, from the weight in the scale, and comparing the remainder with the weight of as much water, as the fhallow cylinder BECHG DF, would, alone, without taking notice of that in the pipe A B, contain; we found the preffure upon CD, fo vaftly greater than could be attributed to the whole quantity of water made ufe of, had it been contain❜d in an uniform cylinder of the fame basis with our inftrument, that we thought it fome fmall confirmation of Stevinus's doctrine; tho' the paradox itself be greatly queftion'd by fome men of learning *. Fig. 16. "Tis now an eftablish'd propofition, that fluids prefs not according to their quantity, but perpendicular altitude; and may be demonftrated thus. Let ABCDFE, be a veffel full of water. Now the column B F, being heavier than the column HG, 'tis plain that if the veffel were open at H, the column GH, would afcend till it balanced the column BF: but fince it cannot afcend at H, the water there must be prefs'd back by the obftacle, with a force equal to the weight of BL; for action and re-action are equal; and all preffure here being reciprocal, the water at G, will prefs against the bottom of the veffel with the fame force. Now, as the weight of the column GH, is added to the former, the force of the water's preffure at G, will be the fame, as if the column GH were of equal height with the column FB, that is, as if it reach'd to M. And the fame reafoning is applicable to all the other columns in the veffel; confequently, the bottom of it ED, will fuftain the fame preffure, as if the veffel were fill'd up to NO. This furprizing property in fluids feems applicable to confiderable purposes; as it fhews us how an exceeding fmall quantity of matter may be made to fupply the place of one infinitely larger. Thus, for inftance, if the tube Fig. 17. AB, of the inverted fyphon ABCD, were clofe fhut, and the smail pipe CD, fill'd with water to D, that little quantity CD, would exert fuch a force upon the larger, ABC, that if the cover of the great orifice were perforated, the water would be made to fly out thereat, as violently, as if part of the little tube, CD, contain'd as much water, as the great one A B. Again, if two cylindrical emboli were fitted to the tubes AB, and CD; the weights laid upon them would be balanced, provided they were proportionable to the capacity of the tubes; that is, fuppofe the capacity of the tube A B, was four times greater than that of CD, one pound laid upon the little embolus would balance the force of four pounds upon the greater. And these experiments might be varied infinite ways. See Clark. Annot, in Robault. p. 41. PA A body, immerfed in a fluid, fuftains a lateral pressure therefrom; which increa- STATICS. Bend a fmall glafs tube at right angles, into a longer and a fhorter part; Fig. 18. then, by fuction, draw up oil of turpentine enough to fill the fhorter, and to rife two or three inches high in the longer; nimbly ftop the orifice, and immerse the lower end of the tube under water; fo that the longer leg, EF, may ftand perpendicularly to the furface A B, and the furface of the oil IK, but a little above the fame A B; when, if the tube be unstopp'd at E, the oil will continue as it was, and, confequently, there's a lateral preffure of the water against the oil contiguous to G; for, nothing else could hinder the perpendicular preffure of the oil in the longer leg, from forcing it out; as will farther appear, by gently raifing the tube, in the fame pofture, towards the top of the water; for, as the lower leg afcends, the oil will be driven out thereat, by the preffure in the other. And that the lateral preffure of the water against the lower orifice is, before the tube be raised, nearly the fame with the perpendicular preffure of a cylinder of water, reaching from the orifice G, to the top of the water; appears from the furface of the oil in the longer leg, always continuing a little above that of the water; as would happen, were a pipe of an equal bore, continued from the orifice G, to the top of the water at H. But, farther, if the fyphon be plunged deeper into the water; the oil, by the lateral preffure thereof, will be gradually driven quite out of the fhorter leg into the longer; and if you immerse it ftill deeper, a cylinder of water will rife in the fame longer leg, and fuftain that of the oil, which is now no longer able to balance the lateral preffure of the water at fo great a depth; whence we find, that water has alfo a lateral preffure against water, which increases proportionably to the depth; for the external fluid, could not, otherwife, impel that of the fhorter leg into the longer, which it does by greatly exceeding the refiftance of the whole cylinder of oil therein. But if the tube be now gently raifed again, and the lateral preffure of the water against the immerfed end thereby diminifhed, the oil will force the water firft out of the longer leg, and then out of the fhorter, till, at length, it leaves the orifice G, and emerges in drops to the furface. Again, when the oil in the tube is a balance for the external water; if its fhorter leg be kept parallel to the furface thereof, and moving backwards and forwards any way at the fame depth therein; the oil in the longer leg will appear at the fame height; from whence 'tis plain, a pillar of water, with a bafis equal to the fide of the body immerfed, may be any where fupported in the containing veffel; and that tho' this imaginary column, as GH, be not included in any firm furface, its lower parts have, by means of the incumbent fluid, a lateral preffure, tending outwards against its ima-. ginary fides; and laftly, that this preffure increafes in proportion to the height of the column of water above them. To conclude, if inftead of Rr 2 holding |