STATICS. the height of the incumbent water was diminish'd, the oil overbalanced 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 ftood directly over it; that is, fuch a column as our paradox requires.

Fig. 12.

Fig. 13.

COROLLARY.

It follows from hence, that water may prefs as regularly upon an immersed body, when not restrained 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 first part of our paradox holds equally true, whether the incumbent water be free or confined in veffels of any irregular shape. 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 sustain 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 orificé 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 sphere. To clear up this matter still 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 fill'd the cavity of the fpherical 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 specific gravities of the two fluids requir'd: 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 ftood 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 spherical cavity E. And thus, if the fphere had been made larger, a small quantity of water in the leg AB, would have fomewhat raised a much greater weight of mercury than its own.

The fecond part of our prefent propofition is proved thus. We took three open glais pipes, of the irregular figures in the fcheme, and plunged them in a glafs vellel of water, the two crooked ones ftanding very oblique to the upper furface thereof, and the higher ends of them all coming

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 fame 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 widenefs 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 those 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 stood 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 slender tube, balances the preffure of the external water, if the tube be barely inclined to the fides of the containing glafs, 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 strait one. Hence we may fairly conclude, that the preffure of fluids contain'd in pipes, muft 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 vessel, and height that of VOL. II. Rr the

Fig. 15.

STATICS, the tube therefrom. The truth of this paradox we attempted to try in the following manner. Having provided a latton veffel ABEH GF, furnish'd with a false wooden bottom C D, which was cover'd with a fine bladder, and oiled about the lower edges, to facilitate its rising 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 also fasten'd a strong 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 string 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 raise the bottom CD alone; deducting that, therefore, from the weight in the fcale, and comparing the remainder with the weight of as much water as the fhallow cylinder BECHGDF 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 bafis with our inftrument, that we thought it fome small confirmation of Stevinus's doctrine; tho' the paradox itself be greatly question'd by fome men of learning.*

Fig. 16.

'Tis now an establifh'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 H G, 'tis plain that if the veffel were open at H, the column G H would afcend till it balanced the column B F; but fince it cannot afcend at H, the water there must be prefs'd back by the obstacle, with a force equal to the weight of B L, 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 F B, 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

AB, of the inverted fyphon Fig. 17.

ABCD, were clofe fhut, and the fmall pipe CD, fill'd with water to D, that little quantity CD would exert fuch a force upon the larger A B C, 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 А В.

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 thefe experiments might be varied infinite ways. See Clark. Annot. in Robault.

P. 41.

PARADOX VII.

A body, immerfed in a fluid, fuftains a lateral preffure therefrom; which increafes with the depth whereto 'tis plunged.

STATICS.

Bend a finall 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 rise 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 posture toward 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 la teral preffure thereof, will be gradually driven quite out of the fhorter leg into the longer; and if you immerse it still 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 againft 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 raised again, and the lateral preffure of the water against the immerfed end thereby diminished, the oil will force the water first 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 imaginary fides; and laftly, that this preffure increases in proportion to the height of the column of water above them. To conclude, if inftead of

Rr 2

hold

STATICS. holding the longer leg of our tube in a perpendicular pofture, it be varioufly inclined to different angles with the furface of the water A B, so that the orifice of the shorter leg F G, may fometimes tend towards the top, and, at others, towards the bottom of the glafs; the oblique preffure of the water depends fo entirely upon its height above the orifice G, that the oil in the longer leg will ftill continue with its furface IK, but little higher than that of A B.

Fig. 19:

SCHOL I A.

1. We once attempted to discover the quantity of this lateral preffure by the following experiment. Having obtain'd a glafs bubble X, with a flender neck, whofe gravity fell very little fhort of that of water; and provided a large wide-mouth'd glafs,exactly fitted with a cork, perforated in the middle, to receive, perpendicularly, a long, flender, glafs pipe, that defcended confiderably below the cork, wherein, alfo, was another hole, near its edge, for a fecond small pipe to pafs thro', whofe upper end reaching only two or three inches into the air, was firmly clofed; then the veffel being filled with water, and the bubble suffer'd to float thereon, the cork was applied, and well cemented to the mouth of the glafs, fo that nothing could pafs in or out of the veffel, except thro' the long flender pipe; all the other vents being carefully ftopped with cement. When things were thus prepared, and the bubble was made to float at a confiderable diftance from the deeply immerfed end of the long tube, we poured water thro' its upper orifice E, which, for want of other vent, afcended back again into the fame cavity; and if the water therein was only permitted to rife as high as I, for inftance, the bubble X, would ftill continue a-float: but if the water rose to K, the bubble prefently fell to the bottom of the veffel, and remain'd thereat while the water ftood to the fame height in the pipe E F ; but if the height were diminish'd, by pouring fome of the water off, the bubble would re-afcend. Hence we fee, that the upper parts of water gravitate on the lower; and that, in a full veffel, the fame thing happens, tho' the former be not directly incumbent on the latter, but at a confiderable diftance from the line of their direction; for the additional cylinder of water KI, acting like any other heavy body upon the water, caufes the bubble X, to fubfide. And, fince the air included in this bubble, was remarkably condenfed, it follows, that the cylinder of water KI, preffed upon the fubjacent fluid in the veflel; for other wife the air in the bubble could not have been comprefs'd: and as this bubble floated on one fide, at a tolerable distance from the pipe E F, and above its lower orifice F; 'tis evident, that the cylinder of water IK, preffes as well laterally, as directly, upon all the bodies that lie beneath its furface. And thus every affignable part of the fides of the veffel, fuftains a preffure proportionable to its depth under water, and its own magnitude: if, therefore, any part were fo weak, as to yield to the weight of the cylinder IK (due allowance being made for the obliquity of the preffure) 'twere unfit to enter the compofition of our glass. Even the cork itself, refifts a preffure proportionable to the