than when it is full of water, as the quantity of matter contained in the compound vessel N M LP is less than the quantity of matter contained in a cylindrical veffel, whose base, is LP, and height LF.

2. The 2d cafe of the propofition is when the vef fel A BOR, fig. 10, is wider at top than bottom. For here alfo the preffure of any fluid upon the bottom, OR, of it, is the fame as in a cylindrical veffel, STOR, of an equal bafe, and filled with the fame fort of fluid to the fame height. For the bottom O R, in either case, sustains just the same quantity of fluid, and confequently the fame quantity of matter. If it is the bottom of a cylinder, then it fuftains no more than the column STOR, because the veffel holds no more. If it be the bottom of an inverted cone, as A BOR, then it fuftains only the fame column; for, though the veffel holds more than this, yet all the reft of the fluid is fupported by the fides A O, BR, and therefore does not prefs on the bottom.

Thus, whether a vessel be narrower or wider at the top than at the bottom, the preffure upon the bottom is the fame as in a cylindrical veffel of the fame base and height; for when it is narrower at the top than at the bottom, though it holds lefs water than the cylindrical one would, yet the preffure is not lefs, because the re-action of the fides fupplies the defect; and when it is wider at the top than at the bottom, though it holds more water than the cylindrical one would hold, yet the preffure is not greater, because the fides fupport the excess.

Let us now confirm by experiment, what we have thus endeavoured to render plain without it. The apparatus, fig. 11, is defigned for this purpofe. It is fometimes called the apparatus of PASCHAL, fometimes the apparatus for illuftrating the HYDROSTATIC PARADOX. It confifts of three veffels, fig. 12, fig. 13, and A B C D, fig. 11, each of which are of the fame fize at bottom and of the fame height, and may be screwed alternately on the brafs barrel E F, fig. 11, in which a pifton flides up and down with ease. One of the veffels, fig. 13, is cylindrical; the other A B CD, fig. 11, is an inverted cone, wider at top than bottom; the third, fig. 12, is a tube fcrewed to a plate, which makes the bottom the fame fize as that of the other two; it has a funnel at top to prevent the water, in making the experiment, from being spilt. First screw the cylindrical veffel to the barrel, pushing down the pifton as low as it will go, then hook the wire of the pifton to the rings from the short ends of the fteelyards GH, IK. Now pour water in the cylinder up to the mark in the infide, and find what weights, fufpended from the longer arms of the fteelyard, will raife the pifton; then take the cylindrical veffel from the barrel. Substitute the veffel ABCD, fig. 11, which is like an inverted cone, in place of the former; fill it with water to the mark, as before, and hook on the wire of the pifton to the fteelyards; and though the quantity of water is now many times greater than what was in the cylinder, yet the fame counterpoife will raise the pifton. Take off the conical veffel, and screw on the tubular one; and though, this holds a much fmaller quantity than either of the former, ftill it requires the fame

counterpoife. The friction of the pifton, being the fame in every case, makes no alteration in the experiment.

To fhow that the lateral preffure is equal to the perpendicular preffure upon a larger scale, and in a manner which relates more to the preceding experiment, we have delineated an apparatus, fig. 1. pl. CLXXXVI. with 3 tubes that communicate with each other. The middle one is a large glafs tube or cylinder, A B; the lower end is firmly cemented into a ftrong brafs hoop; to the fides of this hoop are foldered the brafs tubes G, H, into each of which a glass tube is cemented. One of these, EF, is parallel to the large glass veffel A B; but the other CD is inclined thereto. The inclined tube is fometimes furnished with a joint, that the inclination may be varied as may be necessary.

If we pour water into the tube E F, this will run through G, into the larger veffel A B, and rise therein; and if we continue pouring water until it comes to any given height, as IK, and then leave off, the furface of the water in the fmall tubes E F, CD, will be found at the fame height; the perpendicular altitude is the fame in all the three tubes, however small the one may be in proportion to the other. This experiment clearly proves, that the small column of water ba lances and fupports the large column; which it could not do if the lateral preffures at bottom were not equal to each other. Whatever be the inclination of the tube CD, ftill the perpendicular altitude will be the fame as that of the other tubes, though to that end the column of water must be much longer than thofe in the upright tubes. Hence it is evident, that a small quantity of a fuid may, under certain circumstances, counterbalance any quantity of the fame fluid. Hence alfo it is evident, that in tubes that have a communication, whether they be equal or unequal, fhort or oblique, the fluid always rifes to the fame height. Confequently water cannot be conveyed by means of a pipe that is laid from a refervoir to any place that is higher than the reservoir itself.

The ancients, it has been faid, were ignorant of this principle, and knew not the ufe of pipes for conveying water up hills: but this affertion is not true; they did know the ufe of pipes, but chofe to employ aqueducts in their stead, for reafons we cannot now with certainty account for.

Our next experiment proves, with great clearnefs, the HYDROSTATICA PARADOX, that very great weights may be balanced by a very small weight of water, without its acting to any mecha nical advantage: but, more particularly, it also, proves, that its preffure upwards is equal to its preffure downwards, and all this even to thofe who have no previous knowledge of hydrostatical principles. The apparatus, fig. 2, pl. CLXXXVI. confifts of two large thick boards, CD, E F, connected together by leather, like a pair of bellows; hence it is ufually called the bydrostatic bellows. A long brass pipe is fixed to the bottom board; fo that water being poured in at the top, will pafs between the two boards. We will fuppofe the boards of the apparatus oval; and that the longest diameter is 18 inches, the fhorter one fix teen. Having poured water enough into the bel lows to keep the boards afunder, and put fix half

hundred

SECT. IV. Of the ACTION of FLUIDS on BODIES

Another inftrument has been invented, for proving that the preffure of fluids is in proportion to their perpendicular heights, without any regard to their quantity.

ABCD,fig. 3, pl. CLXXXVI. is a box, at one end of which, as at a, is a groove from top to bottom, for receiving the upright glafs tube I, which is bent to a right angle at the lower end, as at fig. 4; and to that end is tied the end of a large bladder K, fig. 4, which lies in the bottom of the box. Over this bladder is laid the moveable board M, fig. 5, in which is fixed an upright wire. Leaden weights N N, fig. 3, to the amount of 16 lb. with holes in the middle, are put upon the wire, over the board, and prefs upon it with all their force. The bar p is then put on, to fecure the tube from falling, and keep it upright; and then the piece EFG is to be put on, to keep the weights in a horizontal pofition, there being a round hole at e. Within the box are four upright pins, to prevent the board at firft from preffing on the bladder. Pour water into the tube at top; this will run into the bladder and after the bladder has been filled up to the board, continue pouring water into the tube, and the upward preffure of the fluid will raife the board with all the weight upon it, even though the bore of the tube fhould be fo fmall that lefs than an ounce of water would fill it.

Upon this principle mathematicians affert, that the fame quantity of water, however fmall, may produce a force equal to any affignable one, by increafing the height and base upon which it preffes. Dr Goldsmith mentions having feen a strong hogf head split by this method. Aftrong, though small tube of tin, twenty feet high, was inferted in the bung-hole; water was poured in this to fill the bogfhead, and continued till it rofe within about a foot of the top of the tube; the hogfhead then burst, and the water was fcattered about with in credible violence.

As the bottom of a veffel bears a preffure proportional to the height of the liquor, fo likewife do thofe parts of the fides which are contiguous to the bottom, because the preffure of fluids is equal every way; and as the preffure, which the lower parts of a fluid fuftain from the weight of those above them, exerts it felf equally every way, and is likewife proportional to the height of the incumbent fluid, the fides of a veffel muft everywhere fuftain a preffare proportional to their diftance from the upper furface of the liquor. Whence it follows, that in a veffel full of liquor, the fides bear the greateft ftrefs in thofe parts next the bottom; and that the ftrefs upon the fides decreases with the increase of the diftance from the bottom in the fame proportion: fo that in veffels of confiderable height, the lower parts ought to be much strong

IMMERSED in them.

ARCHIMEDES is the firft mathematician we read of (fee his tract De Infidentibus) who made inquiries concerning the finking and floating of bodies in fluids, their relative gravities, their levities, their fituations and pofitions. He was perhaps alfo the first who ever attempted to determine in what proportion bodies differ from one another as to their specific gravities, and this he effected in order to discover the cheat of the work. man who had debased king HIERO'S CROWN; and though the means he employed were certainly much inferior to what would now be used, yet he was fo pleased with his discovery, that not be ing able to contain his joy, like a madman, leaping from the bath naked as he was, he is faid to have run about the streets of Syracufe, crying out Eugna! I have found it! Before we proceed to explain this interefting fubject, fome terms which have only been as yet loosely explained, must be defined.

1. The DENSITY of a body is the QUANTITY OF MATTER which it contains under a GIVEN BULK. The denfity of a body is therefore meafured by the proportion which its quantity of matter bears to its bulk; for, the more numerous the particles of matter are in the fame portion of space, the greater is the denfity of the body, and the fewer the particles the less the density.

2. The SPECIFIC GRAVITY of a body is the WEIGHT OF IT when the BULK is given; or, the specific gravity of a body is its weight compared with another body of the fame magnitude. It is called the specific gravity, because it is the comparative weight of different fpecies or forts of bodies. Thus, if the fpecific gravity of gold is faid to be to that of water as 19 to 1, the meaning is, that, bulk for bulk, or under equal dimenfions, the weight of gold is to that of water as 19 to 1; or that a cubic inch of gold will weigh 19 times as much as a cubic inch of water.

3. The SPECIFIC GRAVITY of BODIES is as their DENSITY, for the fpecific gravity is the weight of a given bulk, and the weight of bodies is as their quantity of matter; therefore the fpecific gravity of a body is as the quantity of matter contained in a given bulk, that is, as its denfity.

4. The SPECIFIC GRAVITY of BODIES is inverfely as their BULK when their WEIGHTS are equal. The specific gravity of bodies is, we have already feen, as their idenfity, and the denfity of bodies is inversely as their bulk when the weights are equal. Thus, if the specific gravity of gold be to that of filver as 19 to 11, and a cylinder of gold 11 inches high weigh a pound, a cylinder of filver having an equal bafe and weighing a pound must be 19 inches high; for fince the fpecific gravities are 19 to 11, the bulks, that is, the heights, must be as thofe gravities inverted, or as 11 to 19. If the fpecific gravity of mercury be to that of water as 14 to 1, and a cylinder of mercury of a certain weight is 30 inches high, then a cylinder of

of water of equal bafe must be 420 times as high; fo that the height of the cylinder of water will be 14 times 30, or 420 inches, or 35 feet.

The MAGNITUDE of a body is expreffed by a number denoting its relation to fome criterion generally ufed, and fimilar to itself, as a cubical inch, foot, &c. The abfolute weight of a body is relative, being expreffed by a number denoting its relation to fome arbitrary or conventional ftandard, as I pound, 1 ounce, of which it is a multiple or aliquot part; and in the fame fort of matter, fuppofed to be homogeneous, it depends upon and varies as the magnitude.

The specific weight or gravity of the fame fpecies of matter, whether its magnitude be great or fmall, as of A, 2A, or 3 A, is the fame, being according to the definition of the weight of a given bulk. The object therefore of specific gravities is to diftinguish different fpecies of matter from each other, in one of their most obvious qualities, weight of matter contained in a given space.

The WEIGHT of any portion of matter is eafily ascertained, but it is not always easy to measure the fpace occupied by a body, or its MAGNITUDE, and in fome inftances it cannot be effected without artificial methods. It is found expedient to employ as a criterion fome pure and homogeneous fubftance, as diftilled water, whose specific gravity is nearly the fame at all times; and by com.paring this with other substances, the ratio of their specific gravity may be discovered; and denoting the specific gravity of water by any number, the numbers expreffing the specific gravities of other bodies are hence obtained.

It follows, from what has been already demonftrated, that when a folid is immersed in a fluid, it is preffed by that fluid on all fides; and that preffure increases in proportion to the height of the Auid above the folid. We may also prove this directly by experiment. Thus, tie a leathern bag to the end of a glafs tube, and fill it with mercury; immerge the bag in water, but fo that the upper or open end of the tube may be always above the furface of the water; the preffure of the water against the bag will raife the mercury in the tube, and the afcent of the mercury will be in proportion to the height of the water above the bag.

When a folid is immersed in a fluid to a great depth, the preffure against the upper part differs very little from the preffure against the under part; whence bodies very deeply immerfed are, as it were, equally preffed on all fides; but a preffure which is equal on all fides may be fuftained by foft bodies without any change of figure, and by very brittie bodics without their breaking. Take a piece of foft wax of an irregular figure, and an egg, and inclofe them in a bladder full of water; place it in a fquare box, and put on a moveable cover, which will bear on the bladder; there may be placed on this cover a weight of 100, or even 150 Ib. without breaking the egg, or any way altering the figure of the wax.

It has been fhewn, that fluids prefs upon bodies to which they are contiguous every way, and on all fides, but the preffure upon each part is not the fame; the altitude of the fluid is everywhere the measure of its force, and the feveral parts of the fame body, being at different depths, muit thus VOL. XI. PART II.

be differently affected; we have therefore to confider which of these impreffions will prevail. It is evident that the lateral preffures all balance each other, being equal, as arifing from equal altitudes of the fluid, and oppofite in their directions; fo that from these the body is no way determined to any motion. But a body immerfed in a fluid is preffed more upwards than it is downwards; for thofe parts of the fluid which are contiguous to the under furface have a greater altitude, and therefore a greater force, than those that are contiguous to the upper furface; the body must therefore be more violently elevated by the former than depreffed by the latter, and would therefore afcend by the excefs of force, were it devoid of gravity. For when a folid body is immersed in a fluid, it preffes down, and endeavours to defcend by the force of its gravity; but it cannot defcend without moving as much of the liquid out of its place as is equal to it in bulk; it is therefore refifted, preffed upwards by a force equal to the weight of as much of the fluid as is equal in magnitude to the bulk of the body; being the difference in weight of two columns of the fluid, whereof one reaches to the upper, the other to the under furface of the body.

We shall illuftrate this by a diagram. When any hard body, as a piece of lead, is immersed in water, the lower part of it, mn, fig. 6. plate CLXXXVI. muft be continually preffed upwards just as much as the water itself in the fame place as the lead is preffed upwards. Now the force with which the water, m n, is preffed upwards, is exactly equal to the force with which it would be preffed downwards if the lead was out of the way; for every part of a fluid is preffed as much upwards as it is downwards. The force with which mn would be preffed downwards if the lead was out of the way, would be equal to the weight of the incumbent column, or of as much water as would fill the whole fpace E Hmn; therefore the force with which m n is preffed upwards, and confequently the force with which the piece of lead is preffed upwards, is equal to the weight of as much water as would fill the whole space Ě H m n, or the whole fpace HP no, if this fpace be taken equal to E Hmn.

Let us next confider the force with which this piece of lead is preffed downwards; this force is juft equal to the weight of as much water as is above it, that is, it is equal to the weight of the column EHrs. The difference therefore of the two preffures will be the difference in weight be tween the 2 columns EH mn, and E Hrs; for the weight of the former is equal to the preffure upwards, and the weight of the latter is equal to the preffure downwards; confequently the preffure upwards will be as much greater than the preffure downward, as the weight of the water E H m n is greater than the weight of the water E Hrs. But the difference between these two weights is just as much as would fill the space rsmn, which the body fills; for juft fo much water added to EHrs, would make it equal to EH mn; confequently the body is preffed more upwards than it is downwards by a force equal to the weight of as much water as would fill the space taken up by the body. In other words, the body is acted upon by two forces in contrary directions, but the Cccc

force