自然哲学的数学原理-32

dium in each of the places thereof, by which a body may describe agiven spiral.From the centripetal force the velocity in each place must be found ;then from the retardation of the velocity the density of the medium isfound, as in the foregoing Proposition.But I have explained the method of managing these Problems in thetenth Proposition and second Lemma of this Book; and will no longerdetain the reader in these perplexed disquisitions. I shall now add somethings relating to the forces of progressive bodies, and to the density andresistance of those mediums in which the motions hitherto treated of, andthose akin to them, are performed.SEC. V.] OF NATURAL PHILOSOPHY. 293SECTION V.Of the density and compression offluids ; and of hydrostatics.THE DEFINITION OF A FLUID.A fluid is any body whose parts yield to any force impressed on it,by yielding, are easily moved among themselves.PROPOSITION XIX. THEOREM XIvAll the parts of a homogeneous and unmovedfluid included in any unmoved vessel, and compressed on every side (setting aside the consideration of condensation, gravity, and all centripetal forces], will beequally pressed on every side, and remain in their places without anymotion arising from that pressure.CASE 1. Let a fluid be included in the sphericalvessel ABC, and uniformly compressed on everyside : 1 say, that no part of it will be moved bythat pressure. For if any part, as L), be moved,all such parts at the same distance from the centreon every side must necessarily be moved at thesame time by a like motion ; because the pressureof them all is similar and equal ; and all othermotion is excluded that does not arise from thatpressure. But if these parts come all of them nearer to the centre, thefluid must be condensed towards the centre, contrary to the supposition.If they recede from it, the fluid must be condensed towards the circumference ; which is also contrary to the supposition. Neither can they movein any one direction retaining their distance from the centre, because forthe same reason, they may move in a contrary direction : but the samipart cannot be moved contrary ways at the same time. Therefore nopart of the fluid will be moved from its place. Q,.E.T).CASE 2. I say now, that all the spherical parts of this fluid are equallypressed on every side. For let EF be a spherical part of the fluid;if thisbe not pressed equally on every side, augment the lesser pressure till it bepressed equally on every side; and its parts (by Case I) will remain intheir places. But before the increase of the pressure, they would remainin their places (by Case 1) ; and by the addition of a new pressure theywill be moved, by the definition of a fluid, from those places. Now thesetwo conclusions contradict each other. Therefore it was false to say thatthe sphere EF was not pressed equally on every side. Q,.E.D.CASE 3. I say besides, that different spherical parts have equal pressures.For the contiguous spherical parts press each other mutually and equallyin the point of contact (by Law III). But (by Case 2) they are pressed onevery side with the same force. Therefore any two spherical parts lot391 THE MATHEMATICAL PRINCIPLES [BoOK II.contiguous, since an intermediate spherical part can touch both, will bepressed with the same force. Q.E.D.CASE 4. I say now, that all the parts of the fluid are every where pressed equally. For any two parts may be touched by spherical parts in anypoints whatever ; and there they will equally .press those spherical parts(by Case 3). and are reciprocally equally pressed by them (by Law III).Q.E.D.CASE 5. Since, therefore, any part GHI of the fluid is inclosed by therest of the fluid as in a vessel, and is equally pressed on every side ; andalso its parts equally press one another, and are at rest among themselves ;it is manifest that all the parts of any fluid as GHI, which is pressedequally on every side, do press each other mutually and equally, and are atrest among themselves. Q.E.D.CASE 6. Therefore if that fluid be included in a vessel of a yieldingsubstance, or that is not rigid, and be not equally pressed on every side,the same will give way to a stronger pressure, by the Definition of fluidity.CASE 7. And therefore, in an inflexible or rigid vessel, a fluid will notSustain a stronger pressure on one side than on the other, but will giveway to it, and that in a moment of time ; because the rigid side of thevessel does not follow the yielding liquor. But the fluid, by thus yielding,will press against the opposite side, and so the pressure will tend on everyside to equality. And because the fluid, as soon as it endeavours to recedefrom the part that is most pressed, is withstood by the resistance of thevessel on the opposite side, the pressure will on every side be reduced toequality, in a moment of time, without any local motion : and from thencethe parts of the fluid (by Case 5) will press each other mutually and equally, and be at rest among themselves. Q..E.D.COR. Whence neither will a motion of the parts of the fluid amongthemselves be changed by a pressure communicated to the external superficies, except so far as either the figure of the superficies may be somewherealtered, or that all the parts of the fluid, by pressing one another more intensely or remissly, may slide with more or less difficulty among themselves.PROPOSITION XX. THEOREM XV.Jf all the parts of a sphericalfluid, homogeneous at equal distancesfromthe centre, lying on. a spherical concentric bottom, gravitate towardsthe centre of the whole, the bottom will sustain the weight of a cylinder, whose base is equal to the superficies of the bottom, and whose altitude is the same with that of the incumbent fluid.Let OHM be the superficies of the bottom, and AEI the upper superficies of the fluid. Let the fluid be distinguished into concentric orbs of3qual thickness, by the innumerable spherical superficies *3PK, CGL : andSEC. V OF NATURAL PHILOSOPHY. 295conceive the force of gravity to act only in theupper superficies of every orb, and the actionsto be equal on the equal parts of all the superficies. Therefore the upper superficies AEis pressed by the single force of its own gravity, by which all the parts of the upper orb,and the second superficies BFK, will (byProp. XIX), according to its measure, beequally pressed. The second superficies BFKis pressed likewise by the force of its owngravity, which, added to the former force,makes the pressure double. The third superficies CGL is, according to itsmeasure, acted on by this pressure and the force of its own gravity besides,which makes its pressure triple. And in like manner the fourth superficies receives a quadruple pressure, the fifth superficies a quintuple, and soon. Therefore the pressure acting on every superficies is not as the solidquantity of the incumbent fluid, but as the number of the orbs reachingto the upper surface of the fluid; and is equal to the gravity of the lowestorb multiplied by the number of orbs : that is, to the gravity of a solidwhose ultimate ratio to the cylinder above-mentioned (when the number ofthe orbs is increased and their thickness diminished, ad infiititum, so thatthe action of gravity from the lowest superficies to the uppermost may besomecontinued) is the ratio of equality. Therefore the lowest superficiessustains the weight of the cylinder above determined. Q..E.D. And by alike reasoning the Proposition will be evident, where the gravity of thefluid decreases in any assigned ratio of the distance from the centre, andalso where the fluid is more rare above and denser below. Q.E.D.COR. 1. Therefore the bottom is not pressed by the whole weight of theincumbent fluid, but only sustains that part of it which is described in theProposition ; the rest of the weight being sustained archwise by the spherical figure of the fluid.COR. 2. The quantity of the pressure is the same always at equal distances from the centre, whether the superficies pressed be parallel to thehorizon, or perpendicular, or oblique ;or whether the fluid, continued upwards from the compressed superficies, rises perpendicularly in a rectilineardirection, or creeps obliquely through crooked cavities and canals, whetherthose passages be regular or irregular, wide or narrow. That the pressureis not altered by any of these circumstances, may be collected by applyingthe demonstration of this Theorem to the several cases of fluids.COR. 3. From the same demonstration it may also be collected (by Prop.XIX), that the parts of a heavy fluid acquire no motion among themselveiby the pressure of the incumbent veight, except that motion which arisesfrom condensation.296 THE MATHEMATICAL PRINCIPLES [BCOK IICon. 4. And therefore if another body of the same specific gravity, incapable of condensation, be immersed in this fluid, it will acquire no motion by the pressure of the incumbent weight: it will neither descend nor .ascend, nor change its figure. If it be spherical, it will remain so, notwithstanding the pressure ;if it be square, it will remain square; and that,whether it be soft or fluid : whether it swims freely in the fluid, or lies atthe bottom. For any internal part of a fluid is in the same state with thesubmersed body ; and the case of all submersed bodies that have the samemagnitude, figure, and specific gravity, is alike. If a submersed body, retaining its weight, should dissolve and put on the form of a fluid, thisbody, if before it would have ascended, descended, or from any pressure assume a new figure, would now likewise ascend, descend, or put on a newfigure ; and that, because its gravity and the other causes of its motionremain. But (by Case 5, Prop. XIX; it would now be at rest, and retainits figure. Therefore also in the former case.COR. 5. Therefore a body that is specifically heavier than a fluid contiguous to it will sink ; and that which is specifically lighter will ascend,and attain so much motion and change of figure as that excess or defect ofgravity is able to produce. For that excess or defect is the same thing as animpulse, by which a body, otherwise in equilibria with the parts of thefluid, is acted on: and may be compared with the excess or defect of aweight in one of the scales of a balance.COR. 6. Therefore bodies placed in fluids have a twofold gravity theone true and absolute, the other apparent, vulgar, and comparative. Absolute gravity is the whole force with which the body tends downwards ;relative and vulgar gravity is the excess of gravity with which the bodytends downwards more than the ambient fluid. By the first kind of gravity the parts of all fluids and bodies gravitate in their proper places ; andtherefore their weights taken together compose the weight of the whole.For the whole taken together is heavy, as may be experienced in vesselsfull of liquor ; and the weight of the whole is equal to the weights of allthe parts, and is therefore composed of them. By the other kind of gravity bodies do not gravitate in their places ; that is, compared with oneanother, they do not preponderate, but, hindering one another s endeavoursto descend, remain in their proper places, as if they were not heavy. Thosethings which are in the air, and do not preponderate, are commonly lookedon as not heavy. Those which do preponderate are commonly reckonedheavy, in as much as they are not sustained by the weight of the air. TheCommon weights are nothing else but the excess of the true weights abovethe weight of the air. Hence also, vulgarly, those things are called lightwhich are less heavy, and, by yielding to the preponderating air, mountupwards. But these are only comparatively lig s &mA not truly so, becausehey descend in racuo. Thus, in water, bodies *>icfc. by their greater orSEC. V.] OF NATURAL PHILOSOPHY. 297less gravity, descend or ascend, are comparatively and apparently heavy orlight ; and their comparative and apparent gravity or levity is the excess.or defect by which their true gravity either exceeds the gravity of thewater or is exceeded by it. But those things which neither by preponderating descend, nor, by yielding to the preponderating fluid, ascend, althoughby their true weight they do increase the weight of the whole, yet comparatively, and in the sense of the vulgar, they do not gravitate in the water. For these cases are alike demonstrated.COR. 7. These things which have been demonstrated concerning gravitytake place in any other centripetal forces.COR. 8. Therefore if the medium in which any body moves be acted oneither by its own gravity, or by any other centripetal force, and the bodybe urged more powerfully by the same force ; the difference of the forces isthat very motive force, which, in the foregoing Propositions, I have considered as a centripetal force. But if the body be more lightly urged bythat force, the difference of the forces becomes a centrifugal force, and is tcbe considered as such.COR. 9. But since fluids by pressing the included bodies do notchange their external figures, it appears also (by Cor. Prop. XIX) that theywill not change the situation of their internal parts in relation to onfanother ; and therefore if animals were immersed therein, and that all sensation did arise from the motion of their parts, the fluid will neither hurtthe immersed bodies, nor excite any sensation, unless so far as those bodiesmay be condensed by the compression. And the case is the same of anysystem of bodies encompassed with a compressing fluid. All the parts ofthe system will be agitated with the same motions as if they were placedin a vacuum, and would only retain their comparative gravity ; unless sofar as the fluid may somewhat resist their motions, or be requisite to conglutinatethem by compression.PROPOSITION XXI. THEOREM XVI.<et the density of any fluid be proportional to the compression, and itsparts be attracted downwards by a centripetal force reciprocally proportional to the distances from the centre : I say, that, if those distances be taken continually proportional, the densities of thefluid atthe same distances will be also continually proportional.Let ATV denote the spherical bottom of the fluid, S the centre, S A, SB.SC, SD, SE, SF, &c., distances continually proportional. Erect the perpendiculars AH, BI, CK, DL, EM, PN, &c., which shall be as the densities of the medium in the places A, B, C, D, E, F : and the specific gravATT RT f^K"ities in those places will be aa -r-, ,-&c., or, which is all one, a&-298 THE MATHEMATICAL PRINCIPLES [BOOK II.AH BI CKATT BC CD Suppose, first, these gravities to be uniformly continuedfrom A to B, from B to C, from C to D, (fee., the decrements in the pointsB, C, D, (fee., being taken by steps. Arid these gravities drawn into the altitudes AB, BC, CD, (fee., willgive the pressures AH, BI, CK, (fee., by which the bottom ATV is acted on (by Theor. XV). Therefore theparticle A sustains all the pressures AH, BI, CK, DJL,(fee., proceeding in infinitum ; and the particle B sustains the pressures of all but the first AH ; and the particle C all but the two first AH, BI ; and so on : andtherefore the density AH of the first particle A is tothe density BI of the second particle B as the sum ofall AH -f- BI + CK + DL, in infinitum, to the sum ofall BI -f- CK -f DL, (fee. And BI the density of the second particle B isto CK the density of the third C, as the sum of all BI -f CK + DL, (fee.,to the sum of all CK -f- DL, (fee. Therefore these sums are proportionalto their differences AH, BI, CK, (fee., and therefore continually proportional (by Lem. 1 of this Book) ; and therefore the differences AH, BI,CK, (fee., proportional to the sums, are also continually proportional.Wherefore since the densities in the places A, B, C, (fee., are as AH, BI,CK, (fee., they will also be continually proportional. Proceed intermissively,and, ex ccquo, at the distances SA, SC, SE, continially proportional,the densities AH, CK, EM will be continually proportional. And by thesame reasoning, at any distances SA, SD, SG, continually proportional,the densities AH. DL, GO, will be continually proportional. Let now thepoints A, B, C. D, E, (fee., coincide, so that the progression of the specif.cgravities from the bottom A to the top of the fluid may be made continual ;and at any distances SA, SD, SG, continually proportional, the densitiesAH, DL, GO, being all along continually proportional, will still remaincontinually proportional. Q.E.D.COR. Hence if the density of the fluid in two places,as A and E, be given, its density in any other place Q,may be collected. With the centre S, and the rectangular asymptotes SQ, SX, describe an hyperbola cutting the perpendiculars AH, EM, QT in , e, and q}as also the perpendiculars HX, MY, TZ, let fall uponthe asypmtote SX, in //, m, and t. Make the areaY////Z to the given area YmAX as the given areaEeqQ to the given area EeaA ; and the line Z produced will cut off theline Q,T. proportional to the density. For if the lines SA, SE, SQ arecontinually proportional, the areas ReqQ., fyaA will be equal, and thenceXSEC. V. OF NATURAL PHILOSOPHY. 299the areas YwYZ. X/zwY, proportional to them, will be also equal ; andthe lines SX, SY, SZ, that is, AH, EM, Q,T continually proportional, asthey ought to be. And if the lines SA, SE, SQ,5 obtain any other orderin the series of continued proportionals, the lines AH, EM, Q,T, becauseof the proportional hyperbolic areas, will obtain the same order in anotherseries of quantities continually proportional.PROPOSITION XXII THEOREM XVII.Let the density of any fluid be proportional to the compression, and itsparts be attracted downwards by a gravitation reciprocally proportional to the squares of the distancesfrom the centre : I say, that ifthe distances be taken in harmonic progression, the densities of thefluid at those distances will be in a geometrical progression.Let S denote the centre, and SA,SB, SC, SD, SE, the distances ingeometrical progression. Erect theperpendiculars AH, BI, CK, (fee.,which shall be as the densities of cthe fluid in the places A, B, C, D,E, (fee., and the specific gravitiesthereof in those places will be asAH BI,^-, (fee. Suppose these SA2 SB 2 SC 2gravities to be uniformly continued, the first from A to B, the second fromB to C, the third from C to I), &c. And these drawn into the altitudesAB, BC, CO, DE, (fec.; or, which is the same thing/into the distances SA,ATT r>T OT7"SB, SC, (fee., proportional to those altitudes, will give -~-r-, ^=5, -~~, (fee..the exponents of the pressures. Therefore since the densities are as th^sums of those pressures, the differences AH BI, BI CK, (fee., of tb,densities will be as the differences of those sums ~-r~, ^, ~~, (fee. Withthe centre S, and the asymptotes SA, S#, describe any hyperbola, cuttingthe perpendiculars AH, BI, CK, (fee., in a, 6, c, (fee., and the perpendiculars H/, I//,, K?#, let fall upon the asymptote Sv, in h, i, k ; and the differences of the densities tu, uw, (fee., will be as A , ^^, (fee. And the SA; SB;rectangles tu X th, uw X uij (fee., or tp, uq, (fee., asthat is, as Aa, Bb, (fee.AH X th BI X ui,(fee. SA SBFor, by the nature of the hyperbola, SA is to AHor St as th to Ar, and therefore pri is equal to Aa. And, by a like SA300 THE MATHEMATICAL PRINC. PLES [BOOK II.reasoning, ^n~~ *s e(lua^ to ^, &c- But Aa> B^> ^c, &cv are continually proportional, and therefore proportional to their differences Aa B&,B6 Cc; &c., therefore the rectangles fy?, nq, &c., are proportional to thosedifferences ; as also the sums of the rectangles tp + uq, or tp + uq -f wto the sums of the differences Aa Cc or Aa Da7. Suppose several ofthese terms, and the sum of all the differences, as Aa F/, will be proportional t? the sum of all the rectangles, as zthn. Increase the numberof terms, and diminish the distances of the points A, B, C, (fee., in iiijinitum,and those rectangles will become equal to the hyperbolic area zthn.and therefore the difference Aa F/ 19 proportional to this area. TakenowTany distances, as SA, SD, SF, in harmonic progression, and the differences Aa Da7, Da1 F/ will be equal ; and therefore the areas thlx,xlnz, proportional to those differences will be equal among themselves, andthe densities St, S:r, Sz, that is, AH, DL, FN, continually proportional.Q.E.D.COR. Hence if any two densities of the fluid, as AH and BI, be given,the area thiu, answering to their difference tu, will be given; and thencethe density FN will be found at any height SF, by taking the area thnz tothat given area thiu as the difference Aa F/ to the difference Aa Eh.SCHOLIUM.By a like reasoning it may be proved, that if the gravity of the particlesof a fluid be diminished in a triplicate ratio of the distances from the centre ;and the reciprocals of the squares of the distances SA, SB, SC, &c., (namely,SA 3 SA 3 SA 3.opt ^e ta^en m an arithmetical progression, the densities AH.BI, CK, &c., will be in a geometrical progression. And if the gravity bediminished in a quadruplicate ratio of the distances, and the reciprocals ofthe cubes of the distances (as ^-r^, SRS sps ^c ^ ^e ta^cn ^ n ai> itnmeticaiprogression, the densities AH, BI, CK, &c., will be in geometrical progression. And so in irtfinitum. Again ;if the gravity of the particles ofthe fluid be the same at all distances, and the distances be in arithmeticalprogression, the densities will be in a geometrical progression as Dr. Halleyhas found. If the gravity be as the distance, and the squares of thedistances be in arithmetical progression, the densities will be in geometrical progression. And so in infinitum. These things will be so, when thedensity of the fluid condensed by compression is as the force of compression ; or, which is the same thing, when the space possessed by the fluid isreciprocally as this force. Other laws of condensation may be supposed,as that the cube of the compressing force may be as the biquadrate of theSEC. V.] OF NATURAL PHILOSOPHY. 301de isity ; or the triplicate ratio of tlie force the same with the quadruplicateratio of the density : in which case, if the gravity he reciprocally as thesquare of the distance from the centre, the density will be reciprocally atthe cube of the distance. Suppose that the cube of the compressing forcebe as the quadrato-cube of the density ; and if the gravity be reciprocallyas the square of the distance, the density will be reciprocally in a sesquiplicateratio of the distance. Suppose the compressing force to be in a duplicate ratio of the density, and the gravity reciprocally in a duplicate ratio of the distance, and the density will be reciprocally as the distance.To run over all the cases that might bo offered would be tedious. But asto our own air, this is certain from experiment, that its density is eitheraccurately, or very nearly at least, as the compressing force ; and thereforethe density of the air in the atmosphere of the earth is as the weight ofthe whole incumbent air, that is, as the height of the mercury in the barometer.PROPOSITION XXIII. THEOREM XVIII.If a fluid be composed of particles mutually flying each other, and thedrnsity be as the compression, the centrifugal forces of the particleswill be reciprocally proportional to tlie distances of their centres. And,vice versa, particlesflying each otli,er, with forces that are reciprocallyproportional to the distances of their centres^ compose an elastic fluid,whose density is as the compression.Let the fluid be supposed to be included in a cubicspace ACE, and then to be reduced by compression into