自然哲学的数学原理-39

the fluid always contrary to the motion of the pendulum in its return : andthe resistance arising from this motion, as also the resistance of the threadby which the pendulum is suspended, makes the whole resistance of a pendulum greater than the resistance deduced from the experiments of fallingbodies. For by the experiments of pendulums described in that Scholium,a globe of the same density as water in describing the length of its semidiameterin air would lose the -3^-0 part of its motion. But by thetheory delivered in this seventh Section, and confirmed by experiments offalling bodies, the same globe in describing the same length would lose onlya part of its motion equal to j-Vir? supposing the density of water to beto the density of air as 8 r>0 to 1. Therefore the resistances were foundgreater by the experiments of pendulums (for the reasons just mentioned)than by the experiments of falling globes ; and that in the ratio of about4 to 3. Bat yet since the resistances of pendulums oscillating in air, water, and quicksilver, are alike increased by like causes, the proportion ofthe resistances in these mediums will be rightly enough exhibited by thSEC. VII.J OF NATUKAL PHILOSOPHY. 355experiments of pendulums, as well as by the experiments of falling bodies.And from all this it may be concluded, that the resistances of bodies, movingin any fluids whatsoever, though of the most extreme fluidity, are, cceterisparibus, as the densities of the fluids.These things being thus established, we may now determine what partof its motion any globe projected in any fluid whatsoever would nearly losein a given time. Let D be the diameter of the globe, and V its velocityat the beginning of its motion, and T the time in which a globe with thevelocity V can describe in vacua a space that is, to the space |D as thedensity of the globe to the density of the fluid ; and the globe projected*Vin that fluid will, in any other time t lose the part , the part1 -p tTVr remaining ; and will describe a space, which will be to that described in the same time in, vacua with the uniform velocity V, as theT + tlogarithm of the number ~ multiplied by the number 2,302585093 isto the number7^, by Cor. 7, Prop. XXXV. In slow motions the resistance may be a little less, because the figure of a globe is more adapted tomotion than the figure of a cylinder described with the same diameter. Inswift motions the resistance may be a little greater, because the elasticityand compression of the fluid do not increase in the duplicate ratio of thevelocity. But these little niceties I take no notice of.And though air. water, quicksilver, and the like fluids, by the divisionof their parts in infinitum, should be subtilized, and become mediums infinitely fluid, nevertheless, the resistance they would make to projectedglobes would be the same. For the resistance considered in the precedingPropositions arises from the inactivity of the matter; and the inactivityof matter is essential to bodies, and always proportional to the quantityof matter. By the division of the parts of the fluid the resistance arisingfrom the tenacity and friction of the parts may be indeed diminished : butthe quantity of matter will not be at all diminished by this division; andif the quantity of matter be the same, its force of inactivity will be thesame ; and therefore the resistance here spoken of will be the sanue, as beingalways proportional to that force. To diminish this resistance, the quantity of matter in the spaces through which the bodies move must be diminished ; and therefore the celestial spaces, through which the globes of theplanets and comets are perpetually passing towards all parts, with theutmost freedom, and without the least sensible diminution of their motion,must be utterly void of any corporeal fluid, excepting, perhaps, some extremely rare vapours and the rays of light.356 THE MATHEMATICAL PRINCIPLES [BoOK 11.Projectiles excite a motion in fluids as they pass through them, and thismotion arises from the excess of the pressure of the fluid at the fore partsof the projectile above the pressure of the same at the hinder parts : andcannot be less in mediums infinitely fluid than it is in air, water, and quicksilver, in proportion to the density of matter in each. Now this excess ofpressure does, in proportion to its quantity, not only excite a motion in thefluid, but also acts upon the projectile so as to retard its motion ; and therefore the resistance in every fluid is as the motion excited by the projectilein the fluid ; and cannot be less in the most subtile aether in proportion tothe density of that aether, than it is in air, water, and Quicksilver, in proportion to the densities of those fluids.SECTION VIII.Of motion propagated through fluids.PROPOSITION XLI. THEOREM XXXII.A pressure is not propagated through a fluid in rectilinear directionsunless ichere the particles of the fluid lie in a right line.If the particles a, b} c, d, e, lie in a right line, the pressure may be indeed directly propagated from a to e ; butthen the particle e will urge the obliquely posited partite) cles / and g obliquely, and those particles / and g willnot sustain this pressure, unless they be supported by theparticles h and k lying beyond them ; but the particlesthat support them are also pressed by them ; and those particles cannotsustain that pressure, without being supported by, and pressing upon, thoseparticles that lie still farther, as / and m, and so on in itiflnitum. Therefore the pressure, as soon as it is propagated to particles that lie out ofright lines, begins to deflect towards one hand and the other, and will bepropagated obliquely in infinitum ; and after it has begun to be propagated obliquely, if it reaches more distant particles lying out of the rightline, it will deflect again on each hand and this it will do as often as itlights on particles that do not lie exactly in a right line. Q.E.D.COR. If any part of a pressure, propagated through a fluid from a givenpoint, be intercepted by any obstacle, the remaining part, which is not intercepted, will deflect into the spaces behind the obstacle. This may bedemonstrated also after the following manner. Let a pressure be propagated from the point A towards any part, and, if it be possible, in rectilinearSEC, Vlil.l OF NATURAL PHILOSOPHY. 57directions ; and the obstacleNBCK being perforated in BC,let all the pressure be interceptedbut the coniform part A PQ, passing through the circular hole BC.Let the cone APQ, be dividedinto frustums by the transverseplants, de, fg, Id. Then whilethe cone ABO, propagating thepressure, urges the conic frustum.degf beyond it on the superficiesde, and this frustum urges the next frustum fgih on the superficies/g", andthat frustum urges a third frustum, and so in infinitum ; it is manifest(by the third Law) that the first frustum defg is, by the re-action of thesecond frustum fghi, as much urged and pressed on the superficies fg, asit urges and presses that second frustum. Therefore the frustum degf iscompressed on both sides, that is, between the cone Ade and the frustumfhig; and therefore (by Case 6, Prop. XtX) cannot preserve its figure,unless it be compressed with the same force on all sides. Therefore withthe same force with which it is pressed on the superficies de,fg, it willendeavour to break forth at the sides df, eg ; and there (being not in theleast tenacious or hard, but perfectly fluid) it will run out, expanding itself,- unless there be an ambient fluid opposing that endeavour. Therefore,by the effort it makes to run out, it will press the ambient fluid, at its sidesdf, eg, with the same force that it does the frustum fylti ; and therefore,the pressure will be propagated as much from the sides df, e~, into thespaces NO, KL this way and that way, as it is propagated from the srptrficies/g- towards PQ,. QJE.D.PROPOSITION XLII. THEOREM XXXIII.All motion propagated through a fluid diverges from a rectilinear pro*gress into ///. unmoved spaces.CASE 1. Let a motion bepropagated from the point Athrough the hole BC, and, if itbe possible, let it proceed in theconic space BCQP according toright lines diverging from thepoint A. And let us first suppose this motion to be that ofwaves in the surface of standingwater ; and let de,fg, hi, kl, &c.,be the tops of the several waves,divided from each other by asany intermediate valleys or hollows. Then, because the water in tht358 THE MATHEMATICAL PRINCIPLES [BOOK I.*ridges of the waves is higher than in the unmoved parts of the fluid KL,NO, it will run down from off the tops of those ridges, e, g, i, I, &c., dy fjhj k, &c., this way and that way towards KL and NO ; and because thewater is more depressed in the hollows of the waves than in the unmovedparts of the fluid KL, NO, it will run down into those hollows out of thoseunmoved parts. By the first deflux the ridges of the waves will dilatethemselves this way and that way, and be propagated towards KL and NO.And because the motion of the waves from A towards PQ is carried on bya continual deflux from the ridges of the waves into the hollows next tothem, and therefore cannot be swifter than in proportion to the celerity ofthe descent ; and the descent of the water on each side towards KL and NOmust be performed with the same velocity ; it follows that the dilatationof the waves on each side towards KL and NO will be propagated with thesame velocity ;is the waves themselves go forward with directly from A toPQ,. And therefore the whole space this way and that way towards KLand NO will be filled by the dilated waves rfgr, shis, tklt, v/nnv, &c.Q.E.I). That these things are so, any one may find by making the experiment in still water.CASE 2. Let us suppose that de, fg, hi, kl, mn, represent pulses successively propagated from the point A through an elastic medium. Conceive the pulses to be propagated by successive condensations and rarefactionsof the medium, so that the densest part of every pulse may occupy aspherical superficies described about the centre A, and that equal intervalsintervene between the successive pulses. Let the lines de, fg. hi, Id, &c..represent the densest parts of the pulses, propagated through the hole BC ;and because the medium is denser there than in the spaces on either sidetowards KL and NO. it will dilate itself as well towards those spaces KL,NO, on each hand, as towards the rare intervals between the pulses ; andthence the medium, becoming always more rare next the intervals, andmore dense next the pulses, will partake of their motion. And because theprogressive motion of the pulses arises from the perpetual relaxation of theden?er parts towards the antecedentrnre intervals; and since the pulses willrelax themselves on each hand towards the quiescent parts of the mediumKL, NO, with very near the same celerity ; therefore the pulses will dilatethemselves on all sides into the unmoved parts KL, NO, with almost thesame celerity with which they are propagated directly from the centre A;and therefore will fill up the whole space KLON. Q.E.D. And we findthe same by experience also in sounds which are heard through a mountaininterposed ; and,*if they come into a chamber through the window, dilatethemselves into all the parts of the room, and are heard in every corner;and not as reflected from the opposite walls, but directly propagated fromthe window, as far as our sense can judge.CASE 3 Let us suppose, lastly, that a motion of any kind is propagated:C. VIII.j OF NATURAL PHILOSOPHY. 369from A through the hole BC. Then since the cause of this propagation isthat the parts of the medium that are near the centre A disturb and agitatethose which lie farther from it; and since the parts which are urged arefluid, and therefore recede every way towards those spaces where they areless pressed, they will by consequence recede towards all the parts of thtquiescent medium; as well to the parts on each hand, as KL and NO,as to those right before, as PQ, ; and by this means all the motion, as soonas it has passed through the hole BC, will begin to dilate itself, and fromthence, as from its principle and centre, will be propagated directly everyway. Q.E.D.PROPOSITION XLIII. THEOREM XXXIV.Every tremulous body in an elastic medium propagates the motion ofthe pulses on every side right forward ; but in a non-elastic mediumexcites a circular motion.CASE. 1. The parts of the tremulous body, alternately going and returning, do in going urge and drive before them those parts of the medium thatlie nearest, and by that impulse compress and condense Nthem ; and in returning suffer those compressed parts to recede again, and expand themselves. Therefore the parts of the medium that lie nearest to the tremulousbody move to and fro by turns, in like manner as the parts of the tremulousbody itself do ; and for the same cause that the parts of this body agitatethese parts of the medium, these parts, being agitated by like tremors, willin their turn agitate others next to themselves ; and these others, agitatedin like manner, will agitate those that lie beyond them, and so on in, infinitum.And in the same manner as the lirst parts of the medium werecondensed in going, and relaxed in returning, so will the other parts becondensed every time they go, and expand themselves every time they return. And therefore they will not be all going and all returning at thesame instant (for in that case they would always preserve determined distances from each other, and there could be no alternate condensation andrarefaction) ; but since, in the places where they are condensed, they approach to, and, in the places where they are rarefied, recede from each other,therefore some of them will be going while others are returning ; and so onin infinitum. The parts so going, and in their going condensed, are pulses,by reason of the progressive motion with which they strike obstacles intheir way; and therefore the successive pulses produced by a tremulousbody will be propagated in rectilinear directions; and that at nearly equaldistances from each other, because of the equal intervals of time in whichthe body, by its several tremors produces the several pulses. And thoughthe parts of the tremulous body go and return .n some certain and determinate direction, yet the pulses propagated from thence through the mediumwill dilate themselves towards the sides, by the foregoing Proposition : anc7360 THE MATHEMATICAL PRINCIPLES [BoOK 11will be propagated on all sides from that tremulous body, as from a common centre, in superficies nearly spherical and concentrical. An exampleof this we have in waves excited by shaking a finger in water, whichproceed not only forward and backward agreeably to the motion of thefinger, but spread themselves in the manner of concentrical circles all roundthe finger, and are propagated on every side. For the gravity of the watersupplies the place of elastic force.Case 2. If the medium be not elastic, then, because its parts cannot becondensed by the pressure arising from the vibrating parts of the tremulousbody, the motion will be propagated in an instant towards the parts wherethe medium yields most easily, that is; to the parts which the tremulousbody would otherwise leave vacuous behind it. The case is the same withthat of a body projected in any medium whatever. A medium yieldingto projectiles does not recede in infinitum, but with a circular motion comesround to the spaces which the body leaves behind it. Therefore as oftenas a tremulous body tends to any part, the medium yielding to it comesround in a circle to the parts which the body leaves ; and as often as thebody returns to the first place, the medium will be driven from the place itcame round to, and return to its original place. And though the tremulousbod} be not firm and hard, but every way flexible, yet if it continue of agiven magnitude, since it cannot impel the medium by its treniors anywhere without yielding to it somewhere else, the medium receding from theparts of the body where it is pressed will always come round in a circle tothe parts that yield to it. Q.E.D.COR. It is a mistake, therefore, to think, as some have done, that theagitation of the parts of flame conduces to the propagation of a pressure inrectilinear directions through an ambient medium. A pressure of thatkind must be derived not from the agitation only of the parts of flame, butfrom the dilatation of the whole.PROPOSITION XL1V. THEOREM XXXV.If water ascend a/id descend alternately in the erected legs KL, MN, ofa canal or pipe ; and a pendulum be constructed whose length betweenthe point of suspension and the centre of oscillation is equal to halfthe length of the ivater in the canal ; I say, that the water will ascendand descend in the same times in which the pendulum oscillates.I measure the length of the water along the axes of the canal and its legs,and make it equal to the sum of those axes; and take no notice of theresistance of the water arising from its attrition by the sides of the canal.Let, therefore, AB, CD, represent the mean height of the water in bothlegs ; and when the water in the leg KL ascends to the height EF, thewater will descend in the leg MN to the height GH. Let P be a pendulou/SEC. Vlll.J OF NATURAL PHILOSOPHY. VJ61body, VP the thread, V the point of suspension, RPQS the cycloid whicLiiL Nthe pendulum describes, P its lowest point, PQ an arc equal to the neightAE. The force with which the motion of the water is accelerated and retarded alternately is the excess of the weight of the water in one leg abovethe weight in the other; and, therefore, when the water in the leg KLascends to EF, and in the other leg descends to GH, that force is doublethe weight of the water EABF, and therefore is to the weight of the wholewater as AE or PQ, to VP or PR. The force also with which the body Pis accelerated or retarded in any place, as Q, of a cycloid, is (by Cor. Prop.LI) to its whole weight as its distance PQ, from the lowest place P to thelength PR of the cycloid. Therefore the motive forces of the water andpendulum, describing the equal spaces AE, PQ, are as the weights to bemoved ; and therefore if the water and pendulum are quiescent at first,those forces will move them in equal times, and will cause them to go andreturn together with a reciprocal motion. Q.E.D.COR. 1. Therefore the reciprocations of the water in ascending and descending are all performed in equal times, whether the motion be more orless intense or remiss.COR. 2. If the length of the whole water in the canal be of 6J feet oiFrench measure, the water will descend in one second of time, and will ascondin another second, and so on by turns in infinitum ; for a pendulumof Sy-j such feet in length will oscillate in one second of time.COR. 3. But if the length of the water be increased or diminished, thetime of the reciprocation will be increased or diminished in the subduplicateratio of the length.PROPOSITION XLY. THEOREM XXXVI.The velocity of waves is in the subduplicate ratio of the breadths.This follows from the construction of the following Proposition.PROPOSITION XLVI. PROBLEM X.Tofind the velocity of waves.Let a pendulum be constructed, whose length between the point of suspension and the centre of oscillation is equal to the breadth of the waves362 THE MATHEMATICAL PRINCIPLES (BOOK 1Land in the time that the pendulum will perform one single oscillation thewaves will advance forward nearly a space equal to their breadth.That which I call the breadth of the waves is the transverse measurelying between the deepestpart of the hollows, or thetops of the ridges. LetABCDEF represent the surface of stagnant water ascending and descending in successive waves ; and let A, C, E, &c., be the tops of the waves ;and let B, D, F, &c., be the intermediate hollows. Because the motion ofthe waves is carried on by the successive ascent and descent of the water,so that the parts thereof, as A, C, E, &c., which are highest at one timebecome lowest immediately after ; and because the motive force, by whichthe highest parts descend and the lowest ascend, is the weight of the elevated water, that alternate ascent and descent will be analogous to the reciprocal motion of the water in the canal, and observe the same laws as to thetimes of its ascent and descent; and therefore (by Prop. XLIV) if thedistances between the highest places of the waves A, C, E, and the lowestB, D, F, be equal to twice the length of any pendulum, the highest partsA, C, E, will become the lowest in the time of one oscillation, and in thetime of another oscillation will ascend again. Therefore between the passage of each wave, the time of two oscillations will intervene ; that is, thewave will describe its breadth in the time that pendulum will oscillatetwice; but a pendulum of four times that length, and which therefore isequal to the breadth of the waves, will just oscillate once in that time.Q.E.LCOR. 1. Therefore waves, whose breadth is equal to 3 7。 French feet,will advance through a space equal to their breadth in one second of time;and therefore in one minute will go over a space of 1S3J feet; and in anhour a space of 11000 feet, nearly.COR. 2. And the velocity of greater or less waves will be augmented ordiminished in the subduplicatc ratio of their breadth.These things are true upon the supposition that the parts of water ascend or descend in a right line; but, in truth, that ascent and descent israther performed in a circle ; and therefore I propose the time denned bythis Proposition as only near the truth.PROPOSITION XLVIL THEOREM XXXVII.Ifpulses are propagated through a fluid, the .ve eral particles of theJluid, goittff and returning with the shortest reciprocal motion, are always accelerated or retarded according to the law of the oscillatingpendulum.Let AB, BC, CD, &c., represent equal distances of successive pulses,ABC the line of direction of the motion of the successive pulses propagatedSEC. VIII.] OF NATURAL PHILOSOPHY.from A to B ; E, F, G three physical points of the quiescent medium situate in the right line AC at equal distances from each other ; Ee, F/, G^,equal spaces of extreme shortness, through which thosepoints go and return with a reciprocal motion in each vibration ; e, </>, y, any intermediate places of the same points ;EF, FG physical lineolae, or linear parts of the mediumlying betAveen those points, and successively transferred intothe places t0, 0y, and ef, fg. Let there be drawn theright line PS equal to the right line Ee. Bisect the samein O, and from the centre O, with the interval OP, describethe circle SIPi. Let the whole time of one vibration; withits proportional parts, be expounded by the whole circumlerenceof this circle and its parts, in such sort, that, whenany time PH or PHS/i is completed, if there be let fall toPS the perpendicular HL or hi, and therebe taken E equal to PL or PI, the physical point E may be found in e. A point,as E, moving acccording to this law witha reciprocal motion, in its going from Ethrough e to e, and returning again throughe to E, will perform its several vibrations with the same degrees of acceleration and retardation with those of an oscillating pendulum. We are now to prove that the severalphysical points of the medium will be agitated with such akind of motion. Let us suppose, then, that a medium hathsuch a motion excited in it from any cause whatsoever, andconsider what will follow from thence.In the circumference PHSA let there be taken the equalarcs, HI, IK, or hi, ik, having the same ratio to the wholecircumference as the equal right lines EF, FG have to BC,the whole interval of the pulses. Let fall the perpendiculars IM, KN, or wi, kn ; then because the points E, F, G aresuccessively agitated with like motions, and perform their en tire vibrationscomposed of their going and return, while the pulse is transferred from Bto C ;if PH or PHS/t be the time elapsed since the beginning of the mo