自然哲学的数学原理-36

these ratios, the efficacy of a particle, falling upon the globe obliquely inthe direction of the right line FBy to move the globe in the direction of itsincidence, is to the efficacy of the same particle falling in the same lineperpendicularly on the cylinder, to move it in the same direction, as BE 2to BC 3. Therefore if in 6E, which is perpendicular to the circular base ofthe cylinder NAO, and equal to the radius AC, we take H equal toBEa-; then 6H will be to 6E as the effect of the particle upon the globe t<?。~i。jthe effect of the particle upon the cylinder. Arid therefore the solid whichis formed by all the right lines 6H will be to the solid formed by all theright lines />E as the effect of all the particles upon the globe to the effectof all the particles upon the cylinder. But the former of these solids is a328 THE MATHEiAlATICAL PRINCIPLES [BooK li.paraboloid whose vertex is C, its axis CA, and latus rectum CA, and thelatter solid is a cylinder circumscribing the paraboloid ; and it is knowrthat a paraboloid is half its circumscribed cylinder. Therefore the wholeforce of the medium upon the globe is half of the entire force of the sameupon the cylinder. And therefore if the particles of the medium are atrest, and the cylinder and globe move with equal velocities, the resistanceof the globe will be half the resistance of the cylinder. Q.E.D.SCHOLIUM.By the same method other figures may be compared together as to theirresistance; and those may be found which are most apt to continue theirmotions in resisting mediums. As if upon the circular base CEBH fromthe centre O, with thy radius OC, and the altitude OD, one would constructa frustum CBGF of a cone, which should meet with less resistance thanany other frustum constructed with the same base and altitude, and goingforwards towards D in the direction of its axis : bisect the altitude OD inU,, and produce OQ, to S, so that QS may be equal to Q,C, and S will bethe vertex of the cone whose frustum is sought.rJWhence, by the bye, since the angle CSB is always acute, it follows, that,if the solid ADBE be generated by the convolution of an elliptical or ovalfigure ADBE about its axis AB, and the generating figure be touched bythreeright lines FG, GH, HI, in the points F, B, and I, so that GH shallbe perpendicular to the axis in the point of contact B, arid FG, HI may beinclined to GH in the angles FGB, BHI of 135 degrees: the solid arisingfrom the convolution of the figure ADFGH1E about the same axis ABwill be less resisted than the former solid; if so be that both move forwardin the direction of their axis AB, and that the extremity B of each goforemost. Which Proposition I conceive may be of use in the building ofships.If the figure DNFG be such a curve, that if, from any point thereof, asN, the perpendicular NM be let fall on the axis AB, and from the givenpoint G there be drawn the right line GR parallel to a right line touchingthe figure in N, and cutting the axis produced in R, MN becomes to GRas GR, 3 to 4BR X GB 2, the solid described, by the revolution of this figureSEC. Vll.J OF NATURAL PHILOSOPHY. 32Sabout its axis AB, moving in the before-mentioned rare medium from Atowards B, will be less resisted than any other circular solid whatsoever,described of the same length and breadth.PROPOSITION XXXV. PROBLEM VII.If a rare medium consist of very small quiescent particles of equal magnitudes, and freely disposed at equal distances from one another : tojind the resistance of a globe moving uniformly forward in thismedium.CASE 1. Let a cylinder described with the same diameter and altitude beconceived to go forward with the same velocity in the direction of its axisthrough the same medium ; and let us suppose that the particles of themedium, on which the globe or cylinder falls, fly back with as great a forceof reflexion as possible. Then since the resistance of the globe (by the lastProposition) is but half the resistance of the cylinder, and since the globeis to the cylinder as 2 to 3, and since the cylinder by falling perpendicularly on the particles, and reflecting them with the utmost force, communicates to them a velocity double to its own; it follows that the cylinder.in moving forward uniformly half the length of its axis, will communicatea motion to the particles which is to the whole motion of the cylinder asthe density of the medium to the density of the cylinder ; and that theglobe, in the time it describes one length of its diameter in moving uniformly forward, will communicate the same motion to the particles ; andin the time that it describes twro thirds of its diameter, will communicatea motion to the particles which is to the whole motion of the globe as thedensity of the medium to the density of the globe. Arid therefore theglobe meets with a resistance, which is to the force by which its whole motion may be either taken away or generated in the time in which it describes two thirds of its diameter moving uniformly forward, as the density of the medium to the density of the globe.CASE 2. Let us suppose that the particles of the medium incident onthe globe or cylinder are not reflected ; and then the cylinder falling perpendicularly on the particles will communicate its own simple velocity tothem, and therefore meets a resistance but half so great as in the formercase, and the globe also meets with a resistance but half so great.CASE 3. Let us suppose the particles of the medium to fly back fromthe globe with a force which is neither the greatest, nor yet none at all, butwith a certain mean force ; then the resistance of the globe will be in thesame mean ratio between the resistance in the first case and the resistancein the second. Q.E.I.COR. 1. Hence if the globe and the particles are infinitely hard, anddestitute of all elastic force, and therefore of all force of reflexion ; thfresistance of the globe will be to the force by which its whole motion may330 THE MATHEMATICAL PRINCIPLES [BOOK I)be destroyed or generated, in the time that the globe describes four thirdparts of its diameter, as the density of the medium to the density of the^lobe.Con. 2. The resistance of the globe, cceteris paribus, is in the duplicateratio of the velocity.CUR. 3. The resistance of the globe, cocteris paribus, is in the duplicateratio of the diameter.COR. 4. The resistance of the globe is, cceteris paribus, as the density ofthe medium.COR, 5. The resistance of the globe is in a ratio compounded of the duplicate ratio of the velocity, arid the duplicate ratio of the diameter, andthe ratio of the density of the medium.COR. 6. The motion of the globe and its resistance may be thus expounded Let AB be thetime in which the globe may, by its resistanceuniformly continued, lose its whole motion.Erect AD, BC perpendicular to AB. J ,et BC bethat whole motion, and through the point C, theasymptotes being AD, AB, describe the hyperbolaCF. Produce AB to any point E. Erect the perpendicular EF meetingthe hyperbola in F. Complete the parallelogram CBEG, and draw AFmeeting BC in H. Then if the globe in any time BE, with its first motion BC uniformly continued, describes in a non-resisting medium the spaceCBEG expounded by the area of the parallelogram, the same in a resistingmedium will describe the space CBEF expounded by the area of the hvperbola;and its motion at the end of that time will be expounded by EF,the ordinate of the hyperbola, there being lost of its motion the part FG.And its resistance at the end of the same time will be expounded by thelength BH, there being lost of its resistance the part CH. All these thingsappear by Cor. 1 and 3, Prop. V., Book II.COR. 7. Hence if the globe in the time T by the resistance R uniformlycontinued lose its whole motion M, the same globe in the time t in aresisting medium, wherein the resistance R decreases in a duplicate/Mratio of the velocity, will lose out of its motion M the part ,.i theTMpart rn . ; remaining ; and will describe a space which is to the space described in the same time t, with the uniform motion M, as the logarithm ofT + tthe number ^. multiplied by the number 2,302585092994 is to thenumber ^ because the hyperbolic area BCFE is to the rectangle BCGEin that proportion.SEC. VII.] OF NATURAL PHILOSOPHY. 331SCHOLIUM.I have exhibited in this Proposition the resistance and retardation ofspherical projectiles in mediums that are not continued, and shewn thatthis resistance is to the force by which the whole motion of the globe may bedestroyed or produced in the time in which the globe can describe two thirdsof its diameter, with a velocity uniformly continued, as the density of themedium to the density of the globe, if so be the globe and the particles ofthe medium be perfectly elastic, and are endued with the utmost force ofreflexion ; and that this force, where the globe and particles of the mediumare infinitely hard and void of any reflecting force, is diminished one half.But in continued mediums, as water, hot oil, and quicksilver, the globe asit passes through them does not immediately strike against all the particles of the fluid that generate the resistance made to it, but presses onlythe particles that lie next to it, which press the particles beyond, whichpress other particles, and so on ; and in these mediums the resistance is diminished one other half. A globe in these extremely fluid mediums meetswith a resistance that is to the force by which its whole motion may bedestroyed or generated in the time wherein it can describe, with that motion uniformly continued, eight third parts of its diameter, as the densityof the medium to the density of the globe. This I shall endeavour to shewin what follows.PROPOSITION XXXVI. PROBLEM VIII.To define the motion of water running out of a cylindrical vessel througha hole made at the bottom.Let AC DB be a cylindrical vessel, AB the mouth p = Q:of it, CD the bottom p irallel to the horizon, EF acircular hole in the middle of the bottom, G thec-?ritre of the hole, and GH the axis of the cylin- Kjcler perpendicular to the horizon. And suppose acylinder of ice APQ,B to be of the same breadthwith the cavity of the vessel, and to have the sameaxis, and to descend perpetually with an uniformmotion, and that its parts, as soon as they touch thesuperficies AB, dissolve into water, and flow( wn by their weight into the vessel, and in theirfall compose the cataract or column of waterABNFEM, passing through the hole EF, and filling up the same exactly.Let the uniform velocity of the descending ice and of the contiguous waterin the circle AB be that which the water would acquire by falling throughthe space IH ; and let IH and HG lie in the same right line ; and through332 THE MATHEMATICAL PRINCIPLES [BOOK Jlthe point I let there be drawn the right line KL parallel to the horizonand meeting the ice on both the sides thereof in K and L. Then the velocity of the water running out at the hole EF will be the same that itwould acquire by falling from I through the space IG. Therefore, byGalih cJ s Theorems, IG will be to IH in the duplicate ratio of the velocity of the water that runs out at the hole to the velocity of the wrater inthe circle AB, that is, in the duplicate ratio of the circle AB to the circleEF ; those circles being reciprocally as the velocities of the water whichin the same time and in equal quantities passes severally through each ofthem, and completely fills them both. We are now considering the velocity with which the water tends to the plane of the horizon. But the motion parallel to the same, by which the parts of the falling water approach toeach other, is not here taken notice of; since it is neither produced bygravity, nor at all changes the motion perpendicular to the horizon which thegravity produces. We suppose, indeed, that the parts of the water coherea little, that by their cohesion they may in falling approach to each otheiwith motions parallel to the horizon in order to form one single cataract,and to prevent their being divided into several : but the motion parallel tothe horizon arising from this cohesion does not come under our presentconsideration.CASE 1. Conceive now the w^hole cavity in the vessel, wrhich encompassesthe falling water ABNFEM, to be full of ice, so that the water may passthrough the ice as through a funnel. Then if the water pass very near tothe ice only, without touching it; or, which is the same tiling, if by reason of the perfect smoothness of the surface of the ice, the water, thoughtouching it. glides over it writh the utmost freedom, and without the le-astresistance; the water will run through the hole EF with the same velocityas before, and the whole weight of the column of water ABNFEM will beall taken up as before in forcing out the water, and the bottom of the vesselwill sustain the weight of the ice encompassing that column.Let now the ice in the vessel dissolve into water ; yet will the efflux ofthe water remain, as to its velocity, the same as before. It will not beless, because the ice now dissolved will endeavour to descend ;it will notbe greater, because the ice. now become water, cannot descend without hindering the descent of other water equal to its own descent. The same forceought always to generate the same velocity in the effluent water.But the hole at the bottom of the vessel, by reason of the oblique motions of the particles of the effluent water, must be a little greater than before,For now the particles of the water do not all of them pass through thehole perpendicularly, but, flowing down on all parts from the sides of thevessel, and converging towards the hole, pass through it with oblique motions : r,r,d in tending downwards meet in a stream whose diameter is a littlesmaller below the hole than at the hole itself : its diameter being to theSEC. V1L! OF NATURAL PHILOSOPHY. 333diameter of the hole as 5 to 6, or as 5^ to 6|, very nearly, if I took themeasures of those diameters right. I procured a very thin flat plate, having a hole pierced in the middle, the diameter of the circular hole beingf parts of an inch. And that the stream of running waters might not beaccelerated in falling, and by that acceleration become narrower, I fixedthis plate not to the bottom, but to the side of the vessel, so us to make thewater go out in the direction of a line parallel to the horizon. Then, whenthe vessel was full of water, I opened the hole to let it run out ; and thediameter of the stream, measured with great accuracy at the distance ofabout half an inch from the hole, was f J- of an inch. Therefore the diameter of this circular hole was to the diameter of the stream very nearlyas 25 to 21. So that the water in passing through the hole converges onall sides, and, after it has run out of the vessel, becomes smaller by converging in that manner, and by becoming smaller is accelerated till it comes tothe distance of half an inch from the hole, and at that distance flows in asmaller stream and with greater celerity than in the hole itself, and thisin the ratio of 25 X 25 to 21 X 21, or 17 to 12, very nearly ; that is, inabout the subdaplicate ratio of 2 to 1. Now it is certain from experiments,that the quantity of water running out in a given time through a circularhole made in the bottom of a vessel is equal to the quantity, which, flowing with the aforesaid velocity, would run out in the same time throughanother circular hole, whose diameter is to the diameter of the former as21 to 25. And therefore that running water in passing through thehole itself has a velocity downwards equal to that which a heavy bodywould acquire in falling through half the height of the stagnant water inthe vessel, nearly. But, then, after it has run out, it is still accelerated byconverging, till it arrives at a distance from the hole that is nearly equal toits diameter, and acquires a velocity greater than the other in about thesubduplicate ratio of 2 to 1; which velocity a heavy body would nearlyacquire by falling through the whole height of the stagnant water in thevessel.Therefore in what follows let the diameter ofthe stream be represented by that lesser hole whichwe called EF. And imagine another plane VWabove the hole EF, and parallel to the plane thereof, to be placed at a distance equal to the diameter of the same hole, and to be pierced throughwith a greater hole ST, of such a magnitude thata stream which will exactly fill the lower hole EFmay pass through it; the diameter of which holewill therefore be to the diameter of the lower hole as 25 to 21, nearly. Bythis means the water will run perpendicularly out at the lower hole ; andthe quantity of the water running out will be, according to the magnitude334 THE MATHEMATICAL PRINCIPLES [BOOK 11of this last hole, the same, very nearly, which the solution of the Problemrequires. The space included between the two planes and the falling streammay be considered as the bottom of the vessel. But, to make the solutionmore simple and mathematical, it is better to take the lower plane alonefor the bottom of the vessel, and to suppose that the water which flowedthrough the ice as through a funnel, and ran out of the vessel through thehole EF made in the lower plane, preserves its motion continually, and thatthe ice continues at rest. Therefore in what follows let ST be the diameter of a circular hole described from the centre Z, and let the stream runout of the vessel through that hole, when the water in the vessel is allfluid. And let EP be the diameter of the hole, which the stream, in falling through, exactly fills up, whether the water runs out of the vessel bythat upper hole ST, or flows through the middle of the ice in the vessel,as through a funnel. And let the diameter of the upper hole ST be to thediameter of the lower EF as about 25 to 21, and let the perpendicular di&tance between the planes of the holes be equal to the diameter of the lesserhole EF. Then the velocity of the water downwards, in running out ofthe vessel through the hole ST, will be in that hole the same that a bodymay acquire by falling from half the height IZ ; and the velocity of boththe falling streams will be in the hole EF, the same which a body wouldacquire by falling from the Avhole height IG.CASE 2. If the hole EF be not in the middle of the bottom of the vessel, but in some other part thereof, the water will still run out with thesame velocity as before, if the magnitude of the hole be the same. Forthough an heavy body takes a longer time in descending to the same depth,by an oblique line, than by a perpendicular line, yet in both cases it acquiresin its descent the same velocity ; as Galileo has demonstrated.CASE 3. The velocity of the water is the same when it runs out througha hole in the side of the vessel. For if the hole be small, so that the interval between the superficies AB and KL may vanish ns to sense, and thestream of water horizontally issuing out may form a parabolic figure; fromthe latus rectum of this parabola may be collected, that the velocity of theeffluent water is that which a body may acquire by falling the height IGor HG of the stagnant water in the vessel. For, by making an experiment, I found that if the height of the stagnant water above the hole were20 inches, and the height of the hole above a plane parallel to the horizonwere also 20 inches, a stream of water springing out from thence wrouldfall upon the plane, at the distance of 37 inches, very nearly, from a perpendicular let fall upon that plane from the hole. For without resistancethe stream would have fallen upon the plane at the distance of 40 inches,the latus rectum of the parabolic stream being 80 inches.CASE 4. If the effluent water tend upward, it will still issue forth withthe same velocity. For the small stream of water springing upward, asSEC. V11.J OF NATURAL PHILOSOPHY. 335cends with a perpendicular motion to GH or GI, the height of the stagnantwater in the vessel; excepting in so far as its ascent is hindered a little bythe resistance of the air : and therefore it springs out with the same velocity that it would acquire in falling from that height. Every particle ofthe stagnant water is equally pressed on all sides (by Prop. XIX., Book II),and, yielding to the pressure, tends always with an equal force, whether itdescends through the hole in the bottom of the vessel, or gushes out in anhorizontal direction through a hole in the side, or passes into a canal, andsprings up from thence through a little hole made in the upper part of thecanal. And it may not only be collected from reasoning, but is manifestalso from the well-known experiments just mentioned, that the velocitywith which the water runs out is the very same that is assigned in thisProposition.CASE 5. The velocity of the effluent water is the same, whether thefigure of the hole be circular, or square, or triangular, or any other figureequalto the circular ; for the velocity of the effluent water does not dependupon the figure of the hole, but arises from its depth below the planeKL.CASE 6. If the lower part of the vessel ABDCB be immersed into stagnant water, and the heightof the stagnant water above the bottom of the vessel be GR, the velocity with which the water thatis in the vessel will run out at the hole EF intothe stagnant water will be the same which thewater would acquire by falling from the heightIR ; for the weight of all the water in the vesselthat is below the superficies of the stagnant waterwill be sustained in equilibrio by the weight of the stagnant water, andtherefore does riot at all accelerate the motion of the descending water inthe vessel. This case will also appear by experiments, measuring the timesin which the water will run out.COR. 1. Hence if CA the depth of the water be produced to K, so thatAK may be to CK in the duplicate ratio of the area of a hole made in anypart of the bottom to the area of the circle AB, the velocity of the effluentwater will be equal to the velocity which the water would acquire by fallingfrom the height KC.COR. 2. And the force with which the whole motion of the effluent wateimay be generated is equal to the weight of a cylindric column of water rwhose base is the hole EF, and its altitude 2GI or 2CK. For the effluentwater, in the time it becomes equal to this column, may acquire, by fallingby its own weight from the height GI, a velocity equal to that with whichit runs out.COR. 3. The weigb t of all the water in the vessel ABDC is to that part。336 THE MATHEMATICAL PRINCIPLES [BOOK IIof the weight which is employed in forcing out the water as the sum ofthe circles AB and EF to twice the circle EF. For let IO be a mean proportional between IH and IG, and the water running out at the hole EFwill, in the time that a drop falling from I would describe the altitude IG,become equal to a cylinder whose base is the circle EF and its altitude2IG; that is, to a cylinder whose base is the circle AB, and whose altitudeis 2IO. For the circle EF is to the circle AB in the subduplicate ratio cfthe altitude IH to the altitude IG ; that is, in the simple ratio of the meanproportional IO to the altitude IG. Moreover, in the time that a dropfalling from I can describe the altitude IH, the water that runs out willhare become equal to a cylinder whose base is the circle AB, and its altitude 2IH ; and in the time that a drop falling from I through H to G describes HG, the difference of the altitudes, the effluent water, that is, thewater contained within the solid ABNFEM, will be equal to the differenceof the cylinders, that is, to a cylinder whose base is AB, and its altitude2HO. And therefore all the water contained in the vessel ABDC is to thewhole falling water contained in the said solid ABNFEM as HG to2HO,that is, as HO + OG to 2HO, or IH + K ) to 2IH. But the weight of allthe water in the solid ABNFEM is employed in forcing out the water ;and therefore the weight of all the water in the vessel is to that part ofthe weight that is employed in forcing out the water as IH + IO to 2IH,and therefore as the sum of the circles EF and AB to twice the circleEF.COR. 4. And hence the weight of all the water in the vessel ABDC isto the other part of the weight which is sustained by the bottom of thevessel as the sum of the circles AB and EF to the difference of the samecircles.COR. 5. And that part of the weight which the bottom of the vessel sustains is to the other part of the weight employed in forcing out the wateras the difference of the circles AB and EF to twice the lesser circle EF, or