as the area of the bottom to twice the hole.COR. 6. That part of the weight which presses upon the bottom is tothe whole weight of the water perpendicularly incumbent thereon as thecircle AB to the sum of the circles AB and EF, or as the circle AB to thfexcess of twice the circle AB above the area of the bottom. For that partof the weight which presses upon the bottom is to the weight of the wholewater in the vessel as the difference of the circles AB and EF to the sumof the same circles (by Cor. 4) ; and the weight of the whole water in thevessel is to the weight of the whole water perpendicularly incumbent onthe bottom as the circle AB to the difference of the circles AB and EF.Therefore, ex ce,quo perturbate, that part of the weight which presses uponthe bottom is to the weight of the whole water perpendicularly incumbentOF NATURAL PHILOSOPHY. 337thereon as the circle AB to the sum of the circles AB and EF. or the excess of twice the circle AB above the bottom.COR. 7. If in the middle of the hole EF there be placed the little circlePQ described about the centre G, and parallel to the horizon, the weightof water which that little circle sustains is greater than the weight of athird part of a cylinder of water whose base is that little circle and itsheight GH. For let ABNFEM be the cataract or column of falling waterwhose axis is GH, as above, and let all the wa- K ^ter, whose fluidity is not requisite for the readyand quick descent of the water, be supposed tobe congealed, as well round about the cataract,as above the little circle. And let PHQ be thecolumn of water congealed above the little circle, whose vertex is H, and its altitude GH.And suppose this cataract to fall with its wholeweight downwards, and not in the least to lieagainst or to press PHQ, but to glide freely byit without any friction, unless, perhaps, just atthe very vertex of the ice, where the cataract at the beginning of its fallmay tend to a concave figure. And as the congealed water AMEC, BNFD,lying round the cataract, is convex in its internal superficies AME, BNF,towards the falling cataract, so this column PHQ will be convex towardsthe cataract also, and will therefore be greater than a cone whose base isthat little circle PQ and its altitude GH; that is, greater than a thirdpart of a cylinder described with the same base and altitude. Now thatlittle circle sustains the weight of this column, that is, a weight greaterthan the weight of the cone, or a third part of the cylinder.COR. 8. The weight of water which the circle PQ; when very small, sustains, seems to be less than the weight of two thirds of a cylinder of waterwhose base is that little circle, and its altitude HG. For, things standingas above supposed, imagine the half of a spheroid described whose base idthat little circle, and its semi-axis or altitude HG. This figure will beequal to two thirds of that cylinder, and will comprehend within it thecolumn of congealed water PHQ, the weight of which is sustained by thatlittle circle. For though the motion of the water tends directly downwards, the external superficies of that column must yet meet the base PQin an angle somewhat acute, because the water in its fall is perpetually accelerated, and by reason of that acceleration become narrower. Therefore,oince that angle is less than a right one, this column in the lower partsthereof will lie within the hemi-spheroid. In the upper parts also it will beacute or pointed; because to make it otherwise, the horizontal motion ofthe water must be at the vertex infinitely more swift than its motion towards the horizon. And the less this circle PQ is, the more acute will22338 THE MATHEMATICAL PRINCIPLES [BOOK IIthe vertex of this column be; and the circle being diminished in infinitn/nthe angle PHQ will be diminished in infinitum, and therefore the column will lie within the hemi-spheroid. Therefore that column is less thanthat hemi-spheroid, or than two-third parts of the cylinder whose base isthat little circle, and its altitude GH. Now the little circle sustains aforce of water equal to the weight of this column, the weight of the ambientwater being employed in causing its efflux out at the hole.COR. 9. The weight of water which the little circle PQ sustains, whenit is very small, is very nearly equal to the weight of a cylinder of waterwhose base is that little circle, and its altitude |GH for this weight is anarithmetical mean between the weights of the cone and the hemi-spheroidabove mentioned. But if that little circle be not very small, but on thecontrary increased till it be equal to the hole EF, it will sustain the weightof all the water lying perpendicularly above it, that is, the weight of acylinder of water whose base is that little circle, and its altitude GH.COR. 10. Arid (as far as I can judge) the weight which this little circlesustains is always to the weight of a cylinder of water whose base is thatlittle circle, and its altitude iGH, as EF2 to EF 2 |PQ2, or as the circle EF to the excess of this circle above half the little circle PQ,, verynearly.LEMMA IV.If a cylinder move uniformly forward in. the direction of its length, theresistance made thereto is not at all changed by augmenting or diminishing- that length ; and is therefore the same with the resistanceof a circle, described with the same diameter, and moving forwardwith the same velocity in the direction, of a right line perpendicular toits plane.For the sides are not at all opposed to the motion ; and a cylinder becomes a circle when its length is diminished in infinitum.PROPOSITION XXXVII. THEOREM XXIX.If a cylinder move uninformly forward in a compressed, infinite, aridnon-elasticfinid, in the direction of its length, the resistance arisingfrom the magnitude of its transverse section is to the force by whichits whole motion may be destroyed or generated, in the time that itmoves four times its length, as the density of the medium to the density of the cylinder, nearly.For let the vessel ABDC touch the surface of stagnant water with itsbottom CD, and let the water run out of this vessel into the stagnant water through the cylindric canal EFTS perpendicular co the horizon ; andlet the little circle PQ, be placed parallel to the horizon any where in theSEC. VII.] OF NATURAL PHILOSOPHY. 339middle of the canal ; and produce CA to K, so K I JLf that AK may be to CK in the duplicate of the-^ jg""eratio, which the excess of the orifice of the canalEF above the little circle PQ bears to the circle AB. Then it is manifest (by Case 5, Case6, and Cor. 1, Prop. XXXVI) that the velocityof the water passing through the annular spacebetween the little circle and the sides of the vessel will be the very same which the water wouldacquire by falling, and in its fall describing thealtitude KG or IG.And (by Cor. 10, Prop. XXXVI) if the breadth of the vessel be infinite,so that the lineola HI may vanish, arid the altitudes IG, HG become equal ;the force of the water that flows down and presses upon the circle will beto the weight of a cylinder whose base is that little circle, and the altitudeiIG, as EF 2 to EF 2 |PQ 2, very nearly. For the force of the waterflowing downward uniformly through the whole canal will be the sameupon the little circle PQ. in whatsoever part of the canal it be placed.I ,et now the orifices of the canal EF, ST be closed, and let the littkcircle ascend in the fluid compressed on every side, and by its ascent let itoblige the water that lies above it to descend through the annular spacebetween the little circle and the sides of the canal. Then will the velocityof the ascending little circle be to the velocity of the descending water asthe difference of the circles EF and PQ, is to the circle PQ; and the velocity of the ascending little circle will be to the sum of the velocities, thatis, to the relative velocity of the descending water with which it passes bythe little circle in its ascent, as the difference of the circles EF and PQ tothe circle EF, or as EF* PQ2 to EF 2. Let that relative velocity beequal to the velocity with v/hich it was shewn above that the water wouldpass through the annular space, if the circle were to remain unmoved, thatis, to the velocity which the water would acquire by falling, and in its falldescribing the altitude IG ; and the force of the water upon the ascendingcirclewill be the same as before (by Cor. 5, of the Laws of Motion) ; thatis, the resistance of the ascending little circle will be to the weight of acylinder of water whose base is that little circle, and its altitude iIG, asEF2 to EF2 iPQ 2, nearly. But the velocity of the little circle willbe to the velocity which the water acquires by falling, and in its fall describing the altitude [G, as EF 2 PQ2 to EF 2.Let the breadth of the canal be increased in wfinitum ; and the ratiosbetween EF 2 PQ2 and EF 2, and between EF 2 and EF 2 iPQ 2.will become at last ratios of equality. And therefore the velocity of thelittle circle wr ill now be the same which the water would acquire in falling,and in its fall describing the altitude IG: and the resistance will become340 THE MATHEMATICAL PRINCIPJ ES [BOOK IT.equal to the weight of a cylinder whose base is that little circle, and itsaltitude half the altitude IG, from which the cylinder must fall to acquirethe velocity of the ascending circle; and with this velocity the cylinder inthe time of its fall will describe four times its length. But the resistanceof the cylinder moving forward with this velocity in the direction of itslength is the same with the resistance of the little circle (by Lem. IV), andis therefore nearly equal to the force by which its motion may be generatedwhile it describes four times its length.If the length of the cylinder be augmented or diminished, its motion,and the time in which it describes four times its lengOth,t will be aug&mentedor diminished in the same ratio, and therefore the force by which the motion so increased or diminished, may be destroyed or generated, will continue the same ; because the time is increased or diminished in the sameproportion ; and therefore that force remains still equal to the resistanceof the cylinder, because (by Lem. IV) that resistance will also remain thesame.If the density of the cylinder be augmented or diminished, its motion,and the force by which its motion may be generated or destroyed in thesame time, will be augmented or diminished in the same ratio. Thereforethe resistance of any cylinder whatsoever will be to the force by which itswhole motion may be generated or destroyed, in the time during which itmoves four times its length, as the density of the medium to the density ofthe cylinder- nearly. Q..E.D.A fluid must be compressed to become continued; it must be continuedand non-elastic, that all the pressure arising from its compression may bepropagated in an instant; and so, acting equally upon all parts of the bodymoved, may produce no change of the resistance. The pressure arisingfrom the motion of the body is spent in generating a motion in the partsof the fluid, and this creates the resistance. But the pressure arising fromthe compression of the fluid, be it ever so forcible, if it be propagated in aninstant, generates no motion in the parts of a continued fluid, produces nochange at all of motion therein ; and therefore neither augments nor lessens the resistance. This is certain, that the action of the fluid arisingfrom the compression cannot be stronger on the hinder parts of the bodymoved than on its fore parts, and therefore cannot lessen the resistance described in this proposition. And if its propagation be infinitely swifterthan the motion of the body pressed, it will not be stronger on the foreparts than on the hinder parts. But that action will be infinitelyswifter, and propagated in an instant, if the fluid be continued and nonelastic.COR. 1. The resistances, made to cylinders going uniformly forward inthe direction of their lengths through continued infinite mediums are in acom AHiESEC. VII.] OF NATURAL PHILOSOPHY- 341ratio compounded of the duplicate ratio of the velocities and the duplicateratio of the diameters, and the ratio of the density of the mediums.COR. 2. If the breadth of the canal be not infinitely increased but thecylinder go forward in the direction of its length through an includedquiescent medium, its axis all the while coinciding with the axis of thecanal, its resistance will be to the force by which its whole motion, in thetime in which it describes four times its length,K............. I... ........Lmay be generated or destroyed, in a ratiopounded of the ratio of EF 2 to EF 2 ionce, and the ratio of EF2 to EF2 PQ,2twice, and the ratio of the density of the mediumto the density of the cylinder.COR. 3. The same thing supposed, and that alength L is to the quadruple of the length ofthe cylinder in a ratio compounded of the ratioEF 2 -- iPQ2 to EF 2once, and the ratio ofEF 2 PQ, 2 to EF 2twice; the resistance ofthe cylinder will be to the force by which its whole motion, in the timeduring which it describes the length L, may be destroyed or generated, asthe density of the medium to the density of the cylinder.SCHOLIUM.In this proposition we have investigated that resistance alone whicharises from the magnitude of the transverse section of the cylinder, neglecting that part of the same which may arise from the obliquity of themotions. For as, in Case 1, of Prop. XXXVL, the obliquity of the motions with which the parts of the water in the vessel converged on everyside to the hole EF hindered the efflux of the water through the hole, so,in this Proposition, the obliquity of the motions, with which the parts ofthe water, pressed by the antecedent extremity of the cylinder, yield to thepressure, and diverge on all sides, retards their passage through the placesthat lie round that antecedent extremity, toward the hinder parts of thecylinder, and causes the fluid to be moved to a greater distance; which increases the resistance, and that in the same ratio almost in which it diminished the efflux of the water out of the vessel, that is, in the duplicate ratioof 25 to 21, nearly. And as, in Case 1, of that Proposition, we made theparts of the water pass through the hole EF perpendicularly and in thegreatest plenty, by supposing all the water in the vessel lying round thecataract to be frozen, and that part of the water whose motion was oblique,and useless to remain without motion, so in this Proposition, that theobliquity of the motions may be taken away, and the parts of the watermay give the freest passage to the cylinder, by yielding to it witli the mostdirect and quick motion possible, so that only so much resistance may re542 THE MATHEMATICAL PRINCIPLES [BoOK II.main as arises from the magnitude of the transverse section, and which isincapable of diminution, unless by diminishing the diameter of the cylinder ;we must conceive those parts of the fluid whose motions are oblique anduseless, and produce resistance, to be at rest among themselves at both extremities of the cylinder, and there to cohere, and be joined to the cylinder.Let ABCD be a rectangle, and letAE and BE be two parabolic arcs, i 1described with the axis AB, and g j^with a latus rectum that is to the .----""space HG, which must be describedby the cylinder in falling, in orderto acquire the velocity with which it moves, as HG to ^AB. Let CF andDF be two other parabolic arcs described with the axis CD, and a latusrectum quadruple of the former; and by the convolution of the figureabout the axis EF let there be generated a solid, whose middle part ABDCis the cylinder we are here speaking of, and whose extreme parts ABE andCDF contain the parts of the fluid at rest among themselves, and concretedinto two hard bodies, adhering to the cylinder at each end like a head andtail. Then if this solid EACFDB move in the direction of the length ofits axis FE toward the parts beyond E, the resistance will be the samewhich we have here determined in this Proposition, nearly ; that is, it willhave the same ratio to the force with which the whole motion of the cylinder may be destroyed or generated, in the time that it is describing thelength 4AC with that motion uniformly continued, as the density of thefluid has to the density of the cylinder, nearly. And (by Cor. 7, Prop.XXXVI) the resistance must be to this force in the ratio of 2 to 3, at theleast.LEMMA V.If a cylinder, a sphere, and a spheroid, of equal breadths be placed successively in the middle of a cylindric canal, so that their axes maycoincide with the axis of the canal, these bodies will equally hinder t^epassage of the water through the canal.For the spaces lying between the sides of the canal, and the cylinder,sphere, and spheroid, through which the water passes, are equal ; and thewater will pass equally through equal spaces.This is true, upon the supposition that all the water above the cylinder,sphere, or spheroid, whose fluidity is not necessary to make the passage ofthe water the quickest possible, is congealed, as was explained above in Cer7, Prop. XXXVI.SEC. VII.] OF NATURAL PHILOSOPHY 343LEMMA VI.The same supposition remaining, the fore- mentioned bodies are equallyacted OIL by the water Jlowing through the canal.This appears by Lein. V and the third Law. For tht water and thebodies act upon each other mutually and equally.LEMMA VILIf the water be at rest in the canal, and these bodies move with equil velocity and the contrary way through the canal, their resistances willbe equal among themselves.This appears from the last Lemma, for the relative motions remain thesame among themselves.SCHOLIUM.The case is the same of all convex and round bodies, whose axes coincidewith the axis of the canal. Some difference may arise from a greater orless friction; but in these Lemmata we suppose the bodies to be perfectlysmooth, and the medium to be void of all tenacity and friction; and thatthose parts of the fluid which by their oblique and superfluous motions maydisturb, hinder, and retard the flux of the water through the canal, are atrest amorg themselves ; being fixed like water by frost, and adhering tothe fore and hinder parts of the bodies in the manner explained in theScholium of the last Proposition : for in what follows we consider the veryleast resistance that round bodies described with the greatest given transverse sections can possibly meet with.Bodies swimming upon fluids, when they move straight forward, causethe fluid to ascend at their fore parts and subside at their hinder parts,especially if they are of an obtuse figure ; and thence they meet with alittle more resistance than if they were acu*-e at the head and tail. Andbodies moving in elastic fluids, if they are obtuse behind and before, condense the fluid a little more at their fore parts, and relax the same at theiihinder parts ; and therefore meet also with a little more resistance than iithey were acute at the head and tail. But in these Lemmas and Propositions we are not treating of elastic but non-elastic fluids; not of bodiesfloating on the surface of the fluid, but deeply immersed therein. Andwhen the resistance of bodies in non-elastic fluids is once known, we maythen augment this resistance a little in elastic fluids, as our air; and inthe surfaces of stagnating fluids, as lakes and seas.PROPOSITION XXXVIII. THEOREM XXX.If a globe move uniformly forward in a compressed, infinite, and no?telastic fluid, its resistance is to the force by which its whole544 THE MATHEMATICAL PRINCIPLES [BOOK IImay be destroyed or generated, in the time that it describes eight thirdparts of its diameter, as the density of the fluid to the density of theglobe, very nearly.For the globe is to its circumscribed cylinder as two to three ; and therefore the force which can destroy all the motion of the cylinder, while thesame cylinder is describing the length of four of its diameters, will destroyall the motion of the globe, while the globe is describing two thirds of thislength, that is, eight third parts of its own diameter. Now the resistanceof the cylinder is to this force very nearly as the density of the fluid to thedensity of the cylinder or globe (by Prop. XXXVII), and the resistance ofthe globe is equal to the resistance of the cylinder (by Lem. V, VI, andVII). Q.E.D.COR. I. The resistances of globes in infinite compressed mediums are ina ratio compounded of the duplicate ratio of the velocity, and the duplicate ratio of the diameter, and the ratio of the density of the mediums.COR. 2. The greatest velocity, with which a globe can descend by itscomparative weight through a resisting fluid, is the same which it mayacquire by falling with the same weight, and without any resistance, andin its fall describing a space that is, to four third parts of its diameter asthe density of the globe to the density of the fluid. For the globe in thetime of its fall, moving with the velocity acquired in falling, will describea space that will be to eight third parts of its diameter as the density ofthe globe to the density of the fluid ; and the force of its weight whichgenerates this motion will be to the force that can generate the same motion, in the time that the globe describes eight third parts of its diameter,with the same velocity as the density of the fluid to the density of theglobe; and therefore (by this Proposition) the force of weight will be equalto the force of resistance, and therefore cannot accelerate the globe.COR. 3. If there be given both the density of the globe and its velocityat the beginning of the motion, and the density of the compressed quiescentfluid in which the globe moves, there is given at any time both the velocity of the globe and its resistance, and the space described by it (by Cor.7, Prop. XXXV).COR. 4. A globe moving in a compressed quiescent fluid of the samedensity with itself will lose half its motion before it can describe the lengthof two of its diameters (by the same Cor. 7).PROPOSITION XXXIX. THEOREM XXXI.If a S lobe move uniformly forward through a fluid inclosed and compressed in a cylindric canal, its resistance is to the force by which itswhole motion may be generated or destroyed, in the time in which itdescribes eight third parts of its dia?netert in a ratio compounded ofEC. VII.] OF NATURAL PHILOSOPHY. 345the ratio of the orifice of the canal to the excess of that orifice abovehalf the greatest circle of the globe ; and the duplicate ratio of theorifice of the canal, to the excess of that orifice above the greatest circleof the globe ; and t/ie ratio of the density of thefluid to the density ofthe globe, nearly.This appears by Cor. 2, Prop. XXXVII, and the demonstration proceeds in the same manner as in the foregoing Proposition.SCHOLIUM.In the last two Propositions we suppose (as was done before in Lem. V)that all the water which precedes the globe, and whose fluidity increasesthe resistance of the same, is congealed. Now if that water becomes fluid,it will somewhat increase the resistance. But in these Propositions thatincrease is so small, that it may be neglected, because the convex superficies of the globe produces the very same effect almost as the congelationof the water.PROPOSITION XL. PROBLEM IX.Tofind by phenomena the resistance of a globe moving through a perfectly fluid compressed medium.Let A be the weight of the globe in vacua, B its weight in the resistingmedium, D the diameter of the globe. F a space which is to fD as the den