Principia Mathematica (Motte Translation, 1848) — part 25 of 45

I have exhibited in this Proposition the resistance and retardation of spherical projectiles in mediums that are not continued, and shewn that this resistance is to the force by which the whole motion of the globe may be destroyed or produced in the time in which the globe can describe two thirds of its diameter, with a velocity uniformly continued, as the density of the medium to the density of the globe, if so be the globe and the particles of the medium be perfectly elastic, and are endued with the utmost force of reflexion ; and that this force, where the globe and particles of the medium are infinitely hard and void of any reflecting force, is diminished one half. But in continued mediums, as water, hot oil, and quicksilver, the globe as it passes through them 'does not immediately strike against all the parti- cles of the fluid that generate the resistance made to it, but presses only the particles that lie next to it, which press the particles beyond, which press other particles, and so on ; and in these mediums the resistance is di- minished one other half. A globe in these extremely fluid mediums meets with a resistance that is to the force by which its whole motion may be destroyed or generated in the time wherein it can describe, with that mo- tion uniformly continued, eight third parts of its diameter, as the density of the medium to the density of the globe. This I shall endeavour to sh|w in what follows.

K

PROPOSITION XXXYL PROBLEM VIIL

To define the motion of water running out of a cylindrical vessel through a hole 7nade at the bottom. Let AC DB be a cylindrical vessel, AB the mouth of it, CD the bottom parallel to the horizon, EF a circular hole in the middle of the bottom, G the centre of the hole, and GH the axis of the cylin- der perpendicular to the horizon. And suppose a a, cylinder of ice APGIB to be of the same breadth with the cavity of the vessel, and to have the same axis, and to descend perpetually with an uniform motion, and that its parts, as soon as they touch the superficies AB, dissolve into water, and flow down by their weight into the vessel, and in their fall compose the cataract or column of water ABNFEM, passing through the hole EF, and filling up the same exactly. Let the uniform velocity of the descending ice and of the contiguous water in the circle AB be that which the water would acquire by falling through the space IH ; and let IH and HG lie in the same right line ; and through

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the point I let there be drawn the right line KL parallel to the horizon, and meeting the ice on both the sides thereof in K and L. Then the ve- locity of the water running out at the hole EF will be the same that it would acquire by falling from I through the space IG. Therefore, by Galileo's Theorems, IG will be to IH in the duplicate ratio of the velo- city of the water that runs out at the hole to the velocity of the water in the circle AB, that is, in the duplicate ratio of the circle AB to the circle EP ; those circles being reciprocally as the velocities of the water which in the same time and in equal quantities passes severally through each of them, and completely fills them both. We are now considering the velo- city with which the water tends to the plane of the horizon. But the mo- tion parallel to the same, by which the parts of the falling water approach to each other, is not here taken notice of; since it is neither produced by gravity, nor at all changes the motion perpendicular to the horizon which the gravity produces. We suppose, indeed, that the parts of the water cohere a little, that by their cohesion they may in falling approach to each other with motions parallel to the horizon in order to form one single cataract, and to prevent their being divided into several : but the motion parallel to the horizon arising from this cohesion does not come under our present consideration.

Case 1. Conceive now the whole cavity in the vessel, which encompasses the falling water ABNFEM, to be full of ice, so that the water may pass through the ice as through a funnel. Then if the water pass very near to the ice only, without touching it ; or, which is the same thing, if by rea- son of the perfect smoothness of the surface of the ice, the water, though touching it, glides over it with the utmost freedom, and without the least resistance ; the water will run through the hole EP with the same velocity as before, and the whole weight of the column of water ABNPEM will be all taken up as before in forcing out the water, and the bottom of the vessel will sustain the weight of the ice encompassing that column.

Let now the ice in the vessel dissolve into water ; yet will the efflux of the water remain, as to its velocity, the same as before. It will not be less, because the ice now dissolved will endeavour to descend ; it will not be greater, because the ice. now become water, cannot descend without hin- dering the descent of other water equal to its own descent. The same force ought always to generate the same velocity in the effluent water.

But the hole at the bottom of the vessel, by reason of the oblique mo- tions 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 the hole perpendicularl}'', but, flowing down on all parts from the sides of the vessel, and converging towards the hole, pass through it with oblique mo- tions ; and in tending downwards meet in a stream whose diameter is a little smaller below the hole than at the hole itself ; its diameter being to the

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diameter of the hole as 5 to 6, or as 5^ to 6|, very nearly, if I took the measures of those diameters right. I procured a very thin flat plate, hav- ing a hole pierced in the middle, the diameter of the circular hole being f parts of an inch. And that the stream of running waters might not be accelerated in falling, and by that acceleration become narrower, I fixed this plate not to the bottom, but to the side of the vessel, so as to make the water go out in the direction of a line parallel to the horizon. Then, when the vessel was full of water, I opened the hole to let it run out ; and the diameter of the stream, measured with great accuracy at the distance of about half an inch from the hole, was |1 of an inch. Therefore the di- ameter of this circular hole was to the diameter of the stream very nearly as 25 to 21. So that the water in passing through the hole converges on all sides, and, after it has run out of the vessel, becomes smaller by converg- ing in that manner, and by becoming smaller is accelerated till it comes to the distance of half an inch from the hole, and at that distance flows in a smaller stream and with greater celerity than in the hole itself, and this in the ratio of 25 X 25 to 21 X 21, or 17 to 12, very nearly; that is, in about the subduplicate ratio of 2 to 1. Now it is certain from experiments, that the quantity of water running out in a given time through a circular hole made in the bottom of a vessel is equal to the quantity, which, flow- ing with the aforesaid velocity, would run out in the same time through another circular hole, whose diameter is to the diameter of the former as 21 to 25. And therefore that running water in passing through the hole itself has a velocity downwards equal to that which a heavy body would acquire in falling through half the height of the stagnant water in the vessel, nearly. But, then, after it has run out, it is still accelerated by converging, till it arrives at a distance from the hole that is nearly equal to its diameter, and acquires a velocity greater than the other in about the subduplicate ratio of 2 to 1 ; which velocity a heavy body would nearly acquire by falling through the whole height of the stagnant water in the vessel.

Therefore in what follows let the diameter of ^ the stream be represented by that lesser hole which we called EF. And imagine another plane Y W above the hole EP, and parallel to the plane there- of, to be placed at a distance equal to the diame- ter of the same hole, and to be pierced through with a greater hole ST, of such a magnitude that a stream which will exactly fill the lower hole EF may pass through it ; the diameter of which hole will therefore be to the diameter of the lower hole as 25 to 21, nearly. By this means the water will run perpendicularly out at the lower hole ; and the quantity of the water running out will be, according to the magnitude

334 THE MATHEMATICAL PRINCIPLES [BoOK 11

of this last hole, the same, very nearly, which the solution of the Problem requires. The space included between the two planes and the falling stream may be considered as the bottom of the vessel. But, to make the solution more simple and mathematical, it is better to take the lower plane alone for the bottom of the vessel, and to suppose that the water which flowed through the ice as through a funnel, and ran out of the vessel through the hole EF made in the lower plane, preserves its motion continually, and that the ice continues at rest. Therefore in what follows let ST be the diame- ter, of a circular hole described from the centre Z, and let the stream run out of the vessel through that hole, when the water in the vessel is all fluid. And let EF be the diameter of the hole, which the stream, in fall- ing through, exactly fills up, whether the water runs out of the vessel by that 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 the diameter of the lower EF as about 25 to 21, and let the perpendicular dis- tance between the planes of the holes be equal to the diameter of the lesser hole EF. Then the velocity of the water downwards, in running out of the vessel through the hole ST, will be in that hole the same that a body may acquire by falling from half the height IZ ; and the velocity of both the falling streams will be in the hole EF, the same which a body would acquire by falling from the whole height IG.

Case 2. If the hole EF be not in the middle of the bottom of the ves- sel, but in some other part thereof, the water will still run out with the same velocity as before, if the magnitude of the hole be the same. For though 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 acquires in its descent the same velocity ; as Galileo has demonstrated.

Case 3. The velocity of the water is the same when it runs out through a hole in the side of the vessel. For if the hole be small, so that the in- terval between the superficies AB and KL may vanish as to sense, and the stream of water horizontally issuing out may form a parabolic figure ; from the latus rectum of this parabola may be collected, that the velocity of tke efiluent water is that which a body may acquire by falling the height IG or HG of the stagnant water in the vessel. For, by making an experi- ment, I found that if the height of the stagnant water above the hole were 20 inches, and the height of the hole above a plane parallel to the horizon were also 20 inches, a stream of water springing out from thence would fall upon the plane, at the distance of 37 inches, very nearly, from a per- pendicular let fall upon that plane from the hole. For without resistance the 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 with the same velocity. For the small stream of water springing upward, as-

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cends with a perpendicular motion to GH or GI, the height of the stagnant water in the vessel ; excepting in so far as its ascent is hindered a little by the resistance of the air ; and therefore it springs out with the same ve- locity that it would acquire in falling from that height. Every particle of the stagnant water is equally pressed on all sides (by Prop. XIX., Book 11), and, yielding to the pressure, tends always with an equal force, whether it descends through the hole in the bottom of the vessel, or gushes out in an horizontal direction through a hole in the side, or passes into a canal, and springs up from thence through a little hole made in the upper part of the canal. And it may not only be collected from reasoning, but is manifest also from the well-known experiments just mentioned, that the velocity with which the water runs out is the very same that is assigned in this Proposition.

Case 5. The velocity of the effluent water is the same, whether the figure of the hole be circular, or square, or triangular, or any other figure equal to the circular ; for the velocity of the effluent water does not depend upon the figure of the hole, but arises from its depth below the plane KL.

j^ Case 6. If the lower part of the vessel ABDC B be immersed into stagnant water, and the height of the stagnant water above the bottom of the ves- sel be GR, the velocity with which the water that is in the vessel will run out at the hole EF into the stagnant water will be the same which the water would acquire by falling from the height IR ; for the weight of all the water in the vessel that is below the superficies of the stagnant water will be sustained in equilibrio by the weight of the stagnant water, and therefore does not at all accelerate the motion of the descending water in the vessel. This case will also appear by experiments, measuring the times in which the water will run out.

Cor. 1. Hence if CA the depth of the water be produced to K, so that AK may be to CK in the duplicate ratio of the area of a hole made in any part of the bottom to the area of the circle AB, the velocity of the effluent water will be equal to the velocity which the water would acquire by fallino- from the height l^C.

Cor. 2. And the force with which the whole motion of the effluent water may be generated is equal to the weight of a cylindric column of water, whose base is the hole EF, and its altitude 2GI or 2CK. For the effluent water, in the time it becomes equal to this column, may acquire, by falling by its own weight from the height GI, a velocity equal to that with which it runs out.

CoR. 3. The weight of all the water in the vessel ABDC is to that part

336 THE MATHEMATICAL PRINCIPLES [BoOK II.

of the weight which is employed in forcing out the water as the sum of the circles AB and EF to twice the circle EF. For let 10 be a mean pro- portional between IH and IG, and the water running out at the hole EF will, 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 altitude 2IG, that is, to a cylinder whose base is the circle AB, and whose altitude is 210. For the circle EF is to the circle AB in the subduplicate ratio of the altitude IH to the altitude IG ; that is, in the simple ratio of the mean proportional 10 to the altitude IG. Moreover, in the time that a drop falling from I can describe the altitude IH, the water that runs out will have become equal to a cylinder whose base is the circle AB, and its alti- tude 2IH ; and in the time that a drop falling from I through H to G de- scribes HG, the difference of the altitudes, the effluent water, that is, the water contained within the solid ABNFEM, will be equal to the diiFerence of the cylinders, that is, to a cylinder whose base is AB, and its altitude 2H0. And therefore all the water contained in the vessel ABDC is to the whole falling water contained in the said solid ABNFEM as HG to 2H0, that is, as HO + OG to 2H0, or IH + 10 to 2IH. But the weight of aU the 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 of the weight that is employed in forcing out the water as IH + 10 to 2IH, and therefore as the sum of the circles EF and AB to twice the circle EF.

Cor. 4. And hence the weight of all the water in the vessel ABDC is to the other part of the weight which is sustained by the bottom of the vessel as the sum of the circles AB and EF to the difference of the same circles.

Cor. 5. And that part of the weight which the bottom of the vessel sus- tains is to the other part of the weight employed in forcing out the water as the difference of the circles AB and EF to twice the lesser circle EF, or as the area of the bottom to twice the hole.

CoR. 6. That part of the weight which presses upon the bottom is to the whole weight of the water perpendicularly incumbent thereon as the circle AB to the sum of the circles AB and EF, or as the circle AB to the excess of twice the circle AB above the area of the bottom. For that part of the weight which presses upon the bottom is to the weight of the whole water in the vessel as the difference of the circles AB and EF to the sum of the same circles (by Cor. 4) ; and the weight of the whole water in the vessel is to the weight of the whole water perpendicularly incumbent on the bottom as the circle AB to the difference of the circles AB and EF. Therefore, ex cequo perturbate, that part of the weight which presses upon the bottom is to the weight of the whole water perpendicularly incumbent

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thereon as the circle AB to the sum of the circles AB and EF, or the ex- cess of twice the circle AB above the bottom.

Cor. 7. If in the middle of the hole EF there be placed the little circle PQ, described about the centre G, and parallel to the horizon, the weight of water which that little circle sustains is greater than the weight of a third part of a cylinder of water whose base is that little circle and its height GH. For let ABNFEM be the cataract or column of falling water whose axis is GH, as above, and let all the wa- j^ x jj

ter, whose fluidity is not requisite for the ready \ ] 5

and quick descent of the water, be supposed to be congealed, as well round about the cataract, as above the little circle. And let PHQ, be the column of water congealed above the little cir- cle, whose vertex is H, and its altitude GH. And suppose this cataract to fall with its whole weio-ht downwards, and not in the least to lie against or to press PHQ, but to glide freely by it without any friction, unless, perhaps, just at*^ spgqp d the very vertex of the ice, where the cataract at the beginning of its fall may 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 towards the cataract also, and will therefore be greater than a cone whose base is that little circle PQ, and its altitude GH ; that is, greater than a third part of a cylinder described with the same base and altitude. Now that little circle sustains the weight of this column, that is, a weight greater than 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, sus- tains, seems to be less than the weight of two thirds of a cylinder of water whose base is that little circle, and its altitude HG. For, things standing as above supposed, imagine the half of a spheroid described whose base is that little circle, and its semi- axis or altitude HG. This figure will be equal to two thirds of that cylinder, and will comprehend within it the column of congealed water PHQ,, the weight of which is sustained by that little circle. For though the motion of the water tends directly down- wards, the external superficies of that column must yet meet the base PQ, in an angle somewhat acute, because the water in its fall is perpetually ac- celerated, and by reason of that acceleration become narrower. Therefore^ since that angle is less than a right one, this column in the lower parts thereof will lie within the hemi-spheroid. In the upper parts also it will be acute or pointed ; because to make it otherwise, the horizontal motion of the water must be at the vertex infinitely more swift than its motion to- wards the horizon. And the less this circle PQ is, the more acute will

22

338 THE MATHEMATICAL PRINCIPLES fBoOK 11.

the vertex of this column be ; and the circle being diminished in infinitum, the angle PHQ, will be diminished in infinitum^ and therefore the co- lumn will lie within the hemi-spheroid. Therefore that column is less than that hemi-spheroid^ or than two-third parts of the cylinder whose base is that little circle, and its altitude GH. Now the little circle sustains a force of water equal to the weight of this column, the weight of the ambient water being employed in causing its efflux out at the hole.

Cor. 9. The weight of water which the little circle PQ. sustains, when it is very small, is very nearly equal to the weight of a cylinder of water whose base is that little circle, and its altitude |GH ; for this weight is an arithmetical mean between the weights of the cone and the hemi-spheroid above mentioned. But if that little circle be not very small, but on the contrary increased till it be equal to the hole EF, it will sustain the weight of all the water lying perpendicularly above it, that is, the weight of a cylinder of water whose base is that little circle, and its altitude GH.

Cor. 10. And (as far as I can judge) the weight which this little circle sustains is always to the weight of a cylinder of water whose base is that little circle, and its altitude |GH, as EF^ to EF=^ — ^^^> or as the cir- cle EF to the excess of this circle above half the little circle PQ,, very nearly.

LEMMA lY,

Jf a cylinder move uniformly forioard in the direction of its lens^thj the resistance made thereto is not at all changed hy augmenting or di- minishing that length ; and is therefore the same with the resistance of a circle, described with the same diameter, and moving foncard with the same velocity in the direction of a right line perpendicidar to its plane.

For the sides are not at all opposed to the motion ; and a cylinder be- comes a circle when its length is diminished in infinitum.

PROPOSITION XXXVII. THEOREM XXIX.

If a cylinder move uninformly forward in a compressed, infinite, and non-elastic fluid, in the direction of its length, the resistance arising from the magnitude of its transverse section is to the force by which its whole motion may be destroyed or generated, in the time that it moves four times its length, as the density of the medium to the den- sity of the cylinder, nearly.

For let the vessel ABDC touch the surface of stagnant water with its hottom CD, and let the water run out of this vessel into the stagnant wa- ter through the cylindric canal EFTS perpendicular to the horizon ; and let the little circle PQ, be placed parallel to the horizon any where in the

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middle of tlie canal ; and produce CA to K, so that AK may be to CK in the duplicate of the ratio, which the excess of the orifice of the canal EF above the little circle PQ, bears to the cir- cle AB. Then it is manifest (by Case 5, Case 6, and Cor. 1, Prop. XXXVI) that the velocity of the water passing through the annular space between the little circle and the sides of the ves- sel will be the very same which the water would acquire by falling, and in its fall describing the altitude KC or IG.

And (by Cor. 10, Prop. XXXVI) if the breadth of the vessel be infinite, so that the lineola HI inay vanish, and the altitudes IG, HG become equal ; the force of the water that flows down and presses upon the circle will be to the weight of a cylinder whose base is that little circle, and the altitude |IG, as EF2 to EF2 — iPQ.% very nearly. For the force of the water flowing downward uniformly through the whole canal will be the same upon the little circle PQ. in whatsoever part of the canal it be placed.

Let now the orifices of the canal EF, ST be closed, and let the little circle ascend in the fluid compressed on every side, and by its ascent let it oblige the water that lies above it to descend through the annular space between the little circle and the sides of the canal. Then will the velocity of the ascending little circle be to the velocity of the descending water a^s the difference of the circles EF and PQ, is to the circle PQ,; and the ve- locity of the ascending little circle will be to the sum of the velocities, that is, to the relative velocity of the descending water with which it passes by the little circle in its ascent, as the difference of the circles EF and PQ to the circle EF, or as EF-'' — PQ^ to EF^. Let that relative velocity be equal to the velocity with v^hich it was shewn above that the water would pass through the annular space, if the circle were to remain unmoved, that is, to the velocity which the water would acquire by falling, and in its fall describing the altitude IG ; and the force of the water upon the ascending circle will be the same as before (by Cor. 5, of the Laws of Motion) ; that is, the resistance of the ascending little circle will be to the weight of a cylinder of water whose base is that little circle, and its altitude |-IG, as EF^ to EF^ ~ |PQ% nearly. But the velocity of the little circle will be to the velocity which the water acquires by falling, and in its fall de- scribing the altitude IG, as EF- — PQ^ to EF^.

Let the breadth of the canal be increased iji ivfinitum ; and the ratios between EF=^ — PQ^ and EF^ and between EF^ and EF^ — iPQ^, will become at last ratios of equality. And therefore the velocity of the little circle will now be the same which the water would acquire in falling, and in its fall describing the altitude IG ; and the resistance will become

340 THE MATHEMATICAL PRINCIPLES [BoOK II.

equal to the weight of a cylinder whose base is that little circle, and its altitude half the altitude IG, from which the cylinder must fall to acquire the velocity of the ascending circle ; and with this velocity the cylinder in the time of its fall will describe four times its length. But the resistance of the cylinder moving forward with this velocity in the direction of its length is the same with the resistance of the little circle (by Lem. IV), and is therefore nearly equal to the force by which its motion may be generated while 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 length, will be augmented or diminished in the same ratiof and therefore the force by which the mo- tion, so increased or diminished, may be destroyed or generated, will con- tinue the same ; because the time is increased or diminished in the same proportion ; and therefore that force remains still equal to the resistance of the cylinder, because (by Lem. IV) that resistance will also remain the same.

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 the same time, will be augmented or diminished in the same ratio. Therefore the resistance of any cylinder whatsoever will be to the force by which its whole motion may be generated or destroyed, in the time during which it moves four times its length, as the density of the medium to the density of the cylinder, nearly. Q.E.D.

A fluid must be compressed to become continued ; it must be continued and non-elastic, that all the pressure arising from its compression may be propagated in an instant ; and so, acting equally upon all parts of the body moved, may produce no change of the resistance. The pressure arising from the motion of the body is spent in generating a motion in the parts of the fluid, and this creates the resistance. But the pressure arising from the com.pression of the fluid, be it ever so forcible, if it be propagated in an instant, generates no motion in the parts of a continued fluid, produces no change at all of motion therein ; and therefore neither augments nor les- sens the resistance. This is certain, that the action of the fluid arising from the compression cannot be stronger on the hinder parts of the body moved than on its fore parts, and therefore cannot lessen the resistance de- scribed in this proposition. And if its propagation be infinitely swifter than the motion of the body pressed, it will not be stronger on the fore parts than on the hinder parts. But that action will be infinitely swifter, and propagated in an instant, if the fluid be continued and non- elastic.

Cor. 1. The resistances, made to cylinders going uniformly forward in the direction of their lengths through continued infinite mediums, are in a

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ratio compounded of the duplicate ratio of the velocities and the duplicate ratio 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 the cylinder go forward in the direction of its length through an included quiescent medium, its axis all the while coinciding with the axis of the canal, its resistance will be to the force by which its whole motion, in the time in which it describes four times its length, may be generated or destroyed, in a ratio com- pounded of the ratio of EF^ to EF^ — iPCl^ once, and the ratio of EF^ to EF^ _ PQ^ twice, and the ratio of the density of the medium to the density of the cylinder.

Cor, 3. The same thing supposed, and that a length L is to the quadruple of the length of the cylinder in a ratio compounded of the ratio EF2 — iPQ,2 tQ EF^ once, and the ratio of EF2 — PQ.^ to EF2 twice; the resistance of the cylinder will be to the force by which its whole motion, in the time during which it describes the length L, may be destroyed or generated, as the density of the medium to the density of the cylinder.

SCHOLIUM.

In this proposition we have investigated that resistance alone which arises from the magnitude of the transverse section of the cylinder, neg- lecting that part of the same which may arise from the obliquity of the motions. For as, in Case 1, of Prop. XXXYL, the obliquity of the mo- tions with which the parts of the water in the vessel converged on every side 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 of the water, pressed by the antecedent extremity of the cylinder, yield to the pressure, and diverge on all sides, retards their passage through the places that lie round that antecedent extremity, toward the hinder parts of the cylinder, and causes the fluid to be moved to a greater distance ; which in- creases the resistance, and that in the same ratio almost in which it dimin- ished the efflux of the water out of the vessel, that is, in the duplicate ratio of 25 to 21, nearly. And as, in Case 1, of that Proposition, we made the parts of the water pass through the hole EF perpendicularly and in the greatest pletity, by supposing all the water in the vessel lying round the cataract to be frozen, and that part of the water whose motion was oblique and useless to remain without motion, so in this Proposition, that the obliquity of the motions may be taken away, and the parts of the water may give the freest passage to the cylinder, by yielding to it with the most direct and quick motion possible, so that only so much resistance may re-

342 THE MATHEMATICAL PKINCIPLES [BoOK II.

main as arises from the magnitude of the transverse section, and which is incapable of diminution, unless by diminishing the diameter of the cylinder ; we must conceive those parts of the fluid whose motions are oblique and useless, and produce resistance, to be at rest among themselves at both ex- tremities of the cylinder, and there to cohere, and be joined to the cylinder.

Let ABCD be a rectauo-le, and let

G H AE and BE be two parabolic arcs, i 1

described with the axis AB, and ^ j^

with a latus rectum that is to the -p,.--' ^ ^"'-..^

space HG, which must be described ""■- I...--'''''

by the cylinder in falling, in order

to acquire the velocity with which it moves, as HG to |AB. Let CF and DF be two other parabolic arcs described with the axis CD, and a latus rectum quadruple of the former; and by the convolution of the figure about the axis EF let there be generated a solid, whose middle part ABDC is the cylinder we are here speaking of, and whose extreme parts ABE and CDF contain the parts of the fluid at rest among themselves, and concreted into two hard bodies, adhering to the cylinder at each end like a head and tail. Then if this solid EACFDB move in the direction of the length of its axis FE toward the parts beyond E, the resistance will be the same which we have here determined in this Proposition, nearly; that is, it will have the same ratio to the force with which the whole motion of the cyl- inder may be destroyed or generated, in the time that it is describing the length 4AC with that motion uniformly continued, as the density of the fluid has to the density of the cylinder, nearly. And (by Cor. 7, Prop. XXXVI) the resistance must be to this force in th'e ratio of 2 to 3, at the least.

LEMMA Y.