SECTION 7^a

The motion of fluids and the resistance encountered by projectiles

Proposition 32, Theorem 26
Let two similar systems of bodies consist of an equal number of particles, and let each of the particles in one system be similar and proportional to the corresponding particle in the other system, and let the particles be similarly situated with respect to one another in the two systems and have a given ratio of density to one another. And let them begin to move similarly with respect to one another in proportional times (the particles that are in the one system with respect to the particles in that system, and the particles in the other with respect to those in the other). Then, if the particles that are in the same system do not touch one another except in instants of reflection and do not attract or repel one another except by accelerative forces that are inversely as the diameters of corresponding particles and directly as the squares of the velocities, I say that the particles of the systems will continue to move similarly with respect to one another in proportional times.

I say that bodies which are similar and similarly situated move similarly with respect to one another in proportional times when their situations in relation to one another are always similar at the end of the times—for instance, if the particles of one system are compared with the corresponding particles of another. Hence the times in which similar and proportional parts of similar figures are described by corresponding particles will be proportional. Therefore, if there are two systems of this sort, the corresponding particles, because of the similarity of their motions at the beginning, will continue to move similarly until they meet one another. For if they are acted upon by no forces, they will, by the first law of motion, move forward uniformly in straight lines. If they act upon one another by some forces and if those forces are as the diameters of the corresponding particles inversely and the squares of the velocities directly, then, since the situations of the particles are similar and the forces proportional, the whole forces by which the corresponding particles are acted upon, compounded of the separate acting forces (by corol. 2 of the laws), will have similar directions, just as if they tended to centers similarly placed among the particles, and those whole forces will be to one another as the separate component forces, that is, as the diameters of the corresponding particles inversely and the squares of the velocities directly, and therefore they will cause corresponding particles to continue describing similar figures. This will be so (by book 1, prop. 4, corols. 1 and 8) provided that the centers are at rest. But if they move, since their situations with respect to the particles of the systems remain similar (because the transferences are similar), similar changes will be introduced in the figures which the particles describe. The motions of corresponding similar particles will therefore be similar until they first meet, and therefore the collisions will be similar and the reflections similar, and then (by what has already been shown) the motions of the particles with respect to one another will be similar until they encounter one another again, and so on indefinitely. Q.E.D.

COROLLARY 1. Hence, if any two bodies that are similar and similarly situated (in relation to the corresponding particles of the systems) begin to move similarly with respect to the particles in proportional times, and if their volumes and densities are to each other as the volumes and densities of the corresponding particles, the bodies will continue to move similarly in proportional times. For the case is the same for the larger parts of both systems as for the particles.

COROLLARY 2. And if all the similar and similarly situated parts of the systems are at rest with respect to one another, and if two of them, which are larger than the others and correspond to each other in the two systems, begin to move in any way with a similar motion along lines similarly situated, they will cause similar motions in the remaining parts of the systems and will continue to move similarly with respect to them in proportional times and thus will continue to describe spaces proportional to their own diameters.

Proposition 33, Theorem 27
If the same suppositions are made, I say that the larger parts of the systems are resisted in a ratio compounded of the squared ratio of their velocities and the squared ratio of the diameters and the simple ratio of the density of the parts of the systems.

For the resistance arises partly from the centripetal or centrifugal forces with which the particles of the systems act upon one another and partly from the collisions and reflections of the particles and the larger parts. Resistances of the first kind, moreover, are to one another as the whole motive forces from which they arise, that is, as the whole accelerative forces and the quantities of matter in corresponding parts, that is (by hypothesis), as the squares of the velocities directly and the distances of the corresponding particles inversely and the quantities of matter in the corresponding parts directly. Thus, since the distances of the particles of the one system are to the corresponding distances of the particles of the other as the diameter of a particle or part in the first system to the diameter of the corresponding particle or part in the other, and since the quantities of matter are as the densities of the parts and the cubes of the diameters, the resistances are to one another as the squares of the velocities, the squares of the diameters, and the densities of the parts of the systems. Q.E.D.

Resistances of the second kind are as the numbers and forces of corresponding reflections jointly. The number of reflections in any one case, moreover, is to the number in any other as the velocities of the corresponding parts directly and the spaces between their reflections inversely. And the forces of the reflections are as the velocities and volumes and densities of the corresponding parts jointly, that is, as the velocities, the cubes of the diameters, and the densities of the parts. And if all these ratios are compounded, the resistances of the corresponding parts are to one another as the squares of the velocities, the squares of the diameters, and the densities of the parts, jointly. Q.E.D.

COROLLARY 1. Therefore, if the systems are two elastic fluids such as air and if their parts are at rest with respect to one another, and if two bodies which are similar and are proportional (with regard to volume and density) to the parts of the fluids and are similarly situated with respect to those parts are projected in any way along lines similarly situated, and if the accelerative forces with which the particles of the fluids act upon one another are as the diameters of the projected bodies inversely and the squares of the velocities directly, then the bodies will cause similar motions in the fluids in proportional times and will describe spaces that are similar and are proportional to their diameters.

COROLLARY 2. Accordingly, in the same fluid a swift projectile encounters a resistance that is very nearly in the squared ratio of the velocity. For if the forces with which distant particles act upon one another were increased in the squared ratio of the velocity, the resistance would be exactly in the squared ratio of the velocity; and thus, in a medium whose parts act upon one another with no forces because they are far apart, the resistance is exactly in the squared ratio of the velocity. Let A, B, and C, therefore, be three mediums consisting of parts that are similar and equal and regularly distributed at equal distances. Let the parts of mediums A and B recede from one another with forces that are to one another as T and V, and let the parts of medium C be entirely without forces of this sort. Then, let four equal bodies D, E, F, and G move in these mediums, the first two bodies D and E in the first two mediums A and B respectively, and the other two bodies F and G in the third medium C; and let the velocity of body D be to the velocity of body E, and let the velocity of body F be to the velocity of body G, as the square root of the ratio of the forces T to the forces V [i.e., as √T to √V]; then the resistance of body D will be to the resistance of body E, and the resistance of body F to the resistance of body G, in the squared ratio of the velocities; and therefore the resistance of body D will be to the resistance of body F as the resistance of body E to the resistance of body G. Let bodies D and F have equal velocities, and also bodies E and G; then, if the velocities of bodies D and F are increased in any ratio and the forces of the particles of medium B are decreased in the same ratio squared, medium B will approach the form and condition of medium C as closely as is desired, and on that account the resistances of the equal and equally swift bodies E and G in these mediums will continually approach equality, in such a way that their difference finally comes out less than any given difference. Accordingly, since the resistances of bodies D and F are to each other as the resistances of bodies E and G, these also will similarly approach the ratio of equality. Therefore, the resistances of bodies D and F, when they move very swiftly, are very nearly equal, and therefore, since the resistance of body F is in the squared ratio of the velocity, the resistance of body D will be very nearly in the same ratio.

COROLLARY 3. The resistance of a body moving very swiftly in any elastic fluid is about the same as if the parts of the fluid lacked their centrifugal forces and did not recede from one another, provided that the elastic force of the fluid arises from the centrifugal forces of the particles and that the velocity is so great that the forces do not have enough time to act.

COROLLARY 4. Accordingly, since the resistances of similar and equally swift bodies, in a medium whose parts (being far apart) do not recede from one another, are as the squares of the diameters, the resistances of equally swift and very quickly moving bodies in an elastic fluid are also very nearly as the squares of the diameters.

COROLLARY 5. And since similar, equal, and equally swift bodies, in mediums which have the same density and whose particles do not recede from one another, impinge upon an equal quantity of matter in equal times (whether the particles are more and smaller or fewer and larger) and impress upon it an equal quantity of motion and in turn (by the third law of motion) undergo an equal reaction from it (that is, are equally resisted), it is manifest also that in elastic fluids of the same density, when the bodies move very swiftly, the resistances they encounter are very nearly equal, whether those fluids consist of coarser particles or are made of the most subtle particles of all. The resistance to projectiles moving very quickly is not much diminished as a result of the subtlety of the medium.

COROLLARY 6. These statements all hold for fluids whose elastic force originates in the centrifugal forces [i.e., forces of repulsion] of the particles. But if that force arises from some other source, such as the expansion of the particles in the manner of wool or the branches of trees, or from any other cause which makes the particles move less freely with respect to one another, then the resistance will be greater than in the preceding corollaries because the medium is less fluid.

Proposition 34, Theorem 28
In a rare medium consisting of particles that are equal and arranged freely at equal distances from one another, let a sphere and a cylinder—described with equal diameters—move with equal velocity along the direction of the axis of the cylinder; then the resistance of the sphere will be half the resistance of the cylinder.

For since the action of a medium on a body is (by corol. 5 of the laws) the same whether the body moves in a medium at rest or the particles of the medium impinge with the same velocity on the body at rest, let us consider the body to be at rest and see with what force it will be urged by the moving medium. Let ABKI, therefore, designate a spherical body described with center C and semidiameter C A, and let the particles of the medium strike the spherical body with a given velocity along straight lines parallel to AC; and let FB be such a straight line. On FB take LB equal to the semidiameter CB, and draw BD touching the sphere in B. To KC and BD drop the perpendiculars BE and LD; then the force with which a particle of the medium, obliquely incident along the straight line FB, strikes the sphere at B will be to the force with which the same particle would strike the cylinder ONGQ (described with axis ACI about the sphere) perpendicularly at b as LD to LB or BE to BC. Again, the efficacy of this force to move the sphere along the direction FB (or AC) of its incidence is to its efficacy to move the sphere along the direction of its determination—that is, along the direction of the straight line BC in which it urges the sphere directly [a direction through the center of the sphere]—as BE to BC. And, compounding the ratios, if a particle strikes the sphere obliquely along the straight line FB, its efficacy to move the sphere along the direction of its incidence is to the efficacy of the same particle to move the cylinder in the same direction, when striking the cylinder perpendicularly along the same straight line, as BE² to BC². Therefore, if in bE, which is perpendicular to the circular base NAO of the cylinder and equal to the radius AC, bH is taken equal to , then bH will be to bE as the effect of a particle upon the sphere to the effect of the particle upon the cylinder. And therefore the solid that is composed of all the straight lines bH will be to the solid that is composed of all the straight lines bE as the effect of all the particles upon the sphere to the effect of all the particles upon the cylinder. But the first solid is a paraboloid described with vertex C, axis CA, and latus rectum CA, and the second solid is a cylinder circumscribed around the paraboloid; and it is known that a paraboloid is half of the circumscribed cylinder. Therefore the whole force of the medium upon the sphere is half of its whole force upon the cylinder. And therefore, if the particles of the medium were at rest and the cylinder and the sphere were moving with equal velocity, the resistance of the sphere would be half the resistance of the cylinder. Q.E.D.

Scholium
By the same method other figures can be compared with one another with respect to resistance, and those that are more suitable for continuing their motions in resisting mediums can be found. For example, let it be required to construct a frustum CBGF of a cone with the circular base CEBH (which is described with center O and radius OC) and with the height OD, which is resisted less than any other frustum constructed with the same base and height and moving forward along the direction of the axis toward D; bisect the height OD in Q, and produce OQ to S so that QS is equal to QC, and S will be the vertex of the cone whose frustum is required.

Note in passing that since the angle CSB is always acute, it follows that if the solid ADBE is generated by a revolution of the elliptical or oval figure ADBE about the axis AB, and if the generating figure is touched by the three straight lines FG, GH, and HI in points F, B, and I, in such a way that GH is perpendicular to the axis in the point of contact B, and FG and HI meet the said line GH at the angles FGB and BHI of 135 degrees, then the solid that is generated by the revolution of the figure ADFGHIE about the same axis AB is less resisted than the former solid, provided that each of the two moves forward along the direction of its axis AB, and the end B of each one is in front. Indeed, I think that this proposition will be of some use for the construction of ships.

But suppose the figure DNFG to be a curve of such a sort that if the perpendicular NM is dropped from any point N of that curve to the axis AB, and if from the given point G the straight line GR is drawn, which is parallel to a straight line touching the figure in N and cuts the axis (produced) in R, then MN would be to GR as GR³ to 4BR × GB². Then, in this case, the solid that is described by a revolution of this figure about the axis AB will, in moving in the aforesaid rare medium from A toward B, be resisted less than any other solid of revolution described with the same length and width.

Proposition 35^a, Problem 7
If a rare medium consists of minimally small equal particles that are at rest and arranged freely at equal distances from one another, it is required to find the resistance encountered by a sphere moving forward uniformly in this medium.

CASE 1. Let a cylinder described with the same diameter and height as before move forward with the same velocity along the length of its own axis in the same medium. And let us suppose that the particles of the medium upon which the sphere or cylinder impinges rebound with the greatest possible force of reflection. Then the resistance of the sphere (by prop. 34) is half the resistance of the cylinder, and the sphere is to the cylinder as 2 to 3, and the cylinder in impinging perpendicularly upon the particles and reflecting them as greatly as possible communicates twice its own velocity to them. Therefore, the cylinder, in the time in which it describes half the length of its axis by moving uniformly forward, will communicate to the particles a motion which is to the whole motion of the cylinder as the density of the medium is to the density of the cylinder; and the sphere, in the time in which it describes the whole length of its diameter by moving uniformly forward, will communicate the same motion to the particles, and in the time in which it describes ⅔ of its diameter it will communicate to the particles a motion which is to the whole motion of the sphere as the density of the medium to the density of the sphere. And therefore the sphere encounters a resistance that is to the force by which its whole motion could be either destroyed or generated, in the time in which it describes ⅔ of its diameter by moving uniformly forward, as the density of the medium is to the density of the sphere.

CASE 2. Let us suppose that the particles of the medium impinging upon the sphere or cylinder are not reflected; then the cylinder, in impinging perpendicularly upon the particles, will communicate its whole velocity to them and thus encounters half the resistance which it met in the former case, and the resistance encountered by the sphere will also be half of what it was before.

CASE 3. Let us suppose that the particles of the medium rebound from the sphere with a force of reflection that is neither the greatest nor nil but some intermediate force; then the resistance encountered by the sphere will also be intermediate between the resistance in case 1 and the resistance in case 2. Q.E.I.

COROLLARY 1. Hence, if the sphere and the particles are infinitely hard without any elastic force and therefore also without any force of reflection, the resistance encountered by the sphere will be to the force by which its whole motion could be either destroyed or generated, in the time in which the sphere describes ⁴/3 of its diameter, as the density of the medium is to the density of the sphere.

COROLLARY 2. The resistance encountered by the sphere, other things being equal, is in the squared ratio of the velocity.

COROLLARY 3. The resistance encountered by the sphere, other things being equal, is in the squared ratio of the diameter.

COROLLARY 4. The resistance encountered by the sphere, other things being equal, is as the density of the medium.

COROLLARY 5. The resistance encountered by the sphere is in a ratio that is compounded of the squared ratio of the velocity and the squared ratio of the diameter, and the simple ratio of the density of the medium.

COROLLARY 6. And the motion of the sphere with the resistance it encounters can be represented as follows. Let AB be the time in which the sphere can lose its whole motion when the resistance is continued uniformly. Erect AD and BC perpendicular to AB. And let BC be the whole motion, and through point C with asymptotes AD and AB describe the hyperbola CF. Produce AB to any point E. Erect the perpendicular EF meeting the hyperbola in F. Complete the parallelogram CBEG, and draw AF meeting BC in H. Then, if the sphere, in any time BE, when its first motion BC is continued uniformly, in a nonresisting medium, describes the space CBEG represented by the area of the parallelogram, it will in a resisting medium describe the space CBEF represented by the area of the hyperbola, and its motion at the end of that time will be represented by the ordinate EF of the hyperbola, with loss of part FG of its motion. And the resistance at the end of the same time will be represented by the length BH, with loss of part CH of the resistance. All of this is evident by book 2, prop. 5, corols. 1 and 3.

COROLLARY 7. Hence, if in time T, when the resistance R is continued uniformly, the sphere loses its whole motion M, then in time t in a resisting medium, when the resistance R decreases in the squared ratio of the velocity, the sphere will lose part of its motion M without loss of part ; and the sphere will describe a space that is to the space described by the uniform motion M, in the same time t, as the logarithm of the number multiplied by the number 2.302585092994 is to the number , because the hyperbolic area BCFE is in this proportion to the rectangle BCGE.

Scholium
In this proportion I have set forth the resistance and retardation encountered by spherical projectiles in noncontinuous mediums, and I have shown that this resistance is to the force by which the whole motion of a sphere could be either destroyed or generated in the time in which the sphere describes ⅔ of its diameter, with a velocity continued uniformly, as the density of the medium is to the density of the sphere, provided that the sphere and the particles of the medium are highly elastic and possess the greatest force of reflecting, and I have shown that this force is half as great when the sphere and the particles of the medium are infinitely hard and devoid of all force of reflecting. Moreover, in continuous mediums such as water, hot oil, and quicksilver, in which the sphere does not impinge directly upon all the particles of the fluid which generate resistance but presses only the nearest particles, and these press others and these still others, the resistance is half as great as in the second case. In extremely fluid mediums of this sort the sphere encounters a resistance that is to the force by which its whole motion could be either destroyed or generated, in the time in which it describes ⁸/3 of its diameter with the motion continued uniformly, as the density of the medium is to the density of the sphere. We will try to show this in what follows.

Proposition 36, Problem 8
To determine the motion of water flowing out of a cylindrical vessel through a hole in the bottom.

Let ACDB be the cylindrical vessel, AB its upper opening, CD its bottom parallel to the horizon, EF a circular hole in the middle of the bottom, G the center of the hole, and GH the cylinder’s axis perpendicular to the horizon. And imagine that a cylinder of ice APQB is of the same width as the interior of the vessel, has the same axis, and descends continually with a uniform motion. Imagine also that its parts liquefy as soon as they touch the surface AB, that when they have turned into water they flow down into the vessel as a result of their gravity, and that in falling these parts form a cataract or column of water ABNFEM and pass through the hole EF and fill it exactly. And let the uniform velocity of the descending ice, as well as that of the contiguous water in the circle AB, be the velocity which the water can acquire in falling and describing by its fall the space IH, and let IH and HG lie in a straight line, and through point I draw the straight line KL parallel to the horizon and meeting the sides of the ice in K and L. Then the velocity of the water flowing out through the hole EF will be that which the water can acquire in falling from I and describing by its fall the space IG. And thus, by Galileo’s theorems, IG will be to IH as the square of the ratio of the velocity of the water flowing out through the hole to the velocity of the water in the circle AB, that is, as the square of the ratio of the circle AB to the circle EF, for these circles are inversely as the velocities of the water passing through them in the same time and with an equal quantity, filling them both exactly. Here it is the velocity of the water toward the horizon that is of concern. And the motion parallel to the horizon by which the parts of the falling water approach one another is not considered here, since it does not arise from gravity or change the motion perpendicular to the horizon that does arise from gravity. Indeed, we are supposing that the parts of the water cohere somewhat and that by their cohesion they approach one another with motions parallel to the horizon as they fall, so that they form only one single cataract and are not divided into several cataracts, but here we are not considering the motion parallel to the horizon arising from that cohesion.

CASE 1. Now suppose that the interior of the vessel around the falling water ABNFEM is filled with ice, so that the water passes through the ice as if through a funnel. Then, if the water does not quite touch the ice, or (what comes to the same thing) if it touches it and, because of the great smoothness of the ice, slides through it with the greatest possible freedom and without any resistance, the water will flow down through the hole EF with the same velocity as before, and the whole weight of the column of water ABNFEM will be used in generating its downflow as before, and the bottom of the vessel will sustain the weight of the ice surrounding the column.

Now let the ice liquefy in the vessel; then the flow of the water will remain the same as before with respect to velocity. It will not be less, since the melted ice will endeavor to descend; and not greater, since the melted ice cannot descend without impeding an equal descent of the original water. The same force ought to generate the same velocity in the flowing water [i.e., since the force is the same, the velocity that it generates will also be the same].

But the hole in the bottom of the vessel, because of the oblique motions of the particles of the flowing water, ought to be a little larger than before. For now the particles of water do not all pass through the hole perpendicularly but, flowing together from all the sides of the vessel and converging into the hole, pass through with oblique motions and, turning their course downward, unite into a stream of water gushing out which is narrower a little below the hole than in the hole itself, its diameter being to the diameter of the hole as 5 to 6, or 5½ to 6½ very nearly, provided that I measured the diameters correctly. At any rate, I obtained a very thin flat plate perforated in the middle, the diameter of the circular hole being ⅝ inch. And so that the stream of water gushing out might not be accelerated in falling and made narrower by the acceleration, I fastened this plate not to the bottom but to the side of the vessel in such a way that the stream went out along a line parallel to the horizon. Then, when the vessel was full of water, I opened the hole so that the water might flow out, and the diameter of the stream, measured as accurately as possible at a distance of about ½ inch from the hole, came out ²¹/40 inch. The diameter of this circular hole, therefore, was to the diameter of the stream very nearly as 25 to 21. Therefore the water in passing through the hole converges from all directions, and after flowing out of the vessel the stream is made narrower by converging and is accelerated by narrowing until it has reached a distance of ½ inch from the hole and at that distance becomes narrower and swifter than it is in the hole itself in the ratio of 25 × 25 to 21 × 21 or very nearly 17 to 12, that is, roughly as the square root of the ratio of 2 to 1. And experiments prove that the quantity of water that flows out in a given time through a circular hole in the bottom of a vessel is the quantity that ought to flow out in the same time, with the velocity mentioned above, not through that hole but through a circular hole whose diameter is to the diameter of that hole as 21 to 25. And thus the flowing water has the downward velocity in the hole itself that a heavy body can acquire very nearly in falling and describing by its fall a space equal to half the height of the water standing in the vessel. But after the water has gone out of the vessel, it is accelerated by converging until it has reached a distance from the hole almost equal to the diameter of the hole and has acquired a velocity that is greater approximately as the square root of the ratio of 2 to 1, which is, as a matter of fact, very nearly the velocity that a heavy body can acquire in falling and describing by its fall a space equal to the whole height of the water standing in the vessel.

In what follows, therefore, let the diameter of the stream be designated by that smaller hole which we have called EF. And suppose that another higher plane VW is drawn parallel to the plane of the hole EF at a distance about equal to the diameter of the hole and pierced by a larger hole ST, and through this let a stream fall that exactly fills the lower hole EF and thus has a diameter which is to the diameter of this lower hole as about 25 to 21. For thus the stream will pass perpendicularly through the lower hole, and the quantity of the water flowing out, depending on the size of this hole, will be very nearly that which the solution of the problem demands. Now, the space which is enclosed by the two planes and the falling stream can be considered to be the bottom of the vessel. But so that the solution of the problem may be simpler and more mathematical, it is preferable to use only the lower plane for the bottom of the vessel and to imagine that the water which flowed down through the ice as if through a funnel and came out of the vessel through the hole EF in the lower plane keeps its motion continually and that the ice keeps its state of rest. In what follows, therefore, let ST be the diameter of a circular hole described with center Z, through which a cataract flows out of the vessel when all the water in the vessel is fluid. And let EF be the diameter of the hole which the cataract fills exactly when falling through it, whether the water comes out of the vessel through the upper hole ST or falls through the middle of the ice in the vessel as if through a funnel. And let the diameter of the upper hole ST be to the diameter of the lower hole EF as about 25 to 21, and let the perpendicular distance between the planes of the holes be equal to the diameter of the smaller hole EF. Then the downward velocity of the water coming out of the vessel through the hole ST will in the hole itself be that which a body can acquire in falling from half of the height IZ; and the velocity of both falling cataracts will, in the hole EF, be that which a body will acquire in falling from the whole height IG.

CASE 2. If the hole EF is not in the middle of the bottom of the vessel, but the bottom is perforated elsewhere, the water will flow out with the same velocity as before, provided that the size of the hole is the same. For a heavy body does descend to the same depth in a greater time along an oblique line than along a perpendicular line, but in descending it acquires the same velocity in either case, as Galileo proved.

CASE 3. The velocity of the water flowing out through a hole in the side of the vessel is the same. For if the hole is small, so that the distance between the surfaces AB and KL vanishes, so far as the senses can tell, and the stream of water gushing out horizontally forms a parabolic figure, it will be found from the latus rectum of this parabola that the velocity of the water flowing out is that which a body could have acquired by falling from the height HG or IG of the water standing in the vessel. Indeed, by making an experiment I found that when the height of the standing water above the hole was 20 inches and the height of the hole above a plane parallel to the horizon was also 20 inches, the stream of water gushing forth would fall upon the plane at a distance of about 37 inches, taken from a perpendicular that was dropped to the plane from the hole. For in the absence of resistance the stream would have had to fall upon the plane at a distance of 40 inches, the latus rectum of the parabolic stream being 80 inches.

CASE 4. Further, if the water flowing out has an upward motion, it comes out with the same velocity. For a small stream of water gushing out ascends with a perpendicular motion to the height GH or GI of the water standing in the vessel, except insofar as its ascent is somewhat impeded by the resistance of the air; and accordingly it flows out with the velocity that it could have acquired in falling from that height. Any one particle of the standing water (by book 2, prop. 19) is pressed equally from all sides and, yielding to the pressure, goes with equal force in every direction, whether it descends through a hole in the bottom of the vessel or flows out horizontally through a hole in its side or comes out into a channel and ascends from there through a small hole in the upper part of the channel. And that the velocity with which the water flows out is that which we have designated in this proposition is not only found by reason but is also manifest from the well-known experiments already described.

CASE 5. The velocity of the water flowing out is the same whether the hole is circular or square or triangular or of any other shape equal in area to the circular one. For the velocity of the water flowing out does not depend on the shape of the hole but on the height of the water in relation to the plane KL.

CASE 6. If the lower part of the vessel ABDC is immersed in standing water, and the height of the standing water above the bottom of the vessel is GR, the velocity with which the water in the vessel will flow out through the hole EF into the standing water will be that which the water can acquire in falling and describing by its fall the space IR. For the weight of all the water in the vessel that is lower than the surface of the standing water will be sustained in equilibrium by the weight of the standing water and thus will not at all accelerate the motion of the descending water in the vessel. This case can also be shown by experiments, by measuring the times in which the water flows out.

COROLLARY 1. Hence, if the height CA of the water is produced to K, so that AK is to CK in the squared 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 water flowing out will be equal to the velocity that the water can acquire in falling and describing by its fall the space KC.

COROLLARY 2. And the force by which the whole motion of the water gushing out can be generated is equal to the weight of a cylindrical column of water whose base is the hole EF and whose height is 2GI or 2CK. For the gushing water, in the time in which it equals this column, can acquire in falling (by its weight) from the height GI the very velocity with which it gushes out.

COROLLARY 3. The weight of all the water in the vessel ABDC is to the part of the weight that is used in making the water flow down as the sum of the circles AB and EF to twice the circle EF. For let IO be a mean proportional between IH and IG; then the water coming out through the hole EF, in the time in which a drop could describe a space equal to the height IG in falling from I, will be equal to a cylinder whose base is the circle EF and whose height is 2IG, that is, to a cylinder whose base is the circle AB and whose height is 2IO, for the circle EF is to the circle AB as the square root of the ratio of the height IH to the height IG, that is, in the simple ratio of the mean proportional IO to the height IG, and in the time in which a drop can describe a space equal to the height IH in falling from I, the water coming out will be equal to a cylinder whose base is the circle AB and whose height is 2IH, and in the time in which a drop describes a space equal to the difference HG between the heights in falling from I through H to G, the water coming out—that is, all the water in the solid ABNFEM—will be equal to the difference between the cylinders, that is, equal to a cylinder whose base is AB and whose height is 2HO. And therefore all the water in the vessel ABDC is to all the water falling in the solid ABNFEM as HG to 2HO, that is, as HO + OG to 2HO, or IH + IO to 2IH. But the weight of all the water in the solid ABNFEM is used in making the water flow down, and accordingly the weight of all the water in the vessel is to the part of the weight that is used in making the water flow down as IH + IO to 2IH and thus as the sum of the circles EF and AB to twice the circle EF.

COROLLARY 4. And hence the weight of all the water in the vessel ABDC is to the part of the weight sustained by the bottom of the vessel as the sum of the circles AB and EF is to the difference between these circles.

COROLLARY 5. And the part of the weight sustained by the bottom of the vessel is to the part of the weight used in making the water flow down as the difference between the circles AB and EF is to twice the smaller circle EF, or as the area of the bottom to twice the hole.

COROLLARY 6. And the part of the weight which alone presses upon the bottom is to the weight of all the water resting perpendicularly on the bottom as the circle AB is to the sum of the circles AB and EF, or as the circle AB is to the amount by which twice the circle AB exceeds the bottom. For the part of the weight which alone presses upon the bottom is to the weight of all the water in the vessel as the difference between the circles AB and EF is to the sum of these circles, by corol. 4; and the weight of all the water in the vessel is to the weight of all the water resting perpendicularly on the bottom as the circle AB is to the difference between the circles AB and EF. Therefore, from the equality of the ratios in inordinate proportion [or ex aequo perturbate], the part of the weight which alone presses upon the bottom is to the weight of all the water resting perpendicularly on the bottom as the circle AB is to the sum of the circles AB and EF, or to the amount by which twice the circle AB exceeds the bottom.

COROLLARY 7. If in the middle of the hole EF there is placed a little circle PQ described with center G and parallel to the horizon, the weight of the water which that little circle sustains is greater than the weight of ⅓ of a cylinder of water whose base is that little circle and whose height is GH. For let ABNFEM be a cataract or column of falling water, with axis GH as above, and suppose that there has been a freezing of all the water in the vessel (around the cataract as well as above the little circle) whose fluidity is not required for the very ready and very swift descent of the water. And let PHQ be the frozen column of water above the little circle, having vertex H and height GH. And imagine that this cataract falls with its whole weight and does not rest or press on PHQ but slides past freely and without friction, except perhaps at the very vertex of the ice, where at the very beginning of falling the cataract begins to be concave. And just as the frozen water (AMEC and BNFD) which is around the cataract is convex on the inner surface (AME and BNF) toward the falling cataract, so also this column PHQ will be convex toward the cataract, and therefore will be greater than a cone whose base is the little circle PQ and whose height is GH, that is, greater than ⅓ of a cylinder described with the same base and height. And the little circle sustains the weight of this column, that is, a weight that is greater than the weight of the cone or of ⅓ of the cylinder.

COROLLARY 8. The weight of the water sustained by the little circle PQ, when it is extremely small, appears to be less than the weight of ⅔ of a cylinder of water whose base is that little circle and whose height is HG. Keeping the same suppositions, imagine that half a spheroid is described, whose base is the little circle and whose semiaxis or height is HG. Then this figure will be equal to ⅔ of that cylinder and will comprehend the column of frozen water PHQ whose weight that little circle sustains. In order that the motion of the water may be straight down, the outer surface of this column must meet the base PQ in a somewhat acute angle, because the water in falling is continually accelerated and the acceleration makes the column become narrower; and since that angle is less than a right angle, the lower parts of this column will lie within the half-spheroid. But higher up, the column will be acute or pointed, for otherwise the horizontal motion of the water at the vertex of the spheroid would be infinitely swifter than its motion toward the horizon. And the smaller the little circle PQ, the more acute the vertex of the column; and if the little circle is diminished indefinitely, the angle PHQ will be diminished indefinitely, and therefore the column will lie within the half-spheroid. That column is therefore less than the half-spheroid, or less than ⅔ of a cylinder whose base is that little circle and whose height is GH. Moreover, the little circle sustains the water’s force equal to the weight of this column, since the weight of the surrounding water is used in making it flow down.

COROLLARY 9. The weight of the water sustained by the little circle PQ, when it is extremely small, is very nearly equal to the weight of a cylinder of water whose base is that little circle and whose height is ½GH. For this weight is an arithmetical mean between the weights of the cone and the said half-spheroid. If, however, the little circle is not extremely small but is increased until it equals the hole EF, it will sustain the weight of all the water resting perpendicularly on it, that is, the weight of a cylinder of water whose base is that little circle and whose height is GH.

COROLLARY 10. And (as far as I can tell) the weight that the little circle sustains always has the proportion to the weight of a cylinder of water whose base is that little circle and whose height is ½GH that EF² has to EF² − ½PQ², or that the circle EF has to the excess of this circle over half of the little circle PQ, very nearly.

Lemma 4
The resistance of a cylinder moving uniformly forward in the direction of its length is not changed by an increase or decrease in length and thus is the same as the resistance of a circle described with the same diameter and moving forward with the same velocity along a straight line perpendicular to its plane.

For the sides of a cylinder offer no opposition to its motion, and a cylinder is turned into a circle if its length is decreased indefinitely.

Proposition 37, Theorem 29
If a cylinder moves uniformly forward in a compressed, infinite, and nonelastic fluid in the direction of its own length, its resistance arising from the magnitude of its transverse section is to the force by which its whole motion can be either destroyed or generated, while it is describing four times its length, very nearly as the density of the medium is to the density of the cylinder.

For if the bottom CD of the vessel ABDC touches the surface of stagnant water, and if water flows out of this vessel into the stagnant water through the cylindrical channel EFTS perpendicular to the horizon, and if the little circle PQ is placed parallel to the horizon anywhere in the middle of the channel, and if CA is produced to K so that CK is to AK in the squared ratio of the circle AB to the amount by which the opening of the channel EF exceeds the little circle PQ, then it is obvious (by prop. 36, case 5, case 6, and corol. 1) that the velocity of the water passing through the annular space between the little circle and the sides of the vessel will be that which the water can acquire in falling and describing by its fall a space equal to the height KC or IG.

And (by prop. 36, corol. 10) if the width of the vessel is infinite, so that the line-element HI vanishes and the heights IG and HG are equal, then the force of the water flowing down into the little circle will be to the weight of a cylinder whose base is that little circle, and whose height is ½IG, very nearly as EF² to EF² − ½PQ². For the force of the water flowing down through the whole channel with uniform motion will be the same upon the little circle PQ in whatever part of the channel it is placed.

Now let the openings EF and ST of the channel be closed, and let the little circle ascend in the fluid compressed on all sides, and by its ascent let it make the upper water descend through the annular space between the little circle and the sides of the channel; then the velocity of the ascending little circle will be to the velocity of the descending water as the difference between 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 (that is, to the relative velocity of the descending water, with which it flows past the ascending little circle) as the difference between the circles EF and PQ is to the circle EF, or as EF² − PQ² to EF². Let that relative velocity be equal to the velocity with which (as shown above) the water passes through the same annular space while the little circle remains unmoved, that is, to the velocity that the water can acquire in falling and describing by its fall a space equal to the height IG; then the force of the water upon the ascending little circle will be the same as before (by corol. 5 of the laws), 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 whose height is ½IG, very nearly as EF² to EF² − ½PQ². And the velocity of the little circle will be to the velocity that the water acquires in falling, and describing by its fall a space equal to the height IG, as EF² − PQ² to EF².

Let the breadth of the channel be increased indefinitely; then those ratios between EF² − PQ² and EF² and between EF² and EF² − ½PQ² will ultimately approach ratios of equality. And therefore the velocity of the little circle will now be that which the water can acquire in falling and describing by its fall a space equal to the height IG, and its resistance will come out equal to the weight of a cylinder whose base is that little circle and whose height is half of the height IG from which the cylinder must fall in order to acquire the velocity of the ascending little circle, and with this velocity the cylinder will, in the time of falling, describe four times its own length. And the resistance of the cylinder, moving forward with this velocity in the direction of its length, is the same as the resistance of the little circle (by lem. 4) and thus is very nearly equal to the force by which its motion can be generated while it is describing four times its length.

If the length of the cylinder is increased or decreased, its motion, and also the time in which it describes four times its length, will be increased or decreased in the same ratio; and thus that force by which the increased or decreased motion, in a time equally increased or decreased, could be generated or destroyed will not be changed and accordingly is under these circumstances still equal to the resistance of the cylinder; for this also remains unchanged, by lem. 4.

If the density of the cylinder is increased or decreased, its motion, and also the force by which the motion can be generated or destroyed in the same time, will be increased or decreased in the same ratio. The resistance, therefore, of any cylinder to the force by which its whole motion could be either generated or destroyed, while it is describing four times its length, will be very nearly as the density of the medium to the density of the cylinder. Q.E.D.

A fluid must be compressed in order to be continuous, and it must be continuous and nonelastic in order that every pressure arising from its compression may be propagated instantaneously and, by acting equally upon all parts of a moving body, not change the resistance. The pressure arising from the body’s motion is of course used in generating the motion of the parts of the fluid and creates resistance. But the pressure arising from the compression of the fluid, however strong it may be, if it is propagated instantaneously, generates no motion in the parts of a continuous fluid, introduces no change of motion at all, and thus neither increases nor decreases the resistance. Certainly the action of a fluid that arises from its compression cannot be stronger upon the back of a moving body than upon the front and thus cannot decrease the resistance described in this proposition; and the action will not be stronger upon the front than upon the back provided that its propagation is infinitely swifter than the motion of the body pressed. And the action will be infinitely swifter and will be propagated instantaneously provided that the fluid is continuous and nonelastic.

COROLLARY 1. The resistances to cylinders that move uniformly forward in the direction of their lengths in infinite and continuous mediums are in a ratio compounded of the squared ratio of the velocities and the squared ratio of the diameters and the ratio of the density of the mediums.

COROLLARY 2. If the breadth of the channel is not increased indefinitely, but the cylinder moves forward in the direction of its own length in an enclosed medium at rest, and meanwhile its axis coincides with the axis of the channel, then the resistance to the cylinder will be to the force by which its whole motion could be either generated or destroyed, in the time in which it describes four times its length, in a ratio compounded of the simple ratio of EF² to EF² − ½PQ² and the squared ratio of EF² to EF² − PQ² and the ratio of the density of the medium to the density of the cylinder.

COROLLARY 3. With the same suppositions, let the length L be to four times the length of the cylinder in a ratio compounded of the simple ratio of EF² − ½PQ² to EF² and the squared ratio of EF² − PQ² to EF²; then the resistance of the cylinder will be to the force by which its whole motion could be either destroyed or generated, while it is describing the length L, as the density of the medium to the density of the cylinder.

Scholium
In this proposition we have investigated the resistance arising solely from the magnitude of the transverse section of a cylinder, without considering the part of the resistance that can arise from the obliquity of the motions. In prop. 36, case 1, the flow of the water through the hole EF was impeded by the obliquity of the motions with which the parts of the water in the vessel converged from all sides into the hole. Similarly, in this proposition, the obliquity of the motions with which the parts of the water pressed by the front end of the cylinder yield to the pressure and diverge on all sides has these effects: it retards the passage of those motions through the places around that front end toward the back of the cylinder, it makes the fluid move to a greater distance, and it increases the resistance in nearly the ratio with which it decreases the flow of the water from the vessel, that is, in the squared ratio of 25 to 21, roughly.

In case 1 of prop. 36 we made the parts of the water pass through the hole EF perpendicularly and in the greatest abundance by supposing that all the water in the vessel that had been frozen around the cataract, and whose motion was oblique and useless, remained without motion. Similarly, in this proposition, in order that the obliquity of the motions may be annulled, and the parts of the water, by yielding with the most direct and rapid motion, may provide the easiest passage to the cylinder, and in order that only the resistance may remain that arises from the magnitude of the transverse section and that cannot be decreased except by decreasing the diameter of the cylinder, it must be understood that the parts of the fluid whose motions are oblique and useless and create resistance are at rest with respect to one another at both ends of the cylinder and cohere and are joined to the cylinder. Let ABDC be a rectangle, and let AE and BE be two parabolic arcs described with axis AB and with a latus rectum that is to the space HG, which is to be described by the falling cylinder while it is acquiring its velocity, as HG to ½AB. Additionally, let CF and DF be two other parabolic arcs, described with axis CD and a latus rectum that is four times the former latus rectum; and by the revolution of the figure about its axis EF, let a solid be generated whose middle ABDC is the cylinder with which we are dealing, and whose extremities ABE and CDF contain the parts of the fluid which are at rest with respect to one another and solidified into two rigid bodies that adhere to the cylinder at the ends as head and tail. Then the resistance to the solid EACFDB moving forward in the direction of its axis FE from F toward E will be very nearly that which we have described in this proposition. That is, the density of the fluid is to the density of the cylinder very nearly in the ratio of this resistance to the force by which the whole motion of the cylinder could be either destroyed or generated, while the length 4AC is being described with that motion continued uniformly. And with this force the resistance cannot be less than in the ratio of 2 to 3, by prop. 36, corol. 7.

Lemma 5
If a cylinder, a sphere, and a spheroid, whose widths are equal, are placed successively in the middle of a cylindrical channel in such a way that their axes coincide with the axis of the channel, these bodies will equally impede the flow of water through the channel.

For the spaces through which the water passes between the channel and the cylinder, sphere, and spheroid are equal; and water passes equally through equal spaces.

This is so on the hypothesis that all the water is frozen which is above the cylinder, sphere, or spheroid, and whose fluidity is not required for the very swift passage of the water, as I have explained in prop. 36, corol. 7.

Lemma 6
With the same suppositions, these bodies are equally urged by the water flowing through the channel.

This is evident by lem. 5 and the third law of motion. Of course, the water and the bodies act equally upon one another.

Lemma 7
If the water is at rest in the channel, and these bodies go through the channel with equal velocity in opposite directions, the resistances will be equal to one another.

This is clear from lem. 6; for the relative motions remain the same with respect to one another.

Scholium
It is the same for all convex round bodies whose axes coincide with the axis of the channel. Some difference can arise from a greater or lesser friction; but in these lemmas we are supposing that the bodies are very smooth, that the tenacity and friction of the medium are nil, and that the parts of the fluid which by their oblique and superfluous motions can perturb, impede, and retard the flow of the water through the channel are at rest with respect to one another as if icebound and adhere to the front and back of the bodies, as I have explained in the scholium to prop. 37. For what follows deals with the least possible resistance of round bodies described with the greatest given transverse sections.

Bodies moving straight ahead in fluids make the fluid ascend in front of them and subside in back of them, especially if they are blunt in shape; and hence they encounter a slightly greater resistance than if they had pointed heads and tails. And bodies moving in elastic fluids, if they are blunt in front and in back, condense the fluid a little more at the front and make it a little less dense at the back; and hence they encounter a slightly greater resistance than if they had pointed heads and tails. But in these lemmas and propositions we are not dealing with elastic fluids but with nonelastic fluids, not with bodies floating on the surface of the fluid but with bodies deeply immersed. And once the resistance of bodies in nonelastic fluids is known, this resistance will have to be increased somewhat for elastic fluids such as air as well as for the surfaces of stagnant fluids such as seas and swamps.

Proposition 38, Theorem 30
The resistance to a sphere moving uniformly forward in an infinite and nonelastic compressed fluid is to the force by which its whole motion could either be destroyed or generated, in the time in which it describes ⁸/3 of its diameter, very nearly as the density of the fluid to the density of the sphere.

For a sphere is to the circumscribed cylinder as 2 to 3, and therefore the force that could take away all the motion of a cylinder, while the cylinder is describing a length of four diameters, will take away all the motion of the sphere while the sphere describes ⅔ of this length, that is, ⁸/3 of its own diameter. And the resistance of the cylinder is to this force very nearly as the density of the fluid to the density of the cylinder or sphere, by prop. 37, and the resistance of the sphere is equal to the resistance of the cylinder, by lems. 5, 6, and 7. Q.E.D.

COROLLARY 1. The resistances of spheres in infinite compressed mediums are in a ratio compounded of the squared ratio of the velocity and the squared ratio of the diameter and the ratio of the density of the mediums.

COROLLARY 2. The greatest velocity with which a sphere, by the force of its own relative weight, can descend in a resisting fluid is that which the same sphere with the same weight can acquire in falling without resistance and describing by its fall a space that is to ⁴/3 of its diameter as the density of the sphere to the density of the fluid. For the sphere in the time of its fall, with the velocity acquired in falling, will describe a space that will be to ⁸/3 of its diameter as the density of the sphere to the density of the fluid; and the force of its weight generating this motion will be to the force that could generate the same motion, in the time in which the sphere describes ⁸/3 of its diameter with the same velocity, as the density of the fluid to the density of the sphere; and thus, by this proposition, the force of its weight will be equal to the force of resistance and therefore cannot accelerate the sphere.

COROLLARY 3. Given both the density of the sphere and its velocity at the beginning of the motion, and also the density of the compressed fluid at rest in which the sphere moves, then by prop. 35, corol. 7, the velocity of the sphere, its resistance, and the space described by it are given for any time.

COROLLARY 4. A sphere moving in a compressed fluid at rest, having the same density as itself, will, by the same corol. 7, lose half of its motion before it has described the length of two of its diameters.

Proposition 39, Theorem 31
The resistance to a sphere moving uniformly forward through a fluid enclosed and compressed in a cylindrical channel is to the force by which its whole motion could be either generated or destroyed, while it describes ⁸/3 of its diameter, in a ratio compounded of three ratios, very nearly: the ratio of the opening of the channel to the excess of this opening over half of a great circle of the sphere, the squared ratio of the opening of the channel to the excess of this opening over a great circle of the sphere, and the ratio of the density of the fluid to the density of the sphere.

This is evident by prop. 37, corol. 2, and the proof proceeds as in prop. 38.

Scholium
In the last two propositions (as in lem. 5) I assume that all the water which is in front of the sphere, and whose fluidity increases the resistance to the sphere, is frozen. If all that water liquefies, the resistance will be somewhat increased. But in these propositions the increase will be small and can be ignored because the convex surface of the sphere has almost the same effect as ice.

Proposition 40, Problem 9
To find from phenomena the resistance of a sphere moving forward in a compressed, very fluid medium.

Let A be the weight of the sphere in a vacuum, B its weight in a resisting medium, D the diameter of the sphere, F a space that is to ⁴/3D as the density of the sphere to the density of the medium (that is, as A to A − B), G the time in which the sphere in falling by its weight B without resistance describes the space F, and H the velocity that the sphere acquires by this fall. Then H will be the greatest velocity with which the sphere can descend by its weight B in the resisting medium, by prop. 38, corol. 2, and the resistance that the sphere encounters while descending with this velocity will be equal to its weight B; and the resistance that it encounters with any other velocity will be to the weight B as the square of the ratio of this velocity to the greatest velocity H, by prop. 38, corol. 1.

This is the resistance that arises from the inertia of matter of the fluid. And that which arises from the elasticity, tenacity, and friction of its parts can be investigated as follows.

Drop the sphere so that it descends in the fluid by its own weight B; and let P be the time of falling, in seconds if the time G is in seconds. Find the absolute number N that corresponds to the logarithm 0.4342944819, and let L be the logarithm of the number , then the velocity acquired in falling will be H, and the space described will be − 1.3862943611F + 4.605170186LF.

If the fluid is sufficiently deep, the term 4.605170186LF can be ignored, and − 1.3862943611F will be the space described, very nearly. These things are evident by book 2, prop. 9 and its corollaries, on the hypothesis that the sphere encounters no other resistance than that which arises from the inertia of matter. But if it encounters another resistance in addition, the descent will be slower, and the quantity of this resistance can be found from the retardation.

So that the velocity and descent of a body falling in a fluid may be found more easily, I have put together the accompanying table, in which the first column denotes the times of descent, the second shows the velocities acquired in falling (the greatest velocity being 100,000,000), the third shows the spaces described in falling in those times (2F being the space that the body describes in the time G with the greatest velocity), and the fourth shows the spaces described in the same times with the greatest velocity. The numbers in the fourth column are , and by subtracting the number 1.3862944 − 4.6051702L, the numbers in the third column are found, and these numbers must be multiplied by the space F in order to get the spaces described in falling. There has been added to these a fifth column, which contains the spaces described in the same times by a body falling in a vacuum by the force of its relative weight B.

Scholium
In order to investigate the resistances of fluids by experiments, I got a square wooden vessel, with an internal length and width of 9 inches (of a London foot), and a depth of 9½ feet, and I filled it with rainwater; and making balls of wax with lead inside, I noted the times of descent of the balls, the space of the descent being 112 inches. A solid cubic London foot contains 76 Roman pounds [troy] of rainwater, and a solid inch of this foot contains ¹⁹/36 ounce of this pound or 253⅓ grains; and a sphere of water described with a diameter of 1 inch contains 132.645 grains in air, or 132.8 grains in a vacuum; and any other ball is as the excess of its weight in a vacuum over its weight in water.

EXPERIMENT 1. A ball which weighed 156¼ grains in air and 77 grains in water described the whole space of 112 inches [when dropped in water] in a time of 4 seconds. And when the experiment was repeated, the ball again fell in the same time of 4 seconds.

The weight of the ball in a vacuum is 156¹³/38 grains, and the excess of this weight over the weight of the ball in water is 79¹³/38 grains. And hence the diameter of the ball comes out 0.84224 inch. That excess is to the weight of the ball in a vacuum as the density of water to the density of the ball, and as ⁸/3 of the diameter of the ball (that is, 2.24597 inches) to the space 2F, which accordingly will be 4.4256 inches. In a time of 1 second the ball will fall in a vacuum by its whole weight of 156¹³/38 grains through 193⅓ inches; and by a weight of 77 grains falling in water without resistance, it will in the same time describe 95.219 inches; and in the time G, which is to 1 second as the square root of the ratio of the space F or 2.2128 inches to 95.219 inches, it will describe 2.2128 inches and will attain the greatest velocity H with which it can descend in water. Therefore the time G is 0.15244 seconds. And in this time G, with that greatest velocity H, the ball will describe a space 2F of 4.4256 inches; and thus in the time of 4 seconds it will describe a space of 116.1245 inches. Subtract the space 1.3862944F or 3.0676 inches and there will remain a space of 113.0569 inches which the ball will describe in falling in water in a very wide vessel in the time of 4 seconds. This space, because of the narrowness of the wooden vessel, must be decreased in a ratio which is compounded of the square root of the ratio of the opening of the vessel to the excess of this opening over a great semicircle of the ball, and of the simple ratio of that same opening to its excess over a great circle of the ball, that is, in the ratio of 1 to 0.9914. When this has been done, the result will be a space of 112.08 inches which the ball should, according to the theory, have very nearly described in falling in water in this wooden vessel in the time of 4 seconds. And it described 112 inches in the experiment.

EXPERIMENT 2. Three equal balls, each of which weighed 76⅓ grains in air and 5¹/16 grains in water, were dropped successively in water, and in a time of 15 seconds each one fell through 112 inches.

By computation the weight of a ball in a vacuum is 76⁵/12 grains; the excess of this weight over the weight in water is 71¹⁷48 grains; the diameter of the ball is 0.81296 inch; ⁸/3 of this diameter is 2.16789 inches; the space 2F is 2.3217 inches; the space that a ball describes in falling by a weight of 5¹/16 grains in the time of 1 second without resistance is 12.808 inches; and the time G is 0.301056 second. The ball, therefore, with the greatest velocity with which it can descend in water by the force of the weight of 5¹/16 grains, will describe in a time of 0.301056 second a space of 2.3217 inches, and in the time of 15 seconds a space of 115.678 inches. Subtract the space 1.3862944F or 1.609 inches, and there will remain a space of 114.069 inches which accordingly the ball ought to describe in falling in the same time in a very wide vessel. Because of the narrowness of our vessel a space of roughly 0.895 inch must be taken away. And thus there will remain a space of 113.174 inches which the ball, according to the theory, should have very nearly described in falling in this vessel in the time of 15 seconds. And it described 112 inches in the experiment. The difference is imperceptible.

EXPERIMENT 3. Three equal balls, each of which weighed 121 grains in air and 1 grain in water, were dropped successively in water, and in times of 46 seconds, 47 seconds, and 50 seconds, fell 112 inches.

According to the theory, these balls should have fallen in a time of roughly 40 seconds. I am uncertain whether their falling more slowly is to be attributed to the smaller proportion of the resistance that arises from the force of inertia in slow motions to the resistance that arises from other causes, or rather to some little bubbles adhering to the ball, or to the rarefaction of the wax from the heat either of the weather or of the hand dropping the ball, or even to imperceptible errors in weighing the balls in water. And thus the weight of the ball in water ought to be more than 1 grain, so that the experiment may be made certain and trustworthy.

EXPERIMENT 4. I began the experiments thus far described in order to investigate the resistances of fluids before formulating the theory set forth in the immediately preceding propositions. Afterward, in order to examine that theory, I obtained a wooden vessel with an internal width of 8⅔ inches and a depth of 15⅓ feet. Then I made four balls out of wax with lead inside, each one weighing 139¼ grains in air and 7⅛ grains in water. And I let them fall in water in order to measure the times of falling, using a pendulum oscillating in half-seconds. When the balls were being weighed, and afterward when they were falling, they were cold and had remained cold for some time, because heat rarefies the wax and by the rarefaction diminishes the weight of the ball in water, and the rarefied wax is not immediately brought back to its original density by chilling. Before they fell, they were entirely immersed in water, so that their descent might not be accelerated at the beginning by the weight of some part projecting out of the water. And when totally immersed and at rest, they were let fall as carefully as possible, so as not to receive some impulse from the hand letting them fall. And they fell successively in the times of 47½, 48½, 50, and 51 oscillations, describing a space of 15 feet 2 inches. But the weather was now a little colder than when the balls were weighed, and so I repeated the experiment on another day, and the balls fell in the times of 49, 49½, 50, and 53 oscillations, and on a third day in the times of 49½, 50, 51, and 53 oscillations. The experiment was made quite often, and the balls for the most part fell in the times of 49½ and 50 oscillations. When they fell more slowly, I suspect that they were retarded by hitting against the sides of the vessel.

Now by computation according to the theory, the weight of a ball in a vacuum is 139⅖ grains; the excess of this weight over the weight of the ball in water is 132¹¹/40 grains; the diameter of the ball is 0.99868 inch; ⁸/3 of the diameter is 2.66315 inches; the space 2F is 2.8066 inches; the space that a ball describes in falling with a weight of 7⅛ grains in the time of 1 second without resistance is 9.88164 inches; and the time G is 0.376843 second. The ball, therefore, with the greatest velocity with which it can descend in water by a force of weight of 7⅛ grains, describes in the time of 0.376843 second a space of 2.8066 inches; in the time of 1 second a space of 7.44766 inches; and in the time of 25 seconds, or 50 oscillations, a space of 186.1915 inches. Subtract the space 1.386294F, or 1.9454 inches, and there will remain the space of 184.2461 inches which the ball will describe in the same time in a very wide vessel. Because of the narrowness of our vessel, decrease this space in a ratio that is compounded of the square root of the ratio of the opening of the vessel to the excess of this opening over a great semicircle of the ball, and the simple ratio of that same opening to its excess over a great circle of the ball, and the result will be the space of 181.86 inches which the ball, according to the theory, should very nearly have described in this vessel in the time of 50 oscillations. And in the experiment it described a space of 182 inches in the time of 49½ or 50 oscillations.

EXPERIMENT 5. Four balls weighing 154⅜ grains in air and 21½ grains in water were dropped often and fell in the times of 28½, 29, 29½, and 30 oscillations, and sometimes 31, 32, and 33, describing a space of 15 feet 2 inches.

By the theory they ought to have fallen in the time of very nearly 29 oscillations.

EXPERIMENT 6. Five balls weighing 212⅜ grains in air and 79½ in water were dropped often and fell in the times of 15, 15½, 16, 17, and 18 oscillations, describing a space of 15 feet 2 inches.

By the theory they ought to have fallen in the time of very nearly 15 oscillations.

EXPERIMENT 7. Four balls weighing 293⅜ grains in air and 35⅞ grains in water were dropped often and fell in the times of 29½, 30, 30½, 31, 32, and 33 oscillations, describing a space of 15 feet 1½ inches.

By the theory they ought to have fallen in the time of very nearly 28 oscillations.

In investigating the reason why some of the balls which were of the same weight and size fell more quickly and others more slowly, I hit upon this: that when the balls were first dropped and were beginning to fall, the side which happened to be heavier descended first and generated an oscillatory motion, so that they oscillated around their centers. For by its oscillations a ball communicates a greater motion to the water than if it were descending without oscillations, and in the process loses part of its own motion with which it should descend; and it is retarded more or retarded less in proportion to the greatness or smallness of the oscillation. Further, the ball always recedes from that side which is descending in the oscillation and, by receding, approaches the sides of the vessel and sometimes strikes against the sides. In the case of heavier balls, this oscillation is stronger, and with larger balls, it agitates the water more. Therefore, in order to reduce the oscillation of the balls, I constructed new balls of wax and lead, fixing the lead into one side of the ball near its surface; and I dropped the ball in such a way that the heavier side, as far as possible, was lowest at the beginning of the descent. Thus the oscillations became much smaller than before, and the balls fell in less unequal times, as in the following experiments.

EXPERIMENT 8. Four balls, weighing 139 grains in air and 6½ in water, were dropped often and fell in the times of not more than 52 oscillations, and not fewer than 50, and for the most part in the time of roughly 51 oscillations, describing a space of 182 inches.

By the theory they ought to have fallen in the time of roughly 52 oscillations.

EXPERIMENT 9. Four balls, weighing 273¼ grains in air and 140¾ in water, were dropped often and fell in the times of not fewer than 12 oscillations and not more than 13, describing a space of 182 inches.

And by the theory these balls ought to have fallen in the time of very nearly 11⅓ oscillations.

EXPERIMENT 10. Four balls, weighing 384 grains in air and 119½ in water, were dropped often and fell in the times of 17¾, 18, 18½, and 19 oscillations, describing a space of 181½ inches. And when they fell in the time of 19 oscillations, I sometimes heard them strike the sides of the vessel before they reached the bottom.

And by the theory they ought to have fallen in the time of very nearly 15⁵/9 oscillations.

EXPERIMENT 11. Three equal balls, weighing 48 grains in air and 3²⁹/32 in water, were dropped often and fell in the times of 43½, 44, 44½, 45, and 46 oscillations, and for the most part 44 and 45, describing a space of very nearly 182½ inches.

By the theory they ought to have fallen in the time of roughly 46⁵/9 oscillations.

EXPERIMENT 12. Three equal balls, weighing 141 grains in air and 4⅜ in water, were dropped several times and fell in the times of 61, 62, 63, 64, and 65 oscillations, describing a space of 182 inches.

And by the theory they ought to have fallen in the time of very nearly 64½ oscillations.

From these experiments it is obvious that when the balls fell slowly (as in the second, fourth, fifth, eighth, eleventh, and twelfth experiments), the times of falling are shown correctly by the theory, but that when the balls fell more quickly (as in the sixth, ninth, and tenth experiments), the resistance was a little greater than in the squared ratio of the velocity. For the balls oscillate somewhat while falling, and this oscillation—in balls that are lighter and fall more slowly—ceases swiftly because the motion is weak, while in heavier and larger balls, because the motion is strong, the oscillation lasts longer and can be checked by the surrounding water only after more oscillations. Additionally, the swifter the balls, the less they are pressed by the fluid in back of them; and if the velocity is continually increased, they will at length leave an empty space behind, unless the compression of the fluid is simultaneously increased. The compression of the fluid, moreover, ought (by props. 32 and 33) to be increased in the squared ratio of the velocity in order for the resistance also to be in a squared ratio. Since this does not happen, the swifter balls are pressed a little less from behind, and because of this diminished pressure their resistance becomes a little greater than in the squared ratio of the velocity.

The theory therefore agrees with the phenomena of bodies falling in water; it remains for us to examine the phenomena of bodies falling in air.

EXPERIMENT 13. ^aFrom the top of St. Paul’s Cathedral in London^a in June 1710, glass balls were dropped simultaneously in pairs, one full of quicksilver, the other full of air; and in falling they described a space of 220 London feet. A wooden platform was suspended at one end by iron pivots, and at the other was supported by a wooden peg. The two balls were placed upon this platform and were let fall simultaneously by pulling out the peg by means of an iron wire extending to the ground, so that the platform, resting on the iron pivots alone, might swing downward upon the pivots and at the same moment a seconds pendulum, pulled by that iron wire, might be released and begin to oscillate. The diameters and weights of the balls and the times of falling are shown in the following table.

However, the observed times need to be corrected. For balls filled with mercury will (by Galileo’s theory) describe 257 London feet in 4 seconds, and 220 feet in only 3 seconds 42 thirds. The wooden platform, when the peg was withdrawn, swung downward more slowly than it should have [i.e., more slowly than in free fall] and as a result impeded the descent of the balls at the start. For the balls were lying upon the platform near its center, and were in fact a little closer to the pivots than to the peg. And hence the times of falling were prolonged by roughly 18 thirds and so need to be corrected by taking away those thirds, especially in the larger balls, which because of the magnitude of their diameters remained a little longer upon the platform as it swung downward. When this has been done, the times in which the six larger balls fell will come out 8 sec. 12 thirds, 7 sec. 42 thirds, 7 sec. 42 thirds, 7 sec. 57 thirds, 8 sec. 12 thirds, and 7 sec. 42 thirds.

Therefore the fifth of those balls filled with air, with a diameter of 5 inches and a weight of 483 grains, fell in the time of 8 sec. 12 thirds, describing the space of 220 feet. The weight of water equal to this ball is 16,600 grains; and the weight of air equal to it is grains, or 19³/10 grains, and thus the weight of the ball in a vacuum is 502³/10 grains, and this weight is to the weight of air equal to the ball as 502³/10 to 19³/10, as is the ratio of 2F to ⁸/3 of the diameter of the ball (that is, 2F to 13⅓ inches). And hence 2F comes out 28 feet 11 inches. The ball in falling in a vacuum, with its whole weight of 502³/10 grains, in the time of one second describes 193⅓ inches as above, and with a weight of 483 grains describes 185.905 inches, and with the same weight of 483 grains also in a vacuum describes the space F, or 14 feet 5½ inches, in the time of 57 thirds 58 fourths, and attains the greatest velocity with which it could descend in air. With this velocity the ball, in the time of 8 sec. 12 thirds, will describe a space of 245 feet 5⅓ inches. Take away 1.3863F, or 20 feet ½ inch, and there will remain 225 feet 5 inches. It is this space, therefore, that the ball should, by the theory, have described in falling in the time of 8 sec. 12 thirds. And it described a space of 220 feet in the experiment. The difference is negligible.

Applying similar computations also to the remaining balls filled with air, I constructed the following table.

EXPERIMENT 14. In July 1719, Dr. Desaguliers made experiments of this sort again, making hogs’ bladders into a round shape by means of a concave wooden sphere, which the moist bladders, inflated with air, were forced to fill; after they were dried and taken out, they were dropped ^bfrom the lantern at the top of the cupola of the same cathedral, that is, from a height of 272 feet,^b and at the same moment a lead ball was also dropped, whose weight was roughly two pounds troy. And meanwhile some people standing in the highest part of St. Paul’s where the balls were released noted the whole times of falling, and others standing on the ground noted the difference between the times of fall of the lead ball and of the bladder. And the times were measured by half-second pendulums. And one of those who were standing on the ground had a clock with an oscillating spring, vibrating four times per second; someone else had another machine ingeniously constructed with a pendulum also vibrating four times per second. And one of those who were standing in the gallery of the cupola had a similar device. And these instruments were so constructed that their motions might begin or be stopped at will. The lead ball fell in a time of roughly 4¼ seconds. And by adding this time to the aforesaid difference between the times, the whole time in which the bladder fell was determined. The times in which the five bladders continued to fall after the lead ball had completed its fall were 14¾ sec., 12¾ sec., 14⅝ sec., 17¾ sec., and 16⅞ sec. the first time, and 14½ sec., 14¼ sec., 14 sec., 19 sec., and 16¾ sec. the second time. Add 4¼ sec., the time in which the lead ball fell, and the whole times in which the five bladders fell were 19 sec., 17 sec., 18⅞ sec., 22 sec., and 21⅛ sec. the first time, and 18¾ sec., 18½ sec., 18¼ sec., 23¼ sec., and 21 sec. the second time. And the times noted from the cupola were 19⅜ sec., 17¼ sec., 18¾ sec., 22⅛ sec., and 21⅝ sec. the first time, and 19 sec., 18⅝ sec., 18⅜ sec., 24 sec., and 21¼ sec. the second time. But the bladders did not always fall straight down, but sometimes flew about and oscillated to and fro while falling. And the times of falling were prolonged and increased by these motions, sometimes by one-half of one second, sometimes by a whole second. The second and fourth bladders, moreover, fell straighter down the first time, as did the first and third the second time. The fifth bladder was wrinkled and was somewhat retarded by its wrinkles. I calculated the diameters of the bladders from their circumferences, measured by a very thin thread wound round them twice. And I compared the theory with the experiments in the following table, assuming the density of air to be to the density of rainwater as 1 to 860, and calculating the spaces that the balls should, by the theory, have described in falling.

Therefore almost all the resistance encountered by balls moving in air as well as in water is correctly shown by our theory, and is proportional to the density of the fluids—the velocities and sizes of the balls being equal.

In the scholium at the end of sec. 6, we showed by experiments with pendulums that the resistances encountered by equal and equally swift balls moving in air, water, and quicksilver are as the densities of the fluids. We have shown the same thing here more accurately by experiments with bodies falling in air and water. For pendulums in each oscillation arouse in the fluid a motion always opposite to the motion of the pendulum when it returns; and the resistance arising from this motion, and also the resistance to the cord by which the pendulum was suspended, made the whole resistance to the pendulum greater than the resistance found by the experiments with falling bodies. For by the experiments with pendulums set forth in that scholium, a ball of the same density as water ought, in describing the length of its own semidiameter in air, to lose of its motion. But by the theory set forth in this seventh section and confirmed by experiments with falling bodies, that same ball ought, in describing that same length, to lose only of its motion, supposing that the density of water is to the density of air as 860 to 1. The resistances therefore were found to be greater by the experiments with pendulums (for the reasons already described) than by the experiments with falling balls, and in a ratio of roughly 4 to 3. But since the resistances to pendulums oscillating in air, water, and quicksilver are increased similarly by similar causes, the proportion of the resistances in these mediums will be shown correctly enough by the experiments with pendulums as well as by the experiments with falling bodies. And hence it can be concluded that the resistances encountered by bodies moving in any fluids that are very fluid, other things being equal, are as the densities of the fluids.

On the basis of what has been established, it is now possible to predict very nearly what part of the motion of any ball projected in any fluid will be lost in a given time. Let D be the diameter of the ball, and V its velocity at the beginning of the motion, and T the time in which the ball will—with velocity V in a vacuum—describe a space that is to the space ⁸/3D as the density of the ball to the density of the fluid; then the ball projected in that fluid will, in any other time t, lose the part of its velocity the part remaining and will describe a space that is to the space described in a vacuum in the same time with the uniform velocity V as the logarithm of the number multiplied by the number 2.302585093 is to the number t/T, by prop. 35, corol. 7. In slow motions the resistance can be a little less, because the shape of a ball is a little more suitable for motion than the shape of a cylinder described with the same diameter. In swift motions the resistance can be a little greater, because the elasticity and the compression of the fluid are not increased in the squared ratio of the velocity. But here I am not considering petty details of this sort.

And even if air, water, quicksilver, and similar fluids, by some infinite division of their parts, could be subtilized and become infinitely fluid mediums, they would not resist projected balls any the less. For the resistance which is the subject of the preceding propositions arises from the inertia of matter; and the inertia of matter is essential to bodies and is always proportional to the quantity of matter. By the division of the parts of a fluid, the resistance that arises from the tenacity and friction of the parts can indeed be diminished, but the quantity of matter is not diminished by the division of its parts; and since the quantity of matter remains the same, its force of inertia—to which the resistance discussed here is always proportional—remains the same. For this resistance to be diminished, the quantity of matter in the spaces through which bodies move must be diminished. And therefore the celestial spaces, through which the globes of the planets and comets move continually in all directions very freely and without any sensible diminution of motion, are devoid of any corporeal fluid, except perhaps the very rarest vapors and rays of light transmitted through those spaces.

Projectiles, of course, arouse motion in fluids by going through them, and this motion arises from the excess of the pressure of the fluid on the front of the projectile over the pressure on the back, and cannot be less in infinitely fluid mediums than in air, water, and quicksilver in proportion to the density of matter in each. And this excess of pressure, in proportion to its quantity, not only arouses motion in the fluid but also acts upon the projectile to retard its motion; and therefore the resistance in every fluid is as the motion excited in the fluid by the projectile, and it cannot be less in the most subtle aether, in proportion to the density of the aether, than in air, water, and quicksilver, in proportion to the densities of these fluids.

SECTION 8

Motion propagated through fluids

Proposition 41, Theorem 32
Pressure is not propagated through a fluid along straight lines, unless the particles of the fluid lie in a straight line.

If the particles a, b, c, d, and e lie in a straight line, a pressure can indeed be propagated directly from a to e; but the particle e will urge the obliquely placed particles f and g obliquely, and those particles f and g will not sustain the pressure brought upon them unless they are supported by the further particles h and k; but to the extent that they are supported, they press the supporting particles, and these will not sustain the pressure unless they are supported by the further particles l and m and press them, and so on indefinitely. Therefore, as soon as a pressure is propagated to particles which do not lie in a straight line, it will begin to spread out and will be obliquely propagated indefinitely; and after the pressure begins to be propagated obliquely, if it should impinge upon further particles which do not lie in a straight line, it will spread out again, and will do so as often as it impinges upon particles not lying exactly in a straight line. Q.E.D.

COROLLARY. If some part of a pressure propagated through a fluid from a given point is intercepted by an obstacle, the remaining part (which is not intercepted) will spread out into the spaces behind the obstacle. This can be proved as follows. From point A let a pressure be propagated in any direction and, if possible, along straight lines; and by the obstacle NBCK, perforated in BC, let all the pressure be intercepted except the cone-shaped part APQ, which passes through the circular hole BC. By transverse planes de, fg, and hi, divide the cone APQ into frusta; then, while the cone ABC, by propagating the pressure, is urging the further conic frustum degf on the surface de, and this frustum is urging the next frustum fgih on the surface fg, and that frustum is urging a third frustum, and so on indefinitely, obviously (by the third law of motion) the first frustum defg will be as much urged and pressed on the surface fg by the reaction of the second frustum fghi as it urges and presses the second frustum. Therefore the frustum degf between the cone Ade and the frustum fhig is compressed on both sides, and therefore (by book 2, prop. 19, case 6) it cannot keep its figure unless it is compressed by the same force on all sides. With the same force, therefore, with which it is pressed on the surfaces de and fg, it will endeavor to yield at the sides df and eg; and there (since it is not rigid, but altogether fluid) it will run out and expand, unless a surrounding fluid is present to restrain that endeavor. Accordingly, by the endeavor to run out, it will press the surrounding fluid at the sides df and eg, as well as the frustum fghi, with the same force; and therefore the pressure will be no less propagated from the sides df and eg into the spaces NO on one side and KL on the other, than it is propagated from the surface fg toward PQ. Q.E.D.

Proposition 42, Theorem 33
All motion propagated through a fluid diverges from a straight path into the motionless spaces.

CASE 1. Let a motion be propagated from point A through a hole BC, and let it proceed, if possible, in the conic space BCQP along straight lines diverging from point A. And let us suppose first that this motion is that of waves on the surface of stagnant water. And let de, fg, hi, kl, . . . be the highest parts of the individual waves, separated from one another by the same number of intermediate troughs. Therefore, since the water is higher in the crests of the waves than in the motionless parts LK and NO of the fluid, it will flow down from e, g, i, l, . . . , and d, f, h, k, . . . , the ends of the crests, toward KL on one side and NO on the other; and since it is lower in the troughs of the waves than in the motionless parts KL and NO of the fluid, it will flow down from those motionless parts into the troughs of the waves. In one case the crests of the waves, and in the other their troughs, are expanded and propagated toward KL on one side and NO on the other. And since the motion of the waves from A toward PQ takes place by the continual flowing down of the crests into the nearest troughs, and thus is not quicker than in proportion to the quickness of the descent, and since the descent of the water toward KL on one side and NO on the other ought to occur with the same velocity, the expansion of the waves will be propagated toward KL on one side and NO on the other with the same velocity with which the waves themselves progress directly from A toward PQ. And accordingly the whole space toward KL on one side and NO on the other will be occupied by the expanded waves rfgr, shis, tklt, vmnv, . . . . Q.E.D. Anyone can test this in stagnant water.

CASE 2. Now let us suppose that de, fg, hi, kl, and mn designate pulses successively propagated from point A through an elastic medium. Think of the pulses as propagated by successive condensations and rarefactions of the medium, in such a way that the densest part of each pulse occupies a spherical surface described about the center A, and that the spaces which come between successive pulses are equal. And let de, fg, hi, kl, . . . designate the densest parts of the pulses, parts which are propagated through the hole BC. And since the medium is denser there than in the spaces toward KL on one side and NO on the other, it will expand toward those spaces KL and NO situated on both sides as well as toward the rarer intervals between the pulses; and thus, always becoming rarer next to the intervals and denser next to the pulses, the medium will participate in their motion. And since the progressive motion of the pulses arises from the continual slackening of the denser parts toward the rarer intervals in front of them, and since the pulses ought to slacken with nearly the same speed into the medium’s parts KL on one side and NO on the other, which are at rest, those pulses will expand on all sides into the motionless spaces KL and NO with nearly the same speed with which they are propagated straight forward from the center A, and thus will occupy the whole space KLON. Q.E.D. We find this by experience in the case of sounds, which are heard when there is a mountain in the way or which expand into all parts of a room when let in through a window and are heard in all corners, being not so much reflected from the opposite walls as propagated directly from the window, as far as the senses can tell.

CASE 3. Finally, let us suppose that a motion of any kind is propagated from A through the hole BC. That propagation does not occur except insofar as the parts of the medium that are nearer to the center A urge and move the further parts; and the parts that are urged are fluid and thus recede in every direction into regions where they are less pressed, and so will recede toward all the parts of the medium that are at rest, the parts KL and NO on the sides as well as the parts PQ in front. And therefore all the motion, as soon as it has passed through the hole BC, will begin to spread out and to be propagated directly from there into all parts as if from an origin and center. Q.E.D.

Proposition 43, Theorem 34
Every vibrating body in an elastic medium will propagate the motion of the pulses straight ahead in every direction, but in a nonelastic medium will produce a circular motion.

CASE 1. For the parts of a vibrating body, by going forward and returning alternately, will in their going urge and propel the parts of the medium that are nearest to them and by that urging will compress and condense them; then in their return they will allow the compressed parts to recede [i.e., to move apart from one another] and expand. Thus the parts of the medium that are nearest to the vibrating body will go and return alternately, like the parts of the vibrating body; and just as the parts of this body acted upon the parts of the medium, so the latter, acted upon by similar vibrations, will act upon the parts nearest to them, and these, similarly acted upon, will act upon further parts, and so on indefinitely. And just as the first parts of the medium condense in going and rarefy in returning, so the remaining parts will condense whenever they go and will expand [i.e., rarefy] whenever they return. And therefore they will not all go and return at the same time (for thus, by keeping determined distances from one another, they would not rarefy and condense alternately), but by approaching one another when they condense and moving apart when they rarefy, some of them will go while others return, and these conditions will alternate indefinitely. And the parts that are going and that condense in going (because of their forward motion with which they strike obstacles) are pulses; and therefore successive pulses will be propagated straight ahead from every vibrating body, and they will be so propagated at roughly equal distances from one another, because of the equal intervals of time in which the body produces each pulse by each of its vibrations. And even if the parts of the vibrating body go and return in some fixed and determined direction, nevertheless the pulses propagated from there through the medium will (by prop. 42) expand sideways and will be propagated in all directions from the vibrating body as if from a common center, in surfaces almost spherical and concentric. We have an example of this in waves, which, if they are produced by a wagging finger, not only will go to and fro according to the finger’s motion but will immediately surround the finger like concentric circles and will be propagated in all directions. For the gravity of the waves takes the place of the elastic force.

CASE 2. But if the medium is not elastic, then, since its parts, pressed by the oscillating parts of the vibrating body, cannot be condensed, the motion will be propagated instantly to the parts where the medium yields most easily, that is, to the parts that the vibrating body would otherwise leave empty behind it. The case is the same as the case of a body projected in any medium. A medium, in yielding to projectiles, does not recede indefinitely, but goes with a circular motion to the spaces that the body leaves behind it. Therefore, whenever a vibrating body goes toward any place [or in any direction], the medium, in yielding, will go with a circular motion to the spaces that the body leaves; and whenever the body returns to its former place, the medium will be forced out and will return to its former place. And even though the vibrating body is not rigid but completely pliant, if it nevertheless remains of a fixed size, then, since it cannot urge the medium by its vibrations in any one place without simultaneously yielding to it in another, that body will make the medium, by receding from the parts where it is pressed, go always with a circular motion to the parts that yield to it. Q.E.D.

COROLLARY. Therefore it is a delusion to believe that the agitation of the parts of flame conduces to the propagation of a pressure along straight lines through a surrounding medium. A pressure of this sort must be derived not only from the agitation of the parts of the flame but from the dilation of the whole.

Proposition 44, Theorem 35
If water ascends and descends alternately in the vertical arms KL and MN of a tube, and if a pendulum is constructed whose length between the point of suspension and the center of oscillation is equal to half of the length of the water in the tube, then I say that the water will ascend and descend in the same times in which the pendulum oscillates.

I measure the length of the water along the axes of the tube and the arms and make it equal to the sum of these axes, and I do not here consider the resistance of the water that arises from the friction of the tube. Let AB and CD therefore designate the mean height of the water in the two arms, and when the water in the arm KL ascends to the height EF, the water in the arm MN will have descended to the height GH. Moreover, let P be a pendulum bob, VP the cord, V the point of suspension, RPQS the cycloid described by the pendulum, P its lowest point, and PQ an arc equal to the height AE. The force by which the motion of the water is alternately accelerated and retarded is the amount by which the weight of the water in one of the two arms exceeds the weight in the other. And thus, when the water in the arm KL ascends to EF, and in the other arm descends to GH, that force is twice the weight of the water EABF and therefore is to the weight of all the water as AE or PQ to VP or PR. Furthermore, the force by which the weight P in any place Q is accelerated and retarded in the cycloid is (by book 1, prop. 51, corol.) to its whole weight as its distance PQ from the lowest place P to the length PR of the cycloid. Therefore the motive forces of the water and the pendulum, describing the equal spaces AE and PQ, are as the weights that are to be moved; and thus, if the water and the pendulum are at rest in the beginning, those forces will move them equally in equal times and will cause them to go and return synchronously with an alternating motion. Q.E.D.

COROLLARY 1. Therefore all the alternations of the ascending and descending water are isochronous, whether the motion is of greater intension or greater remission.^a

COROLLARY 2. If the length of all the water in the tube is 6¹/9 Paris feet, the water will descend in the time of one second and will ascend in another second and will continue to alternate in this way indefinitely. For a pendulum 3¹/18 feet long oscillates in the time of one second.

COROLLARY 3. When the length of the water is increased or decreased, moreover, the time of alternation is increased or decreased as the square root of the length.

Proposition 45, Theorem 36
The velocity of waves is as the square roots of the lengths.

This follows from the construction of the following proposition.

Proposition 46, Problem 10
To find the velocity of waves.

Set up a pendulum whose length between the point of suspension and the center of oscillation is equal to the length of the waves; and in the same time in which the pendulum performs each of its oscillations, the waves as they move forward will traverse nearly their own lengths.

By length of a wave I mean the transverse distance either between bottoms of troughs or between tops of crests. Let ABCDEF designate the surface of stagnant water ascending and descending in successive waves; and let A, C, E, . . . be the crests of the waves, and B, D, F, . . . the troughs in between. Since the motion of the waves is caused by the successive ascent and descent of the water, in such a way that its parts, A, C, E, . . . , which now are highest, soon become lowest, and since the motive force by which the highest parts descend and the lowest ascend is the weight of the elevated water, the alternate ascent and descent will be analogous to the alternating motion of the water in the tube and will observe the same laws with respect to times; and therefore (by prop. 44), if the distances between the highest places A, C, and E of the waves and the lowest, B, D, and F, are equal to twice the length of a pendulum, the highest parts A, C, and E will in the time of one oscillation come to be the lowest, and in the time of a second oscillation will ascend once again. Therefore there will be a time of two oscillations between successive waves; that is, a wave will describe its own length in the time in which the pendulum oscillates twice; but in the same time a pendulum whose length is four times as great, and thus equals the length of the waves, will oscillate once. Q.E.I.

COROLLARY 1. Therefore waves with a length of 3¹/18 Paris feet will move forward through their own length in the time of one second and thus in the time of one minute will traverse 183⅓ feet, and in the space of an hour very nearly 11,000 feet.

COROLLARY 2. And the velocity of waves of greater or smaller length will be increased or decreased as the square root of the length.

What has been said is premised on the hypothesis that the parts of the water go straight up or straight down; but this ascent and descent takes place more truly in a circle, and thus I admit that in this proposition the time has been determined only approximately.

Proposition 47, Theorem 37
If pulses are propagated through a fluid, the individual particles of the fluid, going and returning with a very short alternating motion, are always accelerated and retarded in accordance with the law of an oscillating pendulum.

Let AB, BC, CD, . . . designate the equal distances between successive pulses; ABC the line of motion of the pulses, propagated from A toward B; E, F, and G three physical points in the medium at rest, situated at equal intervals along the straight line AC; Ee, Ff, and Gg very short equal spaces through which those points go and return in each vibration with an alternating motion; ε, φ, γ any intermediate positions of those same points; and EF and FG physical line-elements or linear parts of the medium, put between those points and successively transferred into the places εφ, φγ and ef, fg. Draw the straight line PS equal to the straight line Ee. Bisect PS in O, and with center O and radius OP describe the circle SIPi.

Let the whole circumference of this circle with its parts represent the whole time of one vibration with its proportional parts, in such a way that when any time PH or PHSh is completed, if the perpendicular HL or hl is dropped to PS, and if Eε is taken equal to PL or Pl, then the physical point E is found in ε. By this law any point E, in going from E through ε to e and returning from there through ε to E, will perform each vibration with the same degrees of acceleration and retardation as the oscillating pendulum. It is to be proved that each of the physical points of the medium must move in such a way. Let us imagine, therefore, that there is such a motion in the medium, arising from any cause, and see what follows.

In the circumference PHSh take the equal arcs HI and IK or hi and ik, having the ratio to the whole circumference that the equal straight lines EF and FG have to the whole interval BC between pulses. Drop the perpendiculars IM and KN and also im and kn. Then the points E, F, and G are successively agitated with similar motions and carry out their complete vibrations (consisting of a going and returning) while a pulse is transferred from B to C; accordingly, if PH or PHSh is the time from the beginning of the motion of point E, PI or PHSi will be the time from the beginning of the motion of point F, and PK or PHSk will be the time from the beginning of the motion of point G; and therefore Eε, Fφ, and Gγ will be equal respectively to PL, PM, and PN in the going of the points, or to Pl, Pm, and Pn in the returning of the points. Hence εγ or EG + Gγ − Eε will be equal to EG − LN in the going of the points, and will be equal to EG + ln in their returning. But εγ is the width or expansion of the part of the medium EG in the place εγ; and therefore the expansion of that part in the going is to its mean expansion as EG − LN to EG, and in the returning is as EG + ln or EG + LN to EG. Therefore, since LN is to KH as IM to the radius OP, and KH is to EG as the circumference PHShP to BC, that is (if V is put for the radius of a circle having a circumference equal to the interval between the pulses BC), as OP to V, and since, from the equality of the ratios [or ex aequo], LN is to EG as IM to V, the expansion of the part EG or of the physical point F in the place εγ will be to the mean expansion which that part has in its own first place EG as V − IM to V in the going, and as V + im to V in the returning. Hence the elastic force of point F in the place εγ is to its mean elastic force in the place EG as to in the going, and as to in the returning. And by the same argument the elastic forces of the physical points E and G in the going are as and to ; and the difference between the forces is to the mean elastic force or the medium as to , that is, as to , or as HL − KN to V, provided that (because of the narrow limits of the vibrations) we suppose HL and KN to be indefinitely smaller than the quantity V. Therefore, since the quantity V is given, the difference between the forces is as HL — KN, that is, as OM (because HL − KN is proportional to HK and OM to OI or OP; and HK and OP are given)—that is, if Ff is bisected in Ω, as Ωφ. And by the same argument the difference between the elastic forces of the physical points ε and γ, in the returning of the physical line-element εγ, is as Ωφ. But that difference (that is, the amount by which the elastic force of point ε exceeds the elastic force of point γ) is the force by which the intervening physical line-element εγ of the medium is accelerated in the going and retarded in the returning; and therefore the accelerative force of the physical line-element εγ is as its distance from the midpoint Ω of the vibration. Accordingly, the time (by book 1, prop. 38) is correctly represented by the arc PI, and the linear part εγ of the medium moves by the law previously mentioned, that is, by the law of an oscillating pendulum; and the same is true of all the linear parts of which the whole medium is composed. Q.E.D.

COROLLARY. Hence it is evident that the number of pulses propagated is the same as the number of vibrations of the vibrating body and does not increase as the pulses move forward. For as soon as the physical line-element εγ has returned to its first place, it will be at rest and will not move afterward unless it receives a new motion either by the impact of the vibrating body or by the impact of pulses that are propagated from the vibrating body. It will be at rest, therefore, as soon as the pulses cease to be propagated from the vibrating body.

Proposition 48, Theorem 38
The velocities of pulses propagated in an elastic fluid are as the square root of the elastic force directly and the square root of the density inversely, provided that the elastic force of the fluid is proportional to its condensation.

CASE 1. If the mediums are homogeneous and the distances between pulses in these mediums are equal to one another, but the motion in one medium is more intense, then the contractions and expansions of corresponding parts will be as the motions. In fact, this proportion is not exact. Even so, unless the contractions and expansions are extremely intense, the error will not be perceptible, and thus the proportion can be considered physically exact. But the motive elastic forces are as the contractions and expansions; and the velocities—generated in the same time—of equal parts are as the forces. And thus equal and corresponding parts of corresponding pulses will go and return together through spaces proportional to the contractions and expansions, with velocities that are as the spaces; and therefore the pulses, which advance through their own length in the time of one going and returning and which always succeed into the places of the immediately preceding pulses, will progress in both mediums with an equal velocity, because of the equality of the distances.

CASE 2. But if the distances between pulses, or their lengths, are greater in one medium than in the other, let us suppose that the corresponding parts by going and returning in each alternation describe spaces proportional to the lengths of the pulses; then their contractions and expansions will be equal. And thus if the mediums are homogeneous, those motive elastic forces by which they are agitated with an alternating motion will also be equal. But the matter to be moved by these forces is as the length of the pulses; and the space through which they must move by going and returning in each alternation is in the same ratio. And the time of going and returning is jointly proportional to the square root of the matter and the square root of the space and thus is as the space. But the pulses advance through their own lengths in the times of one going and returning, that is, traverse spaces proportional to the times, and therefore have equal velocities.

CASE 3. In mediums of the same density and elastic force, therefore, all pulses have equal velocities. But if either the density or the elastic force of the medium is intended [i.e., increased], then, since the motive force is increased in the ratio of the elastic force, and the matter to be moved is increased in the ratio of the density, the time in which the same motions as before can be performed will be increased as the square root of the density and will be decreased as the square root of the elastic force. And therefore the velocity of the pulses will be jointly proportional to the square root of the density of the medium inversely and the square root of the elastic force directly. Q.E.D.

This proposition will be clearer from the construction of the following proposition.

Proposition 49, Problem 11
Given the density and elastic force of a medium, it is required to find the velocity of the pulses.

Let us imagine the medium to be compressed, as our air is, by an incumbent weight and let A be the height of a homogeneous medium whose weight is equal to the incumbent weight and whose density is the same as the density of the compressed medium in which the pulses are propagated. And suppose that a pendulum is set up, whose length between the point of suspension and the center of oscillation is A; then, in the same time in which that pendulum performs an entire oscillation composed of a going and a returning, a pulse will advance through a space equal to the circumference of a circle described with radius A.

For with the same constructions as in prop. 47, if any physical line EF, describing the space PS in each single vibration, is urged in the extremities P and S of each going and returning by an elastic force that is equal to its weight, it will perform each single vibration in the time in which it could oscillate in a cycloid whose whole perimeter is equal to the length PS; and this is so because equal forces will simultaneously impel equal corpuscles through equal spaces. Therefore, since the times of the oscillations are as the square root of the length of the pendulums, and since the length of the pendulum is equal to half the arc of the whole cycloid, the time of one vibration would be to the time of oscillation of a pendulum whose length is A as the square root of the length ½PS or PO to the length A. But the elastic force by which the physical line-element EG is urged in its extremities P and S was (in the proof of prop. 47) to its whole elastic force as HL − KN to V, that is (since point K now falls upon P), as HK to V; and that whole force, that is, the incumbent weight by which the line-element EG is compressed, is to the weight of the line-element as the height A of the incumbent weight to the length EG of the line-element; and thus from the equality of the ratios [or ex aequo] the force by which the line-element EG is urged in its places P and S is to the weight of that line-element as HK × A to V × EG, or as PO × A to V² (for HK was to EG as PO to V). Therefore, since the times in which equal bodies are impelled through equal spaces are inversely as the square root of the forces, the time of one vibration under the action of that elastic force will be to the time of the vibration, under the action of the force of weight, as the square root of V² to PO × A, and thus will be to the time of oscillation of a pendulum having a length A as and jointly, that is, as V to A. But in the time of one vibration, composed of a going and returning, a pulse advances through its own length BC. Therefore the time in which the pulse traverses the space BC is to the time of one oscillation (composed of a going and returning) as V to A, that is, as BC to the circumference of a circle whose radius is A. But the time in which the pulse will traverse the space BC is in the same ratio to the time in which it will traverse a length equal to this circumference; and thus in the time of such an oscillation the pulse will traverse a length equal to this circumference. Q.E.D.

COROLLARY 1. The velocity of the pulses is that which heavy bodies acquire in falling with a uniformly accelerated motion and describing by their fall half of the height A. For in the time of this fall, with the velocity acquired in falling, the pulse will traverse a space equal to the whole height A; and thus in the time of one oscillation (composed of a going and returning) it will traverse a space equal to the circumference of a circle described with radius A; for the time of fall is to the time of oscillation as the radius of the circle to its circumference.

COROLLARY 2. Hence, since that height A is as the elastic force of the fluid directly and its density inversely, the velocity of the pulses will be as the square root of the density inversely and the square root of the elastic force directly.

Proposition 50, Problem 12
To find the distances between pulses.

In a given time, find the number of vibrations of the body by whose vibration the pulses are excited. Divide by that number the space that a pulse could traverse in the same time, and the part found will be the length of one pulse. Q.E.I.

Scholium
The preceding propositions apply to the motion of light and of sounds. For since light is propagated along straight lines, it cannot consist in action alone (by props. 41 and 42). And because sounds arise from vibrating bodies, they are nothing other than propagated pulses of air (by prop. 43). This is confirmed from the vibrations that they excite in bodies exposed to them, provided that they are loud and deep, such as the sounds of drums. For swifter and shorter vibrations are excited with more difficulty. But it is also well known that any sounds impinging upon strings in unison with the sonorous bodies excite vibrations in them. It is confirmed also from the velocity of sounds. For since the specific weights of rainwater and quicksilver are to each other as roughly 1 to 13⅔, and since, when the mercury in a barometer reaches a height of 30 English inches, the specific weight of the air and that of rainwater are to each other as roughly 1 to 870, the specific weights of air and quicksilver will be as 1 to 11,890. Accordingly, since the height of the quicksilver is 30 inches, the height of uniform air whose weight could compress our air lying beneath it will be 356,700 inches, or 29,725 English feet. And this height is the very one that we called A in the construction of prop. 49. The circumference of a circle described with a radius of 29,725 feet is 186,768 feet. And since a pendulum 39⅕ inches long completes an oscillation composed of a going and returning in the time of 2 seconds, as is known, a pendulum 29,725 feet or 356,700 inches long must complete an entirely similar oscillation in the time of 190¾ seconds. In that time, therefore, sound will advance 186,768 feet, and thus in the time of one second, 979 feet.

^aBut in this computation no account is taken of the thickness of the solid particles of air, a thickness through which sound is of course propagated instantaneously. Since the weight of air is to the weight of water as 1 to 870, and since salts are nearly twice as dense as water, if the particles of air are supposed to be of roughly the same density as the particles of either water or salts, and if the rarity of air arises from the distances between the particles, the diameter of a particle of air will be to the distance between the centers of the particles roughly as 1 to 9 or 10, and to the distance between the particles as 1 to 8 or 9. Accordingly, to the 979 feet which sound will travel in the time of 1 second according to the above calculation, feet or roughly 109 feet may be added, because of the density of the particles of air; and thus sound will travel roughly 1,088 feet in the time of 1 second.

Additionally, the vapors lying hidden in the air, since they are of another elasticity and another tone, participate scarcely or not at all in the motion of the true air by which sounds are propagated. And when these vapors are at rest, that motion will be propagated more swiftly through the true air alone as the square root of the ratio of the total atmosphere of air and vapor to the matter of the particles of air alone. For example, if the atmosphere consists of 10 parts of true air and 1 part of vapors, the motion of sounds will be swifter as the square root of the ratio of 11 to 10, or in roughly the ratio of 21 to 20, than if it were propagated through 11 parts of true air; and thus the motion of sounds that was found above will have to be increased in this ratio. Thus in the time of 1 second, sound will travel 1,142 feet.

These things ought to be so in the springtime and autumn, when the air is rarefied by the temperate heat and its elastic force is somewhat intended [i.e., increased]. But in winter, when the air is condensed by the cold, and its elastic force is remitted [i.e., decreased], the motion of sounds should be slower as the square root of the density; and alternately, in summer it should be swifter.

It is established by experiments, moreover, that in the time of 1 second sounds advance through more or less 1,142 London feet, or 1,070 Paris feet.

Once the velocity of sounds has been found, the intervals between the pulses can also be found. Sauveur found by making experiments that an open pipe, whose length is more or less 5 Paris feet, produces a sound with the same pitch as the sound of a string that vibrates a hundred times in 1 second. Accordingly, there are more or less 100 pulses in the space of 1,070 Paris feet, the distance which sound travels in the time of 1 second, and thus 1 pulse occupies a space of about 10⁷/10 Paris feet, that is, roughly twice the length of the pipe. Hence it is likely that the lengths of the pulses in the sounds of all open pipes are equal to twice the lengths of the pipes.^a

Furthermore, it is evident from book 2, prop. 47, corol., why sounds immediately cease when the motion of the sonorous body ceases, and why they are not heard for a longer time when we are very far distant from the sonorous bodies than when we are very close. Why sounds are very much increased in megaphones is also manifest from the principles set forth. For every reciprocal motion is increased at each reflection by the generating cause. And the motion is lost more slowly and is reflected more strongly in tubes that impede the expansion of sounds, and therefore is more increased by the new motion impressed at each reflection. And these are the major phenomena of sounds.

SECTION 9

The circular motion of fluids

Hypothesis
The resistance that arises from the friction [lit. lack of lubricity or slipperiness] of the parts of a fluid is, other things being equal, proportional to the velocity with which the parts of the fluid are separated from one another.

Proposition 51, Theorem 39
If an infinitely long solid cylinder revolves with a uniform motion in a uniform and infinite fluid about an axis given in position, and if the fluid is made to revolve by only the impulse of the cylinder, and if each part of the fluid perseveres uniformly in its motion, then I say that the periodic times of the parts of the fluid are as their distances from the axis of the cylinder.

Let AFL be the cylinder made to revolve uniformly about the axis S, and divide the fluid into innumerable concentric solid cylindrical orbs^a of the same thickness by the concentric circles BGM, CHN, DIO, EKP, . . . . Then, since the fluid is homogeneous, the impressions that contiguous orbs make upon one another will (by hypothesis) be as their relative displacements and the contiguous surfaces on which the impressions are made. If the impression upon some orb is greater or less on its concave side than on its convex side, the stronger impression will prevail and will either accelerate or retard the motion of the orb, according as it is directed the same way as its motion or the opposite way. Consequently, so that each orb may persevere uniformly in its motion, the impressions on each of the two sides should be equal and be made in opposite directions. Hence, since the impressions are as the contiguous surfaces and their relative velocities, the relative velocities will be inversely as the surfaces, that is, inversely as the distances of the surfaces from the axis. And the differences between the angular motions about the axis are as these relative velocities divided by the distances, or as the relative velocities directly and the distances inversely—that is, if the ratios are compounded, as the squares of the distances inversely. Therefore, if the perpendiculars Aa, Bb, Cc, Dd, Ee, . . . , inversely proportional to the squares of SA, SB, SC, SD, SE, . . . , are erected to each of the parts of the infinite straight line SABCDEQ and if a hyperbolic curve is understood to be drawn through the ends of the perpendiculars, then the sums of the differences, that is, the whole angular motions, will be as the corresponding sums of the lines Aa, Bb, Cc, Dd, Ee; that is, if, in order to make the medium uniformly fluid, the number of orbs is increased and their width decreased indefinitely, as the hyperbolic areas AaQ, BbQ, CcQ, DdQ, EeQ, . . . , corresponding to these sums. And the times, which are inversely proportional to the angular motions, will also be inversely proportional to these areas. The periodic time of any particle D, therefore, is inversely as the area DdQ, that is (by the known quadratures of curves), directly as the distance SD. Q.E.D.

COROLLARY 1. Hence the angular motions of the particles of the fluid are inversely as the distances of the particles from the axis of the cylinder, and the absolute velocities are equal.

COROLLARY 2. If the fluid is contained in a cylindrical vessel of an infinite length and contains another inner cylinder, and if both cylinders revolve about a common axis, and the times of the revolutions are as the semidiameters of the cylinders, and each part of the fluid perseveres in its motion, then the periodic times of the individual parts will be as their distances from the axis of the cylinders.

COROLLARY 3. If any common angular motion is added to, or taken away from, the cylinder and the fluid moving in this way, then, since the mutual friction of the parts of the fluid is not changed by this new motion, the motions of the parts with respect to one another will not be changed. For the relative velocities of the parts depend upon the friction. Any part will persevere in that motion which is not more accelerated than retarded by the friction on opposite sides in opposite directions.

COROLLARY 4. Hence, if all the angular motion of the outer cylinder is taken away from the whole system of the cylinders and fluid, the result will be the motion of the fluid in the cylinder at rest.

COROLLARY 5. Therefore, if, while the fluid and outer cylinder are at rest, the inner cylinder revolves uniformly, a circular motion will be communicated to the fluid and will be propagated little by little through the whole fluid, and it will not cease to be increased until the individual parts of the fluid acquire the motion defined in corol. 4.

COROLLARY 6. And since the fluid endeavors to propagate its own motion even further, its force will make the outer cylinder also revolve, unless that cylinder is forcibly held in place, and the motion of that cylinder will be accelerated until the periodic times of both cylinders become equal. But if the outer cylinder is forcibly held in place, it will endeavor to retard the motion of the fluid, and unless the inner cylinder conserves that motion by some force impressed from outside, the outer cylinder will cause the motion to cease little by little.

All of this can be tested in deep stagnant water.

Proposition 52, Theorem 40
If a solid sphere revolves with a uniform motion in a uniform and infinite fluid about an axis given in position, and if the fluid is made to revolve by only the impulse of this sphere, and if each part of the fluid perseveres uniformly in its motion, then I say that the periodic times of the parts of the fluid will be as the squares of the distances from the center of the sphere.

CASE 1. Let AFL be a sphere made to revolve uniformly about the axis S, and divide the fluid into innumerable concentric orbs^a of the same thickness by means of the concentric circles BGM, CHN, DIO, EKP, . . . . And imagine the orbs to be solid; then, since the fluid is homogeneous, the impressions that the contiguous orbs make upon one another will (by the hypothesis) be as their relative velocities and the contiguous surfaces on which the impressions are made. If the impression upon some orb is greater or less on the concave side than on the convex side, the stronger impression will prevail and will either accelerate or retard the velocity of the orb, according as it is directed the same way as the motion of the orb or the opposite way. Consequently, so that each orb may persevere uniformly in its motion, the impressions on each of the two sides will have to be equal and to be made in opposite directions. Hence, since the impressions are as the contiguous surfaces and their relative velocities, the relative velocities will be inversely as the surfaces, that is, inversely as the squares of the distances of the surfaces from the center. But the differences in the angular motions about the axis are as these relative velocities divided by the distances, or as the relative velocities directly and the distances inversely—that is, if the ratios are compounded, as the cubes of the distances inversely. Therefore, if to each of the parts of the infinite straight line SABCDEQ there are erected the perpendiculars Aa, Bb, Cc, Dd, Ee, . . . , inversely proportional to the cubes of SA, SB, SC, SD, SE, . . . , then the sums of the differences, that is, the whole angular motions, will be as the corresponding sums of the lines Aa, Bb, Cc, Dd, Ee—that is (if, to make the medium uniformly fluid, the number of orbs is increased and their width decreased indefinitely), as the hyperbolic areas AaQ, BbQ, CcQ, DdQ, EeQ, . . . , corresponding to these sums. And the periodic times, inversely proportional to the angular motions, will also be inversely proportional to these areas. Therefore the periodic time of any orb DIO is inversely as the area DdQ, that is (by the known methods of quadratures of curves), directly as the square of the distance SD. And this is what I wanted to prove in the first place.

CASE 2. From the center of the sphere draw as many infinite straight lines as possible which with the axis contain given angles exceeding one another by equal differences, and imagine the orbs to be cut into innumerable rings by the revolution of these straight lines about the axis; then each ring will have four rings contiguous to it, one inside, another outside, and two at the sides. Each ring cannot be urged equally and in opposite directions by the friction of the inner ring and of the outer ring, except in a motion made according to the law of case 1. This is evident from the proof of case 1. And therefore any series of rings proceeding straight from the sphere indefinitely will be moved in accordance with the law of case 1, except insofar as it is impeded by the friction of the rings at the sides. But in motion made according to this law the friction of the rings at the sides is nil, and thus it will not impede the motion from being made according to this law. If rings equally distant from the center revolved either more quickly or more slowly near the poles than near the ecliptic, the slower rings would be accelerated and the swifter would be retarded by mutual friction, and thus the periodic times would always tend toward equality, in accordance with the law of case 1. This friction, therefore, does not prevent the motion from being made according to the law of case 1, and therefore that law will hold good; that is, the periodic time of each of the rings will be as the square of its distance from the center of the sphere. This is what I wanted to prove in the second place.

CASE 3. Now let each ring be divided by transverse sections into innumerable particles constituting an absolutely and uniformly fluid substance; then, since these sections have no relation to the law of circular motion but contribute only to the constitution of the fluid, the circular motion will continue as before. As a result of this sectioning, all the minimally small rings either will not change the unevenness and the force of their mutual friction or will change them equally. Furthermore, since the proportion of the causes remains the same, the proportion of the effects—that is, the proportion of the motions and periodic times—will also remain the same. Q.E.D.

But since the circular motion, along with the centrifugal force arising from it, is greater at the ecliptic than at the poles, there must be some cause by which each of the particles is kept in its circle; otherwise the matter at the ecliptic would always recede from the center and move on the outside of the vortex to the poles, and return from there along the axis to the ecliptic with a continual circulation.

COROLLARY 1. Hence the angular motions of the parts of the fluid about the axis of the sphere are inversely as the squares of the distances from the center of the sphere, and the absolute velocities are inversely as those same squares divided by the distances from the axis.

COROLLARY 2. If a sphere, in a homogeneous and infinite fluid at rest, revolves with a uniform motion about an axis given in position, it will communicate a motion to the fluid like that of a vortex, and this motion will be propagated little by little without limit, and this motion will not cease to be accelerated in each part of the fluid until the periodic time of each of the parts is as the squares of the distances from the center of the sphere.

COROLLARY 3. Since the inner parts of a vortex, because of their greater velocity, rub and push the outer parts and continually communicate motion to them by this action, and since those outer parts simultaneously transfer the same quantity of motion to others still further out and by this action conserve the quantity of their motion completely unaltered, it is evident that the motion is continually transferred from the center to the circumference of the vortex and is absorbed in that limitless circumference. The matter between any two spherical surfaces concentric with the vortex will never be accelerated, because all of the motion it receives from the inner matter is continually transferred to the outer matter.

COROLLARY 4. Accordingly, for a vortex to conserve the same state of motion constantly, some active principle is required from which the sphere may always receive the same quantity of motion that it impresses on the matter of the vortex. Without such a principle, it is necessary for the sphere and the inner parts of the vortex, always propagating their motion to outer parts and not receiving any new motion, to slow down little by little and cease to be carried around.

COROLLARY 5. If a second sphere were to be immersed in this vortex at a certain distance from the center, and meanwhile by some force were to revolve constantly about an axis given in inclination, then the fluid would be drawn into a vortex by the motion of this sphere; and first this new and tiny vortex would revolve along with the sphere about the center of the first vortex, and meanwhile its motion would spread more widely and little by little would be propagated without limit, in the same way as the first vortex. And for the same reason that the sphere of the new vortex was drawn into the motion of the first vortex, the sphere of the first vortex would also be drawn into the motion of this new vortex, in such a way that the two spheres would revolve about some intermediate point and because of that circular motion would recede from each other unless constrained by some force. Afterward, if the continually impressed forces by which the spheres persevere in their motions were to cease, and everything were left to the laws of mechanics, the motion of the spheres would weaken little by little (for the reason assigned in corols. 3 and 4), and the vortices would at last be completely at rest.

COROLLARY 6. If several spheres in given places revolved continually with certain velocities around axes given in position, the same number of vortices, going on without limit, would be made. For all of the spheres, for the same reason that any one of them propagates its motion without limit, will also propagate their motions without limit, in such a way that each part of the infinite fluid is agitated by that motion which results from the actions of all the spheres. Hence the vortices will not be limited by fixed bounds but will little by little run into one another, and the spheres will be continually moved from their places by the actions of the vortices upon one another, as was explained in corol. 5; nor will they keep any fixed position with respect to one another, unless constrained by some force. And when those forces, which conserve the motions by being continually impressed upon the spheres, cease, the matter—for the reason assigned in corols. 3 and 4—will little by little come to rest and will no longer be made to move in vortices.

COROLLARY 7. If a homogeneous fluid is enclosed in a spherical vessel and is made to revolve in a vortex by the uniform rotation of a sphere placed in the center, and if the sphere and the vessel revolve in the same direction about the same axis, and if their periodic times are as the squares of the semidiameters, then the parts of the fluid will not persevere in their motions without acceleration and retardation until their periodic times are as the squares of the distances from the center of the vortex. No other constitution of a vortex can be stable.

COROLLARY 8. If the vessel, the enclosed fluid, and the sphere conserve this motion and additionally revolve with a common angular motion about any given axis, then, since the friction of the parts of the fluid upon one another is not changed by this new motion, the motions of the parts with respect to one another will not be changed. For the relative velocities of the parts with respect to one another depend upon friction. Any part will persevere in that motion by which the friction on one side does not retard it more than the friction on the other accelerates it.

COROLLARY 9. Hence, if the vessel is at rest, and if the motion of the sphere is given, the motion of the fluid will be given. For imagine that a plane passes through the axis of the sphere and revolves with an opposite motion, and suppose that the sum of the time of the revolution of the plane and the revolution of the sphere is to the time of the revolution of the sphere as the square of the semidiameter of the vessel to the square of the semidiameter of the sphere; then the periodic times of the parts of the fluid with respect to the plane will be as the squares of their distances from the center of the sphere.

COROLLARY 10. Accordingly, if the vessel moves with any velocity either about the same axis as the sphere or about some different axis, the motion of the fluid will be given. For if the angular motion of the vessel is taken away from the whole system, all the motions with respect to one another will remain the same as before, by corol. 8. And these motions will be given by corol. 9.

COROLLARY 11. If the vessel and the fluid are at rest, and if the sphere revolves with a uniform motion, then the motion will be propagated little by little through the whole fluid to the vessel, and the vessel will be driven around unless forcibly constrained, and the fluid and vessel will not cease to be accelerated until their periodic times are equal to the periodic times of the sphere. But if the vessel is constrained by some force or revolves with any continual and uniform motion, the medium will little by little come to the state of the motion defined in corols. 8, 9, and 10, nor will it ever persevere in any other state. But then if, when those forces cease by which the vessel and the sphere were revolving with fixed motions, the whole system is left to the laws of mechanics, the vessel and the sphere will act upon each other by means of the intervening fluid and will not cease to propagate their motions to each other through the fluid until their periodic times are equal and the whole system revolves together like one solid body.

Scholium
In the preceding propositions, I have been supposing the fluid to consist of matter which is uniform in density and fluidity. The fluid is such that a given sphere, set anywhere in it, would with a given motion in a given interval of time be able to propagate similar and equal motions, at distances always equal from itself. Indeed, matter endeavors by its circular motion to recede from the axis of a vortex and therefore presses all the further matter. From this pressure the friction of the parts becomes stronger and their separation from one another more difficult, and consequently the fluidity of the matter is decreased. Again, if there is any place where the parts of the fluid are thicker or larger, the fluidity will be less there, because the surfaces separating the parts from one another are fewer. In cases of this sort, I suppose the deficiency in fluidity to be supplied either by the slipperiness of the parts or by their pliancy or by some other condition. If this does not happen, the matter will cohere more and will be more sluggish where it is less fluid, and thus will receive motion more slowly and will propagate it further than according to the ratio assigned above. If the shape of the vessel is not spherical, the particles will move in paths which are not circular but correspond to the shape of the vessel, and the periodic times will be very nearly as the squares of the mean distances from the center. In the parts between the center and the circumference where the spaces are wider, the motions will be slower, and where the spaces are narrower the motions will be swifter, and yet the swifter particles will not seek the circumference. For they will describe less-curved arcs, and the endeavor to recede from the center will not be less decreased by the decrement of this curvature than it will be increased by the increment of the velocity. In going from the narrower spaces into the wider, they will recede a little further from the center, but they will be retarded by this receding, and afterward in approaching the narrower spaces from the wider ones they will be accelerated, and thus each of the particles will forever alternately be retarded and accelerated. All of this will be so in a rigid vessel. For the constitution of vortices in an infinite fluid can be found by corol. 6 of this proposition.

Moreover, in this proposition I have tried to investigate the properties of vortices in order to test whether the celestial phenomena could be explained in any way by vortices. For it is a phenomenon that the periodic times of the secondary planets that revolve about Jupiter are as the ³/2 powers of the distances from the center of Jupiter; and the same rule applies to the planets that revolve about the sun. Moreover, these rules apply to both the primary and the secondary planets very exactly, as far as astronomical observations have shown up to now. And thus if those planets are carried along by vortices revolving about Jupiter and the sun, the vortices will also have to revolve according to the same law. But the periodic times of the parts of a vortex turned out to be in the squared ratio of the distances from the center of motion, and that ratio cannot be decreased and reduced to the ³/2 power, unless either the matter of the vortex is the more fluid the further it is from the center, or the resistance arising from a deficiency in the slipperiness of the parts of the fluid (as a result of the increased velocity by which the parts of the fluid are separated from one another) is increased in a greater ratio than the ratio in which the velocity is increased. Yet neither of these seems reasonable. The thicker and less-fluid parts, if they are not heavy toward the center, will seek the circumference; and although—for the sake of the proofs—I proposed at the beginning of this section a hypothesis in which the resistance would be proportional to the velocity, it is nevertheless likely that the resistance is in a lesser ratio than that of the velocity. If this is conceded, then the periodic times of the parts of a vortex will be in a ratio greater than the squared ratio of the distances from its center. But if vortices (as is the opinion of some) move more quickly near the center, then more slowly up to a certain limit, then again more quickly near the circumference, certainly neither the ³/2 power nor any other fixed and determinate ratio can hold. It is therefore up to philosophers to see how that phenomenon of the ³/2 power can be explained by vortices.

Proposition 53, Theorem 41
Bodies that are carried along in a vortex and return in the same orbit have the same density as the vortex and move according to the same law as the parts of the vortex with respect to velocity and direction.

For if some tiny part of the vortex is composed of particles or physical points which preserve a given situation with respect to one another and is supposed to be frozen, then this part will move according to the same law as before, since it is not changed with respect to its density, or its inherent force or figure. And conversely, if a frozen and solid part of the vortex has the same density as the rest of the vortex and is resolved into a fluid, this part will move according to the same law as before, except insofar as its particles, which have now become fluid, move with respect to one another. Therefore, the motion of the particles with respect to one another may be ignored as having no relevance to the progressive motion of the whole, and the motion of the whole will be the same as before. But this motion will be the same as the motion of other parts of the vortex that are equally distant from the center, because the solid resolved into a fluid becomes a part of the vortex similar in every way to the other parts. Therefore, if a solid is of the same density as the matter of the vortex, it will move with the same motion as the parts of the vortex and will be relatively at rest in the immediately surrounding matter. But if the solid is denser, it will now endeavor to recede from the center of the vortex more than before; and thus, overcoming that force of the vortex by which it was formerly kept in its orbit as if set in equilibrium, it will recede from the center and in revolving will describe a spiral and will no longer return into the same orbit. And by the same argument, if the solid is rarer, it will approach the center. Therefore, the solid will not return into the same orbit unless it is of the same density as the fluid. And it has been shown that in this case the solid would revolve according to the same law as the parts of the fluid that are equally distant from the center of the vortex. Q.E.D.

COROLLARY 1. Therefore a solid that revolves in a vortex and always returns into the same orbit is relatively at rest in the fluid in which it is immersed.

COROLLARY 2. And if the vortex is of a uniform density, the same body can revolve at any distance from the center of the vortex.

Scholium
Hence it is clear that the planets are not carried along by corporeal vortices. For the planets, which—according to the Copernican hypothesis—move about the sun, revolve in ellipses having a focus in the sun, and by radii drawn to the sun describe areas proportional to the times. But the parts of a vortex cannot revolve with such a motion. Let AD, BE, and CF designate three orbits described about the sun S, of which let the outermost CF be a circle concentric with the sun, and let A and B be the aphelia of the two inner ones, and D and E their perihelia. Therefore, a body that revolves in the orbit CF, describing areas proportional to the times by a radius drawn to the sun, will move with a uniform motion. And a body that revolves in the orbit BE will, according to the laws of astronomy, move more slowly in the aphelion B and more swiftly in the perihelion E, although according to the laws of mechanics the matter of the vortex ought to move more swiftly in the narrower space between A and C than in the wider space between D and F, that is, more swiftly in the aphelion than in the perihelion. These two statements are contradictory. Thus in the beginning of the sign of Virgo, where the aphelion of Mars now is, the distance between the orbits of Mars and Venus is to the distance between these orbits in the beginning of the sign of Pisces as roughly 3 to 2, and therefore the matter of the vortex between these orbits in the beginning of Pisces must move more swiftly than in the beginning of Virgo in the ratio of 3 to 2. For the narrower the space through which a given quantity of matter passes in the given time of one revolution, the greater the velocity with which it must pass. Therefore, if the earth, relatively at rest in this celestial matter, were carried by it and revolved along with it about the sun, its velocity in the beginning of Pisces would be to its velocity in the beginning of Virgo as 3 to 2. Hence the apparent daily motion of the sun in the beginning of Virgo would be greater than 70 minutes, and in the beginning of Pisces less than 48 minutes, although (as experience bears witness) the apparent motion of the sun is greater in the beginning of Pisces than in the beginning of Virgo, and thus the earth is swifter in the beginning of Virgo than in the beginning of Pisces. Therefore the hypothesis of vortices can in no way be reconciled with astronomical phenomena and serves less to clarify the celestial motions than to obscure them. But how those motions are performed in free spaces without vortices can be understood from book 1 and will now be shown more fully in book 3 on the system of the world.