Principia Mathematica (Motte Translation, 1848) — part 22 of 45

Cor. Whence neither will a motion of the parts of the fluid among themselves be changed by a pressure communicated to the external super- ficies, except so far as either the figure of the superficies may be somewhere altered, or that all the parts of the fluid, by pressing one another more in- tensely or remissly, may slide with more or less difficulty among them-

PROPOSITION XX. THEOREM XV.

Jf all the parts of a spherical fluid, homogeneous at equal distances from the centre, lying on a spherical concentric bottom, gravitate towards the centre of the whole, the bottom ivill sustain the weight of a cylin- der, whose base is equal to the superficies of the bottom, and tvhose al- titude is the same with that of the incumbent fluid. Let DHM be the superficies of the bottom, and AEI the upper super- ficies of the fluid. Let the fluid be distinguished into concentric orbs of equal thickness, by the innumerable spherical superficies BFK, CGL ; and

Sec. v.] of natural philosophy. 295

conceive the force of gravity to act only in tlie upper superficies of every orb, and the actions to be equal on the equal parts of all the su- perficies. Therefore the upper superficies AE is pressed by the single force of its own grav- ity, by which all the parts of the upper orb, and the second superficies BFK, will (by Prop. XIX); according to its measure, be equally pressed. The second superficies BFK is pressed likewise by the force of its own

gravity, which, added to the former force, ""' •"'

makes the pressure double. The third superficies CGL is, according to its measure, acted on by this pressure and the force of its own gravity besides, which makes its pressure triple. And in like manner the fourth superfi- cies receives a quadruple pressure, the fifth superficies a quintuple, and so on. Therefore the pressure acting on every superficies is not as the solid quantity of the incumbent fluid, but as the number of the orbs reaching to the upper surface of the fluid ; and is equal to the gravity of the lowest orb multiplied by the number of orbs : that is, to the gravity of a solid whose ultimate ratio to the cylinder above-mentioned (when the number of the orbs is increased and their thickness diminished, ad infinitum, so that the action of gravity from the lowest superficies to the uppermost may be- come continued) is the ratio of equality. Therefore the lowest superficies sustains the weight of the cylinder above determined. Q,.E.D. And by a like reasoning the Proposition will be evident, where the gravity of the fluid decreases in any assigned ratio of the distance from the centre, and also where the fluid is more rare above and denser below. Q,.E.D.

Cor. 1. Therefore the bottom is not pressed by the whole weight of the incumbent fluid, but only sustains that part of it which is described in the Proposition ; the rest of the weight being sustained archwise by the spheri- cal figure of the fluid.

Cor. 2. The quantity of the pressure is the same always at equal dis- tances from the centre, whether the superficies pressed be parallel to the horizon, or perpendicular, or oblique ; or whether the fluid, continued up- wards from the compressed superficies, rises perpendicularly in a rectilinear direction, or creeps obliquely through crooked cavities and canals, whether those passages be regular or irregular, wide or narrow. That the pressure is not altered by any of these circumstances, may be collected by applying the demonstration of this Theorem to the several cases of fluids.

Cor. .3. From the same demonstration it may also be collected (by Prop. XIX), that the parts of a heavy fluid acquire no motion among themselves by the pressure of the incumbent weight, except that motion which arises; from condensation.

296 THE MATHEMATICAL PRINCIPLES [BCOK IL

Cor. 4. And therefore if another body of the same specific gravity, in- capable of condensation, be immersed in this fluid, it will acquire no mo- tion by the pressure of the incumbent weight: it will neither descend nor ascend, nor change its figure. If it be spherical, it will remain so, notwith- standing the pressure ; if it be square, it will remain square ; and that, whether it be soft or fluid ; whether it swims freely in the fluid, or lies at the bottom. For any internal part of a fluid is in the same state with the submersed body ; and the case of all submersed bodies that have the same magnitude, figure, and specific gravity, is alike. If a submersed body, re- taining its weight, should dissolve and put on the form of a fluid, this body, if before it would have ascended, descended, or from any pressure as- sume a new figure, would now likewise ascend, descend, or put on a new figure; and that, because its gravity and the other causes of its motion remain. But (by Case 5, Prop. XtX) it would now be at rest, and retain its figure. Therefore also in the former case.

Cor. 5. Therefore a body that is specifically heavier than a fluid con- tiguous to it will sink ; and that which is specifically lighter will ascend, and attain so much motion and change of figure as that excess or defect of gravity is able to produce. For that excess or defect is the same thing as an impulse, by which a body, otherwise i7i equilibrio with the parts of the fluid, is acted on ; and may be compared with the excess or defect of a weight in one of the scales of a balance.

Cor. 6. Therefore bodies placed in fluids have a twofold gravity ; the one true and absolute, the other apparent, vulgar, and comparative. Ab- solute gravity is the whole force with which the body tends downwards ; relative and vulgar gravity is the excess of gravity with which the body tends downwards more than the ambient fluid. By the first kind of grav- ity the parts of all fluids and bodies gravitate in their proper places ; and therefore their weights taken together compose the weight of the whole. For the whole taken together is heavy, as may be experienced in vessels full of liquor ; and the weight of the whole is equal to the weights of all the parts, and is therefore composed of them. By the other kind of grav- ity bodies do not gravitate in their places ; that is, compared with one another, they do not preponderate, but, hindering one another's endeavours to descend, remain in their proper places, as if they were not heavy. Those things which are in the air, and do not preponderate, are commonly looked on as not heavy. Those which do preponderate are commonly reckoned heavy, in as much as they are not sustained by the weight of the air. The common weights are nothing else but the excess of the true weights above the weight of the air. Hence also, vulgarly, those things are called light which are less heavy, and, by yielding to the preponderating air, mount upwards. But these are only comparatively light, and not truly so, because they descend in vacuo. Thus, in water, bodies which, by their greater or

Sec. Y.] of natural philosophy. 297

less gravity, descend or ascend^ are comparatively and apparently heavy or light : and their comparative and apparent gravity or levity is the excess or defect by which their true gravity either exceeds the gravity of the water or is exceeded by it. But those things which neither by preponder- ating descend, nor, by yielding to the preponderating fluid, ascend, although by their true weight they do increase the weight of the whole, yet com- paratively, and in the sense of the vulgar, they do not gravitate in the wa- ter. For these cases are alike demonstrated.

Cor. 7. These things which have been demonstrated concerning gravity take place in any other centripetal forces.

Cor. 8. Therefore if the medium in which any body moves be acted on either by its own gravity, or by any other centripetal force, and the body be urged more powerfully by the same force ; the difference of the forces is that very motive force, which, in the foregoing Propositions, I have con- sidered as a centripetal force. But if the body be more lightly urged by that force, the difference of the forces becomes a centrifugal force, and is to be considered as such.

Cor. 9. But since fluids by pressing the included bodies do not change their external figures, it appears also (by Cor. Prop. XIX) that they will not change the situation of their internal parts in relation to one another ; and therefore if animals were immersed therein, and that all sen- sation did arise from the motion of their parts, the fluid will neither hurt the immersed bodies, nor excite any sensation, unless so far as those bodies may be condensed by the com.pression. And the case is the same of any system of bodies encompassed with a compressing fluid. All the parts of the system will be agitated with the same motions as if they were placed in a vacuum, and would only retain their comparative gravity ; unless so far as the fluid may somewhat resist their motions, or be requisite to con- glutinate them by compression.

PROPOSITION XXI. THEOREM XYI.

Let the density of any fl,uid he proportional to the compres'sion, a?id its parts he attracted dotvnivards hy a centripetal force reciprocally pro- portional to the distances from the centre: I say, that, if those dis- tances he taken continually proportional, the densities of the fluid at the same distances will he also continually proportional. Let ATY denote the spherical bottoin of the fluid, S the centre, SA, SB, SC, SD, SE, SF, &c., distances continually proportional. Erect the per- pendiculars AH, BI, CK, DL, EM, FN, <fcc., which shall be as the densi- ties of the medium in the places A, B, C, D, E, F ; and the specific grav-

A TT RT Pl^ ities in those places will be as -r-^, ^, -p^, &c., or, which is all one, as

298

THE MATHEMATICAL PRINCIPLES

[Book IL

AH BI OK .

AB ' BCJ' CD' Suppose, first, these gravities to be uniformly continued

from A to B, from B to C, from C to D, &c., the decrements in the points Bj C, D, &c., being taken by steps. And these gravi- ties drawn into the altitudes AB, BC, CD, (fcc, will give the pressures AH, BI, CK, &c., by which the bot- tom ATY is acted on (by Theor. XV). Therefore the particle A sustains all the pressures AH, BI, CK, DL, ~*^ (fee, proceeding in wfinitum ; and the particle B sus- tains the pressures of all but the first AH ; and the par- ticle C all but the two first AH, BI ; and so on : and therefore the density AH of the first particle A is to the density BI of the second particle B as the sum of all AH + BI + CK + DL, in infinitum^ to the sum of all BI + CK + DL, &c. And BI the density of the second particle B is to CK the density of the third C, as the sum of all BI + CK + DL, &c., to the sum of all CK + DL, (fee. Therefore these sums are proportional to their differences AH, BI, CK, (fee, aad therefore continually propor- tional (by Lem. 1 of this Book) ; and therefore the diiferences AH, BI, CK, (fee, proportional to the sums, are also continually proportional. Wherefore since the densities in the places A, B, C, (fee, are as AH, BI, CK, (fee, they will also be continually proportional. Proceed intermis- sively, and, ex ccquo, at the distances SA, SC, SE, continually proportional, the densities AH, CK, EM will be continually proportional. And by the same reasoning, at any distances SA, SD, SG, continually proportional, the densities AH, DL, GO, will be continually proportional. Let now the points A, B, C, D, E, (fee, coincide, so that the progression of the specific gravities from the bottom A to the top of the fluid may be made continual : and at any distances SA, SD, SG, continually proportional, the densities AH, DL, GO, being all along continually proportional, will still remain, continually proportional. Q.E.D.

Cor. Hence if the density of the fluid in two places, as A and E, be given, its density in any other place Q may be collected. With the centre S, and the rectan- gular asymptotes SQ, SX, describe an hyperbola cut- ting the perpendiculars AH, EM, Q,T in a, e, and q, ^ as also the perpendiculars HX, MY, TZ, let fall upon the asypmtote SX, in h, m, and t. Make the area Ym^Z to the given area YfnhJL as the given area Ee^Q, to the given area EeaA ; and the line Zt produced will cut off the line Q.T proportional to the density. For if the lines SA, SE, SQ are continually proportional, the areas Ee^Q,, Ee«A will be equal, and thence

Sec. v.]

OF NATURAL PHILOSOPHY.

299

the areas YmtZ, XhmY, proportional to them, will be also equal ; and the lines SX, SY, SZ, that is, AH,. EM, Q,T continually proportional, as they ought to be. And if the lines SA, SE, SQ,, obtain any other order in the series of continued proportionals, the lines AH, EM, Q,T, because of the proportional hyperbolic areas, will obtain the same order in another series of quantities continually proportional.

PROPOSITION XXII. THEOREM XYII.

Let the density of any fluid be proportional to the co7npression, and its parts be attracted downiuards by a gravitation reciprocally propor- tional to the squares of the distances from the centre : I say, that if the distances be taken in harmonic progression, the densities of the fluid at those distances will be in a geo7netrical progression. Let S denote the centre, and SA,

SB, SO, SD, SE, the distances in ^

geometrical progression. Erect the ^

perpendiculars AH, BI, CK, &c.,

which shall be as the densities of ^

the fluid in the places A, B, C, D, ^

E, cfec.,_ and the specific gravities

thereof in those places will be as

AH BI CK ,

S57' SB^' SC2"' Suppose these

gravities to be uniformly continued, the first from A to B, the second from

B to C, the third from C to D, &c. And these drawn into the altitudes

AB, BC, CD, DE, &c.; or, vi^hich is the same thing, into the distances SA,

ATT RT PT^

SB, SC, (fee, proportional to those altitudes, will give -5-7-, ^^, -^7^, &c.,

oA k5o bO

the exponents of the pressures. Therefore since the densities are as the sums of those pressures, the differences AH — BI; BI — CK, &c., of the

ATT r>T nJT

densities will be as the differences of those sums ^tj 00? 'ar^? ^^- With

loA bJD bO

the centre S, and the asymptotes SA, S:r, describe any hyperbola, cutting the perpendiculars AH, BI, OK, &c., in «, 6, c, &c., and the perpendicu- lars H^, lu, K.Wj let fall upon the asymptote S.r, in A, i, k ; and the dif-

AH BI

ferences of the densities tu, uw, (fee, will be as -^-r, 7^^. &c. And the

rectangles tu X th, uw X ui, &c., or tp, uq^ <fec., as that is, as Aa, B6, &c.

SA' SB' AH X ^A BI X ui

&c.,

SA ' SB

For, by the nature of the hyperbola, SA is to AH

or ^t as th to Aa, and therefore — ^ — is equal to Aa. And, by a like

300 THE MATHEMATICAL PRINCIPLES [BoOK II.

reasoning, — ^^^^ — is equal to Bb, &c. But Aa. Bb, Cc, &c., are continu- ally proportional, and therefore proportional to their differences Aa — Bb, Bb — Cc, &c., therefore the rectangles tp, uq, «fee, are proportional to those differences ; as also the sums of the rectangles tp + uq, or tp -- uq -- wr to the sums of the differences Aa — Cc or Aa — T>d. Suppose several of these terms, and the sum of all the differences, as Aa — F/, will be pro- portional to the sum of all the rectangles, as zthn. Increase the number of terms, and diminish the distances of the points A, B, C, &c., in wfini- turn, and those rectangles will become equal to the hyperbolic area zthuy and therefore the difference Aa — P/ is proportional to this area. Take now any distances, as SA, SD, SF, in harmonic progression, and the dif- ferences Aa — T)d, jyd — F/ will be equal ; and therefore the areas thlx, xlnz, proportional to those differences will be equal among themselves, and the densities S^, S:r, S^^, that is, AH. DL, FN, continually proportional.

aE.D.

CcR. Hence if any two densities of the fluid, as AH and BI, be given, the area thiu, answering to their difference tu, will be given ; and thence the density FN will be found at any height SF, by taking the area thnz to that given area thiu as the difference Aa — Yf to the difference Aa — Bb.

SCHOLIUM.

By a like reasoning it may be proved, that if the gravity of the particles

of a fluid be diminished in a triplicate ratio of the distances from the centre ;

and the reciprocals of the squares of the distances SA, SB, SC, &c., (namely,

SA^ SA^ SA^

^-— , ^-5^, ^T^) be taken in an arithmetical progression, the densities AH,

BI, CK, &c., will be in a geometrical progression. And if the gravity be diminished in a quadruplicate ratio of the distances, and the reciprocals of

. , .. , SA^ SA* SA^ , _

the cubes of the distances (as 5-7-7, ^oi? oT^j ocg.,) be taken m arithmeti-

oA*^ oJj"* oU'^

cal progression, the densities AH, BI, CK, <fcc., will be in geometrical pro- gression. And so m infinitum. Again ; if the gravity of the particles of the fluid be the same at all distances, and the distances be in arithmetical progression, the densities will be in a geometrical progression as Dr. Hal- ley has found. If the gravity be as the distance, and the squares of the distances be in arithmetical progression, the densities will be in geometri- cal progression. And so in infinitum. These things will be so, when the density of the fluid condensed by compression is as the force of compres- sion ; or, whicli is the same thing, when the space possessed by the fluid is reciprocally as this force. Other laws of condensation may be supposed, as that the cube of the compressing force may be as the biquadrate of the

Sec. Y.] of natural philosophy. 301

density ; or the triplicate ratio of the force the same with the quadruplicate ratio of the density : in which case, if the gravity be reciprocally as the square of the distance from the centre, the density will be reciprocally as the cube of the distance. Suppose that the cube of the compressing force be as the quadrato-cube of the density ; dfnd if the gravity be reciprocally as the square of the distance, the density will be reciprocally in a sesqui- plicate ratio of the distance. Suppose the compressing force to be in a du- plicate ratio of the density, and the gravity reciprocally in a duplicate ra- tio of the distance, and the density will be reciprocally as the distance. To run over all the cases that might be oifered would be tedious. But as to our own air, this is certain from experiment, that its density is either accurately, or very nearly at least, as the compressing force ; and therefore the density of the air in the atmosphere of the earth is as the weight of the whole incumbent air, that is, as the height of the mercury in the ba- rometer.

PROPOSITION XXIIL THEOREM XVIII.

If a fluid be composed of particles mutually flying each other ^ and the density be as the compression, the centrifugal forces of the particles will be reciprocally proportional to the distances of their centres. And, vice ^Qi^% particles flying each other, with forces that are reciprocally proportional to the distances of their ceiitres, compose an elastic fluid, whose density is as the compressioji. Let the fluid be supposed to be included in a cubic space ACE, and then to be reduced by compression into a lesser cubic space ace ; and the distances of the par- ticles retaining a like situation with respect to each other in both the spaces, will be as the sides AB, ab of the cubes ; and the densities of the mediums will be re- ciprocally as the containing spaces AB^, ab''. In the plane side of the greater cube ABOD take the square

DP equal to the plane side db of the lesser cube : and, "J ^^

by the supposition, the pressure with which the square ^

DP urges the inclosed fluid will be to the pressure with ^^ Y

which that square db urges the inclosed fluid as the densities of the me- diums are to each other, that is, asaS^ to AB^. But the pressure with which the square DB urges the included fluid is to the pressure with which the square DP urges the same fluid as the square DB to the square DP, that is, as AB^ to ab^. Therefore, ex cequo, the pressure with which the square DB urges the fluid is to the pressure with which the square db urges the fluid as ab to AB. Let the planes FGU, fgh, be drawn through the middles of the two cubes, and divide the fluid into two parts. These parts will press each other mutually with the same forces with which they

302 THE MATHEMATICAL PRINCIPLES [BoOK IL

are themselves pressed by the planes AC, ac, that is, in the proportion of ah to AB : and therefore the centrifugal forces by which these pressures are sustained are in the same ratio. The number of the particles being equal, and the situation alike, in both cubes, the forces which all the par- ticles exert, according to the planes FGH,/§^A, upon all, are as the forces which each exerts on each. Therefore the forces which each exerts on each, according to the plane FG-H in the greater cube, are to the forces which each exerts on each, according to the plane /^A in the lesser cube, as ab to AB, that is, reciprocally as the distances of the particles from each other. Q.E.D.

And, vice versa, if the forces of the single particles are reciprocally as the distances, that is, reciprocally as the sides of the cubes AB, ab ; the sums of the forces will be in the same ratio, and the pressures of the sides DB. db as the sums of the forces ; and the pressure of the square DP to the pressure of the side DB as ah"^ to AB ^. And, ex oiquo, the pressure of the square DP to the pressure of the side db as ab ^ to AB ^ ; that is, the force of compression in the one to the force of compression in the other as the density in the former to the density in the latter. Q,.E.D.

SCHOLIUM.

By a like reasoning, if the centrifugal forces of the particles are recip- rocally in the duplicate ratio of the distances between the centres, the cubes of the compressing forces will be as the biquadrates of the densities. If the centrifugal forces be reciprocally in the triplicate or quadruplicate ratio of the distances, the cubes of the compressing forces will be as the quadrato- cubes, or cubo-cubes of the densities. And universally, if D be put for the distance, and E for the density of the compressed fluid, and the centrifugal forces be reciprocally as any power D" of the distance, whose index is the number n, the compressing forces will be as the cube roots of the power E" -f- ^ whose index is the number ?i + 2 ; and the contrary. All these things are to be understood of particles whose centrifugal forces terminate in those particles that are next them, or are diffused not much further. We have an example of this in magnetical bodies. Their attractive vir- tue is terminated nearly in bodies of their own kind that are next them. The virtue of the magnet is contracted by the interposition of an iron plate, and is almost terminated at it : for bodies further off are not attracted by the magnet so much as by the iron plate. If in this manner particles repel others of their own kind that lie next them, but do not exert their virtue on the more remote, particles of this kind will compose such fluids as are treated of in this Proposition. If the virtue of any particle diffuse itself every way in infinitiim, there will be required a greater force to produce an equal condensation of a greater quantity of the fluid. But whether

Sec. YL] of natural philosophy. 303

elastic fluids do really consist of particles so repelling each other, is a phy- sical question. We have here demonstrated mathematically the property of fluids consisting of particles of this kind, that hence philosophers may take occasion to discuss that question.

SECTION VI.

Of the motion and resistance of funependulous bodies.

PROPOSITION XXIV. THEOREM XIX.

The quantities of matter in fimependidous bodies, lohose centres of oscil- lation are equally distant from the centre of suspension, are hi a ratio compounded of the ratio of the lueights and the duplicate ratio of the tim,es of the oscillations in vacuo.

For the velocity which a given force can generate in a given matter in a given time is as the force and the time directly, and the matter inversely. The greater the force or the time is, or the less the matter, the greater ve- locity will be generated. This is manifest from the second Law of Mo- tion. Now if pendulums are of the same length, the motive forces in places equally distant from the perpendicular are as the weights : and therefore if two bodies by oscillating describe equal arcs, and those arcs are divided into equal parts ; since the times in which the bodies describe each of the correspondent parts of the arcs are as the times of the whole oscillations, the velocities in the correspondent parts of the oscillations will be to each other as the motive forces and the whole times of the oscillations directly, and the quantities of matter reciprocally : and therefore the quantities of matter are as the forces and the times of the oscillations directly and the velocities reciprocally. But the velocities reciprocally are as the times, and therefore the times directly and the velocities reciprocally are as the squares of the times ; and therefore the quantities of matter are as the mo- tive forces and the squares of the times, that is, as the weights and the squares of the times. Q..E.D.

CoR. 1. Therefore if the times are equal, the quantities of matter in each of the bodies are as the weights.

CoE. 2. If the weights are equal, the quantities of matter will be as the squares of the times.

CoR. 3. If the quantities of matter are equal, the weights will be recip- rocally as the squares of the times.

CoR. 4. Whence since the squares of the times, cceteris paribus, are as the lengths of the pendulums, therefore if both the times and quantities of matter are equal, the weights will be as the lengths of the pendulums.

304 THE MATHEMATICAL PRINCIPLES [BoOK 11

Cor. 5. AbcI universally, the quantity of matter in the pendulous body is as the weight and the square of the time directly, and the length of the pendulum inversely.

Cor. 6. But in a non-resisting medium, the quantity of matter in the pendulous body is as the comparative weight and the square of the time directly, and the length of the pendulum inversely. For the comparative weight is the motive force of the body in any heavy medium, as was shewn above ; and therefore does the same thing in such a non-resisting medium as the absolute weight does in a vacuum.

Cor. 7. And hence appears a method both of comparing bodies one among another, as to the quantity of matter in each ; and of comparing the weights of the same body in different places, to know the variation of its gravity. And by experiments made with the greatest accuracy, I have always found the quantity of matter in bodies to be proportional to their weight.

PROPOSITION XXY. THEOREM XX.

Funependuloiis bodies that are, in any medium, resisted in the ratio of the m,ome?its of time, and funependidous bodies that move in a non- resisting Tnedium of the same specific gravity, perform their oscilla- tio?is in a cycloid i?i the same time, and describe proportional parts of arcs together.

Let AB be an arc of a cycloid, which a body D, by vibrating in a non-re- sisting medium, shall describe in any time. Bisect that arc in C, so that C may be the lowest point thereof ; and the accelerative force with which the body is urged in any place D, or d or E, will be as the length of the arc CD, ^ ^ or Cd, or CE. Let that force be ex-

pressed by that same arc ; and since the resistance is as the moment of the time, and therefore given, let it be expressed by the given part CO of the cycloidal arc, and take the arc Od in the same ratio to the arc CD that the arc OB has to the arc CB : and the force with which the body in d is urged in a resisting medium, being the excess of the force Cd above the resistance CO, will be expressed by the arc Od, and will therefore be to the force with which the body D is urged in a non-resisting medium in the place D, as the arc Od to the arc CD ; and therefore also in the place B, as the arc OB to the arc CB. Therefore if two bodies D, d go from the place B; and are urged by these forces ; since the forces at the beginning are as the arc CB and OB, the first velocities and arcs first described will be in the same ratio. Let those arcs be BD and Be?, and the remaining arcs

Sec. VL] of natural philosophy. 305

CD, Odj will be in the same ratio. Therefore the forces, being propor- tional to those arcs CD, Od, will remain in the same ratio as at the be- ginning, and therefore the bodies will continue describing together arcs in the same ratio. Therefore the forces and velocities and the remaining arcs CD. Odj will be always as the whole arcs CB, OB, and therefore those re- maining arcs will be described together. Therefore the two bodies D and d will arrive together at the places C and O ; that which moves in the non-resisting medium, at the place C, and the other, in the resisting me- dium, at the place O. Now since the velocities in C and O are as the arcs CB, OB, the arcs which the bodies describe when they go farther will be in the same ratio. Let those arcs be CE and Oe. The force with which the body D in a non-resisting medium is retarded in E is as CE, and the force with which the body d in the resisting medium is retarded in e, is as the sum of the force Ce and the resistance CO, that is, as Oe; and there- fore the forces with which the bodies are retarded are as the arcs CB, OB, proportional to the arcs CE, Oe ; and therefore the velocities, retarded in that given ratio, remain in the same given ratio. Therefore the velocities and the arcs described with those velocities are always to each other in that given ratio of the arcs CB and OB ; and therefore if the entire arcs AB, ciB are taken in the same ratio, the bodies D and d will describe those arcs together, and in the places A and a will lose all their motion together. Therefore the whole oscillations are isochronal, or are performed in equal times; and any parts of the arcs, as BD, Be?, or BE, Be, that are described together, are proportional to the whole arcs BA, Ba. Q,.E.D.

Cor. Therefore the swiftest motion in a resisting medium does not fall upon the lowest point C, but is found in that point O, in which the whole arc described Ba is bisected. And the body, proceeding from thence to a, is retarded at the same rate with which it was accelerated before in its de- scent from B to O.

PROPOSITION XXVI. THEOREM XXL

Fimependulous bodies, that are resisted in the ratio of the velocity, have their oscillations in a cycloid isochronal. For if two bodies, equally distant from their centres of suspension, de- scribe, in oscillating, unequal arcs, and the velocities in the correspondent parts of the arcs be to each other as the whole arcs ; the resistances, pro- portional to the velocities, will be also to each other as the same arcs. Therefore if these resistances be subducted from or added to the motive forces arising from gravity which are as the same arcs, the differences or sums will be to each other in the same ratio of the arcs ; and since the in- crements and decrements of the velocities are as these differences or sums, the velocities will be always as the whole arcs ; therefore if the velocities axe in any one case as the whole arcs, they will remain always in the same

20

/306 THE MATHEMATICAL PRINCIPLES [BoOX II.

ratio. But at the beginning of the motion, when the bodies begin to de- scend and describe those arcs, the forces, which at that time are proportional to the arcs, will generate velocities proportional to the arcs. Therefore the velocities will be always as the whole arcs to be described, and there- fore those arcs will be described in the same time. Q.E.D.

PROPOSITION XXVII. THEOREM XXII.

If funependtdous bodies are resisted in the duplicate ratio of their velocities, the differences betioeen the times of the oscillations in a re- sisting medium, and the times of the oscillations in a non-resisting medium of the same specific gravity, ivill be prop)ortional to the arcs described in oscillating nearly.

For let equal pendulums in a re- sisting medium describe the unequal arcs A, B ; and the resistance of the _s body in the arc A will be to the resist- ance of the body in the correspondent ^-B part of the arc B in the duplicate ra- tio of the velocities, that is, as AA to BB nearly. If the resistance in the ^ ^ arc B were to the resistance in the arc

A as AB to AA, the times in the arcs A and B would be equal (by the last Prop.) Therefore the resistance AA in the arc A, or AB in the arc B, causes the excess of the time in the arc A above the time in a non-resisting medium ; and the resistance BB causes the excess of the time in the arc B above the time in a non-resisting medium. But those excesses are as the efficient forces AB and BB nearly, that is, as the arcs A and B. Q,.E.D.

Cor. 1. Hence from the times of the oscillations in unequal arcs in a resisting medium, may be known the times of the oscillations in a non- re- sisting medium of the same specific gravity. For the diflference of the times will be to the excess of the time in the lesser arc above the time in a non-resisting medium as the difference of the arcs to the lesser arc.