a lesser cubic space ace ; and the distances of the par- Ftides retaining a like situation with respect to eachother in both the spaces, will be as the sides AB, ab ofthe cubes ; and the densities of the mediums will be reciprocally as the containing spaces AB 3, ab 3. In theplane side of the greater cube ABCD take the squareDP equal to the plane side db of the lesser cube: and,by the supposition, the pressure with which the squareDP urges the inclosed fluid will be to the pressure withwhich that square db urges the inclosed fluid as the densities of the mediums are to each other, that is, asa/>3 to AB 3. But the pressure withwhich the square DB urges the included fluid is to the pressure with whichthe square DP urges the same fluid as the square DB to the square DP,that is, as AB2to ab z. Therefore, ex cequo, the pressure with which thesquare DB urges the fluid is to the pressure with which the square dburges the fluid as ab to AB. Let the planes FGH,/V?, U drawn throughthe middles of the two cubes, and divide the fluid into tw^/ parts, Theseparts will press each other mutually with the same forces with which theyATHE MATHEMATICAL PRINCIPLES [BOOK II.are themselves pressed by the planes AC, ac, that is, in the proportion ofab to AB : arid therefore the centrifugal forces by which these pressuresare sustained are in the same ratio. The number of the particles beingequal, and the situation alike, in both cubes, the forces which all the particles exert, according to the planes FGH,/o7/,, upon all, are as the forceswhich each exerts on each. Therefore the forces which each exerts oneach, according to the plane FGH in the greater cube, are to the forceswhich each exerts on each, according to the plane fgh in the lesser cube,us ab to AB,*that is, reciprocally as the distances of the particles from eachother. Q.E.D.And, vice versa, if the forces of the single particles are reciprocally asthe distances, that is, reciprocally as the sides of the cubes AB, ab ; thesums of the forces will be in the same ratio, and the pressures of the sidesi)B, db as the sums of the forces ; and the pressure of the square DP tothe pressure of the side DB as ab 2 to AB 2. And, ex cequo, the pressure ofthe square DP to the pressure of the side db as ab* to AB 3; that is, theforce of compression in the one to the force of compression in the other asthe 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 reciprocally in the duplicate ratio of the distances between the centres, the cubesof the compressing forces will be as the biquadrates of the densities. Ifthe centrifugal forces be reciprocally in the triplicate or quadruplicate ratioof the distances, the cubes of the compressing forces will be as the quadratocubes,or cubo-cubes of the densities. And universally, if D be put for thedistance, and E for the density of the compressed fluid, and the centrifugalforces be reciprocally as any power Dn of the distance, whose index is thenumber ??, the compressing forces will be as the cube roots of the powerEn + 2. whose index is the number n + 2 ; and the contrary. All thesethings are to be understood of particles whose centrifugal forces terminatein those particles that are next them, or are diffused not much further.We have an example of this in magnetical bodies. Their attractive virtue 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 ironplate, and is almost terminated at it : for bodies further off are not attractedby the magnet so much as by the iron plate. If in this manner particles repelothers of their own kind that lie next them, but do not exert their virtueon the more remote, particles of this kind will compose such fluids as aretreated of in this Proposition, If the virtue of any particle diffuse itselfevery way in inftnitum, there will be required a greater force to producean equal condensation of a greater quantity of the flui 1. But whetherSEC. VI.] OF NATURAL PHILOSOPHY. 303elastic fluids do really consist of particles so repelling each other, is a physical question. We have here demonstrated mathematically the propertyof fluids consisting of particles of this kind, that hence philosophers maytake occasion to discuss that question.SECTION VI.Of the motion and resistance offunependulous bodies.PROPOSITION XXIV. THEOREM XIX.The quantities of matter i/i funependulous bodies, whose centres of oscillation are equally distant from, the centre of suspension, are in a, ratiocompounded of the ratio of the weights and the duplicate ratio of thetimes of the oscillations in vacuo.For the velocity which a given force can generate in a given matter ina 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 velocity will he generated. This is manifest from the second Law of Motion. Now if pendulums are of the same length, the motive forces in placesequally distant from the perpendicular are as the weights : and thereforeif two bodies by oscillating describe equal arcs, and those arcs are dividedinto equal parts ; since the times in which the bodies describe each of thecorrespondent parts of the arcs are as the times of the whole oscillations,the velocities in the correspondent parts of the oscillations will be to eachother as the motive forces and the whole times of the oscillations directly,and the quantities of matter reciprocally : and therefore the quantities ofmatter are as the forces and the times of the oscillations directly and thevelocities reciprocally. But the velocities reciprocally are as the times,and therefore the times directly and the velocities reciprocally are as thesquares of the times; and therefore the quantities of matter are as the motive forces and the squares of the times, that is, as the weights and thesquares of the times. Q.E.D.COR. 1. Therefore if the times are equal, the quantities of matter ineach of the bodies are as the weights.COR. 2. If the weights are equal, the quantities of matter will be as thepquarcs of the times.COR. 3. If the quantities of matter are equal, the weights will be reciprocally as the squares of the times.COR. 4. Whence since the squares of the times, cceteris paribus, are asthe length* of the pendulums, therefore if both the times and quantities ofmatter are equal, the weights will be as the lengths of the pendulums.J04 THE MATHEMATICAL PRINCIPLES [BOOK 11COR. 5. And universally, the quantity of matter in the pendulous bodyis as the weight and the square of the time directly, and the length of thependulum inversely.COR. 6. But in a non-resisting medium, the quantity of matter in thependulous body is as the comparative weight and the square of the timedirectly, and the length of the pendulum inversely. For the comparativeweight is the motive force of the body in any heavy medium, as was shewnabove ; and therefore does the same thing in such a non-resisting mediumas the absolute weight does in a vacuum.COR. 7. And hence appears a method both of comparing bodies oneamong another, as to the quantity of matter in each ; and of comparingthe weights of the same body in different places, to know the variation ofits gravity. And by experiments made with the greatest accuracy, Ihave always found the quantity of matter in bodies to be proportional totheir weight.PROPOSITION XXV. THEOREM XX.Funependulous bodies that are, in, any medium, resisted in the ratio ojthe moments of time, andfunepetidulons bodies that move in a nonresistingmedium of the same specific gravity, perform their oscillations in. a cycloid in the same time, and describe proportional parts ojarcs together.Let AB be an arc of a cycloid, whicha body D, by vibrating in a non-resisting medium, shall describe in anytime. Bisect that arc in C, so that Cmay be the lowest point thereof ; andthe accelerative force with which thebody is urged in any place D, or d orE, will be as the length of the arc CD,pressed by that same arc ; and since the resistance is as the moment of thetime, and therefore given, let it ba expressed by the given part CO of thecycloidal arc, and take the arc Od in the same ratio to the arc CD thatthe arc OB has to the arc CB : and the force with which the body in d isurged in a resisting medium, being the excess of the force Cd above theresistance CO, will be expressed by the arc Od, and will therefore be tothe force with which the body D is urged in a non-resisting medium in theplace 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 placeB, and are urged by these forces ; since the forces at the beginning are asthe arc CB and OB, the first velocities and arcs first described will be inthe same ratio. Let those arcs be BD and Ed, and the remaining arcfSEC. VI. |OF NATURAL PHILOSOPHY. 305CD, Odj will be in the same ratio. Therefore the forces, being proportional to those arcs CD, Od, will remain in the same ratio as at the beginning, and therefore the bodies will continue describing together arcs inthe same ratio. Therefore the forces and velocities and the remaining arcsCD. Od, will be always as the whole arcs CB, OB, and therefore those remaining arcs wLl be described together. Therefore the two bodies D andd will arrive together at the places C and O ; that whicli moves in thenon-resisting medium, at the place C, and the other, in the resisting medium, at the place O. Now since the velocities in C and O are as the arcsCB, OB, the arcs which the bodies describe when they go farther will bein the same ratio. Let those arcs be CE and Oe. The force with whichthe body D in a non-resisting medium is retarded in E is as CE, and theforce with which the body d in the resisting medium is retarded in e, is asthe sum of the force Ce and the resistance CO, that is, as Oe ; and therefore 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 inthat given ratio, remain in the same given ratio. Therefore the velocitiesand the arcs described with those velocities are always to each other inthat given ratio of the arcs CB and OB ; and therefore if the entire arcsAB, aB are taken in the same ratio, the bodies D andc/ will describe thoseaics together, and in the places A and a will lose all their motion together.Therefore the whole oscillations are isochronal, or are performed in equaltimes ; and any parts of the arcs, as BD, Ed, or BE, Be, that are describedtogether, are proportional to the whole arcs BA, B. Q,.E.D.COR. Therefore the swiftest motion in a resisting medium does not fallupon the lowest point C, but is found in that point O, in which the wholearc 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 descent from B to O.PROPOSITION XXVI. THEOREM XXI.Funependulous bodies, that are resisted in the ratio of the velocity, havetheir oscillations in a cycloid isochronal.For if two bodies, equally distant from their centres of suspension, describe, in oscillating, unequal arcs, and the velocities in the correspondentparts of the arcs be to each other as the whole arcs ; the resistances, proportional to the velocities, will be also to each other as the same arcs.Therefore if these resistances be subducted from or added to the motiveforces arising from gravity which are as the same arcs, the differences orsums will be to each other in the same ratio of the arcs ; and since the increments and decrements of the velocities are as these differences or sums,the velocities will be always as the whole arcs; therefore if the velocitiesare in any one case as the whole arcs, they will remain always in the same20306 THE MATHEMATICAL PRINCIPLES [BOOK. 11ratio. But at the beginning of the motion, when the bodies begin to descend and describe those arcs, the forces, which at that time are proportionalto the arcs, will generate velocities proportional to the arcs. Thereforethe velocities will be always as the whole arcs to be described, and therefore those arcs will be described in the same time. Q,.E.D.PROPOSITION XXVII. THEOREM XXII.If fnnependulous bodies are resisted in the duplicate ratio of theirvelocities, the differences between the times of the oscillations in a resisting medium, and the times of the oscillations in a non-resistingmedium of the same specific gravity, will be proportional to the arcsdescribed in oscillating nearly.For let equal pendulums in a resisting medium describe the unequalarcs A, B ; and the resistance of thebody in the arc A will be to the resistance of the body in the correspondentpart of the arc B in the duplicate ratio of the velocities, that is, as, AA toBB nearly. If the resistance in thearc B were to the resistance in the arcA as AB to AA, the times in the arcs A and B would be equal (by the lastProp.) 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-resistingmedium ; and the resistance BB causes the excess of the time in the arc Babove the time in a non-resisting medium. But those excesses are as theefficient 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 aresisting medium, may be knowrn the times of the oscillations in a non- resisting medium of the same specific gravity. For the difference of thetimes will be to the excess of the time in the lesser arc above the time in anon-resisting medium as the difference of the arcs to the lesser arc.COR. 2. The shorter oscillations are more isochronal, and very shortones are performed nearly in the same times as in a non-resisting medium.But the times of those which are performed in greater arcs are a littlegreater, because the resistance in the descent of the body, by which thetime is prolonged, is greater, in proportion to the length described in thedescent than the resistance in the subsequent ascent, by which the time iscontracted. But the time of the oscillations, both short arid long, seems tobe prolonged in some measure by the motion of the medium. For retarded bodies are resisted somewhat less in proportion to the velocity, and accelerated bodies somewhat more than those that proceed uniformly forwards ;SEC. VI.] OF NATURAL PHILOSOPHY. 307because the medium, by the motion it has received from the bodies, goingforwards the same way with them, is more agitated in the former case, andless in the latter; and so conspires more or less with the bodies moved.Therefore it resists the pendulums in their descent more, and in their ascent less, than in proportion to the velocity; and these two causes concurring prolong the time.PROPOSITION XXVIII. THEOREM XXIII.If afunependvlous body, oscillating in a cycloid, be resisted in the rati >of the moments of the time, its resistance will be to the force of gravity as the excess of the arc described in the whole descent above thearc described in the subsequent ascent to twice the length of the pendulum.Let BC represent the arc describedin the descent, Ca the arc described inthe ascent, and Aa the difference ofthe arcs : and things remaining as theywere constructed and demonstrated inProp. XXV, the force with which theoscillating body is urged in any placeD will be to the force of resistance asthe arc CD to the arc CO, which ishalf of that difference Aa. Therefore the force with which the oscillatingbody is urged at the beginning or the highest point of the cycloid, that is,the force of gravity, will be to the resistance as the arc of the cycloid, between that highest point and lowest point C, is to the arc CO ;that is(doubling those arcs), as the whole cycloidal arc, or twice the length of thependulum, to the arc Aa. Q.E.D.PROPOSITION XXIX. PROBLEM VI.Supposing that a body oscillating in a cycloid is resisted in a duplicateratio of the velocity: to find the resistance in each place.Let Ba be an arc described in one entire oscillation, C the lowest pointC OKO ,S P rR Q Mof the cycloid, and CZ half the whole cycloidal arc, equal to the length ofthe pendulum ; and let it be required to find the resistance of the body is30S THE MATHEMATICAL PRINCIPLES [BOOK 1Lany place D. Cut the indefinite right line OQ in the points O, S, P, Q,,so that (erecting the perpendiculars OK, ST, PI, QE, and with the centreO, and the aysmptotcs OK, OQ, describing the hyperbola TIGE cuttingthe perpendiculars ST, PI, QE in T. I, and E, and through the point Idrawing KF. parallel to the asymptote OQ, meeting the asymptote OK i iK, and the perpendiculars ST and QE in L and F) the hyperbolic areaPIEQ may be to the hyperbolic area PITS as the arc BC, described in thedescent of the body, to the arc Ca described in the ascent; and that thearea IEF may be to the area ILT as OQ to OS. Then with the perpendicular MN cut off the hyperbolic area PINM, and let that area be to thehyperbolic area PIEQ as the arc CZ to the arc BC described in the descent. And if the perpendicular RG cut off the hyperbolic area PIGR,which shall be to the area PIEQ as any arc CD to the arc BC describedin the whole descent, the resistance in any place D will be to the force ofORgravity as the area IEF IGH to the area PINM.For since the forces arising from gravity with which the body isurged in the places Z, B, D, a, are as the arcs CZ. CB, CD, Ca and thosearcs are as the areas PINM, PIEQ, PIGR, PITS; let those areas be theexponents both of the arcs and of the forces respectively. Let DC? be avery small space described by the body in its descent : and let it be expressedr>ythe very small area RGor comprehended between the parallels RG, rg ;and produce r<? to //, so that GYlhg- and RGr may be the contemporaneous decrements of the areas IGH, PIGR. And the increment GllhgIEF, or Rr X HG -^ IEF, of the area ~IEF IGH will be , OQ OQIFFto the decrement RGr, or Rr X RG, of the area PIGR, as HG - -ORto RG ; and therefore as OR X HG IEF to OR X GR or OP XPL that is (because of the equal quantities OR X HG, OR X HR ORX GR, ORHK OPIK, PIHR and PIGR + IGH), as PIGR + IGHOR ORIEF to OPIK. Therefore if the area - IEF IGH be called OQY, and RGgr the decrement of the area PIGR be given, the increment ofthe area Y will be as PIGR Y.Then if V represent the force arising from the gravity, proportional tothe arc CD to be described, by which the body is acted upon in D, and Rbe put for the resistance, V R will be the whole force with which thebody is urged in D. Therefore the increment of the velocity is as V Rand the particle of time in which it is generated conjunctly. But the velocity itself is as the contempo] aueous increment of the space described diSEC. VI.J OF NATURAL PHILOSOPHY. 309rectly and the same particle of time inversely. Therefore, since the resistance is, by the supposition, as the square of the velocity, the incrementof the resistance will (by Lem. II) be as the velocity and the increment ofthe velocity conjunctly, that is, as the moment of the space and V Rconjunctly ; and, therefore, if the moment of the space be given, as V11; that is, if for the force V we put its exponent PIGR, and the resistance R be expressed by any other area Z; as PIGR Z. vTherefore the area PIGR uniformly decreasing by the subduction ofgiven moments, the area Y increases in proportion of PIGR Y, andthe area Z in proportion of PIGR Z. And therefore if the areasY and Z begin together, and at the beginning are equal, these, by theaddition of equal moments, will continue to be equal and in like manner decreasing by equal moments, 。vill vanish together. And, vice versa,if they together begin and vanish, they will have equal moments and tealways equal ; and that, because if the resistance Z be augmented, the velocity together with the arc C, described in the ascent of the body, will bediminished ; and the point in which all the motion together with the resistance ceases coming nearer to the point C, the resistance vanishes soonerthan the area Y. And the contrary will happen when the resistance isdiminished.Now the area Z begins and ends where the resistance is nothing, that is,at the beginning of the motion where the arc CD is equal to the arc CB,K /IKO S P /~R Q Mand the right line RG falls upon the right line Q.E ; and at the end ofthe motion where the arc CD is equal to the arc Ca, and RG falls uponthe right line ST. And the area* Y or IEF IGH begins and endsalso where the resistance is nothing, and therefore where IEF andIGH are equal ; that is (by the construction), where the right line RGfalls successively upon the right lines Q,E and ST. Therefore those areasbegin and vanish together, and are therefore always equal. Therefore the areaORIEF IGH is equal to the area Z, by which the resistance is expressed, and therefore is to the area PINM, by which the gravity is expressed, as the resistance to the gravity. Q.E.D.310 THE MATHEMATICAL PRINCIPLES [BOOK 11.COR. 1 . Therefore the resistance in the lowest place C is to the forceOPof gravity as the area ^ ~ IEF to the area PINM.COR. 2. But it becomes greatest where the area PIHR is to the areaIEF as OR to OQ. For in that case its moment (that is, PIGR Y)becomes nothing.COR. 3. Hence also may be known the velocity in each place, as beingin the subduplicate ratio of the resistance, and at the beginning of the motion equal to the velocity of the body oscillating in the same cycloid without any resistance.However, by reason of the difficulty of the calculation by which the resistance and the velocity are found by this Proposition, we have thoughtfit to subjoin the Proposition following.PROPOSITION XXX. THEOREM XXIV.If a right line aB be equal to the arc of a cycloid which an oscillatingbody describes, and at each of its points D the perpendiculars DK beerected, which shall be to the length of the pendulum as the resistanceof the body in the corresponding points of the arc to the force of gravity ; I say, that the difference between the arc described in the wholedescent and the arc described in the whole subsequent ascent drawninto half the sum of the same arcs will be equal to the area BKawhich all those perpendiculars take up.Let the arc of the cycloid, described in one entire oscillation, beexpressed by the right line aB,equal to it, and the arc whichwould have been described in vaciwby the length AB. Bisect AB inC, and the point C will representthe lowest point of the cycloid, andCD Mill be as the force arising from gravity, with which the body in D i,s