urged in the direction of the tangent of the cycloid, and will have the sameratio to the length of the pendulum as the force in D has to the force ofgravity. Let that force, therefore, be expressed by that length CD, andthe force of gravity by the length of the pendulum ; and if in DE youtake DK in the same ratio to the length of the pendulum as the resistancehas to the gravity, DK will be the exponent of the resistance. From thecentre C with the interval CA or CB describe a semi-circle BEeA. Letthe body describe, in the least time, the space Dd ; and, erecting the perpendiculars DE, de, meeting the circumference in E and e, they will be asthe velocities which the body descending in vacuo from the point B wouldacquire in the places D and d. This appears by Prop, LII, Book L LetSEC. VLJ OF NATURAL PHILOSOPHY. 311therefore, these velocities be expressed by those perpendiculars DE, de ;arid let DF be the velocity which it acquires in D by falling from B inthe resisting medium. And if from the centre C with the interval OF wedescribe the circle F/M meeting the right lines de and AB in / and M,then M will be the place to which it would thenceforward, without fartherresistance, ascend, and (//"the velocity it would acquire in d. Whence,also, if FO- represent the moment of the velocity which the body D, in describing the least space DC/, loses by the resistance of the medium ; andCN be taken equal to Cg ;then will N be the place to which the body, ifit met no farther resistance, would thenceforward ascend, and MN will bethe decrement of the ascent arising from the loss of that velocity. DrawF/n perpendicular to dft and the decrement Fg of the velocity DF generated by the resistance DK will be to the incrementfm of the same velocity, generated by the force CD, as the generating force DK to the generating force CD. But because of the similar triangles F////, Fhg, FDC,fm is to Fm or Dd as CD to DF ; and, ex ceqno, Fg to Dd as DK toDF. Also Fh is to Fg as DF to CF ; and, ex ax/uo perturbate, Fh orMN to Do1as DK to CF or CM ; and therefore the sum of all the MN XCM will be equal to the sum of all the Dd X DK. At the moveablepoint M suppose always a rectangular ordinate erected equal to the indeterminate CM, which by a continual motion is drawn into the wholelength Aa ; and the trapezium described by that motion, or its equal, therectangle Aa X |aB, will be equal to the sum of all the MN X CM, andtherefore to the sum of all the Dd X DK, that is, to the area BKVTaO.E.D.COR. Hence from the law of resistance, and the difference Aa of thearcs Ca} CB, may be collected the proportion of the resistance to the gravity nearly.For if the resistance DK be uniform, the figure BKTa will be a rectangle under Ba and DK; and thence the rectangle under ^Ba and Aawill be equal to the rectangle under Ba and DK, and DK will be equal tojAa. Wherefore since DK is the exponent of the resistance, and thelength of the pendulum the exponent of the gravity, the resistance will beto the gravity as 。Aa to the length of the pendulum ; altogether as inProp. XXVIII is demonstrated.If the resistance be as the velocity, the figure BKTa will be nearly anellipsis. For if a body, in a non-resisting medium, by one entire oscillation, should describe the length BA, the velocity in any place D would beas the ordinate DE of the circle described on the diameter AB. Therefore since Ea in the resisting medium, and BA in the non-resisting one,are described nearly in the same times ; and therefore the velocities in eachof the points of Ba are to the velocities in the correspondent points of thelength BA. nearly as Ba is to BA , the velocity in the point D in the re312 THE MATHEMATICAL PRINCIPLES [BJOK 11.sisting medium will be as the ordinate of the circle or ellipsis describedupon the diameter Ba ; and therefore the figure BKVTa will be nearly acellipsis. Since the resistance is supposed proportional to the velocity, le。OV be the exponent of the resistance in the middle point O ; and an ellipsis BRVSa described with the centre O, and the semi-axes OB, OV, willbe nearly equal to the figure BKVTa, and to its equal the rectangle ActX BO. Therefore Aa X BO is to OV X BO as the area of this ellipsisto OV X BO; that is, Aa is to OV as the area of the semi-circle to thesquare of the radius, or as 1 1 to 7 nearly ; and, therefore, T7TAa is to thelength of the pendulum as the resistance of the oscillating body in O toits gravity.Now if the resistance DK be in the duplicate ratio of the velocity, thefigure BKVTa will be almost a parabola having V for its vertex arid OVfor its axis, and therefore will be nearly equal to the rectangle under fBaand OV. Therefore the rectangle under |Ba and Aa is equal to the rectangle fBa X OV, and therefore OV is equal to fAa ; and therefore theresistance in O made to the oscillating body is to its gravity as fAa to thelength of the pendulum.And I take these conclusions to be accurate enough for practical uses.For since an ellipsis or parabola BRVSa falls in with the figure BKVTain the middle point V, that figure, if greater towards the part BRV orVSa than the other, is less towards the contrary part, and is thereforenearly equal to it.PROPOSITION XXXI. THEOREM XXV.If the resistance made to an oscillating body in each of the proportionalparts of the arcs described be augmented or diminished in, a given ratio, the difference between the arc described in the descent and the arcdescribed in the subsequent ascent ivill be augmented or diminished inthe same ratio.For that difference arises fromthe retardation of the pendulumby the resistance of the medium,and therefore is as the whole retardation and the retarding resistance proportional thereto. In theforegoing Proposition the rectan-M isr u c o .-/ n P gle under the right line ^aB andthe difference Aa of the arcs CB, Ca, was equal to the area BKTa, Andthat area, if the length aB remains, is augmented or diminished in the ratio of the ordinates DK ; that is, in the ratio of the resistance and is therefore as the length aB and the resistance conjunctly. And therefore therectangle under Aa and |aB is as aB and the resistance conjunctly, anctherefore Aa is as the resistance. QJE.D.SEC. VI.l OF NATURAL PHILOSOPHY. 313COR. 1. Hence if the resistance be as the velocity, the difference ofthe arts in the same medium will be as the whole arc described : and thecontrary.COR. 2. If the resistance be in the duplicate ratio of the velocity, thatdifference will be in the duplicate ratio of the whole arc : and the contrary.COR. 3. And universally, if the resistance be in the triplicate or anyother ratio of the velocity, the difference will be in the same ratio of the.whole arc : and the contrary.COR. 4. If the resistance be partly in the simple ratio of the velocity,and partly in the duplicate ratio of the same, the difference will be partlyin the ratio of the whole arc, and partly in the duplicate ratio of it: andthe contrary. So that the law arid ratio of the resistance will be thesame for the velocity as the law and ratio of that difference for the lengthof the arc.COR. 5. And therefore if a pendulum describe successively unequal arcs,and we can find the ratio of the increment or decrement of this differencefor the length of the arc described, there will be had also the ratio of theincrement or decrement of the resistance for a greater or less velocity.GENERAL SCHOLIUM.From these propositions we may find the resistance of mediums by pendulums oscillating therein. I found the resistance of the air by the following experiments. I suspended a wooden globe or ball weighing oT^ounces troy, its diameter CJ London inches, by a fine thread on a firmhook, so that the distance between the hook and the centre of oscillation ofthe globe was 10| feet. I marked on the thread a point 10 feet and 1 inchdistant from the centre of suspension and even with that point I placed aruler divided into inches, by the help whereof I observed the lengths of thearcs described by the pendulum. Then I numbered the oscillations iawhich the globe would lose-{- part of its motion. If the pendulum wasdrawn aside from the perpendicular to the distance of 2 inches, and thencelet go, so that in its whole descent it described an arc of 2 inches, and inthe first whole oscillation, compounded of the descent and subsequentascent, an arc of almost 4 inches, the same in 164 oscillations lost j partof its motion, so as in its last ascent to describe an arc of If inches. Ifin the first descent it described an arc of 4 inches, it lost j part of its motion in 121 oscillations, so as in its last ascent to describe an arc of 3^inches. If in the first descent it described an arc of 8, 16, 32, or 64 inches,it lost | part of its motion in 69, 35|, 18|-7 9| oscillations, respectively.Therefore the difference between the arcs described in the first descent andthe last ascent was in the 1st, 2d, 3d, 4th, 5th, 6th cases, }, 1. 1, 2, 4, 8inches respectively. Divide those differences by the number of oscillationsin each case, and in one mean oscillation, wherein an arc of 3 , 7-|, 15, 30314 THE MATHEMATICAL PRINCIPLES [BOOK Jl.60, 120 inches was described, the difference of the arcs described in thedescent and subsequent ascent will be |^, ^{^ e。> T4r; -sji fir parts of aninch, respectively. But these differences in the greater oscillations are inthe duplicate ratio of the arcs described nearly, but in lesser oscillationssomething greater than in that ratio; and therefore (by Cor. 2, Prop. XXXIof this Book) the resistance of the globe, when it moves very swift, is inthe duplicate ratio of the velocity, nearly; and when it moves slowly,somewhat greater than in that ratio.Now let V represent the greatest velocity in any oscillation, and let A,B, and C be given quantities, and let us suppose the difference of the arcs3^to be AV + BV2 + CV2. Since the greatest velocities are in the cycloidas ^ the arcs described in oscillating, and in the circle as | the chords ofthose arcs; and therefore in equal arcs are greater in the cycloid than inthe circle in the ratio of | the arcs to their chords ;but the times in thecircle are greater than in the cycloid, in a reciprocal ratio of the velocity ;it is plain that the differences of the arcs (which are as the resistance andthe square of the time conjunctly) are nearly the same in both curves : forin the cycloid those differences must be on the one hand augmented, withthe resistance, in about the duplicate ratio of the arc to the chord, becauseof the velocity augmented in the simple ratio of the same ; and on theother hand diminished, with the square of the time, in the same duplicateratio. Therefore to reduce these observations to the cycloid, we must takethe same differences of the arcs as were observed in the circle, and supposethe greatest velocities analogous to the half, or the whole arcs, that is, tothe numbers , 1, 2, 4, 8, 16. Therefore in the 2d, 4th, and 6th cases, put1,4, and 1 6 for V ; and the difference of the arcs in the 2d case will becomei 2 * = A + B + C; in the4th case, ^- = 4A + SB + 160 ;in the 6th121 OOjcase, ^ = 16A + 64B -f- 256C. These equations reduced give A =9?0,000091 6, B =-. 0,0010847, and C = 0,0029558. Therefore the differenceof the arcs is as 0,0000916V -f 0,0010847V* + 0,0029558V* : and therefore since (by Cor. Prop. XXX, applied to this case) the re.-ist;mcc of theglobe in the middle of the arc described in oscillating, where the velocityis V, is to its weight as T7TAV -f- T。BV^ + fCV2 to the length of thependulum, if for A, B, and C you put the numbers found, the resistance ofthe globe will be to its weight as 0,0000583V + 0,0007593V* + 0,OJ22169V 2to the length of the pendulum between the centre of suspension and theruler, that is, to 121 inches. Therefore since V in the second case represents 1, in the 4th case 4, and in the 6th case 16, the resistance will be tothe weight of the globe, in the 2d case, as 0,0030345 to 121 ;in the 4th, as0,041748 to 121 ; in the 6th, as 0,61705 to 121.SEC. VI.] OF NATURAL PHILOSOPHY. 315The arc, which the point marked in the thread described in the 6th case,was of 120 Q^,or 119/g inches. And therefore since the radius wasy a121 inches, and the length of the pendulum between the point of suspension and the centre of the globe was 126 inches, the arc which the centre ofthe globe described was 124/T inches. Because the greatest velocity of theoscillating body, by reason of the resistance of the air, does not fall on thelowest point of the arc described, but near the middle place of the wholearc, this velocity will be nearly the same as if the globe in its whole descentin a non-resisting medium should describe 62^ inches, the half of that arc,and that in a cycloid, to which we have above reduced the motion of thependulum; and therefore that velocity will be equal to that which theglobe would acquire by falling perpendicularly from a height equal to theversed sine of that arc. But that versed sine in the cycloid is to that arc62/2 as the same arc to twice the length of the pendulum 252, and therefore equal to 15,278 inches. Therefore the velocity of the pendulum is thesame which a body would acquire by falling, and in its fall describing aspace of 15,278 inches. Therefore with such a velocity the globe meetswith a resistance which is to its weight as 0,61705 to 121, or (if we takethat part only of the resistance which is in the duplicate ratio of the veloc.ty) as 0,56752 to 121.I found, by an hydrostatical experiment, that the weight of this woodenglobe was to the weight of a globe of water of the same magnitude as 55to 97: and therefore since 121 is to 213,4 in the same ratio, the resistancemade to this globe of water, moving forwards with the above-mentionedvelocity, will be to its weight as 0,56752 to 213,4, that is, as 1 to 376^.Whence since the weight of a globe of water, in the time in which theglobe with a velocity uniformly continued describes a length of 30,556inches, will generate all that velocity in the falling globe, it is manifestthat the force of resistance uniformly continued in the same time will takeaway a velocity, which will be less than the other in the ratio of 1 to 376^- ,that is, the rr^-r part of the whole velocity. And therefore in the time37VSGJiat the globe, with the same velocity uniformly continued, would describethe length of its semi-diameter, or 3 T。 inches, it would lose the 3^42 partof its motion.I also counted the oscillations in which the pendulum lost j part of itsmotion. In the following table the upper numbers denote the length of thearc described in the first descent, expressed in inches and parts of an inch ;the middle numbers denote the length of the arc described in the last ascent ; and in the lowest place are the numbers of the oscillations. I giveun account of this experiment, as being more accurate than that in which316 THE MATHEMATICAL PRINCIPLES [BOOK llonly1part of the motion was lost. I leave the calculation to such as aredisposed to make it.First descent ... 2 4 8 16 32 64Last ascent . . , 1| 3 6 12 24 48NoscilL . .374 272 162i 83J 41f 22|I afterward suspended a leaden globe of 2 inches in diameter, weighing26 1 ounces troy by the same thread, so that between the centre of theglobe and the point of suspension there was an interval of 10^ feet, and 1counted the oscillations in which a given part of the motion was lost. Theiirst of the following tables exhibits the number of oscillations in which Jpartof the whole motion was lost ; the second the number of oscillationsin which there was lost 。 part of the same.First descent .... 1 2 4 8 16 32 64Last ascent .... f J 3^ 7 14 28 56Numb, of oscilL . . 226 228 193 140 90^ 53 30First descent .... 1 2 4 8 16 32 64Last ascent .... 1^ 3 6 12 24 4SNunib. of oscill. . .510 518^ 420 318 204 12170Selecting in the first table the 3d, 5th, and 7th observations, and expressing the greatest velocities in these observations particularly by the numbers 1, 4, 16 respectively, and generally by the quantity V as above, therewill come out in ihe 3d observation ~- = A + B + C, in the 5th obser-2 8vation ^ = 4A 4- 8B + 16C. in the 7th observation ^-- == 16A 4- 64B t-,t(j j oU256C. These equations reduced give A = 0,001414, B == 0,000297, C0,000879. And thence the resistance of the globe moving with the velocityV will be to its weight 26^ ounces in the same ratio as 0,0009V +0,000208V* + 0,000659V 2 to 121 inches, the length of the pendulum.And if we regard that part only of the resistance which is in the duplicate ratio of the velocity, it will be to the weight of the globe as 0,000659V 2to 121 inches. But this part of the resistance in the first experiment wasto the weight oi the wooden globe of 572-72 ounces as 0,002217V 2 to 121 ;and thence the resistance of the wooden globe is to the resistance of theleaden one (their velocities being equal) as 57/2- into 0,002217 to 26 Jinto0,000659, that is, as 7|- to 1. The diameters of the two globes were6f and 2 inches, and the squares of these are to each other as 47 and 4,or 11-J-f and 1, nearly. Therefore the resistances of these equally swiftglobes were in less than a duplicate ratio of the diameters. But we havenot yet considered the resistance of the thread, which was certainly veryconsiderable, and ought to be subducted from the resistance of the pendulums here found. I could not determine this accurately, but I found ilSEC. VI.J OF NATURAL PHILOSOPHY. 3 1/greater than a third part of the whole resistance of the lesser pendulum ;and thence I gathered that the resistances of the globes, when the resistance of the thread is subducted, are nearly in the duplicate ratio of theirdiameters. For the ratio of 7} } to 1 , or l(H to 1 is not verydifferent from the duplicate ratio of the diameters 1 L}f to I.Since the resistance of the thread is of less moment in greater globes, Itried the experiment also with a globe whose diameter was ISf inches.The length of the pendulum between the point of suspension and the centre of oscillation was 122| inches, and between the point of suspension andthe knot in the thread 109| inches. The arc described by the knot at thefirst descent of the pendulum was 32 inches. The arc described by thesame knot in the last ascent after five oscillations was 2S inches. Thesum of the arcs, or the whole arc described in one mean oscillation, was 60inches. The difference of the arcs 4 inches. The y1,,- part of this, or thedifference between the descent and ascent in one mean oscillation, is f ofan inch. Then as the radius 10(J| to the radius 122^, so is the whole arcof 60 inches described by the knot in one mean oscillation to the whole arcof 67} inches described by the centre of the globe in one mean oscillation ;and so is the difference | to a new difference 0,4475. If the length of thearc described were to remain, and the length of the pendulum should beaugmented in the ratio of 126 to 122}, the time of the oscillation wouldbe augmented, and the velocity of the pendulum would be diminished inthe subduplicate of that ratio; so that the difference 0,4475 of the arcs described in the descent and subsequent ascent would remain. Then if thearc described be augmented in the ratio of 124 33T to 67}, that difference0.4475 would be augmented in the duplicate of that ratio, and so wouldbecome 1,5295. These things would be so upon the supposition that theresistance of the pendulum were in the duplicate ratio of the velocity.Therefore if the pendulum describe the whole arc of 12433T inches, and itslength between the point of suspension and the centre of oscillation be 126inches, the difference of the arcs described in the descent and subsequentascent would be 1,5295 inches. And this difference multiplied into theweight of the pendulous globe, which was 208 ounces, produces 318,136.Again ;in the pendulum above-mentioned, made of a wooden globe, whenits centre of oscillation, being 126 inches from the point of suspension, described the whole arc of 124 /T inches, the difference of the arcs describedin the descent and ascent was ^^ into ^. This multiplied into thei/wi y^weight of the globe, which was 57-272 ounces, produces 49,396. But I multiply these differences into the weights of the globes, in order to find theirresistances. For the differences arise from the resistances, and are as theresistances directly and the weights inversely. Therefore the resistancesare as the numbers 318,136 and 49,396. But that part of the resistance31 S THE MATHEMATICAL PRINCIPLES [BOOK 1Lof the lesser globe, which is in the duplicate ratio of the velocity, was tothe whole resistance as 0,56752 to- 0,61675, that is, as 45,453 to 49,396 ;whereas that part of the resistance of the greater globe is almost equal toits whole resistance; and so those parts are nearly as 318,136 and 45,453,that is, as 7 and 1. But the diameters of the globes are 18f and 6| ; andtheir squares 351 T9 and 47 J are as 7,438 and 1, that is, as the resistancesof the globes 7 and 1, nearly. The difference of these ratios is scarcegreater than may arise from the resistance of the thread. Therefore thoseparts of the resistances which are, when the globes are equal, as the squaresof the velocities, are also, when the velocities are equal, as the squares ofthe diameters of the globes.But the greatest of the globes I used in these experiments was not perfectly spherical, and therefore in this calculation I have, for brevity s sake,neglected some little niceties; being not very solicitous for an accuratecalculus in an experiment that was not very accurate. So that I couldwish that these experiments were tried again with other globes, of a largersize, more in number, and more accurately formed ; since the demonstration of a vacuum depends thereon. If the globes be taken in a geometricalproportion, as suppose whose diameters are 4, 8, 16, 32 inches; one maycollect from the progression observed in the experiments what would happen if the globes were still larger.In order to compare the resistances of different fluids with each other, 1made the following trials. I procured a wooden vessel 4 feet long, 1 footbroad, and 1 foot high. This vessel, being uncovered, 1 filled with springwater, and, having immersed pendulums therein, I made them oscillate inthe water. And I found that a leaden globe weighing 166| ounces, and indiameter 3f inches, moved therein as it is set down in the following table;the length of the pendulum from the point of suspension to a certainpoint marked in the thread being 126 inches, and to the centre of oscillation 134f inches.The arc described in }the first descent, bya point marked in 。 64 . 32 . 16 . $ . 4 . 2 . 1 . . Jthe thread was 。inches.The arc described in )the last ascent was V 48 . 24 . 12 . 6 . 3 . 1| . . f . T。inches. 。The difference of thearcs, proportional
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