sity of the globe to the density of the medium, that is, as A to A B, Gthe time in which the globe falling with the weight B without resistancedescribes the space P, and H the velocity which the body acquires by thatfall. Then H will be the greatest velocity with which the globe can possibly descend with the weight B in the resisting medium, by Cor. 2, PropXXXVIII ; and the resistance which the globe meets with, when descending with that velocity, will be equal to its weight B ; and the resistance itmeets with in any other velocity will be to the weight B in the duplicate ratio of that velocity to the greatest velocity H, by Cor. 1, Prop. XXXVIII.This is the resistance that arises from the inactivity of the matter ofthe fluid. That resistance which arises from the elasticity, tenacity, andfriction of its parts, may be thus investigated.Let the globe be let fall so that it may descend in the fluid by the weightB ; and let P be the time of falling, and let that time be expressed in seconds, if the time G be given in seconds. Find the absolute number N2Pagreeing to the logarithm 0,4342944819 >and let L be the logarithm ofN 4- 1the number and *^e velocity acquired in falling will bfTHE MATHEMATICAL PRINCIPLES [BOOK 11N i 2PFjj=H, and the height described will be -^ 1 .38629430 IIP +4,6051701S6LF. If the fluid be of a sufficient depth, we may neglect the2PFterm 4,6051 70186LF; and r- 1,3362943611F will be the altitudedescribed, nearly. These things appear by Prop. IX, Book II, and its Corollaries, and are true upon this supposition, that the globe meets with no otherresistance but that which arises from the inactivity of matter. Now if itreally meet with any resistance of another kind, the descent will be slower,and from the quantity of that retardation will be known the quantity ofthis new resistance.That the velocity and descent of a body falling in a fluid might moreeasily be known, I have composed the following table ; the first column ofwhich denotes the times of descent ; the second shews the velocities acquired in falling, the greatest velocity being 100000000: the third exhibits the spaces described by falling in those times, 2F being the space whichthe body describes in the time G with the greatest velocity ; and the fourthgives the spaces described with the greatest velocity in the same times.2PThe numbers in the fourth column are-pnand by subducting the number1,3962944 4,60517021,, are found the numbers in the third column ;and these numbers must be multiplied by the space F to obtain the spacesdescribed in falling. A fifth column is added to all these, containing thespaces described in the same times by a body falling in vacno with theforce of B its comparative weight,SEC. VII.| OF NATURAL PHILOSOPHY. 347。SCHOLIUM*In order to investigate the resistances of lluids from experiments, I procured a square wooden vessel, whose length and breadth on the inside was9 inches English measure, and its depth 9 feet 。 ; this I filled with rainwater: and having provided globes made up of wax, and lead includedtherein, I noted the times of the descents of these globes, the height throughwhich they descended being 112 inches. A solid cubic foot of Englishmeasure contains 76 pounds troy weight of rain water ; and a solid inchcontains if ounces troy weight, or 253 grains: and a globe of water ofone inch in diameter contains 132,645 grains in air, or 132,8 grains invacn.o ; and any other globe will be as the excess of its weight in vacuoabove its weight in water.EXPER. 1. A globe whose weight was 156^ grains in air, and 77 grainsin water, described the whole height of 1 12 inches in 4 seconds. And, uponrepeating the experiment, the globe spent again the very same time of 4seconds in falling.The weight of this globe in vacuo is 156^1 grains; and the excess ofthis weight above the weight of the globe in water is 79^ f grains. Hencethe diameter of the globe appears to be 0,84224 parts of an inch. Then itwill be, as that excess to the weight of the globe in vacuo, so is the densityof the water to the density of the globe; and so is f parts of the diameterof the globe (viz. 2,24597 inches) to the space 2F, which will be therefore4.4256 inches. Now a globe falling in vacuo with its whole weight of156^f grains in one second of time will describe 193| inches ; and fallingin water in the same time with the weight of 77 grains without resistance,will describe 95,219 inches ; and in the time G, which is to one second oftime in the subduplicate ratio of the space P, or of 2,2128 inches to 95,219inches, will describe 2,2128 inches, and will acquire the greatest velocity Hwith which it is capable of descending in water. Therefore the time G is0",15244. And in this time G, with that greatest velocity H, the globewill describe the space 2F, which is 4,4256 inches; and therefore in 4 seconds will describe a space of 1 16,1245 inches. Subduct the space 1,3862944 F,or 3,0676 inches, and there will remain a space of 113,0569 inches, whichthe globe falling through water in a very wide vessel will describe in 4 seconds. But this space, by reason of the narrowness of the wooden vesselbefore mentioned, ought to be diminished in a ratio compounded of the subduplicateratio of the orifice of the vessel to the excess of this orifice abovehalf a great circle of the globe, and of the simple ratio of the same orificeto its excess above a great circle of the globe, that is, in a ratio of 1 to0,9914. This done, we have a space of 112,08 inches, which a globe falling through the water in this wooden vessel in 4 seconds of time oughtnearly to describe by this theory; but it described 112 inches by the experiment.348 THE MATHEMATICAL PRINCIPLES [BOOK IIEXPER. 2. Three equal globes, whose weights were severally 76^- grainsin air, and 5 T^ grains in water, were let fall successively -; and every onefell through the water in 15 seconds of time, describing in its fall a heightof 112 inches.By computation, the weight of each globe in vacuo is 76 T52 grains ;theexcess of this weight above the weight in water is 71 grains J ; the diameter of the globe 0,81296 of an inch; f parts of this diameter 2,1 67Sinches; the space 2F is 2,3217 inches; the space which a globe of 5 T。grains in weight would describe in one second without resistance, 12,80inches, and the time G0",301056. Therefore the globe, with the greatestvelocity it is capable of receiving from a weight of 5^ grains in its descent through water, will describe in the time 0",3L)1056the space of 2,3217inches; and in 15 seconds the space 115,678 inches. Subduct the space1,3862944F, or 1,609 inches, and there remains the space 114.069 inches,which therefore the falling globe ought to describe in the same time, if thevessel were very wide. But because our vessel was narrow, the space oughtto be diminished by about 0,895 of an inch. And so the space will remain113,174 inches, which a globe falling in this vessel ought nearly to describe in 15 seconds, by the theory. But by the experiment it described112 inches. The difference is riot sensible.EXPER. 3. Three equal globes, whose weights were severally 121 grainsin air, and 1 grain in water, were successively let fall; and they fellthrough the water in the times 46", 47", and 50", describing a height oi112 inches.By the theory, these globes ought to have fallen in about 40". Nowwhether their falling more slowly were occasioned from hence, that in slowmotions the resistance arising from the force of inactivity does really beara less proportion to the resistance arising from other causes ;or whetherit is to be attributed to little bubbles that might chance to stick to theglobes, or to the rarefaction of the wax by the warmth of the weather, orof the hand that let them fall; or, lastly, whether it proceeded from someinsensible errors in weighing the globes in the water, I am not certain.Therefore the weight of the globe in water should be of several grains, thatthe experiment may be certain, and to be depended on.EXPER. 4. I began the foregoing experiments to investigate the resistances of fluids, before I was acquainted with the theory laid down in thePropositions immediately preceding. Afterward, in order to examine thetheory after it was discovered, I procured a wooden vessel, whose breadthon the inside was Sf inches, and its depth ] 5 feet and -i. Then I madefour globes of wax, with lead included, each of which weighed 1391grainsin air, and 7。 grains in water. These I let fall, measuring the times of theirfalling in the water with a pendulum oscillating to half seconds. Theglobes were cold, and had remained so some time, both when they wereSEC. V1L] OF NATUKAL PHILOSOPHY. 3-1 *J.reighed and when they were let fall; because warmth rarefies the wax. andby rarefying it diminishes the weight of the globe in the water ; and wax,when rarefied, is not instantly reduced by cold to its former density. Before they were let fall, they were totally immersed under water, lest, by theweight of any part of them that might chance to be above the water, theirdescent should be accelerated in its beginning. Then, when after theirimmersion they were perfectly at rest, they were let go with the greatestcare, that they might not receive any impulse from the hand that let themdown. And they fell successively in the times of 47 J, 48^, 50, and 51 oscillations, describing a height of 15 feet and 2 inches. But the weatherwas now a little colder than when the globes were weighed, and therefore 1repeated the experiment another day ; and then the globes fell in the timesof 49, 49i, 50. and 53; and at a third trial in the times of 49, 50, 51.and 53 oscillations. And by making the experiment several times over, Ifound that the globes fell mostly in the times of 49| and 50 oscillations.When they fell slower, I suspect them to have been retarded by strikingagainst the sides of the vessel.Now, computing from the theory, the weight of the globe in vacno is139| grains; the excess of this weight above the weight of the globe inwater 132|i grains ; the diameter of the globe 0,99868 of an inch :|- partsof the diameter 2,66315 inches; the space 2F 2,8066 inches; the spacewhich a globe weighing 7| grains falling without resistance describes in asecond of time 9,88164 inches; and the time G0",376843 Therefore theglobe with the greatest velocity with which it is capable of descendingthrough the water by the force of a weight of 7} grains, will in the time0",376843 describe a space of 2,8066 inches, and in one second of time aspace of 7,44766 inches, and in the time 25", or in 50 oscillations, the space186,1915 inches. Subduct the space 1,386294F, or 1,9454 inches, andthere will remain the space 184,2461 inches which the globe will describein that time in a very wide vessel. Because our vessel was narrow, let thisspace be diminished in a ratio compounded of the subduplicate ratio of theorifice of the vessel to the excess of this orifice above half a great circle ofthe globe, and of the simple ratio of the same orifice to its excess above agreat circle of the globe ; and we shall have the space of 181,86 inches,which the globe ought by the theory to describe in this vessel in the timeof 50 oscillations, nearly. But it described the space of 182 inches, byexperiment, in 49^ or 50 oscillations.EXPER. 5. Pour globes weighing 154| grains in air, and 21 1 grains inwater, being let fall several times, fell in the times of 28^, 29, 29 , and 30,and sometimes of 31, 32, and 33 oscillations, describing a height of 15 feetand 2 inches.They ought by the theory to have fallen in the time of 29 oscillations,nearly.350 THE MATHEMATICAL PRINCIPLES| BOOK ILEXPER. 6. Five globes, weighing 212f grains in air, and 79^ in water,being several times let fall, fell in the times of 15, 15^, 16, 17, and 18 oscillations, describing a height of 15 feet and 2 inches.By the theory they ought to have fallen in the time cf 15 oscillations,nearly.EXPER. 7. Four globes, weighing 293 f grains in air, and 35| grains inwater, being let fall several times, fell in the times of 29 30, 301 31, 32,and 33 oscillations, describing a height of 15 feet and 1 inch and .By the theory they ought to have fallen in the time of 28 oscillations,nearly.In searching for the cause that occasioned these globes of the same weightand magnitude to fall, some swifter and some slower, I hit upon this ; thatthe globes, when they were first let go and began to fall, oscillated abouttheir centres; that side which chanced to be the heavier descending first,and producing an oscillating motion. Now by oscillating thus, the globecommunicates a greater motion to the water than if it descended withoutany oscillations ; and by this communication loses part of its own motionwith which it should descend; and therefore as this oscillation is greateror less, it will be more or less retarded. Besides, the globe always recedesfrom that side of itself which is descending in the oscillation, and by soreceding comes nearer to the sides of the vessel, so as even to strike againstthem sometimes. And the heavier the globes are, the stronger this oscillation is; and the greater they are, the more is the water agitated by it.Therefore to diminish this oscillation of the globes, 1 made new ones oflead and wax, sticking the lead in one side of the globe very near its surface; and I. let fall the globe in such a manner, that, as near as possible,the heavier side might be lowest at the beginning of the descent. By thismeans the oscillations became much less than before, and the times in whichthe globes fell were not so unequal: as in the following experiments.EXPER. 8. Four globes weighing 139 grains in air, and 6| in water,were let fall several times, and fell mostly in the time of 51 oscillations,never in more than 52, or in fewer than 50, describing a height of 182inches.By the theory they ought to fall in about the time of 52 oscillationsEXPER. 9. Four globes weighing 273^ grains in air, and 140 in water,being several times let fall, fell in never fewer than 12, and never morethan 13 oscillations, describing a height of 182 inches.These globes by the theory ought to have fallen in the time of 1 1 } oscillations, nearly.EXPER. 10. Four globes, weighing 384 grains in air, and 119^ in water,oeing let fall several times, fell in the times of 17f 18, 18^, and 19 oscillations, descril ing a height of 181| inches. And when they fell in the timeSEC. VI1.J OF NATURAL PHILOSOPHY. 351of 19 oscillations, I sometimes heard them hit against the sides of tl.e vessel before they reached the bottom.By the theory they ought to have fallen in the time of 15f oscillations,nearly.EXPER. 11. Three equal globes, weighing 48 grains in the air, and 3||in water, being several times let fall, fell in the times of 43J, 44, 44 1, 45,and 46 oscillations, and mostly in 44 and 45. describing a height of 182*inches, nearly.By the theory they ought to have fallen in the time of 46 oscillationsand f, nearly.EXPER. 12. Three equal globes, weighing 141 grains in air, and 4| inwater, being let fall several times, fell in the times of 61, 62, 63, 64, and65 oscillations, describing a space of 182 inches.And by the theory they ought to have fallen in 641 oscillationsnearly.From these experiments it is manifest, that when the globes fell slowly,as in the second, fourth, fifth, eighth, eleventh, and twelfth experiments;the times of falling are rightly exhibited by the theory but when theglobes fell more swiftly, as in the sixth, ninth, and tenth experiments, theresistance was somewhat greater than in the duplicate ratio of the velocity.For the globes in falling oscillate a little : and this oscillation, in thoseglobes that are light and fall slowly, soon ceases by the weakness of themotion ; but in greater and heavier globes, the motion being strong, it continues longer, and is not to be checked by the ambient water till after several oscillations Besides, the more swiftly the globes move, the less arethey pressed by the fluid at their hinder parts; and if the velocity be. perpetually increased, they will at last leave an empty space behind them,unless the compression of the fluid be increased at the same time. For thecompression of the fluid ought to be increased (by Prop. XXXII andXXXIII) in the duplicate ratio of the velocity, in order to preserve the resistance in the same duplicate ratio. But because this is not done, theglobes that move swiftly are not so much pressed at their hinder parts asthe others; and by the defect of this pressure it comes to pass that theirresistance is a little greater than in a duplicate ratio of their velocity.So that the theory agrees with the phenomena of bodies falling in waterIt remains that we examine the phenomena of bodies falling in air.EXPER. 13. From the top of St. Paul s Church in London, in Juiib1710, there e let fall together two glass globes, one full of quicksilver,the other of air; and in their fall they described a height of 220 Englishfeet. A wooden table was suspended upon iron hinges on one sidi> and theother side of the same was supported by a wooden pin. The twn globeslying upon this table were let fall together by pulling out the pin bjmeans of an iron wire reaching from thence quite down to the ground ;s<352 THE MATHEMATICAL PRINCIPLES [BOOK II,that, the pin being removed, the table, which had then no support but theiron hinges, fell downward, and turning round upon the hinges, gave leaveto the globes to drop off from it. At the same instant, with the same pullof the iron wire that took out the pin, a pendulum oscillating to secondswas let go, and began to oscillate. The diameters and weights of theglobes, and their times of falling, are exhibited in the following table.But the times observed must be corrected;for the globes of mercury (byGalileo s theory), in 4 seconds of time, will describe 257 English feet, and220 feet in only 3"42 ". So that the wooden table, when the pin was takenout, did not turn upon its hinges so quickly as it ought to have done; andthe slowness of that revolution hindered the descent of the globes at thebeginning. For the globes lay about the middle of the table, and indeedwere rather nearer to the axis upon which it turned than to the pin. Andhence the times of falling were prolonged about 18"; and therefore oughtto be corrected by subducting that excess, especially in the larger globes,which, by reason of the largeness of their diameters, lay longer upon therevolving table than the others. This being done, the times in which thesix larger globes fell will come forth 8" 12 ",7" 42% 7" 42 ",7" 57 ",8" 12 "and 7" 42 ".Therefore the fifth in order among the globes that were full of air being5 inches in diameter, and 483 grains in weight, fell in 8" 12 ", describing aspace of 220 feet. The weight of a bulk of water equal to this globe is1 6600 grains; and the weight of an equal bulk of air is l||f- grains, or I9 r3ograins ; and therefore the weight of the globe in vacuo is 502T3?r grains;and this weight is to the weight of a bulk of air equal to the globe as502T;v to 19 T3o- ; and so is 2P to | of the diameter of the globe, that is, to13i inches. Whence 2F becomes 28 feet 11 inches. A globe, falling invacuo with its whole weight of 502T3grains, will in one second of timedescribe 193| inches as above ; and with the weight of 483 grains will describe 185,905 inches; and with that weight 483 grains in vacuo will describe the space F, or 14 feet 5i inches, in the time of 57 "58"", and acquire the greatest velocity it is capable of descending with in the air.With this velocity the globe in 8" 12 " of time will describe 245 feet and5i inches. Subduct 1,3863F, or 20 feet and | an inch, and there remain225 feet 5 inches. This space, therefore, the falling globe ought by theSEC. VIIJ OF NATURAL PHILOSOPHYtheory to describe in 8" 12 ". But* by the experiment it descrioed a spaceof 220 feet. The difference is insensible.By like calculations applied to the other globes full of air, I composedthe following table.EXPER. 14. Anno 1719, in the month of July, Dr. Desaguliers madesome experiments of this kind again, by forming hogs bladders into spherical orbs ; which was done by means of a concave wooden sphere, which thebladders, being wetted well first, were put into. After that being blownfull of air. they were obliged to fill up the spherical cavity that containedthem ; and then, when dry, were taken out. These were let fall from thelantern on the top of the cupola of the same church, namely, from a heightof 272 feet ; and at the same moment of time there was let fall a leadenglobe, whose weight was about 2 pounds troy weight. And in the meantime some persons standing in the upper part of the church where theglobes were let fall observed the whole times of falling ; and others standing on the ground observed the differences of the times between the fallof the leaden weight and the fall of the bladder. The times were measuredby pendulums oscillating to half seconds. And one of those that stoodupon the ground had a machine vibrating four times in one second ; andanother had another machine accurately made with a pendulum vibratingfour times in a second also. One of those also who stood at the top of thechurch had a like machine ; and these instruments were so contrived, thattheir motions could be stopped or renewed at pleasure. Now the leadenglobe fell in about four seconds and i of time; and from the addition ofthis time to the difference of time above spoken of, was collected the 。Vholetime in which the bladder was falling. The times which the five bladdersspent in falling, after the leaden globe had reached the ground, were, tn*efirst time, 14", 12f, 14f, 17 f, and 16J-" ; and the second time, 14i", 14}",14", 19", and 16 J". Add to these 4", the time in which the leaden globewas falling, and the whole times in which the five bladders fell were, thefirst fane, 19* 17", 18J", 22", and 21}"; and the second time, 18f, 18i",ISj", 23{", and 21". The times observed at the top of the church were,the first time, 19 f", 17f , 18f, 22f , and 21f"; and the second time, 19",ISf", ISf, 24". and 211". But the bladders did not always fall directlydown, but sometimes fluttered a little in the air, and waved to and fro, aa354 THE MATHEMATICAL PRINCIPLES [BOOK Jlthey were descending. And by these motions the times of their fallingwere prolonged, and increased by half a second sometimes, and sometimesby a whole second. The second and fourth bladder fell most directly thefirst time, and the first and third the second time. The fifth bladder waswrinkled, and by its wrinkles was a little retarded. I found their diameters by their circumferences measured with a very fine thread wound aboutthem twice. In the following table I have compared the experiments withthe theory ; making the density of air to be to the density of rain-water as1 to 860, and computing the spaces which by the theory the globes oughtto describe in falling.Our theory, therefore, exhibits rightly, within a very little, all the resistance that globes moving either in air or in water meet with ; which^appearsto be proportional to the densities of the fluids in globes of equal velocities and magnitudes.In the Scholium subjoined to the sixth Section, we shewed, by experiments of pendulums, that the resistances of equal and equally swift globesmoving in air, water, and quicksilver, are as the densities of the fluids.We here prove the same more accurately by experiments of bodies fallingin air and water. For pendulums at each oscillation excite a motion in