Principia Mathematica (Motte Translation, 1848) — part 26 of 45

If a cylinder, a sphere, and a spheroid, of equal breadths be placed suc- cessively in the middle of a cylindric canal, so that their axes may coincide ivith the axis of the canal, these bodies ivill equally hinder the passage of the water through the canal.

For the spaces lying between the sides of the canal, and the cylinder, sphere, and spheroid, through which the water passes, are equal ; and the water will pass equally through equal spaces.

This is true, upon the supposition that all the water above the cylinder, sphere, or spheroid, whose fluidity is not necessary to make the passage of the water the quickest possible, is congealed, as was explained above in Cor. 7, Prop. XXXVL

Sec. VIL] of natural philosophy. 343

LEMMA VI.

The same supposition remaining ^ the foi^e-m^e^itioned bodies are equally acted on by the loater Jloiving through the canal. This appears by Lem. V and the third Law. For the water and the bodies act upon each other mutually and equally.

LEMMA VII.

If the water be at rest in the canal, and these bodies move ivith equal ve- locity and the contrary way through the canal, their resistances will be equal amoiig themselves. This appears from the last Lemma, for the relative motions remain the

same among themselves.

SCHOLIUM.

The case is the same of all convex and round bodies, whose axes coincide with the axis of the canal. Some difference may arise from a greater or less friction ; but in these Lemmata we suppose the bodies to be perfectly smooth, and the medium to be void of all tenacity and friction ; and that those parts of the fluid which by their oblique and superfluous motions may disturb, hinder, and retard the flux of the water through the canal, are at rest among themselves ; being fixed like water by frost, and adhering to the fore and hinder parts of the bodies in the manner explained in the Scholium of the last Proposition ; for in what follows we consider the very least resistance that round bodies described with the greatest given trans- verse sections can possibly meet with.

Bodies swimming upon fluids, when they move straight forward, cause the fluid to ascend at their fore parts and subside at their hinder parts, especially if they are of an obtuse figure ; and thence they meet with a little more resistance than if they were acute at the head and tail. And bodies moving in elastic fluids, if they are obtuse behind and before, con- dense the fluid a little more at their fore parts, and relax the same at their hinder parts ; and therefore meet also with a little more resistance than if they were acute at the head and tail. But in these Lemmas and Proposi- tions we are not treating of elastic but non-elastic fluids ; not of bodies floating on the surface of the fluid, but deeply immersed therein. And when the resistance of bodies in non-elastic fluids is once known, we may then augment this resistance a little in elastic fluids, as our air ; and in the surfaces of stagnating fluids, as lakes and seas.

PROPOSITION XXXVIII. THEOREM XXX.

tf a globe move uniformity forioard in a com^pressed, infinite, and non- elastic fluid ^ its resistance is to the force by lohich its whole moiion.:

344 , THE MATHEMATICAL PRINCIPLES [BoOK II.

may be destroyed or generated, in the time that it describes eight third

parts of its diameter, as the density of the fluid to the density of the

globe, very nearly.

For the globe is to its circumscribed cylinder as two to three ; and there^ fore the force which can destroy all the motion of the cylinder, while the same cylinder is describing the length of four of its diameters, will destroy all the motion of the globe, while the globe is describing two thirds of this length, that is, eight third parts of its own diameter. Now the resistance of the cylinder is to this force very nearly as the density of the fluid to the density of the cylinder or globe (by Prop. XXXYII), and the resistance of the globe is equal to the resistance of the cylinder (by Lem. V, YI, and YII). aE.D.

Cor. 1. The resistances of globes in infinite compressed mediums are in a ratio compounded of the duplicate ratio of the velocity, and the dupli- cate ratio of the diameter, and the ratio of the density of the mediums.

Cor. 2. The greatest velocity, with which a globe can descend by its comparative weight through a resisting fluid, is the same which it may acquire by falling with the same weight, and without any resistance, and in its fall describing a space that is, to four third parts^ of its diameter as the density of the globe to the density of the fluid. For the globe in the time of its fall, moving with the velocity acquired in falling, will describe a space that will be to eight third parts of its diameter as the density of the globe to the density of the fluid ; and the force of its weight which generates this motion will be to the force that can generate the same mo- tion, in the time that the globe describes eight third parts of its diameter, with the same velocity as the density of the fluid to the density of the globe ; and therefore (by this Proposition) the force of w eight will be equal to the force of resistance, and therefore cannot accelerate the globe.

CoR. 3. If there be given both the density of the globe and its velocity at the beginning of the motion, and the density of the compressed quiescent fluid in which the globe moves, there is given at any time both the velo- city of the globe and its resistance, and the space described by it (by Cor. 7, Prop. XXXY).

CoR. 4. A globe moving in a compressed quiescent fluid of the same density with itself will lose half its motion before it can describe the length of two of its diameters (by the same Cor. 7).

PROPOSITION XXXIX. THEOREM XXXI.

If a globe move uniformly forward through a- fluid inclosed and com- pressed in a cylindric ca?ial, its resistance is to the force by ivhich its whole motion may be generated or destroyed, in the time in lohich it 'describes eight third 'parts of its diameter, in a ratio co?npounded of

Sec. VIL] of natural philosophy. 345

the ratio of the orifice of the canal to the excess of that orifice above half the greatest circle of the globe ; and the duplicate ratio of the orifice of the canal to the excess of that orifice above the greatest circle of the globe ; and the ratio of the density of thefiuid to the density of the globe, nearly.

This appears by Cor. 2, Prop. XXXVII, and the demonstration pro- ceeds in the same manner as in the foregoing Proposition.

«

SCHOLIUM.

In the last two Propositions we suppose (as was done before in Lem. V) that ail the water which precedes the globe, and whose fluidity increases the resistance of the same, is congealed. Now if that water becomes fluid, it will somewhat increase the resistance. But in these Propositions that increase i^ so small, that it may be neglected, because the convex superfi- cies of the globe produces the very same elFect almost as the congelation of the water.

PROPOSITION XL. PROBLEM IX.

To find by phenomena the resistance of a globe moving through a per- fectly fluid compressed medium,.

Let A be the weight of the globe in vacuo, B its weight in the resisting medium, D the diameter of the globe, P a space which is to |D as the den- sity of the globe to the density of the medium, that is, as A to A — B, G the time in which the globe falling with the weight B without resistance describes the space F, and H the velocity which the body acquires by that fall. Then H will be the greatest velocity with which the globe can pos- sibly descend with the weight B in the resisting medium, by Cor. 2, Prop, XXXVIII'; and the resistance which the globe meets with, when descend- ing with that velocity, will be equal to its weight B ; and the resistance it meets with in any other velocity will be to the weight B in the duplicate ra- tio 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 of the fluid. That resistance which arises from the elasticity, tenacity, and friction of its parts, may be thus investigated.

l^et the globe be let fall so tliat it may descend in the fluid by the weight B ; and let P be the time of falling, and let that time be expressed in sec- onds, if the time G be given in seconds. Find the absolute number N

2P

agreeing to the logarithm 0,4342944819 — , and let L be the logarithm of

N + 1 the number — ^ — ; and the velocity acquired in falling will be

346

THE MATHEMATICAL PRINCIPLES

[Book II.

-— r H, and the height described will be

2PF

G

— 1,386294361 IF +

4,6()5170186LF.

If the fluid be of a sufficient depth, we may neglect the

2PF term 4,6051 70186LF; and -^ — 1,3862943611F will be the altitude

described, nearly. These things appear by Prop. IX, Book 11, and its Corol- laries, and are true upon this supposition, that the globe meets with no other resistance but that which arises from the inactivity of matter. Now if it really meet with any resistance of another kind, the descent will be slower, and from the quantity of that retardation will be known the quantity of this new resistance.

That the velocity and descent of a body falling in a fluid might more easily be known, I have composed the following table ; the first column of which denotes the times of descent ; the second shews the velocities ac- quired in falling, the greatest velocity being 100000000 ; the third exhib- its the spaces described by falling in those times, 2F being the space which the body describes in the time G with the greatest velocity ; and the fourth gives the spaces described with the greatest velocity in the same times.

2P

The numbers in the fourth column are — , and

subductins: the number

1,3862944 — 4,6051702L, are found the numbers in the third column ; and these numbers must be multiplied by the space F to obtain the spaces described in falling. A fifth column is added to all these, containing the spaces described in the same times by a body falling in vacuo with the force of B its comparative weight.

The Times

Velocities of the

The spaces de

The spaces descri-

The spaces de-

P.

body falling

scribed in fall-

bed with the

scribed by fall-

in the fluid.

ing in the fluid.

greatest motion.

ing in vacuo.

0,0()1G

9999929

O.OOOOOlF

0,002F

O.OOOOOlF

0,01G

999967-^"

0,0001F

0,02F

O^OOOlF

0,1G

9966799

0,0099834F

0,2F

O.OIF

0,2G

19737532

0,0397361F

0,4F

o!o4F

0,3G

29131261

0,0886815F

0,6F

0;09F

0,4G

37994896

0,1559070F

0,8F

0,1 6F

0,5G

46211716

0,2402290F

1,0F

0,25F

0,6G

53704957

0,3402706F

1,2F

0,36F

0,7G

60436778

0,4545405F

1,4F

0,49F

0,8G

66403677

0,581507lF

1,6F

0,64F

0,9G

71629787

0.7196609F

1,8F

0,8lF

IG

76159416

0.8675617F

2F

IF

2G

96402758

2,6500055F

4F

4F

3G

99505475

4,6186570F

6F

9F

4G

99932930

6,6143765F

8F

16F

5G

99990920

8,6137964F

lOF

25F

6G

99998771

10,6137179F

12F

36F

7G

99999834

12,6137073F

14F

49F

8G

99999980

14,6137059F

16F

64F

9G

99999997

16,6137057F

18F

81F

lOG

999999991

18,6137056F

20F

lOOF

i

Sec. VIL] of natural philosophy. 347

SCHOLIUM.

In order to investigate the resistances of lluids from experiments, I pro- cured a square wooden vessel, whose length and breadth on the inside was 9 inches English measure, and its depth 9 feet \ ; this I filled with rain- water : and having provided globes made up of wax, and lead included therein, I noted the times of the descents of these globes, the height through which they descended being 112 inches. A solid cubic foot oi English measure contains 7Q pounds troy weight of rain water ; and a solid inch contains |f ounces troy weight, or 253| grains ; and a globe of water of one inch in diameter contains 132,645 grains in air, or 132,8 grains in vacuo ; and any other globe will be as the excess of its weight in vacuo above its weight in water.

ExPER. 1. A globe whose weight was 156^ grains in air, and 77 grains in water, described the whole height of 112 inches in 4 seconds. And, upon repeating the experiment, the globe spent again the very same time of 4 ^conds in falling.

The weight of this globe in vacuo is 156|-f grains; and the excess of this weight above the weight of the globe in water is 79|f grains. Hence the diameter of the globe appears to be 0,84224 parts of an inch. Then it will be, as that excess to the weight of the globe in vacuo, so is the density of the water to the density of the globe ; and so is | parts of the diameter of the globe (viz. 2,24597 inches) to the space 2F, which will be therefore 4,4256 inches. Now a globe falling in vacuo with its whole weight of 156|-f grains in one second of time will describe 193|- inches ; and falling in 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 of time in the subduplicate ratio of the space F, or of 2,2128 inches to 95,219 inches, will describe 2,2128 inches, and will acquire the greatest velocity H with which it is capable of descending in water. Therefore the time G is 0",15244. And in this time G, with that greatest velocity H, the globe will describe the space 2P, which is 4,4256 inches ; and therefore in 4 sec- onds will describe a space of 116,1245 inches. Subduct the space 1,3862944P, or 3,0676 inches, and there will remain a space of 113,0569 inches, which the globe falling through water in a very wide vessel will describe in 4 sec- onds. But this space, by reason of the narrowness of the wooden vessel before mentioned, ought to be diminished in a ratio compounded of the sub- duplicate ratio of the orifice of the vessel to the excess of this orifice above half a great circle of the globe, and of the simple ratio of the same orifice to its excess above a great circle of the globe, that is, in a ratio of 1 to 0,9914. This done, we have a space of 112,08 inches, which a globe fall- ing through the water in this wooden vessel in 4 seconds of time ought nearly to describe by this theory; but it described 112 inches by the ex- periment.

348 THE MATHEMATICAL PRINCIPLES [BoOK 11.

ExPER. 2. Three equal globes, whose weights were severally 76 ^ grains in air, and Oj\ grains in water, were let fall successively ; and every one fell throuo^h the water in 15 seconds of time, describino- in its fall a heio-ht of 112 inches.

By computation, the weight of each globe in vacuo is 76y^2 grains ; the excess of this weight above the weight in water is 71 grains If ; the diam- eter of the globe 0,81296 of an inch; | parts of this diameter 2,16789 inches; the space 2F is 2,3217 inches; the space which a globe of 5j\ grains in weight would describe in one second without resistance, 12,808 inches, and the time G0",301056. Therefore the globe, with the greatest velocity it is capable of receiving from a weight of 5j\ grains in its de- scent through water, will describe in the time 0",301056 the space of 2,3217 inches: and in 15 seconds the space 115,678 inches. Subduct the space 1,3862944F, 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 the vessel were very wide. But because our vessel was narrow, the space ought to be diminished by about 0,895 of an inch. And so the space will remain 113,174 inches, which a globe falling in this vessel ought nearly to de- scribe in 15 seconds, by the theory. But by the experiment it described 112 inches. The difference is not sensible.

ExPER. 3. Three equal globes, whose weights were severally 121 grains in air, and 1 grain in water, were successively let fall ; and they fell through the water in the times 46", 47", and 50", describing a height of 112 inches.

By the theory, these globes ought to have fallen in about 40". Now whether their falling more slowly were occasioned from hence, that in slow motions the resistance arising from the force of inactivity does really bear a less proportion to the resistance arising from other causes ; or whether it is to be attributed to little bubbles that mio^ht chance to stick to the globes, or to the rarefaction of the wax by the warmth of the weatlier, or ' of the hand that let them fall ; or, lastly, whether it proceeded from some insensible 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, that the experiment may be certain, and to be depended on.

ExPER. 4. I began the foregoing experiments to investigate the resistan- ces of fluids, before I was acquainted with the theory laid down in the Propositions immediately preceding. Afterward, in order to examine the theory after it was discovered, I procured a wooden vessel, whose breadth on the inside was 8| inches, and its depth 1 5 feet and |. Then I made four globes of wax, with lead included, each of which weighed 139^ grains in air, and 7} grains in water. These I let fall, measuring the times of their falling in the water with a pendulum oscillating to half seconds. The globes were cold, and had remained so some time, both when they were

Sec. YIL] of natural philosophy. 349

weighed and Avhen they were let fall ; because warmth rarefies the wax, and by 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. Be- fore th-ey were let fall, they were totally immersed under water, lest, by the Aveight of any part of them that might chance to be above the water, their descent should be accelerated in its beginning. Then, when after their immersion they were perfectly at rest, they were let go with the greatest care, that they might not receive any impulse from the hand that let them down. And they fell successively in the times of 47|, 48|-, 50, and 51 os- cillations, describing a height of 15 feet and 2 inches. But the weather was now a little colder than when the globes were weighed, and therefore I repeated the experiment another day ; and then the globes fell in the times of 49, 491 50, and 53 ; and at a third trial in the times of 49i 50, 51, and 53 oscillations. And by making the experiment several times over, I found that the globes fell mostly in the times of 49 1- and 50 oscillations. When they fell slower, I suspect them to have been retarded by striking against the sides of the vessel.

Now, computing from the theory, the weight of the globe in vacuo is 139f grains; the excess of this weight above the weight of the globe in water 132^^ grains ; the diameter of the globe 0,99868 of an inch : | parts of the diameter 2,66315 inches ; the space 2F 2,8066 inches : the space which a globe weighing 7} grains falling without resistance describes in a second of time 9,88164 inches; and the time G0",376843. Therefore the globe with the greatest velocity with which it is capable of descending through the water by the force of a weight of 7|- grains, will in the time 0",376843 describe a space of 2,8066 inches, and in one second of time a space of 7,44766 inches, and in the time 25", or in 50 oscillations, the space 186,1915 inches. Subduct the space 1,386294P, or 1,9454 inches, and there will remain the space 184,2461 inches which the globe will describe in that time in a very wide vessel. Because our vessel was narrow, let this space be diminished in a ratio compounded of the subduplicate ratio of the orifice of the vessel to the excess of this orifice above half a great circle of the globe, and of the simple ratio of the same orifice to its excess above a great 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 time of 50 oscillations, nearly. But it described the space of 182 inches, by experiment, in 49 1- or 50 oscillations.

ExPER. 5. Four globes weighing 154| grains in air, and 21^ graiDS in water, 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 feet and 2 inches.

They ought by the theory to have fallen in the time of 29 oscillations, nearly.

350 THE MATHEMATICAL PRINCIPLES [BoOK 11.

ExPER. 6. Five globes, weighing 212f grains in air, and 79| in water, being several times let fall, fell in the times of 15, 15 1, 16, 17, and 18 os- cillations, describing a height of 15 feet and 2 inches.

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

ExPER. 7. Four globes, weighing 293 1 grains in air, and 35 1 grains in water, being let fall several times, fell in the times of 29| 30, 30i 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 weight and magnitude to fall, some swifter and some slower, I hit upon this ; that the globes, when they were first let go and began to fall, oscillated about their centres ; that side which chanced to be the heavier descending first, and producing an oscillating motion. Now by oscillating thus, the globe communicates a greater motion to the water than if it descended without any oscillations ; and by this communication loses part of its own motion with which it should descend ; and therefore as this oscillation is greater or less, it will be more or less retarded. Besides, the globe always recedes from that side of itself which is descending in the oscillation, and by so receding comes nearer to the sides of the vessel, so as even to strike against them sometimes. And the heavier the globes are, the stronger this oscil- lation is ; and the greater they are, the more is the water agitated by it. Therefore to diminish this oscillation of the globes, I made new ones of lead and wax, sticking the lead in one side of the globe very near its sur- face; 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 this means the oscillations became much less than before, and the times in which the 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 182 inches.

By the theory they ought to fall in about the time of 52 oscillations.

ExPER. 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 more than 13 oscillations, describing a height of 182 inches.

These globes by the theory ought to have fallen in the time of 11| os- cillations, nearly.

ExpER. 10. Four globes, weighing 384 grains in air, and 119|- in water, being let fall several times, fell in the times of 17f 18, IS}, and 19 oscilla- tions, describing a height of 181| inches. And when they fell in the time

Sec. YIL] of natural philosophy. 351

of 19 oscillationSj I sometimes heard them hit against the sides of the ves- sel before they reached the bottom.

By the theory they ought to have fallen in the time of 15|- 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 43|-, 44, 44|, 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 oscillations and f, nearly.

ExPER. 12. Three equal globes, weighing 141 grains in air, and 4| in water, being let fall several times, fell in the times of 61, 62, 63, 64, and 65 oscillations, describing a space of 182 inches.

And by the theory they ought to have fallen in 64^ oscillations, nearly.

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 the globes fell more swiftly, as in the sixth, ninth, and tenth experiments, the resistance was somewhat greater than in the duplicate ratio of the velocity. For the globes in falling oscillate a little ; and this oscillation, in those globes that are light and fall slowly, soon ceases by the weakness of the motion ; but in greater and heavier globes, the motion being strong, it con- tinues longer, and is not to be checked by the ambient water till after sev- eral oscillations. Besides, the more swiftly the globes move, the less are they pressed by the fluid at their hinder parts ; and if the velocity be per- petually increased, they will at last leave an empty space behind them, unless the compression of the fiuid be increased at the same time. For the compression of the fluid ought to be increased (by Prop. XXXII and XXXIIl) in the duplicate ratio of the velocity, in order to preserve the re- sistance in the same duplicate ratio. But because this is not done, the globes that move swiftly are not so much pressed at their hinder parts as the others ; and by the defect of this pressure it comes to pass that their resistance is a little greater than in a duplicate ratio of their velocity.

So that the theory agrees with the phenomena of bodies falling in water. It remains that we examine the phsenomena of bodies falling in air.

ExPER. 13. From the top of St. Paulas Church in London, in June 1710, there were let fall together two glass globes, one full of quicksilver, the other of air ; and in their fall they described a height of 220 English feet. A wooden table was suspended upon iron hinges on one side, and the other side of the same was supported by a wooden pin. The two globes lying upon this table were let fall together by pulling out the pin by means of an iron wire reaching from thence quite down to the ground ; so

352

THE MATHEMATICAL PRINCIPLES

[Book 11.

that, the pin being removed, the table, which had then no support but the iron hinges, fell downward, and turning round upon the hinges, gave leave to the globes to drop off from it. At the same instant, with the same pull of the iron wire that took out the pin, a pendulum oscillating to seconds was let go, and began to oscillate. The diameters and weights of the globes, and their times of falling, are exhibited in the following? table.

The globes filled with mercury.

The globes full of air.

Weights.

Diameters.

rimes 111 falling.

Weights Diameters. ^'^^^J^

908 grains

0,8 of an inch

4"

510 grains

5,1 inches|8"^

983

0,8

4 —

642

5,2 18

866

0,8

4

599 ■

5,1 |8

747

0,75

1 +

515

5,0

8i

808

0,75

4

483

5,0

8^

784 0,75

4- +

641

5,2

8

But the times observed must be corrected ; for the globes of mercury (by Galileo's theory), in 4 seconds of time, will de.scribe 257 English feet, and 220 feet in only 3"42"'. So that the wooden table, when the pin was taken out, did not turn upon its hinges so quickly as it ought to have done; and the slowness of that revolution hindered the descent of the globes at the beginning. For the globes lay about the middle of the table, and indeed were rather nearer to the axis upon which it turned than to the pin. And hence the times of falling were prolonged about IS'" ; and therefore ought to be corrected by subducting that excess, especially in the larger globes, which, by reason of the largeness of their diameters, lay longer upon the revolving table than the others. This being done, the times in which the six 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 being 5 inches in diameter, and 483 grains in weight, fell in 8" 1 2'", describing a space of 220 feet. The weight of a bulk of water equal to this globe is 16600 grains; and the weight of an equal bulk of air is '-11^-5^ grains, or \9j\ grains ; and therefore the weight of the globe in vacuo is 502y\ grains ; and this weight is to the weight of a bulk of air equal to the globe as 502fV to 19fV ; and so is 2F to | of the diameter of the globe, that is, to 13|- inches. Whence 2P becomes 28 feet 11 inches. A globe, falling m vacuo with its whole weight of 502 y\ grains, will in one second of time describe 193| inches as above ; and with the weight of 483 grains will de- scribe 185,905 inches ; and with that weight 483 grains in vacuo will de- scribe the space F, or 14 feet 5| inches, in the time of B7'" 58"", and ac- quire 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 fe^t and 5i inches. Subduct 1,3863F, or 20 feet and \ an inch, and there remain 225 feet 5 inches. This space, therefore, the falling globe ought by the

Sec. YIL]

OF NATURAL PHILOSOPHY.

353

theory to describe in 8" 12'". But by the experiment it described a space of 220 feet. The difference is insensible.

By like calculations applied to the other globes full of air, I composed

the following table.

The weipfhts of the globes.

The diam,9- ters.

The times ol tailing from a height ot 220 feet.

8" 12'"

510 grains

5,1 inches

642

5,2

7 42

599

5,1

7 42

515

5

7 57

483

5

8 12

641

5,2

7 42

The spaces which they would describe by the theory.

226 feet 11 inch

230 227 224 225 230

9

10

5

5

' 7

The excesses.

6 feet 10

7

4

5 10

11 inch, 9 0 5 5 7

ExPER. 14. Anno 1719, in the month of July, Dr. Desagidiers made some experiments of this kind again, by forming hogs' bladders into spheri- cal orbs ; which was done by means of a concave wooden sphere, which the bladders, being wetted well first, were put into. After that being blown full of air, they were obliged to fill up the spherical cavity that contained them ; and then, when dry, were taken out. These were let fall from the lantern on the top of the cupola of the same church, namely, from a height of 272 feet ; and at the same moment of time there was let fall a leaden globe, whose weight was about 2 pounds troy weight. And in the mean time some persons standing in the upper part of the church where the globes were let fall observed the whole times of falling ; and others stand- ing on the ground observed the differences of the times between the fall of the leaden weight and the fall of the bladder. The times were measured by pendulums oscillating to half seconds. And one of those that stood upon the ground had a machine vibrating four times in one second ; and another had another machine accurately made with a pendulum vibrating four times in a second also. One of those also who stood at the top of the church had a like machine ; and these instruments were so contrived, that their motions could be stopped or renewed at pleasure. Now the leaden globe fell in about four seconds and \ of time ; and from the addition of this time to the difference of time above spoken of, was collected the whole time in which the bladder was falling. The times which the five bladders spent in falling, after the leaden globe had reached the ground, were, the firsttime, 14|", 12f", 14f', 17f", andl6f; and the second time, 14i", 141", 14", 19", and 16f". Add to these 4^", the time in which the leaden globe was falling, and the whole times in which the five bladders fell were, the first time, 19", 17", ISf, 22", and 21i"; and the second time, ISf", ISi", 18|-", 231", and 21". The times observed at the top of the church were, the first time, 19f", 17i", 18|", 22}", and 21f"; and the second time, 19", 18f", 18|", 24". and 21|". But the bladders did not always fall directly down, but sometimes fluttered a little in the air, and waved to and fro, as

23

354

THE MATHEMATICAL PRINCIPLES

[Book II.

they were descending. And by these motions the times of their falling were prolonged, a,nd increased by half a second sometimes, and sometimes by a whole second. The second and fourth bladder fell most directly the first time, and the first and third the second time. The fifth bladder was wrinkled, and by its wrinkles was a little retarded. I found their diame- ters by their circumferences measured with a very fine thread wound about them twice. In the following table I have compared the experiments with the theory ; making the density of air to be to the density of rain-water as 1 to 860, and computing the spaces which by the theory the globes ought to describe in fallins:.

The weights of the blad- ders.

128 grams

156

137i

994

The diameters.

5,28 inches

5,19

5,3

The times of falling from a height ol 272 feet.

19"

17

18

22

214

jThe spaces which bj- the theory ought to have been described in those times.

271 feet 11

272 Ok

272 7 277 4 282 0

m,

The difference be tween the theory and the experi- ments.

— 0 ft. 1 in.

-1-0 Ok

-f 0 7

• 5  4 
  

-i- 10 0

Our theory, therefore, exhibits rightly, within a very little, all the re- sistance that globes moving either in air or in water meet with ; which ap- pears to be proportional to the densities of the fluids in globes of equal ve- locities and magnitudes.