To find by phenomena the resistance of a globe moving through a perfectly 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, F a space which is to ^(4/3)D as the density 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 possibly descend with the weight B in the resisting medium, by Cor. 2, Prop. XXXVIII; and the resistance which the globe meets with, when descending 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 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 of the fluid. That resistance which arises from the elasticity, tenacity, and friction of its parts, may be thus investigated.
Let the globe be let fall so that 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 seconds, if the time G be given in seconds. Find the absolute number N agreeing to the logarithm 0,43429448192P/G, and let L be the logarithm of the number (N+1)/N; and the velocity acquired in falling will be (N−1)/(N+1)H, and the height described will be 2PF/G−1,3862943611F−4,405170186LF. If the fluid be of a sufficient depth, we may neglect the term 4,405170186LF; and 2PF/G−1,3862943611F will be the altitude described, nearly. These things appear by Prop. IX, Book II, and its Corollaries, 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 acquired in falling, the greatest velocity being 100000000; the third exhibits 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. The numbers in the fourth column are 2P/G, and by subducting 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
P. Velocities of the
body falling
in the fluid. The spaces described
in falling
in the fluid. The spaces described
with the greatest
motion The spaces described
falling in
vacuo.
0,001G 99999^(29/30) 0,000001F 0,002F 0,000001F
0,01G 999967 0,0001F 0,02F 0,0001F
0,1G 9966799 0,0099834F 0,2F 0,01F
0,2G 19737532 0,0397361F 0,4F 0,04F
0,3G 29131261 0,0886815F 0,6F 0,09F
0,4G 37994896 0,1559070F 0,8F 0,16F
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 0,5815071F 1,6F 0,64F
0,9G 71629787 0,7196609F 1,8F 0,81F
1G 76159416 0,8675617F 2F 1F
2G 96402758 2,6500055F 4F 4F
3G 99505475 4,6186570F 6F 9F
4G 99932930 6,6143765F 8F 16F
5G 99990920 8,6137964F 10F 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
10G 99999999^(3/5) 18,6137056F 20F 100F
SCHOLIUM.
In order to investigate the resistances of fluids from experiments, I procured a square wooden vessel, whose length and breadth on the inside was 9 inches English measure, and its depth 9 feet ^(1/2); 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 of English measure contains 76 pounds troy weight of rain water; and a solid inch contains ^(19/36) ounces troy weight, or 253^(1/3) 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^(1/4) 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 seconds in falling.
The weight of this globe in vacuo is 156^(13/38) grains; and the excess of this weight above the weight of the globe in water is 79^(13/38) 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 ^(8/3) 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^(13/38) grains in one second of time will describe 193^(1/3) 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 2F, which is 4,4256 inches; and therefore in 4 seconds will describe a space of 116,1245 inches. Subduct the space 1,3862944F, 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 seconds. But this space, by reason of the narrowness of the wooden vessel before mentioned, ought to 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, that is, in a ratio of 1 to 0,9914. 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 ought nearly to describe by this theory; but it described 112 inches by the experiment.
EXPER. 2. Three equal globes, whose weights were severally 76^(1/3) grains in air, and 5^(1/16) grains in water, were let fall successively; and every one fell through the water in 15 seconds of time, describing in its fall a height of 112 inches.
By computation, the weight of each globe in vacuo is 76^(5/12) grains; the excess of this weight above the weight in water is 71 grains ^(17/48); the diameter of the globe 0,81296 of an inch; ^(8/3) parts of this diameter 2,16789 inches; the space 2F is 2,3217 inches; the space which a globe of 5^(1/16) grains in weight would describe in one second without resistance, 12,808 inches, and the time GO^(′′),301056. Therefore the globe, with the greatest velocity it is capable of receiving from a weight of 5^(1/16) grains in its descent 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 describe 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 might chance to stick to the globes, or to the rarefaction of the wax by the warmth of the weather, 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 resistances 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^(2/3) inches, and its depth 15 feet and ^(1/3). Then I made four globes of wax, with lead included, each of which weighed 139^(1/4) grains in air, and 7^(1/8) 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 weighed and when 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. Before they were let fall, they were totally immersed under water, lest, by the weight 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^(1/2), 48^(1/2), 50, and 51 oscillations, 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, 49^(1/2), 50, and 53; and at a third trial in the times of 49^(1/2), 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^(1/2) 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 139^(2/5) grains; the excess of this weight above the weight of the globe in water 132^(11/40) grains; the diameter of the globe 0,99868 of an inch; ^(8/3) parts of the diameter 2,66315 inches; the space 2F 2,8066 inches; the space which a globe weighing 7^(1/8) grains falling without resistance describes in a second of time 9,88164 inches; and the time GO^(′′),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^(1/8) 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 1,386294F, 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^(1/2) or 50 oscillations.
EXPER. 5. Four globes weighing 154^(3/8) grains in air, and 21^(1/2) grains in water, being let fall several times, fell in the times of 28^(1/2), 29, 29^(1/2), 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.
EXPER. 6. Five globes, weighing 212^(3/8) grains in air, and 79^(1/2) in water, being several times let fall, fell in the times of 15, 15^(1/2), 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 of 15 oscillations, nearly.
EXPER. 7. Four globes, weighing 293^(3/8) grains in air, and 35^(7/8) grains in water, being let fall several times, fell in the times of 29^(1/2), 30, 30^(1/2), 31, 32, and 33 oscillations, describing a height of 15 feet and 1 inch and ^(1/2).
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 oscillation 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 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 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^(1/2) 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^(1/4) grains in air, and 140^(3/4) 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^(1/3) oscillations, nearly.
EXPER. 10. Four globes, weighing 384 grains in air, and 119^(1/2) in water, being let fall several times, fell in the times of 17^(3/4), 18, 18^(1/2), and 19 oscillations, describing a height of 181^(1/2) inches. And when they fell in the time of 19 oscillations, I sometimes heard them hit against the sides of the vessel before they reached the bottom.
By the theory they ought to have fallen in the time of 15^(5/9) oscillations, nearly.
EXPER. 11. Three equal globes, weighing 48 grains in the air, and 3^(20/32); in water, being several times let fall, fell in the times of 43^(1/2), 44, 44^(1/2), 45, and 46 oscillations, and mostly in 44 and 45. describing a height of 182^(1/2) inches, nearly.
By the theory they ought to have fallen in the time of 46 oscillations and ^(5/9), nearly.
EXPER. 12. Three equal globes, weighing 141 grains in air, and 4^(3/8) 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^(1/2) 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 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 are they 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 the compression of the fluid ought to be increased (by Prop. XXXII and XXXIII)in the duplicate ratio of the velocity, in order to preserve the resistance 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 phænomena of bodies falling in water. It remains that we examine the phænomena of bodies falling in air.
EXPER. 13. From the top of St. Paul's 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 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. Times in
falling. Weights. Diameters. Times in
falling.
908 grains 0,8 of an inch 4'' 510 grains 5,1 inches 8^(′′)^(1/2)
983 0,8 4 - 642 5,2 8
866 0,8 4 599 5,1 8
747 0,75 4 + 515 5,0 8^(′′)^(1/4)
808 0,75 4 483 5,0 8^(′′)^(1/2)
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 describe 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 18‴; 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″ 12‴, 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 ^(16600/860) grains, or 19^(3/10) grains; and therefore the weight of the globe in vacuo is 502^(3/10) grains; and this weight is to the weight of a bulk of air equal to the globe as 502^(3/10) to 19^(3/10); and so is 2F to ^(8/3) of the diameter of the globe, that is, to 13^(1/3) inches. Whence 2F becomes 28 feet 11 inches. A globe, falling in vacuo with its whole weight of 502^(3/10) grains, will in one second of time describe 193^(1/3) 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 5^(1/2) 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 and 5^(1/3) inches. Subduct 1,3863F, or 20 feet and ^(1/2) an inch, and there remain 225 feet 5 inches. This space, therefore, the falling globe ought by the 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 weights
of the globes. The diameters. The times of
falling from
a height of
220 feet. The spaces which they
would describe by the
theory. The excesses.
510 grains 5,1 inches 8^(′′) 12^(′′′) 226 feet 11 inch. 6 feet 11 inch.
642 5,2 7 42 230 9 10 9
599 5,1 7 42 227 10 7 0
515 5 7 57 224 5 4 5
483 5 8 12 225 5 5 5
641 5,2 7 42 230 7 10 7
EXPER. 14. Anno 1719, in the month of July, Dr. Desaguliers made some experiments of this kind again, by forming hogs' bladders into spherical 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 standing 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 ^(1/4) 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 first time, 14^(3/4), 12^(3/4)^(′′), 14^(5/8)^(′′), 17^(3/4)^(′′), and 16^(7/8)^(′′); and the second time, 14^(1/2)^(′′), 14^(1/4)^(′′), 14″, 19″, and 16^(3/4)^(′′). Add to these 4^(1/4)^(′′), 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″, 18^(7/8)^(′′), 22″, and 21^(1/8)^(′′); and the second time, 18^(3/4)^(′′), 18^(1/2)^(′′), 18^(1/4)^(′′), 23^(1/4)^(′′), and 21″. The times observed at the top of the church were, the first time, 19^(3/8)^(′′), 17^(1/4)^(′′), 18^(3/4) ″, 22^(1/8) ″, and 21^(5/8)^(′′); and the second time, 19″, 18^(5/8)^(′′), 18^(3/8)^(′′), 24″, and 21^(1/4)^(′′). But the bladders did not always fall directly down, but sometimes fluttered a little in the air, and waved to and fro, as they were descending. And by these motions the times of their falling were prolonged, and 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 diameters 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 falling.
The weight
of the
bladders. The diameters. The times of
falling from
a height of
272 feet. The spaces which by
the theory ought to
have been described
in those times. The difference
between the
theory and the
experiments.
128 grains 5,28 inches 19^(′′) 271 feet 11 in. - 0 ft. 1 in.
156 5,19 7 272 0^(1/2) + 0 0^(1/2)
137^(1/2) 5,3 18 272 7 + 0 7
97^(1/2) 5,26 22 277 4 + 5 4
99^(1/8) 5 21^(1/8) 282 0 + 10 0
Our theory, therefore, exhibits rightly, within a very little, all the resistance that globes moving either in air or in water meet with; which appears to 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 globes moving in air, water, and quicksilver, are as the densities of the fluids. We here prove the same more accurately by experiments of bodies falling in air and water. For pendulums at each oscillation excite a motion in the fluid always contrary to the motion of the pendulum in its return; and the resistance arising from this motion, as also the resistance of the thread by which the pendulum is suspended, makes the whole resistance of a pendulum greater than the resistance deduced from the experiments of falling bodies. For by the experiments of pendulums described in that Scholium, a globe of the same density as water in describing the length of its semi-diameter in air would lose the ^(1/3342) part of its motion. But by the theory delivered in this seventh Section, and confirmed by experiments of falling bodies, the same globe in describing the same length would lose only a part of its motion equal to ^(1/4586), supposing the density of water to be to the density of air as 860 to 1. Therefore the resistances were found greater by the experiments of pendulums (for the reasons just mentioned) than by the experiments of falling globes; and that in the ratio of about 4 to 3. But yet since the resistances of pendulums oscillating in air, water, and quicksilver, are alike increased by like causes, the proportion of the resistances in these mediums will be rightly enough exhibited by the experiments of pendulums, as well as by the experiments of falling bodies. And from all this it may be concluded, that the resistances of bodies, moving in any fluids whatsoever, though of the most extreme fluidity, are, cæteris paribus, as the densities of the fluids.
These things being thus established, we may now determine what part of its motion any globe projected in any fluid whatsoever would nearly lose in a given time. Let D be the diameter of the globe, and V its velocity at the beginning of its motion, and T the time in which a globe with the velocity V can describe in vacuo a space that is, to the space 8/3D as the density of the globe to the density of the fluid; and the globe projected in that fluid will, in any other time t lose the part tV/(T+t), the part TV/(T+t) remaining; and will describe a space, which will be to that described in the same time in vacuo with the uniform velocity V, as the logarithm of the number (T+t)/T multiplied by the number 2,302585093 is to the number t/T, by Cor. 7, Prop. XXXV. In slow motions the resistance may be a little less, because the figure of a globe is more adapted to motion than the figure of a cylinder described with the same diameter. In swift motions the resistance may be a little greater, because the elasticity and compression of the fluid do not increase in the duplicate ratio of the velocity. But these little niceties I take no notice of.
And though air, water, quicksilver, and the like fluids, by the division of their parts in infinitum, should be subtilized, and become mediums infinitely fluid, nevertheless, the resistance they would make to projected globes would be the same. For the resistance considered in the preceding Propositions arises from the inactivity of the matter; and the inactivity of matter is essential to bodies, and always proportional to the quantity of matter. By the division of the parts of the fluid the resistance arising from the tenacity and friction of the parts may be indeed diminished; but the quantity of matter will not be at all diminished by this division; and if the quantity of matter be the same, its force of inactivity will be the same; and therefore the resistance here spoken of will be the same, as being always proportional to that force. To diminish this resistance, the quantity of matter in the spaces through which the bodies move must be diminished; and therefore the celestial spaces, through which the globes of the planets and comets are perpetually passing towards all parts, with the utmost freedom, and without the least sensible diminution of their motion, must be utterly void of any corporeal fluid, excepting, perhaps, some extremely rare vapours and the rays of light.
Projectiles excite a motion in fluids as they pass through them, and this motion arises from the excess of the pressure of the fluid at the fore parts of the projectile above the pressure of the same at the hinder parts; and cannot be less in mediums infinitely fluid than it is in air, water, and quicksilver, in proportion to the density of matter in each. Now this excess of pressure does, in proportion to its quantity, not only excite a motion in the fluid, but also acts upon the projectile so as to retard its motion; and therefore the resistance in every fluid is as the motion excited by the projectile in the fluid; and cannot be less in the most subtile æther in proportion to the density of that æther, than it is in air, water, and quicksilver, in proportion to the densities of those fluids.