Principia Mathematica (Motte Translation, 1848) — part 23 of 45

Cor. 2. The shorter oscillations are more isochronal, and very short ones are performed nearly in the same times as in a non-resisting medium. But the times of those which are performed in greater arcs are a little greater, because the resistance in the descent of the body, by which the time is prolonged, is greater, in proportion to \h.Q length described in the descent than the resistance in the subsequent ascent, by which the time is contracted. But the time of the oscillations, both short and long, seems to be prolonged in some measure by the motion of the medium. For retard- ed bodies are resisted somewhat less in proportion to the velocity, and ac- celerated bodies somewhat more than those that proceed uniformly forwards j

Sec. VI.]

OF NATURAL PHILOSOPHY.

307

because the medium, by the motion it has received from the bodies, going forwards the same way with them, is more agitated in the former case, and less in the latter ; and so conspires more or less with the bodies moved. Therefore it resists the pendulums in their descent more, and in their as- cent less, than in proportion to the velocity ; and these two causes concur- ring prolong the time.

PROPOSITION XXVIII. THEOREM XXIII.

If a funependulous body, oscillating in a cycloid, be resisted in the ratio of the moments of the tinu, its resistance will be to the force of grav- ity as the excess of the arc described in the whole descent above the arc described in the subsequent ascent to twice the length of the pen- dulum.

Let BC represent the arc described ia the descent, Ga the arc described in the ascent, and Ka the difference of the arcs : and things remaining as they were constructed and demonstrated in Prop. XXV, the force with which the oscillating body is urged in any place D will be to the force of resistance as the arc CD to the arc CO, which is half of that difference ka. Therefore the force with which the oscillating body is urged at the beginning or the highest point of the cycloid, that is, the force of gravity, will be to the resistance as the arc of the cycloid, be- tween that highest point and lowest point C, is to the arc CO ; that is (doubling those arcs), as the whole cycloidal arc, or twice the length of the pendulum, to the arc Aa. Q.B.D.

PROPOSITION XXIX. PROBLEM VI.

Supposing that a body oscillating in a cycloid is resisted in a duplicate ratio of the velocity : to find the resistance in each place. Let Ba be an arc described in one entire oscillation, C the lowest point

\

/-R Q M

of the cycloid, and CZ half the whole cycloidal arc, equal to the length of the pendulum ; and let it be required to find the resistance of the body in

308 THE MATHEMATICAL PRINCIPLES [BoOK IL

any place D, Cut the indefinite right line OQ, in the points O, S, P, Q, so that (erecting the perpendiculars OK, ST, PI, Q,E, and with the centre O, and the aysmptotes OK, OQ, describing the hyperbola TIGE cutting the perpendiculars ST, PI, Q,E in T, I, and E, and through the point I drawing KF, parallel to the asymptote OQ, meeting the asymptote OK ia K, and the perpendiculars ST and QE in L and F) the hyperbolic area PIEQ, may be to the hyperbolic area PITS as the arc BC, described in the descent of the body, to the arc Ca described in the ascent ; and that the area lEF may be to the area ILT as OQ to OS. Then with the perpen- dicular MN cut off the hyperbolic area PINM, and let that area be to the hyperbolic area PIEQ as the arc CZ to the arc BC described in the de- scent. And if the perpendicular RG cut off the hyperbolic area PIGR, which shall be to the area PIEQ as any arc CD to the arc BC described in the whole descent, the resistance in any place D will be to the force of

OR gravity as the area ^=r^ lEF — IGH to the area PINM.

For since the forces arising from gravity with which the body is urged in the places Z, B, D, a, are as the arcs CZ, CB, CD, Ca and those arcs are as the areas PINM, PIEQ, PIGR, PITS ; let those areas be the exponents both of the arcs and of the forces respectively. Let Dd be a very small space described by the body in its descent : and let it be expressed by the very small area RGgr comprehended between the parallels RG, rg ; and produce rg to h, so that GHhg and KGgr may be the contemporane- ous decrements of the areas IGH, PIGR. And the increment GHhg —

Rr Rr OR

^ lEF, or Rr x HG — ^ lEF, of the area ^ lEF —IGH will be

lEF

to the decrement RGgr, or Rr X RG, of the area PIGR, as HG — -pr^

OR

to RG ; and therefore as OR X HG — ^ lEF to OR X GR or OP X

PI, that is (because of the equal quantities OR X HG, OR X HR — OR X GR, ORHK — OPIK, PIHR and PIGR + IGH), as PIGR + IGH — OT? OT?

^ lEF to OPIK. Therefore if the area ^^ lEF — IGH be called

Y, and RGg-r the decrement of the area PIGR be given, the increment of the area Y will be as PIGR — Y.

Then if Y represent the force arising from the gravity, proportional to the arc CD to be described, by which the body is acted upon in D, and R be put for the resistance, V — R will be the whole force with which the body is urged in D. Therefore the increment of the velocity is as V — R and the particle of time in which it is generated conjunctly. But the ve- locity itself is as the contemporaneous increment of the space described di-

Sec. VL]

OF NATURAL PHILOSOPHY.

309

rectly and the same particle of time inversely. Therefore, since the re- sistance is, by the supposition, as the square of the velocity, the increment of the resistance will (by Lem. II) be as the velocity and the increment of the velocity conjunctly, that is, as the moment of the space and V — R conjunctly ; and, therefore, if the moment of the space be given, as V — R ; that is, if for the force V we put its exponent PIGR, and the resist- ance R be expressed by any other area Z, as PIGR — Z.

Therefore the area PIGR uniformly decreasing by the subduction of given moments, the area Y increases in proportion of PIGR — Y, and the area Z in proportion of PIGR — Z. And therefore if the areas Y and Z begin together, and at the beginning are equal, these, by the addition of equal moments, will continue to be equal ; and in like man- ner decreasing by equal momentsj will vanish together. And, vice versa^ if they together begin and vanish, they will have equal moments and be always equal ; and that, because if the resistance Z be augmented, the ve- locity together with the arc Ca, described in the ascent of the body, will be diminished ; and the point in which all the motion together with the re- sistance ceases coming nearer to the point 0, the resistance vanishes sooner than the area Y. And the contrary will happen when the resistance is diminished.

Now the area Z begins and ends where the resistance is nothing, that is, at the beginning of the motion where the arc CD is equal to the arc CB,

and the right line RG falls upon the right line Q,E ; and at the end of the motion where the arc CD is equal to the arc Ca, and RG falls upon

the right line ST.

ryrt

And the area Y or pr-^ lEF

IGH begins and ends

also where the resistance is nothing, and therefore where

OR OGl

lEF and

IGH are equal ; that is (by the construction), where the right line RG falls successively upon the right lines QE and ST. Therefore those areas begin and vanish together, and are therefore always equal. Therefore the area

OR

-_- lEF — IGH is equal to the area Z, by which the resistance is ex- pressed, and therefore is to the area PINM, by which the gravity is ex- pressed, as the resistance to the gravity. GI.E.D.

310

THE MATHEMATICAL PRINCIPLES

[Book 11.

Cor. 1. Therefore the resistance in tlie lowest place C is to the force

OP

of gravity as the area ^^ lEF to the area PINM.

Cor. 2. But it becomes greatest where the area PIHR is to the area lEF as OR to OQ,. For in that case its moment (that is, PIGR — Y) becomes nothing.

Cor. 3. Hence also may be known the velocity in each place, as being in the subduplicate ratio of the resistance, and at the beginning of the mo- tion equal to the velocity of the body oscillating in the same cycloid with- out any resistance.

However, by reason of the difficulty of the calculation by which the re- sistance and the velocity are found by this Proposition, we have thought fit to subjoin the Proposition following.

PROPOSITION XXX. THEOREM XXIV.

Jf a right line aB be equal to the arc of a cycloid which an oscillating body describes, and at each of its points D the perpendiculars DK be erected, which shall be to the length of the pendulum as the resistance of the body in the corresponding points of the arc to the force of grav- ity ; I say, that the difference between the arc described in the ivhole descent and the arc described in the whole subsequmt ascent drawn into half the sum of the sarne arcs ivill be equal to the area BKa lohich all those perpendiculars take up.

Let the arc of the cycloid, de- scribed in one entire oscillation, be expressed by the right line aB, equal to it, and the arc which would have been described in vacuo by the length AB. Bisect AB in C, and the point C will represent -^ the lowest point of the cycloid, and CD will be as the force arising from gravity, with which the body in D is urged in the direction of the tangent of the cycloid, and will have the same ratio to the length of the pendulum as the force in D has to the force of gravity. Let that force, therefore, be expressed by that length CD, and the force of gravity by the length of the pendulum ; and if in DE you take DK in the same ratio to the length of the pendulum as the resistance has to the gravity, DK will be the exponent of the resistance. From the centre C with the interval CA or CB describe a semi-circle BEeA. Let the body describe, in the least time, the space Bd ; and, erecting the per- pendiculars DE, de, meeting the circumference in E and e, they will be as the velocities which the body descending in vacuo from the point B would acquire in the places D and d. This appears by Prop. LII, Book L Let,

MN" ^

Sec. VL] of natural philosophy. 311

therefore, these velocities be expressed by those perpendiculars DE, de ; and let DF be the velocity which it acquires in D by falling from B in the resisting medium. And if from the centre C with the interval CF we describe 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 farther resistance, ascend, and df the velocity it would acquire in d. Whence, also, if ¥g represent the moment of the velocity which the body D, in de- scribing the least space Dd, loses by the resistance of the medium ; and CN be taken equal to C^ ; then will N be the place to which the body, if it met no farther resistance, would thenceforward ascend, and MN will be the decrement of the ascent arising from the loss of that velocity. Draw F'/7^ perpendicular to df, and the decrement ¥g of the velocity DF gener- ated by the resistance DK will be to the increment /m of the same velo- city, generated by the force CD, as the generating force DK to the gener- ating force CD. But because of the similar triangles ¥mf, Fhg, FDC, fm is to Fm or D^ as CD to DF ; and, ex cBqiw, Fg to Dd as DK to DF. Also FA is to Fg as DF to CF ; and, ex cequo perturbate, Fh or MN to Bd as DK to CF or CM ; and therefore tlie sum of all the MN X CM will be equal to the sum of all the T)d X DK. At the moveable point M suppose always a rectangular ordinate erected equal to the inde- terminate CM, which by a continual motion is drawn into the whole length Ka ; and the trapezium described by that motion, or its equal, the rectangle A.a X \aE, will be equal to the sum of all the MN X CM, and therefore to the sum of all the Dd X DK, that is, to the area BKVTa.

aE.D.

Cor. Hence from the law of resistance, and the difference Aa of the arcs Ca, CB, may be collected the proportion of the resistance to the grav- ity nearly.

For if the resistance DK be uniform, the figure BKTa will be a rec- tangle under Bet and DK ; and thence the rectangle under \Ea and Aa will be equal to the rectangle under Ba and DK, and DK will be equal to i-Aa. Wherefore since DK is the exponent of the resistance, and the length of the pendulum the exponent of the gravity, the resistance will be to the gravity as \A.a to the length of the pendulum ; altogether as in Prop. XXVIII is demonstrated.

If the resistance be as the velocity, the figure BKT« will be nearly 'an ellipsis. For if a body, in a non-resisting medium, by one entire oscilla- tion, should describe the length BA, the velocity in any place D would be as the ordinate DE of the circle described on the diameter AB. There- fore since Ba in the resisting medium, and BA in the non-resisting one, are described nearly in the same times ; and therefore the velocities in each of the points of Ba are to the velocities in the correspondent points of the length BA nearly as Ba is to BA , the velocity in the point D in the re-

312

THE MATHEMATICAL PRINCIPLES

[Book II.

sisting medium will be as the ordinate of the circle or ellipsis described upon the diameter Ba ; and therefore the figure BKVTa will be nearly an ellipsis. Since the resistance is su]Dposed proportional to the velocity, let OY be the exponent of the resistance in the middle point O ; and an ellip- sis BRVSa described with the centre O, and the semi-axes OB, OY, will be ne-arly equal to the figure BKYTa, and to its equal the rectangle Aa X BO. . Therefore Aa X BO is to OY X BO as the area of this ellipsis to OY X BO ; that is, Aa is to OY as the area of the semi-circle to the square of the radius, or as 11 to 7 nearly ; and, therefore, yVAa is to the length of the pendulum as the resistance of the oscillating body in O to its gravity.

Now if the resistance DK be in the duplicate ratior of the velocity, the figure BKYTa will be almost a parabola having Y for its vertex and OY for its axis, and therefore will be nearly equal to the rectangle under f Ba and OY. Therefore the rectangle under |Ba and Aa is equal to the rec- tangle f Ba X OY, and therefore OY is equal to |Aa ; and therefore the resistance in O made to the oscillating body is to its gravity as |Aa to the length of the pendulum.

And I take these conclusions to be accurate enough for practical uses. For since an ellipsis or parabola BRYSa falls in with the figure BKYTa in the middle point Y, that figure, if greater towards the part BRY or VSa than the other, is less towards the contrary part, and is therefore nearly equal to it.

PROPOSITION XXXI. THEOREM XXY.

If the resistance inade to an oscillating body in each of the proportional parts of the arcs described be augmented or diminished in a given ra- tio, the difference betioeen the arc described in the descent and the arc described in the subsequent ascent will be augmented or di?ninished in the same ratio.

For that difference arises from the retardation of the pendulum by the resistance of the medium, and therefore is as the whole re- tardation and the retarding resist- ance proportional thereto. In the foregoing Proposition the rectaif- M ]sr ft? c o c/Ti B gle under the right line iaB and

the difference Aa of the arcs CB, Ca, was equal to the area BKTa. And that area, if the length aB remains, is augmented or diminished in the ra- tio of the ordinates DK ; that is, in the ratio of the resistance and is there- fore as the length aB and the resistance conjunctly. And therefore the rectangle under Aa and |aB is as aB and the resistance conjunctly, and therefore Aa is as the resistance. Q.E.D.

Sec. YL] of natural philosophy. 313

Cor. 1. Hence if the resistance be as the yelocity, the difference of the arcs in the same medium will be as the whole arc described : and the contrary.

Cor. 2. If the resistance be in the duplicate ratio of the velocity, that difference 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 any other 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 partly in the ratio of the whole arc, and partly in the duplicate ratio of it : and the contrary. So that the law and ratio of the resistance will be the same for the velocity as the law and ratio of that difference for the length of 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 difference for the length of the arc described, there will be had also the ratio of the increment or decrement of the resistance for a greater or less velocity.

GENERAL SCHOLIUM.

From these propositions we may find the resistance of mediums by pen- dulums oscillating therein. I found the resistance of the air by the fol- lowing experiments. I suspended a wooden globe or ball weighing Sr/j ounces troy, its diameter 6^ London inches, by a fine thread on a firm hook, so that the distance between the hook and the centre of oscillation of the globe was 10|- feet. I marked on the thread a point 10 feet and 1 inch distant from the centre of suspension; and even with that point I placed a ruler divided into inches, by the help whereof I observed the lengths of the arcs described by the pendulum. Then I numbered the oscillations in which the globe would lose | part of its motion. If the pendulum was drawn aside from the perpendicular to the distance of 2 inches, and thence let go, so that in its whole descent it described an arc of 2 inches, and in the first whole oscillation, compounded of the descent and subsequent ascent, an arc of almost 4 inches, the same in 164 oscillations lost I part of its motion, so as in its last ascent to describe an arc of If inches. If in the first descent it described an arc of 4 inches, it lost | part of its mo- tion in 121 oscillations, so as in its last ascent to describe an arc of Sc- inches. If in the first descent it described an arc of 8, 16, 32, or 64 inches, it lost i part of its motion in 69, 35|, 18|-, 9| oscillations, respectively. Therefore the difference between the arcs described in the first descent and the last ascent was in the 1st, 2d, 3d, 4th, 5th, 6th cases, I, |, 1, 2, 4, 8 inches respectively. Divide those differences by the number of oscillations in each case, and in one mean oscillation, wherein an arc of 3|, 7i 15, 30,

314 THE 'MATHEMATICAL PRINCIPLES [BoOK II.

60, 120 inches was described; the difference of the arcs described in the descent and subsequent ascent will be g|e, a^j, j, j, /y, ||- parts of an inch, respectively. But these diiferences in the greater oscillations are in the duplicate ratio of the arcs described nearly, but in lesser oscillations something greater than in that ratio ; and therefore (by Cor. 2, Prop. XXXI of this Book) the resistance of the globe, when it moves very swift, is in tlie 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 arcs

to be AY + BV=^ + CV-. Since the greatest velocities are in the cycloid

as ^ the arcs described in oscillating, and in the circle as | the chords of

those arcs ; and therefore in equal arcs are greater in the cycloid than in

the circle in the ratio of | the arcs to their chords; but the times in the

circle 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 and

the square of the time* conjunctly) are nep.rly the same in both curves : for

in the cycloid those differences must be on the one hand augmented, with

the resistance, in about the duplicate ratio of the arc to the chord, because

of the velocity augmented in the simple ratio of the same; and on the

other hand diminished, with the square of the time, in the same duplicate

ratio. Therefore to reduce these observations to the cycloid, we must take

the same differences of the arcs as were observed in the circle, and suppose

the greatest velocities analogous to the half, or the whole arcs, that is, to

the numbers |, 1, 2, 4, 8, 16. Therefore in the 2d, 4th, and 6 th cases, put

1, 4, and 16 for Y; and the difference of the arcs in the 2d case will become

1 2

-^- = A + B + C; inthe4th case. ^^ = 4A + 8B + 16C ; in the 6th

g

case, Q^ == 16A + 64B -f 256C. These equations reduced give A =~

0,0000916, B = 0,0010847, and C == 0,0029558. Therefore the difference

of the arcs is as 0,0000916Y + 0,0010847Y^ + 0,0029558Y2 : and there- fore since (by Cor. Prop. XXX, applied to this case) the resistance of the globe in the middle of the arc described in oscillating, where the velocity

is Y, is to its weight as y^AY + y^BY^ + f CY^ to the length of the pendulum, if for A, B, and C you put the numbers found, the resistance of

the globe will be to itsweight asO 0000583Y + 0,0007593Y^ + 0,0022169Y2 to the length of the pendulum between the centre of suspension and the ruler, that is, to 121 inches. Therefore since Y in the second case repre- sents 1, in the 4th case 4, and in the 6th case 1 6, the resistance will be to the weight of the globe, in the 2d case, as 0,0030345 to 121 j in the 4th; aa 0;04J748 to 121 ; in the 6th, as 0,6ir05 to 121.

Sec. VI.] OF natural philosophy. 315

The arc, which the point marked in the thread described in the 6th case,

o was of 120 — - q7, or 119/o inches. And therefore since the radius was

121 inches, and the length of the pendulum between the point of suspen- sion and the centre of the globe was 126 inches, the arc which the centre of the globe described was 124/y inches. Because the greatest velocity of the oscillating body, by reason of the resistance of the air, does not fall on the lowest point of the arc described, but near the middle place of the whole arc, this velocity will be nearly the same as if the globe in its whole descent in a non-resisting medium should describe 62^-^2 inches, the half of that arc, and that in a cycloid, to which we have above reduced the motion of the pendulum ; and therefore that velocity will be equal to that which the globe would acquire by' falling perpendicularly from a height equal to the versed sine of that arc. But that versed sine in the cycloid is to that arc 62/2 ^s the same arc to twice the length of the pendulum 252, and there- fore equal to 15,278 inches. Therefore the velocity of the pendulum is the same which a body would acquire by falling, and in its fall describing a space of 15,278 inches. Therefore with such a velocity the globe meets with a resistance which is to its weight as 0,61705 to 121, or (if we take that part only of the resistance which is in the duplicate ratio of the ve- locity) as 0,56752 to 121.

I found, by an hydrostatical experiment, that the weight of this wooden globe was to the weight of a globe of water of the same magnitude as 55 to 97 : and therefore since 121 is to 213,4 in the same ratio, the resistance made to this globe of water, moving forwards with the above-mentioned velocity, will be to its weight as 0,56752 to 213,4, that is, as 1 to 376/0-. Whence since the weight of a globe of water, in the time in which the globe with a velocity uniformly continued describes a length of 30,556 inches, will gaierate all that velocity in the falling globe, it is manifest that the force of resistance uniformly continued in the same time will take away a velocity, which will be less than the other in the ratio of 1 to 376/-o,

that is, the ,. ^ ■ part of the whole velocity. And therefore in the time

that the globe, with the same velocity uniformly continued, would describe the length of its semi-diameter, or 3/g inches, it would lose the 3 3V 2 P^^* of its motion.

I also counted the oscillations in which the pendulum lost | part of its motion. In the following table the upper numbers denote the length of the arc 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 as- cent ; and in the lowest place are the numbers of the oscillations. I give an account of this experiment, as being more accurate than that in which

316 THE MATHEMATICAL PRINCIPLES [BoOK 11.

only I part of tlie motion was lost. I leave the calculation to such as are disposed to make it.

Mrst descent ... 2 4 8 16 32 64

Last ascent . . . 1| 3 6 12 24 48

Nmnh. of oscill. . . 374 272 162i 83|- 41| 22|

I afterward suspended a leaden globe of 2 inches in diameter, weighing 26^ ounces troy by the same thread, so that between the centre of the globe and the point of suspension there was an interval of 10| feet, and I counted the oscillations in which a given part of the motion was lost. The first of the following tables exhibits the number of oscillations in which | part of the whole motion was lost ; the second the number of oscillations in which there was lost { part of the same.

First descent . .

. . 1

2

4

8

16

32 64

Last ascent . .

7 • • 8

4

3i

7

14

28 56

Numb, of oscill.

. . 226

228

193

140

901

53 30

Mrst descent . .

. . 1

2

4

8

16

32 64

Last ascent . , ,

' . f

U

3

6

12

24 48

Nwnb. of oscill.

. . 510

518

420

318

204

121 70

Selecting in the first table the 3d, 5th, and 7th observations, and express- ing the greatest velocities in these observations particularly by the num- bers 1, 4, 16 respectively, and generally by the quantity V as above, there

will come out in the 3d observation ~^ = A + B 4- C, in the 5th obser-

2 . 8

vation qT— = 4A + SB + 16C, in the 7th observation ^ = 16A + 64B -H

256C. These equations reduced give A = 0,001414, B = 0,000297, C = 0,000879. And thence the resistance of the globe moving with the velocity V will be to its weight 26 1 ounces in the same ratio as 0,0009V +

0,00020SY^ + 0,0006597 2 to 121 inches, the length of the pendulum. And if we regard that part only of the resistance which is in the dupli- cate ratio of the velocity, it will be to the weight of the globe as 0,000659 Y^ to 121 inches. But this part of the resistance in the first experiment was to the weight of the wooden globe of 57/2 ounces as 0,002217V^ to 121 ; and thence the resistance of the wooden globe is to the resistance of the leaden one (their velocities being equal) as 57-^^ into 0,002217 to 26^^- into 0,000659, that is, as 7\ to 1. The diameters of the two globes were 6 1 and 2 inches, and the squares of these are to each other as 47^ and 4, or 11 jl and 1, nearly. Therefore the resistances of these equally swift globes were in less than a duplicate ratio of the diameters. But we have not yet considered the resistance of the thread, which was certainly very considerable, and ought to be subducted from the resistance of the pendu- lums here found. I could not determine this accurately, but I found it

Sec. VI.] OF natural philosophy. 317

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 resist- ance of the thread is subducted, are nearly in the duplicate ratio of their diameters. For the ratio of 71 — i to 1 — |, or 10|- to 1 is not very diiferent from the duplicate ratio of the diameters 11|| to 1.

Since the resistance of the thread is of less moment in greater globes, I tried the experiment also with a globe whose diameter was ISf inches. The length of the pendulum between the point of suspension and the cen- tre of oscillation was 122^ inches, and between the point of suspension and the knot in the thread 109^ inches. The arc described by the knot at the first descent of the pendulum was 32 inches. The arc described by the same knot in the last ascent after five oscillations was 28 inches. The sum of the arcs, or the whole arc described in one mean oscillation, was 60 inches. The difference of the arcs 4 inches. The j^ part of this, or the difference between the descent and ascent in one mean oscillation, is f of an inch. Then as the radius 109^ to the radius 122i, so is the whole arc of 60 inches described by the knot in one mean oscillation to the whole arc of 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 the arc described were to remain, and the length of the pendulum should be augmented in the ratio of 126 to 122|-, the time of the oscillation would be augmented, and the velocity of the pendulum would be diminished in the subduplicate of that ratio ; so that the difference 0,4475 of the arcs de- scribed in the descent and subsequent ascent would remain. Then if the arc described be augmented in the ratio of 124 /y to 67}, that difference 0,4475 would be augmented in the duplicate of that ratio, and so would become 1,5295. These things would be so upon the supposition that the resistance of the pendulum were in the duplicate ratio of the velocity. Therefore if the pendulum describe the whole arc of 124/y inches, and its length between the point of suspension and the centre of oscillation be 126 inches, the difference of the arcs described in the descent and subsequent ascent would be 1,5295 inches. And this difference multiplied into the weight of the pendulous globe, which was 208 ounces, produces 318,136. Again ; in the pendulum above-mentioned, made of a wooden globe, when its centre of oscillation, being 126 inches from the point of suspension, de- scribed the whole arc of I2i^\ inches, the difference of the arcs described

126 8

in the descent and ascent was ^q^ i^to ^, This multiplied into the

weight of the globe, which was 57^2 ounces, produces 49,396. But I mul- tiply these differences into the weights of the globes, in order to find their resistances. For the differences arise from the resistances, and are as the resistances directly and the weights inversely. Therefore the resistances are as the numbers 318,136 and 49,396. But that part of the resistance

318 THE MATHEMATICAL PRINCIPLES fSoOK II.

of the lesser globe, which is in the duplicate ratio of the velocity, was to the 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 to its whole resistance ; and so those parts are jiearly as 318,136 and 45,453, that is, as 7 and 1. But the diameters of the globes are 18| and 6|- ; and their squar^ 35 ly^^ and 47 -I- ^ are as 7,438 and 1, that is, as the resistances of the globes 7 and 1, nearly. The difference of these ratios is scarce greater than may arise from the resistance of the thread. Therefore those parts of the resistances which are, when the globes are equal, as the squares of the velocities, are also, when the velocities are equal, as the squares of the diameters of the globes.

But the greatest of the globes I used in these experiments was not per- fectly spherical, and therefore in this calculation I have, for brevity's sake, neglected some little niceties ; being not very solicitous for an accurate calculus in an experiment that was not very accurate. So that I could wish, that these experiments were tried again with other globes, of a larger size, more in number, and more accurately formed ; since the demonstra- tion of a vacuum depends thereon. If the globes be taken in a geometrical proportion, as suppose whose diameters are 4, 8, 1 6, 32 inches ; one may collect from the progression observed in the experiments what would hap- pen if the globes were still larger.

In order to compare the resistances of different fluids with each other, I made the following trials. I procured a wooden vessel 4 feet long, 1 foot broad^ and 1 foot high. This vessel, being uncovered, I filled with spring water, and, having immersed pendulums therein, I made them oscillate in the water. And I found that a leaden globe weighing 166| ounces, and in diameter 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 certain point marked in the thread being 126 inches, and to the centre of oscilla- tion 134f inches. The arc described in '\

the first descent J by 1

a point marked in |^ 64 . 32 . 16 . S . 4 . 2 . 1 , J . J

the thread was

inches. The arc described in )

the last ascent loas V 48 . 24 . 12 . 6 . 3 . 1| . | . f . t-

inches, )

The difference of the '

arcs, proportional [-.g q , ^ i j. j. i _i

to the motion lost, \ ' * ' ' •2*4'8'i¥

was inches. The number of the OS- } 29 ii q 7 v\i 121 13i

dilations in v)ater. \ e 0 • s • • • 4 • 3 • • 3

The number of the os- ) qk , <>q^ kqr

dilations in air. \ ^^^ * '^^^ ' ^^^

Sec, VI.] OF natural philosophy. 319