COR. Hence from the law of refiftance and the dif ference Aa of the arcs Ca, CB may be collected the proportion of the refiftance to the gravity nearly.

For if the refiftance DK be uniform, the figure BKT a will be a rectangle under Ba and DK; and thence the rectangle under Ba and Aa will be equal to the rectangle under Ba and DK, and DK will be equal to Aa. Wherefore fince DK is the exponent of the refiftance, and the length of the pendulum the exponent of the gravity, the refiftance will be to the gravity as a to the length of the pendulum; altogether as in Prop. 28. is demonstrated.

If the refiftance be as the velocity, the figure BKTa will be nearly an ellipfis. For if a body, in a nonrefifting medium, by one entire ofcillation, should defcribe the length BA, the velocity in any place D would be as the ordinate DE of the circle defcribed on the diameter AB. Therefore fince Ba in the refifting medium, and BA in the non-refifting one, are defcribed nearly in the fame times; and therefore the velocities in each of the points of Ba, are to the velocities in the correfpondent points of the length BA nearly as Ba is to BA; the velocity in the point D in the refifting medium will be as the ordinate of the circle or ellipfis described upon the diameter Ba; and therefore the figure BKVTa will be nearly an ellipfis. Since the refiftance is fuppofed proportional to the velocity, let O be the exponent of the refiftance in the middle point O; and an ellipfis BRVS a defcribed with the centre O, and the femiaxes O B, OV will be nearly equal to the figure BKVTa, and to its equal the rectangle AaxBO. Therefore Aax BO is to OVX BO as the area of this ellipfis to OV× BO; that is, Aa is to OV as the area of the femicircle to the fquare of the radius, or as 11 to 7 nearly; and therefore Aa is to the length of the pendulum, as the refiftance of the ofcillating body in to its gravity.

Now

Now if the resistance D K be in the duplicate ratio of the velocity, the figure BKVTa will be almoft a Parabola having for its vertex and O for its axis, and therefore will be nearly equal to the rectangle under Ba and OV. Therefore the rectangle under Ba and Aa is equal to the rectangle Bax OV, and therefore O is equal to Aa: and therefore the resistance in O made to the ofcillating body is to its gravity as a to the length of the pendulum.

+

And I take thefe conclufions to be accurate enough for practical uses. For fince an Ellipfis or Parabola BRVS a falls in with the figure BKVTa in the middle point, that figure, if greater towards the part BRV or VSa than the other, is lefs towards the contrary part, and is therefore nearly equal to it.

PROPOSITION XXXI. THEOREM XXV. If the refiftance made to an ofcillating body in each of the proportional parts of the arcs defcribed be augmented or diminished in a given ratio; the difference between the arc defcribed in the defcent and the arc defcribed in the fubfequent afcent, will be augmented or diminished in the fame ratio.

For that difference arifes from the retardation of the pendulum by the refiftance of the medium, and therefore is as the whole retardation, and the retarding refiftance proportional thereto. In the foregoing Propofition the rectangle under the right linea B and the difference Aa of the arcs CB, Ca was equal to the area BKTa. And that area, if the length a B remains, is augmented or diminished in the ratio of the ordinates DK; that is, in the ratio of the resistance, and is therefore as the length a B and the refiftance con

junctly.

junctly. And therefore the rectangle under Aa and aB is as a B and the refiftance conjun&tly, and therefore Aa is as the refiftance. O.E.D.

COR. I. Hence if the refiftance be as the velocity, the difference of the arcs in the fame medium will be as the whole arc defcribed: and the contrary.

COR. 2. If the refiftance 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 univerfally, if the refiftance be in the triplicate or any other ratio of the velocity, the difference will be in the fame ratio of the whole arc and the contrary.

COR. 4. If the refiftance be partly in the fimple ratio of the velocity, and partly in the duplicate ratio of the fame, 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 refiftance will be the fame for the velocity, as the law and ratio of that difference for the length of the arc.

COR. 5. And therefore if a pendulum defcribe fucceffively unequal arcs, and we can find the ratio of the increment or decrement of this difference for the length of the arc defcribed; there will be had alfo the ratio of the increment or decrement of the refiftance for a greater or lefs velocity.

GENERAL SCHOLIUM.

From these Propofitions, we may find the refiftance of mediums by pendulums ofcillating therein. I found the refiftance of the air by the following experiments. I fufpended a wooden globe or ball weighing 57, ounces Averdupois, its diameter 63 London inches, by a fine thread on a firm hook, fo that the diftance between the hook and the centre of ofcillation of the globe was 10 foot. I marked on the thread a point 10 foot and

I

1 inch diftant from the centre of fufpenfion; and even with that point I placed a ruler divided into inches, by the help whereof I obferved the lengths of the arcs defcribed by the pendulum. Then I number'd the ofcillations, in which the globe would lofe part of its motion. If the pendulum was drawn alide from the perpendicular to the diftance of 2 inches, and thence let go, fo that in its whole descent it defcribed an arc of two inches, and in the firft whole ofcillation, compounded of the defcent and fubsequent ascent, an arc of almost four inches: the fame in 164 ofcillations loft part of its motion, fo as in its laft afcent to defcribe an arc of 14 inches. If in the firft defcent it defcribed an arc of 4 inches; it loft part of its motion in 121 ofcillations, fo as in its laft afcent to defcribe an arc of 3 inches. If in the first defcent it defcribed an arc of 8, 16, 32, or 64 inches; it loft part of its motion in 69, 35, 18, 9 oscillations, refpectively. Therefore the difference between the arcs defcribed in the first descent and the last ascent, was in the 1, 2d, 3d, 4th, 5th, 6th cafe, 4, 1, 1, 2, 4, 8 inches, refpectively. Divide thofe differences by the number of ofcillations in each cafe, and in one mean ofcillation, wherein an arc of 34, 71⁄2, 15, 30, 60, 120 inches was defcribed, the difference of the arcs described in the defcent and fubfequent afcent will be

1

I 4 8 24 parts of an inch, respectively.

656 242 69 71 37 29

But these differences in the greater ofcillations are in the duplicate ratio of the arcs described nearly, but in leffer ofcillations fomething greater than in that ratio; and therefore (by Cor. 2. Prop. 31. of this Book) the refiftance of the globe, when it moves very fwift, is in the duplicate ratio of the velocity, nearly; and when it moves flowly, fomewhat greater than in that ratio.

Let

:

Now let V represent the greatest velocity in any of cillation, and let A, B, and C be given quantities, and let us fuppofe the difference of the arcs to be AV-BV-CV. Since the greatest velocities are in the cycloid as the arcs defcribed in ofcillating, and in the circle as the chords of thofe 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 refiftance and the fquare of the time conjunctly) are nearly the fame, in both curves for in the cycloid those differences must be on the one hand augmented, with the refiftance, in about the duplicate ratio of the arc to the chord, becaufe of the velocity augmented in the fimple ratio of the fame; and on the other hand diminished, with the fquare of the time, in the fame duplicate ratio. Therefore to reduce thefe obfervations to the cycloid, we must take the fame differences of the arcs as were obferved in the circle, and fuppofe the greateft 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 6th cafe, put 1, 4 and 16 for V; and the difference of the arcs in the 2d cafe will become

8

933

3.5/4/

14

121

A--B

=A

= 4A-8B-16C; in

the 6th cafe 16A+64B256C. These equa tions reduced give A=0,0000916, B=0,0010847, and C 0,0029558. Therefore the difference of the arcs is as 0,0000916 V + 0,0010847 V +0,0029558 V2: and therefore fince (by Cor. Prop. 30. applied to this cafe) the refiftance of the globe in the

middle