bodies descending perpendicularly, in the first second of time, is 16,087 feet.-This length of a

velocity with which NE is passed over to the velocity with which FM is paffed over : whence NE, FM, are paffed over in equal times. And fince MQ is equal to EG, the time of descent through M Q will be equal to the time of defcent through EG; fo that the time of defcent through FQ will be equal to the time of descent through NEG. But fince the angle MQG is a right one, FG is greater than FQ; fo that the time through FG will be greater than the time through FQ, or through NEG; and fince DF is greater than DN, the time through DF will be greater than the time through DN. Whence the time of defcent along DF, FG, will be greater than the time of defcent along DN, NG. A heavy body therefore, after its fall from A to D, will defcend from D to G along the curve DEG, in less time than along any other line; confequently the curve ADEGB is the line of fwifteft defcent between the points A and B.

3. Let ADEM, fig. 17, Plate IV. be a cycloid whose base is the horizontal line AG. Through any point D in it draw DQ parallel to the base AG, and cutting the generating circle at N and the axis at Q. Draw the chords GN, NM; through D draw DL perpendicular to the bafe; and draw OE indefinitely near and parallel to LD. Now the indefinitely small part DE of the curve may be confidered as a right line coinciding with the tangent at D, and

fecond pendulum is certainly not mathematically exact, yet it may be considered as fuch for all common purposes; for it is not likely to differ from the truth by more thanth part of an inch.*

XII. The

it may likewise be fuppofed to be defcribed by a body defcending from A, with the fame velocity which the body has acquired by its defcent from A to D; for the acceleration of velocity through that indefinitely small space may be confidered as next to nothing. Now we fhall prove that this cycloid has the property of the above-mentioned curve, viz. that the velocity with which the fmall portion DE is described by a body falling from A, is always proportional DH xa -; (a denoting the axis GM of the cycloid).

to

DE From the above-mentioned properties of the cycloid, the fmall line DE, coinciding with the tangent at D, is parallel to the chord NM, Whence the triangles DHE, NQM, and GMN, are equiangular and of course fimilar; there

fore DE: DH:: GM (a); GN=

DH xa
DE

But GN

is as the volocity which is acquired by the heavy body in its descent from G to Q, or from L to D; viz. as the velocity with which the indefinitely fmall line DE is paffed over; therefore the cycloid, having the property of the above-mentioned curve, is the line of swifteft descent, &c.

*See Mr. Whitehurst's attempt towards obtaining invariable measures of length, capacity, and weight. Also Sir George Shuckburg Evelyn's excellent paper on the standard of weight and measure, in the Philosophical Transactions for the year 1798,

XII. The times in which fimilar vibrations (viz. vibrations through arches of the fame number of degrees) of different pendulums are performed, are as the fquare roots of the lengths of the pendulums.

Thus if the pendulum AB, fig. 18, Plate IV. be four times as long as the pendulum CD, then the time of a vibration of the former will be double the time of a fimilar vibration of the latter. For (by cor. to prop. VII. of this chap.) the vibrations, and of course the femivibrations, being fimilar and fimilarly fituated, the time of the pendulum's descent along the arch GB is to the time of the other pendulum's defcent along the arch HD, as the fquare root of GB is to the fquare root of HD. But the circumferences of circles, or fimilar portions of the circumferences, are as their radii; therefore the fquare roots of fimilar portions of the circumferences are as the fquare roots of the radii; confequently the times of fimilar vibrations are as the fquare roots of the radii, or of the lengths of the pendulums,

Throughout the prefent chapter the force of gravity has been supposed invariable; but when that is not the cafe, as for inftance, when a pendulum, which vibrates near the furface of the earth, is compared with a pendulum on the top of a very high mountain, or with a pendulum which vibrates on an inclined plane; in which cases the action of the gravitating force on the pendulum's is not the fame, then the time of vibration is as the quotient of the

Square root of the length of the pendulum divided by the fquare root of the gravitating force.

This propofition will be found demonstrated in the note.

CHAPTER XI.

OF THE CENTRE OF OSCILLATION, AND CENTRE

OF PERCUSSION.

TH

HE attentive reader muft undoubtedly have remarked, that though in the preceding chapter much has been faid with respect to the length of the pendulum, yet no mention has been made of the point from which that length, or diftance from the point of suspension, should be meafured. The reason of this omiffion is, that the determination of that point, which is called the centre of ofcillation, requires a very particular confideration; fuch indeed as could not without obfcurity be introduced in the preceding chapter. We shall now endeavour to elucidate the nature of that point, and to lay down the methods of determining its fituation or distance from the point of fufpenfion in pendulums of different lengths and shapes.

When

When the pendulum confifts of a spherical body fastened to a string, a person unacquainted with the fubject might at firft fight imagine that the length of the pendulum must be estimated from the point of fufpenfion to the centre of the ball. But this is not the cafe; for in fact the real length of the pendulum is greater than that diftance, the reason of which is, that the spherical body does not move in a ftraight line, but it moves in a circular arch; in confequence of which, that half of it which is fartheft from the point of fufpenfion, runs through a longer space than the other half which is nearer to the point of fufpenfion; hence the two halves of the ball, though containing equal quantities of matter, do actually move with different velocities, therefore their momentums are not equal; and it is in confequence of this inequality that the centre of ofcillation does not lie between the two hemifpheres; that is, in the centre of the ball; but it lies within the lower hemisphere, viz. that which has the greater momentum. Now from this it naturally follows, that if the ball of the pendulum could be concentrated in one point, that point would be the centre of ofcillation; fo that the centre of ofcillation is that point wherein all the matter (and of course the forces of all the particles) of the body or bodies that may be joined together to form a penduluin, may be conceived to be condensed.

The centre of percuffion is that part or point of a pendulous body, which will make the greatest impreffion