each Particle is as its distance from the point D which is the Centre of the Circle, that the fluid in the canal CD defcribes, and therefore the centrifugal forces in counting from the point D muft increase in an Arithmetical

progreffion, the greateft of which is

ny

and

therefore the fumm of all the centrifugal forces

in CD must be equal to

nyy

21

therefore the

weight of the canal OC is p√x2+y2

nyy =pr-nr which equation ex

preffes the nature of the curve that is made by the fection of the earth with a plane through its poles, and by this, the proportion of the axis of the earth to the diameter of the æquator may be easily determined; for when ĈO coincides with OP, then CD or y becomes equal to nothing, and and the equation is pv x2 = pr — 1⁄2 nr or px = pr—nr, and therefore by the 16th of the 6th p has the fame proportion to p- that r has to x or OQ to OD, but p is to p-in as 289 is to 288 or as 578 to 577 which therefore is the proportion of the greateft diameter of the Earth to its leaft; but this is upon fuppofition

that

that gravity is the fame at all diftances from the Centre, but if we will fuppofe that the gravity of bodies without the Earth is in a proportion reciprocal to the fquares of their diftances from the Centre, the gravity of those bodies which are within the Earth will be directly as their diftance, both which do best agree with the obferv'd Phænomena of nature; then will the gravity at the Aquator be to the gravity at the Poles as 689 to 692, which numbers in this Hypothefis do alfo exprefs the proportion of the Diameter of the Earth drawn through its Poles to its Diameter drawn in the plane of the Equator.

It is upon the account of this diminution of gravity, according as we approach the Æquator, that pendulums of the fame length in different Latitudes, take different times, to perform their vibrations; for because the accelerating force of gravity is lefs at the Equator than under any parallel, and under any parallel it is ftill lefs than under another which is nearer the Poles, it do's plainly from thence follow that a body plac'd in the Equator, or in any parallel will take a longer time to defcend through an arch of a given circle, than it would do at the Poles, and the farther a body is removed from the Poles, the longer time it will take to defcend through any given space.

From hence it follows that the length of pendulums which perform their vibrations in

equal

equal times in different Latitudes are directly as the accelerating forces of their gravities. For the time a Body takes to defcend through an Arch of a Cycloid, is to the time it will take to fall through the Axis of the Cycloid always in a given proportion, viz. as the Semiperiphery of a circle is to its Diameter by the 25th Prop. of Hugens Horologium Ofcillatorium; and therefore when the times in which a body defcends through the Axis of two different Cycloids are equal, the times of the defcent through the Cycloids will be alfo equal, but when the times of the defcent through the Axes are equal, these Axes, and confequently the lengths of the pendulum which vibrates in thefe Cycloids are proportional to the accelerating forces of their gravities.

By this, if we know the length of a pendulum which performs its vibrations in a given time, in any one part of the Earth, it is eafy to determine the length of a pendulum which performs its vibrations in the fame time, in any other part of the Earth, as for example: the length of a pendulum which vibrates feconds at Paris is three foot eight lines and an half, let it be required to find the length of a pendulum which vibrates feconds at the Equator. Because the gravity at the Pole is to the gravity at the Equator as 692 is to 689, therefore the decrease of gravity at the Equator is parts of the

whole

whole gravity: but as I have before demonftrated the decrease of gravity at the Equator is to its increase in any other Latitude, as the fquare of the radius is to the fquare of the fine of the Latitude; now the Latitude of Paris being 48°. 45' its fine is 75. 183, and therefore the fquare of the Radius is to the fquare of the fine of the Latitude as 1000000 to 565248, but as 1000000 is to 565248, fo is 3.000 the number which reprefents the decrease of gravity at the Equator to 1. 695, the number which reprefents its increase at Paris which-added to 689 the gravity at the Equator makes 690. 695 the number which will represent the gravity at Paris. But I have already fhewed, that as the gravity at Paris is to the gravity at the Equator, fo is the length of a pendulum which vibrates feconds at Paris to the length of a pendulum which vibrates feconds at the Æquator, that is as 690, 695 to 689, fo is 36, 708 the length of a pendulum at Paris which performs its vibration in a fecond to 36, 616, which therefore is the length of a pendulum which performs its vibrations in a fecond at the Equator: fo that the difference between these two pendulums is parts of an inch which comes pretty near the obfervations of Monf. Richer, who at the Ifland of Cayen, whofe Latitude is 5 degrees, found that a pendulum which vibrates feconds there, was a tenth

part

part of an inch horter than a pendulum which vibrates feconds at Paris.

Thus we fee that the principles and hypothefis and withal their confequences upon which the broad Spheroidical Figure of the Earth is founded do exactly agree with obfervations, and therefore there is no doubt to be made but that the Earth is really of fuch a Figure, and that the hypothefis upon which this Figure is grounded (viz. the diurnal rotation of the Earth and by confequence the centrifugal force of all Bodies upon it) muft be admitted for a true one; fince the different vibrations of Pendulums of the fame length in different Latitudes can depend upon no other cause for the change of Air is not able to produce any fuch effect, for if the Air made really any alterations in the vibrations of a Pendulum it would produce a quite contrary effect, than what is obferved; for Pendulums near the Equator would move fafter than they would do in places of greater Latitude; the Air in the one place being more rarified is much thinner and finer than it is in the other, and therefore gives lets refiftance to Bodies which move in it.

In this reafoning we have fupposed the Earth to have been at first fluid as the Theorift has done before us, but if we will put the cafe that the Earth was at first partly fluid and partly dry as it is at prefent, yet because

we