# Propositions 18-19

##### 9 minutes • 1823 words

## Table of contents

## Proposition 18 Theorem 16= The axes of planets

The axes of the planets are less than the diameters drawn perpendicularly to the axes.

The equal gravitation would give a spherical shape to the planets.

But the circular motion causes the parts to recede from the axis and go to the equator in a fluid state. This enlarges the diameters there and flatten the poles.

Thus, Jupiter’s diameter is shorter between pole and pole than from east to west.

If our earth was not higher about the equator than at the poles, then the seas would subside about the poles, and, rising towards the equator, would lay all things there under water.

## Proposition 19 Problem 3

Find the proportion of the axis of a planet to the diameter perpendicular thereto.

Mr. Norwood measured the distance of between London and York as 99575x feet.

The difference of latitude was 2deg 28'.

He set one degree at 367,196 feet or 57,300 Paris toises.

Mr. Picart, measured an arc of one degree, and 22'55" of the meridian between Amiens and Malvoisine, found an arc of one degree to be 57,060 Paris toises.

Mr. Cassini, the father, measured the distance upon the meridian from the town of Collionre in Roussillon to the Observatory of Paris.

His son added the distance from the Observatory to the Citadel of Dunkirk. The whole distance was 486156.5 toises.

The difference of the latitudes of Collionre and Dunkirk was 8 degrees and 31x.

Hence an arc of one degree is 57,061 Paris toises. From this:

- the Earth’s circumference is 123,249,600 Paris feet
- the Earth’s radius is 19,615,800 Paris feet

In Paris, a heavy body falling in 1 second travels 15 Paris feet, 1 inch, 1 line, as above, that is, 2173 lines J.

The body’s weight is diminished by the weight of the ambient air. The lost weight is x. In this way, the weight falling in a vacuum will be 2,174 lines per second.

A body in every sidereal day of 23h 56’ 4" uniformly revolving in a circle at a distance of 19,615,8OO feet from the center in 1 second. It describes an arc of 143346 feet. The versed sine of which is 0.05236561 feet or 754064 lines.

Therefore, the force with which bodies descend in Paris’ latitude is to the centrifugal force of bodies in the equator arising from the diurnal motion of the earth as 2174 to 7,54064.

The centrifugal force of bodies in the equator is to the centrifugal force with which bodies recede directly from the earth in the latitude of Paris 48deg 50’ 10" in the duplicate proportion of the radius to the cosine of the latitude as 7,54064 to 3,267.

Add this force to the force with which bodies descend by their weight in the latitude of Paris

A body, in the latitude of Paris, falling by its undiminished force of gravity, in 1 second will describe 2177,267 lines, or 15 Paris feet, 1 inch, and 5,267 lines.

The total force of gravity in that latitude will be to the centrifugal force of bodies in the equator of the earth as 2177,267 to 7,54064, or as 289 to 1.

Wherefore if APBQ, represent the figure of the earth, now no longer spherical, but generated by the rotation of an ellipsis about its lesser axis PQ, and ACQqca a canal full of water, reaching from the pole and thence rising to the equator the weight of the water in the leg of the canal ACca will be to the weight of water in the other leg QCcq as 289 Qq to the centre Cc, to 288, because the centrifugal force arising from the circu motion sustains and takes off one of the 289 parts of the weight (in the one leg), and the weight of 288 in the other sustains the rest.

But by if the matter computation (from Cor. 2, Prop. XCI, Book I) I find, that, and its axis PQ, and without of the earth was all uniform, any motion, of the force to AB as diameter the were to 101, gravity in the towards the earth would be to the force of gravity in the same lar

APBQ, the ellipsis about the axis AI3 is to 407 the force of gravity in the same place A, towards the sphere described about the centre C with the But the force of gravity in the place A to radius AC, as 125 to 126.

wards the earth is a mean proportional betwixt the forces of gravity to wards the spheroid and this sphere; because the sphere, by having its diameter PQ, diminished in the proportion of 101 to 100, is transformed into the figure of the earth and this figure, by having a third diameter per and PQ, diminished in the same pro pendicular to the two diameters and the force of gravity in A, is converted into the said spheroid portion, ; AB diminished nearly in the same proportion.

Therefore the force of gravity in A towards the sphere described about the centre C with the radius AC, is to the force of gravity in A towards the earth as 126 to in either case, is

And the force of gravity in the place C with the radius QC, scribed about the centre Q towards the sphere de the force of gravity towards the sphere described about the centre C, with the radius AC, in the proportion of the diameters (by Prop. LXXII, Book I), in the place

100 to 101. If, therefore, we compound those three proportions 126 to 125|. and 100 to 101, into one, the force of gravity in to 125, the place AQ towards the earth will be to towards the earth as 126 is to X 126 X the force of gravity in the place to 125 X 125| X 101 or as

Since (by Cor. 3, Prop. XCI, Book I) the force of gravity in either the of canal ACca, or QCcy, is as the distance of the places from the leg centre of the earth, if those legs are conceived to be divided by transverse., and equidistant surfaces, into parts proportional to the wholes, the weights of any number of parts in the one leg ACca will be to the weights of the same number of parts in the other leg as their magnitudes parallel, and the accelerative forces of their gravity conjunctly, that is, as 10 J to And therefore if the centrifugal 100, and 500 to 501. or as 505 to 501. force of every part in the leg ACca, arising from the diurnal motion, was to the weight of the same part as 4 to 505, so that from the weight of every part, conceived to be divided into 505 parts, the centrifugal force might take off four of those parts, the weights would remain equal in each in an equilibrium. leg, and therefore the fluid would rest

But the centrifugal force of every part is to the weight of the same part as 1 to 289 that is, the centrifugal force, which should be T y parts of the weight, is And, therefore, I say, by the rule of proportion, only |g part thereof. that if the centrifugal force j make the height of the water in the leg

ACca water in the leg QCcq by one T | part of its whole height, the centrifugal force -^jj will make the excess of the height in the leg ACca only ^{^ part of the height of the water in the other leg QCcq ; and therefore the diameter of the earth at the equator, is to its to exceed the height of the diameter from pole to pole as 230 to 229.

And since the mean radius of the earth, according to Picar’s measure, is 19615800 feet, or 3923,16 miles (reckoning 5000 feet to a mile), the earth Paris equator than at the poles by 85472 feet, or 17^- height at the equator will be about 19658600 feet, and at will be higher at the And miles. its the poles 19573000 feet.

If, the density and periodic time of the diurnal revolution remaining the same, the planet was greater or less than the earth, the proportion of the centrifugal force to that of gravity, and therefore also of the diameter be twixt the poles to the diameter at the equator, would likewise remain the But if the diurnal motion was accelerated or retarded in any pro the portion, centrifugal force would be augmented or diminished nearly in the same duplicate proportion and therefore the difference of the diame game.

diminished in the same duplicate ratio very nearly.

If the density of the planet was augmented or diminished in any proportion, the force of gravity tending towards it would also be augmented ters will be increased or or diminished in the same proportion and the difference of the diameters contrariwise would be diminished in proportion as the force of gravity is augmented, and augmented in proportion as the force of gravity is diminished.

Wherefore, since the earth, in respect of the fixed stars, revolves in 23 h 56 but Jupiter in 9 h 56 and the squares of their periodic times are as 29 to 5, and their densities as 400 to 94 the difference of the diameters of Jupiter will be to its lesser diameter as X ^^ X ^Tm to 1; or as 1 to 9 f, nearly. Therefore the diameter of Jupiter from east to west is to diameter from pole to pole nearly as 10 to 9|-.

Therefore since greatest diameter be its lesser diameter lying between the poles will thereto about 3 for the irregular refraction of light, and the apparent diameters of this planet will become 40 and

These things are so very nearly.

the is that uniformly dense. But upon supposition Jupiter body which are now if its

to each other as 11 -j to 10^, body be denser towards the plane of the equator than towards may be to each other as 12 to 11, or 13 to 12, or the poles, its diameters perhaps as 14 to 13.

Cassini observed in the year 1691, that the diameter of Jupiter reaching from east to west is greater by about a fifteenth part than the other diameter.

Mr. Pound with his 123 feet telescope, and an excellent micrometer, measured the diameters of Jupiter in the year 1719, and found them as follow. The Times.

The theory agrees with the phenomena for the planets are more heated by the sun s rays towards their equators, and therefore are a little more condensed by that heat than towards their poles.

The diurnal rotation of the earth causes a reduction of gravity. This is why the earth rises higher there than it does at the poles (supposing that its matter is uniformly dense), will appear by the experiments of pendulums