The orbits of Planets

PROPOSITION XIX. PROBLEM III: find the proportion of the axis of a planet to the diameter, perpendicular thereto

Norwood measurED a distance of 905751 feet of London measure between London and York.

in 1635, and observing the difference of latitudes to be 2° 28′, determined the measure of one degree to be 367196 feet of London measure, that is 57300 Paris toises. M. Picart, measuring an arc of one degree, and 22′ 55″ of the meridian between Amiens and Malvoisine, found an arc of one degree to be 57060 Paris toises. M. Cassini, the father, measured the distance upon the meridian from the town of Collioure in Roussillon to the Observatory of Paris; and his son added the distance from the Observatory to the Citadel of Dunkirk.

The whole distance was 486156½ toises and the difference of the latitudes of Collioure and Dunkirk was 8 degrees, and 31′ 115⁄6″. Hence an arc of one degree appears to be 57061 Paris toises.

From these measures, we conclude that the circumference of the earth is 123249600, and its semi-diameter 19615800 Paris feet, upon the supposition that the earth is of a spherical figure.

In the latitude of Paris a heavy body falling in a second of time describes 15 Paris feet, 1 inch, 17⁄9 line, as above, that is, 2173 lines 7⁄9. The weight of the body is diminished by the weight of the ambient air. Let us suppose the weight lost thereby to be 1⁄11000 part of the whole weight; then that heavy body falling in vacuo will describe a height of 2174 lines in one second of time.

A body in every sidereal day of 23h.56′ 4″ uniformly revolving in a circle at the distance of 19615800 feet from the centre, in one second of time describes an arc of 1433,46 feet; the versed sine of which is 0,05236561 feet, or 7,54064 lines. And therefore the force with which bodies descend in the latitude of Paris 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 48° 50′ 10″ in the duplicate proportion of the radius to the cosine of the latitude, that is, as 7,54064 to 3,267. Add this force to the force with which bodies descend by their weight in the latitude of Paris, and a body, in the latitude of Paris, falling by its whole undiminished force of gravity, in the time of one second, will describe 2177,267 lines, or 15 Paris feet, 1 inch, and 5,267 lines. And 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 Qq to the centre Cc, and thence rising to the equator Aa; 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 to 288, because the centrifugal force arising from the circular 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 computation (from Cor. 2, Prop. XCI, Book I) I find, that, if the matter of the earth was all uniform, and without any motion, and its axis PQ were to the diameter AB as 100 to 101, the force of gravity in the place Q towards the earth would be to the force of gravity in the same place Q towards a sphere described about the centre C with the radius PC, or QC, as 126 to 125. And, by the same argument, the force of gravity in the place A towards the spheroid generated by the rotation of the ellipsis APBQ about the axis AB is to the force of gravity in the same place A, towards the sphere described about the centre C with the radius AC, as 125 to 126. But the force of gravity in the place A towards the earth is a mean proportional betwixt the forces of gravity towards 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 perpendicular to the two diameters AB and PQ diminished in the same proportion, is converted into the said spheroid; and the force of gravity in A, in either case, is 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 125½. And the force of gravity in the place Q towards the sphere described about the centre C with the radius QC, is to the force of gravity in the place A towards the sphere described about the centre C, with the radius AC, in the proportion of the diameters (by Prop. LXXII, Book I), that is, as 100 to 101. If, therefore, we compound those three proportions 126 to 125, 126 to 125½, and 100 to 101, into one, the force of gravity in the place Q towards the earth will be to the force of gravity in the place A towards the earth as 126 × {\displaystyle \scriptstyle \times } 126 × {\displaystyle \scriptstyle \times } 100 to 125 × {\displaystyle \scriptstyle \times } 125½ × {\displaystyle \scriptstyle \times } 101; or as 501 to 500.

Since (by Cor. 3, Prop. XCI, Book I) the force of gravity in either leg of the canal ACca, or QCcq, is as the distance of the places from the centre of the earth, if those legs are conceived to be divided by transverse, parallel, 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 and the accelerative forces of their gravity conjunctly, that is, as 101 to 100, and 500 to 501, or as 505 to 501. And therefore if the centrifugal 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 leg, and therefore the fluid would rest in an equilibrium. 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 4⁄505 parts of the weight, is only 1⁄289 part thereof. And, therefore, I say, by the rule of proportion, that if the centrifugal force 4⁄505 make the height of the water in the leg ACca to exceed the height of the water in the leg QCcq by one 1⁄100 part of its whole height, the centrifugal force 1⁄289 will make the excess of the height in the leg ACca only 1⁄289 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 diameter from pole to pole as 230 to 229. And since the mean semi-diameter of the earth, according to Picart’s mensuration, is 19615800 Paris feet, or 3923,16 miles (reckoning 5000 feet to a mile), the earth will be higher at the equator than at the poles by 85472 feet, or 17⅒ miles. And its height at the equator will be about 19658600 feet, and at 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 betwixt the poles to the diameter at the equator, would likewise remain the same. But if the diurnal motion was accelerated or retarded in any proportion, the centrifugal force would be augmented or diminished nearly in the same duplicate proportion; and therefore the difference of the diameters will be increased or diminished in the same duplicate ratio very nearly. And if the density of the planet was augmented or diminished in any proportion, the force of gravity tending towards it would also be augmented 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 23h.56′, but Jupiter in 9h.56′, and the squares of their periodic times are as 29 to 5, and their densities as 400 to 94½, the difference of the diameters of Jupiter will be to its lesser diameter as 29 5 × 400 94 1 2 × 1 229 {\displaystyle \scriptstyle {\frac {29}{5}}\times {\frac {400}{94{\frac {1}{2}}}}\times {\frac {1}{229}}} to 1, or as 1 to 9⅓, nearly. Therefore the diameter of Jupiter from east to west is to its diameter from pole to pole nearly as 10⅓ to 9⅓. Therefore since its greatest diameter is 37″, its lesser diameter lying between the poles will be 33″ 25‴. Add thereto about 3″ for the irregular refraction of light, and the apparent diameters of this planet will become 40″ and 36″ 25‴; which are to each other as 11⅙ to 10⅙, very nearly. These things are so upon the supposition that the body of Jupiter is uniformly dense. But now if its body be denser towards the plane of the equator than towards the poles, its diameters may be to each other as 12 to 11, or 13 to 12, or perhaps as 14 to 13.

And 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. Greatest diam. Lesser diam. The diam. to each other. Day. Hours. Parts Parts January 28 6 13,40 12,28 As 12 to 11 March 6 7 13,12 12,20 13¾ to 12¾ March 9 7 13,12 12,08 12⅔ to 11⅔ April 9 9 12,32 11,48 14½ to 13½ So that the theory agrees with the phænomena; 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.

Moreover, that there is a diminution of gravity occasioned by the diurnal rotation of the earth, and therefore 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 related under the following Proposition.