# The Mathematical Principles of Natural Philosophy (1846)/BookIII-Phaenomena

PHÆNOMENA, OR APPEARANCES.
PHÆNOMENON I.
    That the circumjovial planets, by radii drawn to Jupiter's centre, describe areas proportional to the times of description; and that their periodic times, the fixed stars being at rest, are in the sesquiplicate proportion of their distances from, its centre.
This we know from astronomical observations. For the orbits of these planets differ but insensibly from circles concentric to Jupiter; and their motions in those circles are found to be uniform. And all astronomers agree that their periodic times are in the sesquiplicate proportion of the semi-diameters of their orbits; and so it manifestly appears from the following table.
The periodic times of the satellites of Jupiter.
1$^d$.18$^h$.27′.34″. 3$^d$.13$^h$.13′ 42″. 7$^d$.3$^h$.42′ 36″. 16$^d$.16$^h$.32′ 9″.
The distances of the satellites from Jupiter's centre.
| From the observations of | 1 | 2 | 3 | 4 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$semi-diameter of Jupiter. |
| Borelli | 5⅔ | 8⅔ | 14 | 24⅔ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$semi-diameter of Jupiter. |
| Townly by the Microm. | 5,52 | 8,78 | 13,47 | 24,72 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$semi-diameter of Jupiter. |
| Cassini by the Telescope | 5 | 8 | 13 | 23 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$semi-diameter of Jupiter. |
| Cassini by the eclip. of the satel. | 5⅔ | 9 | 14+$^{23}$⁄$_{60}$ | 25+$^3$⁄$_{10}$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$semi-diameter of Jupiter. |
| From the periodic times | 5,667 | 9,017 | 14,384 | 25,299 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,$semi-diameter of Jupiter. |
Mr. Pound has determined, by the help of excellent micrometers, the diameters of Jupiter and the elongation of its satellites after the following manner. The greatest heliocentric elongation of the fourth satellite from Jupiter's centre was taken with a micrometer in a 15 feet telescope, and at the mean distance of Jupiter from the earth was found about 8′ 16″. The elongation of the third satellite was taken with a micrometer in a telescope of 123 feet, and at the same distance of Jupiter from the earth was found 4′ 42″. The greatest elongations of the other satellites, at the same distance of Jupiter from the earth, are found from the periodic times to be 2′ 56″ 47‴, and 1′ 51″ 6‴.
The diameter of Jupiter taken with the micrometer in a 123 feet telescope several times, and reduced to Jupiter's mean distance from the earth, proved always less than 40″, never less than 38″, generally 39″. This diameter in shorter telescopes is 40″, or 41″; for Jupiter's light is a little dilated by the unequal refrangibility of the rays, and this dilatation bears a less ratio to the diameter of Jupiter in the longer and more perfect telescopes than in those which are shorter and less perfect. The times in ​which two satellites, the first and the third, passed over Jupiter's body, were observed, from the beginning of the ingress to the beginning of the egress, and from the complete ingress to the complete egress, with the long telescope. And from the transit of the first satellite, the diameter of Jupiter at its mean distance from the earth came forth 37⅛″. and from the transit of the third 37⅜″. There was observed also the time in which the shadow of the first satellite passed over Jupiter's body, and thence the diameter of Jupiter at its mean distance from the earth came out about 37″. Let us suppose its diameter to be 37¼″ very nearly, and then the greatest elongations of the first, second, third, and fourth satellite will be respectively equal to 5,965, 9,494, 15,141, and 26,63 semi-diameters of Jupiter.
PHÆNOMENON II.
    That the circumsaturnal planets, by radii drawn to Saturn's centre, describe areas proportional to the times of description; and that their periodic times, the fixed stars being at rest, are in the sesquiplicate proportion of their distances from its centre.
For, as Cassini from his own observations has determined, their distances from Saturn's centre and their periodic times are as follow.
The periodic times of the satellites of Saturn.
1$^d$.21$^h$.18′ 27″. 2$^d$.17$^h$.41′ 22″. 4$^d$.12$^h$.25′ 12″. 15$^d$.22$^h$.41′ 14″. 79$^d$.7$^h$.48′ 00″.
The distances of the satellites from Saturn's centre, in semi-diameters of its ring.
| From observations | 1+$^{19}$⁄$_{20}$. | 2½. | 3½. | 8. | 24. |
| From the periodic times | 1,93. | 2,47. | 3,45. | 8. | 23,35. |
The greatest elongation of the fourth satellite from Saturn's centre is commonly determined from the observations to be eight of those semi-diameters very nearly. But the greatest elongation of this satellite from Saturn's centre, when taken with an excellent micrometer in Mr. Huygens' telescope of 123 feet, appeared to be eight semi-diameters and $^7$⁄$_{10}$ of a semi-diameter. And from this observation and the periodic times the distances of the satellites from Saturn's centre in semi-diameters of the ring are 2,1. 2,69. 3,75. 8,7. and 25,35. The diameter of Saturn observed in the same telescope was found to be to the diameter of the ring as 3 to 7; and the diameter of the ring, May 28–29, 1719, was found to be 43″; and thence the diameter of the ring when Saturn is at its mean distance from the earth is 42″, and the diameter of Saturn 18″. These things appear so in very long and excellent telescopes, because in such telescopes the apparent magnitudes of the heavenly bodies bear a greater proportion to the dilatation of light in the extremities of those bodies than in shorter telescopes. ​If we, then, reject all the spurious light, the diameter of Saturn will not amount to more than 16″.
PHÆNOMENON III.
    That the five primary planets, Mercury, Venus, Mars, Jupiter, and Saturn, with their several orbits, encompass the sun.
That Mercury and Venus revolve about the sun, is evident from their moon-like appearances. When they shine out with a full face, they are, in respect of us, beyond or above the sun; when they appear half full, they are about the same height on one side or other of the sun; when horned, they are below or between us and the sun; and they are sometimes, when directly under, seen like spots traversing the sun's disk. That Mars surrounds the sun, is as plain from its full face when near its conjunction with the sun, and from the gibbous figure which it shews in its quadratures. And the same thing is demonstrable of Jupiter and Saturn, from their appearing full in all situations; for the shadows of their satellites that appear sometimes upon their disks make it plain that the light they shine with is not their own, but borrowed from the sun.
PHÆNOMENON IV.
    That the fixed stars being at rest, the periodic times of the five primary planets, and (whether of the sun, about the earth, or) of the earth about the sun, are in the sesquiplicate proportion of their mean distances from the sun.
This proportion, first observed by Kepler, is now received by all astronomers; for the periodic times are the same, and the dimensions of the orbits are the same, whether the sun revolves about the earth, or the earth about the sun. And as to the measures of the periodic times, all astronomers are agreed about them. But for the dimensions of the orbits, Kepler and Bullialdus, above all others, have determined them from observations with the greatest accuracy; and the mean distances corresponding to the periodic times differ but insensibly from those which they have assigned, and for the most part fall in between them; as we may see from the following table.
The periodic times with respect to the fixed stars, of the planets and earth revolving about the sun, in days and decimal parts of a day.
| 10759,275. | 4332,514. | 686,9785. | 365,2565. | 224,6176. | 87,9692. |
The mean distances of the planets and of the earth from the sun.
| According | to Kepler | 951000. | 519650. | 152350. |
| ” | to Bullialdus | 954198. | 522520. | 152350. |
| ” | to the periodic times | 954006. | 520096. | 152369. |
| According | to Kepler | 100000. | 72400. | 38806. |
| ” | to Bullialdus | 100000. | 72398. | 38585. |
| ” | to the periodic times | 100000. | 72333. | 38710. |
As to Mercury and Venus, there can be no doubt about their distances from the sun; for they are determined by the elongations of those planets from the sun; and for the distances of the superior planets, all dispute is cut off by the eclipses of the satellites of Jupiter. For by those eclipses the position of the shadow which Jupiter projects is determined; whence we have the heliocentric longitude of Jupiter. And from its heliocentric and geocentric longitudes compared together, we determine its distance.
PHÆNOMENON V.
    Then the primary planets, by radii drawn to the earth, describe areas no wise proportional to the times; but that the areas which they describe by radii drawn to the sun are proportional to the times of description.
For to the earth they appear sometimes direct, sometimes stationary, nay, and sometimes retrograde. But from the sun they are always seen direct, and to proceed with a motion nearly uniform, that is to say, a little swifter in the perihelion and a little slower in the aphelion distances, so as to maintain an equality in the description of the areas. This a noted proposition among astronomers, and particularly demonstrable in Jupiter, from the eclipses of his satellites; by the help of which eclipses, as we have said, the heliocentric longitudes of that planet, and its distances from the sun, are determined.
PHÆNOMENON VI.
    That the moon, by a radius drawn to the earth's centre, describes an area proportional to the time of description.
This we gather from the apparent motion of the moon, compared with its apparent diameter. It is true that the motion of the moon is a little disturbed by the action of the sun: but in laying down these Phænomena, I neglect those small and inconsiderable errors.