by William M. Malisoff,

[original emphases throughout; the paper is presented below in essentially three separate parts].

"Apropos of A. E. Caswell's suggested law, namely, "the mean distances of the planets from the sun are proportional to the squares of simple integral numbers," the writer wishes to point out the following corrections, extensions and other new laws.

(1) The percentage deviation
from proportionality to the *squares* of the integers is double that
indicated by him.

(2) Since the earth's distance is taken as a standard in all measurements, one would expect a good reason for not assuming its distance to correspond to a perfect square of an integer (in this case 5). If this is

done the deviations from the above law are as high as 12 per cent.

(3) One would expect similar relations
to hold for the satellites of the planets. For the satellites of Mars the
ratio 5** ^{2}** : 8

(4) The writer would point
out a relation that depends strictly on the square root of the distance
of a planet from the sun or a satellite from its planet. It is the velocity,
which varies inversely as the square root of the distance from the axis
of revolution. For the planetary system one could then state as a law.*
The velocities of the planets are inversely in proportion to simple
integral numbers*. Thus:

PLANETS |
Period |
Velocity |
30.3/Velocity |
Integer |

Mercury | 0.2408 | 10.1006 | 3 | 3 |

Venus | 0.6152 | 7.3872 | 4.1 | 4 |

Earth | 1.000 | 6.2832 | 4.83 | 5 |

Mars | 1.88 | 5.0924 | 5.95 | 6 |

Jupiter | 11.86 | 2.7563 | 11.0 | 11 |

Saturn | 29.46 | 2.0344 | 14.9 | 15 |

Uranus | 84.01 | 1.4346 | 21.1 | 21 |

Neptune | 164.6 | 1.1464 | 26.5 | 27 |

(5) Another law may be stated,
as a consequence of Kepler's third law and the distance relation, namely,
*the periods of the planets are proportional to the cubes of simple integral
numbers*. The same integers as above are involved.

(6) In this connection the
writer would bring to the attention of American scientists an effort by
Viktor Goldschmit to elucidate some of the numerical regularities in the
distances of the planets and satellites from their axes of revolution.
The journal in which it occurs is not generally known. He observes the
distances of the planets to be closely in the sequence 1/13, 1/7, 1/5,
1/3, 1, 2, 4, 6. The four larger planets are considered to have condensed
together before the group of the four smaller ones. A mathematical treatment
strictly analogous to the phenomena of *standing waves in sound*,
the distribution of lines of spectra, the progress of crystallization and
similar phenomena gives the same law of harmonic relations of distances
not only for the planets but also for satellites. The harmonic sequences
are as follows:

Condensations | 0 | 1/3 | 1/2 | 2/3 | 1 | 3/2 | 2 | 3 |

Large Planets | 0 | -- | 1/2 | -- | 1 | -- | 2 | 3 |

Small planets | 0 | 1/3 | -- | 2/3 | 1 | -- | 2 | -- |

Jupiter's Satellites | 0 | -- | 1/2 | 2/3 | 1 | -- | 2 | -- |

Uranus' Satellites | 0 | -- | 1/2 | 2/3 | 1 | 3/2 | -- | -- |

Saturn inner satellites | 0 | -- | 1/2 | 2/3 | 1 | 3/2 | 2 | -- |

Saturn outer satellites | 0 | -- | -- | -- | 1 | (6/5) | -- | 3 |

Earth - Moon | 0 | -- | -- | -- | 1 | -- | -- | -- |

The sequences are brought into line by transformations derived from considerations of the dominance of certain positions, as the 0, 1 and infinity in condensation.

(7) The writer would further approach the question of the regularities of the spacing of the planets from another basic point of view. In brief, considering the accurate correspondence of the velocity of a planet or satellite inversely to the square root of its distance from the axis of revolution, we may perceive the propagation of a wave of velocity at the initiation of revolution to follow the law of a logarithmic spiral. We should then expect the distances of the planets as well as their velocities to be represented as the radii vectors of a logarithmic spiral. The law we would propose is:

log d = nk, or *e*^{nk}
= d

where *d* is the distance, *k *is a constant and

log v = k ' - nk/2

where *k *' is another constant and *n *the *same simple
integer as above. *It follows also that a similar logarithmic relation exists for
the periods, that is,

If

log p = 3/2 nk + k"** **

where* k" *is still another constant and

Similar relations may be derived for centripedal force and the like, but enough has been given here to indicate the underlying principle."

*William M. Malisoff, 1929
*