Second letters from Sir John Herschel

No. 5I did not intend to have entered into the question with you as a matterof correspondence, but as I think you do not quite clearly perceive thepoint at issue, I will try to put it more distinctly than it would seem I
have done.
M. Comte contends that he has shewn a numerical coincidence to subject
between the periods of the planets as given by observation & as calculationon the nebular hypothesis.
Now the nebular hyp. makes two assumptions. 1st (assumption A)
that the sun shrank in dimension, not arbitrarily, but according tocertain determinate physical laws depending on radiation of heat
& the constitution of its materials- its form, motion, & distributionof its strata at each instant following from those laws by purely
dynamical considerations. (N B These laws I contend are utterly
unknown, nor can I conceive any means whereby they can everbecome known).
2d (Assumption B) That if the sun shrank in succession to
the size of each planet & orbit is abandoned a portion of its exterior
which became the planet, leaving it revolving in the orbit where
it now is. & of course in the period it now has or then had ―― & thatfor every planet in succession.
Assumption B is equivalent to that of the <???> of aplanetary rotation between a superficial particle at the suns
equator & the general attracting mass. In it assumes the centrifugal
to be equal to the centripetal force, so that the superficial portions
shall not press upon the interior body, & shall consequently
be left behind as planetary molecules, as that body continues to shrink.
If therefore we calculate from assumption B, the period (P) of the sunrotation at the equator where its radius (R) was equal to that of the orbit ofa given planet, we necessarily arrive at a result (P =2𝜋3∙ 𝑅2 )𝑀√

identical with the derived period of the planet (which equation is whatI <???> by Kepler's law) ; provided we suppose the sun at all times tohave been spherical. And if we take into consideration the oblateness,& put m for the mass which placed at the sun's center should attract
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as the oblate mass M does ―― then P =32𝜋∙ 𝑅2 will be the period:
𝑀√which will agree neatly with the planetary periods (say within a45th part) if the ratio

1√𝑚
do not exceed, by more than45√𝑀To argue therefore from any exact or approximate coincidence
between the results of such a calculation, & the actual planetaryperiods, is arguing in a vicious circle ―― It is tautology.
Now this is what I contend M. Comte does in effect. The legitimate& logical course would be to shew that Assumption B is a necessary
result from assumption A & that therefore the calculated periods
flow, not from B as an arbitrary assumption, but from A. Now to do
this on correct dynamical principles is not practicable by reason
1st of our ignorance of the Physical laws alluded to (which must
influence the result, as is easily shewn by trying a few cases) and
2d by reason of the utter unmanageability of the analysis (we shouldhave assumption upon assumption to make ―― as to the law of coding―― of internal density of friction of gaseous shells reordering one
within another in different times, & innumerable other such.)
M. Comte avowedly shies the difficulty & substitutes in placeof a legitimate attack on the problem, a short cut.
This short cut consists in "combining" the Huygenian equation
Centrifugal force (C) =𝑉2
𝑅

with "the law of Gravity”.

or which is the same thing C = L𝜋 2 ∙𝑅

𝑃2The Law of Gravity is ―― that the sun if a sphere attracts asuperficial equatorial particle with a force =spheroid with a force =𝑚

𝑅2

𝑀

𝑅2& if an oblate

where m is an equivalent mass placedat the centre calculable when all the illiptricities & densities ofthe strata are known ―― in other words it is the m above spoken of.
Therefore M. Comte’s short cut consists in "combining" the twoequations. C (= centrifugal force) =Gravity) =𝑉2
𝑅= 4𝜋 2 ∙𝑚𝑅2

𝑅

𝑃2and G (=There are many ways in which two equations may be "combined."2

1st. We may "combine" them by putting C = G. This is in fact arbitrarily& without any or <???> reason to make the assumption B ―― & is thevicious circle I complain of.

2d. We may combine them in some other way ―― but then C
will not be = G; the centrifugal force will not balance gravity& the nebular hypothesis is at an end.
The following is a resumé of my argument
1.

To calculate the periods from assumption B alone is tautologicalor a vicious circle. &2.

Assumption B cannot be legitimately deduced from assumption

A, by reason. 1st of physical data being wanting, & 2dly by reasonof the analytical difficulties of the problem.
3.

M. Comte pretends to have deduced assumption B from A,or else the periods in question direct from A.
4.

But his process amounts to assuming B, & can amountto nothing else, for centrifugal force can have nothing to do withthe question except in so far as it goes to diminish the gravity ―― &if it diminishes it to any point short of destroying it altogether,
the nebular hypothesis is set aside.
If what is above said about m as distinct from M appears
to puzzle the argument, it may be omitted without injuring its
validity. Let the gravity to the sun’s center be what it will. To
assume that C = G is to assume the planetary rotation & toassume that C is not = G is to destroy the nebular hypothesis.
It was incumbent on M. Comte to prove C = G through the mediumof assumption A, not to assume it ―― i.e. to prove his short cut bymeans of the very considerations it was designed to avoid.

I have left myself no room for entering into the other subject
alluded to in your letter, & it will take me some little timeto <???> my scattered ideas on the subject.
One word ―― in reference to a point mentioned in your letter.
You say "and if it can be shewn that the present rotation of the sun
takes place in the same time in which it ought to take place
supposing that proposition & the nebular hypothesis to be true, would
not that be of considerable weight" &c ??
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Doubtless it would ―― but let us see how the fact stands .
The Sun actually revolves in 25

12days.

A planet at the sun's surface would revolve in 2 h 46 m 48 s.
I see no signs of a "numerical coincidence" here at least.
16 July 1845

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