English
translation of a 1799 essay by

Pierre
Simon Laplace^{†}

*Proof of the theorem, that the attractive force of a heavenly
body could be so large, that light could not flow out of it. ***‡**

* *

(1)
If *v* is the velocity, *t *the
time and s space which is uniformly moving during this time, then, as is well
known, *v* = *s/t.*

* *

(2)
If the motion is not uniform, to obtain the value of *v* at any instant one has to divide the elapsed space d*s* and this time interval d*t* into each other, namely *v *= d*s*/d*t*, since the velocity over an infinitely
small interval is constant and thus the motion can be taken as uniform.

(3)
A continuously working force still strive to change the velocity. This change
of the velocity, namely d*v,* is
therefore the most natural measure of the force. But as any force will produce
double the effect in double the time, so we must divide the change in velocity
d*v* by the time d*t* in which it is brought about by the force **P**, and one thus obtains a general expression for the force **P**, namely

_{}

Now
if d*t* is constant.

_{}

accordingly _{}

‡ This theorem, that a luminous body
in the universe** **of the same** **density
as** **the earth whose diameter is 250
times larger than that of the sun, can by its attractive power prevent its
light rays from reaching us, and that consequently the Largest bodies in the
universe could remain invisible to us, has been stated by Laplace in his *Exposition du Système du Monde , *Part
II, p. 305, without proof. Hero is the proof. Cf. A.G.E. May 1798, p. 603. v.
Z.

(4)
Let the attractive force of a body *= M; *a
second body, for example a particle of light, finds itself at distance *r; *the action of the force *M *on this light particle will be—*M/rr*; the negative sign occurs because
the action of *M* is opposite to the
motion of the light.

(5)
Now according to (3) this force also equals
dd*r*/d*t*^{2}, hence

_{}

* *

Multiplying
by d*r, _{}*

integrating, _{}

where
C is a constant quantity. or

_{}

Now by (2) d*r*/d*t*
is the velocity *v*, accordingly

_{} _{}

holds,
where *v *is the velocity of the light
particle at the distance *r*.

(6)
To now determine the constant *C*, let *R *be the radius of the attracting body,
and *a* the velocity of the light at
the distance *R, *hence on the surface
of the attracting body; then one obtains from (5) *a*^{2}* = *2*C+*2*M/R
, *therefore 2*C = a ^{2} ‑
*2

* _{}*

* *

(7) Let *R*' be the radius of another attracting
body, its attractive power be *iM*, and
the velocity of the light at a distance *r
*be *v*', then according to the
equation in (6)

* _{}*

(8) If one
makes *r *infinitely large, the last
term in the previous equation vanishes and one obtains

_{}

The distance
of the fixed stars is so large, that this assumption is justified.

(9)
Let the attractive ponder of the second body be so large that light cannot
escape from it; this can be expressed analytically in the following way: the
velocity *v*' of the light is equal to
zero. Putting this value of *v*' in the
equation (8) for *v*', gives an
equation from which the mass *iM *for
which this occurs can be derived. One has therefore

_{} or _{}

(10) To determine *a*, let the first attracting body be the sun; then *a* is the velocity of the sun’s light on
the surface of the sun. The attractive
power of the sun is however so small in comparison with the velocity of light, that
one can take this velocity as uniform.
From the phenomena of aberration it appears that the earth travels at
20”_{}in its path while the light travels from the sun to the
earth, accordingly: let V be the average velocity of the earth in its orbit,
then on has a : V = radius (expressed in seconds) : 20”_{} = 1 : tang. 20”_{}.

(11)
My assumption made in *Expos. du Syst. du Monde. *Part II,

p.
305, *is R' = *250*R*. Now the mass changes as the volume of the attracting body
multiplied by its density the volume, as the cube of the radius. accordingly
the mass as the cube of the radius multiplied by the density. Let the density
of the sun = 1; that of the second body = *r*
; then

* M* : *iM*
= 1 *R*^{3} : *r**R*’^{3}
= 1 *R*^{3} : *r*
250^{3} *R*^{3}

or
1 : i = 1 : *r*(250)^{3}

or i = (250)^{3}*r*

(12) One substitutes the
values of I and R’ in the equation *a*^{2}=2*iM*/*R’*,
and thus obtains

_{}

_{}

(
13) To obtain *r*,
one must still determine *M*. The force *M* of the sun is equal at a distance *D* to *M/D*^{2}*. *Let *D*
be the average

distance
of the earth, V the average velocity of the earth; then this force is also
equal** **to *V*^{2}/*D* (see
Lande's *Astronomy, *III, §3539). Hence
*M*/*D*^{2}* = V*^{2}*/D *or M = *V*^{2}*D*. Substituting this in the equation
(12) for *r*
gives

_{},

_{} according to (10)

_{}
tan average apparent radius of O

from which *r*
is approximately 4, or as large as the density of the earth**.**

** **

D.
W. Dewhirst adds:

The *Allgemeine
geographische Ephemerides *was a journal founded by F. X. von Zach, of which
51 volumes were published between 1796 and 1816. The footnote (‡) is a
translation of that added by von Zach to the original paper which is however
not very helpful to the modern reader.

There are no less than 10 different editions
of Laplace’s *Exposition du Système du
.Monde *published between 1796 and 1835, some in one quarto volume and some
in two volumes octavo. In the earlier editions the 'statement without proof'
comes a few pages before the end of Book 5, Chapter 6, though Laplace removed
the specific statement. from later editions.

The reference by von Zach to *A.G.E.* May 1798, p. 603, seems to be a
mistake on von Zach's part: he was perhaps intending to refer to *A.G.E.* Vol. I, p. 89, 1798 where there
is an extensive essay review of the first edition of Laplace's *Exposition du Système du M*onde.