VII. 
On
the Means
of discovering the Distance,
Magnitude, &c. of the Fixed Stars, in consequence of the
Diminution
of the Velocity of their Light, in case such a Diminution should
be found to take place in any of them, and such other Data should
be procured from Observations, as would be farther necessary for
that Purpose.
By the Rev. John
Michell,
B. D. F.R.S.
In a Letter to Henry
Cavendish,
Esq.
F.R.S. and A. S. 
Read
November 27, 1783.
Thornhill,
May 26, 1783.
DEAR
SIR,
THE method,
which I mentioned to you when I was
last in London,
by which it might perhaps be possible to find the distance, magnitude,
and weight of some of the fixed stars, by means of the diminution
of the velocity of their light, occurred to me soon after I wrote
what is mentioned by Dr. PRIESTLEY
in his History
of Optics, concerning the diminution of the
velocity of
light in consequence of the attraction of the sun; but the extreme
difficulty, and perhaps impossibility, of procuring the other
data necessary for this purpose appeared to me to be such objections
against the scheme, when I first thought of it, that I gave it
then no further consideration. As some late observations, however,
begin to give us a little more chance of procuring some at least
of these data, I thought it would not be amiss, that astronomers
should be apprized of the method, I propose (which as far as I
know, has not been suggested by any one else) lest, for want of
being aware of the use, which may be made of them, they should
neglect to make the proper observations, when in their power;
I shall therefore beg the favour of you to present the following
paper on this subject to the Royal
Society.
I am,
&c.
THE
very
great number of stars that have been
discovered to be double, triple, &c. particularly by Mr. HERSCHEL
*
, if we apply the
doctrine of chances, as I have heretofore done in my
"Enquiry into the probable Parallax, &c. of the
Fixed Stars," published in the Philosophical
Transactions for the year 1767, cannot leave a doubt with
any one, who is properly aware of the force of those arguments,
that by far the greatest part, if not all of them, are systems
of stars so near to each other, as probably to be liable to be
affected sensibly by their mutual gravitation; and it is therefore
not unlikely, that the periods of the revolutions of some of these
about their principals (the smaller ones being, upon this hypothesis,
to be considered as satellites to the others) may some time or
other be discovered.
 2.
Now the apparent
diameter of any
central body, round which any other body revolves, together with
their apparent distance from one another, and the periodical time
of the revolving body being given, the density of the central
body will be given likewise. See Sir ISAAC NEWTON's
Prin. b. III. pr.
VIII. cor. I.
 3.
But the density
of
any central body
being given, and the velocity any other body would acquire by
falling towards it from an infinite height, or, which is the same
thing, the velocity of a comet revolving in a parabolic orbit,
at its surface, being given, the quantity of matter, and consequently
the real magnitude of the central body, would be given likewise.
 4.
Let us now
suppose
the particles
of light to be attracted in the same manner as all other bodies
with which we are acquainted; that is, by forces bearing the same
proportion to their vis
inertiae,
of which there can be no reasonable doubt, gravitation being,
as far as we know, or have any reason to believe, an universal
law of nature. Upon this supposition then, if any one of the fixed
stars, whose density was known by the abovementioned means, should
be large enough sensibly to affect the velocity of the light issuing
from it, we should have the means of knowing its real magnitude,
&c.
 5.
It has been
demonstrated by Sir ISAAC
NEWTON, in the 39th
proposition
of the first book of his Principia,
that if a right line be drawn, in the direction of which a body
is urged by any forces whatsoever, and there be erected at right
angles to that line perpendiculars every where proportional to
the forces at the points, at which they are erected respectively,
the velocity acquired by a body beginning to move from rest, in
consequence of being so urged, will always be proportional to
the square root of the area described by the aforesaid perpendiculars.
And hence,
 6.
If such a body,
instead of beginning
to move from rest, had already some velocity in the direction
of the same line, when it began to be urged by the aforesaid forces,
its velocity would then be always proportional to the square root
of the sum or difference of the aforesaid area, and another area,
whose square root would be proportional to the velocity which
the body had before it began to be so urged; that is, to the square
root of the sum of those areas, if the motion acquired was in
the same direction as the former motion, and the square root of
the difference, if it was in a contrary direction. See cor.2
to the aforesaid proposition.
 7.
In order to find,
by the foregoing
proposition, the velocity which a body would acquire by falling
towards any other central body, according to the common law of
gravity, let C in the
figure (tab.
III. ) represent the centre of the central body,
towards which
the falling point is urged, and let CA
be a line drawn from the point C,
extending
infinitely towards A.
If then the line RD be
supposed to represent the force,
by which the falling body would be urged at any point D,
the velocity which it would have acquired by falling from an infinite
height to the place D
would be the
same as that which it would acquire by falling from D
to C with the force RD,
the area of the infinitely extended hyperbolic space ADRB,
where RD is always
inversely proportional
to the square of DC,
being equal to
the rectangle RC
contained between
the lines RD and CD.
From hence we may draw the following corollaries.
 8.
Cor.
1.
The central body DEF
remaining the
same, and consequently the forces at the same distances remaining
the same likewise, the areas of the rectangles RC, rC
will always be inversely as the distances of the points D,
d,
from C,
their sides RD, rd
being inversely in the duplicate ratio of the sides CD,
cd:
and therefore, because the
velocity of a body falling from an infinite height towards the
point C, is always in
the subduplicate
ratio of these rectangles, it will be in the subduplicate ratio
of the lines CD, Cd,
inversely. Accordingly the velocities of comets revolving in parabolic
orbits are always in the subduplicate ratio of their distances
form the sun inversely; and the velocities of the planets, at
their mean distances (being always in a given ratio to the velocity
of such comets, viz.
in the
subduplicate ratio of 1
to 2)
must necessarily observe the same law likewise.
 9. Cor.
2.
The magnitude of the central body remaining the same, the velocity
of a body falling towards it from an infinite height will always
be, at the same distance from the point C,
taken any where without the central body, in the subduplicate
ratio of its density; for in this case the distance Cd
will remain the same, the line rd
only being increased or diminished in the proportion of the density,
and the rectangle rC
consequently increased or diminished in the same proportion.
 10.
Cor.
3.
The density of the
central body remaining the same, the velocity of a body falling
towards it from an infinite height will always be as its semidiameter,
when it arrives at the same proportional distance from the point C;
for
the weights, at the surfaces
of different sphaeres of the same density are as their respective
semidiameters; and therefore the sides RD
and CD, or any other
sides rd
and Cd,
which are in a given
ratio to those semidiameters, being both increased or diminished
in the same proportion, the rectangles RC
or rC
will be increased or diminished in the duplicate ratio of the
semidiameter CD, and
consequently
the velocity in the simple ratio of CD.
 11.
Cor.
4. If the velocity of a body falling
from an infinite
height towards different central bodies is the same, when it arrives
at their surfaces, the density of those central bodies must be
in the duplicate ratio of their semidiameters inversely; for
by the
last cor. the density of the central
body remaining the same, the rectangle RC
will be in the duplicate ratio of CD;
in order therefore that the rectangle RC
may always remain the same, the line RD
must be inversely, as CD,
and consequently
the density inversely, as the square of CD.
 12.
Cor.
5. Hence the quantity of matter
contained in those
bodies must be in the simple ratio of their semidiameters directly;
for the quantity of matter being always in a ratio compounded
of the simple ratio of the density, and the triplicate ratio of
their semidiameters, if the density is in the inverse duplicate
ratio of the semidiameters, this will become the direct triplicate
and inverse duplicate, that is, when the two are compounded together,
the simple ratio of the semidiameters.
 13.
The velocity a
body
would acquire
by falling from an infinite height towards the sun, when it arrived
at his surface, being, as has been said before in article 3d,
the same with that of a comet revolving in a parabolic orbit in
the same place, would be about 20.72
times greater than that of the earth in its orbit at its mean
distance from the sun; for the mean distance of the earth from
the sun, being about 214.64 of
the
sun's semidiameters, the velocity of such a comet would be greater
at that distance than at the distance of the earth from the sun,
in the subduplicate ratio of 214.64
to 1, and the velocity
of the comet
being likewise greater than that of planets, at their mean distances,
in the subduplicate ratio of 2
to 1;
these, when taken together, will make the full subduplicate ratio
of 429.28 to 1,
and the square root of 429.28
is 20.72,
very nearly.
 14.
The same result
would have been
obtained by taking the line RD
proportional
to the force of gravity at the sun's surface, and DC
equal to his semidiameter, and from thence computing a velocity,
which should be proportional to the square root of the area RC
when compared with the square root of another area, one whose
sides should be proportional to the force of gravity at the surface
of the earth; and the other should be, for instance, equal to 16
feet, 1 inch,
the space a body would
fall through in one second of time, in which case it would acquire
a velocity of 32 feet, 2 inches
per
second. The velocity thus found compared with the velocity of
the earth in its orbit, when computed from the same elements,
necessarily gives the same result. I have made use of this latter
method of computation upon a former occasion, as may be seen in
Dr. PRIESTLEY's History
of
Optics, p787, &c. but I have rather chosen to take
the velocity from that of a comet, in the article above,
on account of its greater simplicity, and its more immediate connexion
with the subject of this paper.
 15.
The velocity of
light, exceeding
that of the earth in its orbit, when at its mean distance from
the sun, in the proportion of about 10,310
to 1, if we divide 10,310
by 20.72, the quotient 497,
in round numbers, will express the number of times, which the
velocity of light exceeds the velocity a body could acquire by
falling from an infinite height towards the sun, when it arrived
at his surface; and an area whose square root should exceed the
square root of the area RC,
where RD
is supposed to represent the force of gravity at the surface of
the sun, and CD is
equal to his semidiameter,
in the same proportion, must consequently exceed the area RC
in the proportion of 247,009,
the square
of 497 to 1.
 16.
Hence, according
to article
10,
if the semidiameter of a sphaere of the same density with the
sun were to exceed that of the sun in the proportion 500
to 1, a body falling
from an infinite
height towards it, would have acquired at its surface a greater
velocity than that of light, and consequently, supposing light
to be attracted by the same force in proportion to its vis inertiae,
with other bodies, all light emitted from such a body would be
made to return towards it, by its own proper gravity.
 17.
But if the
semidiameter of a sphaere,
of the same density with the sun, though the velocity of light
emitted from such a body, would never be wholly destroyed, yet
would it always suffer some diminution, more or less, according
to the magnitude of the said sphaere; and the quantity of this
diminution may be easily found in the following manner: Suppose S
to
represent the semidiameter of
the sun, and aS to
represent the semidiameter
of the proposed sphaere; then, as appears from what has been shewn
before, the square root of the difference between the square of 497
S
and the square of aS
will be always proportional to the ultimate remaining velocity,
after it has suffered all the diminution, it can possibly suffer
from this cause; and consequently the difference between the whole
velocity of light, and the remaining velocity, as found above,
will be the diminution of its velocity. And hence the diminution
of the velocity of light emitted from the sun, on account of it's
gravitation towards that body, will be somewhat less than a 494,000dth
part of the velocity which it would have had if no such diminution
had taken place; for the square of 497
being 247,009, and the
square of 1
being 1, the
diminution of the velocity
will be the difference between the square root of 247,009,
and the square root of 247,008,
which
amounts, as above, to somewhat less than one 494,000th
part of the whole quantity.
 18.
The same effects
would likewise
take place, according to article
11,
if
the semidiameters were different from those mentioned in the
two last articles, provided the density was greater or less in
the duplicate ratio of those semidiameters inversely.
 19.
The better to
illustrate this matter,
it may not be amiss to take a particular example. Let us suppose
then, that it should appear from observations made upon some of
those double stars above alluded to, that one of the two performed
its revolution about the other in 64
years,
and that the central one was of the same density with the sun,
which it must be, if its apparent diameter, when seen from the
other body, was the same as the apparent diameter of the sun would
be if seen from a planet revolving round him in the same period:
let us further suppose, that the velocity of the light of the
central body was found to be less than that of the sun, or other
stars whose magnitude was not sufficient to affect it sensibly,
in the proportion of 19
to 20.
In this case then, according to article 17,
the square root of 247,009 SS
must
be to the square root of the difference between 247,009
SS and aaSS
as 20
to 19. But the squares
of 20
and 19 being 400
and 361, the quantity 247,009
SS must therefore be to the difference between this
quantity
and aaSS in the same
proportion, that
is as 247,009 to 222,925.62;
and aaSS must
consequently be equal
to 24,083.38 SS,
whose
square root
is 155.2 S nearly, or,
in round numbers, 155
times the diameter of the sun,
will be the diameter of the central star sought.
 20.
As the squares of
the periodical
times of bodies, revolving round a central body, are always
proportional
to the cubes of their mean distances, the distance of the two
bodies from each other must therefore, upon the foregoing suppositions,
be sixteen times greater in proportion to the diameter of the
central body, than the distance of the earth from the sun in proportion
to his diameter; and that diameter being already found to be also
greater than that of the sun in the proportion of 155.2
to 1, this distance
will consequently
be greater than that of the earth and sun from each other in the
proportion of 16 times 155.2,
that is 2483.2 to 1.
 21.
Let us farther
suppose, that from
the observations, the greatest distance of the two stars in question
appeared to be only one second; we must then multiply the number 2483.2
by 206,264.8,
the number of seconds in the radius of a circle, and the product 512,196,750
will shew the number of
times which such a star's distance from us must exceed that of
the sun. The quantity of matter contained in such a star would
be (155.2)³ or 3,738,308
times as much as that contained in the sun; its light, supposing
the sun's light to take up 8'.7''
in
coming to the earth, would, with its common velocity, require 7,900
years
to arrive at us, and 395
years more on account of the diminution of that velocity;
and supposing such a star to be equally luminous with the sun,
it would still be very sufficiently visible, I apprehend, to the
naked eye; notwithstanding its immense distance.
 22.
In the elements
which I have employed
in the above computations, I have supposed the diameter of the
central star to have been observed, in order to ascertain its
density, which cannot be known without it; but the diameter of
such a star is much too small to be observed by any telescopes
yet existing, or any that it is probably in the power of human
abilities to make; for the apparent diameter of the central star,
if of the same density with the sun, when seen from another body,
which would revolve round it in 64
years,
would be only the 1717th
part
of the distance of those bodies from each other, as will appear
from multiplying 107.32,
the number
of times the sun's diameter is contained in his distance from
the earth, by 16, the
greater proportional
distance of the revolving body, corresponding to 64
years instead of 1.
Now the 1717th
part of a second must
be magnified 309,060
times in order
to give it an apparent diameter of three minutes; and three minutes,
if the telescopes were mathematically perfect, and there was no
want of distinctness in the air, would be but a very small matter
to judge of. *
 23.
But though there
is
not the least
probability that this element, so essential to be known, in order
to determine with precision the exact distance and magnitude of
a star, can ever be obtained, where it is in the same circumstances,
or nearly the same, with those above supposed, yet the other elements,
such as perhaps may be obtained, are sufficient to determine the
distance, &c. with a good deal of probability, within some
moderate limits; for in whatever ratio the real distance of the
two stars may be greater or less than the distance supposed, the
density of the central star must be greater or less in the sixth
power of that ratio inversely; for the periodic time of the revolving
body being given, the quantity of matter contained in the central
body must be as the cube of their distance from each other. See Sir I. Newton's
Prin. b. 3d. pr. 8th. cor
3d. But the quantity of matter in different bodies, at
whose surfaces the velocity acquired by falling from an infinite
height is the same, must be, according to art. 12,
directly as their semidiameters; the semidiameters therefore
of such bodies must be in the triplicate ratio of the distance
of the revolving body; and consequently their densities, by art. 11,
being in the inverse duplicate ratio of their semidiameters,
must be in the inverse sextuplicate ratio of the distance of the
revolving body. Hence if the real distance should be greater or
less than that supposed, in the proportion of two or three to
one, the density of the central body must be less or greater,
in the first case, in the proportion of 64,
or in the latter of 729
to 1.
 24.
There is also
another circumstance,
from which perhaps some little additional probability might be
derived, with regard to the real distance of a star, such as that
we have supposed; but upon which however, it must be acknowledged,
that no great stress can be laid, unless we had some better analogy
to go upon than we have at present. The circumstance I mean is
the greater specific brightness which such a star must have, in
proportion as the real distance is less than that supposed, and vice
versa;
since, in order
that the star may appear equally luminous, its specific brightness
must be as the fourth power of its distance inversely; for the
diameter of the central star being as the cube of the distance
between that and the revolving star, and their distance from the
earth being in the simple ratio of their distance from each other,
the apparent diameter of the central star must be as the square
of its real distance from the earth, and consequently, the surface
of a sphaere being as the square of its diameter, the area of
the apparent disc of such a star must be as the fourth power of
its distance from the earth; but in whatever ratio the apparent
disc of the star is greater or less, in the same ratio inversely
must be the intensity of its light, in order to make it appear
equally luminous. Hence, if its real distance should be greater
or less than that supposed in the proportion of 2
or 3 to 1,
the intensity of its light must be less or greater, in the first
case, in the proportion of 16,
or,
in the latter of 81 to 1.
 25.
According to
Mons. BOUGER
(see his Traité d'Optique)
the
brightness of the sun exceeds that of a wax candle in no less
a proportion than that of 8000
to 1.
If therefore the brightness of any of the fixed stars should not
exceed that of our common candles, which, as being something less
luminous than wax, we will suppose in round numbers to be only
one 10,000dth
part as bright
as the sun, such a star would not be visible at more than an 100dth
part of the distance, at which it would be visible, if it was
as bright as the sun. Now because the sun would still appear,
I apprehend, as luminous, as the star Sirius,
when removed to 400,000
times his present
distance, such a body, if no brighter than our common candles,
would only appear equally luminous with that star at 4000
times the distance of the sun, and we might then begin to be able,
with the best telescopes, to distinguish some sensible apparent
diameter of it; but the apparent diameters of the stars of the
less magnitudes would still be too small to be distinguishable
even with our best telescopes, unless they were yet a good deal
less luminous, which may possibly however be the case with some
of them; for, though we have indeed very slight grounds to go
upon with regards to the specific brightness of the fixed stars
compared with that of the sun at present, and can therefore only
form very uncertain and random conjectures concerning it, yet
from the infinite variety which we find in the works of the creation,
it is not unreasonable to suspect, that very possibly some of
the fixed stars may have so little natural brightness in proportion
to their magnitude, as to admit of their diameters having some
sensible apparent size, when they shall come to be more carefully
examined, and with larger and better telescopes than have been
hitherto in common use.
 26.
With regard to
the
sun, we know
that his whole surface is extremely luminous, a very small and
temporary interruption sometimes from a few spots only excepted.
This universal and excessive brightness of the whole surface is
probably owing to an atmosphaere, which being luminous throughout,
and in some measure also transparent, the light, proceeding from
a considerable depth of it, all arrives at the eye; in the same
manner as the light of a great number of candles would do, if
they were placed one behind another, and their flames were sufficiently
transparent to permit the light of the more distant ones to pass
through those that were nearer, without any interruption.
 27.
How far the same
constitution may
take place in the fixed stars we don't know; probably however
it may do so in many; but there are some appearances with regard
to a few of them, which seem to make it probable, that it does
not do so universally. Now, if I am right in supposing the light
of the sun to proceed from a luminous atmosphaere, which must
necessarily diffuse itself equally over the whole surface, and
I think there can be very little doubt that this is really the
case, this constitution cannot well take place in those stars,
which are in some degree periodically more and less luminous,
such as that in Collo Ceti,
&c.
It is also not very improbable, that there is some difference
from that of the sun, in the constitution of those stars, which
have sometimes appeared and sometimes disappeared, of which that
in the constellation of Cassiopeia
is a notable instance. And if those conjectures are well founded
which have been formed by some philosophers concerning stars of
these kinds, that they are not wholly luminous, or at least not
constantly so, but that all, or by far the greatest part of their
surfaces is subject to considerable changes, sometimes becoming
luminous, and at other times being extinguished; it is amongst
the stars of this sort, that we are most likely to meet with instances
of a sensible apparent diameter, their light being much more likely
not to be so great in proportion as that of the sun, which, if
removed to four hundred thousand times his present distance would
still appear, I apprehend, as bright as Sirius,
as I have observed above; whereas it is hardly to be expected,
with any telescopes whatsoever, that we should ever be able to
distinguish a welldefined disc of any body of the same size with
the sun at much more than ten thousand times his distance.
 28.
Hence the
greatest
distance at which
it would be possible to distinguish any sensible apparent diameter
of a body as dense as the sun cannot well greatly exceed five
hundred times ten thousand, that is, five million times the distance
of the sun; for if the diameter of such a body was not less than
five hundred times that of the sun, its light, as has been shewn
above, in art.16.
could never arrive at
us.
 29.
If there should
really exist in
nature any bodies, whose density is not less than that of the
sun, and whose diameters are more than 500
times the diameter of the sun, since their light could
not arrive at us; or if there should exist any other bodies of
a somewhat smaller size, which are not naturally luminous; of
the existence of bodies under either of these circumstances, we
could have no information from light; yet, if any other luminous
bodies should happen to revolve about them we might still perhaps
from the motions of these revolving bodies infer the existence
of the central ones with some degree of probability, as this might
afford a clue to some of the apparent irregularities of the revolving
bodies, which would not be easily explicable on any other hypothesis;
but as the consequences of such a supposition are very obvious,
and the consideration of them somewhat beside my present purpose,
I shall not prosecute them any farther.
 30.
The diminution of
the velocity of
light, in case it should be found to take place in any of the
fixed stars, is the principal phaenomenon whence it is proposed
to discover their distance, &c. Now the means by which we
may find what this diminution amounts to, seems to be supplied
by the difference which would be occasioned in consequence of
it, in the refrangibility of the light, whose velocity should
be so diminished. For let us suppose with Sir ISAAC
NEWTON
(see his Optics, prop.
VI. paragr. 4 and 5) that the refraction of light is
occasioned
by a certain force impelling it towards the refracting medium,
an hypothesis which perfectly accounts for all the appearances.
Upon this hypothesis the velocity of light in any medium, in whatever
direction it falls upon it, will always bear a given ratio to
the velocity it had before it fell upon it, and the sines of incidence
and refraction, in consequence of this, bear the same ratio to
each other with these velocities inversely. Thus, according to
this hypothesis, if the sines of the angles of incidence and
refraction,
when light passes out of air into glass, are in the ratio of 31
to 20, the velocity of
light in the
glass must be to its velocity in air in the same proportion of 31
to 20.
But because the areas, representing the forces generating these
velocities, are as the square of the velocities, see art. 5.
and 6.
these areas must be to each other as 961 to 400.
And if 400 represents
the area which
corresponds to the force producing the original velocity of light, 561,
the difference between 961
and 400 must represent
the area corresponding
to the additional force, by which the light is accelerated at
the surface of the glass.
 31.
In art.
19.
we supposed, by way of example, the velocity of the light of some
particular star to be diminished in the ratio of 19
to 20, and it was
there observed, that
the area representing the remaining force which would be necessary
to generate the velocity 19,
was therefore
properly represented by (361 / 400)dth
parts of the area, that should represent the force that would
be necessary to generate the whole velocity of light, when
undiminished.
If we then add 561,
the area representing
the force by which the light is accelerated at the surface of
the glass, to 361, the
area representing
the force which should have generated the diminished velocity
of the star's light, the square root of 922,
their sum, will represent the velocity of the light with the diminished
velocity, after it has entered the glass. And the square root
of 922 being 30.364,
the sines of incidence and refraction of such light out of air
into glass will consequently be as 30.364
to 19, or what is
equal to it, as 31.96
to 20 instead of 31
to 20, the ratio of
the sines of incidence
and refraction, when the light enters the glass with its velocity
undiminished.
 32.
From hence a
prism,
with a small
refracting angle, might perhaps be found to be no very inconvenient
instrument for this purpose: for by such a prism, whose refracting
angle was of one minute, for instance, the light with its velocity
undiminished would be turned out of its way 33'',
and with the diminished velocity 35''.88
nearly, the difference between which being almost 2''.53'''
would be the quantity by which the light, whose velocity was
diminished,
would be turned out of its way more than that whose velocity was
undiminished.
 33.
Let us now be
supposed to make use
of such a prism to look at two stars, under the same circumstances
as the two stars in the example abovementioned, the central one
of which should be large enough to diminish the velocity of its
light one twentieth part, whilst the velocity of the light of
the other, which was supposed to revolve about it as a satellite,
for want of sufficient magnitude in the body from whence it was
emitted, should suffer no sensible diminution at all. Placing
then the line, in which the two faces of the prism would intersect
each other, at right angles to a line joining the two stars; if
the thinner part of the prism lay towards the same point of the
heavens with the central star, whose light would be most turned
out of its way, the apparent distance of the stars would be increased 2''.53'''
and consequently become 3''.53'''
instead of 1''. only,
the apparent
distance supposed above in art. 21. On
the contrary, if the prism should be turned half way round, and
its thinner part lye towards the same point of the heavens with
the revolving star, their distance must be diminished by a like
quantity, and the central star therefore would appear 1''.53'''
distant from its place near three times the whole distance between
them.
 34.
As a prism might
be
made use of
for this purpose, which should have a much larger refracting angle
than that we have proposed, especially if it was constructed in
the achromatic way, according to Mr. DOLLOND's
principles, not only such a diminution, as one part in twenty,
might be made still more distinguishable; but we might probably
be able to discover considerably less diminutions in the velocity
of light, as perhaps a hundredth, a twohundredth, a fivehundredth,
or even a thousandth part of the whole, which, according to what
has been said above, would be occasioned by sphaeres, whose diameters
should be to that of the sun, provided that they were of the same
density, in the several proportions nearly of 70, 50,
30,
and 22 to 1
respectively.
 35.
If such a
diminution of the velocity
of light, as that above supposed, should be found really to take
place, in consequence of its gravitation towards the bodies from
whence it is emitted, and there should be several of the fixed
stars large enough to make it sufficiently sensible, a set of
observations upon this subject might probably give us some considerable
information with regard to many circumstances of that part of
the universe, which is visible to us. The quantity of matter contained
in many of the fixed stars might from hence be judged of, with
a great degree of probability, within some moderate limits; for
though the exact quantity must still depend upon their density,
yet we must suppose the density most enormously different from
that of the sun, and more so, indeed, than one can easily conceive
to take place in fact, to make the error of the supposed quantity
of matter very wide of the truth, since the density, as has been
shewn above in art.
11. and 12.
which is necessary to produce the same diminution in the velocity
of light, emitted from different bodies, is as the square of the
quantity of matter contained in those bodies inversely.
 36.
But though we
might
possibly from
hence form some reasonable guess at the quantity of matter contained
in several of the fixed stars; yet, if they have no luminous satellites
revolving about them, we shall still be at a loss to form any
probable judgement of their distance, unless we had some analogy
to go upon for their specific brightness, or had some other means
of discovering it; there is, however, a case that may possibly
occur, which may tend to throw some light upon this matter.
 37.
I have shewn in
my Enquiry
into the probable Parallax, &c. of the Fixed Stars,
published in the Philosophical
Transactions
for the year 1767, the extremely great probability there is, that
many of the fixed stars are collected together into groups; and
that the Pleiades in
particular constitute
one of these groups. Now of the stars which we there see collected
together, it is highly probable, as I have observed in that paper,
that there is not one in a hundred which does not belong to the
group itself; and by far the greatest part, therefore, according
to the same idea, must lie within a sphaere, a great circle of
which is of the same size with a circle, which appears to us to
include the whole group. If we suppose, therefore, this circle
to be about 2°. in
diameter, and
consequently only about a thirtieth part of the distance at which
it is seen, we may conclude, with the highest degree of probability,
that by far the greatest part of these stars do not differ in
their distances from the sun by more than about one part in thirty,
and from thence deduce a sort of scale of the proportion of the
light which is produced by different stars of the same group or
system in the Pleiades
at least; and,
by a somewhat probable analogy, we may do the same in other systems
likewise. But having yet no means of knowing their real distance,
or specific brightness, when compared either with the sun or with
one another, we shall still want something more to form a farther
judgement from.
 38.
If, however, it
should be found,
that amongst the Pleiades,
or any other
like system, there are some stars that are double, triple, &c.
of which one is a larger central body, with one or more satellites
revolving about it, and the central body should likewise be found
to diminish the velocity of its light; and more especially, if
there should be several such instances met with in the same system;
we should then begin to have a kind of measure both of the distance
of such a system of stars from the earth, and of their mutual
distances from each other. And if several instances of this kind
should occur in different groups or systems of stars, we might
also, perhaps, begin to form some probable conjectures concerning
the specific density and brightness of the stars themselves, especially
if there should be found any general analogy between the quantity
of the diminution of the light and the distance of the system
deduced from it; as, for instance, if those stars, which had the
greatest effect in diminishing the velocity of light should in
general give a greater distance to the system, when supposed to
be of the same density with the sun, we might then naturally conclude
from thence, that they are less in bulk, and of greater specific
density, than those stars which diminish the velocity of light
less, and vice versa.
In like
manner, if the larger stars were to give us in general a greater
of less quantity of light in proportion to their bulk, this would
give us a kind of analogy, from whence perhaps we might form some
judgement of the specific brightness of the stars in general;
but, at all adventures, we should have a pretty tolerable measure
of the comparative brightness of the sun and those stars, upon
which such observations should be made, if the result of them
should turn out agreeable to the ideas above explained.
 39.
Though it is not
improbable, that
a few years may inform us, that some of the great number of double,
triple stars, &c. which have been observed by Mr. HERSHEL,
are systems of bodies revolving about one another, especially
if a few more observers, equally ingenious and industrious with
himself could be found to second his labours; yet the very great
distance at which it is not unlikely many of the secondary stars
may be placed from their principals, and the consequently very
long periods of their revolutions *,
leave very little room to hope that any great progress can be
made in this subject for many years, or perhaps some ages to come;
the above outlines, therefore, of the use that may be made of
the observations upon the double stars, &c. provided the
particles
of light should be subject to the same law of gravitation with
other bodies, as in all probability they are, and provided also
that some of the stars should be large enough sensibly to diminish
their velocity, will, I hope, be an inducement to those, who may
have it in their power, to make these observations for the benefit
of future generations at least, how little advantage soever we
may expect from them ourselves; and yet very possibly some observations
of this sort, and such as may be made in a few years, may not
only be sufficient to do something, even at present, but also
to shew, that much more may be done hereafter, when these observations
shall be come more numerous, and have been continued for a longer
period of years.
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