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PHILOSOPHICAL

TRANSACTIONS,

OF THE

ROYAL SOCIETY

LONDON.

FOR THE YEAR MDCCXCVIH.

M ' 7

PART 1.

LONDON,

SOLD BY PETER ELMSLY,
PRINTER TO THE ROYAL SOCIETY.

MDCCXCVIII.

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ADVERTISEMENT.

The Committee appointed by the Royal Society to direct the pub-
lication of the Philosophical Transactions , take this opportunity to
acquaint the Public, that it fully appears, as well from the council-
books and journals of the Society, as from repeated declarations which
have been made in several former Transactions , that the printing of
them was always, from time to time, the single act of the respective
Secretaries, till the Forty-seventh Volume : the Society, as a Body,
never interesting themselves any further in their publication, than by
occasionally recommending the revival of them to some of their Se-
cretaries, when, from the particular circumstances of their affairs, the
Transactions had happened for any length of time to be intermitted.
And this seems principally to have been done with a view to satisfy
the Public, that their usual meetings were then continued, for the im-
provement of knowledge, and benefit of mankind, the great ends of
their first institution by the Royal Charters, and which they have ever
since steadily pursued.

But the Society being of late years greatly enlarged, and their com-
munications more numerous, it was thought advisable, that a Com-
mittee of their members should be appointed to reconsider the papers
read before them, and select out of them such as they should judge
most proper for publication in the future Transactions ; which was
accordingly done upon the 26th of March, 1752. And the grounds

A 2

z * n

of their choice are, and will continue to be, the importance and sin-
gularity of the subjects, or the advantageous manner of treating them;
without pretending to answer for the certainty of the facts, or pro-
priety of the reasonings, contained in the several papers so published,
which must still rest on the credit or judgment of their respective
authors.

It is likewise necessary on this occasion to remark, that it is an esta-
blished rule of the Society, to which they will always adhere, never to
give their opinion, as a Body, upon any subject, either of Nature or
Art, that comes before them. And therefore the thanks, which are
frequently proposed from the Chair to be given to the authors of such
papers as are read at their accustomed meetings, or to the persons through
whose hands they receive them, are to be considered in no other light
than as a matter of civility, in return for the respect shewn to the So-
ciety by those communications. The like also is to be said with re-
gard to the several projects, inventions, and curiosities of various
kinds, which are often exhibited to the Society ; the authors whereof,
or those who exhibit them, frequently take the liberty to report, and
even to certify in the public news-papers, that they have met with the
highest applause and approbation. And therefore it is hoped, that no
regard will hereafter be paid to such reports, and public notices ; which
in some instances have been too lighdy credited, to the dishonour of
the Society.

CONTENTS.

*

I. The Bakerian Lecture. Experiments upon the Resistance of

Bodies moving in Fluids. By the Rev. Samuel Vince, A. M.
F. R. S. Plumian Professor of Astronomy and Experimental
Philosophy in the University of Cambridge. Page 1

II. Experiments and Observations , tending to show the Compo-

sition and Properties of Urinary Concretions. By George
Pearson, M. D. F. R. S. p .15

III. On the Discovery of four additional Satellites of the

Georgium Sidus. The retrograde Motion of its old Satel-
lites announced; and the Cause of their Disappearance at
certain Distances from the Planet explained. By William
Herschel, LL.D. F. R. S. p. 47

IV. An Inquiry concerning the Source of the Heat which is

excited by Friction. By Benjamin Count of Rumford,
F. R. S. M. R. I. A. p. 80

V. Observations on the Foramina Thebesii of the Heart. By

Mr. John Abernethy, F. R. S. Communicated by Everard
Home, Esq. F. R. S. p. 103

VI. An Analysis of the earthy Substance from New South IFales,

called Sydneia or Terra Australis. By Charles Hatchett, Esq.
F. R. S. p. 1 1 o

C Vi 3

VII. Abstract of a Register of the Barometer , Thermometer ,

and Rain , at Lyndon, in Rutland, for the Tear 1796. By
Thomas Barker, Esq. Communicated by Mr. Timothy Lane,
F. R. S. p 1^Q

VIII. An Account of some Endeavours to ascertain a Standard of

JF eight and Measure . By Sir George Shuckburgh Evelyn,
Bart. F. R. S. and A. S. p_ 19^

IX. A new Method of computing the Value of a slowly con-

verging Series , of which all the Terms are affirmative . By
the Rev. John Hellins, F. R. S. and Vicar of Potter’s- Pury,
in Northamptonshire. In a Letter to the Rev. Dr. Maske-
lyne, F. R. S. and Astronomer Royal. p. 183

APPENDIX.

Meteorological Journal kept at the Apartments of the Royal So-
ciety, by Order of the President and Council.

/

PHILOSOPHICAL

TRANSACTION S.

I. The Bakerian Lecture. Experiments upon the Resistance oj
Bodies moving in Fluids. By the Rev. Samuel Vince, A. M.
F. R. S. Plumian Professor of Astronomy and Experimental
Philosophy in the University of Cambridge.

Read November 9, 1797.

In a former Paper upon the Motion of Fluids, I stated the
difficulties to which the theory is subject, and showed its in-
sufficiency to determine the time of emptying vessels, even in
the most simple cases ; I also proved, by actual experiments,
that, in many instances, there was no agreement between their
results and those deduced from theory. The great difference
between the experimental and theoretical conclusions, in most
of the cases which respect the times in which vessels empty
themselves through pipes, necessarily leads us to suspect the
truth of the theory of the action of fluids under all other cir-
cumstances. In the doctrine of the resistances of fluids, we
see strong reasons to induce us to believe, that the theory can-
not generally lead us to any true conclusions. When a body
mdccxcviii. B

2

Mr. Vince's Experiments on the Resistance

moves in a fluid, its particles strike the body; and, in our theo-
retical considerations, after this action, the particles are sup-
posed to produce no further effect, but are conceived to be, as
it were, annihilated. But, in fact, this cannot be the case;
and what we are to allow for their effect afterwards, is beyond
the reach of mere theoretical investigation. Whatever theory
therefore we can admit, must be that which is founded upon
such experiments as include in them every principle which is
subject to any degree of uncertainty. We must therefore have
recourse to experiments, in order to establish any conclusions
upon which we may afterwards reason. In the paper above
mentioned, I described a machine to find the resistances of
bodies moving in fluids ; since which time, I have made a va-
riety of experiments with it, upon bodies moving both in air
and water, and I have every reason to be satisfied of its great
accuracy. In this paper, I propose to examine the resistance
which arises from the action of non-elastic fluids upon bodies.

This subject divides itself into two parts ; we may consider
the action of water at rest upon a body moving in it, or we
may consider the action of the water in motion upon the body
at rest. We will first give the result of our experiments in the
former case, and compare them with the conclusions deduced
from theory. Now the radius of the axis of the machine made
use of in these experiments was 0,2117 in. the area of the four
planes was 3,73 in. the distance of their centres of resistance
from the axis was 7,57 in. and they moved with a velocity of
o,66 feet in a second. The first column of the following table
exhibits the angles at which the planes struck the fluid ; the
second column shows the resistance by experiment, in the
direction of their motion, in Troy ounces; the third column

3

of Bodies moving in Fluids .

gives the resistance by theory, assuming the perpendicular re-
sistance to be the same as by experiment ; the fourth column
shows the power of the sine of the angle to which the resistance
is proportional.

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The fourth column was thus computed : Let s be the sine
of the angle to radius unity, r the resistance at that angle, and
suppose r to vary as then iw: r :: 0,2321 : r, hence, sK

= 5^7? and consequently m = i and, by sub-

stituting for r and 5 their several corresponding values, we get
the respective values of m, which are the numbers in the fourth
column. Now the theory supposes the resistance to vary as the
cube of the sine; whereas, the resistance decreases from an
angle of 90°, in a less ratio than that, but not as any constant
power of the sine, nor as any function of the sine and cosine,
that I have yet discovered. Hence, the actual resistance is al-
ways greater than that which is deduced from theory, assuming
the perpendicular resistance to be the same; the reason of
which, in part at least, is, that in our theory we neglect the

B 2

4 Mr , Vince’s Experiments on the Resistance

whole of that part of the force which, after resolution, acts
parallel to the plane; whereas (from the experiments which
will be afterwards mentioned), it appears that part of that
force acts upon the plane; also, the resistance of the fluid
which escapes from the plane, into the surrounding fluid, may
probably tend to increase the actual resistance above that which
the theory gives, in which that consideration does not enter; but,
as this latter circumstance affects the resistance at all angles,
and we do not know the quantity of effect which it produces,
we cannot say how it may affect the ratio of the resistances at
different angles.

In theory, the resistance perpendicular to the planes is sup-
posed to be equal to the weight of a column of fluid, whose
base = 3,73 in. and altitude =? the space through which a body
must fall to acquire the velocity of o ,66 feet; now that space
is 0,08124 in. consequently the weight of the column = 0,1598
Troy oz. ; but the actual resistance was found to be = 0,2321 oz.
Hence, the actual resistance of the planes : the resistance in our
theory :: 0,2321 ; 0,1598, which is nearly as 3 : 2.

I am aware that experiments have been made upon the re-
sistances of bodies moving in water, which have agreed with
our theory. An extensive set was instituted by D’Alembert,
Condorcet, and Bossut, the result of which very nearly
coincided with theory, so far as regards the absolute quantity
of the perpendicular resistance. Their experiments were made
upon floating bodies, drawn upon the fluid by a force acting
upon them in a direction parallel to the surface of the fluid.
There can be no doubt but that these experiments were very
accurately made. The experiments here related were also re-
peated so often, and with so much care, and the results always

of Bodies moving in Fluids. 5

agreed so nearly, that there can be no doubt but that they give
the actual resistance to a very considerable degree of accuracy.
In our experiments, the planes were immersed at some depth
in the fluid ; in the other case, the bodies floated on the surface ;
and I can see no way of accounting for the difference of the
resistances, but by supposing that, at the surface of the fluid,
the fluid from the end of the body may escape more easily
than when the body is immersed below the surface ; but this,
I confess, appears by no means a satisfactory solution of the
difficulty. The resistances of bodies descending in fluids mani-
festly come under the case of our experiments.

Two semi-globes were next taken, and made to revolve with
their flat sides forwards. The diameter of each was 1,1 in. the
distance of the centre of resistance from the axis was 6, 22 in.
and they moved with a velocity of 0,542 feet in a second ; and
the resistance was found to be 0,08339 oz. by experiment. By
theory, the resistance is 0,05496 oz. ; hence, the resistance by
experiment : the resistance by theory : : 0,08339 : 0,05496,
agreeing very well with the abovementioned proportion. But,
when the spherical sides moved forwards with the same velo-
city, the resistance was 0,034 oz- Hence, the resistance on
the spherical side of a semi-globe : resistance on its base ::
0,034 : 0,08339; but this is not the proportion of the resist-
ance of a perfect globe to the resistance of a cylinder of the
same diameter, moving with the same velocity, because the
resistance depends upon the figure of the back part of the
body.

I therefore took two cylinders, of the same diameter as the
two semi-globes, and of the same weight; and, giving them
the same velocity, I found the resistance to be 0,07998 oz. ;

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therefore the resistance on the flat side of a semi-globe : the
resistance of a cylinder of the same diameter, and moving
with the same velocity : : 0,08339 : 0,07998. This difference
can arise only from the action of the fluid on the back side
of the semi-globe, moving with its flat side forwards, being less
than that on the back of the cylinder, in consequence of which
the semi-globe suffered the greater resistance. The resistance
of the cylinders, thus determined directly by experiment, agrees
very well with the foregoing experiments. The resistance,
cceteris paribus , varies as the square of the velocity very nearly,
and may be taken so for all practical purposes, as I find by
repeated experiments, made both upon air and water, in the
manner described in my former paper. Hence, for different
planes, the resistance varies as the area x the square of the ve-
locity. Now the resistance of the planes whose area was 3,73 in.
moving with a velocity of o ,66 feet in a second, was found to
be = 0,2321 oz. Also, the area of the two cylinders was 1,9 in.
and their velocity was 0,542 feet in a second ; to find, there-
fore, the resistance of the cylinders from that of the planes,
we have o ,66* x 3,73 : 0,542* x 1,9 :: 0,2321 oz : 0,07973 oz.
for the resistance on the cylinders, differing but a very little
from 0,07998 oz. the resistance found from direct experiment.

Now, to get the resistance on a perfect globe, we must con-
sider, that when the back part is spherical, the resistance is
greater than when it is flat, in the ratio of 0,08339 : 07998 ;
hence, the resistance on a globe : the resistance on a semi-
globe in the same ratio ; but the resistance on the semi-globe
was 0,034 oz. hence, 0,07998 : 0,08339 : : 0,034 oz. : 0,0354
oz. the resistance of a globe ; consequently, the resistance of a
globe : the resistance of a cylinder of the same diameter, mov-

’ a

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of Bodies moving in Fluids. 7

ing with the same velocity in water :: 0,0354 : 0,07998 :: 1 :
2,23.

We proceed next to compare the actual resistance of a globe
with the resistance assumed in our theory. In the first place,
the absolute quantity of resistance has been found to be greater
than that which we use in theory, in the ratio of 0,2321 : 0,1598;
but, by theory, the resistance of the globe : the resistance of
the cylinder :: 1 : 2, or as 1,115 : 2,23; hence, by theory,
we make the resistance of the globe too great, in the ratio of

3.115 : 1 » and it is too small, from the former consideration,
in the ratio of 0,1598 : 0,2321; therefore the actual resistance
of the globe : the resistance in theory :: 0,2321 : 0,1598 x

1.115 :: 0,2321 : 0,1782, which is nearly in the ratio of 4 : 3.
T.hus far we have considered the resistance of bodies moving
in a fluid ; we come next to consider the action of a fluid in
motion upon a body at rest.

A vessel 5 feet high was filled with a fluid, which could be
discharged by a stop-cock, in a direction parallel to the horizon.
The cock being opened, the curve which the stream described
was marked out upon a plane set perpendicular to the horizon ;
and, by examining this curve, it was found to be a very accu-
rate parabola, the abscissa of which was 13,85 in. and the ordi-
nate was 50 in. hence, the latus rectum was 180,5 in. one- fourth
of which is 45,1 in. which is the space through which a body
must fall to acquire the velocity of projection ; hence, that ve-
locity was 189,6 in. in a second. And here, by the by, we
may take notice of a remarkable circumstance. The depth of
the cock below the surface of the fluid was 45,1 in. hence, the
velocity of projection was that which a body acquires in falling
through a space equal to the whole depth of the fluid ; whereas,

8

Mr. Vince's Experiments on the Resistance

through a simple orifice, the velocity would have been that
which is acquired in falling through half the depth ; the pipe
of the stop-cock therefore increased the velocity of the fluid in
the ratio of 1 : </¥, and gave it the greatest velocity possible ;
the length of the pipe was 3 in. and the area of the section
0,045 in. ; also, the base of the vessel was a square, the side of
which was 12 inches.

The area of the section of the pipe may be found very accu-
rately, in the following manner. The vessel being kept con-
stantly full, receive the quantity of fluid run out in any time t",
and then weigh it, by which we shall be able to get the quan-
tity in cubic inches. Now if v = the velocity of the fluid when
it issues from the pipe, a = the area of the section of the pipe,
l = the length of the cylinder of water run out, whose base
= a , and m = the quantity of fluid discharged in t" ; then
v ; l : : 1" : t", hence, l — vt; but a l = m; therefore avtz=m;
hence, a = In the present instance, t = 20, m = 170,63

cubic inches, v = 189,6 ; hence, a = 0,045.

Let AB C D (fig. 1. Tab. I.) be a solid piece of wood, upon
which are fixed two upright pieces, rs, tu ; between these, a
flat lever eac is suspended, in a perpendicular position, on the
axis x y, and nicely balanced ; and let a be a point directly
against the middle of the axis, in a line perpendicular to the
plane of the lever. This apparatus is placed against the stop-
cock, at the distance of about 1 inch, and, when the water is let
go, let us suppose the centre of the stream to strike the lever
perpendicularly at e ; take a c = a e, and, on the opposite side to
that at which the stream acts, fasten a fine silk string at c, and
bring it over a pulley p , and adjust it in a direction perpendi-
cular to the plane of the lever, and, at the end which hangs

9

of Bodies moving in Fluids.

down, fix a scale Q, the weight of which is to be previously
determined. All the apparatus being thus adjusted, open the
stop-cock, and let the fluid strike the lever, and put such weight
into the scale as will just keep the lever in its perpendicular
situation, and that weight, with the weight of the scale, must
be just equivalent to the action of the fluid. Thus we get the
perpendicular effect of the water. Now incline the plane of the
lever, at any angle, to the direction of the stream, and adjust
the string perpendicular to the plane, as before ; then put such
a weight into the scale as will keep the lever perpendicular to
the horizon, whilst the fluid acts upon it, and you get that part
of the effect of the fluid which acts perpendicular to the plane.
In this manner, when the fluid acts oblique to the plane, we
get the perpendicular part of the force. The second column of
the following table shows this effect, by experiment, for every
i oth degree of inclination shown in the first column ; and the
third column shows the effect, by theory, from the perpendicular
force, supposing it to vary as the sine of inclination.

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MDCCXCVIII.

C

io Mr. Vince’s Experiments on the Resistance

It appears from hence, that the resistance varies as the sine
of the angle at which the fluid strikes the plane ; the difference
between the theory and experiment being only such as may be
supposed to arise from the want of accuracy to which the ex-
periments must necessarily be subject.

Let us now first consider, what the whole perpendicular
resistance by experiment is, when compared with that by
theory. Now, by theory, the resistance is equal to the weight
of a column of the fluid, whose base = 0,045 in. and alti-
tude = 45,1 in. and the weight of that column is = 1 oz.
1 dwt. logrs. Hence, the resistance by theory : the resistance
, . 2 '$$:■/ 3 by experiment : : 1 oz. x dwt logrs. : 1 oz. 17 dwts. lagrs.

: : 514 : 900.

In the next place, let us examine what is this resistance, com-
pared with the resistance of a plane moving in a fluid. We here
prove, that the resistance of the fluid in motion acting on the
plane at rest : the resistance by theory : : 900 : 514; and we
have before proved, that the resistance by theory : the resist-
ance of a plane body moving in a fluid : : 1598 : 2321 ; hence,
the resistance of a fluid in motion upon a plane at rest : the re-
sistance of the same plane, moving with the same velocity, in a
fluid at rest : : goo x 1598 : 5H x 2321 : : 1438200 : 1192954
: : 6:5 nearly. Now we know that the actual effect on the
plane must be the same in both cases; and the difference, I
conceive, can arise only from the action of the fluid behind
the body, in the latter case, there being no effect of this kind
in the former case. For, in respect to the pressure before the
body, that will probably be the same in both cases ; for there
is a pressure of the column of the spouting fluid, acting against

11

of Bodies moving in Fluids.

the particles which strike the body at rest, similar to the action
of the fluid before the body, upon the particles which strike
the body moving in the fluid. Hence, the resistance of the
planes moving in the fluid, with the velocity here given, is
diminished about one fifth part of the whole, by the pressure
behind the body ; but, with different velocities, this diminution
must increase as the velocity increases.

The effect of that part of the force which acts perpendicular
to the plane being thus established, we proceed next to exa-
mine, what part of the whole force which acts parallel to the
plane, is effective. To determine which, the axis w v (fig. 2.)
was fixed perpendicular to the plane of the lever abed, and the
ends of the axis were conical, and laid in conical holes; and the
thread from which the scale was hung was fixed to the edge
at e, and acted perpendicular to it and the weight drew the
lever in the direction e s, contrary to that in which the fluid
tends to move the lever, and it acted at the same perpendicular
distance from the axis below, as the fluid acted above it. Let
x?nz be a line parallel to the horizon, when the lever is per-
pendicular to it, and which passes through the centre of the
stream ; and let xmz be also the direction of that part of the
force which acts parallel to the plane. This apparatus being
adjusted, the experiments were made for every tenth degree of
inclination ; and here a circumstance took place, for which I
can give no satisfactory reason. Having gone through the ex-
periments once, and noted the results, I repeated them ; and,
to my great surprise, I found all the second results to be very
different from the first. The experiments were therefore re-
peated again, and the results were still different. Being certain
that the experiments were very accurately made each time, I

C 2

12

rot// 3

Mr. Vince's Experiments on the Resistance

was totally at a loss to conjecture to what circumstance this
difference of results was owing. By repeating however the
experiments, and observing at what point of the line xmz the
centre of the stream acted, I discovered that the effect varied
by varying that point ; that it was greatest when the stream
struck the lever as near as it could to x ; less when it struck it
at the middle m ; and least when it struck it as near as it could
to z, notwithstanding the stream acted at the same perpendi-
cular distance from the axis in each case, and the parallel part
of the force always acted in the line x m z. At the angles 8o°,
70°, 6o°, the fluid striking as near as it could to the edge
gave the lever a motion, not in the direction x m z, but in the
opposite direction z m x, as appeared by taking away the scale.

I have therefore marked such results with the sign , the

motion produced being then in a direction opposite to that
which ought to have been produced, by that part of the force
of the stream which acts parallel to the plane of the lever. The
forces which are here put down, are those which take effect in
a direction parallel to the plane of the lever, for every tenth
degree of inclination ; the perpendicular force being 1 oz.
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3

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3

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At

3°°

inch <

Middle m

7

2

'&SS

4

v Edge x

12

15

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y

4

r Edge %

4

16

At

20°

incl. <

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0

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l Edge x

11

12

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It is a remarkable circumstance, that the effect of the fluid
at z increased regularly as the angle decreased ; for, though I
did not measure the negative effects, I could plainly perceive
that that was the case ; whereas, the effects at m and x in-
creased to about the middle of the quadrant, and then de-
creased. At io°, the obliquity was such, that the section of
the stream extended very nearly from one side of the lever to
the other.

As it appears by experiment, that the velocity of the fluid
flowing out of the vessel was equal to the velocity which a body
acquires in falling down the altitude of the fluid above the ori-
fice, the square of the velocity must be in proportion to that
altitude. To find therefore, in this case, whether the resistance
varied as the square of the velocity, I let the water flow per-

14 Mr. Vince's Experiments , Be c.

pendicularly against the plane (fig. 1.) at different depths,
and I always found the resistances to be in proportion to
the depths, and therefore in proportion to the square of the
velocity, agreeing with what takes place when the body moves
in the fluid.

£ 15 3

II. Experiments and Observations , tending to show the Co?npo -
sit mi and Properties of Urinary Concretions. By George
Pearson, M. D. F. R. S.

Read December 14, 17 97.

I. Historical Observations.

Urinary concretions have obtained their denominations, like
most other things, from their obvious properties. According-
ly, in our language, they are popularly known by the names
Stone and Gravel, or Sand, from their resemblance to the states
of earth so named : and we find names of the same import in
other languages, such as a Aog, (Aret/eus;) a iQtauns, (Celius
Aurelianus;) tyotfipos, (Aret^eus;) a iQiXia, (various au-
thors;) Calculus, (Celsus and Pliny;) Sabulum , (various
authors.) In other languages, and especially in those now
spoken, it is unnecessary to notice names which have the same
meaning.

The notion very generally entertained, of the nature of uri-
nary concretions, consisted with the terms, till the last twenty
years ; although the experiments of Slare, Frederic Hoff-
man, and Hales, long before showed that these substances
commonly consist of animal matter. Galen indeed imagined
that (pxeypot, or viscid animal matter, is the basis of animal
concretions; but, in his days, earth was believed to be the
basis of animal matter. Alkaline medicines were, however,
employed by the Greek physicians, in diseases from calculi.

16 Dr. Pearson’s Experiments and Observations

The experiments of the alchemists also made it appear, that
earth was only a part of the matter of concretions. It was
probably the observation of the deposition and crystallization
of saline bodies, which suggested the notion of urinary calculi,
being of the nature of tartar. Such was the opinion of Basil
Valentine, and after him of Hochener, better known by the
name of Paracelsus ; but, whether the latter adopted the de-
nomination Duelecb from its import, or from caprice, has not
been explained. Van Helmont, a century after his proto-
type Paracelsus, being struck with the experiment in which
he discovered the concretion of salts in distilled urine by al-
cohol, was led to depart from his adored master s opinion,
with respect to the nature of calculi ; although he acknow-
ledges the merit of Paracelsus, in having discovered the sol-
vent Ludus, (a calcareous stone also called Septarium,) which
Van Helmont says is preferable to alkaline lixivium. He also
says, that when the archeus spirit of urine meets with a vola-
tile earthy spirit, and does not act in a due manner, a con-
cretion will be formed ; but, in a healthy state, although all
urine contains the matter of urinary calculi, no concretion can
take place, because the archeus, or vital power of the bladder,
counteracts its formation.

As to the kind of earth composing calculi, the only distinc-
tion of earths, till about the last half century, was into absor-
bent and non-absorbent ; but, since the absorbent earths were
distinguished into calcareous, magnesia, and alumine or clay,
the calcareous was considered to be the earth of urinary con-
cretions ; apparently however for no other reason but its ob-
vious properties, and its extensive diffusion through the whole
animal kingdom.

on the Composition of Urinary Concretions . 17

At length, viz. in 177 6, the experiments of the wonderful
Scheele were published in Sweden, but were scarcely known
in this country till 1785. These experiments exploded the
opinion of the earthy nature of calculi, and substituted that of
their consisting of a peculiar acid, resembling the succinic,
and of a gelatinous matter, without any earth. Afterwards
about of their weight of lime was found by Bergman ;
which, for a cause now well known, had eluded the acuteness
of Scheele. Although the experiments of Scheele were con-
fessedly unquestionable, and were ably supported by the learned
Bergman, some very eminent chemists, having obtained dif-
ferent results by their own experiments, adopted a different
opinion of the composition of these concretions. The im-
mortal, and ever to be deplored, Lavoisier supposed these sub-
stances to consist of acidulous phosphate of lime and animal
matter, many of them being partially fusible ; but still it was
the unrivalled Scheele who discovered, that the urine of healthy
persons contains superphosphate, or acidulous phosphate, of
lime ; and who also indicated the experiment which verified
his opinion, that phosphate of lime is the basis of bone.

Experiments have been likewise made, for the most part in a
rather desultory way, and most of them by persons but little
practiced in chemical inquiries, which at least afford evidence,
that urinary concretions are very different, with respect to the
proportion of the ingredients in their composition, and per-
haps also in kind. M. Fourcroy, who however must not
be classed with inexperienced chemists, I believe first obtained
prussic acid by fire, and by nitric acid, from these concretions ;
and showed that they sometimes contain phosphate of ammo-
niac and of soda; which may be dissolved out of them by

MDCCXCVIII. D

18 Dr. Pearson's Experiments and Observations

water. M. Fourcroy also says, he found magnesia in the
intestinal calculus of a horse ; which calculus was a triple com-
bination, of one part of phosphate of ammoniac, two parts of
magnesia, and one of water, besides traces of animal and
vegetable matter.

Dr. Link, in a very elaborate dissertation, published at Got- '
tingen, in 1788, on urine and calculi, concludes that urinary
concretions consist of phosphoric acid, lime, ammoniac, oil, the
bases of different kinds of gazes, together with the acid sublimate
of Scheele, although he did not succeed in obtaining it.

It is a proof of Dr. Black's sagacity, that he should have
been able to perceive, from a few experiments, that urinary
concretions consisted of animal matter and the earth of bone,
before the composition of this earth was demonstrated by Gahn.

In this historical sketch it should be noticed, that alkaline
substances, though used by the Greek physicians, and after-
wards by the alchemical physicians, appear to have been laid
aside by the regular practitioners, for a century or two prece-
ding their revival, by the famous Mrs. Stephens, in 1720.
Her prescription brought into vogue the theory of these me-
dicines operating by their causticity. The successful use, by
Mr. Colborne, of potash saturated with carbonic acid, ac-
cording to the discovery of Bewley and Bergman, and the
still further improvement in practice, from the use of soda, as
well as potash, super-saturated with carbonic acid, by the dis-
covery of a peculiar method by Mr. Schweppe, have com-
pletely refuted the theory of the agency of alkalies on the
principle of causticity.

It appears, from the preceding brief history, as well as from
the confession of the latest and best writers, that the experi-

on the Composition of Urinary Concretions. ig

ments hitherto made, rather “afford indications of what re-
“ mains to be done, than furnish demonstrations of the nature
“ of animal concretions.” It is also too obvious to need expla-
nation, that more efficacious and innocent practice, in diseases
from these concretions, can only be discovered by a further
investigation of their properties. It is with this view, as well
as for the sake of chemical philosophy, that I think it my duty
to submit to the Society some of the observations I have
made, in the course of inquiry on this subject.

file observations which I shall now offer, are principally
on a substance, which my experiments inform me is very ge-
nerally a constituent of both urinary and arthritic concretions.
It is a substance obtained by dissolving it out of these concre-
tions, by lye of caustic fixed alkali, and precipitating it from the
solution by acids. In this way, Scheele separated this matter ;
but he did not consider its importance, nor of course at all inves-
tigate its properties. He does not even seem to have been aware
that it was a distinct constituent part of the urinary concretion :
foi, when he 1 elates the experiment of precipitating matter
from the nitric solution of calculus by metallic salts, no dis-
tinction is made between the precipitations in this experiment,
and that in the former; yet we can now show, that in the one
case the precipitate is a peculiar animal oxide, and in the
other they are metallic phosphates. As Scheele obtained an
acid sublimate, it has been imagined by some writers, that the
precipitate by any acid (even by the carbonic) from the alkaline
menstruum, was an acid ; the same as that obtained by subli-
mation, and which, in the new system of chemistry, has been
denominated lithic acid. The following experiments show that
these substances are different species of matter.

D 2

2 TO -

C\

f , • a a' <

Ox

O'fJJ $

20 Dr. Pearson’s Experiments and Observations.

II. Experiments.

250 grains of a white, smooth, laminated, urinary calculus,
and the same quantity of a nut-brown one, with an uneven
surface, both of which were of a roundish figure, were pul-
verized together.* 300 grains of these pulverized calculi were
triturated with three ounces and a half, by measure, or five
ounces, by weight, of lye of caustic soda. The mixture be-
came thick, and copiously emitted ammoniacal gaz. After
digestion for a night, and then boiling, with the addition of
five ounces of pure water, I obtained, by filtration, five ounces
of clear colourless liquid. Boiling water was repeatedly poured
upon the strainer, till what passed through it was almost taste-
less, and remained clear, on the addition of diluted sulphuric acid.

( a ) The matter remaining on the strainer, being dried, was
an impalpable, white, tasteless, heavy powder, which weighed
96 grains.

( b ) The five ounces of filtrated liquid, having been set apart,
on standing, deposited a white, opaque, granulated, soap-like
matter, from a colourless clear liquid. The liquid being de-
canted, the deposit was dried, and was then an opaque, brittle,
soap-like matter, which dissolved readily in water, giving a
clear but not viscid solution, and tasting weakly of soda. This
soap-like matter weighed 280 grains.

(c) The decanted liquor, ( b ,) being mixed with the above
filtrated liquors, on evaporation to three ounces, afforded no
deposit on standing, although it was a very heavy and soapy

* The object of these experiments being principally to investigate the properties of
one of the constituent parts of urinary concretions, which part was previously deter-
mined (by the test of nitric acid,) to exist in both these, it can be no objection to the
experiments, that I made use of a mixture of two calculi.

f

3 -3 £"3 3

on the Composition of Urinary Concretions. 2 1

liquid to the feel ; but, on adding diluted sulphuric acid gra-
dually, till it ceased to become turbid, a sediment was deposited,
which was a very light, white, impalpable powder, in weight,
when dried, 2 6 grains. The liquid from which this powder
was precipitated, being evaporated, afforded nothing but sul-
phate of soda, and a few grains of crystals, which seemed to be
phosphate of soda. There was also a blackish matter, which
burnt like horn, or other animal matter, and did not leave a
pink or rose-coloured matter, on evaporating the solution of it
in nitric acid to dryness, but left a carbonaceous residue;
whereas, the white precipitate, so treated, afforded a beautiful
pink matter.

( d ) 250 grains of the soap-like matter ( b ) being dissolved
in eight ounces of pure water ;

1. A little of this solution, further diluted by one ounce of
water, grew milky on adding a few drops of nitric acid, but
became less so on standing. On adding more nitric acid, and
heating it, the mixture became quite clear : by adding a few
drops of lye of caustic soda, a very slight curdy appearance
took place.

2. On adding, to the same diluted solution, a little of the
diluted sulphuric or muriatic acid, milkiness ensued, and re-
mained, although the acids were added till the mixture was
extremely sour. O11 adding lye of caustic soda, much more
than to saturate the superabundant acid, the mixture became
clear again; and, on adding the acids a second time, the milki-
ness was reproduced. It was found that the milkiness could
be produced and destroyed, or clearness be produced, by the
alternate addition of the acid and alkali, for an unlimited num-
ber of times. If the nitric acid however was used, at length

'23/ /

'02 7 3^ ’OtS 4

'0?t

22 Dr. Pearson's Experiments and Observations

no milkiness could be induced. If carbonate of soda was added,
in place of the caustic soda, the mixture could not be made clear.

3. Lime water was rendered turbid by this solution, but I
neglected to examine the precipitated matter.

4- A little of the solution, with the addition of a few drops of
concentrated nitric acid, being evaporated to dryness, sometimes
a pink, and at other times a blood-red, or rose-coloured matter
was left ; which, by further application of fire, became black.

3. Carbonic acid, digested and shook with this solution, did
not render it turbid.

6. To the whole of the remaining solution was added diluted
sulphuric acid, to saturate the alkali. On standing, a copious
precipitate took place, from a clear liquid ; which precipitate,
being washed and dried, was a mass of very light, mica-like,
whitish crystals, amounting to 123 grains. It was estimated
that the solution used in the Experiments 1. — 5. would have
produced 12 grains, and that the 30 grains of soap-like matter,
(6,) not decompounded, would have yielded about 14 grains more.

( e ) The precipitate, ( d , 6.)

1. Had no taste, nor smell, and did not dissolve in the mouth.

2. About one part of it only dissolved in 800 parts of boil-
ing water; which solution did not redden paper stained with
turnsole, nor the solution and tincture of this test; neither did it
change turnsole paper, reddened by acid, to a blue colour. On
cooling, the greatest part of what had been dissolved was de-
posited, in a crystallized state, equally on the sides and bottom
of the vessel. This crystallized matter had the properties
abovemen tioned ( d .). Boiling water was found to dissolve a
much greater proportion of urinary stone, and also of gravel ,
than of this precipitate.

on the Composition of Urinary Concretions. 23

3 • Lye of mild potash, or subcarbonate of potash, being
dropped into the solution ( e , 2.) with its crystallized deposit,
the crystals at first seemed to dissolve; but, on standing, a
great part of the matter was deposited, and the liquid remained
turbid.

4. The precipitate being boiled with lye of carbonate of
soda, more seemed to be dissolved than in pure water; but
the solution was not clear, and, on evaporating it nearly to
dryness, and pouring cold water upon it, on a paper strainer,
scarcely any thing but the soda passed through with the water ;
the precipitate remaining behind on the paper. The result
was the same, when this experiment was made with a lye of
carbonate of ammoniac. The result was also the same, with
water in which red oxide of mercury had been boiled ; which
was also boiled with this precipitate, and filtrated after cooling.

5. A little of the precipitate being triturated with quick-
lime, hot water was poured upon it. The filtrated liquor gave
the precipitate back again, on adding muriatic acid.

6. The precipitate exposed* to flame, with the blowpipe,
turned black, emitted the smell of burning animal matter, and
evaporated or burnt away without any signs of fusion ; stain-
ing the platina spoon black.

7. Five grains of the precipitate, in half an ounce of water,
were left to stand in a warm room, during the months of Au-
gust and September last, without any signs of putrefaction ap-
pearing, or any obvious change taking place.

8. Twenty-four ounces of boiling water were saturated
with the precipitate, and divided into six portions ; from each
of which, on cooling, most of it again precipitated.

The first portion, on boiling with a little lye of carbonate of

* o // ,3

2 4 Dr. Pearson’s Experiments and Observations

soda, (the pneumatic apparatus being affixed,) discharged no
carbonic acid into lime water; but a transparent solution was
produced, and, on cooling, very little was precipitated.

The second portion was, in the same manner, boiled in a little
lye of caustic soda ; which gave a transparent solution on cool-
ing, without any precipitation.

The third portion being boiled with lime water, very little
more seemed to be dissolved than in pure water.

The fourth portion being boiled with 4 grains of subphos-
phate of lime, or calcined bone, no more seemed to be dis-
solved on account of this addition.

Nor was more dissolved in the fifth portion, by the addition
of 4 grains of phosphate of lime, made by dropping phosphoric
acid into lime water.

And the result was the same with the sixth portion, to which
were added 4 grains of superphosphate of lime, made by add-
ing phosphoric acid to lime water, so as just to make a clear
solution, and then evaporating the solution.

9. Urine seemed to dissolve, or at least to suspend, a greater
quantity of the precipitate than mere water ; so likewise did wa-
ter with a little sulphate of soda.

10. The precipitate did not render solution of hard soap
at all curdy; but, on adding the precipitate to solution of,
sulphuret of potash, it became very turbid.

11. The precipitate produced a strong effervescence, even
in the cold, with nitric acid, but the fumes were not those of
nitrous acid : there was a clear solution, which, on evaporation
to dryness, afforded black matter, surrounded by a pink, or
blood-red margin.

12. The substance, with sulphuric acid, turned black, and

on the Composition of Urinary Concretions, 25

emitted fumes copiously, which were scarcely those of sulphu-
reous acid ; and, on evaporation, a black mark only was left.

13. I first digested, and then boiled, in water, the preci-
pitate with prussiate of iron ; but the filtrated liquor afforded
no precipitation with sulphate of iron.

14. Two drachms, by measure, of nitric acid, of the spe-
cific gravity of 1,35, were poured upon 7 grains of the precipi-
tate. A violent effervescence took place, which was soon suc-
ceeded by a complete solution.

A few drops of this solution, being evaporated on glass,
left a black mark, surrounded by a pink margin. A few drops
of nitric acid being evaporated from this residue, nothing but
a still less black mark, and a few red spots remained.

Nitric acid being added a third time, nothing but a black
mark, still smaller, remained ; which entirely disappeared, on
evaporating this acid from it a fourth time.

I found that a few drops of this solution, so diluted that they
did not contain the or even a much smaller part, of a grain
of the precipitate, on evaporation, left a pink stain on glass.

The whole of the rest of the solution was distilled in a
very low temperature, so that a drop only fell about every half-
minute, till a thick brownish sediment remained, with a red
margin. A similar distillation was performed, with the distilled
liquor, a second time, when there remained a little whitish thick
matter. On a third distillation, as before, with the distilled
liquor, towards the close white fumes arose, and about half a
drachm of liquid, which now remained in the retort, being left
to stand, prismatical crystals, decussating each other, were
- formed. They had a sharp taste, but were scarcely sour;

MDCCXCVIII. E

‘?7A 2

'olG

‘ooo on /

26 Dr. Pearson’s Experiments and Observations

were very soluble in the mouth, and evaporated in white
fumes, leaving a very slight black stain,

15. Twenty grains of the precipitate were introduced into
a tube, | of an inch wide in the bore, sealed by melting at one
extremity ; which extremity was coated, and the tube was fitly
bent for retaining sublimate, and collecting gaz. The tempe-
rature, from the fire applied, was at first very low, but was
gradually increased, so as to make the coated part, containing
the charge, red hot. At first, the precipitate turned black, and
a little water appeared. Secondly, gaz came over, which had
the smell of empyreumatic liquor cornu cervi. Thirdly, a
brown sublimate appeared, and gaz as before, but also with
prussic acid gaz. Fourthly, black matter, staining the tube, as
if from tar, or animal oil. On cooling, there was found a resi-
due, of nearly three grains, of pure carbon. The sublimate
was principally carbonate of ammoniac ; the rest was animal
oil. The gaz discharged was nearly half its bulk, or 5 cubic
inches by measure, carbonic acid ; and the remaining 5 cubic
inches were nitrogen gaz, containing prussic acid and empy-
reumatic oil.

I treated in the same manner, the same quantity of reddish
crystals, deposited spontaneously from urine. The result was
not very different from that of the former experiment. The
gaz was more offensive, smelling like putrid urine, and the
carbonaceous residue was more copious, and contained lime
and phosphoric acid ; at least the lixivium of it became white,
on dropping into it oxalic acid ; and it became slightly curdy,
on adding lime water.

I treated in the same manner, some quite round and smooth

on the Composition of Urinary Concretions. 27

concretions, of the size of black pepper seeds. The products
were the same as the former, but the gaz was still more offen-
sive, and in smaller quantity; and the carbonaceous matter
was more copious.

I, in the same way, subjected to experiment 20 grains of a
nut-brown light calculus, which I had previously ascertained
to contain the matter above described, which was precipitated
from caustic soda by acids. The products were of the same
kind as the former; but I could find no trace of phosphoric acid
in the residue, which I did of lime, and the gaz was less offen-
sive. The carbonaceous residue was not, in weight, 3 grains.

It will be proper, before I proceed further, to point out
some of the more obvious conclusions from the above experi-
ments.

1. It appears that at least one half of the matter of the
urinary concretions subjected to the above experiments united
to caustic soda, and was precipitated from it by acids. (II.
a — d.)

2. This precipitate does not indicate acidity to the most de-
licate tests; ( e , 2.) and, as it is inodorous, tasteless, ( e , 1.)
scarcely soluble in cold water, ( e , 2.) does not unite to the
alkali of carbonate of potash, of soda, or of ammoniac, (<*,3,
4.) nor to oxide of mercury, (*,4.) nor to the lime of lime wa-
ter, ( e , 8.) nor decompound soap, ( e , 10.) or prussiate of iron,

( e , 13.) and, as its combination with caustic soda resembles
soap, more than any double salt known to consist of an
acid and alkali, this precipitate does not belong to the genus
acids.

3. As this precipitate could not be sublimed, without being
decompounded, like animal matter, (<?, 15.) and also for the

• E 2

'0A3 S

• 00 h

28 Dr. Pearson’s Experiments and Observations

reasons mentioned in the last paragraph, it cannot be the
same thing as the acid sublimate of Scheele, or the succinic
acid. t

4. As it does not appear to be putrescible, nor form a viscid
solution with water, it cannot be referred to the animal muci-
lages.

5. On account of its manner of burning in the air, under
the blowpipe, ( e , 6.) and its yielding, on exposure to fire in
close vessels, the distinguishing products of animal matter,
(especially ammoniac and prussic acid,) as well as on account
of its affording a soap-like matter with caustic soda, this preci-
pitate may be considered as a species of animal matter ; and,
from its composition being analogous to that of the substances
called, in the new system of chemistry, animal oxides , it belongs
to that genus. Its peculiar and specific distinguishing properties
are, imputrescibility, facility of crystallization, insolubility in cold
water , and, that most remarkable property of all others, pro-
ducing a pink or red matter, on evaporation of its solution in
nitric acid. *

I do not avail myself of various other conclusions in this
place, because they relate especially to the agency of medicines
for preventing and removing concretions ; and of course do not
properly fall within the views of the Royal Society.

Having found the above precipitate to be an oxide, and not,
as is commonly supposed, an acid, I thought it probable that,

* It is much to be wished that we possessed equally delicate tests ©f the other species
of animal matter, which are confounded together, although, from their obvious pro-
perties, there is reason to believe they are of very different kinds, as is the case with
the matter of the brain, liver, voluntary muscles, mucus, &c. Mr. Hun per has dis-
covered a distinguishing specific property of pus, and one is here indicated for the
oxide of urinary concretions.

on the Composition of Urinary Concretions. &g

like other analogous oxides, it was acidifiable, and I suspected
that I had really rendered it into the acid state, by the nitric
acid; which, in the above experiments, ( e , 14.) had imparted
oxygen to it, and thereby rendered it soluble, deliquescent,
pungent, and volatile. This change also would account for
the nitric solution not affording the precipitate.

In order to obtain, for examination, an adequate quantity of
this supposed acid, the following experiments were instituted,
with the three acids (viz. the oxymuriatic, the nitro-muriatic,
and the nitric,) which can acidify oxides analogous to the
present one.

Experiment 1. Twenty-five grains of the above animal oxide,
(for so I will now venture to call it,) and three ounces of nitric
acid, of the specific gravity of 1,25, were put into a retort, and
the hydro-pneumatic apparatus was adjoined.

At a very low temperature, a clear solution was made. First,
soon after the solution began to boil, 23 ounces, by measure,
of colourless gaz came over, which were succeeded (secondly,)
by white fumes, filling the apparatus, and 23 ounces more of
gaz. Thirdly, a white sublimate ascended, and there was a
strong smell of prussic acid. The sublimate was very readily
washed out, being very soluble, and tasted pungent or sharp,
but not sour. Fourthly, the distillation being renewed, more
white sublimate appeared, but only 3 ounces more of gaz came
over; and then the retort only contained a dark-brown solid
matter.

The first portion of gaz, viz. 23 ounces, consisted of about
equal bulks of carbonic acid and atmospherical air. The se-
cond portion, viz. 23 ounces, was two-thirds of its bulk
carbonic acid, and the rest nitrogen gaz. The third portion,

go Dr. Pearson's Experiments and Observations

or 3 ounces, was atmospherical air, with a little carbonic
acid.

Nitric acid was poured, in the same quantity as before, into
the retort. An effervescence immediately took place, which was
succeeded by a transparent solution. The distillation yielded
gaz of the same kind as before, but in smaller quantity, with
white fumes, and white sublimate. When only about 4, drachms,
by measure, of liquid remained in the retort, a little of it was
evaporated; and, when reduced to a solid matter, it turned
black, and took fire, leaving a carbonaceous residue ; but, be-
fore this, a margin of beautiful pink matter appeared.

Nitric acid was poured, as before, into the retort, for the third
time, but very little gaz ascended, and much less white fumes
than before. The distillation proceeded, till about one drachm-
measure of liquid remained in the retort: this being left to
stand, prismatic crystals were formed in a very small quantity
of liquid. These crystals did not taste sour, but sharp, and they
reddened turnsole-paper. Adding a little soda to a part of them,
to see whether I could form a neutral salt, I was surprised by
the extrication of ammoniac. To another portion of crystals I
added sulphuric acid, which disengaged nitric acid. A third
portion of crystals, being exposed over a lamp, wholly evapo-
rated, without leaving a mark behind. The remaining matter
in the retort being examined, was found to be nitrate of am-
moniac. It was plain that the nitric acid had, by parting with
oxygen to the carbon of the oxide, formed carbonic acid. The
carbon being thus carried off, of course the nitrogen and hy-
drogen of the oxide uniting produce ammoniac; which, uniting
with the redundant nitric acid, composes nitrate of ammoniac ;
but great part of the nitrate of ammoniac was carried off in the

on the Composition of Urinary Concretions. 31

vapour state, exhibiting white fumes, and sublimate, as above
observed.

The mode of making the experiments with the other acids
was of course different from the former experiment.

Experiment 11. Twenty -five grains of the above animal oxide,
and half an ounce of water, were put into a bottle capable of
containing three pints ; a stream of oxymuriatic acid gaz, from
manganese and muriatic acid, was made to pass into the bottle,
and upon the charge, for twelve hours ; and, for twenty-four
hours more, oxymuriatic gaz kept issuing, but in smaller quan-
tity, and circulating through the bottle. The oxide, by this time,
was completely dissolved. Upon adding lime to a little of the
solution of it, ammoniac was disengaged ; and, upon adding
sulphuric acid, there was a disengagement of oxymuriatic acid.
On evaporation, however, I obtained nothing but muriate of
ammoniac, with which was mixed a little manganese.

In this experiment, I could not doubt that the carbon had
been carried off, in the state of carbonic acid, by the oxygen of
the oxymuriatic acid; and thus ammoniac was compounded,
from the union of the two remaining constituent parts of the
oxide, viz. the nitrogen and hydrogen. The oxymuriatic acid,
united to the ammoniac, parted with oxygen, and became mu-
riatic acid during evaporation; hence, muriate of ammoniac was
formed.

Experiment 111. The above experiment was repeated, only
the gaz was nitro-muriatic gaz, from a mixture of nitric and
muriatic acids. The result was the same as in the last experi-
ment, except that the product was a mixture of nitrate, and
muriate, of ammoniac.

'0,$7 /

Dr. Pearson’s Experiments and Observations

I made other experiments of the same kind, but their results
were so nearly the same as those above related, that I shall not
give an account of them. By the unexpected issue of these ex-
periments, all my hopes of acidifying the animal oxide were
exploded; but lam indebted to that pursuit, for the curious
discovery of the change of the most common basis of urinary
concretions, (the animal oxide,) into ammoniac and carbonic
acid, by the oxygen of the above acids ; which will be found
extremely important, as it enables us to interpret many phas-
nomena, in a variety of cases beside the present. It now ap-
pears, that the inflammation mentioned in one of the above
experiments, (and which also happened in several others,) on
evaporation of the nitric solution of the animal oxide, was from
the nitrate of ammoniac, the nitrum fiammans of the old che-
mists, compounded in those experiments. This inflammation
takes place sometimes, on evaporation of nitric solutions, both
of urinary concretions, and of urine itself evaporated to the state
of soft extract, on account of the ammoniac already existing in
these substances. The composition of ammoniac also explains
the disappearance of the whole matter of some sorts of urinary
concretions, a very small residue of black matter excepted, by
repeated affusion and evaporation of nitric acid, from the solu-
tion of them in this menstruum.

It remains for me to give an account of the 96 grains of
powdery matter left on the paper strainer, ( a ;) which are
the insoluble portion, in lye of caustic soda, of 300 grains of
urinary concretions.

1 . A small portion of the insoluble matter, being exposed to
flame with the blowpipe, did not turn black, nor yield any

on the Composition of Urinary Concretions. 33

smell of animal matter; but it became whiter, and I could just
agglutinate the powder into one mass, although I was unable
to render it fluid.

2. The filtrated liquid, from a little of the matter boiled in
water, became very turbid and white with oxalic acid : with
lime water it grew barely curdy; and it did not alter the colour
of turnsole, or of violet juice.

3. The matter dissolved completely in muriatic acid, and
also in nitric acid, without effervescence.

This nitric solution, having been evaporated, to carry off
most of the free acid, instantly became very curdy on the addi-
tion of lime water.

It grew thick and white on adding sulphuric acid, yielding
a copious precipitate of sulphate of lime. One portion of the
supernatant liquor upon this precipitate, on evaporation, afforded
an extract-like matter; which readily melted, as phosphoric
acid does when it is mixed with a little earthy matter. To the
other portion of this supernatant liquor was added liquid caus-
tic ammoniac, producing a precipitate which afforded no sul-
phate of magnesia with sulphuric acid.

From these experiments it appears, that the above 96 grains
of insoluble matter consisted of phosphate of lime. Accord-
ingly, the 300 grains of urinary concretions examined, appear
to contain,

grains.

Peculiar animal oxide - - - ~ 17 5

Phosphate of lime - - - 96

Ammoniac, (and most probably phosphoric acid united to

the ammoniac,) water, and common mucilage of urine,

which were not collected and weighed, by estimation 2 9

MDCCXCVIII.

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300

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34 ft: Pearson's Experiments and Observations

I shall next relate some experiments, made in order to ob-
tain the acid sublimate of Scheele, or lithic acid of the new
system of chemistry.

• v ^ 100 grains of an urinary concretion, which had been pre-

viously found to contain principally the above animal oxide,
were introduced into a tube \ of an inch wide ; which was
sealed at one end by fusion, and which also was fitly bent for
collecting sublimate, and obtaining gaz. The sealed end was
coated and exposed to fire, first to a low temperature, and gra-
dually to a very elevated one.

1 . Gaz was discharged, which had the smell of burning bone.

2. Water appeared boiling immediately over the charge,
which seemed to be burning, and was turned black.

3. Gaz was discharged, of the smell of empyreumatic liquor
cornu cervi , and about half a drachm of this liquor was in the
upper part of the tube.

4. A brown sublimate of carbonate of ammoniac appeared
in the cold part of the tube; but in the hotter part, near the
charge, was tar-like matter, and the gaz discharged had a very
offensive smell of empyreumatic animal oil, with which was
mixed that of prussic acid.

The coated part of the tube was kept red hot, for some time
after gaz ceased to come over.

The quantity of gaz amounted to 24 ounces, by measure:
it consisted of nearly 16 ounces of carbonic acid gaz, and the
rest was air, with a larger proportion of nitrogen gaz than is
contained in atmospheric air.

5. There was a residue of 30 grains, almost pure carbon;

* q ‘i y and 10 grains of heavy black and brown matter, a little above

the coated part of the tube. In this last mentioned matter

on the Composition of Urinary Concretions. 35

were many small white spicula . At about half an inch above
the carbonaceous residue, dark gray matter had been raised,
which weighed 15 grains.

This sublimed gray matter did not contain any ammoniac,
nor throw down any prussiate of iron, with sulphate of iron.
It reddened turnsole paper and tincture. It dissolved in caustic
soda ; from which solution muriatic acid precipitated nothing ;
for, although on dropping it into the solution milkiness ap-
peared, the liquid soon grew clear again.

Ten grains of this sublimate dissolved in four ounces of boil-
ing water; which being evaporated to half an ounce, there was,
on cooling, a copious deposit of white spicula* The sublimate
had a sharp, but not sour taste. Being boiled in muriatic acid, and
also in nitric, it did not dissolve at all ; but remained, on evapo-
ration to dryness, in the same state as before; and it must be par-
ticularly observed, that it left no red or pink matter, on evapo-
rating the nitric acid from it. Sulphuric acid did not act upon it
in the cold; but, when heated, it dissolved it, without efferves-
cence, from which solution nothing was precipitated by caustic
soda ; on evaporating it to dryness, black fumes arose, leaving
behind only a black stain. This sublimed matter did not render
lime water turbid. Boiled in muriatic acid, so as to carry off all
but a very little free acid, on the addition of lime water there
was no turbid appearance, but milkiness ensued on adding
oxalic acid.

The spicula, in the 10 grains of sublimate above mentioned,
seemed to be of the same nature as the matter just described.

* From the deposition of these spicula by cooling, and from many of the following
properties, they appear to be analogous to benzoic acid.

F 2

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3 6 Dr. Pearson’s Experiments and Observations

The whole of this sublimate amounted, by estimation, to 1 8
grains ; and I apprehend it is the acid sublimate of Scheele.

The sublimate of carbonate of ammoniac amounted to 20
grains; and it was black empyreumatic animal oil which
stained the tube.

This experiment was repeated, on 120 grains of a nut-brown,
very light, urinary concretion. The result was not very diffe-
rent from that of the former experiment, except that the gaz
contained a portion of hydrogen gaz. There were 30 grains
of the above described spicula , principally mixed with carbo-
naceous matter : they were light, and had only a very slight
sharp and bitter taste.

The experiment repeated a third time, with 80 grains of
urinary concretion, afforded 15 grains of the white spicula above
described, mixed with carbonaceous matter. These 1 found
did dissolve in a large proportion of muriatic acid; which solu-
tion yielded them, on evaporation, in the same state as before.
Under the flame applied by the blowpipe, they first melted,
and then evaporated, without any smell ; leaving a slight black
mark. Turnsole was reddened by these spicula.

In a fourth experiment, I found the white spicula contained
in the carbonaceous matter united, on boiling, with carbonate
of soda, as well as with caustic soda ; but, as before, muriatic
acid precipitated nothing from the solution. These spicula
could not be dissolved in nitric acid ; nor did the solution of
them in water become turbid with oxalic acid. Their taste was,
as before, rather bitter and sharp than sour. A very suffocat-
ing smell issued forth, on breaking the tube used in this expe-
riment, but it was not from sulphur, nor from prussic acid.

on the Composition of Urinary Concretions. 37

These experiments afford evidence of the wide difference be-
tween the animal oxide above described and the acid sublimate
of Scheele.*

If this conclusion be allowed to be just, it will be necessary
to give a name to this urinary animal oxide. Agreeably to the
principles of the new chemical nomenclature, the nafne should
be Lithic oxide . But the term lithic is a gross solecism ; and I
trust that philological critics will find the name onric or uric
oxide perfectly appropriate ; for, if it be thought objectionable,
on account of the existence of the matter in arthritic as well
as urinary concretions, still philology will allow its admission,
as in other similar cases, koit sjo^i/ ; it being found in greater
abundance, by far, in the urinary passages than in other situa-
tions, and therefore falling under common observation, as an
ingredient of the urine. If, however, the term lithic oxide, or
any other denomination, shall obtain acceptance, I shall very
willingly adopt it.

It requires no sagacity, in a person acquainted with the facts
of the preceding experiments, to perceive that they are appli-
cable to a variety of uses in chemical investigation, and in the prac-
tice of physic. The latter I of course take no notice of in this
place; but, relative to the former uses, I shall particularly point
out, that we are now able not only to detect, in the easiest man-
ner, the presence of the minutest proportion of the above animal
oxide in urinary concretions, and also in other substances, but
even to determine its proportion to the other constituent parts,

From these experiments, it now appears very doubtful whether the lithic acid of
Scheele exists as a constituent of urinary concretions, or is compounded, in conse-
quence of a new arrangement taking place, of the elementary matters of the concretion,
by the agency of fire; but it is demonstrated, that the urinary animal oxide is really a
constituent part, and even a principal one, of almost all human urinary calculi.

38 Dr. Pearson’s Experiments and Observations

in the space of a few minutes, in most cases, and in all in a
very little time, without any other apparatus than nitric acid,
a round bottomed matrass or glass dish, and a lamp. By this
method, I have, in a general way, examined above 300 speci-
mens of concretions, of the human subject and other animals,
principally urinary ones ; and also many from other parts, par-
ticularly those from the joints. For these opportunities I am

<•

beholden to several professional gentlemen ; whose willingness
to furnish me with specimens, I shall have much satisfaction in
acknowledging on a future occasion. At present, I must ac-
knowledge my obligations to Mr. Heaviside, in whose museum
I found between 700 and 800 specimens. The liberal possessor
of this treasure offered me, what I could not have taken the
liberty of requesting, namely, permission to break off pieces
from any of the articles, for experiment. Mr. Edward Howard
did me the honour to take upon himself the task of writing
down the reports, and otherwise assisted me.

At this time I shall only mention,

1. That out of 200 specimens of urinary calculi, not more
than six did not contain the animal oxide above described, i. e.
about 32 out of 33 contained it.

2. That the proportion of this oxide was very different; va-
rying from — -g-Q (exclusive of water,) to but, for the most
part, varying between -8q% and *

3. That the common animal mucilage of urine is frequently
found in concretions, in very different proportions ; but is per-
haps never a principal constituent part of them.

* In some urinary concretions, the interior part contained this oxide, and the exterior
part had none of it. On the contrary, in other urinary concretions, the exterior part
contained it, and the interior part did not.

on the Composition of Urinary Concretions. gg

4. That the above animal oxide was not found in the urinary
concretions, or any other concretions, of any animal but the
human kind.

5. That this animal oxide was found also in human arthritic
calculi, but not in those of the teeth, stomach, intestines, lungs,
brain, &c.

P. S. I think proper to subjoin a few experiments, made
after the preceding paper was written, which afford evidence
of the truth of some of my conclusions, and enable us to ex-
plain several properties of animal concretions.

I. On an Urinary Concretion from a Dog.

This calculus may be said to be a great curiosity, for it is
probably the only specimen in London. I owe the opportu-
nity of examining it to Mr. H. Leigh Thomas, who met with
it in the course of his dissections ; and therefore we have
unquestionable authority, that the concretion was really from
the urinary bladder of a dog. It is worthy to be noticed, that
the animal appeared to be in perfect health.

This concretion is of an oval figure ; is three inches and
three quarters in length, and three inches in breadth ; is white
as chalk ; its surface is rough and uneven. Being sawed
through longitudinally, no nucleus was found, nor was it lami-
nated, but near the centre it was radiated, and contained shi-
ning spicula . In other parts it was, for the most part, compact
and uniform in its texture. It weighed nearly ten ounces and
a half. Its specific gravity was found to be greater than that of
human urinary concretions, in general ; which I have learned
by experiments is also the case with urinary and intestinal

I

‘34 2 ^

'0A5 *

'/■$& >
Vf3 '

40 Dr. Pearson’s Experiments and Observations.

concretions of other brute animals, especially with those of the
horse. -

The specific gravity of the present calculus was 1,7.

That of one from the urinary bladder of the human subject,
of the sort called mulberry calculus, and which consisted almost
entirely of uric oxide, was 1,609.

That of another human urinary concretion, of the same com-
position as the former, but quite smooth, extracted by Mr. Ford,
was 1,571.

1. The present calculus of the dog had no taste, nor smell,
till exposed to fire.

2. Under the blowpipe it first became black, and emitted
the smell of common animal matter ; it next smelt strongly of
empyreumatic liquor cornu cervi; and, afterburning some time,
became inodorous, and white, and readily melted, like super-
phosphate of lime.

3. On trituration with lye of caustic soda, there was a copious
discharge of ammoniac.

4. It dissolved, on boiling in nitric acid : the solution was
clear and colourless ; and, on evaporation to dryness, left a re-
sidue of white bitter matter , which, under the blowpipe emitted,
weakly, the smell of animal matter.

5. Upon distilling a mixture of 150 grains of this concretion
pulverized and two pints and a half of pure water, to three
ounces, the distilled liquid was found to contain nothing but a
little ammoniac. The three ounces of residuary liquid, being fil-
trated and evaporated, yielded 20 grains of phosphate of ammo-
niac, with a little animal matter ; and the residuary undissolved

matter amounted to 67 grains.

6. These 67 grains, being triturated with four ounces of caustic

on the Compositions of Urinary Concretions. 41

soda lye, discharged very little ammoniac. On distilling this
mixture to one ounce, a very small proportion only of ammo-
niac was found in the distilled liquid. The residuary ounce of
alkaline liquid was filtrated, and mixed with the water of elu-
triation of the undissolved matter. One half of those liquids,
on evaporation to dryness, afforded a dark brown matter,
amounting to 20 grains, which consisted of phosphate of lime • ■]. <

and animal matter. To the other half of the alkaline liquids
was gradually added muriatic acid, which occasioned a deposit,
in small proportion, of matter that dissolved in nitric acid,
but which, on evaporation to dryness, left behind only a
brownish matter, consisting of phosphate of lime and animal
matter.

7. The residuary insoluble substance in caustic lye, (b.)
under the blowpipe, first turned black, and then grew white,
but could not be melted.

By diluted sulphuric acid it was decompounded. On the ad-
dition of nitrate of mercury, to the filtrated liquid, it yielded
phosphate of mercury; and, with oxalic acid, it afforded oxalate
of lime ; but no sulphate of magnesia was found remaining after
these precipitations were produced.

These experiments fully demonstrate, that the above con-
cretion of a dog contained none of the uric or lithic oxide
above described, but that it consisted, principally at least, of
phosphate of lime, phosphate of ammoniac, and animal matter.

The present instance leads me to explain the reason of the
fusibility of calculi. This is demonstrated, by the above expe-
riments, to depend upon the discharge and decomposition of
the ammoniac of the phosphate of ammoniac, during the burn-
ing away of the animal matter; hence the residuary phosphoric

mdccxcviii, G

4,2 Dr. Pearson's Experiments and Observations

acid readily fuses, and, uniting to the phosphate of lime, com-
poses superphosphate of lime, a very fusible substance.

The phosphate of ammoniac being dissolved out by water,
or caustic alkaline lye, the remaining matter is infusible, being
phosphate of lime.

A very hard, brittle, and blackish intestinal calculus of a
dog, from Mr. Wilson, was found to be of greater specific
gravity than human urinary calculi, and to have the same
composition as that of the dog above described.

This also was found to be the composition of a white, smooth,
round, intestinal calculus of a horse, the specific gravity of
which was 1,791.

The same composition was discovered, on examining a very
hard, gray, brittle, laminated, quadrilateral concretion, said to
be from the urinary bladder, but which, I think, was more pro-
bably from the intestines, of a horse.

II. On a Calculus from the urinary Bladder of a Rabbit.

This is also a curiosity, being the only instance I have seen.
I am likewise indebted to Mr. Thomas for this specimen,
which he very kindly sent me, fitted up as a preparation, in-
cluded in the bladder itself. Mr. Thomas found this concretion,
on dissecting a perfectly healthy and very fat rabbit.

This specimen is spherical, and of the size of a small nut-
meg. It is of a dark brown colour, has a smooth surface, is
hard, brittle, and heavy. When broken, it appeared to consist
of concentric laminas. Its specific gravity was 2.

1. Under the blowpipe it grew black, and emitted the smell
of animal matter while burning ; at last it ceased to emit any

on the Composition of Urinary Concretions . 43

smell ; and, urged with the intensest fire, showed no signs of
fusibility.

2. It readily dissolved, with effervescence, like marble, in
both muriatic and nitric acids, giving clear solutions.

3. The nitric solution (2.) being evaporated partly to dry-
ness, and partly to the consistence of extract, the dry residuary
matter was white ; and the extract-like matter, which was bitter,
could not be fused under the blowpipe; but, when brought to
the state of a powder, the particles of it were made to cohere
loosely together into one mass.

4. On dropping sulphuric acid into the muriatic solution, (2.)
turbidness, and a copious white precipitation, immediately en-
sued, from the composition of sulphate of lime.

From these experiments it is warrantable to conclude, that
the above urinary calculus of a rabbit consisted principally of
’ carbonate of lime and common animal matter, with, perhaps,
a very small proportion of phosphoric acid : it certainly con-
tained no uric oxide.

I examined, in the same mafmer, a concretion which was
said to be from the stomach of a monkey; but I have not evi-
dence of its origin equally satisfactory as that of the two last
calculi. Its composition was found to be similar to that of the
calculus of the rabbit, viz. carbonate of lime and animal matter.
Its obvious properties were also the same ; it was of the size
of the largest nutmeg.

III. On urinary Concretions of the Horse.

I examined several specimens in cabinets, said to be vesical
calculi of the horse, and found none of them to contain the
uric oxide above described ; but that they consisted (as well

G 2

44 Dr. Pearson’s 'Experiments and Observations

as the calculi from the stomach and intestines of the same ani-
mal) of phosphate of lime, phosphate of ammoniac, and com-
mon animal matter, which melted like superphosphate of lime,
after burning away the animal matter and ammoniac. As
these, and some other experiments, seemed to concur in esta-
blishing an important truth, I thought it necessary to examine
an urinary concretion of a horse, which, from its figure and
size, was unquestionably from the kidney of that animal ; for I
have found by experience, that one cannot depend entirely on
the accounts in cabinets, nor indeed, sometimes, on the asser-
tions of persons who collect specimens.

1. This concretion, which Dr. Baillie was so good as to
give me, was of a blackish colour, was very brittle and hard,
and had no smell or taste. It felt heavier than human urinary
calculi.

/

2. Under the blowpipe it became quite black, and emitted
the smell, weakly, of common animal matter. It was reduced
very little in quantity, and showed no appearances of fusibility,
after being exposed for a considerable time to the most intense
fire of the blowpipe.

3. Muriatic acid dissolved this concretion, with effervescence,
yielding a clear solution; which, on evaporation to dryness,
left a black and bitter residue.

4. 'A little of the residue (3.) being boiled in pure water, to
the filtrated liquor superoxalate of potash was added ; which
occasioned a very turbid appearance, and copious white preci-
pitation.

5. Nitric acid also readily dissolved this concretion, with ef-
fervescence. The solution being evaporated, partly to dryness,
and partly to the consistence of an extract, the dry residuary

on the Composition of Urinary Concretions . 45

matter was white and bitterish, and the extract-like part
showed no signs of fusibility under the intensest fire of the
blowpipe.

6. A little of the concretion, being triturated with lye of
caustic soda, emitted no smell of ammoniac.

From these experiments it appears, that this calculus, like
the former one from a rabbit, consists of carbonate of lime and
common animal matter.

A renal calculus of a horse, in Mr. Heaviside’s collection,
appeared, on examination, to consist of carbonate of lime and
common animal matter.

Another specimen, however, of renal calculus of a horse, in
the same collection, marked No. 3. was found to consist of phos-
phate of lime, phosphate of ammoniac, and common animal
matter. It was fused under the blowpipe.

The specimen marked No. 8. in the same collection, which
was said to be a vesical calculus of a horse, appeared to consist
of the three ingredients just mentioned.

I have met with two instances of a deposit of a prodigious
quantity of matter in the urinary bladder of horses, which had
not crystallized, or even concreted : it amounted, in one speci-
men, which was given to me by Dr. Marshall, to several
pounds weight; and in the other, which is in the possession of
Mr. Home, to about 45 pounds. Its composition was, princi-
pally, carbonate of lime and common animal matter. *

I have not found any instance of human urinary calculi of a

* Since this paper was read, Mr. Blizard has been so attentive as to send me
another specimen of the same kind of deposit as those here mentioned. It now ap*
pears probable, that such deposits frequently take place, although I believe they have
not been noticed before.

4 6 Dr. Pearson's Experiments and Observations, &c.

similar composition to that of the rabbit, and those of horses
above described, which consist of carbonate of lime and animal
matter ; and I believe that human urinary calculi very rarely
occur of a similar composition to those of the dog and horses
above mentioned, which were found to consist of phosphate of
ammoniac, phosphate of lime, and animal matter, without con-
taining uric oxide.

The difference in the constitution of urinary concretions may
depend on the difference of the urinary organs of different ani-
mals, on the food and drink,* and on the various diseased and
healthy states of the urinary organs.

I have not found the uric oxide in the urinary concretions of
any phytivorous animal ; but, whether it would be formed in
the human animal when nourished merely by vegetable matter,
must be determined by future observations. In the mean time,
it is warrantable to conclude, from analogy, that it would not,
and the application of this fact to practice is obvious ; but I
now purposely avoid making any practical inferences, until I
can, at the same time, state a number of facts I have collected,
relative both to concretions and to the urine itself.

* I found the stomach-concretion called Oriental Bezoar, to consist merely of ve-
getable matter ; as did the intestinal concretion of a sheep.

C 47 3

III. Oil the Discovery of four additional Satellites of the
Georgium Sidus. The retrograde Motion of its old Satel-
lites announced ; and the Cause of their Disappearance at
certain Distances from the Planet explained . By William
Herschel, LL.D. F. R. S.

Read December 14, 1797.

jtIaving been lately much engaged in improving my table*
for calculating the places of the Georgian satellites, I found it
necessary to recompute all my observations of them. In look-
ing over the whole series, from the year of the first discovery
of the satellites in 1787 to the present time, I found these ob-
servations so extensive, especially with regard to a miscella-
neous branch of them, that I resolved to make this latter part
the subject of a strict examination.

The observations I allude to relate to the discovery of four
additional satellites : to surmises of a large and a small ring, at
rectangles to each other : to the light and size of the satellites :
and to their disappearance at certain distances from the planet.

In this undertaking, I was much assisted by a set of short
and easy theorems I had laid down for calculating all the par-
ticulars respecting the motions of satellites; such as, finding
the longitude of the satellite from the angle of position, or the
position from the longitude : the inclination of the orbit from
the angle of position and longitude : the apogee : the greatest

48 Dr. Herschei/s Discovery of four additional

elongation; and other particulars. Having moreover calcu-
lated tables for reduction : for the position of the point of great-
est elongation; and for the distance of the apogee, or opening
of the ellipsis ; and also contrived an expeditious application of
the globe for checking computations of this sort, I found many
former intricacies vanish.

By the help of these tables and theorems, I could examine
the miscellaneous observations relating to additional satellites,
on a supposition that their orbits were in the same plane with
the two already known, and that the direction of their motion
was also the same with that of the latter.

And here I take an opportunity to announce, that the motion
of the Georgian satellites is retrograde.

This seems to be a remarkable instance of the great variety
that takes place among the movements of the heavenly bodies.
Hitherto, all the planets and satellites of the solar system have
been found to direct their course according to the order of the
signs : even the diurnal or rotatory motions, not only of the
primary planets, but also of the sun, and six of their seconda-
ries or satellites, now are known to follow the same direction;
but here we have two considerable celestial bodies completing
their revolutions in a retrograde order.

I return to the examination of the miscellaneous observations,
the result of which has been of considerable importance, and
will be contained in this paper. The existence of four addi-
tional satellites of our new planet will be proved. The obser-
vations which tend to ascertain the existence of rings not ap-
pearing to be satisfactorily supported, it will be proper that
surmises of them should either be given up, as ill founded, or
at least reserved till superior instruments can be provided, to

49

Satellites of the Georgium Sidus , See,

throw more light upon the subject. A remarkable phenome-
non, of the vanishing of the satellites, will be shewn to take
place, and its cause animadverted upon.

I shall now, in the first place, relate the observations on
which these conclusions must rest for support, and afterwards
join some short arguments, to shew that my results are fairly
deduced from them.

For the sake of perspicuity, I shall arrange the observations
under three different heads ; and begin with those which relate
to the discovery of additional satellites.

A great number of observations on supposed satellites, that
were afterwards found to be stars, or of which it could not be
ascertained whether they were stars or satellites, for want of
clear weather, will only be related. For, to enter into the par-
ticular manner of recording these supposed satellites, or to give
the figures which were delineated to point them out, would take
up too much time, and be of no considerable service to our present
argument. It ought however to be mentioned, that nearly the
same precaution was taken with all the related observations as,
it will be found, was used in those that are given in the words of
the journals that contain them. The former will be distinguished
under the head Reports , the latter under that of Observations.

Investigation of additional Satellites.

Reports.

Feb. 6 , 1782. A very faint star was pointed out as probably
a satellite, but Feb. 7 and 8 was found remaining in its former
situation.

March 4, 1783. A satellite was suspected, but March 8 was
found to be a star.

H

MDCCXCVIII.

50 Dr. Herschel’s Discovery of four additional

April 5, 1783. A suspected satellite was delineated, but the
6th it was seen remaining in its former place.

Nov. 19, 1783. A supposed satellite was marked down, but
no opportunity could be had to account for it afterwards.

Nov. 16, 1784. Supposed 1st and 2d satellites were pointed
out, but not accounted for afterwards.

Many other fruitless endeavours for the discovery of satellites
were made; but, finding my instrument, in the Newtonian
form, not adequate to the undertaking, the pursuit was partly
relinquished. The additional light however which I gained, by
introducing the Front- view in my telescope, soon after gave me
an opportunity of resuming it with more success.

Jan. 11, 1787. Three supposed satellites were observed: a
first, a second, and a third. Jan. 12, the 1st and 2d were gone
from the places in which I had marked them, but the 3d was
remaining, and therefore was a fixed star. *

Jan. 14. A supposed 3d satellite was delineated, but on the
17th it was found to be a star.

Jan. 17. Supposed 3d, 4th, and 5th satellites were marked,
but were found remaining in their former places on the 1 8th.

Jan. 24. Supposed 3d and 4th satellites were noted, but the
weather proving bad on the succeeding nights, till February 4,
they were lost in uncertainty.

Feb. 4. A 3d satellite was marked, but not being afterwards
accounted for remains lost.

Feb. 7. A supposed 3d satellite was proved to be a star the 9th.

Feb. 10. Supposed 3d and 4th satellites have not been after-
wards accounted for.

* It has already been shewn. In a former paper, that the removed satellites were
those two which now are sufficiently known.

Satellites of the Georghm Sidus , &c. 51

Feb. 13. Supposed 3d, 4th, and 5th satellites proved stars the
16th.

Feb. 16. A 3d satellite proved a star the 17th.

Feb. 19. Supposed 3d and 4th satellites were proved to be
stars the same evening, by being left in their places, while the
planet was moving on.

Feb. 22. The supposed 3d and 4th of the 19th were seen
remaining in their former places ; and new 3d, 4th, and 5th
satellites were marked ; but these were lost through bad wea-
ther, which lasted till March 4.

March 5. A supposed 3d satellite proved to be a star the 7th.

March 7. The position of a 3d was taken, and a 4th also
marked ; but March 8 they were both proved to be fixed stars.

October 20. A very small star was seen near the planet, but
lost, for want of opportunity to account for it.

March 13, 1789. The positions of 3d and 4th satellites were
taken, but the 14th they were found to be stars.

March 16. Supposed 3d and 4th satellites were well laid
down, but March 20 were found to be stars.

March 2 6. The places of supposed 3d and 4th satellites were
ascertained, but no opportunity could be had of deciding whe-
ther they were stars or satellites.

Dec. 15. A supposed 3d satellite was accurately delineated,
but proved to be a star the 16th.

Observations.

“ Jan. 18, 1790. 6h5i'.* A supposed 3d satellite is about

* All the times have been corrected so as to be true, sidereal ; but are only given
here to the nearest minute.

H 2

53 Dr. Herschei/s Discovery of four additional

“ 2 diameters of the planet following ; excessively faint, and
“ only seen by glimpses.”

“ 7h 57' . I cannot perceive the 3d.”

Reports.

Jan. 18, 1790. A supposed 4th satellite was described, but
was found to be a star the 19th.

Jan. 20. A 3d satellite was perceived, and its angle of posi-
tion ascertained ; but was afterwards lost, for want of opportu-
nity to examine its place again.

Observations.

“ Feb. 9, 1790. 6h 28'. There is a supposed 3d satellite, in a
“ line with the planet and the 2d satellite.”

“ 6h 40'. Configuration of the Georgian planet and satel-
“ lites.” See Tab. II. fig. 1.

« Clouds prevent further observations.”

“ Feb. 11. The supposed 3d satellite of the 9th of February
“ I believe is wanting ; at least I cannot see it, though the wea-
“ ther is very clear, hut windy.”

“ Feb. 12. The supposed 3d satellite of the 9th is not hi
« the place where I saw it that night.”

Reports.

Feb. 11, 1790. Supposed 3d and 4th satellites were laid down,
but on the 12th they were both found remaining in their for-
mer places.

Feb. 16. A 3d satellite was delineated, but on the 17th it
proved to be a star.

Satellites of the Georgium Sidus, &c, 53

March 5. Supposed 3d and 4th satellites were laid down,
but on the 8th were seen remaining in their places.

Feb. 4, 1791. A 3d satellite was marked, but has not been
accounted for afterwards.

Feb. 5. Supposed 3d, 4th, and 3th satellites were delineated,
but no opportunity could afterwards be found to ascertain their
existence.

March 5. Supposed 3d, 4th, and 5th satellites were put down.
They could not be seen March 6, but were proved to be small
stars the 7 th.

Feb. 12, 1792. A third satellite was delineated, but was left
behind by the planet the same evening, and also seen in its
former place the next night.

Feb. 13. A 3d satellite was put down, but proved to be a
star the 14th.

Feb. 20. The position of a 3d satellite was taken, but 4 hours
after was found to be left behind by the planet. It was also
seen in its former place Feb. 21.

Feb. 26. A 3d satellite, between the planet and 2d, was ob-
served ; which, 3h 3 7' afterwards, was thought to be left behind,
but was so faint as hardly to be perceivable. A fourth was also
put down. Neither of them have been accounted for after-
wards.

March 8, 1793. The position of a supposed 3d satellite was
taken, but the next day it was found to be a star.

March 9. A supposed 3d satellite was observed, at 5 or 6
times the distance of the 1st, but was not accounted for after-
wards.

March 14. Supposed 3d and 4th satellites were observed, but
no opportunity could be had afterwards to see them again.

54

Dr. Herschel’s Discovery of four additional

Observations.

“ Feb. 25, 1794. With 320, there is a small star fig. 2,

“ about 1 5 degrees north preceding the planet ; and another 6,

“ about 30 degrees north preceding : also one c , directly pre~

ceding. There is a very small fourth star d, making a trape-
“ zium with the other three- ; and two more ef preceding this
“ 4th star, are in a line with it.”

“ Feb. 2 6. The stars, in figure 2, marked/*? d a , are in a line.

“ There is a star g, at rectangles to f e da: the perpendicular
“ falls upon d: it is towards the south. There is also a star 5,

“ north of fe da; but it is too faint to admit of a determination
“ of its place : I can only see it now and then by imperfect
“ glimpses.”

“ Feb. 28. 6h 40'. The stars f e d a of the 26th are in their
“ places, c is in the place where I have marked it. The star
“ g is in the place where I marked it. I see also the very
“ small star b.”

“ 6h 50'. There is a very small star k, but not so small as bs
“ very near to, and north following/, which I did not see on
“ the 26th. It is not quite half way between / and e, but
« nearer to / than to e. It makes an obtuse triangle with /
“ and e.”

“ qh The motion of the planet this evening, since the
“ first observation, is very visible.”

ioh 7'. I cannot perceive the star k. The weather is not
“ so clear as it was.”

« ioh 2i/. I cannot perceive the star k in the place where it
“ was 6h 50'.”

“ March 4, 1794. Power 320. 6k 46'. The stars ab c def g _

55

Satellites of the Georgium Sidus, &c.

“ of Feb. 28, fig. 3. are in their places, but I cannot see the
small star k. The evening is not very clear/'

“ 9h 51' • I cannot see the star k”

“ ioh 25'. I suppose a, in figure 4, to be the star towards
“ which the planet is moving."

“cab are in a crooked line.

“ c ef are nearly in a line ; / is a little preceding.

“ c d e form a triangle.

“ There is a small star h , preceding d.

“ There is an exceeding small star k , in the line b kg, but a
little preceding and nearer b.

“ a b c are large stars.

“deg are also pretty large.

“ f and h are small. Power 157.

“ With 320, there is also a very small star l, near d, forming
“ an isosceles triangle h d /, on the preceding side."

“ March 5. 7h 39'. Power 320. The stars abcdefghkl
“ are in the places where they were marked last night."

“ 9h 37' • There is a very small star n, south of g; another m,
“ preceding g ; and a third 0, south following g”

“ ioh 19'. I suspect a very small star, south following the
“ planet, at one-third of the distance of the 1st satellite ; but
“ cannot verify it with 480. With 600, the same suspicion
“ continues."

“ March 7. 9^ 48'. The stars abed ef g hkl are in their
“ places."

“ nmo are in their places."

“ The planet has passed between the stars ef, pretty near

“ to/."

56 Dr. Herschel's Discovery of four additional

Reports.

March 21, 1 794. Power 320. A small star was suspected
south of the planet, or about 85° south following. It could not
be verified with 480, nor with 600 ; and was even supposed, to
have been a deception ; but the 2 2d was found remaining in
the place where the planet had left it.

Observations.

“ March 2 6, 17 94. gh 35'. With 480, I see the 1st satellite
e‘ much better than with 320. I suspected, with 320, a 3d sa-
“ tellite, directly north of the planet, a little farther off than the
“ 1st, and this power almost verifies the suspicion.” See
figure 5. (Tab. III.)

“ gh 44', With 600, I still suspect the same, but cannot sa~
“ tisfy myself of the reality.”

“ 1 ih 32'. I see the supposed 3d satellite perfectly well now.
“ It is much smaller than the 1st, and in a line with the planet
“ and the 1st; so that probably it is a fixed star; since it pre-
“ ceded the 1st, when I saw it before, I think more than the
“ quicker motion of the 1st satellite would account for. If it
“ be a fixed star, it makes almost a rectangular triangle with qr ,
« the shorter leg being 3d r ; or it is almost in a line with q
“ and n ”

“ N. B. The lines in the description are truer than in the
“ figure, as the latter is only intended to point out the stars
“ in question.”

“ March 27. 8h 37'. Power 320. The same small star, ob-
ec served last night at nh 32', is gone from the place where I
“ saw it. From its light last night, compared to r, which

hi

Satellites of the Georgium Sidus , &c.

(i to-night is very near the planet, and scarcely visible, I am
<e certain that it must be bright enough to be perceived imme-
“ diately, if it were in the place pointed out by my descrip-
“ tion."

“ ioh 20'. The planet is considerably removed from the
u star r.

“ nh4i'. I had many glimpses of small stars or supposed
“ satellites : one of them in a place agreeing with the 3d satel-
“ lite of last night, (supposing it to have moved with the planet; )
“ that is, a little farther off’ and after the 1st. Another pre-
“ ceding the 1st, but nearer. Some others south, at a good dis-
“ tance ; but not one of them could I see for any constancy.
“ They were only lucid glimpses/'

Reports.

March 27, 1794. A supposed 4th satellite was delineated,
but proved to be a star the 28th.

Observations.

“ March 4, 1 796. Configuration of the Georgian planet and
“ fixed stars for 1 oh 3'/’ See fig. 6.

“ March 5. 911 50'. I suspected a very small star between c
“ and h , which was not there last night. I had a pretty cer-
“ tain glimpse of it. It is in a line from the planet towards/;
“ power 320. With 600, 1 see the satellite better than before ;

but cannot perceive the suspected small star."

“ ioh 17'. The air is remarkably clear at present, but I can-
“ not perceive the suspected star."

“ March 9. 1 ih 23'. As the probability of other satellites is,
“ that they revolve in the same plane with the 1st and 2d, I

mdccxcviii. I

58 Dr. Herschei/s Discovery of four additional

“ chiefly look for them in the direction of their orbits, which
“ is now nearly a straight line/'

“ April 5, 1 796. There is no star in the line of the transverse,
“ that can be taken for a satellite : the evening is very beauti-
“ ful, and I examined that line with 300, at a distance ; and
“ with 600, within the orbits of the two satellites/'

“ March 23, 1797. Three very small stars O P Q, are in the
“ path of the planet ; they form an obtuse triangle."

“ March 25. nh 4/. A very bright star S, at almost the dis-
“ tance of the field of view, is a little south of the path of the
“ planet. It has a small north preceding star T, which points
“ to two more V W, towards the north."

“ Between the triangle of March 23d and the four last men-
“ tioned stars, is a very small star X."

“ March 28. ioh52'. I see the stars ST VWX of March
“ 25th."

“ 1 ih 25'. From X towards the triangle O P Q of March 23d,
“ is an exceeding small star Y, about four times the distance
“ of the 2d satellite, and nearly in the line of the greatest elonga-
“ tion. I do not remember to have seen it the 25th."

“ nh 41'. The distance of Y from X is about \ of the dis-
“ tance of X from the triangle. It requires much attention to
“ see it ; but I have a very complete view of it, by drawing
“ the planet just out of the field, and the star X almost on
“ the preceding side."

Arguments upon the Reports and Observations.

From the reports of the great number of supposed satellites,
compared with the select observations which are given at length,
it must be evident that the method of looking for difficult

69

Satellites of the Georgium Sidus, &c.

objects, and of marking them down by lines and angles, with
every other possible advantage for finding them again, has
been completely understood and put in practice. So guarded
against deceptions, we cannot but allow, that even a single
glimpse of a very small star is a considerable argument in fa-
vour of its existence. What I call verifying a suspicion, which
is generally done with a higher power than that which caused
the suspicion, is obtaining a steadier view of the existence
of the object in question ; that is, to see it in such a man-
ner as to be able to fix an eye upon it, and to compare it with
other surrounding objects ; and thus to be able to ascertain its
relative situation with those other objects, in a satisfactory
manner.

An hiterior Satellite.

The observation of Jan. 18, 1790, says, “ a supposed 3d satel-
i( lite is about two diameters of the planet following/’ There is
not the least doubt expressed about the existence of the satellite,
or object in question, which therefore must be looked upon as as-
certained. Now, the angle of the greatest elongation of the Geor-
gian satellites, by my new tables, at the time of observation, was
8i°33/ N.F. Therefore, the angle of the apogee was 8° 27' S.F. ;
and since, by observation, the satellite was “ following,” without
any mention of degrees being made, we may admit it to have
been not far from the parallel; suppose 11 or 12 degrees S.F. In
this case, the satellite would be in the apogee about the time of
the 2d observation, at 7*57'; which says, “ I cannot perceive
the satellite. But it will be shewn hereafter, when I come to
treat of the vanishing of the satellites, that it would become
invisible in this situation. Indeed, without the supposition of

I 2

6o Dr. Herschel’s Discovery of four additional

the satellite's coming to the apogee, it might easily happen that
the least change in the clearness of the air, during a time of
ih 5', which elapsed between the first and second observation,
might render an object invisible, which, as the first observa-
tion says, was “ excessively faint, and could only be seen by
glimpses."

From the observed distance, which is put at “ 2 diameters
of the planet," we may conclude what would be the distance
of its greatest elongation. For, 2 diameters from the disk of
the planet give 2^ from the centre. Now, the distance of the
apogee at this time, by my tables, was ,64, supposing that of
the greatest elongation 1 ; therefore we have the radius of its

u

orbit

2,5 X 4", 1 2
,64

16", i,

This calculation is not intended to determine precisely the
distance of the satellite, but only to shew that its orbit is more
contracted than that of the 1st, and that consequently it is an
interior satellite.

If any doubt should be entertained about the validity of this
observation, we have a second, and very striking one, of March
5, 1794; where an interior satellite was suspected south fol-
lowing the planet, at one-third of the distance of the 1st.
March 4, when a description was made of the stars, as in
figure 4, this satellite was not in the place where it was ob-
served the 5th. And, by an examination of the same stars
March 7, it appears, that even the smallest stars n m 0, of the
5th, were seen in their former places, but not the satellite.
The observation therefore must be looked upon as decisive with
regard to its existence. If any doubt should arise, on account
of the suspicion not being verified with 480, I must remark,
that being used to such imperfect glimpses, it has generally

Satellites of the Georgium Sidus , &c. 6 1

turned out, even when I have given up as improbable the ex-
istence of a supposed satellite seen in that manner, that it has
afterwards nevertheless been discovered that a small star re-
mained in the place where the satellite had been suspected to
be situated. An instance of this may be seen in the report of
the observations that were made March 21 and 22, 1794. Be-
sides, in the present case, it is additionally mentioned, that the
same object was examined with a power of 600, which conti-
nued the suspicion.

From the assigned place of this satellite, at-Jof the distance
of that of the first, it appears that this observation belongs to
the interior satellite of Jan. 18, 1790, which has already been
examined. The 1st satellite was this evening at its greatest
elongation, one-third of which is about 11". The apogee dis-
tance of a satellite whose greatest distance is 16", 1 would have
been 6", 1 on the day of our observation; but, not being come to
the apogee, by many degrees, it could not be so near the planet.

For the sake of greater precision, let us admit that the satel-
lite was exactly south following ; that is, 45 degrees from the
parallel, and 45 from the meridian; then, by calculation, a
satellite whose orbit is at i6",i from the planet, would, in the
situation now admitted, have been 7",i from its centre, which
might coarsely be rated at ± of the distance of the first. But
the estimation of 1 1" is probably more accurate than that in
the 1st observation, where 2 diameters are given. And, by
calculating from this quantity, we find that the greatest elonga-
tion distance of the satellite is 25", 5 ; now, putting 2^ diame-
ters in the first observation, instead of 2, the distance deduced
from it will come out 19", 3; which is certainly an agreement

6 s Dr. Herschel’s Discovery of four additional

sufficiently near to admit both observations to belong to the
same satellite.

March 27, 1794. We find a third observation, -which will
assist in supporting the two former ones. A glimpse of a sa-
tellite is mentioned, which was preceding the 1st, but nearer
the planet. The position of the 1st satellite the same evening
was, by measuring, found to be 6V, 1 N.F. which is still a
considerable way from its greatest elongation; but our new
satellite preceded it, and was therefore more advanced in its
orbit, or nearer its greatest distance ; and yet the observation
says, that it was not so far from the planet as the 1st; not-
withstanding this latter was in a more contracted part of its
orbit. It follows therefore that this was also an interior satel-
lite. Now, since we may allow these three observations to be-
long to the same, we ought not to make a distinction; but
admit, as sufficiently established, the existence of at least one
interior satellite of our new planet.

An intermediate Satellite.

March 26, 1794. A satellite was suspected, directly north
of the planet. At first it could not be verified, but was seen
perfectly well afterwards. It was supposed that probably it
might be a star, but this was left undecided. The observation
of March 27th however removes all doubt upon the subject ; as
it fully affirms that the small star observed the 26th, at nh32',
was gone from the place in which it was the day before. Such
strong circumstances are mentioned in confirmation, that we
cannot hesitate placing this among the list of existing satellites.
It was not the interior satellite of Jan. 18, 179° f°r both the

Satellites of the Georgium Sidas, &c. 63

1st and 2d known satellites were in full view March 26th ; see
figure 5. and the observation places this new one in a line
drawn from the planet continued through the 1st; with the
remark, that it was a little farther from the planet than the 1st.
The 2d was then near its greatest southern elongation, and we
may see from the figure, as well as from the above description,
that the orbit of this new satellite is situated between the orbits
of the other two.

We have a second observation of the same satellite March 27,
1 794; where, among the glimpses of additional satellites at
nh4i/, is mentioned “ one in a place probably agreeing,with
<c the new satellite of March 26th,” which, by its motion, must
have been carried forward, so as to be where the observation of
the 27th says it was, namely, “ a little farther off* and after the
“ 1st that is, at a little greater distance from the planet than
the 1st, and not so far advanced in its orbit as that satellite.
This amounts not only to an additional proof, but even an-
nounces the recognition of the satellite, and its motion in the
course of one day.

An exterior Satellite.

Feb. 9, 179°- A new satellite was seen, in a line with
the planet and the 2d satellite. See figure 1. To convince
us that this was not a fixed star, we have the observations
of two other nights, the 11th and 12th of February, where the
removal of it, from the place in which it was Feb. 9, is clearly
demonstrated. As it was in a line continued from the planet
through the second satellite, its orbit must evidently be of a
greater dimension than that of the 2d; I shall therefore put it
down as an exterior satellite.

64 Dr. Herschel’s Discovery of four additional

Most likely this satellite also was seen among the supposed
satellites south of the planet, March 27, 1 794 ; where we find
mention made of “ some others south, at a good distance/*
In that case, this will make a second observation.

We have a third observation of the same new satellite March
5, 1796 ; when a very small star was seen, in a place where the
evening before there had been none ; as appears by the confi-
guration of the 5th of March. See figure 6. At the time of the
observation, the planet was come to the longitude of the place
where the star was perceived to be; which agrees with the idea
of its having been brought to that situation by the planet. It
may be objected, that the star could not be verified with a power
of 600; but here we have more than a bare suspicion of the
satellite, for the observation says, “ I had a pretty certain
“ glimpse of it and this appears also from the assigned
place of the star at the intersection of two given lines. For,
such a delineation could not have been made, without having
perceived it with a considerable degree of steady vision. Its
distance, to judge by the description, will agree sufficiently
with the two foregoing observations of this new exterior sa-
tellite.

The most distant Satellite.

On Feb. 28, 1794, a star was perceived where on the 26th
there was none. This star was larger than a very small star
- which was observed the 26th, not far from the place of the
new supposed satellite ; and a configuration having been made
expressly, by way of ascertaining what stars might afterwards
come into a situation where they could be mistaken for satel-
lites, our new star or satellite would not have been omitted.

Satellites of the Georgium Sidus , &c. 65

when a smaller one very near it was scrupulously recorded.
The motion of the planet, in 3 hours and 3 minutes, is men-
tioned as very visible. The place of the star, which was a new
visitor this evening, was very particularly delineated, at 6h 50'.
From its situation, it is evident the motion of the planet must
have carried this star, if it was one of its satellites, towards the
large star f figure 3 ; in the light of which a dim satellite
would be lost. This accordingly happened ; for at ioh 7' and
ioh 21' it was no longer visible. The direction of the planet's
motion is plainly pointed out, by the place of the planet
March 2d.

With respect to the orbit of this satellite, it appears, from its
situation near the apogee, where it was seen, that its distance
was to that of the second satellite, which was then near its
greatest elongation, as 8 to 5. And, since the apogee distance,
on the day of observation, was only ,37, we have its greatest
elongation as — to 5; that is, as 21,6 to 5, or above 4 to 1.
From which we may conclude, that its orbit must lie consi-
derably without the before mentioned exterior satellite of Feb.
9> i79°-

We have a second observation of it March 27, 1794 ; which,
though not very strong, yet adds confirmation to the former.
For that evening, which was uncommonly fine, other satellites,
south, at a good distance, were perceived. This must relate
principally to our present satellite, which may certainly be said
to be at a good distance from the planet, and which, by that
time, was probably in the southern part of its orbit, and near
its greatest elongation.

There is a third observation, March 28, 1797, which probably
also belongs to this satellite. For the exceedingly small star Y,

MDCCXC VIII. • K

66' Dr. Herschel's Discovery of four additional

which is mentioned as not having been seen the 25th, when
the delineation of the stars was made, will agree very well with
the two former observations ; and, being near the greatest elon-
gation, the distance of this satellite is well pointed out, and
agrees remarkably well with the calculation of the first obser-
vation of it.

It remains now only to be mentioned, that in such delicate
observations as these of the additional satellites, there may
possibly arise some doubts with those who are very scrupulous;
but, as I have been much in the habit of seeing very small and
dim objects, I have not been detained from publishing these
observations sooner, on account of the least uncertainty about
the existence of these satellites, but merely because I was in
hopes of being able soon to give a better account of them, with
regard to their periodical revolutions. It did not appear satis-
factory to me to announce a satellite, unless I could, at the same
time, have pointed out more precisely the place where it might
be found by other astronomers. But, as more time is now already
elapsed than I had allowed myself for a completion of the theory
of these satellites, I thought it better not to defer the communi-
cation any longer.

The arrangement of the four new and the two old satellites
together will be thus :

First satellite, the interior one of Jan. 18, 1790.

Second satellite, the nearest old one of Jan. 11, 1787.

Third satellite, the intermediate one of March 26, 1 794,.

Fourth satellite, the farthest old one of Jan. 11, 1787.

Fifth satellite, the exterior one of Feb. 9, 1790.

Sixth satellite, the most distant one of Feb. 28, 1794.

Satellites of the Georgium Sidns , &c.

6y

Observations and Reports te?iding to the Discovery of one or
more Rings of the Georgian Planet , and the flattening of its
polar Regio?is. *

“ Nov. 13, 1782. 7-feet reflector, power 460. I perceive no
“ flattening of the polar regions.”

“ April 8, 1783. I surmise a polar flattening.”

“ Feb. 4, 1787. 20-feet reflector, power 300. Well defined;
“ ntf appearance of any ring ; much daylight.”

“ March 4. I begin to entertain again a suspicion that the
planet is not round. When I see it most distinctly, it appears
“ to have double, opposite points. See figure 7. Perhaps a
“ double ring ; that is, two rings, at rectangles to each other.”
March 5; The Georgian Sidus not being round, the telescope
was turned to Jupiter. I viewed that planet with 157, 300, and
480, which shewed it perfectly well defined. Returning to the
Georgian planet, it was again seen affected with projecting
points. Two opposite ones, that were large and blunt, from
preceding to following ; and two others, that were small and
less blunt, from north to south. See figure 7,

March 7. Position of the great ring R, from 7o°S.P. to 70*
N.F. Small ring r, from 2o°N.P. to 2o°N.F. 600 shewed R
and r. 800 R and r. 1200 R and r.

“ March 8. R and r are probably deceptions.”

“ Nov. 9. The suspicion of a ring returns often when I ad-
“ just the focus by one of the satellites, but yet I think it has
no foundation.”

Feb. 22, 1789. A ring was suspected.

The observations are distinguished from the reports by marks of quotation,
(“ ”■)

68 Dr. Herschei/s Discovery of four additional

“ March 16. 7h 37'. I have turned my speculum 90° round.
“ A certain appearance, owing to a defect which it has con-
“ tracted by exposure to the air since it was made, is gone
“ with it ; (see fig. 9 and 10 ;) but the suspected ring remains
in the place where I saw it last.

“ 7h 50'. Power 471 shews the same appearance rather
“ stronger. Power 589 still shews the same/’

“ Memorandum. The ring is short, not like that of Saturn.
“ It seems to be as in figure 8 ; and this may account for the
“ great difficulty of verifying it. It is remarkable that the two
“ ansce seem of a colour a little inclined to red. The blur oc-
“ casioned by the fault of the speculum is, to-night, as repre-
“ sented in figure 9. The other evening it was as in figure 10 ;
“ and the ring is likewise as it was the same evening.”

“ March 20. 7h 53/ When the satellites are best in focus,
“ the suspicion of a ring is the strongest.”

“ Dec. 15. The planet is not round, and I have not much
“ doubt but that it has a ring.”

“ Feb. 26, 1792. 6h34'. My telescope is extremely distinct;
“ and, when I adjust it upon a very minute double star, which
« is not far from the planet, I see a very faint ray, like a ring
“ crossing the planet, over the centre. This appearance is of
“ an equal length on both sides, so that I strongly suspect it
“ to be a ring. There is, however, a possibility of its being an
“ imperfection in the speculum, owing to some slight scratch :
“ I shall take its position, and afterwards turn the speculum on
“ its axis.”

“ 8h 39'. Position of the supposed ring fromN.P. to S.F.”

« 9h 56'. I have turned the speculum one quadrant round ;

' 44 but the appearance of the very faint ray continues where it

Satellites of the Georgium Sidus , &c. 6*9

« was before, so that the defect is not in the speculum, nor is
« it in the eye-glass. But still it is now also pretty evident
“ that it arises from some external cause ; for it is now in the
“ same situation, with regard to the tube, in which it was 3-J
“ hours ago : whereas, the parallel is differently situated, and
“ the ring, of course, ought to be so too.”

“ March 5, 1792. I viewed the Georgian planet with a newly
“ polished speculum, of an excellent figure. It shewed the pla-
“ net very well defined, and without any suspicion of a ring.
“ I viewed it successively with 240, 300, 480, 6 00, 800, 1200,
“ and 2400 ; all which powers my speculum bore with great dis-
“ tinctness. I am pretty well convinced that the disk is flat—
“ tened.” The moon was pretty near the planet.

“ Dec. 4, 1793. 7-feet reflector, power 287. The Georgian
tc planet is not so well defined as, from the extraordinary dis-
“ tinctness of my present 7-feet telescope, it ought to be. There
“ is a suspicion of some apparatus about the planet.”

“ Feb. 26, 1794. 20-feet reflector, power 480. The planet
“ seems to be a little lengthened out, in the direction of the
“ longer axis of the satellites' orbits.”

“ April 21, 1795. 10-feet reflector, power 400. The telescope
“ adjusted to a neighbouring star, so as to make it perfectly
“ round. The disk of the planet seems to be a little elliptical.
“ With 600, also adjusted upon the neighbouring star, the disk
“ still seems elliptical.”

Remarks upon the foregoing Observations.

With regard to the phenomena which gave rise to the sus-
picion of one or more rings, it must be noticed, that few spe-
cula or object-glasses are so very perfect as not to be affected

7° Dr. Herschel’s Discovery of four additional

with some rays or inequalities, when high powers are used,
and the object to be viewed is very minute. It seems, how-
ever, from the observations of March i6‘, 1789, and Feb. 2 6,
1792, that the cause of deception, in this case, must be looked
for elsewhere. It has often happened, that the situation of the
eye-glass, being on one side of the tube, which brings the ob-
server close to the mouth of it, has occasioned a visible defect
in the view of a very minute object, when proper care has not
been taken to keep out of the way ; especially when the wind
is in such a quarter as to come from the observer across the
telescope. The direction of a current of air alone may also
affect vision. Without, however, entering further into the dis-
cussion of a subject that must be attended with uncertainty, I
will only add, that the observation of the 26th seems to be very
decisive against the existence of a ring. When the surmises
arose at first, I thought it proper to suppose, that a ring might
be in such a situation as to render it almost invisible ; and that,
consequently, observations should not be given up, till a suffi-
cient time had elapsed to obtain a better view of such a sup-
posed ring, by a removal of the planet from its node. This
has now sufficiently been obtained in the course of ten years ;
for, let the node of the ring have been in any situation what-
soever, provided it kept to the same, we must by this time
have had a pretty good view of the ring itself. Placing there-
fore great confidence on the observation of March 5, 1792,
supported by my late views of the planet, I venture to affirm,
that it has no ring in the least resembling that, or rather those,
of Saturn.

The flattening of the poles of the planet seems to be suffi-
ciently ascertained by many observations. The 7-feet, the

Satellites of the Georgium Sidus, &c. 7l

10-feet, and the 20-feet instruments, equally confirm it; and
the direction pointed out Feb. 2 6, 1794, seems to be conform-
able to the analogies that may be drawn from the situation of
the equator of Saturn, and of Jupiter.

This being admitted, we may without hesitation conclude,
that the Georgian planet also has a rotation upon its axis, of a
considerable degree of velocity.

Reports and Observations relating to the Light and Size of the
Georgian Satellites , and to their vanishing at certain Dis-
tances from the Planet.

Jan. 14, 1787. A star was put down, as a supposed very faint
satellite; but, on the 17th, the planet being removed, it ap-
peared nearly as bright as two considerable stars that had also
been noted.

“ Jan. 17. The 1st satellite is the smallest in appearance.”

“ Jan. 24. The 2d satellite is brighter than the first.”

“ Feb. 9, 1787. The 1st satellite is larger than the second.”
Feb. 10. The planet was supposed to go to a triangle of
pretty bright stars. The 1 ith it was between them, and the
stars of the triangle were so dim, that, had they not been seen
before, they might have been supposed to be satellites.

“ Sept. 19, 1787. 4h 24'. I can still see the satellites, though
“ daylight is already very strong : they are fainter than the
" faintest of Saturn’s satellites.” *

“ Feb. 22, 1791. I cannot perceive the 1st satellite, probably
u owing to its nearness to the planet.”

“ March 2, 1791. The 1st satellite is hardly to be seen ; I

* Five satellites of Saturn were only known at that time.

72 Dr. Herschel’s Discovery of four additional

“ have however had several perfect glimpses of it. It seems to
“ be about the most contracted part of its orbit.”

March 6. The supposed 3d and 4th satellites of March 5th
were imagined to have been gone from their former places;
but were seen the 7th, with this memorandum. “ I mistook
them last night for other stars, they being so large that I did
not know them again.”

“ March 9. The 2d satellite is nearer the planet than the
“ first, and on that account appears smaller.”

“ Dec. 9, 1791. I do not perceive the 1st satellite.”

“ Feb. 13, 1792. 6h The 3d supposed satellite of last
“ night is a considerable star ; not much less than b”

When the supposed third was pointed out the night before,
it is said to be smaller than the 1st and 2d satellites. By the
figure, it did not exceed the distance of the 2d ; and b is called
a pretty large star.

Feib. 20, 1792. The 2d satellite, being at a great distance, was
mistaken for a pretty large star, till about four hours after,
when its motion along with the planet was perceived.

“ Feb. 21, 1792. 7h36'. I cannot see the 2d satellite. By
“ calculation, it should be about 8°, 6 S.F. and I suspect it to
“ be there, but cannot get the least assurance.”

“ March 15, 1792. I cannot see the 1st satellite with 300;
nor with 480 ; nor with 600.”

“ March 19. 8h 35'. I cannot see the 2d satellite with 300.
“ With 480 I see it very well. I see it also with 800 ; and very
“ well with, 1200. With 2400 and 4800 the satellite cannot be
“ seen ; but there seems to be a whitish haziness coming on.”
March 4, 1794. The 1st satellite could not be seen.

73

Satellites of the Georgium Sidns , &c.

March 7. The 1st was invisible.

March 17. Both 1st and 2d were invisible.

March si. The 1st was invisible, though looked for with
all the powers of the instrument.

March 22. The 2d was hardly visible.

March 23. The 2d was not to be seen.

March 2 b. The 1st was but just visible.

March 5, 1796. The 2d was invisible.

April 4, 1796. The 1st was invisible.

“ March 17, 1797. Power boo. Neither of the satellites are
“ visible to-night ; with 300 I cannot see them. The night is
a very beautiful, and I have a field bar to hide the planet ; but,
“ notwithstanding this, I cannot see either of the satellites."

March si. The 1st satellite was invisible.

March 23. The 2d was invisible. The 1st could not be seen
immediately, but, having been informed where exactly to look
for it, according to my calculation of its place, it was perceived;
and with boo seen very well.

March 23. Both satellites were invisible.

Remarks on the foregoing Observations.

From the observations of Jan. 14, Feb. 10, March b, 178 7,
and Feb. 13, 1792, it appears, that all very small stars, when
they come near the planet, lose much of their lustre. Indeed,
every observation that has been recorded before, of supposed
satellites that have been proved to be stars afterwards, has fully
confirmed this circumstance; for they were always found to be
considerable stars, and their being mistaken for satellites was
owing to their loss of light when near the planet. This would
hardly deserve notice, as it is well known that a superior light

MDCCXCVIII. L

74 Dr. Herschel’s Discovery of four additional

will obstruct an inferior one ; but some circumstances which
attend the operation of the affections of light upon the eye,
when objects are very faint, are so remarkable, that they must
not be passed over in silence.

After having been used to follow up the satellites of Saturn
and Jupiter, to the very margin of their planets, so as even to
measure the apparent diameter of one of Jupiter's satellites by
its entrance on the disk,* I was in hopes that a similar oppor-
tunity would soon have offered with the Georgian satellites :
not indeed to measure the satellites, but to measure the planet
itself, by means of the passage of the satellite over its disk.
I expected also to have settled the epochs of the satellites,
from their conjunctions and oppositions, with more accuracy
than I have yet been able to do, from their various positions
in other parts of their orbits. A disappointment of obtaining
these capital advantages deserves to have its cause investigated ;
but, first of all, let us cast a look upon the observations.

The satellites, we may remark, become regularly invisible,
when, after their elongation, they arrive to certain distances
from the planet. In order to find what these distances are, we
will take the first observation of this kind, as an example.

Feb. 22, 1791, the first satellite could not be seen. Now, by
my lately constructed tables, its longitude from the apogee, at
the time of observation, was 204,5 degrees ; that is, 24,5 de-
grees from the most contracted part of its orbit, on the side that
is turned to us, which, as its opposite is called the apogee, I
shall call the perigee. By my tables also for the same day, we
have the distance of the apogee from the planet, which is ,60 ;
supposing the greatest elongation distance to be 1. This

* See Phil. Trans, for 1797, Part II. page 335.

Satellites of the Georgium Sidus , See. 75

being given, we may find an easy method of ascertaining the
distance of the satellite, when it is near the apogee or perigee :
for it will be sufficiently true for our purpose to use the fol-
lowing analogy. Cosine of the distance of the satellite from
the apogee or perigee is to the apogee distance from the planet,
as the greatest elongation is to the distance of the satellite from
the planet. When the ellipsis is very open, this theorem will
only hold good in moderate distances from the apogee or peri-
gee ; but, when it is a good deal flattened, it will not be con-
siderably out in more distant situations : and it will also be
sufficiently accurate to take the natural cosine from the tables
to two places of decimals only. When this is applied to our
present instance, we have ,91 for the natural cosine of 24,5
degrees ; and the distance of the satellite from the planet will
come out ,6 x 33 = 21", 8.

,gi

By this method, it appears that the satellite, when it could
not be seen, was nearly 22" from the planet.

We must not however conclude, that this is the given dis-
tance at which it will always vanish. For instance, the same
satellite, though hardly to be seen, was however not quite in-
visible March 2, 1791. Its distance from the planet, computed
as before, was then only -6-* 33-. = 19", 8 .

The clearness of the atmosphere, and other favourable cir-
cumstances, must certainly have great influence in observations
of very faint objects ; therefore, a computation of all the ob-
servations where the satellites were not seen, as well as a few
others where they were seen, when pretty near the apogee or
perigee, will be the surest way of settling the fact. The result
of these computations is thus.

L 2

7 6 Dr. Herschei/s Discovery of four additional

First satellite invisible.

Second satellite invisible.

1791. Feb. 22

at

21,8

1792.

Feb. 21

at

*3>3

Dec. 19

at

16,9

’794.-

March 17

at

20,7

1792. March 15

at

18,4

March 23

at

17.9

1794. March 4

at

00

1796.

March 5

at

9’3

March 7

at

12,5

March 17

at

6,3

March 17

at

17,0

March 23

at

6, 2

March 21

at

*5>5

March 25

at

8.7

April 4

at

8,5

1797. March 17

at

4>8

March 21

at

4>6

March 25

at

4>8

First satellite visible.

Second satellite visible.

1791. March 2 at

i^8

1794-

March 22

at

17*5

1794. Feb. 2 6

at

14A

Thus, having the observations and calculated distances under -
our inspection, we find that both the satellites became always
invisible when they were near the planet : that the ist was ge-
nerally lost when it came within i8/; of the planet, and the 2d
at the distance of about 20" In very uncommon and beautiful
nights, the 1st has once been seen at 13", 8, and the 2d at 17", 3;
but at no time have they been visible when nearer the planet,

I shall now endeavour to investigate the cause which can

Satellites of the Georgium Sidns, &c. 77

render small stars and satellites invisible at so great a distance
as 18 or 20".

A dense atmosphere of the planet would account for the de-
falcation of light sufficiently, were it not proved that the satellites
are equally lost, whether they are in the nearest half of their
orbits, or in that which is farthest from us. But, as a satellite
cannot be eclipsed by an atmosphere that is behind it, a sur-
mise of this kind cannot be entertained. Let us then turn our
view to light itself, and see whether certain affections between
bright and very bright objects, contrasted with others that
take place between faint and very faint ones, will not explain
the phenomena of vanishing satellites.

The light of Jupiter or Saturn, for instance, on account of
its brilliancy, is diffused, almost equally, over a space of several
minutes all around these planets. Their satellites also, having
a great share of brightness, and moving in a sphere that is
strongly illuminated, cannot be much affected by their various
distances from the planets. The case then is, that they have
much light to lose, and comparatively lose but little.

The Georgian planet, on the contrary, is very faint ; and the
influence of its feeble light cannot extend far, with any degree
of equality. This enables us to see the faintest objects, even
when they are only a minute or two removed from it. The
satellites of this planet are very nearly the dimmest objects that
can be seen in the heavens ; so that they cannot bear any con-
siderable diminution of their light, by a contrast with a more
luminous object, without becoming invisible. If then the sphere
of illumination of our new planet be limited to 18 or 20", we
may fully account for the loss of the satellites when they come

78 Dr. Herschei/s Discovery of four additional

within its reach ; for they have very little light to lose, and lose
it pretty suddenly.

This contrast, therefore, between the condition of the Georgian
satellites and those of the brighter planets, seems to be suffi-
cient to account for the phenomenon of their becoming invi-
sible.

We may avail ourselves of the observations that relate to the
distances at which the satellites vanish, to determine their re-
lative brightness. The 2d satellite appears generally brighter
than the ist; but, as the former is usually lost farther from the
planet than the latter, we may admit the ist satellite to be
rather brighter than the 2d. This seems to be confirmed by
the observation of March 9, 1791 ; where the 2d appeared to be
smaller than the 1st, though the latter was only 25" from the
planet, while the other was 30", 8.

The first of the new satellites will hardly ever be seen other-
wise than about its greatest elongations, but cannot be much
inferior in brightness to the other two ; and, if any more in-
terior satellites should exist, we shall probably not obtain a sight
of them; for the same reason that the inhabitants of the Georgian
planet perhaps never can discover the existence of our earth,
Venus, and Mercury.

The 2d new or intermediate satellite is considerably smaller
than the 1st and 2d old satellites. The two exterior, or 5th and
6th satellites, are the smallest of all, and must chiefly be looked
for in their greatest elongations.

Periodical Revolutions of the new Satellites.

It may be some satisfaction to know what time the four

Thilos. TransMX) C CX C V1L1, Tab. Jlp.

b

JO.

orj, fc .

79

Satellites of the Georgium Sidus, &c.

additional satellites probably employ in revolving round their
planet. Now, as this can only be ascertained with accuracy by
many observations, we must of course remain in suspense, till a
series of them can be properly instituted. But, in the mean
time, we may admit the distance of the interior satellite to be
2 5", 5, as our calculation of the estimation of March 5, 1794,
gives it; and from this we compute that its periodical revolution
will be 5 days, 2 1 hours, 25 minutes.

If we place the intermediate satellite at an equal distance
between the two old ones, or at 38", 57, its period will be 10
days, 23 hours, 4 minutes.

By the figure of Feb. 9, 1790, it seems that the nearest ex-
terior satellite is about double the distance of the farthest old
one ; hence, its periodical time is found to be 38 days, 1 hour,
49 minutes.

The most distant satellite, according to the calculation of
the observation of Feb. 28, 1794, is full four times as far from
the planet as the old 2d satellite ; it will therefore take at least
107 days, 16 hours, 40 minutes, to complete one revolution.

It will hardly be necessary to add, that the accuracy of these
periods depends entirely upon the truth of the assumed dis-
tances ; some considerable difference, therefore, may be expect-
ed, when observations shall furnish us with proper data for
more accurate determinations.

Slough, near Windsor,
September i, 1797.

C 80 3

IV. An Inquiry concerning the Source of the Heat which is
excited by Friction . By Benjamin Count of Rumford,
F.R.S. M.R.I.A.

Read January 25, 1798.

It frequently happens, that in the ordinary affairs and occupa-
tions of life, opportunities present themselves of contemplating
some of the most curious operations of nature ; and very interest-
ing philosophical experiments might often be made, almost with-
out trouble or expence, by means of machinery contrived for
the mere mechanical purposes of the arts and manufactures.

I have frequently had occasion to make this observation ; and
am persuaded, that a habit of keeping the eyes open to every
thing that is going on in the ordinary course of the business
of life has oftener led, as it were by accident, or in the play-
ful excursions of the imagination, put into action by contem-
plating the most common appearances, to useful doubts, and
sensible schemes for investigation and improvement, than all
the more intense meditations of philosophers, in the hours ex-
pressly set apart for study.

It was by accident that I was led to make the experiments
of which I am about to give an account ; and, though they are
not perhaps of sufficient importance to merit so formal an in-
troduction, I cannot help flattering myself that they will be
thought curious in several respects, and worthy of the honour
of being made known to the Royal Society.

Count Rumford's Inquiry , See. 81

Being engaged, lately, in superintending the boring of can-
non, in the workshops of the military arsenal at Munich, I
was struck with the very considerable degree of heat which a
brass gun acquires, in a short time, in being bored ; and with
the still more intense heat (much greater than that of boiling
water, as I found by experiment,) of the metallic chips sepa-
rated from it by the borer.

The more I meditated on these phenomena, the more they
appeared to me to be curious and interesting. A thorough in-
vestigation of them seemed even to bid fair to give a farther
insight into the hidden nature of heat; and to enable us to
form some reasonable conjectures respecting the existence, or
non-existence, of an igneous fluid: a subject on which the opi-
nions of philosophers have, in all ages, been much divided

In order that the Society may have clear and distinct ideas
of the speculations and reasonings to which these appearances
gave rise in my mind, and also of the specific objects of phi-
losophical investigation they suggested to me, I must beg leave
to state them at some length, and in such manner as I shall
think best suited to answer this purpose.

, From zvhence comes the heat actually produced in the mecha-
nical operation above mentioned ?

Is it furnished by the metallic chips which are separated by
the borer from the solid mass of metal ?

If this were the case, then, according to the modern doc-
trines of latent heat, and of caloric, the capacity for heat of the
parts of the metal, so reduced to chips, ought not only to be
changed, but the change undergone by them should be suffi-
ciently great to account for all the heat produced.

But no such change had taken place; for I found, upon
>idccxcviii. M

8s Count Rum ford’s Inquiry concerning

taking equal quantities, by weight, of these chips, and of thin
slips of the same block of metal separated by means of a fine
saw, and putting them, at the same temperature, (that of boil-
ing water,) into equal quantities of cold water, (that is to say,
at the temperature of 59°-^ F.) the portion of water into which
the chips were put was not, to all appearance, heated either
less or more than the other portion, in which the slips of metal
were put.

This experiment being repeated several times, the results
were always so nearly the same, that I could not determine
whether any, or what change, had been produced in the metal,
in regard to its capacity for heat , by being reduced to chips by
the borer.*

From hence it is evident, that the heat produced could not

..7 ►* ^ *

* As these experiments are important, it may perhaps be agreeable to the Society
to be made acquainted with them in their details.

One of them was as follows :

' v To 4590 grains of water, at the temperature of 59°! F. (an allowance as compen-
sation, reckoned in water, for the capacity for heat of the containing cylindrical tin
■'2 ’2 ^ vessel, being included,) were added 101 6\ grains of gun-metal in thin slips, separated

from the gun by means of a fine saw, being at the temperature of 2 io° F. When they
had remained together 1 minute, and had been well stirred about, by means of a small
rod of light wood, the heat of the mixture was found to be — 63°.

From this experiment, the specific heat of the metal, calculated according to the rule
given by Dr. Crawford, turns out to be = 0.1100, that of water being = 1.0000.

An experiment was afterwards made with the metallic chips, as follows :

To the same quantity of water as was used in the experiment above mentioned, at
the same temperature, (viz. 59°|>) and in the same cylindrical tin vessel, were now
42 ' ; put 10 1 61 grains of metallic chips of gun-metal, bored out of the same gun from

which the slips used in the foregoing experiment were taken, and at the same tempe-
rature (2100). The heat of the mixture, at the end of 1 minute, was just 63°, as
before; consequently the specific heat of these metallic chips was = 0.1100. Each
of the above experiments was repeated 3 times, and always with nearly the same
results.

the Source of the Heat excited by Friction. 83

possibly have been furnished at the expence of the latent heat
of the metallic chips. But, not being willing to rest satisfied
with these trials, however conclusive they appeared to me to
be, I had recourse to the following still more decisive experi-
ment.

Taking a cannon, (a brass six-pounder, ) cast solid, and rough
as it came from the foundry, (see fig. 1. Tab. IV.) and fixing
it (horizontally) in the machine used for boring, and at the
same time finishing the outside of the cannon by turning, (see
fig. 2. ) I caused its extremity to be cut off ; and, by turning
down the metal in that part, a solid cylinder was formed, *]—
inches in diameter, and 9^ inches long; which, when finished,
remained joined to the rest of the metal (that which, properly
speaking, constituted the cannon, ) by a small cylindrical neck,
only 2| inches in diameter, and 3-^ inches long.

This short cylinder, which was supported in its horizontal
position, and turned round its axis, by means of the neck by
which it remained united to the cannon, was now bored with
the horizontal borer used in boring cannon ; but its bore, which
was 3.7 inches in diameter, instead of being continued through
its whole length (9.8 inches) was only 7.2 inches in length;
so that a solid bottom was left to this hollow cylinder, which
bottom was 2 .6 inches in thickness.

This cavity is represented by dotted lines in fig. 2 ; as also

in fig. 3. where the cylinder is represented on an enlarged
scale.

This cylinder being designed for the express purpose of ge-
nerating heat by friction , by having a blunt borer forced against
its solid bottom at the same time that it should be turned round
its axis by the force of horses, in order that the heat accumu-

M 2

3'/

b

j'sh

3' OS 3

8 4 / 1 to '

2 V

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o-3osb

6J,

J/ff £.#<Ul

22 3 '23$ <Vt'
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84 Count Rumford's Inquiry concerning

lated in the cylinder might from time to time be measured, a.
small round hole, (see d , e , fig. 3.) 0.37 of an inch only in dia-
meter, and 4.2 inches in depth, for the purpose of introducing
a small cylindrical mercurial thermometer, was made in it, on
one side, in a direction perpendicular to the axis of the cylinder,
and ending in the middle of the solid part of the metal which
formed the bottom of its bore.

The solid contents of this hollow cylinder, exclusive of the
cylindrical neck by which it remained united to the cannon,
were 385 f cubic inches, English measure; and it weighed
113.131b. avoirdupois : as I found, on weighing it at the end
of the course of experiments made with it, and after it had
been separated from the cannon with which, during the expe-
riments, it remained connected, *

Experiment No. 1.

This experiment was made in order to ascertain how much
heat was actually generated by friction, when, a blunt steel
borer being so forcibly shoved (by means of a strong screw)
against the bottom of the bore of the cylinder, that the pressure

* For fear I should be suspected of prodigality in the prosecution of my philoso-
phical researches, I think it necessary to inform the Society, that the cannon 1 made
use of in this experiment was not sacrificed to it. The short hollow cylinder which
was formed at the end of it, was turned out of a cylindrical mass of metal, about 2 feet
in length, projecting beyond the muzzle of the gun, called in the German language
the verlorner kopf, (the head of the cannon to be thrown away,) and which is
represented in fig. 1 .

This additional projection, which is cut off before the gun is bored, is always cast
with it, in order that, by means of the pressure of its weight on the metal in the lower
part of the mould, during the time it is cooling, the gun may be the more compact
in the neighbourhood of the muzzle; where, without this precaution, the metal would
be apt to be porous, or full of honeycombs.

the Source of the Heat excited by Friction. 85

against it was equal to the weight of about loooolb. avoirdu-
pois, the cylinder was turned round on its axis, (by the force
of horses,) at the rate of about 32 times in a minute.

This machinery, as it was put together for the experiment,
is represented by fig. 2. W is a strong horizontal iron bar,
connected with proper machinery carried round by horses, by
means of which the cannon was made to turn round its axis.

To prevent, as far as possible, the loss of any part of
the heat that was generated in the experiment, the cylinder
was well covered up with a fit coating of thick and warm
flannel, which was carefully wrapped round it, and defended
it on every side from the cold air of the atmosphere. This
covering is not represented in the drawing of the apparatus,
fig. 2.

I ought to mention, that the borer was a flat piece of hardened
steel, 0.63 of an inch thick, 4 inches long, and nearly as wide
as the cavity of the bore of the cylinder, namely, 3^- inches.
Its corners were rounded off at its end, so as to make it fit
the hollow bottom of the bore ; and it was firmly fastened to
the iron bar (m) which kept it in its place. The area of the
surface by which its end was in contact with the bottom of the
bore of the cylinder was nearly 2\ inches. This borer, which
is distinguished by the letter n , is represented in most of the
figures.

At the beginning of the experiment, the temperature of the
air in the shade, as also that of the cylinder, was just 6 o° F.

At the end of 30 minutes, when the cylinder had made 960
revolutions about its axis, the horses being stopped, a cylin-
drical mercurial thermometer, whose bulb was t3~q of an inch
in diameter, and 3^ inches in length, was introduced into the

Qeelii

Oz

/(DO 600 •

1/

8 6 Count Rumford’s Inquiry concerning

hole made to receive it, in the side of the cylinder, when the
mercury rose almost instantly to 130°.

Though the heat could not be supposed to be quite equally
distributed in every part of the cylinder, yet, as the length of
the bulb of the thermometer was such that it extended from
the axis of the cylinder to near its surface, the heat indicated
by it could not be very different from that of the mean tempera-
ture of the cylinder ; and it was on this account that a thermo-
meter of that particular form was chosen for this experiment.

To see how fast the heat escaped out of the cylinder, (in or-
der to be able to make a probable conjecture respecting the
quantity given off by it, during the time the heat generated
by the friction was accumulating,) the machinery standing still,
I suffered the thermometer to remain in its place near three
quarters of an hour, observing and noting down, at small in-
tervals of time, the height of the temperature indicated by it.

Thus at the end of The heat’ as shown

X HUS, UL LllC CHU VI the thermometer, was

4 minutes - 126°

after 5 minutes, always reckoning from the
first observation,
at the end of 7 minutes -

12 -

14

16

20

24

28

31

3 7i

and when 41 minutes had elapsed

125

1230

120°

119°

1180

11 6°

1*5°

114°

1130

112°

111°

110°

87

the Source of the Heat excited by Friction .

Having taken away the borer, I now removed the metallic
dust, or rather scaly matter, which had been detached from the
bottom of the cylinder by the blunt steel borer, in this experi-
ment ; and, having carefully weighed it, I found its weight to
be 837 grains Troy.

Is it possible that the very considerable quantity of heat that
was produced in this experiment (a quantity which actually
raised the temperature of above 113 lb. of gun-metal at least
70 degrees of Fahrenheit's thermometer, and which, of course,
would have been capable of melting 6^-lb. of ice, or of causing fos] ‘
near 51b. of ice-cold water to boil,) could have been furnished
by so inconsiderable a quantity of metallic dust ? and this merely
in consequence of a change of its capacity for heat ?

As the weight of this dust (837 grains Troy) amounted to
no more than ^-^th part of that of the cylinder, it must have
lost no less than 94,8 degrees of heat, to have been able to have
raised the temperature of the cylinder 1 degree; and conse-
quently it must have given off 66360 degrees of heat, to have
produced the effects which were actually found to have been
produced in the experiment !

But, without insisting on the improbability of this supposi-
tion, we have only to recollect, that from the results of actual
and decisive experiments, made for the express purpose of as-
certaining that fact, the capacity for heat, of the metal of which
great guns are cast, is not sensibly changed by being reduced to
the form of metallic chips, in the operation of boring cannon ;
and there does not seem to be any reason to think that it can
be much changed, if it be changed at all, in being reduced to
much smaller pieces, by means of a borer that is less sharp.

i.

/ y/3 / /

/y/zi*

88“ Count Rumford's Inquiry concerning

If the heat, or any considerable part of it, were produced in
consequence of a change in the capacity for heat of a part of
the metal of the cylinder, as such change could only be super-
ficial, the cylinder would by degrees be exhausted ; or the
quantities of heat produced, in any given short space of time,
would be found to diminish gradually, in successive experi-
ments. To find out if this really happened or not, I repeated
the last-mentioned experiment several times, with the utmost
care ; but l did not discover the smallest sign of exhaustion
in the metal, notwithstanding the large quantities of heat ac-
tually given off.

Finding so much reason to conclude, that the heat generated
in these experiments, or excited , as I would rather choose to
express it, was not furnished at the expence of the latent heat
or combined caloric of the metal, I pushed my inquiries a step
farther, and endeavoured to find out whether the air did, or
did not, contribute any thing in the generation of it.

'Experiment No. 2.

As the bore of the cylinder was cylindrical, and as the iron
bar, (m,) to the end of which the blunt steel borer was fixed,
Was square, the air had free access to the inside of the bore,
and even to the bottom of it, where the friction took place by
which the heat was excited.

As neither the metallic chips produced in the ordinary course
Of the operation of boring brass cannon, nor the finer scaly
particles produced in the last mentioned experiments by the
friction of the blunt borer, showed any signs of calcination, I
4lid not see how the air could possibly have been the cause

the Source of the Heat excited by Friction. 8cj

of the heat that was produced ; but, in an investigation of this
kind, I thought that no pains should be spared to clear away
the rubbish, and leave the subject as naked and open to inspec-
tion as possible.

In order, by one decisive experiment, to determine whether
the air of the atmosphere had any part, or not, in the genera-
tion of the heat, I contrived to repeat the experiment, under cir-
cumstances in which it was evidently impossible for it to produce
any effect whatever. By means of a piston exactly fitted to the
mouth of the bore of the cylinder, through the middle of which
piston the square iron bar, to the end of which the blunt steel
borer was fixed, passed in a square hole made perfectly air-
tight, the access of the external air, to the inside of the bore of
the cylinder, was effectually prevented. (In fig. 3. this piston
0 P ) ^ seen jin its place; it is likewise shown in fig. 7 and 8.)

I did not find, however, by this experiment, that the exclu-
sion of the air diminished, in the smallest degree, the quantity
of heat excited by the friction.

rheie still remained one doubt, which, though it appeared
to me to be so slight as hardly to deserve any attention, I was
however desirous to remove. The piston which closed the
mouth of the bore of the cylinder, in order that it might be
air-tight, was fitted into it with so much nicety, by means of
its collars of leather, and pressed against it with so much force,
that, notwithstanding its being oiled, it occasioned a consider-
able degree of friction, when the hollow cylinder was turned
round its axis. Was not the heat produced, or at least some
part of it, occasioned by this friction of the piston ? and, as
the external air had free access to the extremity of the bore,
where it came in contact with the piston, is it not possible that

MDCCXC VIII. N

go Count Rumford's Inquiry concerning

this air may have had some share in the generation of the heat
produced ?

Experiment No. 3.

A quadrangular oblong deal box, (see fig. 4.) water-tight,
* \ , y '0 1 English inches long, 9/^ inches wide, and g-p^ inches deep,

(measured in the clear,) being provided, with holes or slits in
the middle of each of its ends, just large enough to receive, the
one, the square iron rod to the end of which the blunt steel
borer was fastened, the other, the small cylindrical neck which
joined the hollow cylinder to the cannon; when this box (which
was occasionally closed above, by a wooden cover or lid moving
on hinges,) was put into its place; that is to say, when, by
means of the two vertical openings or slits in its two ends, (the
upper parts of which openings were occasionally closed, by means
of narrow pieces of wood sliding in vertical grooves,) the box
( g , h , i, k, fig. 3. ) was fixed to the machinery, in such a manner
that its bottom (z, k,) being in the plane of the horizon, its axis
coincided with the axis of the hollow metallic cylinder ; it is
evident, from the description, that the hollow metallic cylinder
would occupy the middle of the box, without touching it on either
side, (as it is represented in fig. 3. ;) and that, on pouring water
into the box, and filling it to the brim, the cylinder would be
completely covered, and surrounded on every side, by that fluid.
And farther, as the box was held fast by the strong square iron
rod, (m,) which passed, in a square hole , in the centre of one of
its ends, ( a , fig. 4.) while the round or cylindrical neck, which
joined the hollow cylinder to the end of the cannon, could turn
round freelv on its axis in the round hole in the centre of the
other end of it, it is evident that the machinery could be put

9*

the Source of the Heat excited by Friction.

in motion, without the least danger of forcing the box out of
its place, throwing the water out of it, or deranging any part
of the apparatus.

Every thing being ready, I proceeded to make the experiment
I had projected, in the following manner.

The hollow cylinder having been previously cleaned out, and
the inside of its bore wiped with a clean towel till it was quite
dry, the square iron bar, with the blunt steel borer fixed to the
end of it, was put into its place ; the mouth of the bore of the
cylinder being closed at the same time, by means of the circu-
lar piston, through the centre of which the iron bar passed.

This being done, the box was put in its place, and the join-
ings of the iron rod, and of the neck of the cylinder, with the
two ends of the box, having been made water-tight, by means
of collars of oiled leather, the box was filled with cold water,

{viz. at the temperature of 6o°. ) and the machine was put in
motion.

The result of this beautiful experiment was very striking,
and the pleasure it afforded me amply repaid me for all the
trouble I had had, in contriving and arranging the complicated
machinery used in making it

The cylinder, revolving at the rate of about 32 times in a
minute, had been in motion but a short time, when I perceived,
by putting my hand into the water, and touching the outside
of the cylinder, that heat was generated ; and it was not long

before the water which surrounded the cylinder began to be
sensibly warm.

At the end of 1 hour I found, by plunging a thermometer into
the water in the box, (the quantity of which fluid amounted to
i877lb. avoirdupois, or 2* wine gallons,) that its temperature 300 7 , 7: ; 7 H<»<

N 2

9 2 Count Rumforp’s Inquiry concerning

had been raised no less than 4 y degrees ; being now 1 07° of
Fahrenheit's scale.

When 30 minutes more had elapsed, or 1 hour and 30 mi-
nutes after the machinery had been put in motion, the heat of
the water in the box was 1420.

At the end of 2 hours, reckoning from the beginning of the
experiment, the temperature of the water was found to be raised
to 178°.

At 2 hours 20 minutes it was at 200°; and at 2 hours 30 mi-
nutes it ACTUALLY BOILED !

It would be difficult to describe the surprise and astonish-
ment expressed in the countenances of the by-standers, on
seeing so large a quantity of cold water heated, and actually
made to boil, without any fire.

Though there was, in fact, nothing that could justly be con-
sidered as surprising in this event, yet I acknowledge fairly
that it afforded me a degree of childish pleasure, which, were I
ambitious of the reputation of a grave philosopher , I ought most
certainly rather to hide than to discover.

The quantity of heat excited and accumulated in this expe-
riment was very considerable ; for, not only the water in the
box, but also the box itself, (which weighed i5^1b.) and the
hollow metallic cylinder, and that part of the iron bar which,
being situated within the cavity of the box, was immersed in
the water, were heated 150 degrees of Fahrenheit's scale;
viz. from 6o° (which was the temperature of the water, and of
the machinery, at the beginning of the experiment,) to 210°*
the heat of boiling water at Munich.

The total quantity of heat generated may be estimated with
some considerable degree of precision, as follows :

93

the Source of the Heat excited by Friction.

,, i , , , Quantity of ice-cold water which,

Of the heat excited there appears to with the given quantity of heat,

‘ might have been heated 180 de-

have been actually accumulated, grees, or made to boii.

J In avoirdupois weight.

In the water contained in the wooden box,

1 8|db.'avoirdupois, heated 150 degrees, name- ^

ly, from 6o° to 210° F. - - 15.2

In 113.131b. of gun-metal, (the hollow cylinder,)
heated 150 degrees ; and, as the capacity for heat of
this metal is to that of water as 0.1100 to 1.0000, this
quantity of heat would have heated 12^ lb. of water
the same number of degrees - 10.37

In 36.75 cubic inches of iron, (being that part of
the iron bar to which the borer was fixed which en-
tered the box,) heated 150 degrees; which may be
reckoned equal in capacity for heat to i.2ilb. of water 1.01
N. B. No estimate is here made of the heat accu-
mulated in the wooden box, nor of that dispersed
during the experiment.

Total quantity of ice-cold water which, with the heat
actually generated by friction, and accumulated in 2

hours and 30 minutes, might have been heated 1 80 de-

grees, or made to boil - - - 26.58

From the knowledge of the quantity of heat actually pro-
duced in the foregoing experiment, and of the time in which it
was generated, we are enabled to ascertain the velocity of its
production , and to determine how large a fire must have been,
or how much fuel must have been consumed, in order that, in
burning equably, it should have produced by combustion the
same quantity of heat in the same time.

In one of Dr. Crawford's experiments, (see his Treatise
on Heat, p. 321,) 371b. 7 oz. Troy,== 181320 grains, of water,

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94 Count Rumford’s Inquiry concerning

»

were heated 2TY degrees of Fahrenheit’s thermometer, with
the heat generated in the combustion of 26 grains of wax.
This gives 382032 grains of water heated 1 degree with 2 6
grains of wax; or 1 grains °f water heated 1 degree, or
=— 81.631 grains heated 180 degrees, with the heat gene-
rated in the combustion of 1 grain of wax.

The quantity of ice-cold water which might have been heated
1 80 degrees, with the heat generated by friction in the before-
mentioned experiment, was found to be 26.581b. avoirdupois,
= 188060 grains; and, as 81.631 grains of ice-cold water re-
quire the heat generated in the combustion of 1 grain of wax, to
heat it 180 degrees, the former quantity of ice-cold water, namely
188060 grains, would require the combustion of no less than
2303.8 grains (= 4t8qOZ. Troy) of wax, to heat it 180 degrees.

As the experiment (No. 3.) in which the given quantity of
heat was generated by friction, lasted 2 hours and 30 minutes,
= 150 minutes, it is necessary, for the purpose of ascertaining
how many wax candles of any given size must burn together, in
order that in the combustion of them the given quantity of heat
may be generated in the given time, and consequently with the
same celerity as that with which the heat was generated by fric-
tion in the experiment, that the size of the candles should be
determined, and the quantity of wax consumed in a given time
by each candle, in burning equably, should be known.

Now I found by an experiment, made on purpose to finish
these computations, that when a good wax candle, of a mode-
rate size, -J of an inch in diameter, burns with a clear flame,
just 49 grains of wax are consumed in 30 minutes. Hence it
appears, that 245 grains of wax would be consumed by such
a candle in 150 minutes; and that, to burn the quantity of

95

the Source of the Heat excited by Friction.

wax (= 2303.8 grains) necessary to produce the quantity of
heat actually obtained by friction in the experiment in ques-
tion, and in the given time, (150 minutes,) nine candles , burn-
ing at once, would not be sufficient ; for, 9 multiplied into 245
(the number of grains consumed by each candle in 150 mi-
nutes) amounts to no more than 2205 grains; whereas the
quantity of wax necessary to be burnt, in order to produce the
given quantity of heat, was found to be 2303.8 grains.

From the result of these computations it appears, that the
quantity of heat produced equably, or in a continual stream,
(if I may use that expression,) by the friction of the blunt steel
borer against the bottom of the hollow metallic cylinder, in the
experiment under consideration, was greater than that produced
equably in the combustion of nine wax candles, each J of an
inch in diameter, all burning together, or at the same time,
with clear bright flames.

As the machinery used in this experiment could easily be
carried round by the force of one horse, (though, to render the
work lighter, two horses were actually employed in doing it,)
these computations show further how large a quantity of heat
might be produced, by proper mechanical contrivance, merely
by the strength of a horse, without either fire, light, combus-
tion, or chemical decomposition ; and, in a case of necessity,
the heat thus produced might be used in cooking victuals.

But no circumstances can be imagined, in which this method
of procuring heat would not be disadvantageous ; for, more
heat might be obtained by using the fodder necessary for the
support of a horse, as fuel.

As soon as the last mentioned experiment (No. 3.) was
finished, the water in the wooden box was let off* and the box

9«

Count Rumford's Inquiry concerning

removed ; and the borer being taken out of the cylinder,
the scaly metallic powder, which had been produced by the
friction of the borer against the bottom of the cylinder, was
collected, and, being carefully weighed, was found to weigh
4145 grains, or about m oz. Troy.

As this quantity was produced in 2-j hours, this gives 824
grains for the quantity produced in half an hour.

In the first experiment, which lasted only half an hour , the
quantity produced was 837 grains.

In the experiment No. 1, the quantity of heat generated, In
half an hour , was found to be equal to that which would be
required to heat 51b. avoirdupois of ice-cold water 180 degrees,
or cause it to boil.

According to the result of the experiment No. 3, the heat
generated in half an hour , would have caused 5.311b. of ice-cold
water to boil. But, in this last-mentioned experiment, the heat
generated being more effectually confined, less of it was lost ;
which accounts for the difference of the results of the two ex-
periments.

It remains for me to give an account of one experiment
more, which was made with this apparatus. I found by the
experiment No. 1. how much heat was generated when the
air had free access to the metallic surfaces which were rubbed
together. By the experiment No. 2, I found that the quan-
tity of heat generated was not sensibly diminished when the
free access of the air was prevented; and, by the result of
No. 3, it appeared that the generation of the heat was not
prevented, or retarded, by keeping the apparatus immersed in
water. But as, in this last-mentioned experiment, the water,
though it surrounded the hollow metallic cylinder on every

the Source of the Heat excited by Friction* g’j

side, externally, was not suffered to enter the cavity of its
bore, (being prevented by the piston,) and consequently did
not come into contact with the metallic surfaces where the
heat was generated; to see what effects would be produced
by giving the water free access to these surfaces, I now made
the

Experiment No. 4.

The piston which closed the end of the bore of the cylinder
being removed, the blunt borer and the cylinder were once
more put together; and the box being fixed in its place, and
filled with water, the machinery was again put in motion.

There was nothing in the result of this experiment that ren-
ders it necessary for me to be very particular in my account of
it. Heat was generated, as in the former experiments, and, to
all appearance, quite as rapidly ; and I have no doubt but the
water in the box would have been brought to boil, had the ex-
periment been continued as long as the last. The only circum-
stance that surprised me was, to find how little difference was
occasioned in the noise made by the borer in rubbing against
the bottom of the bore of the cylinder, by filling the bore with
water. This noise, which was very grating to the ear, and
sometimes almost insupportable, was, as nearly as I could
judge of it, quite as loud, and as disagreeable, when the sur-
faces rubbed together were wet with water, as when they were
in contact with air.

By meditating on the results of all these experiments, we
are naturally brought to that great question which has so

often been the subject of speculation among philosophers;
namely,

MDCCXCVIII. 0

98 Count Rum ford’s Inquiry concerning

What is heat ? — Is there any such thing as an igneous
fluid, ? — Is there any thing that can with propriety be called
caloric f

We have seen that a very considerable quantity of heat may
be excited in the friction of two metallic surfaces, and given
off in a constant stream or flux, in all directions , without inter-
ruption or intermission, and without any signs of diminution,
or exhaustion.

From whence came the heat which was continually given off
in this manner, in the foregoing experiments ? Was it furnished
by the small particles of metal, detached from the larger solid
masses, on their being rubbed together ? This, as we have al-
ready seen, could not possibly have been the case.

Was it furnished by the air ? This could not have been the
case; for, in three of the experiments, the machinery being kept
immersed in water, the access of the air of the atmosphere was
completely prevented.

Was it furnished by the water which surrounded the ma-
chinery ? That this could not have been the case is evident ;
first, because this water was continually receiving heat from
the machinery, and could not, at the same time, be giving to,
and receiving heat from, the same body ; and secondly , because
there was no chemical decomposition of any part of this water.
Had any such decomposition taken place, (which indeed could
not reasonably have been expected, ) one of its component elas-
tic fluids (most probably inflammable air) must, at the same
time, have been set at liberty, and, in making its escape into
the atmosphere, would have been detected ; but, though I fre-
quently examined the water, to see if any air bubbles rose
up through it, and had even made preparations for catching

99

the Source of the Heat excited by Friction,

them, in order to examine them, if any should appear, I could
perceive none; nor was there any sign of decomposition of
any kind whatever, or other chemical process, going on in
the water.

Is it possible that the heat could have been supplied by means
of the iron bar to the end of which the blunt steel borer was
fixed ? or by the small neck of gun-metal by which the hollow
cylinder was united to the cannon ? These suppositions appear
more improbable even than either of those before mentioned ;
for heat Was continually going off’ or out of the machinery , by
both these passages, during the whole time the experiment
lasted.

And, in reasoning on this subject, we must not forget to
consider that most remarkable circumstance, that the source
of the heat generated by friction, in these experiments, ap-
peared evidently to be inexhaustible.

It is hardly necessary to add, that any thing which any in-
sulated body, or system of bodies, can continue to furnish with-
out limitation , cannot possibly be a material substance ; and it
appears to me to be extremely difficult, if not quite impossible,
to form any distinct idea of any thing, capable of being excited,
and communicated, in the manner the heat was excited and
communicated in these experiments, except it be motion.

I am very far from pretending to know how, or by what
means, or mechanical contrivance, that particular kind of mo-
tion in bodies, which has been supposed to constitute heat, is
excited, continued, and propagated, and I shall not presume
to trouble the Society with mere conjectures ; particularly on
a subject which, during so many thousand years, the most

O 2

loo Count Rumford's Inquiry concerning

enlightened philosophers have endeavoured, but in vain, to
comprehend.

But, although the mechanism of heat should, in fact, be one
of those mysteries of nature which are beyond the reach of
human intelligence, this ought by no means to discourage us,
or even lessen our ardour, in our attempts to investigate the
laws of its operations. How far can we advance in any of the
paths which science has opened to us, before we find ourselves
enveloped in those thick mists which, on every side, bound the
horizon of the human intellect ? But how ample, and how in-
teresting, is the field that is given us to explore !

Nobody, surely, in his sober senses, has ever pretended to
understand the mechanism of gravitation ; and yet what sub-
lime discoveries was our immortal Newton enabled to make,
merely by the investigation of the laws of its action !

The effects produced in the world by the agency of heat,
are probably just as extensive, and quite as important, as
those which are owing to the tendency of the particles of
matter towards each other; and there is no doubt but its
operations are, in all cases, determined by laws equally im-
mutable.

Before I finish this paper, I would beg leave to observe,
that although, in treating the subject I have endeavoured to
investigate, I have made no mention of the names of those
who have gone over the same ground before me, nor of the
success of their labours ; this omission has not been owing to
any want of respect for my predecessors, but was merely to
avoid prolixity, and to be more at liberty to pursue, without
interruption, the natural train of my own ideas.

the Source of the Heat excited by Friction.

xoi

/ DESCRIPTION OF THE FIGURES (Tab. IV.)

*

Fig. 1. shows the cannon used in the foregoing experiments,
in the state it was in when it came from the foundry.

Fig. 2. shows the machinery used in the experiments No. 1,
and No. 2. The cannon is seen fixed in the machine used
for boring cannon. W is a strong iron bar, (which, to save
room in the drawing, is represented as broken off,) which bar,
being united with machinery (not expressed in the figure)

that is carried round by horses, causes the cannon to turn
round its axis.

m is a strong iron bar, to the end of which the blunt borer
is fixed; which, by being forced against the bottom of the
bore of the short hollow cylinder that remains connected by
a small cylindrical neck to the end of the cannon, is used in
generating heat by friction.

Fig. 3. shows, on an enlarged scale, the same hollow cylin-
der that is represented on a smaller scale in the foregoing
figure. It is here seen connected with the wooden box (g, h,
i, k,) used in the experiments No. 3, and No. 4, when this
hollow cylinder was immersed in water.

p, which is marked by dotted lines, is the piston which
closed the end of the bore of the cylinder.

n is the blunt borer seen side- wise.

d, e , is the small hole by which the thermometer was in-
troduced, that was used for ascertaining the heat of the cylin-
der. To save room in the drawing, the cannon is represented
broken off near its muzzle; and the iron bar, to which the blunt
borer is fixed, is represented broken off at m.

102 Count Rumforels Inquiry ,

I ig. 4. is a perspective view of the wooden box, a section
of which is seen in the foregoing figure, (see^, h , i, k , fig. 3.)

Fig. 5 ar*d 6. represent the blunt borer n, joined to the iron
bar m, to which it was fastened.

Fig. 7 and 8. represent the same borer, with its iron bar,
together with the piston which, in the experiments No. 2 and
No. 3, was used to close the mouth of the hollow cylinder.

OfkoJ <-TKs i/i--.

Th ilo-s-. Tran-y^sSfo C CXC V I WJTab. YST. /' .joz .

J I -H— H~ 1 I 1-H--K

C 103 3

V. Observations on the Foramina Thebesii of the Heart. By
Mr* John Abernethy, F. R. S. Communicated by Everard
Home, Esq. F. R. S.

Read February 1, 1798.

A s the investigation of the resources of nature in the animal
oeconomy, for the maintenance of health, and the prevention of
disease, cannot but be interesting to the philosopher as well
as to the physician, I therefore am induced to submit to the
Society the following observations.

There is a remarkable contrivance in the blood vessels which
supply the heart, not to be met with in any other part of the
body, and which is of great use in the healthy functions of
that organ, but which is particularly serviceable in preventing
disease of a part so essential to life.

A distended state of the blood vessels must always impede
their functions, and consequently be very detrimental to the
health of the part which they supply ; but, as the cavities of the
heart are naturally receptacles of blood, a singular opportunity
is afforded to its nutrient vessels, to relieve themselves when sur-
charged, by pouring a part of their contents into those cavities.
Such appears to be the use of the foramina by which injec-
tions, thrown into the blood vessels of the heart, escape into
the cavities of that organ ; and which were first noticed by
Vieussens, but, being more expressly described by Thebesius*
generally bear the name of the latter author.

Mr, Abernethy's Observations on

Anatomists appear to have been much perplexed concerning
these foramina Tbebesii ; even Haller, Senac, and Zinn,
were sometimes unable to discover them ; which suggested an
idea, that when an injection was effused into the cavities of the
heart, the vessels were torn, and that it did not escape through
natural openings. When these foramina were injected, they
were found under various circumstances, as to their size and
situation ; and Haller observed, that the injection, for the
most part, escaped into the right cavities of the heart. It also
remains undetermined, whether these foramina belong both to
the arteries and veins, or respectively to each set of vessels.

It is from an examination of these openings in diseased sub-
jects, that a solution of such difficulties may probably be ob-
tained. Whoever reflects on the circumstances under which
the principal coronary vein terminates in the right auricle of
the heart, will perceive that an impediment to the flow of
blood through that vessel must occasionally take place; but
the difficulty will be much increased, when the right side of
the heart is more than ordinarily distended, in consequence of
obstruction to the pulmonary circulation. Indeed it seems pro-
bable, that such an obstruction, by occasioning a distended
state of the right side of the heart, and thus impeding the
circulation in the nutrient vessels of that organ, would as ne-
cessarily occasion corresponding disease in it, as an obstruction
to the circulation in the liver occasions disease in the other ab-
dominal viscera, were it not for some preventing circumstances,
which I now proceed to explain.

Having been attentive to some very bad cases of pulmonary
consumption, from a desire to witness the effects of breathing
medicated air in that complaint, I was led to a more particular

the Foramina Thebesii of the Heart. 105

examination of the heart of those patients who died. In these
cases, I found, that by throwing common coarse waxen injec-
tion into the arteries and veins of the heart, it readily flowed
into the cavities of that organ ; and that the left ventricle was
injected in the first place, and most completely. When the
ventricle was opened, and the effused injection removed, the
foramina Thebesii appeared both numerous and large, and dis-
tended with the different coloured wax which had been im-
pelled into the coronary arteries and veins. Upon eight com-
parative trials, made by injecting the vessels of hearts taken
from subjects whose lungs were either much diseased, or in a
perfectly sound state, I found, that in the former, common in-
jection readily flowed, in the manner which I have described,
into all the cavities of the heart, but principally into the left
ventricle ; whilst, in many of the latter, I could not impel the
least quantity of such coarse injection into that cavity.

This difference in the facility with which the cavities of the
heart can be injected from its nutrient vessels, was observed
by most anatomists, though they did not advert to the cir-
cumstances on which it depended. Haller's recital of his
own observations, and of those of others on this subject, so
well explain the facts which I have stated, that I shall take
the liberty of quoting the passage, in order further to illustrate
and authenticate them. He says, “ Si per arterias liquorem in-
“ jeceris, perinde in dextra auricula, sinuque et ventriculo dextro,
“ et in sinu atque thalamo sinistro guttulas exstillabunt ; ssepe
“ quidem absque mora, alias difficilius, et nonnunquam omnino,
<c uti continuo dicemus, et mihi, et SENACo,et clarissimo Zinnio,
“ nihil exsudavit.”— Physiol. Tom. I. page 382.

As it seems right that the blood which had been distributed
mdccxcviii. P

icb Mr. Abernethy’s Observations on

9

by the coronary arteries, and which must have lost, in a greater
or less degree, the properties of arterial blood, should not be
mixed with the arterial blood which is to be distributed to every
part of the body, but ought rather to be sent again to the
lungs, in order that it may re-acquire those properties ; we
therefore perceive why, in a natural state of the heart, the
principal foramina Thebesii are to be found in the right cavi-
ties of that organ. However, as, even in a state of health, those
cavities are liable to be uncommonly distended, in consequence
of muscular exertion sometimes forcing the venous blood into
the heart faster than it can be transmitted through the lungs,
there seems to arise a necessity for similar openings on the left
side ; but these, in their natural state, though capable of emit-
ting blood, and of relieving the plethora of the coronary ves-
sels, are not of sufficient size to give passage to common waxen
injections. Yet, when there is a distended state of the right
cavities of the heart, which is almost certainly occasioned by a
diseased state of the lungs, these foramina leading into the left
cavities then become enlarged, in the manner that has been
already described ; and thus the plethoric state of the nutrient
vessels of the heart, and the consequent disease of that impor-
tant organ, are prevented.

The preceding remarks will, I think, sufficiently explain the
cause of the variety in the size and situation of these foramina,
which also appear to belong both to the arteries and veins ; be-
cause, the injection which was employed was too coarse to pass
from one set of vessels to the other, and yet the different co-
loured injections passed into the cavities of the heart unmixed.

There is yet another mode by which diseases of the heart,
that would otherwise so inevitably succeed to obstruction in the

the Foramina Thehesii of the Heart , 107

pulmonary vessels, are avoided ; and which I next beg leave
to explain.

Having formerly been much surprised to find the heart so
little affected, when the lungs were greatly diseased, and ob-
serving, in one or two instances, that the foramen ovale was
open, I was led to pay more particular attention to the state of
that part ; and I have found this to be almost a constant occur-
rence in those subjects where pulmonary consumption had for
some time existed previous to the person’s decease. I took no-
tice of this circumstance thirteen times in the course of one year;
and, in several instances, the aperture was sufficiently large
to admit of a finger being passed through it. Now, as the
septum auricularum is almost constantly perfect in subjects
whose lungs are healthy, I cannot but conclude, that the re-
newal of the foramen ovale is the effect of disease: nor will the
opinion appear, on reflection, improbable ; for the opening be-
comes closed by the membranous fold growing from one edge
of it, till it overlaps the other, and their smooth surfaces being
kept in close contact, by the pressure of the blood in the left
auricle, they gradually grow together. But, should there be a
deficiency of blood in the left auricle, and a redundance in the
right, the pressure of the latter on this membranous partition,
will so stretch and irritate the uniting medium, as to occasion
its removal ; and thus a renewal of the communication between
the auricles will again take place.

From these observations it is natural to suppose, that in
those men, or animals, who are accustomed to remain long un-
der water, this opening will either be maintained or renewed :
yet on this circumstance alone the continuance of their life
does not depend ; for we now have sufficient proof, that if the

P 2

io8 Mr. Abernethy’s Observations on

blood is not oxygenated in the lungs, it is unfit to support the
animal powers. There is an experiment related by Buffon,
the truth of which, I believe, has not been publicly contro-
verted, and which tends greatly to misrepresent this subject.
He says, that he caused a bitch to bring forth her puppies un-
der warm water ; that he suddenly removed them into a pail of
warm milk ; that he kept them immersed in the milk for more
than half an hour; and that when they were taken out of it, all
the three were alive. He then allowed them to respire about
half an hour, and again immersed them in the warm milk,
where they remained another half hour ; and, when taken out,
two were vigorous, but the third seemed to languish : this sub-
mersion was again repeated, without apparent injury to the
animals.

This experiment is so directly contrary to what we are led
to believe from all others, and also to the information derived
from cases which frequently occur in the practice of midwifery,
(in which, an interruption to the circulation through the umbi-
lical chord occasions the death of the foetus,) as to make me sus-
pect its truth : I was therefore induced to examine what would
happen in a similar experiment. I did not indeed cause the
bitch to bring forth her puppies in water; but immersed a
puppy, shortly after its birth, under water which was of the
animal temperature. It lost all power of supporting itself in
about 60 seconds, and would shortly have perished, had I not
removed it into the air. Neither could I, by repeating this ex-
periment, so accustom the animal to the circulation of unoxy-
genated blood, as to lengthen the term of its existence in such
an unnatural situation. I thought that a dog might have been
made a good diver in this way; but, having satisfied myself

the Foramina Thehesii of the Heart . 109

that this could not be done, without greatly torturing the ani-
mal, I did not choose to prosecute so cruel an experiment.

Young animals, indeed, retain their irritability for a consi-
derable time, so that they move long after they have been
plunged beneath water ; and may even, on this account, reco-
ver after they are taken out. But the manner in which Buf-
fon has related his experiment seems to imply, that the circu-
lation of the blood, and other functions of life, were continued
after the animals had been excluded from the air. I am con-
vinced that the poor dog who was the subject of my experiment
would have been beyond recovery in a few minutes.

Those animals who are accustomed to remain long under
water, probably first fill their lungs with air, which may, in a
partial manner, oxygenate their blood during their submersion.
The true statement of this subject may probably be, that the
circulation of venous blood will destroy most animals in a very
short space of time ; but that custom may enable others to en-
dure it, with very little change, for a longer period.

1 11° }

VI. An Analysis of the earthy Substance from New South Walest
called Sydneia or Terra Australis. By Charles Hatchett, Esq,

F, R . S,

Read February 8, 1798.

The late ingenious Josiah Wedgwood, Esq. F.R.S. published,
in the Philosophical Transactions for the year 1790, an account
of some analytical experiments on a mineral substance from
Sydney Cove, in New South Wales.*

This substance, Mr. Wedgwood describes to be composed of
a fine white sand, a soft white earth, some colourless micaceous
particles, and also some which were black, resembling black
mica, or black lead.

Nitric acid did not appear to act on any part of this earthy
substance; and even a portion on which sulphuric acid had
been boiled to dryness, afforded afterwards, when edulcorated
with water, only a few flocculi, which Mr. Wedgwood con-
ceived to be aluminous earth.

The muriatic acid, during digestion, seemed to act as little
as the two preceding acids ; but, upon water being poured in,
to wash out the remaining portion, the liquor instantly became
white as milk, with a fine white curdy substance intermixed ;
the concentrated acid having, in the opinion of the author,

* Philosophical Transactions, Vol. LXXX. Part II. page 306.

Mr. Hatchett’s Analysis , &c. 1 1 1

extracted something which the simple dilution with water pre-
cipitated.

The remaining part was repeatedly digested with muriatic
acid, and treated with water, as before, till the milky appear-
ance was no longer produced.

The properties of this white precipitate, Mr. Wedgwood states
to be as follows.

ist. It is only soluble in boiling concentrated muriatic acid.

2dly. It is precipitated by water, in the form of a white earth;
Which may again be dissolved by boiling muriatic acid.

gdly. When nitric acid is mixed with the muriatic solution
of this earth, there is no appearance of a precipitate ; not even
when water is added, provided that the nitric acid exceeds, or
nearly approaches, the quantity of muriatic acid.

4thly. The earth is precipitated by the alkalies.

^thly. The muriatic solution does not crystallize by evapo-
ration ; but becomes a butyraceous mass, which soon liquefies
on exposure to the air.

6thly. The butyraceous mass is not corrosive to the taste ;
and is even less pungent than the combination of calcareous
earth with the same acid.

7thly. Heat approaching to ignition disengages the acid from
the butyraceous mass, in white fumes, and a white substance
remains.

8thly. The white precipitated earth is fusible per se, in from
142 ° to 156° of Mr. Wedgwood’s thermometer, and it is thus
distinguished from all the other primitive earths.

And, 9thly. This precipitate cannot be reduced to a metallic
state, when exposed to heat with inflammable substances.

From these properties, Mr. Wedgwood says, that although

112 Mr. Hatchett’s Analysis of the earthy Substance

he cannot absolutely determine whether this substance belongs
to the class of earths, or that of metallic substances, yet he is
inclined to refer it to the former.

Professor Blumenbach, of Gottingen, in his Manual of Na-
tural History, published in 1791, also mentions that he had exa-
mined a portion of this earthy substance, by means of muriatic
acid, after the manner of Mr. Wedgwood, and that he had ob-
tained a slight precipitate by the addition of water. *

In consequence of these experiments, the mineralogists
throughout Europe admitted the white precipitated substance
to be a primitive earth; and we accordingly find, in all the
systematical works on mineralogy published since the above-
mentioned period, that it is arranged as a distinct genus, under
the names of Sydneia , Australa, Terra Australis , and Austral
Sand.

The extreme scarcity of this substance prevented the che-
mists in general from examining more minutely into the nature
of this new primitive earth, till Mr. Klaproth, in the second
volume of his Additions to the Chemical Knowledge of Mineral
Bodies, gave to the public a memoir entitled, A Chemical Exa-
mination of the Austral Sand, ■f*

In this memoir, Mr. Klaproth says, that he had received from
Mr. Haidinger, of Vienna, two samples of this substance; one
of which had a considerable quantity of black shining particles
intermixed with it, which, although regarded by many as gra-
phite or plumbago, he was inclined to believe to be Eisenglimmer
or micaceous iron ore.

The other contained much less of these black or dark grey

*

* ILmdbucb der Naturgeschichte, p, 567 and 568.

f Beitriige zur Cbemiscben Kenntniss der Miner alkor per . — Zweiter Rand# p. 66.

called Sydneia or Terra Australis . 113

particles, and, as he considered it to be more pure than the
former, he subjected it to the fgllowing experiments.

1. It was digested at three different times with concentrated
muriatic acid, in a boiling heat, and the acid was afterwards fil-
trated through paper. The solution was then mixed by degrees
with pure water, which did not however produce any precipi-
tate, even when warmed.

Carbonate of potash caused some flocculi to fall, which, edul-
corated and dried, weighed 3.25 grains.

This precipitate was dissolved in diluted sulphuric acid, and
left a small portion of siliceous earth; after which the solution,
by evaporation, afforded crystals of alum.

2. The residuum of the muriatic solution was mixed with
three times the weight of potash, and exposed to a red heat.
Muriatic acid was then poured on the mass, and the insoluble
gelatinous residuum was edulcorated on a filter; and, after a
red heat, weighed 19.50 grains, which proved to be siliceous
earth.

3. The muriatic solution, with prussiate of potash, afforded a
blue precipitate ; the ferruginous part of which was about one
quarter of a grain.

T The solution was then saturated with carbonate of pot-
ash, and some alumine was precipitated; which, after a red

heat, weighed 8.50 grains, and with sulphuric acid formed
alum.

Siliceous earth, alumine, and iron, appeared therefore to be
the only ingredients of this substance ; but, as Mr. Klaproth
had no more than thirty grains to examine, he could not extend
his experiments.

From those above related, he is of opinion that the existence
mdccxcviii. O

i 14, Mr. Hatchett’s Analysis of the earthy Substance

of this primitive earth may be much doubted, and that this
doubt can only be removed in the course of time, by other
analyses.

Mr. Klaproth concludes his memoir by saying, that the
substance examined by him was undoubtedly the genuine
austral sand, as Mr. Hai dinger had received it from Sir
Joseph Banks, when he was in London.

Mr. Nicholson, however, in the 9th Number of his Journal
of Natural Philosophy, &c. (p. 410.) published on the 1st of
December, 1797, questions much, whether the substance exa-
mined by Mr. Klaproth was the same as that examined by
Mr. Wedgwood ; and, after having contrasted their experi-
ments, says, “ hence it seems fair to conclude that the two
“ minerals were not the same, however this may have hap-
“ pened ; and that the existence of the new fusible earth of
“ Wedgwood stands on the same evidence as before, namely *
“ his experiments, which have not yet been repeated, that I
“ know of.”

Some of Mr. Nicholson’s objections to the experiments of
Mr. Klaproth, being founded principally on some difference
in the external characters of the substance examined by him?
and the one examined by Mr. Wedgwood, are such as very
naturally occur; but the following pages will, I believe, prove
that Mr. Klaproth’s experiments were made on that which
might be justly regarded as the Sydneia or austral sand.

In 1796, the Right Hon. Sir Joseph Banks, P. R. S. favoured
me with a specimen of the Sydneia , which had been lately
brought to England ; a portion of this I soon after examined,
in a cursory manner, by muriatic acid, but did not obtain any
precipitate when water was added to the filtrated solution.

ii 3

called Sydneia or Terra Australis.

Upon mentioning this circumstance, and expressing a desire
to examine this substance with more accuracy, Sir Joseph
Banks, with his usual readiness to promote every scientific
inquiry, not only permitted me to take specimens from diffe-
rent parts of the box which contained the earth already men-
tioned, but (that every doubt might be obviated) gave me about
300 grains which remained of the identical substance examined
by Mr. Wedgwood.

Upon these the following experiments were made ; and, to
distinguish them, I shall call the first, No. 1, and that examined
by Mr.WEDGWooD, No. 2.

§. 2.

Analysis of the Sydneia, No. 1.

The Sydneia , No. 1, is in masses and lumps, of a pale greyish
white, intermixed with a few particles of white mica, and also
occasionally with some which are of a dark grey, resembling
graphite or plumbago.

It easily crumbles between the fingers, to a powder nearly
impalpable, which has rather an unctuous feel.

Small fragments of vegetable matter are also commonly
found intermixed with it; and the general aspect is that of an
earthy substance which has been deposited by water.

EXPERIMENT 1.

400 Grains were put into a glass matrass, and one quart of
distilled water being added, the whole was boiled to one fourth.

The liquor was then filtrated, and a portion being examined
by the re-agents commonly used, afforded no trace of matter

O 2

n6 Mr. Hatchett’s Analysis of the earthy Substance

in solution. The remainder was then evaporated, without leav-
ing any residuum.

♦

EXPERIMENT 2.

About 200 grains of the earth, rubbed to a fine powder, were
put into a glass retort, into which I poured three ounces of
concentrated pure muriatic acid. The retort was placed in sand,
and the acid was distilled, till the matter in the retort remained
dry. Two ounces of muriatic acid were again poured on it, and
distilled as before, till only one fourth remained. The whole
was then put into a matrass, which was placed in an inclined
position, so that when the earth had subsided, the liquor might
be decanted, without disturbing the sediment.

When it had remained thus for 12 hours, the acid was care-
fully poured into a glass vessel ; but, as I observed that it was
not so perfectly transparent as before it had been thus employed,
I suffered it to remain 24, hours, but did not perceive any sedi-
ment. Half of this liquor was diluted with about twelve parts
of distilled water, and, after a few hours, a very small quantity
of a white earth subsided.

This however did not appear to me to be a precipitate caused
by a change in the chemical affinities, but rather an earthy
matter which had been suspended in the concentrated acid,
and afterwards deposited, when the liquor was rendered less
dense by the addition of water. To ascertain this, I poured the
remaining portion of the concentrated liquor on a filter of four
folds : it passed perfectly transparent, and, although diluted with
twenty-four parts of water, it remained unchanged, and as pel-
lucid as before. I now filtrated the former portion, and added
it to that already mentioned.

called Sydneia or Terra Australis . 117

It was then evaporated to dryness, and left a pale brownish
mass, which was dissolved again, by digestion, in the smallest
possible quantity of muriatic acid.

Water was added, in a very large proportion, to this solution,
without producing any effect ; I then, by prussiate of potash,
precipitated a quantity of iron, which was separated by a filter.

The clear solution was then saturated with lixivium of car-
bonate of potash, and a white precipitate was produced, which
was collected and edulcorated. This, when digested with di-
luted sulphuric acid, was dissolved; and the superfluous acid
being driven off by heat, boiling water was poured on the re-
siduum, and completely dissolved it.

To this solution some drops of lixivium of potash were
added, and, by repeated evaporations, the whole formed crys-
tals of alum.

From the above experiment it appeared, that the muriatic
acid had only dissolved some alumine and iron ; but, in order
to satisfy myself more completely in respect to the component
parts of this substance, I made the following analysis.

Analysis . A. 400 grains were put into a glass retort, which
was then made red-hot during half an hour. Some water came
over, and the earth afterwards weighed 380.80 grains, so that
the loss amounted to 13.20 grains. The greater part of this
loss was occasioned by the dissipation of the water imbibed by
the earth ; to which must be added, the loss of weight caused
by the combustion of a small portion of vegetable matter.

B. The 380.80 grains were rubbed to a fine powder, and
being put into a glass retort, 1470 grains of pure concentrated
sulphuric acid were added. The retort was then placed in a
small reverberatory, and the fire was gradually increased, till

1 1 8 Mr. Hatchetts Analysis of the earthy Substance

the acid was distilled over : it was then poured back on the
matter in the retort, and distilled as before, till a mass nearly
dry remained.

On this, boiling distilled water was repeatedly poured, until
it no longer changed the colour of litmus paper, and was de-
void of taste. The undissolved portion was then dried, and
made red-hot; after which it weighed 281 grains.

C. I now mixed the 281 grains with 300 grains of dry car-
bonate of potash, and exposed the mixture to a strong red heat,
in a silver crucible, during four hours. The mass was loose,
and of a greyish white : it was softened with water, and, being
put into a retort, sulphuric acid was added to a considerable
excess. The whole was then distilled to dryness, and, when
a sufficient quantity of boiling water had been added, it was
poured on a filter, and the residuum was well washed ; it was
then made red-hot, and afterwards weighed 274.7 5 grains.

D. The solutions of B and C were added together, and were
much reduced by evaporation. Pure ammoniac was then em-
ployed to saturate the acid, and a copious loose precipitate, of
a pale yellowish colour was produced ; which, collected, edul-
corated, and made red-hot, weighed 103.70 grains.

E. The filtrated liquor of D was again evaporated, and
carbonate of potash being added, a slight precipitation of earthy
matter took place; which, by the test of sulphuric acid, proved
to be some alumine which had not been precipitated in the
former experiment : this weighed 1.20 grain,

F. The 103.70 grains of D were completely dissolved when
digested with nitric acid, excepting a small residuum of silice-
ous earth, which weighed 0.90 grain.

G. The nitric solution was evaporated to dryness, and a

called Sydneia or Terra Australis. 119

second portion of the same acid was added, and in like man-
ner evaporated. The residuum was then made red-hot, and
digested with diluted nitric acid, which left a considerable por-
tion of red oxide of iron. The solution was again evaporated,
and the residuum, being treated as before, again deposited some
oxide of iron, much less in quantity than the former.

The whole of the oxide was then heated with wax in a por-
celain crucible, was taken up by a magnet, and weighed 26.50
grains.

H. The nitric solution of G was saturated with ammoniac,
and a loose white precipitate was formed ; which, edulcorated
and made red-hot, weighed 76 grains.

I. These 76 grains were dissolved when digested with di-
luted sulphuric acid ; and, when the excess of acid had been ex-
pelled by heat, the saline mass was dissolved in boiling water.
To this solution I added some lixivium of potash, and, by gra-
dual and repeated evaporations, obtained the whole in regular
octoedral crystals of alum.

K. The 274.75 grains of C now alone remained to be exa-
mined. They appeared to consist of siliceous earth, mixed with
the dark grey shining particles already mentioned ; but, as I
shall describe, in the following experiments, the process by
which these were separated, I shall now only say that they
amounted to 7.50 grains.

L. The earth with which the abovementioned particles were
mixed weighed 267.25 grains. This earth was white, and arid
to the touch : when melted with two parts of soda, it formed a
colourless glass ; and, with four parts of the same it dissolved
in water, and formed a liquor silicnm : it was therefore pure
siliceous earth or silica.

120 Mr. Hatchett's Analysis of the earthy Substance

The substance here examined was composed therefore of the
following ingredients.

Pure siliceous earth or silica

’ (F-
lL.

grains.

°-9°

267.25

Alumine -

- fE-

LH.

1.20

76

Oxide of iron

G.

26.50

Dark grey particles

K.

7-5o

Water and vegetable matter

A.

19.20

398-55

The foregoing analysis was repeated several times, and al-
ways with similar results ; excepting, that as I had taken the
specimens from different parts of a large quantity, I found that
the proportions of the ingredients were not constantly the same:
that of the siliceous earth, for example, was sometimes greater,
and the alumine and iron proportionably less. Some specimens
were also nearly or totally destitute of the dark grey shining
particles ; in short, every circumstance was such as might be
expected from a mixed substance, which, from the nature of its
formation, cannot have the ingredients in any fixed proportion.*
As this substance agreed in its general characters, for the
greater part, with that described by Mr. Wedgwood, and as
it was indisputably brought from the same place, there appeared
every reason to believe that the nature of both was the same;

* The description given by Mr. Klaproth convinces me, that his experiments
were made on a portion of this substance. Moreover, when my late friend Mr. Hai-
dinger was in London, I gave him some of this earth for his collection; so that,
whether Mr. Klaproth made his experiments on that which had been received by
Mr. Haidinger from Sir Joseph Banks, or from myself, it is not less certain that
he operated on that which might be regarded as the genuine Sydneia.

called Sydneia or Terra Australis. 121

but, to obviate as much as possible any doubt or objection, I
determined to repeat the experiments, and the analysis, on that
portion which remained of the identical substance examined by
Mr. Wedgwood, and which from that period had been reserved
by Sir Joseph Banks, who kindly favoured me with it for this
purpose.

§• 3-

Analysis of the Sydneia, No. 2.

This substance, as has already been mentioned, consists of
a white transparent quartzose sand, a soft opaque white earth,
some particles of white mica, and a quantity of dark lead-grey
particles, which have a metallic lustre.

The Sydneia, No. 2, appears chiefly to differ from No. 1, by
being more arenaceous, and by a larger proportion of the dark
grey particles. Many experiments, similar to those made on
No. 1, already described, were made on this substance, with
pure concentrated muriatic acid ; but, as none of these afforded
any appearance of a precipitate by the means of water, I do not
think it necessary to enter into a circumstantial account of them,
and shall proceed therefore to the analysis.

A. 100 grains were exposed to a red heat, in a glass retort,
and, after half an hour, were found to have lost in weight 2.20
grains.

B. The 97.80 grains which remained were mixed with 300
grains of dry carbonate of potash, and the mixture was exposed
to a strong red heat, in a crucible of silver, during three hours.

When cold, the mass was softened with water, and was put
into a glass matrass. I then added three ounces of pure con-

MBCCXCVIII, R

122

Mr. Hatchett’s Analysis of the earthy Substance

centrated muriatic acid, and digested it for two hours in a
strong sand heat. Boiling water was then added, and the
whole being poured on a filter, the residuum was edulcorated,
dried, and made red-hot ; it then weighed 85.50 grains.

C. The filtrated solution was evaporated to one fourth, and
pure ammoniac being added, a precipitate was formed, which,
after a red heat, weighed 10.70 grains.

D. One ounce of muriatic acid was poured on the 10.70
grains, in a matrass, which was then heated. The whole of
the 10.70 grains was dissolved, excepting a small portion of
siliceous earth, which weighed 0.30 grain.

E. The muriatic solution was then reduced by evaporation,
to about one fourth ; to which I added a large quantity of dis-
tilled water, which did not however produce any change. I
then gradually added a solution of pure crystallized prussiate of
potash, and heated the liquor till the whole of the iron was pre-
cipitated; after which, ammoniac precipitated a loose white
earth, which, edulcorated and made red-hot, weighed 7.20 grains.
The iron precipitated by the prussiate may therefore be esti-
mated at 3.20 grains.

F. The 7.20 grains of the white earth were digested with
sulphuric acid, and, after the excess of acid had been expelled
by heat, boiling water was poured on the saline residuum.
The solution was then gradually evaporated, with the addition
of a small portion of lixivium of potash, and afforded crystals
of alum, without a trace of any other substance.

G. I - now proceeded to examine the 85,50 grains of B,
These appeared to consist of siliceous earth, or fine particles of
quartz, mingled with a considerable quantity of the dark grey
shining particles.

called Sydneia or Terra Australis. 123

Mr. Wedgwood was of opinion that these were a peculiar
species of plumbago or graphite. Professor Blumenbach, on
the contrary, regards them as molybdaena : and Mr. Klaproth
believes them to be eisenglimmer or micaceous iron ore.

When rubbed between the fingers, they leave a dark grey
stain, and the feel is unctuous, like that of plumbago, or mo-
lybdaena : the traces which they make on paper also resemble
those of the abovementioned substances, but the lustre of the
particles approaches nearer to that of molybd^na.

In order therefore to determine whether or not they consisted
totally or partially of molybdaena, I put the 85.50 grains into
a small glass retort, and added two ounces of concentrated ni-
tric acid. The retort was then placed in a sand heat, and the
distillation was continued, till the matter remained dry. I he
acid was then poured back into the retort, and distilled as be-
fore ; but I did not observe that the grey particles had suffered
any change, nor were nitrous fumes produced, as when mo-
lybdaena is thus treated.

To be more certain, however, I digested pure ammoniac on
the residuum ; and, having decanted it into a matrass, I eva-
porated it to dryness, without perceiving any vestige of oxide
of molybdaena, or indeed of any other substance.

It was evident therefore that molybdaena was not present;
and, as the general external characters and properties corre-
sponded with those of plumbago, I was inclined to believe that
these were particles of that substance, and not micaceous iron,
as Mr. Klaproth imagined. To determine this, the following
experiment was made.

H. 200 grains of pure nitre in powder were mixed with the
85.50 grains, and the mixture was gradually projected into a

R 2

c\

C*

r

324 Mr. Hatchett’s Analysis of the earthy Substance

crucible, made strongly red-hot. A feeble detonation took
place at each projection ; and, after a quarter of an hour had
elapsed, the crucible was removed.

When cold, the mass was porous and white, without any ap-
pearance of the dark grey particles. Boiling water was poured
on it, and the whole being put into a matrass, one ounce of
muriatic acid was added, and digested with it in a sand heat.
By evaporation it became gelatinous : it was then emptied on a
biter, and, being well washed, dried, and made red-hot, weighed
75-25 grains.

The appearance of this was that of a white earth, arid to
the touch. When melted with two parts of soda, a colourless
glass was formed ; and, with four parts of the same, it was so-
luble in water, and produced liquor silicum ; it was therefore
pure siliceous earth.

1. The filtrated liquor was saturated with ammoniac, and,
upon being heated, a few brownish flocculi were precipitated,
which, when collected and dried, weighed 0.40 grain. This pre-
cipitate was dissolved in muriatic acid, and was again precipi-
tated by prussiate of potash, in the state of Prussian blue.

The liquor from which the flocculi of iron had been separated
was then examined, by adding carbonate of potash, and lastly,
by being evaporated to dryness ; but it no longer afforded any
earthy or metallic substance : so that, by the process of detona-
tion with nitre, the 85.50 grains afforded 75.25 grains of pure
siliceous earth, with 0.40 grain of iron ; and, as the dark grey
substance was destroyed, excepting the 0.40 grain of iron above-
mentioned, and as 9.85 grains of the original weight of 85.50
grains were dissipated, there can be no doubt but that this sub-
stance, amounting to 10.25 grains, was carburet of iron or

33T33

called Sydneia or Terra Australis . 125

plumbago ; especially as some experiments which I purposely
made, on that from Keswick in Cumberland, were attended
with similar results.

It is also evident, that these particles could not be ei sen-
glimmer or micaceous iron, as nitre has little or no effect on
that substance, when projected into a heated crucible.

In a subsequent experiment on the same, the crucible was
removed immediately after the last projection, and I then ob-
served that an effervescence, with a disengagement of carbonic
acid, took place, upon the addition of the muriatic acid, as is
usual when pure plumbago is decomposed by nitre, and that
less of the gelatinous matter w^as formed by evaporation.

The cause of this difference was evidently the duration of
the red heat ; for, in the first instance, the alkali developed by
the decomposition of the nitre had time to unite with the sili-
ceous earth, so as, when dissolved, to form liquor silicum ; but,
in the second experiment, a portion of alkali remained com-
bined with the carbonic acid, produced by the carbon of the
decomposed plumbago.

The produce of 100 grains by this analysis was,

grains.

Silica -

[ D. 0.30
lH. 75.25

Alumine -

F. 7.20

* OJ

Oxide of iron

E. 3.20

♦ A)

Graphite or plumbago

I. 10.25

' O

Water -

A. 2.20_

r>
Jj C/

^8.^0

Mr. Wedgwood says, that sulphuric acid cannot dissolve
the precipitated earth, and has but little effect on the mixed

'Q/JlTi 4

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h f A Q.
*7 / *^f / V

t ***. - '! A >

*/) sy f *7 /

i 26 iVfr. Hatchett’s Analysis of the earthy Substance

substance, even when distilled to dryness ; but, from the pre-
ceding experiments, I had reason to believe that the aluminous
earth and iron would be separated by reiterated distillation; I
therefore repeated the analysis in the following manner.

A. 100 grains of the earth were put into a glass retort, upon
which 400 grains of pure concentrated sulphuric acid were
poured. The retort was placed in a small reverberatory, and
the lire was continued till a dry mass remained. 400 grains
of the acid were again poured in, and distilled as before. Upon
the dry mass, boiling water was poured, and the whole was
then emptied on a filter, and edulcorated. The residuum, after
a red heat, weighed 87.75 grains, and consisted of siliceous
earth, mixed with some mica, and with particles of plumbago.

B. The filtrated solution, by ammoniac, afforded a precipi-
tate, which weighed 9.50 grains; and, being examined, as in the
former experiment, yielded 6.50 grains of alumine, and g grains
of oxide of iron.

The plumbago was separated from the siliceous matter, in
the manner already described, and amounted to about 10 grains.

By this analysis I obtained.

It appears therefore that the Sydneian earth, when treated
with sulphuric acid, is capable of being for the greater part

Second Analysis of the Sydneia, No. 2.

grains.

Silica and mica
Alumine
Oxide of iron
Plumbago

77*75

6.50

3

10

97-25

called Sydneia or Terra Australis. 127

decomposed; and Mr. Wedgwood probably did not succeed,
because his process was in some respect different, or that the
distillation was not sufficiently repeated.

I have not thought it necessary to be more circumstantial in
the account of this second analysis, as the operations were si-
milar to those of the former.

§• 4-

These experiments prove, that the earthy substance called
Sydneia or terra australis, consists of siliceous earth, alumine,
oxide of iron, and black lead or graphite.

The presence of the latter appears to be accidental, and it
probably was mixed with the other substances at the time when
they were transported, and deposited, by means of water; for
this appears evidently to have been the case, from the general
characters of this mixed earthy substance.

The quartz and mica, which are so visible, indicate a granitic
origin ; and the soft white earth has probably been formed by
a decomposition of feldt spar, such as is to be seen in many
places, and particularly at St. Stephen’s, in Cornwall. The gra-
nitic sand which covers the borders of the Mer de Glace, at Cha-
mouni, in Savoy, also much resembles the terra australis, ex-
cepting that the feldt spar is not in a state of decomposition :
in short, the general aspect, and the analysis, concur to prove,
that the Sydneia has been formed by the disintegration and de-
composition of granite, or gneiss.

Mr. Wedgwood’s experiments are so circumstantial, that
had I only examined the earth last brought to England, I should
have supposed, with Mr. Nicholson, that I had operated on a
different substance ; but, as I had an opportunity to examine,

128 Mr. Hatchett’s Analysis of the earthy Substance

by analysis, a portion of the same earth on which Mr. Wedg-
wood made his experiments, and as I received it from Sir
Joseph Banks, the same gentleman who had furnished Mr.
Wedgwood with it, no suspicion can be entertained about its
identity.

Some of the experiments which I have related, and which
prove that some of the finer earthy particles remained suspended
in the concentrated muriatic acid, and were precipitated when
the acid was diluted with water, appear in some measure to ac-
count for the mistake which has been made, in supposing that
a primitive earth, before unknown, was present ; but this alone
will not account for many of the other properties mentioned by
Mr. Wedgwood, such as,

ist. The repeated and exclusive solubility in the muriatic
acid, and subsequent precipitation by water.

2dly. The butyraceous mass which was formed by evapo-
ration.

And, 3dly. The degree of fusibility of the precipitated earth.

These indeed I can by no means explain, but by supposing
that the acids used by Mr. Wedgwood were impure. This
supposition appears to be corroborated by a passage in Mr.
Wedgwood’s paper, where he says, “ here the Prussian lixi-
« vium, in whatever quantity it was added, occasioned no pre-
“ cipitation at all, (only the usual bluishness arising from the
<f iron always found in the common acids .”)* Now if (as it
seems from this expression) Mr. Wedgwood employed the
common acids of the shops, without having previously exa-
mined and purified them, all certainty of analysis must fall,
as the impurity of such acids is well known to every practical

* Philosophical Transactions, Vol. LXXX. Part II. p. 313.

called Sydneia or Terra Australis . 129

chemist ; but, whether this was the cause, or not, of the effects
described by Mr. Wedgwood, I do not hesitate to assert, that
the mineral which has been examined does not contain any
primitive earth, or substance possessing the properties ascribed
to it, and consequently, that the Sydneian genus, in future,
must be omitted in the mineral system.

Mpccxcvm.

S

C *3° 3

VII. Abstract of a Register of the Barometer , Thermometer ,
ifoz’tt, #£ Lyndon, m Rutland, /or the Tear 1 796. By
Thomas Barker, Fsg. Communicated by Mr. Timothy Lane,
F. F. 5.

Read February 15, 1798.

Barometer.

Thermometer,

Rain.

In the House.

Abroad.

Highest.

Lowest.

Mean.

High.

Low.

Mean.

High.

Low.

Mean.

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sep.

Oct.

Nov.

Dec.

Morn.

Aftern.

Morn.

Aftern.

Morn.

Aftern.

Morn.

Aftern.

Morn.

Aftern.

Morn.

Aftern.

Morn.

Aftern.

Morn.

Aftern.

Morn.

Aftern.

Morn.

Aftern.

Morn.

Aftern.

Morn.

Aftern.

Inches.
29 >77

29.87

2 9>97

29.88

29>7 3
29,84

z9>73

29,95

29,83

30,07

29,86

29,98

Inches.

28,47

28,53

29.22
28,67
28,33
29,05
28,91

29.23
29,03
28,75
28,69
28,75

Inches.

29,17

32

57

59
27

47

32

60

5°

45

36

3i

0

50

52
45 1

51
48

5i

59

61

55

56

59

70

67

671

651

68*

65

68

55 *

56
51!

53
44*
4 6f

0

41

4if

38
34

33

34

44

45

47

48
5 °f

52
50
54f
57
5 81

53

54
43f
44f
39f

39
28

295

0

45

46

42

42f
4i
42J-
5 °f
52

51

52
55
57
58*
6of
61

63

59f

6x

49
5 °f

43
43 f

34

35

0

52f

55

49

5*f

45

57

52

72

52

65

60

80

64f

77

62J

77

63

77

59

6if

52f

57

48

49

0

32

34

3*f

32

26

34

36

45
40

46

56
49f
59
49

57

56

29

26

28I

Hf

25

0

4*f

47

38

43

36

45

45
57f

46

56

54

67

56f

68*

56

69

54

66

43

Slz

38

43

30

34

Inches.

i>955

1,643

0,376

0,649

2,839

0,927

5,646

1,120

1,891

1,320

2,048

1,668

22,082

Mr, Barker's Register of the Barometer , &c. 131

The year began with a remarkably open winter, sometimes
quite warm and pleasant, and several times thunder. It was
showery at. first, then dry and fine; but the end of January
and beginning of February were wet, yet still open and mild,
and more dry afterward ; but colder, and inclined to frost, the
end of February, and in March, the middle of which was again
mild and fine, and not windy, but frosty toward the end. The
last days of March, and beginning of April, were dry and
pleasant ; a good seed time, and calm ; but rain was wanted
toward the. end, which came plentifully the end of April, and
beginning of May. The season in general cool, and the latter
half more dry; too much so in the south of England, for grass
and hay were scarce there; but, in this country, both grass and
corn came on well, and continued to do so all June, which was
of a moderate heat, with a mixture of wet and dry ; frequent
but moderate winds, and calm at the end. There was plenty
of hay this year; but, through a very wet and windy July, a
good deal of it was not well got. The crops of grain were al-
most all good, and the moist July made the beans and pease
remarkably so. The harvest, though threatening at first, was
in general very well got ; the ’weather being chiefly fair, and
rather hot, with some rain at times, kept the grass in a grow-
ing state, of which there was plenty left upon the ground
against winter.

The autumn was in general very fine and pleasant ; for the
most part fair, with few frosty mornings, till near the end of
November ; when a severer season began, and continued very
hard frost the first third of December ; then an imperfect break
for some days, but not so as to take the frost all away. It re-
turned again as hard as before, and continued another third

S 2

132 Mr. Barker's Register of the Barometer ,

of the month, till the two last days ; when a thaw came on*
with very warm and remarkably wet air, though no great
quantity of rain fell. It was agreed every where, that Decem-
ber 24, at night, was much the coldest time this winter; in some
places the thermometer was down below o, at 4 or 8 ; but 1
did not happen to look at it just at that time, so that I never
saw it so low as 14.

C JS3 3

VIII. An Account of some Endeavours to ascertain a Standard of
Weight and Measure. By Sir George Shuckburgh Evelyn,
Bart. F. R. S. and A. S.

Read February 22, 1798.

§• i-

Having for some years turned my thoughts to the considera-
tion of an invariable and imperishable standard of weight and
measure, as being a thing, in a philosophical view, highly de-
sirable, and likely to become extremely beneficial to the public,
I had, so early as the year 1780, taken up the idea of an uni-
versal measure, from whence all the rest might be derived, by
means of a pendulum with a moveable centre of suspension,
capable of such adjustments, as to be made to vibrate any num-
ber of times in a given interval; and, by comparison of the
difference of the vibrations with the difference of the lengths of
the pendulum, (which difference alone might be the standard
measure,) to determine its positive length, if that should be
thought preferable, under any given circumstances ; by which
means, all the difficulties arising in determining the actual
centre of motion and of oscillation, which have hitherto so
much embarrassed these experiments, would be gotten over.

(§.2.) I made several computations of the probable accuracy
that might be expected from such an experiment, and was sa-
tisfied with their result. But, not seeing clearly how such a
pendulum could be connected to a piece of mechanism, to

134 Sir George Shuckburgh Evelyn's Endeavours

number the vibrations without affecting them, I dropped the
idea for that time. I learnt, however, some time afterwards,
that Mr. John Whitehurst, a very ingenious person, had been
in pursuit of the same object with better success, and had con-
trived a machine fully corresponding to his expectations and
my wishes. This he afterwards explained to the world, in a
pamphlet, entitled, “ An Attempt to obtain Measures of Length,
“ &c. from the Mensuration of Time, or the true Length of
c< Pendulums;” published in 1787. Mr. Whitehurst having
therein done alffthat related to the standard measure of length,
and suggested that of weight, it appeared to me that it remained
only to verify and complete his experiments.

(§.3.) For this purpose, by the kind assistance of my friend
Dr. G. Fordyce, who, at Mr. Whitehurst’s death, had pur-
chased his apparatus, I was furnished with the very machine
with which Mr. Whitehurst had made his observations. I
also procured to be made, by Mr. Troughton, a very excellent
beam-compass or divided scale, furnished with microscopes and
micrometer, for the most exact observations of longitudinal mea-
sure : as also a very nice beam or hydrostatic balance, sensible
with the of a grain, when loaded with 61b. Troy at each
end. Mr. Arnold made me one of his admirable time-keepers,
in order to carry time from my sidereal regulator in my obser-
vatory, with which it was adjusted, to the room wherein I had
fixed Mr. Whitehurst’s pendulum; and who, having taken a
journey from London into Warwickshire, was so good as to
assist in the beginning of these experiments. Thus equipped,
I went to work in the latter end of August, 1 796, when the
temperature was about 6o°, first to examine the length of the
pendulum ; when, to my great mortification, I found that the

to ascertain a Standard of Weight and Measure. 1 35

thin wire, of which the rod consisted, was too weak to support
the ball in a state of vibration ; and that, after 15 or 20 hours
action, it repeatedly broke. The same misfortune attended my
trials with three other different sorts of wires that I had obtained
from London. Whether this accident happened from any rust
in the old wire, or from want of due temper in the new, or
from its being too much pinched between the cheeks,* I cannot
tell : I can only observe, that all the wires that I used were con-
siderably heavier, and therefore probably stronger, than what
Mr. Whitehurst mentions, viz. 3 grains in weight for 80 inches
in length ; nay, mine proceeded as far as from 5 to 6 grains for
that length, and yet I could never get it to support the ball during
the whole period of my experiment. This being the case, and
being in the country, far removed from the manufactory of this
fine wire, I was reluctantly compelled to relinquish this part of
the operation to some more favourable opportunity. In the
mean while, however, I thought it desirable to measure the dif-
ference of the lengths of Mr. Whitehurst's pendulum from
his own observations; for, very fortunately, the marks that he
had made on the brass vertical ruler of his machine were still
visible; and this interval, which he calls “ 59,892 inches,” I
determined, on my divided scale made by Troughton, from
Mr. Bird's standard, to be = 59,89358 inches, from a mean of
four different trials in the temperature of 64°; that mean differ-
ing from the extremes only = ,0003 inch.

(§.4.) By this examination, if I have not verified, I have
at least preserved, Mr. Whitehurst's standard; and, for the
present, I shall consider this measure of the difference of the
length of the two pendulums, vibrating 42 and 84 times in a

* C,C, fig. 1. of Plate II. in Mr. Whitehurst’s pamphlet.

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minute of mean time, as correct. On this presumption, I shall
proceed to the examination of weight.

(§.5.) From the opinion of different skilful persons, with
whom I have conferred, as well as from the result of my own
considerations, I am inclined to believe there is hardly any body
in nature, with which we are familiarly acquainted, that is of so
simple and homogeneous a quality as pure distilled water, or so
fit for the purposes of this inquiry ; and I have concluded, that
if the weight of any quantity of water, whose bulk had been
previously measured by the abovementioned scale, could be
obtained, under a known pressure* and temperature of the at-
mosphere, we should be in possession of a general standard of
weight.

(§. 6.) With this view, I directed Mr. Troughton to make,
in addition to the very sensible hydrostatic balance before men-
tioned, a solid cube of brass, whose sides were 5 inches ; and
also a cylinder of the same metal, 4 inches in diameter, and 6
high. From St. Thomas's hospital, by the favour of Dr. For-
dyce, I procured 3 gallons of distilled water. With these I
made the following observations ; but, before I relate the ex-
periments, I will describe the apparatus.

Mr. Whitehurst’s machine for measuring the pendulum
has been sufficiently explained in his pamphlet mentioned
above ; my divided scale, which was a new instrument, was
as follows.

• I do not here mean to infer any opinion respecting the compressibility of water ;
but only to say, that where water, or any thing else, is weighed in air, the density of
that medium, as shewn by the barometer and thermometer, must be known, in order
to make allowances fo-r it, if necessary.

to ascertain a Standard of Weight and Measure. 137

(§. 7.) Description of the Beam Compass , or divided Scale of

equal Parts.

a b, (Tab. V. fig. 1.) is a block or beam of mahogany,
6 feet 3 inches long, 6 inches deep, and 5 wide, upon which
are laid two brass rulers, c d e, and f g, each divided into
60 inches, and tenths. The former of these, called the Scale,
is, for a time, kept immoveable by the finger-screws c e d, and
is furnished with very fine hair-line divisions, intended to be
viewed only by the microscopes h, i : the latter, called the
Beam, has no motion but by means of the screw g , and bears
stronger divisions upon it, with which the sliding pieces or in-
dexes, at k and m , may readily be compared by the naked eye,
and is intended only to set the microscopes, or rather the wires,
in their focus, to the required distance nearly, viz. tc within T—
or of an inch. The microscopes are compound, and similar
to those described by the late General Roy, in his account of
his large theodolite. (See Phil. Trans. Vol. LXXX.) The one
at h, contains only cross wires fixed in its focus ; the other at
i, has a micrometer also, by means of which its cross wires
may be moved to the right or left, over the image of the divi-
sions of the scale, any given space, not exceeding inch ;
and the quantity so moved may be measured by the divisions
on the screw head, passing under the index at 0. The divisions
on these rules have been called inches and tenths : it was not
necessary that they should be more than equal parts ; but they
were in fact laid down by Mr. Troughton, from a scale of the
late excellent artist Mr. J. Bird, who had divided into inches
several scales of different lengths ; one of which, 42 inches
long, belonged to the late General Roy ; a second, of 5 feet,
was purchased by Alexander Aubert, Esq. and a third, of

mdccxcviii. T

138 Sir George Siiuckburgh Evelyn’s Endeavours

90 inches, which is now the property of the Royal Society, is
kept in their archives, and is said to have been used by Mr.
Bird, in dividing his large mural quadrants.* Besides these, he
made two standards of three feet, by order of the House of
Commons, of which I shall speak more hereafter. The mode
of using this instrument is as follows.

(§. 8.) Let the object to be measured be supposed to be
about six inches, and let it be desired to compare it with the
interval between the 20th and the 26th division on the scale c d:
move by hand the microscope h , with its sliding plate, until the
division of the index at k coincide with the division of 20 inches
on the rule jfg ; then move by hand also the microscope i, with
its sliding plate and appendage l m n 0 , until the index division
near m coincide with 26 inches on fg: the axes of the micro-
scopes, or centres of their cross wires, will be at the approximate
distance of 6 inches. To correct this, examine if the wires of
b correspond with a division on c d ; if not, move the rule f g
backward or forward, by the screw g, till they do, then will
the microscope h be adjusted. Now examine if the wires in i
cover exactly a division ; if they do so, the true interval of 6
inches between the microscopes is obtained; if not, move
the microscope i a little, by means of the screw /, till they do,
dnd both the microscopes will be adjusted : then remove the
rule c e d from its place, by taking out the screws ced , and
place the object to be measured in its room, at the same time
taking care that it be exactly in the focus of the object glass of
the microscope, in such manner that one extremity may cor-
respond with the wires in the microscope h ; that done, if the
other extremity coincide with the wires in i, the dimension
of the object is exactly 6 inches ; if not, restore the coincidence*

* A farther account of these scales is given in the Appendix,

to ascertain a Standard of Weight and Measure. 139

by turning the micrometer screw n , and the divisions at 0 will
give the difference, in loooths and io,oooths of an inch, -j- or
— 6 inches.

Ql ecef'

7fi >

(§. 9.) Description of the Hydrostatic Balance.

abed , (Tab. VI.) is a box, which contains the whole ap-
paratus when not in use, and when used serves as a foot to
the hollow brass pillar efg h, which is fixed into it by the four
screws at the bottom e and/. This pillar contains another
within it, which is raised up and down about inch, by
means of the screw x. no is the beam, 27 inches long, and
3,9 inches wide in its greatest diameter ; each arm of which is
made hollow and conical, for strength and lightness : through
the centre, at m , passes the axis of motion, the ends of which,
when used, are suffered to fall gently upon two crystal planes,
which are set horizontally by means of the spirit levels k, l, and
the screws underneath the box, at c and b. The ends of this
axis are of hardened steel, of a wedge-like shape, and reduced
to a fine edge, viz. to an angle of about 40°, so as to move
upon the planes with very little friction, and at the same time
so hard as (with due care in using) to be in no danger of be-
ing blunted : to prevent which, the inner pillar has a motion
upwards, as has been said, by the screw x, and, by means of
a semicircular arm at its upper extremity, lifts the beam off its
bearings, when it is not used, or is greatly loaded. This axis
is placed carefully at right angles to the beam; and, by means
of two small brass springs that press gently at the ends, is.
brought always to have the same bearing upon the crystal ; so
that no error need be feared from a small deviation from the
right-angular position of the axis to the beam, should any such

T 2

140 Sir George Shuckburgh Evelyn's Endeavours

exist; and, from its shape and quality, it maybe considered as in-
flexible in any ordinary experiments. At p is a small adjusting
screw, which raises or depresses a weight within, and with it,
in consequence, the centre of gravity of the whole beam ; by
this means, the motion on its centre may be brought to almost
any required degree of sensibility. Should the centre of gra-
vity be raised above the centre of motion, the beam would turn
over ; if it be in that centre, the beam would stand any where
indifferently, without any vibration ; if it be placed much below
it, the vibration would be too quick, and its sensibility not suffi-
cient: it is therefore brought, by the screw />, a very small quan-
tity below the centre of motion, so as to describe one vibration
in 40 or 50 seconds; the sensibility is then fully sufficient.
At each end of the beam are circular boxes, n and 0, through
which pass the steel centres, from whence are suspended the
scale-pans q and r : these centres resemble, in some degree,
those at m, but have their chamfered or angular edges upwards,
and thereon hang the hooks (3, to which are affixed the links a,
and to them the three silken lines of the scale. Each of these
centres has a motion in its respective box, by means of two
small adjusting screws ; that in 0 laterally, and that in n ver-
tically ; the former to make the two arms of the beam of an
equal length, the latter to bring the three points of suspension
of the beam and scales into a right line. At the extremity of
the boxes are fixed two needle points or indexes, which play
against the ivory scale of divisions at s and t. These divisions,,
although they do not, indeed they cannot, shew any definite
weight, are nevertheless very useful in making the adjustments,
and even in weighing to the small fractions of a grain, u v are
two steady plates, that are raised or depressed by the wooden nut

to ascertain a Standard of Weight and Measure. 141

w, to check the vibrations of the scales q and r, and bring them
more speedily to an equilibrium, y z is a table, whereon the
whole is placed, to raise it to a height convenient for experiments.

To use with this beam, I had three sets of weights made, viz.

The 1st set or series of 15 weights, rising in a duplicate pro-
gression from 1 to 1 6384 grains, viz.

No. Grains.

3 =

4 =

5 =

6 =

7 =

8 =

9 =

10 =

11 =

12 =

*3 =

14 =

15 =

1

2
4
8

- 16
32
64

- 128

2 56

- 512
1024

- 2048

- 40 96

- 8192

- 16384

Fractions
of a
Grain.

1

3 2

__i_

I 6

J_

8

X

4-

I

o ' 00 n
O' 004 0

O ' oo 7 /

0 * /&

0 *03 &

0 -of 3f
0 ' M a 2
0 * 20 2 S
0 S' /

/'//O'!

r$405
4’ Qzt/
Qy3f>10

78 :7Z4 -

/J rt /

The 2d series of weights, in an arithmetical order, as follow, viz.

Grains-

Grains.

Grains.

Grains.

Decimal Fractions of a Grain, viz.

xoolh Grain.

T enths .

1

IO

IOO

IOOO

,01

,10

2

20

200

2000

,02

,20

3

30

• » ft

« • • •

>03

>3°

4

40

400

4000

,04

,4°

5

50

• • •

• 9 • •

,50

6

60

600

6000

,06

,60

7

70

• * •

... *

>°7

,7o

8

80

800

8000

,08

^80

9

90

• • •

•

10,000

,09

> 9 0

V. B. The fractions of a grain are

20,000

made of fine wire flatted.

. •$£/? {

J; i G

• £/

r

3

C

3

r7/, ' /y »- 7

/ -
t.y r • ’.4-*

'oooa r
'OOOi .?

’OOl

000 2 £
006? r 1

'oon

/

to ascertain a Standard of Weight and Measure. 141

wy to check the vibrations of the scales q and r, and bring them
more speedily to an equilibrium, y z is a table, whereon the
whole is placed, to raise it to a height convenient for experiments.
To use with this beam, I had three sets of weights made, viz.
The 1st set or series of 15 weights, rising in a duplicate pro-

n _ o .

JfiPt mAh { {.<? ' j A 1

0

r\

J

/

//■■ A-

A -i ,-a

■a

•r r A-

r

07r

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2 = 0'Q0/\

3- 0 '006&

4- 0 VOf/

d - 0 ’O//4
£ = 0 '0/3J

/ =. d 'qiG

8 - 0 mot%Z

Q =■ i*k020f

A,

/

9*/

v.,

r* "7

{ *:-> ft a «> e,
()%-

JL = #'00228 2-/7 A 2 ' \ f 4,
f - 0‘ 00/, rj// ? 8 f 1 -

- 0 ‘oo 68 f y// % -;i

- 0 'ooq 1 4 2$

=r ' af/4 i$ $// 4

- b’Q/'bJf/tS 57

- ^ ^ iT ^ 0 a

= 807/42

= *01O?7JA 25T

>

r

,3

"2

*•*

gT

/ i

>

c>

?7‘ It.

/A A /

j? ,i* ^ « .

; v •

/

'/

7 2

/ •/ A! *

~ fL. ",

.'r / -/

- .. w .y j ts4i_ 4

4 * *

1

10

100

1000

,01

,10

£

20

200

2000

,02

,20

3

30

• • a

« 1 « •

>03

,3°

4

40

400

4000

,04

,4°

5

50

• • •

• 0 • •

,°5

,50

6

60

600

6000

,ob

,bo

7

70

• • •

,07

,7°

8

80

800

8000

,08

„8o

9

90

• » •

10,000

20,000

,09

V 5. The fra<
made of fine

,9°

:tions of a g
wire flatted

-y / K

V<W/ { ' 4 /

■ t J r a.

.At , (? +* y

r ‘ ,

r

3

V/4.

Cf A

/ , ,• '71

7? - ^

9 00 on -<r

W02 £
r

’00// .5

142 Sir George -Shucks urgh Evelyn’s Endeavov ™

flounce 1 ; 7/

!

The 3d set consists
of a weight of

2 ounces . ;

1 *

4 ounces ‘/Troy. =

I 8 ounces

I , ■ v

(_ 1 pound j ' . 7

(§. 10.) abed and abed, (Tab. VII. fig. 1.) is the brass
cube of 5 inches that has been mentioned, suspended in its
own scale, by means of four fine wires, from the arm 0 of the
beam, Tab. VI. by taking away the common scale a r. The cube
rests upon a cradle or cross, three arms of which are seen at^ h i,
and by this means may be weighed either in air or water, by
immersion into the large glass vessel gh, Tab. VII. fig. 3.

At fig. 2. is seen the cylinder abed and abed, four inches in
diameter, and five high, slung in another cradle, part of which
is seen at gb hi, supported by four wires from the point/.

In fig. 3. is seen a sphere of brass d, 6 inches in diameter,
slung in a cradle a be, by three wires* from the links/, sus-
pended in a glass jar,/ containing near four gallons of water,
whose temperature is shewn by a thermometer at e.

* These wires were of such a size that 91 inches weighed 20,71 grains, con-
sequently 1 inch — 0,2276 grain, and the three wires =: 0,6828 grain ; and their
specific gravity being 8,7, their loss of weight, by sinking 1 inch in water, would be
— 0,0785 grain. This correction it may be necessary hereafter to attend to.
f The glass jar is made somewhat conical, being in

Which is in ale gallons - - — 3j° — 1 5 y quarts.

It may also be noted, that 1 inch in depth of the water near the top is = 113 cubic
inches, which is equal to the exact bulk of the sphere, as will be seen hereafter.

Diameter at top
Ditto at bottom
Mean ditto
Mean height within
Contents in cubic inches

inches,

12,0

to ascertain a Standard of Weight and Measure. 143

(§.11.) It was necessary to measure the exact size, and cor-
rectness of figure, of this sphere. For this purpose was made a
wooden gauge orframe a be £?£,(Tab.V. fig. 2.) in which the sphere
was placed, upon semicircular pieces within, lined with green
cloth to prevent bruising it : upon this frame was placed a brass
square klmn , whose sides were about inch in length more

than the diameter of the sphere. This square, by raising or
lowering the screws or s, was easily made to coincide with a
plane passing through the centre of the sphere, p is a micro-
meter screw, the interior extremity of which is brought just to
touch the surface of the sphere, while the opposite side bears
gently against the interior side of the frame at 0 ; and, by turn-
ing the sphere round, so as to present different diameters to
these points of contact, any variety in the diameter may be
seen by the index /, and plate q, divided into io,oooths inch.
To render this operation more convenient, three great circles
were drawn with a pencil upon the sphere, at 90° distance from
each other, (the two former were traced by the artist in the
lathe, while the sphere was making, and the third was drawn
from them, ) and each was divided into 8 equal parts. The im-
mediate result of these experiments would only give the diffe-
rences, and not the absolute quantity, of the diameter ; for this
purpose, a brass ruler r, fig. 3. was made, of such a length
as just to go within the brass frame klmn ; and, being sub-
stituted in the place of the sphere, could easily be compared
with any given diameter, and afterwards measured with the
divided scale, fig. 1. With these instruments I made the fol-
lowing observations, August 31, 1 796, the thermometer being
at

144 Sir George Shuckburgh Evelyn's Endeavours

(§. 12.) Examination of the Dimensions of the Brass Cube, by

The microscope and micrometer being both adjusted, as well
with respect to their focus,* as to the value of the micrometer
scale, the cross wires in their focus were removed to a distance

being at 27, and the latter at 32, inches,) and then correctly ad-
justed to this interval on the divided scale. I must observe,
indeed, that the value of the micrometer scale was not exactly
ten revolutions of the screw to YJo inch, as Mr. Troughton
designed ; but this measure by the screw,-f- from 6 trials, was
deficient by — 0,0002 inch ; viz. two ten-thousandths of an
inch were to be added to each tenth of an inch measured by
the micrometer, and so in proportion for a less quantity ; but
this correction is hardly worth notice.

microscope and micrometer

Interval of ditto, on another part of the 1 ^ ^ ^

>20 and 31 = 5,0000 —

scale, viz. J

Ditto, ditto 25 and 30 = 5,0001 -J-

Means of the divided Scale.

from each other of five inches nearly on the beam, (the former

On the scale.

The interval of the cross wires in

Inch.

* The focal length of the object lens is

The distance of the cross wires from the object lens
The focal length of the combined eye glass

2,00

f One Revolution of the screw of the micrometer was
Each grand division, of which there were ten ' -
These again subdivided into five, each became
And half a division, which is very visible, is

Whence the magnifying power of the microscope becomes — 14,2 times,
nno Revolution of the screw of the micrometer was - == inch.

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To the Independent Freeholders of the $3$$ 94$

County of Middlesex* !7T£ff3 7^

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W1'

GeNTLEMEHj

The friendship and acquaintance which
my recent engagements as the Agent ot your worthy repre-
sentative, Sir Francis Burdett, enabled me to form with a
numerous and respectable Body of the Freeholders of Middlesex,
indusS me to Solicit the favor of your Votes and Interest to
succeed the late Mr. Walter, as one of the Coroners of your
County. f

I am fully sensible of the important

an Office, which is almost the only one that is left to the Elgcnpi^
of the freeholders ; but great as I feel those Duties, I trust mat I
shall be able, by uniting integrity with diligence, to fulfil dhem
to the satisfaction of the bounty.

7 ' -

The arduous Contest for the rights of the Free-
holders, in which I am engaged on the part of your represen-
tative must necessarily occupy so much of my Time, as to render
it impossible that I should make a personal Canvas; but I trust
a!nd hope, that my determination to forego every personal Interest
of my own, which comes in competition with your .Rights, and
the success of your Representative, will form a suffici‘e*n£ Excuse

\K

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I ana. Gentlemen, /•'iGiAj1

Your very obedient Humble Servant,

J. AUGUSTUS BONNEY

Gray’s Inn,
March 26, I8QI.

J. Snieeton, Printer, 148, St. Martin’s Lane.

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to ascertain a Standard of Weight and Measure . J45

I therefore say, this interval was 5 inches correctly, to within
less than the twenty thousandth part of an inch, on this scale.

Measurement of the Cube , viz. of the Side 1. (See the Figure.)

\

Inches. Inches.

From a to 6 = 5 — , 0114 therefore = 4,9886

a to c — 5 — ,0115 -

c to d = 5 — ,0105
b to d = 5 — ,0113 -

The Side 2.

From a to b = 5 — ,0106
a to c = 5 — ,0098
c to d = 5 — ,0102 -

b to d~ 5 — ,0112

= 4>9885
= 4>9895
= 4>9887.

= 4>9894
= 4>9902
= 4>9898
= 4,9888

Mean.
Inches.

>= 4,98882

4’98955

Height of the Cube , from Side 1 to Side 2.

* Inches.

Inches.

From a to a = 5 — ,0110
b to b = 5 — ,0105
c to c = 5 — ,0107
d to d = 5 — ,0108

= 4.9890’
= 4.9895
= 4.9893
= 4.989a.

>= 4.98925*

* It cannot escape notice, that all these measures were something less than 5 inches,
the quantity proposed : it arose from this, Mr. Trouchton informs me, he was more

u

MDCCXCVIII.

146 Sir George Shuckburgh Evelyn’s Endeavours

(§• 13- ) Now the three foregoing mean measures of the side
JJ'ZJOSZ- /?4' /ywf foe. of the cube, multiplied into each other, will give == 124,18917
o* Jbn cubic inches, for the contents of the brass cube ; which must be

very near the truth ; for, if not, let us suppose the error, in ta-
/ /24- / 8^2052 ^ king each of these measurements, to be half a thousandth of an

inch, which is much greater than is probable, viz. = part
of the side of the cube; and let us suppose each of these errors
to lie the same way, which is also very improbable ; in that
case, the error in determining the solid content would be only
of the whole; in the above instance, about o,oq cubic inch;
but, more probably, the error does not amount to half this
quantity.

(§. 14.) Examination of the Cylinder.

The micrometer and microscope of the divided scale (Tab. V.
fig. 1.) being removed till their cross wires were four inches
distant, viz. from 54 inches to 58 inches, and the thermo-
meter at 6 20, I observed, of the end or base of the cylinder.
No. 1.

solicitous to obtain a true figure, than the exact size; neither of which however were
very important, as both were to be proved by the mode I have adopted. What was
important, was to have the sides true planes ; and these were examined, as I am in-
~ j /formed, by the reflected image of the moon, seen through a large telescope, the mens

of which would be altered, if the surface were either hollow or convex.

to ascertain a Standard of Weight and Measure. 147

Inches. Inches. Mean.

The diameter ah = 4 — ,0027 = 3,9973 1 Inches-

1 2>9974&

c d = 4 ,0024 = 3,9970 J .

End 2. of the Cylinder .

Inches. Inches.

The diameter ^6 = 4 — ,0014 = 3,9986 j
t j = 4 — ,0029 = 3,9971 j

3’99785

IS s

TTG

Inches.

>= 5>995°zr

Height of the Cylinder .

The microscope and micrometer being placed respectively
at 52,1 inches and 38,1 inches, viz. at the interval of exactly 6
inches on the scale, I found

Inches. Inches. Mean.

The height from a to a = 6 — ,0049 = 3,995 1

b to b == 6 — ,0047 = 5’9953

c to r = 6 — ,0047 = 5>9953

d to d z= 6 — ,0054!

+ a r 58 >= 5>9944

Repeated - j g ^

Now the mean diameter of the cylinder having been found

Inches.

at the end 1 = 3,99745 .a_

at the end 2 = 3,99785 JB
The factor for the square of thed
diameter of a circle, to find the >= 0,7854 <*r o /4S3J8/ (03

area, being, as is well known,]

And the height of the cylinder =5,9950 0* rr, ft+r SyqsO'ZS’
The above four quantities, multiplied into each other, give
for the contents of this cylinder, in inches, = 74,94823; and
this result may be taken at least as correct as that of the cubes
viz. to about the third place of decimals.

+ 7£-74f2/2(j Vc"; f J //7f ///jn 7 7 / 1?777* /ft?

if ef^Oef/J faust insert r tty* 7 / //ey/<9 /ft Sus*.

Ofr>f 7X7 ex (ft // sf) ^ 72^7 c/teff // f*/ faff

/ft j/f/f) fryt/t-r-e/ ^f/f^ fcts&y c/fe*

u.

148 Sir George Shuckburgh Evelyn's Endeavours

(§.15.) Having adjusted the beam of the balance, (Tab, VI.)
with respect to the length of its arms, its centre of gravity, and
the three points of suspension of the beam and scales, and
having examined the weights, I proceeded to the remaining
parts of this experiment.

Sept. 2d, 1 796. The balance beam adjusted by the screw p>
till the vibrations were so slow as to require more than 50 se-
conds of time for each, yiso grain appeared to move the index
through three divisions* of the scale 5 and t, = f inch, when
the beam was not loaded; but, when the beam was loaded
with 16384 grains, or near 31b. Troy, grain was equal
only to o\ division-f of the same scale.

(§. 16.) Sept. 4th. The thermometer being at 63°, and the
barometer at 29,36 inches.

* 20 divisions are — 1,0 inch.

+ That is, the beam was sensible with 7-555000 Part t^ie w^°^e weight. Mr.

' #

Harris’s beam, with which he and Mr. Bird made their observations on the Ex-
chequer weights, turned with part of the whole weight, and was consequently

only ~ part so sensible as this. See “ The Report of the Committee of the House of
<< Commons in 1758, to inquire into the original Standards of Weights and Measures
« in this Kingdom, and to consider the Laws relating thereto.” See also a second
Report 11^1759 ; both of which contain a vast deal of useful information on this sub-
ject, extending through fifty folio pages, and are to be found in the zd volume of
Reports, from 1737 to 1767. A bill was brought in, in consequence, but afterwards
dropped; and it is much to be lamented, that this inquiry did not go to the full length
of an act of parliament. Note farther, the largest of the beams, of which there are
four of different sizes, now made use of in the dutchy court of Lancaster, for the
actual sizing of the weights of the kingdom, is about 3 feet long, and is moveable
with about 30 grains, when 561b. avoirdupois are in each scale, viz. about yj^oo Par£
of the whole.

The weight of the counterpoh
pan or scale for weighing
in air, was -

J

to ascertain a Standard of Weight and Measure . 149

>

grains,

413>4°

= 968,4s

413>4°

965,74

To which, add the weight of the common-'
pan with the silk lines, on the left arm
of the beam, and marked with x, the
common right-hand pan having been
removed - -

And the whole weight of the pan or ap-"
paratus for weighing the cube in air , >
becomes J

(S. 17.) The counterpoise to the pan orl

\ o i s l I j ox. grs.

scale for weighing the cylinder in air, >= 1 72,34 = 552,34
was found -

To, which, add the weight of the com-
mon pan on the left arm, as before
And the whole weight of the pan or scale"!
for weighing the cylinder in air, be- >
comes J

Note, in the preceding and such like experiments, the com-
mon right-hand scale being removed, and the left-hand scale
being always used, and always the same weight, viz. 413,40
grains, when either the cube, or cylinder, or any large body, is
weighed, notice need only be taken of the counterpoise weights,
viz. 555,02 grains, or 552,34 grains, respectively ; and these
are to be deducted from the general amount of all the weights
in the left-hand scale, marked x ; but it certainly would have
been more convenient to have had single weights, ready ad-
justed, for these counterpoises, both in air and in water. These,
though at first omitted, have since been supplied.

(§. 18.) The counterpoise to the scale for the cube, j &rain8‘

in distilled water, with the heat of 6i° - J

To this, add the weight of the common scale, as before = 413,40

17

1 50 Sir George Shuckburgh Evelyn’s Endeavours

1 % ,

e 7

il

44,1,68 " 7

'Wt

ra ;

TWJM3

ip

■If 2

£

1.

ah

8,7

!/<?

And we have the whole weight of the scale for the 1 £rains-

, . & 1=856,15 /

cube in water - J

But the weight in air having already been found - = 968,40
The difference of the weights - = 112,25

Gives for the specific gravity of this brass - = 8,63

(§. 19.) The counterpoise to the scale for the cylin-
der, in the same water, with the same heat -
To this, add the weight of the common scale, as before = 413,40
And the whole weight of the scale for the cylinder, ]

855,08

in water, becomes j

Its weight in air has already been found = 965,7 4

The difference of these weights - = 110,66

Gives for the specific gravity of this brass - == 8,78

The mean specific gravity of this brass and brass-
wire may therefore be put at about
N. B. The tables of specific gravity give that of wrought
brass from 8,00 to 8,20. It was necessary to ascertain the
specific gravity of the brass wire, to make the correction men-
tioned in the note to §. 10.; for, as it was highly probable, that
in experiments with this hydrostatic balance, the scales for the
cube and cylinder would occasionally be immersed to different
depths in the water, and their weights would be altered, as
more or less of the wires by which they were suspended re-
mained out of the water ;

I accordingly found, that 80 inches in length of thisl grainj
wire, used in the scales for the cube and cylinder, )>=6,i6
weighed in air - - - j

And consequently, 1 inch would be = 0,077 grain, and four
wires of 1 inch = ,308 grain ; which, divided by the specific
gravity, viz. fj, would give 0,0354 grain, for the correction of

f

to ascertain a Standard of Weight and Measure . 151

every inch that the scale was sunk lower in the water ; and so
in proportion.

(§. 20.) Experiment of the Cube of Brass weighed in Air.

The cube was suspended to the right arm of the beam, by
the scale belonging to it, and the left scale pan, with the
mark x, was hung at the other end of the beam, in which
were placed the following weights,* made by E. Troughton.

grains.

viz. No. 15 of 16384
14- 8192
13 - 4096
12 - 2048
11 - IO24
9 - 256

84,82

Inch.

The total weight of 1
the cube in air J

the barom. being at 29,0
the therm. - at 62°,o

£7 ‘AA 4/ A'

' I? • /r A .* ..

/ O

/) ' a

0 f * X ft A

W L> Q / • 1

0 •

0 # $ 7

r

7

(§.21.) Experiment of the Weight of the Cylinder in Air.

grains.

No.- 15 of 16384
13 - 4096
11 - 1024

grains. 53’37

Buta counterpoise of 555,02^

having been used,
by mistake, in-
stead of - 552,34

H *1,557.37
)*= . 2,68

Add this excess

2,68

And the total weight of the cy- 1

= 21,560,05

inch.

f the barom. at 29,0

finder is - J " [ the therm. at62°,o

* This scale contained also 555,02 grains, being the weight or counterpoise to the
scale for the cube.

4 <7*.

■j j 0 u a

. n -1

2

u} ‘ 3 & 2 28

/ 1 * & a 0

9

IQ

152 Sir George Shuckburgh Evelyn's Endeavours

■ \

i

(§. 22.) The Cube weighed in distilled Water.

Sept. 5. Put into the left scale, the counterpoise f 300 1 srain9

for the water scale

1 100J

= 400,00

The cube, with its scale, was then immersed in the water.

I then restored the equilibrium, by putting into the
opposite or left-hand common scale, Mr. Trough-
ton’s weights, No. 10.

(The barom. standing at 29,47 inches,
the therm, at 6o°,2.)

grains.

= 512,00
200,

3°>

3.70

But a counterpoise of

grams.

400

745. 7°

having been taken, by mistake, instead of 442,75 ^ ^

Deduct the di fference, whi ch was so left out=42 ,75 j —

The apparent weight of the cube in water becomes = 702,95

Add the correction * for the loss of weight of the 4I

wires, by immersion 2^ inches deeper than >= + >08

when the counterpoise was adjusted
And the true corrected weight of the cube in water,
with 6o°,2 of heat, becomes

7°3’°3

* When the cube was immersed, the water in the glass jar stood inches higher
than when the counterpoise for this water-scale was adjusted, and found to be 442,75
grains ; (see §. 18.) and 1 inch of alteration in the height of the water having appeared
to be = 0,0354 grain in weight, (§. 19.) 2| inches will be = 0,078 grain; and somuch
must be added, to correct for the loss of weight, in the four wires, that suspended the
scale and cube in water. When the cube was immersed, the surface of the water
stood 1,5 inch below the top of the glass jar, and 9,7 inch below the centre of the
beam, or index point.

When the cube was in the water, the beam was clearly, sfnsible yith T’6 of a grain.

to ascertain a Standard of Weight and Measure. 153

>►=•441,7

(§. 23.) Experiment of the Cylinder in distilled Water .

Sept. 5. The thermometer being at from 6o‘,2 to 6o°,5, and
the barometer 29,47 inches,

grains,
f QOO

Put into the left scale pan, the counterpoise |

to the water-scale for the cylinder - j i

J L 4d>7 J.

The cylinder, with its water-scale, was immersed in water.
I then restored the equilibrium, by putting into the left scale,

grains.

= 2048
= 256
200
30

IO

4

1,10

Mr. Troughton's weights. No. 12

No. 9

Weight of the cylinder in water
Add the correction for the loss of weight of
the four wires, by being l-J inch deeper
immersed in the water, than when the j

2549, to

>= + 0,05

*

counterpoise was adjusted j

Corrected weight of the cylinder in water

= 2549,15

ft

/

'?$ Jttt

W) 4-
' 1 A 0

( t

0 «? ? ? $ ^
riS! 0

> V. V ■7,. /. 4?

1 A),
l

• r* ,1 ‘

* In order that ibis and some other corrections may be the more easily applied,
I have computed the 3 following tables, to be used whenever great accuracy ig
required.

MDCCXCVHL

X

i54< Sir George Shuckburgh Evelyn’s Endeavours

After this experiment, I discovered that some small bubbles
of air had insinuated themselves between the cylinder and the

Table I. Shewing the expansion of cast brass, both in length and solidity, and also of
water, in solidity, by the effect of heat : the former is derived from Mr. Sm Eaton’s
experiments t (Phil. Trans. \ ol. XLV1I1.) and the latter from some of my own, when
I was a resident member of the University of Oxford.

Degrees

Expansion of Brass.

Expansion of Water.

of Heat.

In Length.

In solidity.

In solidity.

0

Millionth Parts.

Millionth Parts.

Millionth Parts,

I

1

3

165

2

2

6

33°

3

3

9

495

+

4

1 2

b6o

5

16

S25

6

6

*9

99 0

7

7

22

1 *55

8

8

25

1 320

9

9

28

14S5

10

10,4

3 1

1650

Table IT. Shewing the Correction for
the Wires, or the Diminution of the
Weight of the Water-Scales, by Im-
mersion in Water.

Urt

A

it

Bylmmer-
vion inWi-
ler.

The 4 Wires
of the Cube, or
Cylinder.

The 3 Wires
of the Sphere
lose

b>‘ Mr~

. — — ~~~

Inches,

Grains.

Grains.

•0000 bo

'Off 0/739

I

— 0.035

— 0,078

'aoc/oH t.

W’3fi

2

— 0,071

— OA57

' j* a/t *>y\x]

1 * V* / l

3

— 0,106

-0,235

7 f 7

4

— 0,142

-0,314

* & &0Jl Aj) j

'ooosro

5

— 0,177

— 0,392

‘t/tojjSd *.7

‘ OO JO? h ,

6

— 0,212

— 0,471

•

4 00 1 IS '

7

— 0,248

— 0,549

‘00 >£46 *

’ S

8

— 0,283

— 0,628

4 iC‘JO/$

•T V_

9

-0,319

— 0,706

e o<) 5 0 if /

‘ 00 J 7^4

10

— o,354

-0,785

^ ^ /I J k. j^o n

* u i-W i-v A.

20

— 0,708

- 1,570

eft/

K, B. 80 inches in length, of the wires') grs.
for the scales for ihe cube and cylin- y 6,i6
der weigh - - - - J

therefore i inch will be ,077 grain, ) Q

and 4 wires of 1 inch - ) °'3°

Also, at inches of the wire for the >

sphere weigh - 5' zo,f1

and 1 inch — 0,227, and 3 wires of ) __

1 1 inch - - - 5 °’

and the specific gravity of the wire is — 8,7

* r ‘ £* n ic

•* 7 n /f " '•> /)

V4/33 i

l

Table III. Shewing the Correction of the
Weight of the Sphere in Air, on Account
of the Weight, or Heat, of the Atmo-
sphere.

Barometer

Correction.

Therm.

Correction.

Inch. T*5.

Grains.

O

Grains.

z9’5

0,00

50

0,00

1

— ,12

I

-f 0,10

2

,23

2

o,zo

3

>35

3

0,30

4

,47

4

0,40

5

>5«

5

0,50

6

,70

6

0,60

7

,82

• 7

0,70

8

,94

8

0,80

9

1,05

9

0,90

10

Si 7

10

1,00

N. B. If the barometer is below ag^ inches, or the
thermometer below 500, use the contrary signs.

Water being taken as heavier than air, as 836 : t, (see
Observations in Savoy, Phil. Trans, for 1777,) the
barometer being at 29,27, and thermometer 510, a
sphere of air equal in bulk, to the brass sphere, viz.
— 1 13^ cubic inches, would weigh, when thebarom.
was 29,5 inch, and the therm. 50° = 34,57 grains;
and 1 cubic inch of such air - 2= 0,304

This correction will serve for any other body whose
bulk is known.

0 7/27 '’OOJ'IZJ

eccft 4 ' 0004s j
c co 8 'ooodS

'00/3?

* ‘

J ffj

* f

\ 0 3 / c

A * ft
v V C jL

$ j /V

1 JO/ s
\M2
\ao\

\oa20f

• 'to*? g

6f

7 ' 7

V -07SJ

V /

*

fiztst

‘ coen

T'^iC\3

to ascertain a Standard of Weight and Measure . 155

scale in which it hung ; these therefore were removed, and the
experiment repeated, as follows :

Weights
as before

grains.

No. 12 — 2048
No. 9 =

<

The buoyancy of'
the air bubbles
being removed.

Add the correction'
for the loss of
weight in the
wires, as beforej
And the more exact 'i
weight of the cylin-
der in water becomes

= +{

v= +

256
200
30
10
4

1,10
3

1,07
2553’ 1 7

0,05

r= 2553,22

Note, when

the cube
the cylinder'

" with the temperature 6o°,g

inches.

and the barometer 29,47

is weighed in water, its 'i mches‘
centre was below the 1=
surface of the water J ^’7

that is, the cylinder was the deepest by - - =1,2

The repetition of this experiment shews how necessary it is

to attend to the most trifling circumstances : there were not

more than three or four of these particles of air, and those not

larger than a small pin’s head. Moreover, it may be noted,

the distilled water in which these experiments were made,

being afterwards examined with my (Martin’s) hydrometer,

in the heat of 6o°£, weighed on that scale = 1,0005; so that

I see no reason for di ffidence in the quality of the water.

X 2

^/r~f A* 'A?

MS 3 ’22 'h3Si}3 3

4'

•4T-7/4

. /)/? /i /

/

) 6 /

3 56 Sir George Shuckburgh Evelyn's Endeavours

(§. 24.) A Synopsis of the preceding Experiments.

■ V

Cube.

Therm .

Cylinder.

Therm.

Barom.

* Ci * 0 ' 11 \

Indies.

O

O

Inches,

Contents (true to T&§55) m inches

124,18917

6l

74,94826

62

Weight in air, true to 0,02 grain

grains.

21560,05

62

'33&/3 /

32084,82

62

29,00

£0 ba 2 4

Weight in water, true to 0,10 grain

703,03

60,2

2553,22

60,5

29>47

f

Weight of an equal bulk of water, 1
true to 0, 1 2 grain, or T^ooo J

3*38i>79

19006,83

Weight of a cubic inch of water, "1
from these experiments - J

252,694

253,600 *

The diversity in the result of these two experiments is de-
serving of notice, and must be explained. It may proceed from
two causes, which we will now inquire into. But first it may
be observed, that the accuracy in measuring the dimensions of
these two bodies, as well as the precision in weighing them,
has, I think, been such as to put out of all doubt this part of
Tie experiment. From whence then does this difference arise?
Either of two causes may be suspected ; viz. the pressure of
the water against the sides of these two bodies altering their
volumes, which, it may be presumed, would have a greater ef-
fect on the cube, from its figure, than on the cylinder, and In
a direction agreeable to this difference; that is, it would dimi-
nish the capacity of the cube more than that of the cylinder,
and thus make the apparent weight of a cubic inch less in the
experiment of the cube. But also we see, that the cylinder was

'if 8 IS
Siyoj
■ i '/ 1 G<j

* The weight of a cubic inch of common or rain water has been reckoned
about 253 grains., sometimes — 253,33 grains, at others 253,18. But authors do not
seem to have agreed in what they meant by common water, rain water, pump water,
spring water, and distilled water ; for occasionally they are all confounded, and made
to pass for each other; and sufficient notice seems not to have been taken of the
temperature to which these weights were assigned. See Martin’s Pbilosopbia Bri-
tannica. Lewis’s Philosophical Commerce of Arts. Chambers s Dictionary, by Dr.
Rees. &c. &c.

• s

/sf

g yp £ A ?

A

jr ^ ; 7

*/

7^4 f

-

y

c.

<6* *

z

/ r* —

4 /n <•?- r*

r A

// r /

(\>

J

/ (Y/

^ v

/

y- :

/

/ /%£

A y\

?4'$2b' 7

J/-

,.y/j

/

4

is

,

97/823

to ascertain a Standard of Weight and Measure. 157

weighed at a greater depth, by 1,2 inch, than the cube, below
the surface of the water. Now, if it be true that water is com-
pressible,* it will become denser, from its weight, at different
depths, and this circumstance would act in the same way with
that just mentioned; viz. would make the apparent weight of a
cubic inch less from the experiment of the cube than the cylin-
der, which we see is the fact.

(§. 25.) In order to dissipate these doubts, I caused a very accu-
rate hollow brass sphere to be made, of about six inches diameter,
and of such thickness of metal, viz. 0,13 inch, as to be very little
heavier than water, and yet of such strength as, together with
its form, to resist any probable change of bulk by the pressure

of water.

This sphere, which has already been mentioned, (§. 10.)
was examined in the following manner. The six-inch move-
able bar r, (Tab. V. fig. 3.) of the gauge, was compared with
the divided scale of inches, fig. 1 . The microscopes being ad-
justed to exactly six inches, or the interval between 26 inches
and 32 inches, and the bar placed under them, the excess above
6 inches was found to be as follows, by the micrometer, n 0.

1st trial,
inches.

6+>0055

>°°53

,0036'

5oo57

>-

thermom.
64°, o

2d. trial, after re-adjustment,
inches.

h + ,0055 ~

,0032
,0055 >

,c°34

,0032

therm.
6*4°, o

Mean of the 1st trial = 6 ,00330 6 ,0033b

Mean of the 2d trial = 6 ,00336

Mean of both, orl

length of bar

k=6

J

1 0
,oo343j>in the temperature of 64°

* See Mr. Canton’s Experiment in the Phil. Trans. Vol. LII,

158 Sir George Shuckburgh Evelyn's Endeavours

(§. 26.) The bar was then placed in the rectangular gauge
k l m n, fig. 2. in the direction p 0 ; and the end of the micro-
, meter screw brought to bear against it repeatedly, so as to
touch without force, or considerable pressure; and the divi-
sions* cut by the index, on the micrometer plate of the gauge,
were as follow :

Trial x sf.
Division on the
micrometer.

%

63

70

66

Trial 2d.
Division on the
micrometer.

64 1

64D

therm.

62

therm.

65

>b2°,o

65

63

62E

02°, O

66 >

63

624-

63,3+

64,2

Trial 3d.
Division on the
micrometer.

thermom.

6 2C

'3

Mean = 66

The mean of these three means is 64,5, with the temperature
6q°, 1.

(§. 27.) The bar was now removed from the gauge, and
the sphere put there in its place ; and, by means of the three
great circles, each of which was divided into 8 equal parts, nine
several diameters of the sphere were taken, as follow :

Div. of
microm.

f 44

diam. I ^

Div. of

Div. of

Div. of

microm.

microm.

microm.

r4°i

r4°'

[41 1

diam.

50

therm.

diam.

42

therm.

diam.

49

therm’

AB ^

47

GH ■

45

'62 °,5

CD s

43

to

O

»

42

2

42

42

l46j

l44j

l44tj

Mean=

=45

42,6

43.9

I K

therm.

1 47 62°,5

451 0

46]

45.6

* Each thread, of this screw is = T|T inch, and each revolution of the screw is di-
vided into 100 ; so that every division on the micrometer plate is — totoo inch.

f In all these experiments with the gauge, the figures on the micrometer plate in-
crease as the screw goes forward ; viz. the higher numbers indicate a less interval
or diameter.

to ascertain a Standard of Weight and Measure. 159

f45

The above four mean dimensions may be called equa- J 42 ,6

torial, viz.

The mean of which is

I 4 3>9
14 5>6

- 44’3

diam.

Div. of micr.
r

44

46

therm.

EF ^4 >62°, 5

Q

.45.

Mean 44,8

Div. of micr.

42~

^ therm.

1,2. <4>5>6q°,6
4°

41

diam.

42,6

diam.

3>4

Div. of micr
40

44

K 41 >

40

45

therm.
62°, 8

415

These three last dimensions, together with the 1st TA B = 45
of the preceding set, may be called meridional J EF = 44,8

1,2 = 42,6

being in a circle at right angles to the former,
viz.

3,4 = 41,1

— 43’ 4

The mean of which is -

and differs from the former not quite ~-Q inch.

In another great circle, 90° from the preceding, comprising
the diameters already taken, E F and C D, at the intersection
of the two former circles, were taken

Div. of micr

41
44

4 44 >(

41

14° J

The diameters

5 taken as before/^'’ ^

c DJ L43’9

- - 42’5

7 $ - - - 41’0

The diameter

a (3

therm.

63°

diameter

y S

Div. of micr.

"40

45

< 42

40

42

therm.

63°, 1

Mean

= 43,0, which is that of another great circle

160 Sir George Shugkburgh Evelyn's Endeavours

or meridian, at right angles to the former; from whence it will
be seen, that not one of the three circles differs from another more
than about inch.

io»ooo

The preceding 9 mean dimensions of the diameter, collected,
are

A B = 45,
CD = 43,9
GH = 42,6
IK = 45,6
EF = 44,8
1,2 = 42,6

the mean of which is = 43,7
in the temperature - 62°, 6

3,4 =41,1

« Q = 4 2>5

=41,0.

Now the import of the foregoing experiments is this, that
when the mean diameter of the sphere is holden between the
points of contact of the gauge, near 0 and p, the index of the mi-

crometer shews - - - — 43,7 divis.

but, when the bar r is placed there, it shews = 64,5

the difference is = 20,8

and by so much is the bar shorter than the diameter of the

SPhere- . _ inches.

These divisions, 20,8, are equal to (§. 2 6.) - 0,00202

and the length of thebar has already (§. 25.) been found=6, 00543
therefore the true diameter of the sphere becomes =6, 00745
which quantity I think must be true to within inch.

(§. 28.) The cube of this diameter, 6,00745 inches x ,5236,
as is well known, will give the contents of the sphere in cubic
Ranches, viz. = 113,5194 inches, which must be very near the
truth : for, if not, let it be supposed that the inaccuracy in the
measurement, or the irregularities in the figure of this sphere.

to ascertain a Standard of Weight and Measure . 161

should be such as to amount to T™o inch, and these so many,
without balancing each other, as to produce a spheroidical form,
one of whose diameters should exceed the other by -j-£oo inc^ ’
in that case, the error in the assumed solid would not exceed
— JL_ part of the whole ; and this is a position infinitely too ex-
travagant to be admitted, when we recollect, that this diameter
has been probabty taken to within a tenth part of that error.

(§. 29.) The weight of this sphere, in air and in water, comes
next under our consideration ; the experiments for which were
as follow, made June 12, 1797; the barometer being at 29,74
inches, and the thermometer, in air, at 67°.

"Experiment the 1st.

The weight of the sphere in air, the counterpoise, 1
or weight of the scale or cradle, a bcf, (Tab. VII. I
fig. 3.) in which the sphere hung, being allowed j '
for*, so that this was the net weight - J
The sphere and scale suspended in
water, with its centre 5,6 inches
belowthe surface, and the heat 66°

Deduct the counterpoise, or weighty
of the scale, in water, with the ’
same heat of 66°, and same depth’f*

below the surface - j

The difference is the net weight of the sphere in
water, of the temperature 66°, which, deducted
from its weight in air

Leaves the weight of a bulk of water = the sphere,
in the temperature 66°, and 5,6 inches below
the surface -----

Troy grains.

28722,64

grains.

3°3>i7

253’3*

49>85

28672,79

* The weight of this scale, with its 3 wires, in air, was — 276,10 grains,
f The sphere having been weighed in the same depth of water that the counter-
poise to the scale was determined in, no correction for the greater or less immersion of
the scale-wires was here necessary ; which however will sometimes be the case. See
§. 29. and table II. of correction, §. 23.

MDCCXCVIII. Y

162 Sir George Shuckburgh Evelyn's Endeavours

Experiment the 2 d. June 16, 1797.

The barometer being at 30,13 inches, and the thermom. at 68e

gi dins.

Weight of the sphere, together with the scale, in air 29265,91
Deduct the weight of the scale, or counterpoise, 1

in air - - - J

Remains the total net weight of the sphere in air = 28721,88
And, to reduce this to the same state of the
atmosphere as the preceding observation,
viz. 29,74 inches of the barometer, add the
correction for 0,39 inch (see table, §. 23. )J
Also the correction for T of the thermometer = -f-
And the net weight of the sphere, in an atmo-1

-v — a ^ 1 J

>= +

544>°3

,88

,4^

,08

sphere of 29,74 inches, and heat of 67°, becomes j*
Weight of the sphere, with its scale,
in water, 3,7 inches below the sur
face, and the thermometer at j
66°, 1

28722,42

grains.

484>7°

J

From thence deduct the weight ofl
the scale in water - j

435,09

49,61

49,81

The net weight of the scale in wa-|

ter becomes - j

To which, add the correction for the
wires of the scale being immersed
2,53 inches deeper now, than when >= 0,20

its weight in water was determin-
ed (see table, §. 23.) - j

And the corrected net weight, in water, is
Which, deducted from its weight in air, leaves "
the weight of a bulk of water = the sphere,
in temperature 66°, 1
Correction for o°,i of heat*

And the true corrected weight of a bulk of wa-
ter equal to the sphere, reduced to the baro-
meter = 29,74, and therm. 66°, o, becomes

* One degree difference of heat in the water will alter the weight of the sphere in
water, or the weight of the bulk of water equal to it, — 4,54 grains ; so that, by far
the greatest source of error, in these experiments, lies in the difficulty of exactly know-
ing, and preserving, the temperature of the water.

+

28672,61

,45

■= 28673,06

to ascertain a Standard of Weight and Measure. 163

Experiment the 3 d. June 16, 1797

The true net weight of the sphere in air, re- '
duced to a state of the barometer of 29,74 '►=
inches, and therm. 67°, as in last experiment _
Weight of the sphere, together with 'j grains,

its scale, in water, 6,8 inches below >= 484,20
the surface; the thermom. at 66°, 1 „

Deduct the weight of the scale in water 435,09

The difference is the net weight off|

the sphere in water, of the tem- 49,11
perature 66°, 4

To which, add the correction for the
wires of the scale being immersed
5,5 inches deeper now, than when > -j- ,44

its weight in water was determin-
ed (see table, §. 23.)

The corrected net weight, in water, becomes
Which, deducted from its net weight in air,
leaves the weight of a bulk of water = the
sphere, and 6 inches below the surface, with
the heat of 66°, 4 - .

Correction for o°,4 of heat (see table, §. 23.)

The true corrected weight of a bulk of water
= the sphere, in the heat of 66°, o, and with
a pressure of the barometer of 29,74 inches,
and 6 inches below the surface - j

grains.

28722,42

■

49,55

r

28672,87

=+

1

l,8l

>—

28674,68

(§. 30.) Results of the Observations of the Sphere collected .

Correct weight of a bulk of water = sphere, the
barom. being at 29,74 inches, therm. 66°, o.

grains.

08672,79

By the 1 st observation
2d observation
3d observation

Mean of all

28673,06

28674,68

28673,51

At a depth be-
low the surface
of the water,
inches.

5.6

3.7

6.8

5.37

Y a

25?'41Z* r7M =■
4 3 £' /S 3 v(o . U? 45/

164 Sir George Shuckburgh Evelyn’s Endeavours

Which, I think, may fairly be presumed to be within 1 part
in 50,000 of the truth.

(§. 31.) Now the contents of this sphere having already
(§. 28.) been found to be = 113,519 cubic inches;

= 252,587 grains, will be the weight of a cubic inch of dis-
tilled water, under the circumstances above mentioned, by Mr.
Troughton’s weights.*

I think it may now be concluded, that the variety in the
experiments of the cylinder and the cube, (§. 24.) does not
proceed from the different depths 'f in the water, at which they
were made ; at least, that the pressure of 3 inches, in perpen-
dicular height of water, does not render that fluid more dense
bv -1- part, which may be reckoned an insensible quantity ;

J 20,000 l ; J

but that this variety did proceed from a difference in the yield-
ing of the sides of the cube and the cylinder. And lastly, I
hope it may be trusted, that the weight of a bulk of water

* But, as will appear hereafter, (§.41.) these weights are too light, when com-
pared with the standard in the House of Commons, by about x in 1523,92, the
correction therefore, for this difference, would be rr 0,165 grain, to be deducted
from - - - - ' 2 5 2^5 87 grains.

— ,165

And the weight of a cubic inch of distilled water, in grains of
the parliamentary standard, will be

}=

252,422

+ Bv means of an alteration and addition to my apparatus, since the experiment

T y

/ 1 * Cj .vtA

M*

tAf&i

/ (JD ts* O

U J U9 "

/'3/4'HA

±u‘

rfb2- ■

A o.rrct-

y // /A

7

) > •

9

abovementioned was made, I have been able to repeat it at greater depths below the
surface of the water, viz. when the centre of the sphere was 5 inches, 13 inches, and 21
inches, below, without any appearance of water having a sensible difference of density at
different depths. The vessel I used for this purpose was of wood, 32 inches high, and
12 square, containing 16 gallons, with two sides of plate-glass, to admit the light ; and
the wires by which the sphere was suspended were 45 inches long, and stronger than
before, viz. 100 inches of the single wire weighed 7.4,14 grains; and due allowance
was made for the different weight of the scale and wires, in air and water, from actual
experiment.

.r-*? 7

A

r

to ascertain a Standard of Weight and Measure. 165

= the sphere, has been determined to within of the whole,
and probably to within half that quantity.

(§. 32.) Having then, through the means of Mr. White-
hurst’s observations, and of his own instrument, ascertained
the length of his proposed standard, in the latitude of London,
113 feet above the level of the sea*, under a density of the at-
mosphere corresponding to 30 inches of the barometer, and 6o°
of the thermometer, which is full as satisfactory, for all practical
purposes, as if it had been done in vacuo which I conceive
to be nearly impossible ; and, having determined the weight of
any given bulk of water, compared with this common measure;
I believe it now only remains, to ascertain the proportion of this
common measure and weight, to the commonly received mea-
sures and weights of this kingdom.

(§. 33-) ^ is perfectly true, that if I chose to indulge in fan-
ciful speculation, 1 might neglect these comparisons, as an un-
philosophical condescension to modern convenience, or to an-
cient practice, and might adopt some more magnificent integer
than the English pound or fathom; such as the diameter or cir-
cumference of the world, &c. &c. and, without much skill in
the learned languages, and with little difficulty, I might ape
the barbarisms of the present day. But in truth, with much
inconvenience, I see no possible good in changing the quan-
tities, the divisions, or the names of things of such constant
recurrence in common life ; I should therefore humbly submit
it to the good sense of the people of these kingdoms at least, to

* The height, as I have been informed, of the room of Mr. Whitehurst’s obser-
vations.

f It is perfectly true, that this supposes the experiment to be made with a pendu-
lum similar to Mr. Whitehurst’s.

1 66 Sir George Shuckburgh Evelyn's Endeavours

preserve, with the measures, the language of their forefathers.
I would call a yard a yard, and a pound a pound, without any
other alteration than what the precision of our own artists may
obtain for us, or what the lapse of ages, or the teeth of time,
may have required.

(§< 34.) The difference of the length of the two pendu-
lums, from Mr. Whitehurst's observations, appearing to be
59,89358 inches, on Mr. Troughton's scale ; and a cubic inch
of distilled water, in a known state of the atmosphere, having
been found to weigh 252,587 Troy grains, according to the
weights of the same artist, it remains only to determine the
proportions of these weights and measures, to those that have
been usually, or may be fitly, considered as the standards of
this kingdom ; and herein a small discrepancy between them-
selves, in these authoritative standards, will have no influence
on the general conclusion I propose to draw ; which is, not so
much to say what has been the standard of Great Britain, as
what it shall be henceforward, and may be immutably so ; and
which shall differ but a very small quantity, and that an as-
signable one, from those that have been in use for two or three
hundred years past. By these means, no inconvenience would
be produced from change of terms, or subdivisions of parts, or
from sensible deviation from ancient practice: all that will be
done, will be to render that certain and permanent, which has
hitherto been fluctuating, or liable to fluctuation. To give effect
and energy to these suggestions, is the province of another
power.

(§. 35.) The chief standards of longitudinal measure, as far
as I can learn, that carry any authority with them, are those
preserved in the Exchequer; in the House of Commons ; at the

to ascertain a Standard oj Weight and Measure. 167

Royal Society; and in the Tower. The first alone, indeed,
bear legal authority, and have been in use for more than 200
years ; the last is considered as a copy of them, and is not used
for sizing generally. The two remaining ones are of modern
date; and, although they do not carry with them at present any
statuteable authority, yet, from the high reputation and acknow-
ledged care of the artists who made them, (the celebrated Mr.
George Graham, and Mr. John Bird,) are undoubtedly en-
titled to very great respect ; and are probably derived from a
mean result of the comparisons of the old and discordant ones
in the Exchequer. I shall begin with that of Mr. Graham,
which contains also the length of the Tower standard laid down
upon it; will proceed then to Mr. Bird’s, and finally conclude
with those at the Exchequer.

(§. 36.) May 5, 1797. I went to the apartments of the Royal
Society, at Somerset House, and, with the ready assistance of Mr.
Gilpin, at the kind instance of Sir Joseph Banks, I made the
following observations on Mr. Graham’s* -brass standard yard,
made in 1742. This scale is about 42 inches long, and half an
inch wide, containing three parallel lines engraven thereon, on
the exterior and ulterior of which are three divisions, expressing
feet ; with the letter E at the last division, and, by a memoran-
dum preserved with it in the archives of the Society, is said to
signify English measure, as taken from the standard in the
Tower of London. That with the letter F denoting the length
of the half of the French toise; put on here, by the authority and
under the inspection of the Royal Academy of Sciences, then

* This rod was not made by Mr. Graham, but, at his instance, procured by him
from Mr. Jonathan Sisson, a celebrated artist of that time. See Phil. 1 rans„.
VoL XL II.

168 Sir George Shuckburgh Evelyn’s Endeavours

subsisting at Paris, to whom it was sent in 1742, for the pur-
pose of comparing the French and English measures. The
middle line, marked Exch. of the three abovementioned, de-
notes, as is supposed, the standard yard from the Exchequer.

(§. 37.) This bar of Mr. Graham’s had been previously
laid together with my scale divided by Mr. Troughton, for
twenty-four hours, to acquire the same temperature; they
were also of the same metal, and, by placing it under my
microscopes, adjusted to the interval between 10 and 46
inches, I found the interval on the Tower standard exceed

Inches.

mine, by —

0,00127
,00135
,00128 ^

= the total length therefore 36,00130
inches, the thermometer at 6o°,8.

Mean = ,00130

The interval on the line marked Exch. was shorter than

Inches.

mine by

— ,0066
,00 66
,0068 >

= the total length = 35,9933
inches, the therm, at 6o°,6.

,0067^

And the Paris half-toise, which had been supposed by the Aca-
demy to be = 38,355 English inches, was found, compared

Inches.

with mine, to be = 38,356 1

>3563

>3 539

Inches.

>Mean = 38,3561 *.

• Dr. Maskelyne says, this standard yard of Mr. Graham's was inch
longer on three feet than Mr. Bird’s divided scale, which he generally made use of
in all his operations of dividing ; and, from one made conformably to this of Mr.
Bird’s, Mr. TROucHTONdivided my scale of 60 inches. This remark seems to

to ascertain a Standard of Weight and Measure . 169

Inches.

The 1st of the preceding observations giving - - 36,0013

The 2d ... 35,9933

The mean length of Mr. Graham's standard becomes 35,9973*

agree with my ist and 3d comparison, but not with the intermediate one. See Phil.
Trans, for 1768, p. 324.

As I am now upon the subject of foreign measure, it may not be quite out of place
to say a word on the length of the ancient Roman foot, which I am enabled to do
with some precision.

Some years ago, when I was in Italy, I had several opportunities of ascertaining the
length of this measure, by actual examination of the Roman foot rules, of which I
have met with nine, viz. two in the Capitol at Rome ; one in the Vatican; five in the
Museum at Portici, near Naples; and lastly, one in the British Museum, sent from
Naples by Sir Willi am Hamilton. They were all of brass, except one half-foot, of
ivory, with a joint in the middle, resembling our common box or ivory rules : and, by
reference to my journal kept at that time, I find the mean result from all the nine
rules, viz. by taking both the whole and the parts of each, (for they were divided into
12 inches, and also into i6ths, or digits,) gave, for the length of the old Roman
foot, in English inches, correspondent to Mr. Bird’s measure, — 11,6063.

In confirmation also of this conclusion, and agreeably to the idea of Mons. De la
Cond amine, in the “ Journal of his Tour to Italy”, I took the dimensions of seve-
ral ancient buildings, viz. the interior diameter of the temple of Vesta ; the width of
the arch of Severus ; the door of the Pantheon ; and the width of the base of the
quadrilateral pyramid of Cestius, which, it is curious to observe, I found exactly 100
old Roman feet, and 125 feet high. This I do not remember to have seen noticed by
any former traveller.

The mean result of these experiments gave me - 11,617 English inches.

Ditto, as before, from the rules - 11,606 ditto.

The mean of the two modes of determination is - 1 1,612 ditto.

I may add, that in the Capitol is a stone, of no very ancient date however, let into
the wall, on which is engraven the length of several measures, from whence I took the
following :

The ancient Roman foot, = 11,635 English inches.

The modern Roman palm, — 8,82 ditto.

The ancient Greek foot, — 12,09 ditto.

z

MDCCXC VIII.

170 Sir George Shuckburgh Evelyn's Endeavours

(§. 38.) From the information in the report of a committee
of the House of Commons, that sat in the year 1758, I learnt
that Mr. Bird's parliamentary standard had been in the cus-
tody of some of its officers, but of whom nobody knew : how-
ever, under the authority of the speaker, who was so good as
to furnish me with a room in his house, to make the compari-
sons in, I at last discovered this valuable original in the very
safe keeping of Arthur Benson, Esq. Clerk of the Journals
and Papers, and which, I believe, had never seen the light for
five-and- thirty years before. It is a brass rod or bar, about 39
inches long, and 1 inch square, inclosed in a mahogany frame,

inscribed “ Standard 1 758" ; at each extremity of it is

Geo. 2d.

a gold pin, of about ~ inch in diameter, with a central point,
and these points are distant = 36 inches. It bears, however,
no divisions ; but there was found with it, in another box, a
scale divided into 3 6 inches, with brass cocks at the extremities,
for the purpose of sizing or gauging other scales or rules by.
Besides these, I found another standard, in size, and in all re-
spects, similar to the last, inscribed 1760, having been made for
another committee, that sat in that year ; this also was accom-
panied with a similar divided scale of 36 inches.

These bars being too thick to be conveniently placed under
the microscopes of my instrument, the interval of 36 stan-
dard inches was laid down on my scale with a beam-compass,
two fine points made, and, compared with Troughton's di-
visions, was = 36,00023 inches ; the thermometer being at 64°.
I then examined the other standard, marked “ Standard, 1760",

to ascertain a Standard of Weight and Measure. 171

and found it to agree exactly with that of 1758 ; at least it did
not differ from it more than ,0002 inch*.

(§• 39') I was now to examine the old standards kept in
the Exchequer : these Mr. Charles Ellis, Deputy Chamber-
lain of the Tally Court at the receipt of the Exchequer, was so
good as to supply me with ; viz. the standard yard of the 30th
of Eliz. 1588, and also the standard ell of the same date.
These are what have been constantly used, and are indeed the
only ones now in use, for sizing measures of length^. They
are made of brass, about 0,6 inch square, and are very rudely
divided indeed, into halves, quarters, eighths, and sixteenths ;
the lines being two or three hundredths of an inch broad, and
not all of them drawn square, or at right angles to the sides of
the bar, so that no accuracy could possibly be expected from
such measures. However, the middle point of these transverse
lines, between the sides of the bar, was taken as the intended
original division ; and these divisions, such as they were, were
transferred, by a dividing knife, to the reverse side of my
brass scale made by Mr. Troughton, the thermometer be-
ing at 63°; and, at my leisure afterwards, I found as follows.

The ends of these venerable standards having been bruised
a little, or rounded, in the course of so many years’ usage, I
conceived a tangent to be drawn to the most prominent part,
which was about the centre or axis of the bar, and this point

* These quantities then being so small, I shall consider them as wholly insensible;
and shall say, that Mr. Bird’s parliamentary standards of 3 feet exactly correspond
with Mr. Trou gh ton’s scale.

+ There was also a standard yard of Henry VII. but of very rude workmanship
indeed ; now quite laid by, and at what time last used, no information remains : but
of this more hereafter.

Z2

172 Sir George Shuckburgh Evelyn’s Endeavours

being referred to Troughton’s scale, between 6 and 42 inches,
the entire yard of 1 588, measuring from one extremity to the
other, was found to be shorter than this, by — ,007 inch : but
these comparisons will be better exhibited in a table.

Exchequer standard of 1588.

Difference

from

Trough-

ton.

Length in
• Inches.

Difference
on 36
inches.

Mean difference
on 36 inches.

Entire yard

Inch.
— ,007

35,993

— ,007

iyard, from 2410 42 inch.

+ >°®3

18,063

+ ,126

Inch.

J yard, from 15 to 42 inch.
|-yard, from 104 to 42 in.

— ,008
-f- ,022

26,992

3:1,522

— ,011
-fr ,025

>= + 0,015

y|-yard, from 8^ to 42 in.
Entire ell, from 2 to 47

— ,°55

33

pi

0

*\

1

-

inch. -

\ ell, from 2 to 244 inch.

— ,036
+ >°32

22,532

— ,020
+ ,052

from 2 to g#! inch.

+ >ot7

33>767

— b ,018

> — — }~ 0,016

-b, from 2 to 41,37$ inch.
h from 2 to 44,1875

— ,001

39>374

— ,001

Viz. the Exche-
quer measure is
by so much the

j inch.

+ ,°55

42,239

+ >°43

longer, or aboui
1 in 2322.

(§• 40.)' It appears then, from the above table, that the an-
cient standards of the realm differ very little from those that
have been made by Mr. Bird, or Mr. Troughton, and conse-
quently, even in a finance view, (if one might look so far for-
ward,) nothing need be apprehended, of loss in the customs, or
excise duties, by the adoption of the latter.

to ascertain a Standard of Weight and Measure . 1 73

(§.41.) I shall now endeavour to shew the proportion of the
weights that I have used, compared with the standards that
were made by Mr. Harris, Assay Master of the Mint, under
the orders* of the House of Commons, in the year 1758. They
are kept in the same custody with Mr. Bird’s scales of length,
and appear to have been made with great care, as a mean re-
sult from a great number of comparisons of the old weights in
the Exchequer, which have been detailed at length in that re-
port. Mr. Harris having been of opinion that the Troy pound
was the best integer to adopt, as the standard of weight, I ven-
ture to conclude that this was the most accurate, and most to
be depended upon, of all the various weights and duplicates
that he made for the use of this committee ; for he made them
of 1, 2, 4, 8, 1 6, lb. and of 1, 2, 3, 6 ounces. It will there-
fore be sufficient for my purpose, to compare the 1 and 2 pounds
Troy, and their duplicates, with the weights of Mr. Trough-
ton.

I did this, June 2d, 1737; the barometer being at 29, 72 inches,
and thermometer 67°.

Troughton’s weights.

The standard weight of 1 Troy pound, or! ib. grains. grains,

576ograins, marked 1758, kept at the j = 1 3,75"! 7g„

House of Commons, in a small box by [ ,74/ b

itself, by Mr. Benson, weighed - j

A duplicate of the preceding, kept with ~\ = 1 3,70! ^

some other weights, in a box marked BJ ,6jj **

The mean weight of the Troy pound, from these
two - = 57%715

* See the report referred to in the note of page 148.

174 ' George Shuckburgh Evelyn’s Endeavours

Tsoughton’s Weights,
grains.

~ lOOOO 1

rains*

The two-pounds weight, from the
House of Commons, kept in a
deal box, marked A

i

=<<

L

1000

400

100

20

7

0,84

>=11527,84

r 10000

A duplicate of the last mentioned 2 lb.
weight, preserved in a deal box,
marked B

The thermometer now stood at 68°

1000
400
100
20

7

o ,55

>= 11527,55

Therefore the mean weight of 2 lb. Troy, from then
two last trials, is - j" 11527>70

And consequently 1 lb. becomes - 5763,85

But, from the examination of the two single pound!

weights, as above, 1 pound is - - j ^^3>7 1

Therefore the mean of all is - - = 5763,78

That is, Mr.TRouGHTON’s weights are too light by 0,6562

grain on 1000 grains, or 1 in 1523,92 grains.

(§. 42.) In conclusion, it appears then that the difference of
the length of two pendulums, such as Mr. Whitehurst used,
vibrating 42 and 84 times in a minute of mean time, in the la-
titude of London, at 113 feet above the level of the sea, in the
temperature of 6o°, and the barometer at 30 inches, is = 59,89358
inches of the parliamentary standard ; from whence all the mea-
sures of superficies and capacity are deducible.

That, agreeably to the same scale of inches, a cubic inch of
pure distilled water, when the barometer is 29,74 inches, and

to ascertain a Standard of Weight and Measure . 175

thermometer at 66°, weighs 252,422 parliamentary grains ; from
whence all the other weights may be derived.

As a summary of what has been done, I hope it may now be
said, that we have attained these three objects ;

1st. An invariable, and at all times communicable, measure
of Mr. Bird's scale of length, now preserved in the House of
Commons ; which is the same, or agrees within an insensible
quantity, with the ancient standards of the realm.

2dly. A standard weight of the same character, with reference
to Mr. Harris's Troy pound.

3dly. Besides the quality of their being invariable, (without
detection,) and at all times communicable, these standards will
have the additional property of introducing the least possible
deviation from ancient practice, or inconvenience in modern
use.

(§. 43.) Before I close this Paper, after having said so much
on the subject of weights and measures, it may not be impro-
per to add a few words upon a topic that, although not imme-
diately connected, has some affinity to it; I mean the subject
of the prices of provisions, and of the necessaries of life, &c. at
different periods of our history, and, in consequence, the de-
preciation* of money. Several authors have touched inciden-
tally upon this question, and some few have written professedly
upon it ; but they do not appear to me to have drawn a distinct
conclusion from their own documents. It would carry me in-
finitely too wide, to give a detail of all the facts I have collected;:
I shall therefore content myself with a general table of their

* The various changes that have taken place, by authority, in different reigns, either
in the weight or alloy of our coins, are allowed for in the subsequent table.

17 6 Sir George Shuckburgh Evelyn's Endeavours

results, deduced from taking a mean rate of the price of each
article, at the particular periods, and afterwards combining
these means, to obtain a general mean for the depreciation at
that period; and lastly, by interpolation, reducing the whole
into more regular periods, from the Conquest to the present
time : and, however I may appear to descend below the dig-
nity of philosophy, in such oeconomical researches, I trust I
shall find favour with the historian, at least, and the antiquary.

[To face p> 1 76.

P

A Table exhibiting the
from the Conquest to
added, the Mean Appn

in Decimals, at different Periods,
y inferred therefrom. To which is
<fng the present Century, at shorter

Periods, deduced by Int

Mean Appre-
ciation by
Interpolation.

A. D.

lO^O

26

YlOO

34

1 150

43

1200

31

I25O

60

I3OO

68

1350

77

1400

83

1450

88

1500

94

J55°

100

l600

144

1 650

188

1675

210

1700

238

1720

257

1740

287

1750

3M

1760

342

1770

384

1780

427

1790

49s

J795

531

i8oo\

nearly j

56<*

Year of
our Lord

Wheat,
per Bushel.

.ciation of Money, according
to the Price of

Horse.

12 mis-
:cllanc-
3us Ar-
tides.

Meat.

Day

Labour.

Mean of
all.

s. d.

£. s. d.

1050

O 2|

117 6

* 89

42

2 6

1150

0 4t

0 12 ,5

1250

1 7i

111 0

1330

1 10±

0 18 4

43

56

73

77

1430

1 3

| 133°

1 10i

220

100

lOO

lOO

100

100

1600

4 °Y

1623

4 11

"16*30

5 6

1675

4 6’

c*
£ 0

0

0

2 39

1 66

188

210

l— l

O

O

4 9i

1720

4 4‘i'

1740

3 8

10 0 0

476

434

2 66

230

287

I76O

3 9i

14 0 0

667
1 ' "

49 2

400

275

342

1780

4 3t

- 1

17 95

7 10

IQ O O
9° 4

752

511

436

53 1

Besides most of the old chror ref iosum^ 1st and 2d edit. Liber Garderobee, in
1299. The Sketch of the Estahsehold, in divers Reigns, from Edward III. to
King William and Queen Mann the Year 1000 to 176?, by Mr. Combrune*
fol. Lond. by T, Longman, -

[To face p. 17 6.

A Table exhibiting the Prices of various Necessaries of Life, together with that of Day Labour, in sterling Money, and also in Decimals, at different Periods,
from the Conquest to the present Time, derived from respectable Authorities; with the Depreciation of the Value of Money inferred therefrom. To which is
added, the Mean Appreciation of Money, according to a Series of Intervals of 50 Years, for the first 600 Years; and, during the present Century, at shorter

Periods, deduced by Interpolation.

Year of
our Lord

THE PRICES OF VARIOUS ARTICLES AT DIFFERENT TIMES.

Wheat,
per Bushel.

Miscellaneous Articles.

Mean
Depre-
ciation
from
these 12
Articles.

Beef and
Mutton,
per lb.

Labour
in Hus-
bandry,
per Day.

Cattle in Husbandry.

Poultry.

Butter,
per lb.

Cheese,
per lb.

Ale, per
Gallon.

Small

Beer,

per

Gallon.

to the Price of

Morse.

Ox.

Cow.

Sheep,

Hog.

Goose.

Hen.

Cock.

Wheat.

12 mis-
cellane-
ous Ar-
ticles.

Meat.

Day

Labour.

Mean of
all.

s* d.

£■ s- d ■

£■ s ■ d-

£■ s. d

£■ s ■ d-

£■ s. d.

s. d.

s. d.

s. d.

d.

d.

s. d.

d.

d. qr.

s. d.

1050

O 2T

1 17 6

* 89

076

20

060

37

0 1 3

z9

0 2 0

36

42

io

42

2 6

115°

O 4!

012 5

0 4 8t

018

03°

0 3

0 2

1250

1 7\

111 0

107

0170

017

1 0

0 3

° 4 i

135°

1 io±

018 4

43

14b

66

O 17 2

106

027

61

0 2 6'
45

09

75

0 2

24

0 St
3 1

56

0 3

100

5b

75

77

1430

i 5

1 15 8

0 15 6

0 4 lii

051

O 6T

0 Si

1 55°

1 10|-

2 2 0

100

1 i6‘ 7

1 00

O l6 O

IOO

0 4 3t

IOO

056*

IOO

1 O

IOO

0 8|

IOO

1 O

IOO

5

IOO

2

ICO

0 1 i

IOO

1

IOO

IOO

1 °i

0 4

100

100

100

100

100

lb’oo

4 °T

0 4

2

1 2

0 6

ib‘25

4 11

2 O

1 6

0 6i

1650

5 6

0 4

2

1^75

4 6

5 10 °

25O

3 6 °

184

2170

345

011 0

256

O

tH n

0

3 0

3 00

1 3

182

1 3

Iz5

4t

90

Q

ICO

0 8
53°

2 —
2

250

239

1 3i

0 7i

246

239

16b

188

210

1700

4 9i

0 10

3

1720

4 4'a

1 0

3

2 2

0 8

1740

3 8

10 0 0

476

800

437

00

00 „

-f- -vj

0

160

6oz

1 15 0
634

3 6

35°

1 6

218

1 6

150

9

180

3i

175

1 0

800

3

300

434

3 0

0 10

597

434

2 66

250

287

1760

3 9i

1400

667

oc

-f- M

v? O

O

700

874

170

626

1 15 0

634

5 0
500

1 10

2^6

1 10
'83

10

200

Si

262

1 2

930

3

3°°

492

4 2

0 11

2°3

492

400

2 75

342

1780

4 5i

1 2

1795

7 10

IOO O

9°4

16 8 0

890

16 8 0

2000

1 18 0

882

5 80

i960

3 0

300

1 6

218

1 6

150

11i

230

5

25O

1 2i

969

H

275

75z

5 3

1 si

426

752

511

43b

531

Mean Appre-
ciation by
Interpolation.

A. D.

1030

26

llOO

34

1150

43

1200

51

I25O

60

I3OO

68

1350

77

1400

83

1450

88

1500

94

1550

1 GO

l6‘00

144

1650

l88

ib75

210

1700

238

1720

257

x74°

287

1750

3H

1760

342

1770

384

1780

427

1790

49b

1795

53 1

i8oo\

nearly j

5b 2

* The small figures denote the price in decimals, whereof those for the year 1550 may be taken for the integer, viz. 100.

Besides most of the old chronicles and historians, the following books were consulted, in constructing the above table ; viz. Bishop Fleetwood’s Chronicon Pretiosum, 1st and 2d edit. Liber Garderobm, in
1299. The Sketch of the Establishments of this Kingdom, temp. Ed. III. ct seqq. by J. Bree, 1791. Collection of Ordinances and Regulations of the Royal Household, in divers Reigns, from Edward III. to
King William and Queen Mary, Lond. 1790, 4m. The nth volume of the Archaologia. An Enquiry into the Prices of Wheat and other Provisions in England, from the Year 1000 to 1765, by Mr. Combrune,
fol. Lond. by T. Longman, 1768. Dr. Smith’s Wealth of Nations. Sir James Steuart’s Political CEconomy ; and Dr. Henry’s History.

to ascertain a Standard of Weight and Measure. 177

APPENDIX.

(§. 44.) Since the writing of the preceding Memoir, I have
had an opportunity of examining three other scales, divided
into inches, or equal parts, of considerable authority in this
country, having been executed by the late Mr. J. Bird. I have
also compared the old standard in the Exchequer, of the time
of Hen. VII. and which is considered to be the most ancient
authority of this sort now subsisting : these observations, I
flatter myself, the Royal Society will be desirous of possessing.

(§. 45.) The first of the abovementioned scales belonged to
the late General Roy, and was purchased by him at Mr.
Short’s sale, the celebrated optician ; it was used by him in
his operations of measuring a base line on Hounslow Heath.
(See Phil. Trans. Vol. LXXV.) It was originally the property
of Mr. G. Graham, has the name of Jonathan Sisson en-
graven upon it, but is known to have been divided by Mr.
Bird, who then worked with old Mr. Sisson. It is 42 inches
long, divided into tenths, with a vernier of 100 at one end, and
of 50 at the other, giving the subdivisions of 5ooths,and loooths,
of an inch.

(§. 46.) The second is in the possession of Alexander Aubert,
Esq. and formerly belonged to Mr. Harris, of the Tower;
contains 60 inches, divided into loths, with a vernier, like that
of the preceding. It is one inch broad, and 0,2 thick.

(§. 47.) The third was presented by Alexander Aubert,
Esq. and the late Admiral Campbell, Mr. Bird’s executors, to
the Royal Society, in whose custody it now remains. It con-
sists of a brass rod, 92,4 inches long, 0,57 inch broad, and

mdccxcviii. A a

178 Sir George Shuckburgh Evelyn's Endeavours

0,3 inch thick ; bearing a scale of 90 inches, or equal parts,
each subdivided into 10, with a vernier at the commencement,
being a scale of 100 divisions to 101 tenths. This has been
called Mr. Bird’s own scale, viz. made for his own use; and
was the instrument with which he is said to have laid off the
divisions of his 8-feet mural quadrants. It is probable that
Mr. Bird made many more of these scales, now in the hands
of private persons, (one of which, indeed, I saw at the Presi-
dent De Saron’s, many years ago, at Paris,) but those have
not come to my knowledge.

(§. 48.) In comparing General Roy’s (Bird’s) scale with
Mr. Proughton’s, I found 42 inches of the former were
= 42,00010 inches ouTroughton’s; (the thermometer 5 1°,7;)
3 6 inches were consequently = 36,00008.

And 12 inches on the 1st foot were equal'

to the 12 inches from 12 to 24 on > — ,0003
Troughton’s scale

The 2d foot - - -f ,0006

The 3d foot - - ,0004

The last foot >0oo6

The mean foot, therefore, in General Roy’s scale, 1 “

taken from four different feet, compared with 1
Troughton’s, between the 12th and 24th inch, f 12>00012
is as 12 to - - _ _ J

That is, General Roy’s scale is longest on 1 foot by

so much, and longer on 3 feet by - ,00036

And the greatest probable error from the inequality

in the divisions is about - - ,0005

And the mean probable error about - - ,0003

(§. 49.) Mr. Aubert’s scale, compared with Mr. Trough-
ton’s, was as follows : 58 inches were equal to 57,9982 inches

Inches.

: 11 >9997

12,0006

12,0006

to ascertain a Standard of Weight and Measure. ijg

on Troughton’s; (thermometer at 5i°,o;) viz. Mr. Bird’s
measure was shortest ,0018 ; or, shortest on 3 6 inches = ,0012,
And 12 inches, or 1st foot, on Mr. inches. 1

Aubert’s

2d foot,
3d foot,
4th foot,
5th foot,

Therefore the mean foot is

on Mr. Trough-
ton’s scale, from
6 inches to 18
inches ; the ther-
mometer being at
50°,o.

= 1 1 ,9999
= 12,0003

1 *>9996>

12,0019
12,0006
12,0005

The greatest error in this scale appears to be about = ,00 12
And the mean probable error - = ,0006.

(§. 50.) The Royal Society’s scale, compared, was as follows :
58 inches on Mr. Bird’s were equal to 57,99912 inches on Mr.

Troughton’s : (thermometer 5o°,5)
viz. Mr. Bird’s measure was shortest ,00088
Or shorter on 36 inches - - ,00054

32 inches on the same were equal to - 31,99967
viz. Mr. Bird’s was shortest by - ,00033

Or, on 36 inches, by - ,00037

The mean of these two comparisons is ,00045
And, by so much, is Mr. Bird’s scale shorter, in three feet, than
Troughton’s.

And 12 inches, or 1st foot, of the inches. '

Royal Society’s scale, is = 12,00013

2d foot of ditto = 1 1,99957
3d foot of ditto = 12,00027 v ton’s scale;
4th foot of ditto = 11,99990 r 'T '

5th foot of ditto = 12,00063
6th foot of ditto = 11,99823
7th foot of ditto = 12,00000

The mean of these seven feet is = 11,99982

And the greatest error in these divisions = ,0008

And the mean probable error - =,0004.

A a 2

on Trough-
ton’s scale;
the thermo-
meter at 510.

1 80 Sir George Shuckburgh Evelyn's Endeavours

(§. 51.) Lest, however, it should be suspected, that Mr.
Troughton's scale, with which I have made these comparisons,
is not sufficiently correct for this apparent preference, I will
now give the result of my examination of that scale, from one
end to the other. I set the microscopes to an interval of nearly
6 inches, correctly speaking, it was 6,00013 inches, taken from
a mean of the whole scale ; and, comparing this interval suc-
cessively, I found as follows.

Inches. Inches.

Inches.

Error, or difference
from the mean.

viz. from o to 6

= 6,00023* ~

-f >00012

6 to 12

= 6,00013 -

,00000

12 to 18

= 6,00020

- + ,00007

1 8 to 24

= 6,00000

— ,00013

24 to 30

= 6,00007

- ,00006

30 to 3 6

= 6,00033

4” ,00020

3 6 to 42

= S>9998° ~

- — ,00033

42 to 48

= 6,00020

-[■ ,00007

48 to 34 -

== 6,00010

— ,00003

34 to 60

= 6,00023

4" ,00010

Mean of all -

= 6,00013.

From whence it appears, that the greatest probable error,
without a palpable mistake, in Mr. Troughton's divisions,
is = ,00033 inch ’ against which, the chance is 9 to 1 ; and

* It is not pretended, that in this and the foregoing observations, the quantity of any
interval can be determined to the precision of the one hundred thousandth part of an
inch ; but it is presumed, that with the assistance of the microscopes, the ten thou-
sandth part of an inch becomes visible ; and, as a mean is taken from 3 or 4 times
reading off the micrometer at each trial, it has been deemed not unreasonable to set
down the quantities to five places of decimals.

to ascertain a Standard of JVeight and Measure . 181

the mean probable error = ,00016; and that it is 4 to 1 the
error doth not exceed ~^-0 inch.

This accuracy is about three times as great as that of Mr.
Bird's scales, and about equal to that of the divisions of my
equatorial instrument, made by Mr. Ramsden, in 1791. See
Phil. Trans, for 1793.

(§. 52.) I now proceed to the examination of the standard
rod of Henry VII. which is an octangular brass bar, of about 4
an inch in diameter, with one of the sides rudely divided, into
halves, thirds, quarters, eighths, and sixteenths ; and the first
foot into inches. Each end is sealed with a crowned old Eng-

lish H,

and from hence is concluded to be of the

time of King Henry VII. viz. about 1490, but is now become
wholly obsolete, since the introduction of the standard of Queen
Elizabeth ; but such as it is, I have thought proper to examine

it, and find as follows :

Inches.

On this rod, 4, or the 1st foot, is

equal to 11,973 ouTroughton's.

the 2d foot is

11,948

the 3d foot is

- 12,047

Error, or difference
Difference. on 3 feet.

The mean foot is

11,989

-,on — ,033

4- yard, or 1 8 inches

= i7>946

— ,054 —,108

4 yard, or 24 inches

= 23,921

— >079 —>n8

4 yard, or 27 inches -

- = 26,937

— ,063 — ,084

■J- yard, or 3 14 inches

= 3R443

— >°57 — >0 65

yard, or 334 inches

= 33>66'5

— ,085 —,091

Entire yard, or 36 inches

- = 35.966

— ,034 —,034

And the mean yard -

= 35.924

Mean ,0 76

And, by so much, Mr. Troughton’s measure

is longest.

182 Sir George Shuckburgh Evelyn’s Endeavours^ Sic.

And the probable error, in the divisions of this old standard, is
about inch.

(§• 53*) It may now be desirable to see the comparative
lengths of these various standards and scales, reduced to one
and the same measure, viz. Mr. Troughton’s.

Diffe- Probable error
rence. in divisions.

“”,076 ,03

+ .015 ,04

+ ,016 ,04

+ ,032
-j-,014

36 inches, on a mean, of Hen. VI I. standard
of 1490, are equal to -

■ of standard yard of Eliz. of i 5 88

1660

— of standard ell of ditto, of 1588
— *of yard-bed of Guildhall, about

-*of ell-bed of ditto, about 1660

Inches on
Troughton’s.

35>924

36.015

36.016

36,032

36,014

*of standard of clock- makers’

company, 1671 - - 35,972 — ,028

#of the Tower standard, by Mr.

Rowley, about 1720 - 36,004 +,004

— ofG Raham’s standard, by Sis-
son, of 1742, viz. line E - =36,0013 +,0013

— of ditto, ditto, viz. line Exch. =35,9933 — -,0067

of Gen. Roy’s'

(Bird’s) scale -
of Mr. Au-

bert’s, ditto, ditto
■of Royal So-

ciety’s ditto, ditto

all made, pro-
v bably, between
the years 1745
and 1760.

p= 36,00036 -f- ,00036 ,0003
<j =35,99880 —,00120 ,0006'

l=35>99955 >0004

of Mr. Bird’s parliamentary

standard, of 1758 - =36,00023 -j-,00023

— — of Mr. Troughton’s scale,

in i 796 - _ - =36,00000 ,00000 ,0001.

From hence it appears, that the mean length of the standard

yard, taken from the seven first instances in this table, agrees

with the quantity assumed by Mr. Bird, or Mr. Troughton, to

within T-ooo inch, but that the latter is the longest.

* These four quantities are taken from Mr. Graham’s account, in the Phil. Trans.
Vol. XLII.

I’haojr.Tmw.lsR} C ('XCV\Il.7v6.V:/J.j

f/tilo-w 7'mns. M 1 ) C CKCVJ117«AVJ,/4/.''

C 183 ]

IX. A new Method of computing the Value of a slowly con-
verging Series, of which all the Terms are affirmative. By
the Rev. John Hellins, F. R. S. and Vicar of Potter’s-Pury,
in Northamptonshire. In a Letter to the Rev. Dr. Maske-
lyne, F. R. S. and Astronomer Royal.

Read March 8, 1798.

REV. SIR, Potter’s-Pury, February 8, 1798.

That several of the most curious and difficult problems in
physical astronomy have hitherto been solved only by means
of slowly converging series, is a truth which you are well ac-
quainted with, and which may be seen in the works of the late
learned Euler, and others, on that subject. Of this kind of
series is the following, viz. ax -f bxz-\-cx*- f dx*+ &c. ad
infinitum , when all the terms are affirmative, and a, h, c, &c.
differ but little from each other, and x is but little less than
1 ; to obtain the value of which, to seven places of figures, by
computing the terms as they stand, and adding them together,
is a very laborious and tiresome operation ; and therefore some
easier method of obtaining it is very desirable. About five
years ago, the consideration of this matter was recommended
to me by Mr. Baron Maseres, (who not only employs his own
leisure in explaining and improving the higher parts of the
mathematics, but also encourages others who make the same
laudable use of their leisure,) to whom I then communicated
the method of computation explained in the paper inclosed in

184 Mr. Hellins's new Method of computing

this letter. As this method is general, for all slowly conver-
ging series of the form abovementioned, (which is generally
allowed to be the most difficult,) I am induced to present it to
you, requesting that, if it meets with your approbation, you
will communicate it to the Royal Society.

I am,

Rev. Sir, &c.

JOHN HELLINS.

P. S. I need not observe to you, that it is not requisite to
the summation of the series mentioned in this letter, that b
should be less than a , c less than b, d less than c, & c. but only
that the first, second, third, &c. differences of these coefficients
should be a series of decreasing quantities : for, you well know,
there are series of that form, which arise in physical astronomy,
of which the coefficients are actually a diverging series, and yet
the sum of the whole is a finite quantity. And the same thing
is evident, from the bare inspection of the theorem which I
shall presently use.

1. The computing of the value of the series ax -}• bx*- j- cx1 * 3

+ dx*-\- &c. ad infinitum , in which all the terms are affirma-

tive, and the differences of the coefficients a, b, c, & c. are but
small, though decreasing, quantities, and x is but little less
than 1, is (as has been before observed) a laborious operation,
and has engaged the attention of some eminent mathemati-
cians, both at home and abroad, whose ingenious devices on
the occasion entitle them to esteem. Of the several methods of

the Value of a slowly converging Series. 185

obtaining the value of this series, which have occurred to me,
the easiest is that which I am now to describe, by which the
business is reduced to the summation of two, three, or more
series of this form, viz . ax — bx*-\- cxz — dx*, &c. and one
series of this form, viz. pxn + qx'n-\- rx3”-f &c. where n is
= 4, B, lb, 32, or some higher power of 2. The investigation
of this method is as follows.

2. The series ax -|- bxr-\- ex' -[- dx* -f ex'+fx6-^ &c. is
evidently equal to the surirof these two series, viz.

ax — bx't-{- ex' — dx*-\- exs — fx s, &c.

* + 2bx* * 4- * 4“ 2fx6> &c-

of which, the value of the former is easily attainable, by the
method so clearly explained, and fully illustrated, by Mr. Baron
Maseres, in the Philosophical Transactions for the year 1777 ;
and the latter, although it be of the same form with the series
first proposed, yet has a great advantage over it, since it .con-
verges twice as fast. Upon this principle, then, we may pro-
ceed to resolve the series 26a:* -j- Qdx*-{- vf x* 4- %hx*-\- skx1*
2W.r,14- &c- into the two following:

2 bx*— <zdx*-^ 2/ir6 — vkx10 — 2,mxlx, &c.

* 4- 4^x4 * 4" 4 hx* -* 4“ 4 nix", &c.

where, again, the value of the one may easily be computed ;
and the other, although it be -of the same form with the series
at first proposed, yet converges four times as fast. And, in this
manner we may go on, till we obtain a series of the same form
with the# series at first proposed, which shall converge 8, lb,
32, b4, -&c. times as fast, and consequently a few terms of it
will be all that are requisite.

An example, to illustrate this method, may be proper,
which therefore is here subjoined.

MDccxcvm. B b

i8 6

Mr. Hellins’s new Method of computing

Xz

3. Let it be proposed to find the value of the series x -{- ~

+ 7‘+'7 + 7 + T+ &c‘ a ^ infinitum> when x —

4. In order to obtain the sum of this series, with the less
work, it will be requisite to compute a few of the initial terms,,
as they stand. For, if we begin the operation with computing

the value of x — - — |- ~ > &c- by the differential series be-

fore mentioned *, the values of D', D", D"', &c. will be — , diL.y

2 2-3

&c . respectively, i. e. j-, j-, &c. which is a series de-

creasing so very slowly, that the only advantage obtained by
this transformation of the series is in the convergency of the
powers of — — instead of the powers of x, which indeed is

I I* X

very great ; for, x being = ~, -j— is = ^ ; so that the new

series i . ^4+ &c. although = -

i . i . ^j3 — i . -fQ4, &c. yet converges more than seven

times as swiftly^. But, if we begin the work by computing the
first eight terms of the series, as they stand, and then compute

the value of -L 77 — -77+’ &Ct a ^ hy the same

theorem, the values of D', D", D'", &c. will be
9 10 i i "ii3 &Cm is a series decreasing, for a great number

* The theorem best adapted to this business is the following; viz. ax — bx^-{-

D" *3 D1" .r4

, ax D' x*

cx 3 — dx*, &c. r= — f-

+

4- &c. D' being — a

l+x * (i+^)a * (i+.r)3 r (i+^)4
— b, D"— a — 2 b+c, W— a — * 36 + 3c — d, &c.

See Scriptores Logaritbmici, Vol. III. p. 290, where b , c , d, &c. denote the same
quantities that a, b, c, &c. do here.

f is — 0.4736842, and— I is = 0.4782969.
19 1Q|

the Value of a slowly converging Series. i 87

of its terms, much more swiftly than the powers of and, in
the first seven terms, much more swiftly than the powers of

tV- The value °f the Series f ■ To — To • Sf + TTjfe)’

&c. is therefore = the series

X 2

I

~9

j

19 * 9.10

-l+-

IQ 1 o.

9_

*9

+

2*3

9.10.1 1

-f- &c. the first seven terms of which con-

9.10.x i. 12

verge above fourteen times as swiftly as the other ; or, in other
words, the first seven terms of it will give a result much nearer
the truth than a hundred terms of the other. And if, instead of
the first eight terms of the proposed series, the first twenty-
four terms were computed, as they stand, and then the value of

,T . X X* . X3

the series — ? A

25 26 1 27

X*

28’

&c. by its equivalent, - — — : -|_

25.26 (1+*)3

+

25.26.27 (i-fx

\3

+

2-3

25.26.27.28 (1+*)*

rapid decrease of the coefficients

25* 25.26 25.26.27*

25 l+x

-f- &c. the
&e. com-

pounded with the decrease of the powers of (in the pre-
sent case = the powers of T9^,) produces such a very swiftly
converging series, that eight terms of it will give the result true
to eleven places of decimals.

It may be further remarked, for the sake of my less expe-
rienced readers, (for whose information this article is chiefly
intended,) that the second term of the last series is produced
by multiplying the first by ^ ; the third, by multiplying

the second by ■— * the fourth, by multiplying the third by

28 ' bjb ’ anc* so on- therefore, the first term be called P,
the second Q, the third R, the fourth S, &c. we shall have.

6b 2

18B Mr. Hellins's new Method of computing

L

26

X

I +x’
X

T+x*

fO,.

27

3*_

28

X

I+X**

which form is well adapted to arithmetical calculation. And,,
that a similar form for that purpose may always be obtained,
when the coefficients of the proposed series are the reciprocals
of any arithmetical progression, has been shewn by Mr. Simp*
son, in his Mathematical Dissertations, p. 64.

Having premised these observations, I now proceed to the
arithmetical work ; in which I shall use, and explain, such other
devices as have occurred to me for facilitating it.

5. Since the work of computing the value of the proposed
series is so much shortened, by first finding the sum of a mo-
derate number of initial terms, and then resolving the remain-
ing terms into several series, in the manner described in Art. 2,
I will begin with computing the first twenty-four terms of it ;
and, to facilitate this part of the work, I will separate these
terms into three parts ; viz.

4"

+

X

+ '

x1

2

+ •

X3

_ 3

+ •

X *

4

+ •

Xs

5

+

a:8

6

4

Xr

7

4-

X*

8

x 9

4

• +

x”

■4

x "

■ +

X13

■ +

xI+

•4

x15

■+

_xg_

9

IO

1 1

12

13

*4

1.5

16

x‘7

4

x,?

4~

X19

4-

X 10

+ •

X 11

+ •

X**

4“ ■

x 13

X **

17

18

l9

20

21

22

23

24.

which are evidently equal to

x

X

4-

4c1

4.

X3

4-

x+

4- ■

X5

I_ .

X6

4-

X7

2

3

4

5

1

6

7

X

~9~

■4-

x1

10

4--

X3

1 1

4-

x *
12

4--

Xs

13

4-

X6

*4

4 ■

X7

7T

X

•4-

xa

■ 4"

X3

•4

X 4

i .

X5

,4.

X6

4

X7

* 7

1 8

19

30

~r

31

22

23

+ ■

+i
+

8 *

ill

6 1*

£_

H

the Value of a slowly converging Series. 189

which three parts, for the sake of reference, may be called A,
Bx8, and Cx ,6. It is evident also, that the numerical values of
the first eight powers of x are all that are required for finding
the sum of these twenty-four terms of the series proposed : for
Bxs+ Cx16 is = x3 (B 4- Cxs ); and this product -{- A = the
sum required*.

The numerical values of the first eight powers of x are
these :

x = 0-9,
x*= 0’8i,
x3— 0729,
x4= 06561,

«ZS= 0-59049,

Z6= 0-531441,,
x7= 0-4782 969,
xs= 0-4304672 1>;

from which values of the powers of x, the values of A, B, and
C, continued to twelve places of decimals, are easily obtained,
and are as follow.

* The same thing might have been obtained from the first four powers of x, or from
the first six powers of as might likewise the sum of 32, 36, 40, or any other num-
ber of terms that is a multiple of 4 or 6. But the number 8 is here chosen, for the

sake of a subsequent use that will be made of the value of **.

19^ Mr. Hellins's new Method of computing
x — o 9
T = °'4°5
7 = °-243

JC4

— = 0*16402,5
J = 0-11809,8
T = 0*08857,35

X7

y = 0-06832,8l285,7l

Y = 0*05380,84012,50
The sum is 2*04083,30298,21 = A.

*

— = 0*1

9

~ = 0*081

10

77 = 0*06627,27272,73
77 = 0*05467,50000,00
7 = 0*04542,23076,92
“ = 0*03796,00714,29
~ = 0*03188,64600,00

xs s

~ == 0*02690,42006,25

The sum is 0*44412,07670,19 = B.

the Value of a slowly converging Series .

191

“ = 0*05294,11764,71

— = 0-04500,00000,00

3

■^=0-03836,84310,53

^4

— = 0-03280,50000,00
^ = 0-02811,85714,29
~ — 0-02415,64090,91
“ = 0*02079,55173,91
^=0-01793,61337,50

The sum is 0-26012,12291,85 = C.

We may now quickly find the sum of the first twenty-four

terms of the series proposed, by two multiplications by x\ and

two additions, as was pointed out in the former part of this
article; but, since the labour of these two separate multipli-

cations may be saved, I shall now proceed to compute the va-
lue of the remaining terms of the proposed series.

6. I he remaining terms are -j- -|- -f-

-|- &c. ad infinitum , which, being disposed according to

the form mentioned in Art. 2, are evidently equal to x^x these

r *

two series

<|

I

25

^ — j—

v2, v 3 v4

A j ^ A I

26 2-7 28 *

2.X1 * * * * 6 * *

27

* +

28

2X*

IF

29

— , &c.

30 9

L * 1 26

the value of the first of which is very easily attainable, in the
manner shewn above, in Art. 4. The arithmetical work will
stand thus ;

ig% Mr. Hellins's new Method of computing

P —-Jf * ~Tf = 0-01894,73684,2
2 = ~k"-* ~7f = 000034,51949,7
R = 4r X = 0-00001,2X121.0

27 19

S = •— x-—* = 0-00000,06147,1

T =-^- = 0-00000,00401,6

U = — - x = 0-00000,00031,7
Tr 6U 9

V = X — = 0-00000, 00002yQ

31 19

W= dJL. x — = 0-00000,00000,3

32 19 ^

The sum of these terms is 0-01930,53338,5, which, for the sake
of reference, call a. Then we have ax^= apx + ■“

— -£L 4-

- 28 >

29

uO

30 *

2X°

~

4

10

7. The part of the proposed series which now remains to be
computed, is x** x : - + '

—

2X

~

34 * 6

ad infinitum , or x^x : — (-

+ -^r + ^r +

X

.10

16

17

+

18

4- &c. ad infinitum , which may also be divided into two series,
in the manner above shewn ; but, to obtain a swifter conver-
gency in the next, as well as the succeeding applications of the

differential series, it will be convenient first to compute the va-
lue of four terms at the beginning of this series ; which we may

quickly do, since all the powers of x requisite in that part of
the calculation are ready at hand, being set down in Art, 5.
The work will stand thus :

the V %lue of a slowly converging Series .

m

X*

~

X*

H

*5

16

-

°’03542>94000>°

And the sum is 0-17150,5578 6,5, which call D» Then will

D*M=*“* (-f +-Sr+^-'+4)*

8. It now remains to find the value of x : — + 4-

17 * 18 1

a x1^ X1^ x3*®

~7f * “ + “7 r + "77" + &c- ad infinitum , or of its equal, x

the sum of these two series,

X

X*

t7 rr + — -

“* * +•

£l

20

► -v.8

+

.10

21

v\t

22

zx

* +^T+ &C.

* + is ■ ■ 20

Here, again, the value of the first series may easily be computed,
by the theorem referred to in Art. 4, it being:

17 l+XX

^ — -

1 17. 1

7.18

-C , 1-2 # XT , I-2-3 o x , fj .

(l + ^)a"^" I7. l8. X9 (i-fXT)3 > I7.I8. 19.2O ’ (l + Xr)4 * ^C‘ JI'

numbers,

C €

MDCCXCVIIL

194 Mr. Hellins's new Method of computing

X

.

1

X

81

181

> •—

l7

P

81

18

X

181

3

X

81

19

1 8 1

jR_

81

20

A

181

JL

X

81

21

181

5T

81

22

✓N

1 8 1

6U

x •

81

23

1 8 1

1

7V

V ■

81

24

181

8W

81

— 1

z5

X '

181

—

0-00000,00020,9

0*00000,00002,7

0-00000,00000,4

And the sum of these terms is 0.02701,19108,2 = C.

9. To find the value of : —• -f +

ad infinitum, or its equal, x : "1“ “7 T 4“ "77" +

*

fid' infinitum , which is all that now remains of the proposed
series, it will be expedient, first, to compute the two initial
terms of the series, and then to separate the remaining terms
of it into two parts, as has been done in the preceding articles.
These two terms are

oft

= 0-07290,00000,0

— = 0*04304,67210,0

And their sum is 0-11594,67210,0 = EJ.

the Value of a slowly converging Series. 195

• Xxt

10. There now remains, of the proposed series, a:,4x : -pj- -f-

„I5 r!0 ri4 ra8 „3a

-77- + — + "7T+TT + “ + infinitum , =/x

the sum of these two series.

„8

XI

* -f

12

2,r3

12

_L ___ __

' X3

+ 2X

—

14

.16

L _L_

4. 8 I?

* +7?+

14 • 16

Here, likewise, the value of the series which has the signs -f-
and — alternately, is easily obtained by a swiftly converging
series, the terms of which are set down here below, the decimal

= 7^7 being used in the calculation, for the

value of

sake of facility.

I

II

p

12

__

*3

3R

H

4S

*5

ST

16

6U

17

7V

18

8W

*9

9X

S = x 0-39617,17287,6 = 0-00000,61523,3
T ^ "TT x 0-39617,17287,6 = 0-00000,06499,7

cT

U = x 0-39617,17287,6 = 0.00000,00804,7

V = x 0-39617,17287,6 = 0-00000,00112,5
W=— - x 0-39617,17287,6 = 0.00000,00017,3

x 0.39617,17287,6 = 0-00000,00002,9

Y = -^- x 0-39617,17287,6 = 0-00000,00000,5

20

The sum of these terms is

0-03728,40092,2 == y.

C c 2

1 96

Mr. Hellins's new Method of computing

11. The part now remaining, of the proposed series, is x**

x

zxa

12

+ ^ + ^ +

ZX

32

+ + &c, ad infinitum ,

22

14 1 l6 1 18

which may very conveniently be resolved into x*s x these two

r 1

senes<

I

I 6

+

x°

X

16

7

*

+

x

.24-

+

ZX

16

9

* -f*

r32

10

+

.r40

1 1

~4-8

- — , &c.

I *

zx

32

10

C48
12
2„t43

12

L 1 8

the value of the first of which will quickly be obtained, by means

of the theorem which has been repeatedly used in the preceding

operations. The terms of it are as below, in which the decimal

is used, for the reason mentioned in

value of — : — r =

43 °46721

1 -f x° 143046721

the preceding article.

1

~6~ —

0-16666,66667

P = -y- X 0-30092,770l8 = 0-04298,96717

P

Q = — x 0-30092,77018 = 0-00161,70979
R = x 0-30092,77018 = 0-00010,81399
S = x 0-30092,77018 = 0-00000,97627
T = x 0*30092,77018 = 0*00000,10683
U = x 0-30092,77018 = 0*00000,01339

V = ■£-¥- x 0.30092,77018 = 0*00000,00186

^ 3'

TT

W = d— x 0-30092,77018 = 0-00000,00028
X = x 0*30092,77018 = 0-00000,00005

Y = x 0-30092,77018 = 0*00000,00001
The sum of these terms is 0-21139,25631 = t

the Value of a slowly converging Series. 197

12. There now remains, of the proposed series, only x4%
x : "T — 1“ “j~“ 4 -77- + + &c. ad infinitum , the value of

which, or of its equal, x4S x : — — h “7 — h *7 — h — - + &c.

may easily be obtained without any transformation, since the
powers of x16 (= the powers of 0*18530,20188,9) decrease
swifter than the powers of The numerical values of these
terms are as below.

— - = 0*04632,55047
= 0*00686,73676

^48

— == 0*00106,04476

.r118

1 1

12

■r160

13

H

xlgz

~~

0*00016,84313

0*00002,73093

0*00000,44982

0*00000,07502

0*00000,01264

0*00000,00215

0*00000,00037

0'C0000,00006

0*00000,00001

The sum of these terms is 0*05445,44611' = F.

Mr, Hellins’s new Method of computing

13. The values of the several parts, into which the proposed
series has been resolved, being now so far obtained that we
have only to multiply each by its proper factor, viz. the nu-
merical value of x8, x16, x3\ &c. and add the products together,
to get the sum of it ; this, therefore, is now to be performed.
And, in this part of the calculation, several multiplications may
be saved, and no larger factor than xs be used, by attending to
the method described by Sir Isaac Newton, in his Tract De
Analysi per JEquationes infinitas ; p. 10. of Mr. Jones’s edition
of Sir Isaac s Tracts ; or p. 270. Vol. I. of Bishop Horsley’s
edition of all his works. The manner in which this is to be
done will appear, by collecting the several parts from the pre-
ceding Articles, and exhibiting them in one view, thus :

A-f Bx8-f Cx16+ 7+~Dx r+E~xx3*+ ryx*°+ J+F^

a:43 = the sum of the proposed series. Now,

1st. Calling ^ + F, z', and multiplying by x*, we have z’ x*
= 0-26584,70242 x 0-43046,721 = 0-11443,84267,9.

2dly. Putting y -\ - z' xs = z", and multiplying by x\ we get
z,rx*= 0-15172,24360,1 x 0-43046,721 = 0-06531,15337,2.

3dly. Putting £ -j- E -j- z"x8= z'", and multiplying by xss we
get z'"xs= 0-20827,01655,4 x 0-43046,721 = 0-08965,34770,9.

4thly. Putting u -f- D -j- z"'xs =£iv, and multiplying by a:8, we
get 2lvu:8= 0-28046, 43895, 9x0.43046, 72 i = o- 12073, 07232, 90.

5thly. Putting C -f zivxs = zy, and multiplying by a8, we get
zyx* = 0-38085,19524,75 x 0-43046,721 = 0-16394,42774,05.

6thly. Putting B -f £v x* = zyi , and multiplying by x\ we get
%vi xs = 0-60806,50444,24 x 0*43046,721 = 0-26175,20631,73.

Lastly, to this product add A - = 2*04083,30298,21,

and we have the value of the series proposed = 2*30258,50929,94,
which is true to twelve places of figures.

the Value of a slowly converging Series. igg

14. It may not be improper to remark, that this degree of
accuracy is much greater than is requisite, even in astronomy;
for which, as well as for most other purposes, six or seven
places of figures are sufficient: and, to that degree of exactness
the value of the proposed series might have been obtained, by
less than a fourth part of the labour that has been taken above.
But it was my intention to show, that the value of a very slowly
converging series, of which all the terms are affirmative, may,
by the method now described, be computed to ten or twelve
places of figures, in the space of a few hours.

• '

'

-

, . r '

■

» V

■ ' - Vi : . ■

'

y

-

:

1

■

*

• ■ /. • - : ■ : '

-

s ' •-

s ■

»■ i ■

■

,

.

/

; ' ,

-

-

-

*

!

METEOROLOGICAL JOURNAL,

KEPT AT THE APARTMENTS

or TKI

ROYAL SOCIETY,

BY ORDER OP THE

PRESIDENT AND COUNCIL.

C ® 2

METEOROLOGICAL JOURNAL
for January, 1797.

Six’s

Time.

Therm.

Therm.

Barom.

Hy-

Rain.

Winds.

Therm.

without.

within.

ero-

least and

1 7Q7

T X r r, n 4-

x J y /

greatest

ter.

v v earner.

Heat.

H.

M.

O

0

Inches.

Inches.

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Jan, 1

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8

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29»74

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s

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47

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29,91

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0,025

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30

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3°

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30,48

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9

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26

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34

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29,70

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32

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14

36

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9°

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*5

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16

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0,244

NE

I

Cloudy.

36

2

O

36

5i

30,16

81

NE

I

Fine.

C.3 3

METEOROLOGICAL JOURNAL

for January, 1797.

1797

Six’s
Therm,
least and
greatest
Heat.

Time,

Therm.

without.

Therm.

within.

Barom.

Hy-

gr°-

me-

ter.

Rain.

Winds.

Weather.

H.M.

O

O

Inches.

Inches.

Points.

Str.

Jan. 17
18

*9

zo

21

22

23

24

25

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27

28

29

30

31

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25

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2 O

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8 0
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8 0

z 0

26

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38
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43
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41
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3°

38

35

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43

39
43

43
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45,5

44

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51

48

5°

48

5 1
5°

54

53

55

54

5 5>5
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3 °’3*
3°’35
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30,16
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30,04

29,92

29,98

30,24

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29,89

29,53

29,61

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3°, 1 7
30,16

29,70

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85

85
87

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89

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86
87
85
87

85

86

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0,040

SW

ssw

ssw

ssw

ssw

ssw

ssw

ssw

ssw

SW

SW

SSW

s

s

s

w

ssw

ssw

E

SE

SE

SSE

s

s

SW

WNW

WSW

WSW

S

w

I

I

I

I

I

I

I

I

I

1

2
2
2
2
2
2
I
I
I
I
I

1

2
2
2

2,

2

2

2

2

Fair.

Fine.

Cloudy.

Rain.

Foggy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Fair.

Cloudy.

Fair.

Fine.

Cloudy.

Cloudy.

Fair.

Fair.

Rain. r“uch
Cloudy.1 Ultmght’
Rain.

Fine.

Fair.

Cloudy.

Rain rMuch wind
Fair.' L last night.

a 2

C 4 3

METEOROLOGICAL JOURNAL
for February, 1797.

Six’s

Time.

Therm.

Therm.

Barom.

Hy-

Rain.

Winds.

Therm.

without.

within.

gro-

t nc\n

least and

me-

1 /y/

greatest

ter.

Heat.

H.

M.

0

O

Inches.

Inches.

Points.

Str.

Feb. i

0

36

7

O

44

53

29,96

88

0,068

SW

2

Cloudy.

50

2

O

5°

56

29,99

86

WNW

2

Cloudy.

2

46

7

O

46

54>5

30,27

87

W3W

2

Cloudy.

5°

2

O

5°

57

3°>32

85

SW

2

Cloudy.

3

45

7

O

45 >5

56

3°’37

82

SW

1

Cloudy.

49

2

O

49

57

3°>37

79

SSW

1

Cloudy.

4

37>S

7

O

38>5

56

3°>36

85

NW

I

Cloudy.

40

2

O

40

57

30,41

84

NNW

I

Cloudy.

5

35

7

O

35

55

30,50

84

E

I

Cloudy.

36

2

O

36

55

3°>54

85

E

I

Cloudy.

6

3z

7

O

32,5

52

3°>53

85

SE

I

Cloudy.

37

2

O

37

54

3<>*55

82

NW

I

Cloudy.

7

32

7

O

33

52

30^55

82

SE

I

Cloudy.

35

2

O

35

53

30,55

85

SE

1

Cloudy.

8

3i

7

O

31

52

30.47

85

SW

I

Cloudy.

35

2

O

32

52

30,48

83

SW

1

Cloudy.

9

32

7

O

35

5i

30.59

82

SW

1

Cloudy.

4i

2

O

41

53

30,61

80

SSE

I

Fair.

IO

3i,5

7

O

3M

52

30,62

83

SE

I

Cloudy.

33

2

O

33

53

30.57

82

E

I

Cloudy.

1 1

29

7

O

30

5i

30,44

84

S

1

Cloudy.

36

2

O

36

52

30,33

85

SSE

I

Cloudy.

12

35

7

O

37

5i

30,08

87

SW

I

Cloudy.

46

2

O

46

52

30,04

85

SSW

I

Fair.

*3

38

7

O

4i

5i

29,81

87

0,023

S

2

Rain.

45

2

0

45

53

29,62

81

s

2

Cloudy.

14

36

7

O

36

5i

29.44

84

0,128

wsw

2

Fair.

42

2

0

42

53

29.37

76

wsw

2

Cloudy.

15

29

7

0

29>5

5i

29,82

79

w

I

Fair.

37

2

0

37

52

29,98

69

NW

2

Fine.

16

25

7

0

25

5°

30,36

78

WNW

1

Fine.

39

2

0

39

51

30,41

72

w

1

Fair.

METEOROLOGICAL JOURNAL

for February, 1797.

1797

Six’s
Therm,
least and
greatest
Heat.

Time.

Therm.

without.

Therm.

within.

Barom.

Hy-

gro-

me-

ter.

Rain,

Winds.

Weather.

H.

M.

0

0

Inches.

Inches ,

Points.

Str.

Feb. 17

0

27^5

7

O

27,5

50

30,46

78

E

I

Fine.

4L5

2

O

4M

54

30,4°

73

ESE

I

Fine.

18

29

7

O

3 >

5i

30,4°

82

E

I

Fair.

46

2

O

46

54

30,4°

70

SSE

I

Fine.

19

29

7

O

29

5i

30,48

81

ENE

I

Fair.

44

2

0

43,5

54-5

30,50

80

S

I

Fine.

20

28

7

O

28

52

3°’43

81

NE

I

Fair.

44

2

O

44

55

3°»37

77

EN E

I

Fine.

21

26

7

O

26

5L5

30,40

82

EN E

I

Foggy.

46

2

O

46

55

30,40

82

E

I

Fine.

22

30

7

O

33

5L5

30,42

83

E

I

Fair.

49>5

2

O

49.5

56

3°, 44

78

SE

1

Fine.

23

3i

7

O

3i

5^

30,42

84

NE

I

Cloudy,

49

2

O

46,5

54

30,41

81

NE

I

Foggy.

24

29

7

O

29

54

3°>39

81

S

I

Fair.

46

2

O

45

55

30.38

81

S

I

Foggy.

25

3l>5

7

O

32

54

30.36

80

E

1

Fair.

5°

2

O

50

56

30.34

80

E

I

Fine.

26

34

7

O

35

54

30.31

84

E

I

Cloudy.

47

2

O

47

55

30.31

83

E

I

Fair.

27

32,5

7

0

33,5

54

30,41

77

NE

I

Cloudy.

43

2

0

42

56

30,41

67

E

I

Fine.

28

24,5

7

0

25

53

30,25

83

NE

I

Foggy.

42

2

0

4L5

56

30.15

75

E

I

Fine.

-

c 6 n

METEOROLOGICAL JOURNAL

for March, 1797.

1797

Six’s
Therm,
least and
greatest
Heat.

Time.

Therm.

without.

Therm.

within.

Barom.

Hy-

gro-

me-

ter.

Rain.

Winds.

Weather.

H.

M.

O

O

Inches.

Inches.

Points.

Str.

Ma^. 1

O

31

7

O

3l

53

30,02

81

ENE

I

Fair.

40

z

O

39

56

29,98

76

ENE

2

Fine.

z

28

7

O

3i

53

29,92

83

E

I

Cloudy.

44

2

O

44

55

29,90

77

E

I

Fine.

3

32

7

O

33

53>5

29,83

81

E

I

Fair.

53

2

O

53

57

29,81

69

SSE

I

Fine.

4

35

7

O

35

55

29,72

78

E

I

Fine.

44

2

O

43>5

55

29,69

74

E

2

Hazy.

S

33

7

O

33

53

29,60

77

E

2

Fair.

4L5

2

O

41

57

29,54

71

E

2

Fair.

6

36

7

O

35

53

29,71

84

NE

2

Cloudy.

45

2

O

45

56

29,86

81

WM W

2

Cloudy.

7

33

7

O

33

53

85

sw

I

Foggy.

4g,5

O

48

57

3<M9

74

s

I

Fair.

8

36

7

O

36

54

30,02

86

0.335

NE

I

Rain.

42

z

O

4I>5

56

3°->°3

85

N

I

Cloudy.

9

34

7

O

35

54

3°,°3

86

0,176

E

I

Cloudy.

42

2

O

41

55

30,05

78

NE

I

Cloudy.

10

3**5

7

O

33

53

29,86

80

NE

2

Cloudy.

38>5

2

O

37

55

29,84

74

NE

2

Cloudy.

1 1

32

7

O

32,5

52

29,78

76

NE

2

Cloudy.

42

2

O

42

54

29,83

70

NE

2

Fair.

IZ

32

7

O

32

52

30,00

74

NE

2

Cloudy.

38

2

O

38

53

30,00

7i

NE

2

Fair.

13

3L5

7

O

3i>5

5i

30,08

75

ENE

Z

Cloudy.

40,5

2

O

40

53

30,12

7i

NE

2

Fair.

14

33

7

O

32

5 US

30,11

77

NE

2

Cloudy.

45

2

O

45

54

3°»°5

75

NE

2

Cloudy.

15

35>5

7

O

35’5

52

30,03

79

ENE

I

Cloudy.

47

2

O

46,5

54

30,02

72

E

2

Fine.

16

3°*5

7

O

32

52

29,98

83

NE

2

Cloudy.

47

2

O

47

54

29,97

70

NE

2

Cloudy.

/

METEOROLOGICAL JOURNAL

for March

1797-

Six’s

Time.

Therm.

Therm.

Barom.

Hy-

Rain.

Winds.

Therm,
least and

without.

within.

gro-

1 797

me-

Weather.

greatest

Ler.

Heat.

H.

M.

O

0

Inches.

Inches.

Points.

Str.

Mar.17

O

34

7

O

34

53

3°-°3

75

NE

2

Cloudy.

1 8

42

2

O

42

53

30,06

70

NE

2

Cloudy.

31

7

O

33

53

30,16

79

NE

2

Cloudy.

43

2

O

43

54

30,18

69

NE

2

Cloudy.

1 9

, 34

7

O

34

53

30,16

74

NE

I

Cloudy.

20

44

2

O

44

54

30,20

70

NE

1

Cloudy.

32-5

7

O

35

52

30,20

82

NE

I

Cloudy.

21

45

2

O

45

54

30,28

72

NE

I

Cloudy.

27- 5

7

O

29

5 2

3°-42

75

NE

I

Cloudy.

Fair.

22

43

2

O

43

57

30,42

67

E

I

27,5

7

O

29,5

52

3°,42

75

SE

I

Fair. '•

47

2

O

47

56

30,38

64

Sb. E

I

Fair.

z3

3»

7

O

32

52

30-24

73

SSW

I

Fair.

24

54

2

O

53

54

30,11

60

SSW

2

Hazy.

38

7

O

39

54

30,06

84

SSW

I

Cloudy.

25

54

2

O

54

56

29,94

74

SSW

2

Cloudy.

45

7

O

45

54

29,66

86

s

2

Cloudy.

26

52

2

O

52

56

29,61

75

w

2

Cloudy.

38

7

O

42

55

29,82

82

0,076

sw

2

Cloudy.

27

49

2

O

48

57

z9-7o

76

0,068

SSW

2

Cloudy.

43

7

O

44

55

29-54

84

SE

1

Cloudy.

28

5°

2

0

49

56

29,49

80

SE

2

Rain.

40

7

O

42

55

29,44

84

0,042

SE

2

Cloudy.

29

5 1

2

O

5i

57

29,49

72

SE

2

Cloudy.

38

7

O

39

55

29-55

82

E

I

Foggy.

54

2

O

53-5

59

29,60

72

sw

1

Fair.

3°

37

7

O

39

56

29,76

81

sw

1

Cloudy.

3i

44

2

O

44

57

29,80

82

NE

I

Cloudy.

34

7

0

36

56

29,85

83

0,080

sw

1

Fair.

48

2

0

0

48

57

29,82

77

SSW

2

Cloudy.

C 8 3

METEOROLOGICAL JOURNAL

for April, 1797.

1 797

Six’s
Therm,
east and

Time.

Therm.

without.

Therm.

within.

Baropi.

Hy-

gro-

me-

ter.

Rain.

greatest

Heat.

H.

M.

O

O

Inches.

Inches.

Apr. j

0

41

7

O

42

57

29,75

84

52

z

O

5 1

58

29,70

75

0,061

2

41

7

O

4i

57

29,55

83

5Z

2

O

S2

58

29>45

81

3

35

7

O

35

56

29,10

84

0,210

39

2

O

39

57

29,13

84

4

34

7

O

37

55

29*23

84

0,520

5i

2

O

5°

57

29,40

70

5

40

7

O

40

55

29’55

85

44

2

O

44

56

29*53

80

6

36*5

7

O

38

55

29*57

8z

47

2

O

47

56

29,57

76

7

37

7

O

37

55

29,68

83

48

2

O

48

57

29*75

77

8

40

7

O

40

56

29*97

84

48

2

O

48

57

29,98

81

9

38

7

O

39

55

30,12

83

5°

2

O

5°

57

3°*ii

76

10

40

7

O

42

55

3°*°7

8z

47

2

O

44

57

29*97

77

1 1

40

7

O

45

55

29*77

81

5°

,2

O

46

57

29,76

79

12

4°, 5

7

O

42,5

56

29,82

86

63

2

O

61,5

59

29,80

72

13

45

7

O

46

58

29*95

84

54

2

O

54

59

29,89

79

14

42,5

7

O

43

58

29*94

85

°*3 97

48

2

O

48

58

29,93

83

IS

42,5

7

O

44

58

29,87

87

0,052

56

2

O

55

60

29,83

69

1 c

42

7

O

43

58

29*79

80

56

2

O

54

60

29*79

73

Winds.

Points. Str,

s

w

sw

w

NE

NE

E

E

NE

NE

NE

NE

NE

NE

E

NE

NE

NE

NE

NE

NE

NE

NE

ESE

NE

NE

NE

E

E

S

NW

W

Weather.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Rain.

Rain.

Fine.

Hazy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Rain.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

Cloudy.

■>

C 9 3

METEOROLOGICAL JOURNAL

for April, 17 97.

Six’s

Time.

Therm.

Therm.

Barom.

h7-

Rain.

Winds.

Therm.

without.

within.

gro-

I7Q7

least and

me-

Weather.

* / y /

greatest

ter.

Heat.

H.

M.

O

O

Inches.

Inches .

Points.

Str.

A pi*. 17

O

38

7

O

40

58

29,77

84

W

1

Fair.

58

2

O

58

59

29,76

65

w

1

Cloudy.

18

40

7

O

42

57

29,78

70

NW

1

Fair.

56

2

O

56

59

29,80

63

sw

1

Fair.

*9

37

7

0

42

58

29>93

77

NE

1

Fair.

56

2

O

55

61

29,96

65

NE

1

Fine.

20

36

7

O

40

58

3°’ 1 3

75

E

1

Fine.

57

2

O

57

60

30,08

65

E

1

Fine.

21

40

7

O

43

58

29,89

80

E

1

Cloudy.

59

2

O

59

59

29,83

80

E

1

Cloudy.

22

46

7

O

47

58

29>77

83

E

1

Cloudy.

61

2

O

61

61

29,78

68

E.

1

Fair.

23

45

7

O

47

59

29,94

82

S

1

Fine.

64

2

O

63

61

29,94

64

s

1

Fair.

24

47

7

O

5 1

60

29,94

77

SSE

1

Fair.

65

2

O

65

61

29,94

64

SSW

1

Cloudy.

25

45

7

O

46

60

30,00

82

SW

1

Cloudy.

62

2

O

61

62

3°’°3

67

SSW

1

Fair.

26

42

7

O

47

•60

30,05

76

sw

r

Fair.

56

2

O

5 1

58

29,86

7>

SSE

1

Rain.

27

44

7

O

45

59

29,46

82

0,394

NW

2

Cloudy.

53

2

O

5 1

60

29,32

78

NW

2

Cloudy.

28

42

7

O

43

59

z9>74

79

NE

2

Cloudy.

5 5 >5

2

O

55

61

29’75

7i

NE

1

Fair.

29

42

7

O

47 '

60

29,65

80

S

1

Cloudy.

57

2

O

55

61

29,52

76

SSW

1

Rain.

3°

44

7

O

44

59

29,38

81

0,225

SSW

1

Cloudy.

58

2

O

57

60

29’39

67

.

NW

1

Cloudy.

b

C 10 3

METEOROLOGICAL JOURNAL

for May, 1797.

1797

Six’s
Therm,
east and
greatest
Heat.

Time.

Therm.

without.

Therm.

within.

Barom.

Hy-

51-0-

mc-

ier.

Rain.

Winds.

Weather.

H.

M.

0

O

Inches.

Inches.

Points.

Str.

May 1

0

43

7

O

44

58

z9’54

77

NW

I

Cloudy.

57

2

O

54

6l

29,66

71

NW

I

Fair.

2

42

7

0

48

59

29,76

84

0,022

S

2

Rain.

57

2

O

55

59

29,64

80

s

2

Rain.

3

42

7

O

46

59

29,61

82

0,202

ssw

2

Fine.

5b

2

O

56

61

29,62

73

sw

2

Fair.

4

43

7

O

4 6

59

29,56

83

0,072

s

2

Rain.

53

2

O

5i

61

29,38

84

s

2

Rain.

5

39

7

O

45

58

29,49

76

0,198

ssw

2

Cloudy.

56

2

O

52

61

29,46

75

s

2

Rain.

6

41

7

O

45

59

29,70

80

0,l6o

sw

I

Fine.

57

2

O

55

60

29,79

73

NW

1

Cloudy.

7

42

7

O

44

59

29,91

82

0,196

E

I

Cloudy.

47

2

O

47

60

29,91

80

NE

I

Cloudy.

8

37

7

O

42

58

30,06

75

0,070

NE

I

Fair.

52

2

O

52

61

3°>°3

69

NE

I

Fair.

9

36

7

O

4i

58

29,88

79

NE

2

Cloudy.

49

2

O

48

58

29,77

77

NE

2

Cloudy.

10

34

7

O

40

56

29,61

90

0,265

NE

2

Rain.

49

2

O

47

58

29,61

84

NE

2

Cloudy.

1 1

4U5

7

O

42

56

29,58

85

NE

I

Cloudy.

5°

2

0

5°

57

29,55

77

JNE

I

Cloudy.

12

42

7

0

45

57

29,58

82

E

I

Cloudy.

60

2

0

59

59

29,58

70

E

I

Fair.

13

42

7

0

46

58

29,88

78

0,068

SSW

2

Fair.

60

2

0

59

61

29,98

65

s

2

Fair.

14

42

7

0

47

58

3°07

73

NW

I

Fair.

61

2

0

61

60

30,19

66

E

I

Hazy.

15

43

7

0

47

59

30,08

80

E

1

Cloudy.

62

2

0

61,5

61

30,00

7i

E

1

Fine.

16

47

7

0

53

60

29,91

81

E

1

Fair.

67

2

0

67

62

29,85

74

E

I

Fair.

\

C n 3

METEOROLOGICAL JOURNAL

for May, 1797.

1 797

Six’s
Therm,
least and
greatest
Heat.

Time.

Therm.

without.

Therm.

within.

Barom.

Hy-

gr°-

me-

ter.

Rain.

Winds.

Weather.

H.

M.

O

O

Inches.

Inches.

Points.

Str.

May vj

O

53

7

O

56

6l

29,86

76

s

2

Cloudy.

63

2

O

6l

62

29,80

83

s

2

Rain.

18

5Z

7

O

54

62

29,91

83

0,115

s

2

Cloudy.

68

2

O

67

62

29,96

77

ssw

2

Cloudy.

l9

5 1

7

O

55

62

29,99

81

s

I

Cloudy.

73

2

O

7r

65

29,90

72

ESE

2

Fair.

20

59

7

O

61

64

29,67

76

SSW

2

Cloudy.

68

2

O

68

65

29,77

67

SSW

2

Fair.

21

52

7

O

54

63

29,91

76

SSW

2

Fair.

68

2

O

68

64

29,95

66

ssw

2

Fine.

22

49

7

O

S2

63

30,08

75

ssw

2

Fair.

66

2

O

66

64

30,17

61

sw

I

Fine.

23

48

7

O

53

63

3°->33

72

w

I

Hazy.

69

2

O

68

64

3°>33

65

SE

I

Fair.

24

48

7

O

54

64

30,28

75

E

I

Fair.

7i

2

O

70

65

30,20

63

E

2

Fine.

25

50 ■

7

O

58

64

30,08

73

E

2

Fair.

79

2

O

78

68

30^3

68

SSE

2

Fine.

26

61

7

O

64

68

29,95

76

w

I

Fair.

73

2

O

73

68

29,96

66

SSW

1

Cloudy.

27

5°

7

O

54

66

30,09

75

SSW

2

Fair.

7i

2

O

71

67

30,09

65

sw

2

Fair.

28

50

7

O

54

65

29,98

76

s

I

Cloudy.

70

2

O

70

66

29,83

66

SSE

2

Fair.

29

5i

7

O

54

65

29,70

74

ssw

2

Fair.

62

2

O

62

64

29,59

76

s

2

Rain.

3°

49

7

O

5°

63

29,77

81

0,068

NE

2

Cloudy.

60

2

0

59

62

29,99

72

NE

2

Cloudy.

3i

46

7

O

53

62

30,10

74

ssw

2

Hazy.

65

2

O

65

62

30,00

69

sw

2

Cloudy.

b 2

C 12 3

METEOROLOGICAL JOURNAL
for June, 17 97.

Six’s

Time.

Therm.

Therm.

Barom.

Hy-

Rain.

Winds.

Therm.

without.

within.

gro-

least and

me-

Weather*

1/97

greatest

ter.

Heat.

H.

M.

0

O

Inches.

Inches.

Points.

Str.

June 1

O

52

7

O

54

62

29»7I

83

SW

2

Cloudy.

65

z

O

64

62

29,67

75

sw

2

Cloudy.

2

49

7

O

53

62

29,78

71

0,295

E

1

Cloudy.

69

2

O

.69

63

29,76

66

s

1

Cloudy.

<->

J

5 1

7

O

54

62

29,36

85

O

0

ssw

1

Rain.

56

2

O

48

61

29,52

80

WNW

1

Rain.

4

40

7

O

45

60

29,82

73

0,225

W

2

Fine.

63

2

O

63

60

29,90

66

w

2

Cloudy.

5

46

7

O

5°

60

30,04

73

NE

1

Fair.

63

2

O

62

61

3°,I4

64

NE

1

Fine.

6

43

7

O

46

60

30,29

76

NE

1

Fine.

64

2

O

63

61

30,27

65

NE

1

Fine.

7

45

7

O

53

60

30,01

78

W

1

Cloudy.

62

2

O

58

61

29,89

80

NW

1

Rain.

8

45

7

O

5o

60

29,78

75

SW

1

Fine.

64

2

O

63

60

29,71

65

sw

1

Fair.

9

46

7

O

49

59

29U5

78

0,210

NE

1

Cloudy.

6 1

2

O

60

61

29,84

70

E

1

Cloudy.

10

47

7

O

53

59

30,06

77

0,060

NE

2

Fair.

62

2

O

60

60

30, 1 8

70

NE

2

Fair.

1 (

44

7

O

52

60

29,98

73

N

, 1

Fine.

69

2

O

66

60

29,88

66

N

1

Cloudy.

1 2

53

7

O

55

60

29,66

81

0,050

SW

1

Fair.

65

2

O

60

61

29,66

77

w

1

Rain.

13

49

7

O

53

61

29,68

80

0,218

E

1

Cloudy.

65

2

O

65

62

29,68

68

E

1

Fair.

H

50

7

0

54

61

29,74

78

W

1

Cloudy.

69

2

O

68

62

29,74

70

w

1

Fair.

*5

5i

7

O

53

62

29,76

82

0,045

w

1

Cloudy.

64

2

O

64

62

29,80

79

N

1

Cloudy.

16

50

7

O

53

62

30,04

78.

0,185

NE

1

Cloudy.

66

2

O

65

62

30,08

69

NE

1

Cloudy.

C 13 3

METEOROLOGICAL JOURNAL

for June, 1 797.

1797

Six’s
Therm,
least and
greatest
Heat.

Time.

Therm.

without.

Therm.

within.

Barom.

Hy-

gro-

me-

ter.

Rain.

Winds.

Weather.

H.

M.

O

O

Inches.

Inches.

Points.

Str.

Junei7

O

53

7

O

55

62

3°,i6

75

s

1

Fair.

71

2

O

69

63

30,16

70

sw

1

Cloudy.

18

51

7

O

57

62

30,15

73

E

1

Fine.

73

2

O

73

63

30, 10

67

E

1

Fair.

l9

49

7

O

56

62

29,94

76

E

1

Fine.

7i

2

O

70

63

29,84

66

E

1

Cloudy.

20

52

7

O

56

63

29,77

80

E

1

Fair.

71

2

O

68

65

29>7l

?o

ESE

1

Cloudy.

21

51

7

O

54

63

29,68

74

0,673

E

1

Cloudy.

65

2

O

63

64

29,64

74

E

1

Cloudy.

22

53

7

O

55

63

29,54

81

0,088

E

1

Cloudy.

67

2

O

63

63

29,44

69

ESE

2

Cloudy.

23

5i

7

O

53

63

29’34

84

0,280

ESE

1

Rain.

63

2

O

62

63

29’39

81

W

1

Cloudy.

24

52

7

O

54

62

29,75

82

°’535

SW

1

Cloudy.

65

2

O

65

63

29,84

76

WSW

1

Cloudy.

z5

52

7

O

53

62

3°j°3

83

O

0

SW

1

Rain.

69

2

O

68

63

3°,°4

66

SW

1

Fair.

26

5°

7

O

53

62

30,08

80

SW

1

Fair.

66

2

O

65

64

30,07

68

SW

1

Fair.

27

50

7

0

53

62

30,02

81

0,022

N

1

Cloudy.

67

2

O

65

64

29>93

7i

NE

1

Cloudy.

28

5i

7

O

54

62

29,99

79

E

1

Cloudy.

70

2

O

70

64

30,00

63

SE

1

Fair.

29

50

7

O

53

63

29,87

82

0,135

W

1

Rain.

66

2

O

64

63

29,82

67

NE

1

Fair.

3°

5o

7

0

54

63

30,06

77

0,062

NE

1

Cloudy.

67

2

0

64

63

30,10

69

SE

1

Cloudy.

C h 3

METEOROLOGICAL JOURNAL

for July, 1797..

1797

Six’s
Therm,
least and
greatest
Heat.

Time.

Therm ,
without.

Therm.

within.

Barom,

Hy-

gro-

me-

ter.

Rain.

Winds.

Weather.

H.

M.

O

0

Inches.

Inches.

Points.

Str.

July 1

0

54

7

O

56

63

30,07

78

sw

I

Cloudy.

70

2

O

68

63

30,01

74

s

I

Cloudy.

2

5i

7

O

56

63

30,00

81

sw

I

Cloudy.

72

2

O

69

64

30,03

71

sw

2

Fair.

3

54

7

O

55

63

29,91

80

sw

2

Cloudy.

6 4

2

O

61

63

29>74

80

s

2

Cloudy.

4

48

7

O

55

63

29*91

78

0,101

w

2

Fair.

69

2

O

67

64

29,94

69

w

2

Cloudy.

5

54

7

O

55

63

29,76

80

0,020

ssw

2

Rain.

65

2

O

63

63

29,64

83

ssw

2

Cloudy.

6

5i

7

O

56

62

29,51

79

0,063

ssw

2

Cloudy.

66

2

O

64

63

29,51

74

WNW

2

Cloudy.

7

5 2

7

O

57

62

29,72

79

w

2

Cloudy.

68

2

O

65

62

29,76

71

w

I

Cloudy.

8

54

7

O

57

63

29,77

80

NW

I

Cloudy.

67

2

O

66

63

29,88

75

NW

I

Cloudy.

9

54

7

O

58

63

3°*I4

78

NE

I

Fine.

75

2

O

74

64

30,19

68

N

I

Cloudy.

10

58

•7

0

60

64

30,25

82

sw

I

Cloudy.

81

2

0

78

66

30,24

72

s

I

Fair.

1 1

59

7

0

61

66

80

0,082

s

I

Fine.

80

2

0

79

68

3°»°3

67

s

I

Fine.

12

57

7

0

59

66

29,92

78

s

2

Cloudy.

74

2

0

73

67

29,92

70

s

2

Cloudy.

13

55

7

0

58

66

29,92

77

SE

I

Fair.

76

2

0

75

68

29,88

69

SE

I

Fine.

H

61

7

0

66

68

29,84

82

0,1 10

E

I

Fine.

85

2

0

84

69

29,87

70

SE

I

Fine.

15

60

7

0

66

69

29,90

74

E

I

Cloudy.

82

2

0

81

7i

29,92

68

E

I

Hazy.

1 6

59

7

0

64

7o

30,04

79

SW

I

Fine.

85

2

0

84

73

29,99

67

SSW

I

Fair.

C 3

METEOROLOGICAL JOURNAL

for July, 1797.

1 797

Six’s
Therm,
least and
greatest
Heat.

Time.

Therm.

without.

Therm.

within.

Barom.

Hy-

gro-

me-

ter.

Rain.

Winds.

W eather.

H.

M.

0

O ~

Inches.

Inches.

Points.

Str.

July 17

O

62

7

O

65

71

29,95

77

0,420

s

I

Fine.

"Much than-

84

2

O

83

73

29,96

66

s

I

Fine.

lightning

18

56

7

O

6 1

7i

30,07

75

s

I

Fine.

_ last night.

76

2

O

76

72

30,07

64

sw

I

Fair.

19

57

7

O

60

70

3°,°7

77

ssw

2

Cloudy.

73

2

O

73

70

30,02

68

SSW

2

Fair.

20

59

7

O

59

69

30,00

81

0,125

s

I

Rain.

7i

2

O

69

69

29,98

76

ssw

I

Fair.

21

5i

7

O

56

68

30, 1 1

78

0,102

sw

I

Fair.

7i

2

O

7i

69

30,14

66

sw

I

Fair.

22

5i

7

O

55

66

30,16

77

w

I

Cloudy.

63

2

O

63

66

3°> *5

75

w

I

Cloudy

ii

23

5°

7

O

56

64

30,21

81

w

I

Fine.

75

2

O

75

68

30*23

66

w

I

Fair.

24

55

7

O

63

66

30,20

78

w

1

Cloudy.

80

2

O

79

69

3°>*3

66

w

I

Fine.

25

56

7

O

59

67

30,1 1

82

sw

I

Cloudy.

80

2

O

79

73

30,10

7 1

w

I

Fine.

26

60

7

0

63

69

^0,07

80

sw

1

Fine.

85

2

O

83

73

30,04

66

ssw

I

Fine.

27

63

7

O

66

7i

29,98

75

w

I

Hazy.

80

2

O

80

74

29,99

68

w

I

Fair.

28

61

7

O

64

69

30,04

74

N

I

Hazy.

75

2

O

74

72

30,04

68

SSE

I

Hazv.

29

58

7

O

65

7i

29>93

80

E

1

Cloudy.

78

2

O

76

73

29,86

70

E

I

Fair.

3°

62

7

O

62

72

29,61

87

0,265

NE

I

Rain.

76

2

O

75

72

29,69

72

S

2

Fair.

3i

59

7

O

62

68

29,74

77

S

2

Cloudy.

73

2

O

73

70

29,74

69

s

2

Fair.

C 16 3

METEOROLOGICAL JOURNAL
for August, 1797.

Six’s

Time.

Therm.

Therm.

Barom.

Hy-

Rain.

Winds.

Therm.

without.

within.

gro-

I 707

feast and

me-

Weather.

1 /y /

greatest

ter.

Heat.

H.

M.

0

0

Inches.

Inches.

Points .

Str.

Aug. 1

O

58

7

O

58

69

29,72

84

0,123

SE

I

Rain.

72

2

O

70

69

29,60

76

S

2

Cloudy.

2

53

7

O

57

68

29,76

80

0,142

SSW

1

Fine.

70

2

0

65

69

z9’74

74

SSW

I

Rain.

3

54

7

O

58

67

29>74

77

0,190

SSW

2

Cloudy.

69

2

O

66

67

29>74

78

sw

2

Rain.

4

53

7

O

54

66

29,81

81

°’I33

SSW

2

Cloudy.

7i

2

O

69,5

67

29,85

68

w

I

Fair.

5

55

7

O

57

67

29,97

79

0,080

w

I

Fair.

7i

2

O

65

67

29,99

7 1

w

I

Rain.

6

50

7

O

53

66

30,09

83

0,083

SSW

I

Fine.

70

2

O

68

67

3°, 10

73

SSW

I

Fair.

7

51

7

O

55

65

30,09

78

0,038

SSW

I

Fine.

70

2

O

70

66

30,09

66

SSW

I

Cloudy.

8

55

7

O

58

65

z9>93

78

ESE

I

Fair.

76

2

O

76

66

29,79

67

SSE

2

Fine.

9

55

7

0

57

67

z9>74

73

SW

2

Fair.

7l

2

O

69

67

29,81

68

sw

z

Cloudy.

10

54

7

O

57

66

29>93

77

0,030

sw

I

Cloudy.

69>5

2

0

68

66

29,94

7°

sw

I

Cloudy.

1 1

58

7

0

58

66

29,72

80

SSW

I

Rain.

64

2

0

63

66

29,68

80

s

2

Cloudy.

12

5°

7

0

53

65

29,78

80

0,057

s

I

Fine.

72

2

0

7i

67

29,78

68

sw

I

Cloudy.

13

49

7

0

53

65

29,78

78

sw

1

Cloudy.

70

2

0

69

65

29,81

66

sw

2

Cloudy.

53

7

0

56

65

29,90

82

0,113

sw

I

Fine.

73

2

0

73

66

29,98

67

sw

1

Fair.

*5

53

7

0

57

65

29,96

75

s

I

Cloudy.

71

2

0

69

65

■ 29,88

73

s

2

Cloudy.

16

56

7

0

57

66

29»77

83

0,1 10

NE

i

Cloudy.

:

68

2

0

68

65

29,83

75

s

1

Cloudy.

C 17 3

- T

METEOROLOGICAL JOURNAL

for August, 1797.

Six’s

Time.

Therm.

Therm,

Barom.

Hy-

Rain.

Winds.

Therm,
least and

without.

within.

gro-

1797

me-

Weather.

greatest

Iieat.

H.

M;

O

O

Inches.

ter.

Inches.

Points.

Str.

Aug. 1 7

0

51

*7

/

O

55

65

2 9>97

80

s

1

Fine.

18

72

2

0

7i

67

29,91

70

SE

1

Fair.

58

72

7

2

0

0

58

69

66

66

29,48

29,60

88

69

o,I95

E

S

1

2

Cloudy.

Fair.

l9

. 55

7

0

58

67

29,66

82

SSE

1

Fair.

70

2

0

70

67

29,64

68

S

1

Fair.

20

53

7

0

58

66

29,85

81

0,170

SW

1

Cloudy.

66

2

0

61

66

29,84

80

S

1

Rain.

21

54

7

0

56

65

29,86

80

0,156

SW

2

Fair.

70

2

0

69

68

29,87

69

WSW

2

Fine.

22

48

7

0

53

63

30,07

78

WSW

1

Fine.

67

2

0

67

66

30,10

67

NW

1

Cloudy.

23

5°

7

0

52

64

30,14

81

NW

1

Cloudy.

67

2

0

67

6J

30,14

73

NW

1

Cloudy.

24

52

7

0

56

64

3°, 1 8

81

0,025

W

1

Cloudy.

71

2

0

70

65

30,18

70

SW

1

Cloudy.

25

56

7

0

58

64

30,04

83

0,074

SE

1

Cloudy.

26

65

2

0

64

65

30,01

78

S

1

Cloudy.

48

7

0

52

64

30,02

82

S

1

Fine.

7i

2

0

7l

66

29,95

73

S

1

Fair.

27

60

7

0

61

65

29,83

86

0,150

S

1

Rain.

28

72

2

0

72

67

29,83

75

S

2

Fair.

59

7

0

61

66

29,80

86

0,095

S

2

Rain,

70

2

0

70

66

29,80

81

S

2

Cloudy.

29

57

7

0

62

66

29,88

80

s

2

Cloudy.

72

2

0

7i

67

29,81

78

s

2

Cloudy.

3°

59

7

0

59

67

29,70

86

CO

0

s

1

Fair.

69

2

0

69

67

29>77

70

s

1

Fair.

31

52

7

0

57

66

29,90

82

SSE

1

Cloudy.

70

2

0

7°

66

29,86

7i

S

1

Cloudy.

C

I

C 18 3

METEOROLOGICAL JOURNAL

for September, 1 797.

1797

Six’s
Therm,
least and
greatest
Heat.

Time.

Therm.

without.

Therm.

within.

Barom.

Hy-

gro-

me-

ter.

Rain.

Winds.

Weather.

H.

M.

0

0

Inches.

Inches.

Points.

Str.

Sept. 1

O

57

7

O

60

66

29’53

86

0,075

E

1

Cloudy.

7i

2

O

69

67

29,45

78

8E

2

Cloudy.

2

53

7

O

53

66

29,61

84

0,840

WJN W

1

Rain.

62

2

O

62

66

29,80

75

WNW

1

Cloudy.

3

44

7

O

47

63

30,10

80

s w

1

Fine.

64

2

O

64

64

3°> * 3

67

NW

I

Cloudy.

4

45

7

O

5Z

64

3°>I3

77

E

I

Cloudy.

64

2

O

64

64

3°>n

67

sw

I

Cloudy.

5

44

7

0

46

64

30,14

80

w

I

Fine.

68

2

O

68

64

30,07

67

w

I

Fair.

6

54

7

O

55

62

29,83

81

0,1 12

s

2

Rain.

68

2

O

68

63

29>79

78

sw

2

Cloudy.

7

55

7

O

55

63

29>74

89

0,55°

s

2

Cloudy.

69

2

O

69

64

29,70

70

sw

2

Fair.

8

5J

7

O

52

64

29,84

77

w

1

Cloudy.

63

2

O

63

64

29,84

66

ssw

I

Fair.

9

47

7

O

49

63

29,67

77

0,145

sw

2

Fine.

62

2

O

62

63

29,84

65

w

2

Fair.

10

42

7

O

45

62

30,07

78

w

1

Fine.

62

2

O

62

63

29,98

70

w

1

Fair.

1 1

5°

7

O

51

62

29G7

80

E

I

Rain.

59

2

O

58

62

29,04

90

E

I

Rain.

12

5°

7

O

51

61

29,51

81

0,505

SSW

2

Fine.

60

2

O

58

61

29,57

73

SW

2

Fair.

13

52

7

O

53

6 1

29,80

83

0,030

SSW

2

Fine.

66

2

O

64

63

29>73

71

S

2

Fair.

14

55

7

O

57

62

29,61

82

0,275

s

2

Cloudy.

62

2

O

62

62

29,54

78

S

2

Cloudy.

15

48

7

O

5°

61

29,78

81

0,088

S

1

Fine.

64

2

O

64

61

29,96

75

w

I

Fair.

16

47

7

O

49

61

30,13

82

ssw

1

Fair.

64

2

O

64

61

30,05

73

s

2

Cloudy.

C 19 3

METEOROLOGICAL JOURNAL

for September, 1797.

1 797

Six’s
Therm
leas: anc
greatest
Heat.

Time.

1

Therm.

without

Therm.

within.

Barom,

Hy-
gro-
in e-
ter.

Rain.

Wind*.

Weather.

H.

M

0

O

Inches.

Inches.

Points.

Str.

Sept.17

0

55

7

O

58

61

29,67

90

0,203

SVV

2

Cloudy.

68

2

O

68

62

z9>73

78

s w

1

Cloudy.

18

5i

7

O

52

61

29,62

82

0,050

j SbW

1

Fine.

64

2

O

64

62

29,62

74

ssw

1

Fair.

19

49

7

O

52

61

29,61

82

0,032

E

1

Cloudy.

60

2

O

57

61

29->43

81

E

1

Rain.

20

49

7

O

5i

60

29,41

83

0,390

S

2

Cloudy.

59

z

O

57

61

29H3

76

S

2

Fair.

21

46

7

O

47

60

29,72

85

S

1

Fine.

63

2

O

°3

62

29,79

70

S

1

Fine.

22

42

7

O

43

60

29,89

84

E

1

Hazy.

61

2

O

61

6l

29,91

73

E

1

Cloudy.

23

5°

7

O

5i

60

29,80

88

NE

1

Cloudy.

65

2

O

65

62

Z9’7I

75

E

1

Fine.

24

55

7

O

I5

61

29,60

87

SW

1

Cloudy.

62

2

O

62

62

29>57

80

E

1

Cloudy.

25

53

7

O

I3

6l

29’55

88

0,270

N

1

Cloudy.

60

2

O

60

62

29,64

87

N

1

Rain.

26

55

7

O

I5

61

29,75

90

0,410

NE

1

Cloudy.

63

2

O

62

62

29,71

84

NE

1

Cloudy.

27

53

7

O

53

6l

29,69

86

ESE

1

Cloudy.

63

2

O

62

62

29j73

79

E

1

Cloudy.

23

52

7

O

54

6l

29,90

87

NE

1

Foggy.

64

2

O

64

62

29,94

81

NE

1

4azy.

29

54

7

O

55

62

29,96

88

E

1

Cloudy.

65

2

O

64

62

29,86

79

S

1

Cloudy.

3°

53

7

O

54

6 1

29’54

88

0,085

S

2

lain.

61

2

O

60

61

29,58

72

s

1

2

Cloudy.

c 2

C 20 D

METEOROLOGICAL JOURNAL

for October, 1797.

1 797

Six’s
Therm,
ea.-f and
greatest
Heat.

Time.

Therm.

without.

Therm.

within.

Barom.

Uy-

=>ro-

me-

Ram.

Winds.

Weather.

H.

M.

O

O

Inches.

ter.

Inches.

Points.

Str.

Oct. 1

O

50

7

0

52

6l

29,85

84

sw

I

Cloudy.

57

2

O

57

6l

29,96

75

w

I

Fair.

2

46

7

0

46

6l

30,02

84

E

I

Fair.

62

2

O

62

6l

30,00

77

S

1

Cloudy.

3

53

7

O

54

6l

29,99

86

0,063

E

I

Fair.

63

2

O

62

6l

29,96

75

SE

I

Fair.

4

53

7

O

53

61

30,04

86

NE

I

Cloudy.

61

2

O

61

62

30,08

77

NE

1

Fair.

5

5°

7

O

5i

6l

30,12

86

NE

I

Cloudy.

59

2

O

59

60

30,09

82

NE

I

Cloudy.

6

52

7

O

52

6l

29,97

87

0,065

NE

I

Cloudy.

57

2

O

57

60

29,91

84

NE

I

Rain .

7

44

7

O

45

60

29,93

83

NE

I

Fine.

58

2

O

57

60

29,95

82

NE

I

Fair.

8

44

7

O

46

59

30,00

84

W

I

Cloudy.

61

2

O

60

60

29,96

82

s

2

Cloudy.

9

5 1

7

O

52

59

30,02

87

wsw

I

Cloudy.

61

2

O

60

60

30,07

73

NNE

I

Fair.

10

49

7

O

50

59

30,27

73

NE

I

Cloudy.

58

2

O

58

58

30,28

67

NE

I

Cloudy.

1 1

50

7

O

50

58

3°,3J

80

NW

I

Cloudy.

57

2

O

57

58

3°,3i

84

NW

I

Cloudy.

12

46

7

O

49

58

3°,3I

83

NW

I

Cloudy.

57

2

O

57

58

30,25

78

NW

I

Cloudy.

13

48

7

0

48

58

30,25

82

NW

I

Fair.

57

2

O

57

58

3°, *4

72

NW

I

Fair.

14

46

7

O

47

58

29,93

81

0,035

NW

I

Fine.

53

2

O

52

57

30,08

73

NW

I

Fine.

15

39

7

O

42

57

30,19

81

SW

I

Cloudy.

58

2

O

57

57

30,16

79

SW

I

Cloudy.

16

48

7

O

5i

57

30A4

85

s

I

Cloudy.

58

2

O

58

58

30,12

78

ssw

I

Cloudy.

C 21 3

METEOROLOGICAL JOURNAL
for October, 1797.

Six’s

Time.

Therm.

Therm.

Barom.

Hy-

Rain.

Winds,

Therm.

without.

within.

gro-

1797

least and

me-

greatest

ter.

Heat.

H.

M.

O

O

Inches.

Inches.

Points.

Str.

Oct. 17

O

50

7

O

53

57

3°,°3

85

sw

I

Cloudy.

60

2

O

59

58

29,99

88

sw

I

Cloudy.

18

5°

7

O

53

57

29,57

83

s

2

Cloudy.

56

2

O

55

59

29,57

70

w

2

Fair.

*9

39

7

O

4°

56

29.62

79

sw

1

Fair.

5°

2

O

49

58

29,63

74

sw

I

Fine.

20

42

7

O

43

57

29»37

89

0,385

NE

1

Cloudy.

4 6

2

O

45

57

29,30

83

NE

I

Cloudy.

21

36

7

O

36

5 6

29,38

83

SW

1

Fine.

48

2

O

48

57

29,40

7i

sw

I

Fine.

22

35

7

O

36

56

29,48

82

ssw

1

Cloudy.

48

2

O

48

57

29,5 1

80

SSE

2

Cloudy.

23

42

7

O

44

55

29’53

86

°»145

E

1

Rain.

45

2

O

44

55

29,54

87

ESE

1

Rain.

24

42

7

O

44

54

29,45

90

0,923

E

1

Cloudy.

5i

2

O

5i

56

29’33

83

E

I

Cloudy.

25

42

7

O

42

55

29,12

87

0,050

SE

I

Rain.

48

2

O

48

56

29,05

80

WSW

I

Cloudy.

26

38

7

O

39

55

29,10

86

0,240

N

I

Cloudy.

46

2

O

45

56

29,38

83

W

I

Cloudy.

27

35

7

O

37

54

29,76

8?

N

I

Fair.

46

2

O

46

55

29,80

83

NE

I

Cloudy.

28

35

7

O

35

54

29,96

85

NE

I

Fair.

47

2

O

47

54

29,94

79

NE

2

Cloudy.

29

3<5

7

O

42

53

29’73

81

NE

2

Cloudy.

42

2

0

42

54

29,57

85

NE

I

Rain.

3°

4i

7

O

42

53

29,60

85

0,095

NE

2

Cloudy.

46

2

0

46

54

29,63

83

NE

2

Cloudy.

3i

4i

7

0

42

53

z9>77

81

NE

2

Cloudy.

46

2

0

46

54

29,90

75

NE

2

Cloudy.

C 22 J

METEOROLOGICAL JOURNAL

for November, 17 97.

J797

Six’s
Therm,
least and
greatest
Heat.

Time.

Therm.

without.

Therm.

within.

Barom.

Hy-

gro-

me-

ter.

Rain.

Winds.

Weather.

H.

M.

O

O

Inches.

Inches.

Points.

Str.

Nov. 1

0

34

7

O

35

52

3°>i8

81

NE

2

Fine.

47

2

O

47

54

30,27

73

NE

2

Fair.

2

3i

7

O

32

52

3°,23

82

W

I

Cloudy.

44

2

O

44

53

30,09

85

sw

I

Cloudy.

3

42

7

0

44

53

29,89

88

sw

2

Cloudy.

52

2

O

5 2

55

29,84

81

sw

I

Cloudy.

4

44

7

O

44

54

29,70

83

sw

I

Cloudy.

48

2

O

48

56

29,57

83

E

I

Cloudy.

5

44

7

O

45

54

29,41

87

0,102

E

I

Rain.

55

2

O

55

58

29’43

89

S

2

Cloudy.

6

5°

7

O

5°

55

29,88

89

SE

I

Fair.

57

2

O

57

59

29,96

86

E

I

Fine.

7

49

7

O

48

57

30,11

91

E

I

Foggy.

55

2

O

55

59

3°’17

89

E

J

Fair.

8

47

7

O

47

58

3°>39

91

E

I

Foggy.

54

2

O

53

59

3°, 41

89

E

I

Fair.

9

41

7

O

42

57

30,48

90

E

I

Foggy.

48

2

O

47

58

30,46

89

E

I

Cloudy.

10

44

7

O

44

57

3°G9

88

E

I

Cloudy.

52

2

O

50

57

30,39

82

E

I

Fair.

I X

40

7

O

40

56

30,38

88

E

I

Fine.

52

2

O

52

59

30,35

80

E

I

Fair.

12

40

7

O

40

56

30,24

88

Foggy.

45

2

O

45

57

30,20

88

W

I

Foggy.

13

34

7

O

34

56

30,11

86

NE

I

Foggy.

51

2

O

5*

58

30,07

87

NE

I

Fair.

H

45

7

O

5°

57

30,04

90

S

2

Cloudy.

53

2

O

53

58

30,11

81

SW

2

Cloudy,

IS

43

7

O

44

57

30,09

88

0,102

sw

- I

Fine.

5i

2

O

51

59

30,15

78

wsw

I

Fair.

16

4i

7

O

42

57

3oA5

84

wsw

I

Cloudy.

52

2

O

52

58

30,16

83

NE

I

Cloudy.

C 23 3

METEOROLOGICAL
for November,

JOURNAL

1797-

Six’s

Time,

Therm.

Therm.

Barom.

Hy-

Rain.

Winds.

Therm.

without.

within.

gro-

1 7Q7

least and

me-

Weather.

greatest

ter.

Heat.

H.

M.

0

O

Inches.

Inches .

.Points.

Str.

Nov. 1 7

O

35

7

0

35

57

3°*32

85

NE

I

Fair.

44

2

O

44

57

30,28

79

NE

1

Fine.

18

33

7

O

37

56

29,86

84

0,060

SW

I

Rain.

46

2

0

44

57

29,76

84

NE

1

Cloudy.

*9

3i

7

O

32

53

29,88

84

NE

2

Snow.

37

2

0

37

55

29,96

81

NE

2

Fair.

20

27

7

O

3°

53

29,97

83

W

1

Cloudy.

43

2

O

38

54

29,77

85

SSW

I

Rain.

21

34

7

O

35

52

z9’73

83

0,060

NW

1

Fine.

42

2

O

40

53

29,71

79

NW

I

Fine.

22

38

7

O

39

52

29,22

80

WNW

2

Cloudy.

45

2

O

45

53

29,14

73

WNW

2

Fair.

23

3°

7

O

32

5i

29>43

86

SW

I

Cloudy.

36

2

O

36

5i

29,47

84

SSW

I

Cloudy.

24

27

7

O

27

49

29,64

85

SSW

1

Fair.

36

2

O

36

52

29,68

80

N

I

Cloudy.

25

28

7

O

3°

50

29,78

82

NE

X

Cloudy.

45

2

O

39

52

29,78

80

E

I

Cloudy.

26

44

7

O

53

52

29,62

9*

O

00

0

S

2

Cloudy.

55

2

O

55

54

29,66

9°

S

2

Cloudy.

27

49

7

O

5°

54

29’74

88

0,102

SW

2

Fair.

52

2

O

52

56

29,83

81

SW

I

Hazy.

28

47

7

O

47

54

29,82

90

0,020

SW

I

Rain.

47

2

O

44

56

29,78

91

NE

I

Rain.

29

35

7

O

35

53

29,78

88

0,502

NE

I

Rain.

48

2

O

37

53

29,68

87

NE

I

Cloudy.

3°

36

7

O

40

52

29,44

90

0,145

SW

I

Cloudy. 1

49

2

O

49

56

29,46

90

SW

2

Cloudy.

|

i

C H 3

METEOROLOGICAL JOURNAL

for December, 1797.

r

1797

Six’s
Therm,
least and
greatest
Heat.

Time.

Therm.

without.

Therm.

within.

Barom.

Hy-

gro.

me-

ter.

Rain, 1

Winds.

Weather.

H. M.

O

O

Inches.

Inches.

Points.

Str.

Dec. 1

2

3

4

5

6

7

8

9

10

1 1

12

13
H

15

16

0

45

51

41

47
3i
38
29

52

42
54
5°

48

33
4i

34
5°

37

43

36
40

34

3i

43
34

44

38
44

37
5i
48

53

8 O
2 O

8 0

2 O

8 0

2 O
8 0

2 O

8 0

2 0

8 0

2 0

8 0

2 O

8 0

2 0

8 0

2 O

8 0

2 0
8 0

2 0

8 0

2 0
8 0

2 O

8 0

2 O

8 0

2 O

8 0

2 0

48

48

41
47
31
38

3°

42

45

54

5°

45

35

36

45

37

42

3 8
4°

33

34
40

43

35

43

38

44
38
47

49
53

54
57
53

55

53

54
5i
53
53

55

55
57
53
57

53

54
5i

56
53

55

51
53

50

52
49

53

52

54

53

54
54

57

29’35

29.27
29,20
29,23
29,60
29,78
30,02
29,85
29,87

30.01
29,65
29,85

3°, 16

3°* 1 5

30.02

29*79

29,62

29,67

29*34

29*33

29*53

29,65

29,36

29.11

29*33

29.28
29,07

29.12
29,50
29*44
29,5°
29,48

91

88

80

78

83
78

84
84
88

83
88
70
82

7 6
78
80

84
80
86
77

85

85
84
89
88
88
89

86
86
89
89
87

0,058

0,315

0,1 IO
0,020

0,075

0,625

0,105

s

s

s

sw

wsw

NW

NE

E

WSW

sw

w

WNW

WNW

WNW

NW

SW

sw

s

ssw

wsw

WNW

NW

s

s

s

s

s

s

ssw

s

s

s

2

2

2

2

1

1

1

1

1

1

1

1

1

1

1

2
1
1
1
1
1

1

2
2

1

2
1
1

1

2
2
2

Cloudy.

Rain.

Fine.

Fair.

Fair.

Cloudy.

Fair.

Cloudy.

Fair.

Fair.

Rain.

Fine.

Fair.

Cloudy.

Cloudy.

Cloudy.

Fair.

Fine.

Rain.

Fair.

Cloudy.

Fair.

Rain.

Rain.

Fine.

Rain.

Fair.

Fair.

Cloudy.

Rain._

Rain.

Cloudy.

C 25 3

METEOROLOGICAL
for December,

JOURNAL

1797-

Six’s

Time.

Therm.

Therm.

Barom.

Hy-

Rain.

Winds.

Therm.

without.

within.

8TO-

least and

I 707

Weather.

k /y /

greatest

ter.

Heat.

H.

M.

O

O

Inches.

Inches.

Points.

Str.

Dec. 17

O

53

8

O

53

56

29,28

85

s

3

Cloudy.

5°

2

O

48

58

29,44

82

ssw

2

Rain.

1 8

44

8

O

46

56

29,88

88

0,020

s

2

Cloudy.

52

2

O

52

58

29,84

85

s

2

Cloudy.

*9

5°

8

O

52

56

29,98

83

SE

2

Cloudy.

56

2

0

56

58

30,02

83

SSE

2

Fair.

20

5 1

8

O

5i

57

30,17

88

SW

i

Cloudy.

54

2

O

54

60

30,22

87

W

1

Cloudy.

21

34

8

O

36

57

3°>32

85

NE

1

Foggy.

45

2

O

44

57

30,24

84

E

1

Cloudy.

22

43

8

O

44

57

30,04

89

0,210

SE

1

Rain.

49

2

0

49

59

29,95

89

SE

1

Cloudy.

23

40

8

O

40

54

29,87

89

Foggy.

4°

2

O

39

58

29,88

88

NE

1

Cloudy.

24

35

8

O

35

56

3°, 1 8

88

SW

1

Cloudy.

37

2

0

37

57

30,24

88

WNW

1

Cloudy.

25

32

8

O

32

54

30.45

87

W

1

Fair.

38

2

O

38

57

30.45

86

WSW

1

Fair.

26

33

8

O

47

55

30,34

91

0,068

SW

1

Cloudy.

5°

2

O

5°

57

30,33

9i

SW

1

Cloudy.

27

47

8

0

47

55

30,32

89

SW

1

Cloudy.

5°

2

O

50

58

30,32

87

WSW

1

Cloudy.

28

i 1

8

O

33

54

30,46

86

SW

1

Fair.

42

2

O

4i

56

30,43

81

WNW

1

Fair.

29

40

8

O

42

54

30,20

85

w

1

Cloudy.

47

2

0

46

56

29,95

84

w

1

Cloudy.

3°

41

8

O

41 -

53

29,80

76

0,005

w

2

Fair.

46

2

O

44

56

29,93

75

NW

1

Fair.

3i

38

8

O

44

54

29,63

78

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1

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47

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46

55

29,62

76

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The quicksilver in the bason of the barometer is 81 feet above the level of low water spring tides at

Somerset-house.

PHILOSOPHICAL

TRAN SAC T IONS

OF THE

ROYAL SOCIETY

OF

LONDON.

FOR THE YEAR MDCCXCVIII.

PART If.

LONDON,

SOLD BY PETER ELMSLY,
PRINTER TO THE ROYAL SOCIETY.

MDCCXCVIII.

. ^

.

.8TH3THOD

1 A. *d'S -«8"vio\w\\3Q%\<1 K ZX

.Z H A .yiS. tboowJA

' ••• ’■> ; ^iroi^vO" - > zv* .IX

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* • ' ; t ;• ■ & 5\ ^

<\ ••.. 1 )iC. -adi \o sdi i-,a *>>?\hO sdt'\o kK JI’X

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v .. ■ v vuwwoO .^o^uX .noaliW -^'Tt.t.;i 7A XI .VtasH

. 177. 7 ,V\ < • 7 v .-•• jj -yT

. i-. - . <\'.o^U> 51 ■ v: -a. - ■-.' • ■■r^y:?, • iO ^Xvy.^K 7 ! A

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7 A .*ie& ,«>[nfi8 dqs«ol .stoK X^/II

^vo7/,e^a^l .vS. j&suAtav>$3L ^ i^ooSI s&t :^0 . 7 "

.

, Z ,5\ A .& XI ,9n :

> * .1 >

-

'

p *

CONTENTS.

X. A disquisition on the Stability of Ships. By George

Atwood, Esq. F. R. S. p. 201

XI. Quelques Remarques d’Optique , principalement relatives a

la Reflexibilite des Rayons de la Lumiere. Par P. Prevost,
Professeur de Philosophie a Geneve , de V Academie de Berlin ,
de la Societe des Curieux de la Nature , et de la Societe Roy ale
d* Edimbourg. Communicated by Sir Charles Blagden, Knt.
F. R. S. p. 311

XII. An Account of the Orifice in the Retina of the human Eye ,

discovered by Professor Soemmering. To which are added ,
Proofs of this Appearance being extended to the Eyes of other
Animals. By Everard Home, Esq. F. R. S. p. 332

XIII. A Description of a very unusual Formation of the human

Heart. By Mr. James Wilson, Surgeon. Communicated by
Matthew Baillie, M. D. F. R. S. p. 346

XIV. Account of a singular Instance of atmospherical Refraction.
In a Letter from William Latham, Esq. F.R.S. and A.S. to
the Rev. Henry Whitfeld, D.D. F.R.S. and A.S. p. 357

XV. Account of a Tumour found in the Substance of the human
Placenta. By John Clarke, M. D. Communicated by the
Right Hon. Sir Joseph Banks, Bart. K. B. P. R. S. p. 361

XVI. On the Roots of Equations. By James Wood, B.D. Fellow
of St. Johns College , Cambridge . Communicated by the Rev.
Nevil Maskelyne, D . D. F. R. S. and Astronomer Royal.

P- 3 69

C iv 3

XVII. General Theorems , chiefly Pori sms, in the higher Geo-
metry. By Henry Brougham, Jun. Esq. Communicated by
Sir Charles Blagden, Knt. F. R. S. p. 378

XVIII. Observations of the diurnal Variation of the Magnetic
Needle , in the Island of St. Helena ; with a Continuation of
the Observations at Fort Marlborough , in the Island of Su-
matra. By John Macdonald, Esq. In a Letter to the Right
Hon. Sir Joseph Banks, Bart. K. B. P. R. S. p. 397

XIX. On the Corundum Stone from Asia. By the Right Hon.

Charles Greville, F. R. S. p. 403

XX. An Inquiry concerning the chemical Properties that have

been attributed to Light. By Benjamin Count of Rumford,
F. R. S. M. R. I. A. p. 449

XXI. Experiments to determine the Density of the Earth. By

Henry Cavendish, Esq. F.R.S. and A.S. p. 469

XXII. An improved Solution of a Problem in physical Astro-

nomy ; by which , swiftly converging Series are obtained, which
are useful in cornputing the Perturbations of the Motions of
the Earth , Mars, and Venus, by their mutual Attraction. To
which is added an Appendix, containing a?i easy Method of
obtaining the Sums of many slowly converging Series which
arise in taking the Fluents of binomial Surds, &c. By the
Rev. John Hellins, F. R. S. Vicar of Potter s Pury, in North-
amptonshire. In a Letter to the Rev. Nevii Maskelyne, D. D.
F. R. S. and Astronomer Royal. p. 527

XXIII. Account of a Substance found in a Clay -pit ; and of
the Effect of the Mere of Diss, upon various Substances im-
mersed in it. By Mr. Benjamin Wiseman, of Diss, in Nor-
folk. Com?nunicated by John Frere, Esq. F. R. S. With an
Analysis of the Water of the said Mere. By Charles

\

C v 3

Hatchett, Esq. F. 11. S. In a Letter' to the Right Hon . Sir
Joseph Banks, Bart . K. B. P. R. S. p.

XXIV. A Catalogue of Sanscnta Manuscripts presented to the
Royal Society by Sir William and Lady Jones. By Charles

p. 582

Wilkins, Esq. F. R . S.

Presents received by the Royal Society, from November, 1797
to June, 1798. p. 595

Index . p ggg

I

ERRATA

In Mr. Atwood’s Disquisition on the Stability of Ships.

Page 213, lines 8 and 9, dele ct situated between wind and water* according to a
technical expression.”

SB

Page 2Z3, 1. 22, for -f SB, read -{ .

3

Flu,* xy3 + P3 , a , r . yz + vl
247, 1, 9, for — —t—, read fluent of x x- .

3 P P

252, 1. 25, after point V, add produce XL in directum, and in the line so pro-

254, 1. 22, after the words c< proportion of 3 to 5,” instead of the comma

insert a colon.

Page 268, 1. 8, for DD'G'GA, read DD'G'GD.

duced.

PHILOSOPHICAL

TRANSACTIONS.

X. A Disquisition on the Stability of Ships , By George

Atwood, Esq . F.R.S .

Read March 8, 1798.

The stability of vessels, by which they are enabled to carry
a sufficient quantity of sail, without danger or inconvenience,
is reckoned amongst their most essential properties ; although
the wind may, in one sense, be said to constitute the power
by which ships are moved forward in the sea, yet, if it acts on
a vessel deficient in stability, the effect will be to incline the
ship from the upright, rather than to propel it forward : sta-
bility is therefore not less necessary than the impulses of the
wind are, to the progressive motion of vessels. This power has
also considerable influence in regulating the alternate oscil-
lations of a ship in rolling and pitching; which will be smooth
and equable, or sudden and irregular, in a great measure, ac-
cording as the stability is greater or less at the several angles
of inclination from the upright. From constantly observing
that the performance of vessels at sea depends materially on
their stability, both navigators and naval architects must, at all
MDCCXCVI II. D d

so 2 Mr. Atwood's Disquisition on

times, be desirous of discovering in what particular circum-
stances of construction this property consists, and according to
what laws the st?K is affected by any varieties that may
be given to their forms, dimensions, and disposition of con-
tents ; which are determined partly according to the skill and
judgment of the constructor, and partly by adjustments after
the vessel has been set afloat.

Little more than a century has now elapsed, since the theory of
mechanics was first applied to the construction and management
of vessels ; whatever principles had been previously adopted, for
regulating their forms and equipment, as well as for directing
them in the ocean, were the result of experience and observation
alone : a mode of arriving at truth, however advantageous in
many respects, yet not entirely to be relied on in this instance,
for explaining satisfactorily, and reducing to system, pheno-
mena depending on the intricate combination of causes which
influence a vessel's motion, and equilibrium, at sea. The theory
of mechanics is known to explain all effects that can arise from
the action of forces, however complicated, of which the quan-
tities and directions are defined with sufficient precision. This
science, having been greatly extended, and successfully em-
ployed, by Sir Isaac Newton, in the investigation of causes
requiring the most profound research, would naturally be re-
sorted to, for a solution of many difficulties that occur in the
theory of naval architecture, which could not be obtained from
any other mode of considering this subject. The practice of
ship building having been many ages antecedent to the dis-
covery of the theory of mechanics, one object of theoretic
inquiry must necessarily be, to explain the principles of con-
struction and management which experience and practical

the Stability of Ships. %og

observation have previously discovered ; distinguishing those
which are founded in truth and right practice, from others
which have been the offspring of vague and capricious opinion,
misinterpretation of facts, and unfounded conjecture, by which,
phenomena arising in the practice of navigation are often at-
tributed to causes entirely different from those by which they
are really governed. It is also the object of mechanic theory
to investigate, from the consideration of any untried plans of
construction, what will be the effect thereof on the motion of
vessels at sea ; also to suggest new combinations, by which
the approved qualities of vessels may be extended, their faults
amended, or defects supplied. These several objects, and others
connected with them, have employed the attention of many
eminent theorists, by whose discoveries naval architecture has
been greatly benefited ; yet the progress made toward esta-
blishing a general theory, founded on the laws of motion, has
not been adequate to what might be expected from the abi-
lities of the writers on this subject, and the laborious attention
they have bestowed upon it. Although all results deduced by
strict geometrical inference from the laws of motion, are
found, by actual experience, to be perfectly consistent with
matter of fact, when subjected to the most decisive trials, yet,
in the application of these laws to the subject in question, dif-
ficulties often occur, either from the obscure nature of the
conditions, or the intricate analytical operations arising from
them, which either render it impracticable to obtain a solu-
tion, or, if a result is obtained, it is expressed in terms so
involved and complicated, as to become in a manner useless,
as to any practical purpose. These imperfections in the theory
of vessels, are amongst the causes which have contributed to

Dd s

204 Mr. Atwood's Disquisition on

retard the progress of naval architecture, by increasing the
hazard of failure in attempting to supply its defects by expe-
riment ; for, when no satisfactory estimate can be formed
from theory, of the effects likely to ensue from adopting any
alteration of construction that may be proposed, doubts must
necessarily arise respecting its success or failure, which can be
resolved only by having recourse to actual trial : a species of
experiment rarely undertaken under the impressions of uncer-
tain success, when the objects of it are so costly, and otherwise
of so much importance. To the imperfections of theory, may
also be attributed that steady adherence to practical methods,
rendered familiar by usage, which creates a disposition to re-
ject, rather than to encourage, proposals of innovation in the
construction of vessels : the defects or inconveniences which
are known, and have become easily tolerable by use, or may
perhaps be the less distinctly perceived for want of comparison
with more perfect works of art, being deemed preferable to
the adoption of projected improvements, attended by the dan-
ger of introducing evils, the nature and extent of which cannot
be fully known. These are amongst the difficulties and dis-
advantages which have concurred in rendering the progress of
improvement, in the art of constructing vessels, extremely slow,
and have left many imperfections in this practical branch of
science, which still remain to be remedied. In respect to the
theory of vessels, it would be giving that term too narrowed
a meaning, to consider it as derived solely from the laws of
mechanics ; every notion or opinion which may be applied to
explain satisfactorily the phenomena depending on a vessel's
construction and qualities, so as to infer the consequences of
given conditions, independently of actual trial, whether it ori~

the Stability of Ships. 205

ginates from experience alone, or from investigations founded
on the laws of motion, is to be regarded as forming a part of
this theory, in which, a constant reference to practice is so
essentially necessary. For, although many principles are de-
ducible from the laws of mechanics, which it is probable that
no species of experiment, or series of observation, however
long continued, would discover, yet there are others, no less
important, which have been practically determined with suf-
ficient exactness, the investigation of which it is scarcely pos-
sible to infer from the laws of motion ; the complicated and
ill defined nature of the conditions, in particular instances,
rendering analytical operations founded on them liable to un-
certainty. Since the practice of naval architecture depends so
materially on the knowledge of the causes which influence
the motion of vessels at sea, much benefit may probably be
derived from the extension of well founded principles, both by
attentive observation of the qualities of vessels, compared with
their construction, as well as by investigation of the effects
arising from particular modes of construction, depending on
the laws of statics and mechanics, whenever the conditions
admit of inferring principles which are clear and satisfactory,
and easily applicable in practice. With a view to these ob-
jects, so far as regards the theory of stability, the ensuing
Disquisition has been written.

When a ship, or other floating body, is deflected from its
quiescent position, the force of the fluid’s pressure operates to
restore the floating body to the situation from which it has been
inclined. This force is distinctly described, in a treatise written
by the most celebrated geometrician of ancient times, who
uses the following argument for demonstrating the position in
which a parabolic conoid will float permanently in given cir-

20 6 Mr. Atwood's Disquisition on

cumstances. To shew that this solid will float with the axis
inclined to the fluid's surface at a certain stated angle, depend-
ing on the specific gravity and dimensions of the solid, he de-
monstrates,* that if the angle should be greater than that which
he has assigned, the fluid's pressure will diminish it ; and that,
if the angle should be less, the fluid’s pressure will operate to
increase it, by causing the solid to revolve round an axis which
is parallel to the horizon. It is an evident consequence, that
the solid cannot float quiescent with the axis inclined to the
fluid’s surface, at any angle except that which is stated. The
force which is shewn in this proposition, to turn the solid, so
as to alter the inclination of the axis to the horizon, is the
same with the force of stability; the quantity or measure of
which, Archimedes does not estimate ; nor was it necessary
to his purpose, since the alteration of inclination required to
establish the quiescent position, may be produced either in a
greater or less time, without affecting his argument. It
does not appear, that this method of determining the float-
ing positions of bodies was afterwards extended to infer
similar conclusions in respect to solids of any other forms,
nor to determine any thing concerning the inclination or
equilibrium of ships at sea, which require the demonstration,
not only that a force exists, in given circumstances, to turn
the vessel round an axis, but also the magnitude or precise
measure of that force. M. Bouguer, in his treatise intitled
“ Traite du Navire •f has investigated a theorem for esti-
mating the exact measure of the stability of floating bodies.
This theorem, in one sense, is general, not being confined to
bodies of any particular form ; but, in respect to the angles of

* Archimedes de iis qua in bumido vebuniur. + Livr. ii. sect. 2. chap. 8.

the Stability of Ships. 207

Inclination, it is restrained to the condition that the inclina-
tions from the upright shall be evanescent, or, in a practical
sense, very small angles. In consequence of this restriction,
the rule in question cannot be generally applied to ascertain
the stability of ships at sea ; because the angles to which they
are inclined, both by rolling and pitching, being of consider-
able magnitude, the stability will depend, not only on the con-
ditions which enter into M. Bouguer's solution, but also on
the shape given to the sides of the vessel above and beneath
the water-line or section, of which M. Bouguer's theorem
takes no account. But it is certain that the quantity of sail a
ship is enabled safely to carry, and the use of the guns in
rough weather, depend in a material degree 011 the form of
the sides above and beneath the water-line ; this observation
referring to that portion of the sides only which may be im-
mersed under, or may emerge above, the water's surface, in
consequence of the vessel's inclination ; for, whatever portion
of the sides is not included within these limits, will have no
effect on the vessel's stability, the centres of gravity, volume
of water displaced, and other elements not being altered. By
the water-section is meant, the plane in which the water's sur-
face intersects the vessel, when floating upright and quiescent ;
and the termination of this section in the sides of the vessel is
termed the water-line. A general theorem for determining
the floating positions of bodies is demonstrated in a former
paper, inserted in the Phil. Trans, for the year 1 796, and ap-
plied to bodies of various forms : the same theorem is there
shewn to be no less applicable to the stability of vessels, taking
into account the shape of the sides, the inclination from the
upright, as well as every other circumstance by which the

so8 Mr. Atwood's Disquisition on

stability can be influenced. To infer, from this theorem, the
stability of vessels in particular cases, the form of the sides,
and the angle of inclination from the perpendicular, must be
given. These conditions admit of great variety, considering
the shape of the sides, both above the water-line and beneath
it ; for we may first assume a case, which is one of the most
simple and obvious ; this is, when the sides of a vessel are pa-
rallel to the plane of the masts, both above and beneath the
water-line ; or, secondly, the sides may be parallel to the masts
under the water-line, and project outward, or may be inclined
inward, above the said line ; or they may be parallel to the
masts above the water-line, and inclined either inward or out-
ward beneath it ; some of these cases, as well as those which
follow, being not improper in the construction of particular
species of vessels, and the others, although not suited to prac-
tice, will contribute to illustrate the general theory. The
sides of a vessel may also coincide with the sides of a wedge,
inclined to each other at a given angle ; which angle, formed
at an imaginary line, where the sides, if produced, would inter-
sect each other, may be situated either under or above the
water's surface. To these cases may be added, the circular
form of the sides, and that of the Apollonian or conic para-
bola. The sides of vessels may also be assumed to coincide
with curves of different species and dimensions, some of which
approach to the forms adopted in the practice of naval archi-
tecture, particularly in the larger ships of burden. And lastly,
the shape of the sides may be reducible to no regular geome-
trical law ; in which case, the determination of the stability,
in respect to a ship's rolling, requires the mensuration of the
ordinates of the vertical sections which intersect the longer

209

the Stability of Ships.

axis at right angles ; similar mensurations are also required
for determining the stability, in respect to the shorter axis,
round which a vessel revolves in pitching. In order to de-
scribe distinctly these several cases, the variation of the sec-
tions, both in form and magnitude, from head to stern of the
vessel, has not been considered ; the sections being supposed
equal and similar figures, such as they in reality are, near the
greatest section of a ship, growing smaller, and altering their
form, toward the head and stern. But, before this alteration
can be taken into account, it is necessary first to ascertain the
stability corresponding to a vessel or segment, in which the
sections are equal and similar figures ; from which determi-
nation, the stability is inferred which actually exists, when the
form and magnitude of the sections alter continually, from one
extremity of the vessel to the other. The consideration of
the cases which have been here stated, with inferences and
observations thereon, is the subject of the ensuing pages ; ill
which, if any ideas are suggested which may be at all useful
in the practice of naval architecture, or may contribute to
remove imperfect or erroneous notions which have been en-
tertained respecting a principal branch of it, the intention of
the Author will be accomplished.

Let WBCOFAH (Tab. VIII. fig. i.) represent a vertical
section of a vessel floating quiescent and upright, and inter-
sected by the water's surface in the line BA : BCOFA will be
the area immersed under water. Suppose the vessel to be in-
clined from the perpendicular, through the angle ASH, so
that the intersection of the vessel by the water's surface, which
before coincided with BA, shall now coincide with the line

mdccxcviii. • Ee

210 Mr. Atwood's Disquisition on

CH : the area under water will now be COFAH, equal to the
area BCOFA.

Let the section WBCOFAH, and all the other vertical sec-
tions intersecting the longer axis at right angles, be assumed
similar and equal figures, projected on the plane WBOAH : in
consequence, the area BOA will be to the area ASH, as the
entire volume immersed is to the volume immersed by the
vessel's inclination. Moreover, if E is the centre of gravity of
the area BOA, that point will truly represent the centre of
gravity of the volume immersed, when the vessel is upright :
if the centre of gravity of the immersed area COFAH, when
the vessel is inclined, should be situated at Q, that point will
also coincide with the centre of gravity of the corresponding
displaced volume. For these reasons, the spaces BOA, ASH,
COFAH, will be denominated, in the following pages, indif-
ferently, areas or volumes.

Let G be the centre of gravity of the vessel, by which term,
the vessel and its contents, of every kind, are always under-
stood to be implied. Through G, draw GU parallel to CH :
and through Q, draw QZ perpendicular to C H. When the
ship is inclined round the longer axis, through the angle ASH,
the fluid’s pressure acts in the direction of the vertical line
QZ, with a force equal to the vessel's weight ; and the sta-
bility or effect of this force, to turn the vessel round an axis
passing through G, perpendicular to the plane BOA, will be
greater or less, according to the magnitude of the line G Z,
or distance from the axis at which the force of pressure acts.
In the same vessel, the weight not being altered, the stability,
at different angles of inclination from the upright, will be truly
measured by the line G Z ; and, in different vessels, or in the

211

the Stability of Ships .

same vessel differently laden, the stability will be measured
by the weight of the vessel and the line G Z jointly. The
weight of any vessel (including the lading) is equal to the
weight of water displaced by it ; which will be obtained by
measuring the solid contents of the„ displaced volume, and
from knowing the weight of a given portion of sea water, such
as a cubic foot, which weighs 64 pounds avoirdupois. The
vessel's weight being thus obtained, the determination of the
stability, whatever be its form or inclination from the upright,
requires only that the line G Z shall be known, or the pro-
portion which it bears to some given line, for instance, the
line B A, shall be ascertained.

A general method of constructing this line is demonstrated
in the Phil. Trans, for the year 1 796, but is there principally
applied to the floating position of bodies ; its use in inves-
tigating the stability of vessels is incidentally mentioned, and
in general terms, rather than as being itself a subject of dis-
quisition. This theorem is founded on supposing the centres
of gravity of the several volumes BOA, COFH, ASH, BSC,
(fig. 1.) to be given in position ; an assumption allowable in
demonstrating a general theorem : but, in applying it to the
stability of particular vessels, it becomes necessary that the
positions of these points should be absolutely found, and the
results combined with the other conditions, to infer the mea-
sure of stability ; a determination which, in some cases, is
attended with much difficulty, and in others, is not prac-
ticable by any direct methods ; an instance, amongst mamr
that might be mentioned, in which the particular application
is more difficult than the general demonstration of propo-
sitions. The following constructions and investigations are

E e 2

212

Mr. Atwood's Disquisition on

principally inferred from the general theorem for ascertaining
the stability of floating bodies ; which is here subjoined, to
avoid the necessity of future references, as well as for the
purpose of stating more distinctly the observations which fol-
low it.

Let M (fig. i.) be the centre of gravity of the volume ASH,
which has been immersed under water, and let I be the centre
of gravity of the volume BSC, which has emerged above the
water's surface, in consequence of the vessel's inclination ;
through the points M and I, draw the lines ML, IK, per-
pendicular to the line CH, which coincides with the water's
surface when the vessel is inclined : through E, the centre of
gravity of the displaced volume BOA, draw EV parallel and
equal to KL, and through G draw GU parallel and GR perpen-
dicular to CH ; according to the theorem, the line ET will be
determined by the following proportion. As the total volume
displaced BOA is to SAH, the volume immersed in consequence
of the inclination, so is KL or EV to ET ; and, since the angle
EGR is equal to the vessel's inclination ASH, and the dis-
tance GE is supposed to be given, the line ER will be known ;
because ER is to GE as the sine of the angle EGR to radius;
ER being subtracted from ET will leave RT or GZ, equal
to the measure of the vessel’s stability.

Suppose the line K L to be denoted by the letter b : let
the volume ASH be represented by A, and the volume BOA
by V. Then, according to the theorem, since V : A : : : : b :
ET, it follows that ET = and if GE is put = d, and 5
= the sine of the angle to which the vessel is inclined, radius
being = 1, ER will be = ds ; and the measure of the ves-
sel's stability RT or GZ = y — d s.

the Stability of Ships.

213

Through the points C and H, (fig. 1.) let the lines CF,
WH, be drawn parallel to BA. The position of the points
M and I, the magnitude of the line KL, and the areas or
volumes ASH, BSC, being the same, whatever alteration
may take place in the volume V, or the entire volume dis-
placed, the quantity KL x area ASH or b A will remain
the same : and, since the line ET= it will follow, that
the zone WHFC, situated between wind and water, (ac-
cording to a technical expression,) not being altered, ET will
be in the inverse proportion of V, or the total volume dis-
placed. If, therefore, the shape of the vessel under the line CF
should be any how changed, so as to coincide with another
figure, suppose C cfF, (fig. 2.) instead of COF, (fig. 1.) the
volume C cf F being equal to the volume COF, the line ET
will be the same in both cases. In consequence of this change
of figure, the position of the point E, (fig. 1. ) or centre of gra-
vity of the volume BOA, may be situated higher or lower in
the line OD ; yet, if the centre of gravity G is so adjusted by
ballast, or other means, that the distance GE shall be the
same, the stability of each vessel, BCOA (fig. 1.) and BC cf A
(fig. 2.) will be perfectly the same, when inclined to the same
angle ASH from the upright. It must also be observed,
that since ET is always greater in the same proportion in
which the volume immersed BOA is less, the zone WH CF
being both in magnitude and form the same, having found
by construction or calculation the value of the line E T cor-
responding to any given volume displaced, suppose V =
BCOA, (fig. 1.) the line E t corresponding to any other mag-
nitude of volume displaced, suppose z> = BCV w l FA, (fig. 2.)
will be immediately inferred ; for, since V . v ; : E# : E T, it

214 Mr. Atwood's Disquisition on

follows that E t == ET-X or because ET = by substitu-

tion, E£ = — . For these reasons, the determination of sta-
bility does not require that the form of the entire volume
displaced should be given, but the form only of the zone
WCHF, (fig. 1. and 2.) including the angle of the vessel's
inclination ASH; these conditions, together with’ the mag-
nitude of the immersed volume, and the distance between the
two centres of gravity G and E, are sufficient for finding
the measure of stability, at any given angle of inclination
from the upright.

/

CASE I.

The sides of a vessel are parallel to the plane of the masts,
both above and beneath the water-line.

QBCOAH (fig. 3 ) coincides with the vertical section of
a vessel when it floats upright and quiescent, and is inter-
sected by the water's surface in the line B A ; the sides Q C,
H D, are parallel to each other, and to the plane of the masts
W O, and are therefore perpendicular to B A. G is the centre
of gravity of the vessel ; Y represents the magnitude of the
volume immersed under the water ; the centre of gravity of
this volume is situated at E. Suppose the vessel to be in-
clined from its quiescent position through any given angle,
it is required to express, by geometrical construction, the mea-
sure of the vessel's stability, when thus inclined. Bisect B A
in the point S, and through S draw C S H, inclined to B A, at
the given angle of the vessel's inclination from the upright.
Bisect BC in F, and AH in N ; and join SF and SN. In the
line SF take SI to SF as 2 to 3 ; also, in the line SN, take

the Stability of Ships. 215

SM to SN as 2 to 3. Through the points I and M, draw IK,
ML, perpendicular to C H. Through the point E, draw EV
parallel and equal to KL. In the line EV, take ET to EVS
in the proportion which the volume ASH bears to the entire
volume displaced- Through G, draw GU parallel to C H;
and through T, draw TZ perpendicular to GU. GZ is the
measure of the vessel's stability. The demonstration of this
construction evidently follows from the general theorem.

From this construction, the value of GZ, or measure of the
vessel's stability, may be investigated analytically, and ex-
pressed in general terms. Through G,*draw GR perpendi-
cular to E V. Let B A = t, GE=^, the angle AS H = S ;
radius = 1. The rules of trigonometry give the following de-
terminations. AN = : SN = ~ xv/4 + tang/ S.

Also, as S N : H N : : sine NHS : sin. N S H, or ~ x

4

+ tang.1 S : * x s ; : cos. S. : sin. N S H. Wherefore

sin.NSH= =J!-4— ; ;

V/4+ tang.aS

2 4- sec.2, S -f cos.2 S
4 -J- tang.2 S

consequently cos. NSH = ^

V4

cos.1 N S H = 4 + tang.- S - sin.- S

44- tang.2 S

sec S 4- cos/sf ,,

4 4. tang'.* s' (because 2 x
sec. S 4- cos. S

4- tang.2S

cos.S x sec.S =s 2)
. And since by construe-
tion, SM-|SN, and SN = ~ + tang.1 S,SM =

~ x /1 + tang. 1 S, and SL=|x ^4, + tang.1 S x

sec. S 4- cos. S t — — —■ —

^- 4- = — x sec. S -|- cos. S : and the triangles S L M,

S I K being similar and equal, K L = 2 S L : Wherefore

= y x sec, S -|- cos. S = EV. The area of the triangle

AS H == — — 8 -g; S representing the volume immersed by the
vessel's inclination ; and by construction.

21 6

Mr. Atwood's Disquisition on

As V : volume ASH: : E V : E T, or

y . L-.x. ^-ng' s : : y x sec. S -j- cos. S : ET ; this will give

the value of E T = si . and because

E R : E G : : sin. S : 1, and E G = d, it follows, that
ER = d x sin. S ; and therefore R T, or the measure of the

vessel's stability GZ = - -■ x cos. S + sec. S — d x sin. S.

To exemplify this determination by referring to a particu-
lar case, let the vessel's breadth at the water's surface, or BA,
be divided into 100 equal parts, and let GE be 13 thereof;
so that t = 100, and d = 13. Suppose the inclination of the
vessel from the perpendicular, or ASH, to be 150, = S ; and
let the area BCODA, representing the volume displaced,
be equal to a square of which the side is = 60 ; so that the
area V shall = 3600 : then, referring to the solution, we
obtain

cos. S -|- sec. S = 2.0012

Also <-x-t-7's. = = 3.1013

ET = 2.0012 x 3.10190 = 6.2063

d x sin.S = 13 x sin .15° = 3.3646

measure of stability, or G Z = 2.8417
It appears by this result, that when the vessel has been
inclined from the upright through an angle of 150, the direc-
tion of the fluid's pressure, acting to restore the quiescent
position, will pass at a distance estimated horizontally from
the axis = 2.84, when the breadth B A = 100. And this will
be true, whatever be the length of the axis.

The fluid's pressure is the weight of water displaced, the
magnitude of which depends both on the area of the vertical

the Stability of Ships, 217

sections, and length of the axis : suppose this weight to be
1000 tons ; according to the preceding determination, the sta-
bility of the vessel, when inclined from the upright to an angle
of 150, will be a pressure equal to the weight of 1000 tons, act-
ing at a distance of parts of the breadth B A from the
axis, to restore the vessel to the position from which it has
been inclined. This force is the same as if a pressure of

* °°°-* 2— = 56.8 tons, should be applied to turn the vessel at
the distance of 50 from the axis : if therefore the wind, or other
equivalent power, should act on the sails of the vessel with a
force of 56.8 tons, at the mean or average distance of 50, or ~
the breadth BiV from the axis, to incline the ship, the force of
stability will just balance it, so as to preserve an equilibrium ;
the vessel continuing inclined from the upright at the angle of
150. If the wind's force should be less, the inclination must
necessarily be diminished ; if greater, it must be increased,
until the two forces balance each other. Here it is to be ob-
served, that the force of the wind is estimated in a direction
which is perpendicular to the plane of the masts *

* In this and the following numerical examples, in order to bring into comparison
the effect of giving different forms to the sides of vessels, their weights, and all the
other conditions (the figure of the sides excepted) on which the stability depends, are
assumed to be the same. The measures of stability are compared, both by the relative
distances from the axis at which ,a given pressure, equal to the vessel’s weight, acts to
turn the ship round the longer axis, and by the relative equivalent weights which act
at a given distance from the axis. By the latter method, the proportions of stability
are perhaps more distinctly expressed than by the former, although both are essen-
tially the same.

The mechanical force employed to incline a vessel from the upright, through any
given angle, for the purpose of examining and repairing -the bottom of a ship, is to
ibe ascertained from the theorems here given for expressing the measures of stability,

jwdccxcvili. F f

2l8

Mr. Atwood's Disquisition on

CASE II.

The sides of a vessel project outward above the water-line*
and are parallel to the masts under the water-line.

The line B A (fig. 4.) represents the intersection of the
water's surface with the vessel, when floating upright. The
lines PC, QW, are parallel to each other, and to the line XO,
which coincides with the plane of the masts, and bisects the line
BA in the point D ; B C and AW, which are parallel to the
plane of the masts, coincide with the sides of the vessel under
the water-line ; and BY, AH, which project outwards from
the plane of the masts, at the angle QAH, or YBP, are the
sides of the vessel above the water-line. C H represents the
intersection of the water's surface with the vessel, when in-
clined from the perpendicular, through a given angle OPO =
ASH. The distance G E, between the centres of gravity of
the vessel and of the volume displaced, and the magnitude of
that volume being supposed known, and the angle QAH, at
which the sides AH, BY, are inclined to the plane of the masts,
being also known, it is required to ascertain, by geometrical
construction, the measure of the vessel's stability, when the

which is exactly equal to the force to be applied for that purpose. Another method
of inclining a vessel (well adapted for making experiments on this subject) is, by ap-
plying a timber at right angles to the plane of the masts. If a weight be affixed to
one of its extremities, from having given the weight so applied, and its distance from
the plane of the masts, together with the other conditions which determine stability,
the angle of inclination, through which the ship will be inclined, may be determined
by the theorems in these pages. The same inferences may be obtained, from having
given the weights and spaces through which the guns are run out on one side, and
drawn in on the other, instead of the weight affixed, according to the method last
described.

the Stability of Ships.

** 9

vessel is inclined from the perpendicular, through an angle
equal to the angle OPQ.

At whatever angle the vessel may be inclined from the per-
pendicular, the total volume immersed must always remain the
same, while the vessel's weight continues unaltered. Where-
fore, the volume which has been immersed, or ASH, must be
equal to the volume BSC, which has emerged from the water,
in consequence of the vessel's inclination. For this reason, and
because the side AH projects outward, while the side BC is
parallel to the plane of the masts, it must necessarily happen,
that the point S will not in this case bisect the line BA, as it
did in the preceding construction, blit will be removed nearer
to the side AH, which has been immersed by the inclination
of the vessel. Previously, therefore, to any consideration of
the stability, it will be necessary to define the position of the
point S in the line AB, so that a line CH, being drawn
through S, at a given angle of inclination to AB, equal to that
of the ship's inclination from the perpendicular, shall cut off
the area ASH equal to the area BSC.

Let the given angle of inclination be OPO equal to ASH
(fig. 4.) the angle QAH, at which the sides of the vessel are
inclined outward from the plane of the masts above the water-
line, is supposed to be given : this angle -[- go° will be the angle
SAH, which is therefore a known quantity : the remaining
angle SHA, in the triangle ASH, will likewise be known.

Through the extremity B of the line BA, (fig. 5.) equal to
the vessel's breadth at the water-line, draw the indefinite line
BU inclined to BA, at an angle ABU equal to OPQ : in BU,
take any point O, and, in the line BO, set off BD to BO, as the
cosine of the angle ABU to radius. In the line BD, take BE to
BD, as the sine of the angle BAU is to radius : also take BF to

F f 2

220

Mr. Atwood's Disquisition on

BO, as the sine of the angle AUB to radius. Let BG be taken
a geometrical mean proportional between the lines BF and BE ;
and from the point G, in the direction of the line GU, set offGZ
equal to BF : join AZ ; and, through the point G, draw GS
parallel to ZA, intersecting BA in the point S. Through S,
draw the line CH parallel to BU : the area ASH will be equal
to the area BSC.

Since, in the triangles ASH, BSC, the angle ASH is equal
to the angle BSC, the areas of the triangles will be in a ratio
compounded of the ratios of the sides, including the equal
angles ; that is, the area of the triangle ASH, will be to the
area of the triangle BSC in the ratio of SA x SH to SB x
SC. By the construction, the angle ASH = the angle ABU
= OPQ ; and the angle AHS = the angle AUB : also, by
construction,

BO : BD : : rad. : cos. ASH.

Also BD : BE : : rad. : sin. SAH.

And BF : BO : : sin, AHS : rad.

Joining these ratios BF : BE : : sin. AHS x rad. : cos. ASH x sin. SAH.

But, by the construction, and by the similarity of the triangles,
BGS, BZA, BF or GZ : BE* : : BF* : BG* : : GZ* : BG* : : SA* : SB* :

Wherefore SA* : SB* : : sin. AHS x rad. : cos. ASH x sin. SAH.

And by trigonometry SH : SA : : sin. SAH : sin. AHS ;

and SB : SC : : cos. ASH : rad.

Joining these ratios SA x SH : SB x SC : : rad. : rad.

But the area ASH is to the area BSC as SA x SH to SB x
SC ; consequently, the area ASH is equal to the area BSC.

To proceed with the construction of the second case.
Through the point S, (fig. 4.) determined by the preceding
construction, draw the line CH inclined to BA at the angle

* Because the ratio of BE to BG is equal to the ratio of BG to BF, by the construe-
tion, it will follow that the ratio of BE to BF is double the ratio of BE to BG.

221

the Stability of Ships.

ASH, equal to the given angle OPQ: when the vessel is inclined
from the perpendicular through this angle, it will be inter-
sected by the water's surface coinciding with the line C H.
Bisect B C in F, and AH in N ; and join S F, S N : take S I
to S F, as 2 to 3 ; and S M to S N, in the same proportion.
Through I and M, draw the lines IK, ML, perpendicular to
C H. Through the centre of gravity of the vessel G, draw
G U parallel to CH; and through the centre of gravity E, of
the displaced volume BOA, draw E V parallel and equal to
KL; and in EV take ET to E V, in the same proportion
which the volume ASH bears to the entire volume displaced
B OA. Through T, draw T Z perpendicular to G U. G Z is
the measure of the vessel's stability.

To obtain an analytical value of the line GZ, for brevity,
let the sine of the angle ASH be denoted by s, when radius
is = l, make sin. HAS = a ; sin. AH S == b ; sin. S C B = c.
Let GE = rf. Also, let the entire volume displaced = V. By
the rules of trigonometry, it is found that

* SL = — x V 4 + s'x i—% I V x

3 V ^ 1 b* h

Also S K = — x cos. ASH 4- sec. ASH.

3 1

p f . i, ac tt S B2 x tang. ASH

The area SCB or ASH = -

2

Wherefore ET x j- *+?■* W+^^+

+ Xt^-ASH x cos. AS H sec. ASH. If the breadth BA

* When the angle SAH- go, r — cP rf o ; and h — cos. S ; in which case, if c is
SA Z F s+

put— cos. S, SL r= — ■ x 4- + but 4 + — s =4 + tang.2 S x sin.2 S — 4 +
3 c c

tang.2 S —tang.2 S x cos.2 S — 2 4- sec.2 S 4- cos.2 S ~ cos. S -f sec. sj1. Wherefore SL —

SA ___

— - x cos. S sec. S,

3^

222

Mr. Atwood’s Disquisition on

be represented by the letter t, it is inferred, from the construc-
tion in p. 220, that SA = —I* ^4= and SB = . The

Vb-^V'ac Vh-\-Vac

value of the line ET having been thus determined, if ER = d x
sin. ASH or d s be subtracted from it, the result will be GZ,
the measure of the vessel's stability.

Suppose the sides BY, AH, (fig. 4.) to project outward, at an
angle of 150 inclination to the parallel sides BC, AW, so as to
make the angle SAH = 105°, Let the vessel's inclination from
the upright be the angle ASH — 15°; and therefore AHS = 6o°,
and SCB = 750. Let the breadth BA or t = 100 equal parts, of
which d or GE = 13. Then, by calculating from the analy-
tical values just determined, it is found that KL = SL -{- SK~
68,017: the area ASH = 347.44, and the entire volume
immersed V, being, as in the former case, = 3600, E T —

-■’°I = 6.37. And, since ER or d x sin. ASH is =
3.36, if the latter value be subtracted from the former, the re-
sult will be GZ = 3-2i, or the measure of the vessel's stability.

The force of stability, to restore the vessel to the upright po-
sition, will be precisely the vessel’s weight, or fluid's pressure,
acting in the direction of a vertical line, which passes at a dis-
tance of 3.20 from the axis, estimated in a horizontal direction.
And this force is equivalent to, and will counterbalance, —
parts of the vessel's weight, applied to act in a contrary direc-
tion, at the distance of 50 from the said axis. So that, if the
vessel's weight should be 1000 tons, the force of stability would

balance a weight or force of — 21 *qi00° = 64.2 tons, applied to
act at the distance 50 from the axis.

the Stability of Ships .

223

CASE III.

The sides of a vessel are inclined inward above the water-
line, and are parallel to the plane of the masts under the water-
line.

AH, BY, (fig. 6.) are the sides of a vessel inclined inward
above the water-line BA, at an angle HAQ = YBP from the di-
rection of the sides AW, BC, under the water-line, which are
parallel to each other, and to the plane of the masts. Suppose
the vessel to be inclined from the upright, through an angle
== O P Q. By the construction, (p. 220.) draw the line CH
intersecting BA, in a point S, at an angle ASH equal to
the given angle OPQ; so that the area ASH shall be equal
to the area BSC. When the vessel has been inclined through
the given angle OPQ, it will be intersected by the water's
surface in the line CH. The construction of the line GZ, or
measure of the vessel's stability, is the same as in the preceding
case.

Let the sine of ASH = 5 ; sin. SAH == a ; sin. SHA == h ;
sin. S C B == c to rad. = 1. Also let GE = d.

From the rules of trigonometry, it is inferred that

If, therefore, the total volume immersed is made = V, the
value of the line ET will be

+ SB x cos. ASH -f- sec. ASH.

The area SBC or ASH = — X-Hng- ASH.

ET ^ sb

X tang. ASH / ,

Sv- * ^ 4 +

4S X '/ I — CL'

h ~~

Mr. Atwood’s Disquisition on

224

+ SB-X'"^,,AS^' x cos* aSH + sec. ASH ; in which expres-
sion S A = and SB = \ t being = the

v b -\-V ac v h -\-V ac 0

breadth BA.

The value of ET having been thus obtained, if ER = d x
sine ASH be subtracted from it, there will remain the val^ie of
GZ, the measure of the vessel’s stability.

Suppose the vessel’s inclination from the perpendicular, or
ASH, to be = 150, let the inclination of the sides inward above
the water-line, from the direction of the parallel sides under the
water, or HAQ = 15°; therefore SAH == 750, and SHA = 90°,
making BA = t, and, applying these conditions to the analyti-
cal value just determined, it is found that KL = 65.530; the
area ASH = 323.42 ; and the volume immersed, or V, being

assumed = 3600, as in the preceding cases, ET =

= 5.89. Subtracting from this. Eft .= 3.36, there will remain
GZ = 2.53, or the measure of stability. If the vessel’s weight
should be 1000 tons, the force of stability will be 1000 tons.

acting to turn the vessel at a distance of parts of the breadth

B A from the axis ; which is equal to a force or weight of

jooo^x M3 _ tons, acting to turn the vessel at a distance of
50 from the axis.

CASE iv.

The sides of a vessel project outwards, and at equal incli-
nations to the plane of the masts, both above and beneath the
water-line.

BA (Tab. IX. fig. 7.) is the breadth of the vessel, and coin-

the Stability of Ships.

22 5

r cides with the water’s surface, when the vessel floats upright.
XE denotes the plane of the masts, bisecting BA in the point S.
PU, QW, are lines drawn through the extremities of the line BA,
and perpendicular to it, and therefore parallel to EX : the sides
of the vessel above the water-line, AH, BY, are inclined outward
from the plane of the masts, at an angle QAH — PBY ; and
BC, AD, are the sides under the water-line, also inclined out-
ward from the plane of the masts, at an angle = DAW =
CBU = QAH. G and E represent the centres of gravity of
the vessel, and of the volume displaced, as in the former cases.
To construct the measure of stability, corresponding to any
given angle of inclination from the upright,

Through the point S, which bisects the line B A, draw the
line CH inclined to BA, at the angle ASH, equal to the given
angle of inclination from the upright. Since, by the condi-
tions of this problem, the triangles ASH, BSC, are similar and
equal figures, it follows, that when the vessel is inclined from
the perpendicular through the angle ASH, it will be inter-
sected by the water’s surface, in the direction of the line C H.
The subsequent part of this construction is similar to those of
the preceding cases, as sufficiently appears by inspection of the
figure.

Let the breadth of the vessel at the water’s surface, or B A
= t: put the sine of the angle ASH == s, sine S AH = a, sine
SHA = h = radius = 1, GE = d. Then the area ASH, or

BSC = and, if the total volume immersed is put = V, the

, ■»

measure of the vessel’s stability, or GZ, will be == — x

Gg

MDCCXCVIII.

22 6 Mr. Atwood's Disquisition on

Let BA, or t = 100, d = 13, ASH = 15°, SAH = 105°, V
=3600, as in the former cases : then, s = sine 150, a =sine 103°,
h = sine 6 o°; by referring to the solution, GZ = 3.59; and
the stability will be the weight of the vessel, suppose 1 000 tons,
acting at the distance 3.59 from the axis, to turn the vessel;
which force is equivalent to a weight of 71.7 tons, applied at
the distance of 50 from the axis.

CASE V.

The sides of a vessel are inclined inward, and at equal
angles of inclination to the plane of the masts, both above and
beneath the water-line.

BA (fig. 8.) is the breadth of the vessel coinciding with
the water's surface, when floating upright. XE represents the
plane of the masts, bisecting BA in the point S. UP, WQ,
are lines drawn through the extremities of the line BA, parallel
to XE. BY, AH, are the sides of the vessel above the water-
line, inclined inward to the plane of the masts, at the angle
QAH = YBP. BC, AD, are the sides under the water-line,
inclined inward to the plane of the masts, at the angle DAW
or CBU, which are equal to HAQ or YBP. The other condi-
tions are as in the former cases. Through the point S, draw
the line CH inclined to BA, at the angle ASH, equal to the
vessel's inclination from the upright. Since the triangles ASH,
BSC, are similar and equal figures, it follows, that when the
vessel is inclined to the angle ASH, it will be intersected by
the water’s surface in the line CH. The remaining part of
this construction is similar to that of the preceding cases.

The same notation being adopted with that which was used

the Stability of Ships .

in the preceding case, by referring to trigonometrical properties,
it is found that the measure of stability, or

GZ

t3 sa
24V/J

x s/ 4 s x

I —h% 4^ X ^ 1 •

d s.

Let t = 100, d—13, the inclination of the sides inward, or
HAQ= if, ASH = if, SAH = 7 f, SHA == 90°: by calcu-
lating from these data, it is found that GZ = 2.21.

If the vessel's weight should be 1000 tons, the stability will
be this weight, acting to turn the vessel at the distance 2.21
from the axis ; which is equivalent to a force of 44.2 tons, ap-
plied at the distance of 50 from the axis.

case vr.

The sides of a vessel coincide with the sides of an isosceles
wedge, (fig. 9.) meeting, if produced, in an angle BWA,
which is beneath the water’s surface.

Supposing the sides to be continued till they meet, the ver-
tical sections will be equal isosceles triangles. BAW repre-
sents one of these triangles, BA being coincident with the
water s surface, and cutting off the line BW equal to AW.
'The angle WBA = WAB is supposed to be given. If the
vessel should be inclined from the perpendicular, so that the
water’s surface shall coincide with the line C H, the point of
intersection S must be so situated, that the area or volume im-
mersed, in consequence of the inclination, that is, ASH, shall
be equal to the area or volume SBC, which has emerged from
the water. Previously, therefore, to the construction of this
case, the position of the point S is to be geometrically deter-
mined, according to the conditions required.

228 Mr. Atwood’s Disquisition on

Let BWA (fig. 10.) represent a vertical section of the ves-
sel. Through the extremity B of the line BA, draw BO in-
clined to BA, at the angle ABO, equal to the vessel’s inclina-
tion from the upright. In this line, take any point R, and in
B R take B I to B R, as the sine of the angle WBR to radius.
Also take BF to BR as the sine of BRW to radius; and let
B G be a geometrical mean proportional between the lines B F
and B I ; from the point G, set off GZ equal to BF ; join Z A,
and, through G, draw GS parallel to ZA; and, through S,
draw CH parallel to BZ. The area ASH will be equal to the
area SBC.

By the construction, the angle ARB =AHS, and the angle
WCH = WBR ;

also BR : BI : : rad. : sine SCB,
and BF : BR : : sine AHS : rad.

Joining these ratios, BF : BI : : sine AHS : sine SCB.

By the construction, and the similarity of the triangles
BGS, BZA.

BF : BI : : BF2 : BG2 : : GZ1 : BG2 : : SA2 : SB2.

Wherefore SA2 : SB2 : : sine AHS : sine SCB

By trigonometry, SH : SA : : sine SAH : sine AHS

and SB : SC : : sine SCB : sine SAH = sine SBC

Joining these ratios, SA x SH : SB x SC : : 1 : 1.

Therefore SA x SH = SB x SC.

But the angle ASH being equal to the angle BSC, the area
of the triangle ASH will be to the area of the triangle BSC, as
SA x SH is to SB x SC ; and, since SA x SH is equal to SB
x SC, the area of the triangle ASH is equal to the area of the
triangle SBC.

22 9

the Stability of Ships.

The point S having been thus determined, (fig. 9.) if the
line CH is drawn through it, inclined to BA at an angle equal
to the vessel's inclination from the upright, the water s surface
will coincide with the line CH.

To proceed with the construction of this case ; bisect BA
in D. (% 9-) and join WD : let G represent the centre of
gravity of the vessel, and E the centre of gravity of the volume
displaced, when the vessel floats upright. Let M and I be the
centres of gravity of the triangles SAH, SBC ; and ML, IK,
lines drawn perpendicular to CH, through the points M and I
respectively. Through G, draw GU parallel to CH ; and,
through E, draw EV parallel and equal to KL. In EV, take
ET to EV as the area ASH is to the area representing the
total volume immersed. Through T, draw TZ perpendicular
to GU. GZ will be the measure of the vessel's stability.

As in the preceding cases, let BA be denoted by the letter t ,
and put the sine of ASH == s, sine SAH = a, sine SHA = h ,
sine SCB = c ; the total volume immersed = V.

By trigonometry, S L

s a /
T x \Z

, slxi — bz . 4s X V 1 __a*

4H — +

b

And, since the area ASH = SA *sa — SB and V is the area

2 h 2 c

representing the entire volume immersed, the measure of sta-
bility, or

g-^rj SA3 x / , szx i—bz . 4sxv'i—

OZ = -6 vT- x V' 4 + — M 1- -

6Vb
SB3 x s«

b

+ x v//4 +

SlXI-CJ

45 X v' I— O1

— d s.

In which expression, S A = and SB = .

Vb+Vc Vh+Vc

23° Mr. Atwood’s Disquisition on

Let the sides ot a vessel be plane surfaces, inclined to each
other at an angle of 30° ; the vessel’s inclination from the up-
right = 15°; BA== £==ioo ; GE = <i = i3; angle SAH =
105 ; AHS = 6o°: BCS = c)o0. By calculating the value of
the line GZ, according to the solution just given, it is found

,v 4. SA3 X s a y

j*X i — hz . isxv'i-a2

4 “r ~T£ F i

b

and

SB3 x 5 a
6Vc

X \r.

4 +

sz X 1 — cz

4s X x —c

= 3-1155
= 3-i°75

Sum of these values
d s

Finally, the measure of the vessel’s stability.

6.2230

3-3%

or

■f

SA3 X s a
Wb~

SB3 x s a
6Vc

xv/

x

4 H l— + ± r

4 +

sz X I

4s X ^ 1

d s — . GZ — 2.858

If the weight of the vessel should be icoo tons, the force of
stability will be equivalent to that weight of pressure, acting at
the distance of 2.85 from the axis ; or the weight of 57.0 tons,
acting at the distance of 50 from the axis.

If the sides should be inclined at an angle of 6o°, instead of
30°, the measure of stability will be 2.92 ; and the effort to turn
the vessel equal to 1000 tons, acting at the distance 2.92, or
58.4 tons acting at the distance of 50 from the axis.

The sides of vessels are not unfrequently formed so as to
coincide with the sides of an isosceles wedge, or are so little
curved as to approximate nearly to that figure, at least so far
as that portion of the sides extends which may be immersed in,
or may emerge from, the water, by the vessel’s inclination. The
preceding solution being expressed in terms which are rather

231

the Stability of Ships .

complicated, another solution is subjoined, by which the mea-
sure of stability is exhibited in more simple terms. The inves-
tigation is troublesome ; but the conciseness of the result, and
the readiness with which it is applied to practical cases, com-
pensate for the difficulty of obtaining it.

Let the isosceles triangle BAF (Tab. X. fig. 11.), represent
a vertical section of the vessel ; the base of which, BA, coin-
cides with the water’s surface, when the vessel floats upright.
Bisect BA in D, and join FD. Let G be the centre of gravity of
the vessel, and take FE to FD as 2 to 3 ; E will be the centre of
gravity of the immersed volume when the vessel floats upright.
Draw the line CH, * intersecting the line BA at an angle
ASH, equal to the given angle of the vessel’s inclination from
the perpendicular, and cutting off the area ASH equal to
the area BSC. When the vessel is inclined through the angle
ASH, the line of intersection with the water’s surface will co-
incide with CH. Bisect CH in the point N, and join FN :
take FQ to FN as 2 to 3. Q is the centre of gravity of the
area CFH, representing the volume immersed when the vessel
is inclined. Through Q, draw QM perpendicular to CH ; and*
through G, draw GZ perpendicular to QM. GZ is evidently
the measure of the vessel’s stability.

To obtain an analytical value of the line GZ, through O,
draw OQP parallel to CH ; through G, draw GR parallel to
QM ; and, through E, draw ET perpendicular to QM. In this
investigation it will be expedient, first, to express in general
and known terms the line FW ; secondly, the line WQ, which
is to MW as the sine of the vessel’s inclination to radius : this
will give the value of MW, which being added to WF before

* By the construction, p. 228.

232 Mr. Atwood's Disquisition on

found, the sum will be the line FM; from which, if FE, or ~ of
FD, be subtracted, there will remain the line ME ; which is
to ET as radius is to the sine of the inclination EMT, or
ASH. ET will therefore be expressed in known terms ; from
which, if ER be subtracted, the remaining line will be RT,
or GZ, the measure of the vessel's stability, analytically ex-
pressed.

By the construction, the area FBA is equal to the area
FCH ; and, since the area BAF is to the area IKF in the
same * proportion which the area FCH bears to the area FOP,
it follows, that the area FIK is equal to the area FOP. Also,
because CN is equal to NH, and OP is parallel to CH, it fol-
lows, that OQ is equal to QP. For brevity, let the angle KYP,
or ASH, be denoted by the letter S; FPO = FHC by P;
POF = HCF by O ; also let the angle PFO be made = F.

Because the areas IFK, PFO, are equal,

FOxFPx sineF __ p £z x tallg i radius being = 1 : wherefore,

:FE-XtangdF=FE^.^F because

FOx sineF

FP , substitution FP = ^xsec^FxsineO

me O ’ J sine P

and therefore FP = FE x sec. ^-F x \ / sin--S- : but sine F WP =s

" SifiG ir

cos. S.

Wherefore, FW : FP : : sine P : cos. S; or
FW : FE x sec. {Fx \/h~, : sine P : cos. S ; consequently,

2 -v sme p

FP =
FO =

FW =

FE x sec. |Fx^ sine O x sine P
cos. S

By investigation,^ founded on the rules of trigonometry, it

* Each of these proportions being as 9 to 4.

f See Appendix.

the Stability of Ships.

*33

appears that y/ sine O x sine P

V i — tang.1 A F x tang.2 S #
sec. A F X sec. S

which quantity being substituted instead of y/ sine O x sine P,
in the value of FW just found, the result will be

FW == FE x s/ 1 — tang.1 \ F x tang.1 S.

It is found also, from trigonometrical rules, that
* WO = FE X ttng.^Fxfang.Sxj^ and since

v'j—tang.1 A Fxtang.1 S

WQ : WM : : sine S : rad. we have

WM= ^xtang.^Fxtang.Sxsec.S or
sine SxV'i — rang.2, A / x tang.2, S

— SeC. s, WM = FE x ta.ig»_|Fxsec^s_

Sme 15 v i — tang.2 a ft x tang.1 S

and, since

FW=FE x y/ 1— tang.ai F xtang.1 S, and FM=WF+WM,

we obtain the value of FM == FE x

sec.1 a F

v' i — tang.1 A F x tang1 S

Therefore ME = FM — FE = FE x

sec.1 A F

V i — tang.1 \ F x tang.1 S

~ h

and ET = FE x sine S x

sec.1 A F

— 1.

V i — tang.1 A F x tang 1 S

This value of ET is inferred from supposing the area BFA to

r

4 x tang, a F

represent the entire volume immersed, and which =
t being equal to the line BA.

If, the sides BC, AH, remaining the same, the figure and
magnitude of the immersed volume should be changed, so as to
be represented by any other quantity V-f, the line ET will be
increased or diminished in the inverse proportion of the en-
tire volumes immersed, that is

e "

as V :

4 x tang, a F

: : FE x sine S x

sec.1 a, F

V" i_ tang.1 A Fxtang.1 S

l : ET.

And, since FE =

3 tang, a F ’

* See Appendix.
MDCCXCVIII.

f See pages 213 and 214.

Hh

Q34<

Mr. Atwood’s Disquisition on

t3 X sine S see.2, A F

~ izVxtaniTp X ~ i;

and the measure of the vessel’s stability, expressed in general
and known terms, will be

*

GZ =

t3 x sine S

12 V x tang.- \ F V i — tang.2, |Fx tang.2, S

Vv'hen the angle of inclination S is evanescent, or in a prac-
tical sense very small, the expression becomes

t3 x sine S , . 0 . . , , . .

~—y a x sine S, agreeing with the solution

sec.2, -£ F

i — d x sin. S.

GZ =

given by M. Euler -f in this particular case.

If the inclination of the sides BF, AF, should be evanescent,
the sides will become parallel to each other, and to the masts,
both above and beneath the water-line ; a case which has al-
ready been solved J : and consequently, the solution of case 1 .
ought to agree with that which has been just given for the
stability, when the two sides are inclined at a given angle,
assuming that angle as evanescent. Assuming, therefore, the
angle BFA evanescent, and S of any finite magnitude in the
general value of GZ, above determined, we have

v/i - tang. ~fF "tang/ S = 5 , and

sec.2, \ F

2 + 2 x fang2 1 F + tang.2 \ F x sec.2, f F x tang.2, S #

vT — tang.2, y F x tang.2, S

and therefore §GZ = -
— d x sin. S.

t3 x sine S

2V x tang.2, \ F

tang.2 A F x tang.2 S + zx tang.2, -§F

* This expression for the measure of stability, is evidently more simple, and better
adapted to practical application, than that which is inserted in page 229. The pre-
sent result might perhaps be obtained by more concise methods : the investigation
here given is the best that occurred to the author, after repeatedly endeavouring to
discover some other, requiring fewer trigonometrical calculations.

f Theory of the Construction and Properties of Vessels, chap. viii. J Case 1.

i3xsin.S 2 + 2 x tang.2 4 F + tang. 2-|- F x sec. 2i F x tang.2 S

s h I “ — - — - X ; II .III -I . 1 ■■ — I ,

1 2V x tang.2 |F 2

or because sec. | F n,

ET

t3 x sin. S 2 + tang.2 S
X ■ ■ «

1.2V

2

the Stability of Ships.

235

or GZ = -?*si”s

24V

or GZ = -^a-"g- S

x tang.1 S 2 — d x sin. S,

2^v x cos. S -f- sec. S — d y sin. S,

which is the measure of stability, when the inclined sides AF,
BF, become parallel, the angle F vanishing. But this quantity
is the measure of stability when the sides are parallel, as de-
termined by direct investigation * ; by which agreement the
consistency of the two solutions is evinced.

To exemplify the general solution for the case of the sides
inclined at a given angle, suppose the angle BFA to be 30°=
F, let S = 150, AB — t — 100, GE = d — 13, V = 3600.
From the analytical value of the line GZ, we obtain

t 3 x sine S

12V xtang.2 yj?

sec.2, i F

- — 1

v' x —tang.3, i F x tang.2 S

83.4464 x .07457
d x sin. S

= 83.4464
= .07457

= 6.223
= 336*5

and GZ, the measure of stability = 2.858,
precisely agreeing with the result calculated by the solution, in
pages 229 and 230, which has no apparent similitude or rela-
tion to the value for stability, as expressed according to this
last investigation, which is

GZ =

t 3 X sin. S

sec.2 \ F

d x sin. S.

12V xtang.2 \ F v' j — tang.* \ Fi x tang.2 S

According to the solution in page 229, the measure of sta-
bility is

GZ

SA3 ysa

6Vb

x V<

. S2XI — /ja . 4s X^ I— a1

4 d n 1 z

+

SB3 ysa
6Vc

x y/

4 +

s2Xi — c2 4s x^ 1— a1

ds ,

* Case x.

H h 2

'236 Mr. Atwood’s Disquisition on

in which value 5 = sin. S ; a = sin. SAH ; c = sin. SCB ;

SA = and SB = 11 miSht not- PerhaPs-

be easy to deduce either of these values from the other, or to
demonstrate their equality, otherwise than by the separate in-
vestigations from which they have been inferred; and yet
these quantities are not approximations to equality, but are
strictly and mathematically equal.

CASE VII.

The sides of a vessel are coincident with the sides of a wedge,
meeting, if produced, at an angle which is above the water’s
surface.

The sides of a vessel are represented by the lines qb , cdy
(fig. 12.) inclined at an angle, so as, if produced, to meet at
the point w above the water’s surface, which is coincident with
ba; the lines wa, wb , are assumed equal. Suppose the vessel to
be inclined from the perpendicular through any given angle;
let a line ch be drawn, intersecting the line ba at the given
angle of inclination, and cutting* off the area ash equal to the
area bsc : when the vessel is inclined to the given angle from the
upright, the water’s surface will be coincident with the line ch.
Let m and i represent centres of gravity of the areas ash, bsc ,
respectively, and let the line kl be constructed as in the former
cases. Let^ be the centre of gravity of the vessel, situated in the
line we, which is drawn perpendicular to and bisects ba, and
let e be the centre of gravity of the volume displaced ; making
ev parallel and equal to Ik, take et to ev as the area bsc is to
the area representing the entire volume immersed. Through g,
draw gu parallel to ch, and, through t, draw tz perpendicu-
lar to gu. gz will be the measure of the vessel’s stability.

9 Page 228.

the Stability of Ships. 237

From this construction, the following proposition is to be
inferred.

The sides of a vessel are plane surfaces, represented, (fig. 9.)
when produced, by the equal lines AW, BW, which meet in the
point W, beneath the water-line. The sides of another vessel
(fig. 12.) are also plane surfaces inclined to each other at the
same angle as in the former case, and represented by the equal
lines aw, bzv , which meet at the point w above the water-line :
suppose the breadth of both vessels to be equal at the water-
line, and the angle BWA = the angle bwa ; if the distances
between the centres of gravity of the vessels and of the im-
mersed volumes are equal, and the weights of the vessels are
also equal, the proposition affirms, that the stabilities of the
two vessels, when inclined to the same angle from the upright,
will always be equal.

Since the line BA — b a, and the angle BAW = the angle
baw, (fig. 9 and 12.) by the conditions of the proposition, if the
angle BAW be applied over the angle baw, the point A coinci-
ding with the point a , it follows, that the point W, and the point
B, must coincide with the point w and the point b respectively;
and, since the lines BA, ba , are divided in the points S, s, on the
same conditions, namely, so that the lines CH, c h, shall be in-
clined to BA, and b a, at the same angle, and shall cut off the
areas ASH, ash, equal respectively to the areas BSC, b sc; it
must follow, that when the line AB is applied so as to coincide
with the line a b, the point S will coincide with the point s ;
and the angle ASH being equal to the angle a s h, by the sup-
position, the line SH will be equal to the line s h ; and the
triangle ASH will be equal and similar to the triangle a s h
The centres of gravity of these triangles, therefore, or the points
M and m, will coincide, as will also the lines ML, m l , which

238 Mr. Atwood’s Disquisition on

are drawn through these points perpendicular to CH and ch.
The line SL will therefore coincide with the line si, and is equal
to it. In the same manner, it is proved that the line SK is
equal to the line s k ; consequently, KL is equal to k 1 And
since, by construction, the area ASH is equal to the area
BSC, and the area ash equal to the area bsc; and, on appli-
cation of the figure AWB to the figure awb, the triangle ASH
coincides with the triangle ash, it follows, that the four areas
ASH, a s h, BSC, bsc, are all equal.

But ET* — KLx area ASH ^ g _ klxvohimeasb

total volume immersed5

total volume immersed ’

and, since KL — kl, and the volume ASH = the volume ash,
KL x volume ASH = kl x volume ash ; and the entire volume
immersed being the same in both vessels, by the supposition, it
follows that ET — et.

This equality between the lines ET, e t, is independent of
the position of the centres of gravity of the vessels, G, g, and
also of the position of the centres of gravity, E, e, in the lines
WD, wd. If the distances of GE, ge, should be equal, since the
angles of inclination from the upright, or EGR, egr, are equal
by the supposition, it follows that the sines of those angles to
equal radii must be equal, or ER = er. Subtracting, therefore,
ER from ET, and e r from e t, the remaining lines RT, r t ,
must be equal, or G Z = g z. The stability, therefore, of a
vessel, the sides of which are inclined to an angle under the
water’s surface, is equal to the stability of the vessel of which
the sides are inclined to an angle which is above the water’s
surface : the breadth at the water-line, and the other condi-
tions, being the same in both vessels.

This proposition is not confined to the case here demon-

* Fig. 9. Seepage 212.

t Fig. 12. See page 212.

the Stability of Ships. 239

strated, being equally true, whatever figure be given to the sides ;
and whether they are plane or curved, provided the sides under
the water-line in one vessel are similar and equal, and similarly
disposed, in respect of the water-line, to the sides of the other
vessel above the water-line. QC, HO, (fig. 13.) represent the
sides of a vessel projecting outward above the water-line, and
inclined inward under the water-line. Suppose the vessel to
be inclined from the upright through any given angle, and let
Cfl be supposed drawn inclined to the line BA at the given
angle, and cutting off the area ASH equal to the area SBC :
when the vessel is inclined, the waters surface will coincide
with the line CH.

Let the sides QC, OH, be conceived to revolve round the line
BA as an axis, through 180°; the position of the sides will be re-
versed, as represented in fig. 14: the sides which projected out-
ward above the water-line (fig. 13.) equally project outward un-
der the water-line in fig. 14. and are similarly situated in respect
to the water-lines BA, ba. In like manner, the sides which are
inclined inward under the water-line, in fig. 13. are equally in-
clined inward above the water-line in fig. 14. ; and are also simi-
larly situated in respect to that line. If M, I, are the centres of
gravity of the areas ASH, BSC, and m, i, the centres of gravity
of the areas ash , bsc, as in the former cases, and perpendicular
lines be drawn through them, ML, IK, and ml, ik ; by argu-
ments similar to those which were used to demonstrate the pre-
ceding proposition, it will be evident that the lines KL, k l, are
equal; also that the areas ASH, BSC, ash , bsc , are all equal:
and,, by proceeding to construct the measures of stability corre-
sponding.to the two cases, it will appear that GZ =gz ; the
weight of both vessels, and consequently the entire volumes im-
mersed under water, being the same. The conclusion is, that, the

H° Mr. Atwood’s Disquisition on

other conditions remaining the same, if the position of the sides
should be reversed, in the manner described in the proposition,
the stability, at equal angles of inclination, will remain the same.

It may be proper in this place to remark, that the metacentric
curve, described by M. Bouguer,* and M. Clairbois,-)- and ap-
plied to the preceding cases, does not appear to have any relation
to the stability of vessels, except in the single point where the
curve intersects the vertical axis ; and therefore can be applica-
ble only in the case when the angle of the vessel’s inclination
from the upright is evanescent. Let FBC, DAH, (fig. 15. ) repre-
sent the sides of a vessel, BA coinciding with the water’s sur-
face when the vessel floats upright : bisect BA in S, and draw
1SE perpendicular to BA. Let E be the centre of gravity of the
volume immersed. Suppose the vessel to be inclined through
a very small angle AS a, so that the water's surface shall now
coincide with the line ba\ and let the centre of gravity of the
volume immersed be Q. Through Q, draw the line QWz per-
pendicular to ba, intersecting the line IE in the point W. This
point is called by M. Bouguer the metacentre. One of the prin-
cipal properties of this point is, that whenever the centre of gra-
vity of the vessel is situated beneath it, any where in the line WE,
(suppose at G,) the vessel will float permanently, with the line
IE vertical; but that, if the centre of gravity is placed above the
metacentre, suppose at g, the vessel will overset, from that po-
sition ; for, drawing GZ, gz, perpendicular to Qz, if the vessel
should be inclined through a small angle AS a, so as to immerse
the portion of the side A a, the force of pressure acting in the
direction of the line QZ, to turn the vessel round an axis passing
horizontally through G, will elevate the parts adjacent to A, so as
to restore the upright position : whereas, if the centre of gravity

* Trade du Navire, p. 270. j- Clajrkqis, p. 289, et seq.

the Stability of Ships. 241

should be placed above the metacentre, suppose at g, the same
force of the fluid's pressure, by turning the vessel round an axis
passing through g, must immerse further the portion of the side
A a; and this immersion, being continued, will cause the vessel to
overset. Another property of this point has been demonstrated
by M. Euler, * and other authors ; which is, that when the
angles of a vessel's inclination are evanescent, or very small,
the effect of stability, to restore the vessel to the upright po-
sition, will be as the sine of the angle of inclination GWZ -f*
and the line WG jointly: at the same small angles of incli-
nation, the stability of different vessels will be in proportion
to the line WG, or distances of the metacentre above the
centre of gravity.

Let the curve EQ q (Tab. XI. fig. 16.) represent the line traced
by the successive centres of gravity of the immersed volumes,
while the vessel is inclined from the upright through any angle
ASH. M. Bouguer demonstrates, that a tangent to this curve in
any point Q, will be parallel to the water's surface CH, corre-
sponding to that point : if, therefore, through any two adjacent
points Q and q, in the curve EQ q, lines QM, q N, are drawn per-
pendicular to the lines CH, c hf respectively, the intersection of
those lines in the point X will be the centre of curvature, and
XQ, X q , will be the radii of a circle, which has the same cur-
vature with the curve EQ in the point Q. For the same rea-
sons, the line WE (fig. 15, 16.) is the radius of a circle which
has the same curvature with the curve EQ in the point E. The
point W has been denominated the metacentre corresponding
to the upright position of the vessel, when the line WGE is
perpendicular to the water's surface. M. Bouguer denomi-

* Theory of the Construction of Vessels, chap. 8, book i. f To radius — x.

MDCCXCVIII. I i

242 Mr . Atwood’s Disquisition on

nates the point X the metacentre corresponding to the position
when the vessel has been inclined from the upright through the
angle ASH; and the curve WX is termed the metacentric
curve, being the line traced by the successive metacentres, or
intersections, of the lines QM, q N, drawn perpendicular to the
lines in which the vessel is intersected by the water’s surface*
while it is gradually inclined. Consequently, according to this
construction, the metacentric curve WX is the evolute, of
which the curve EQ q is the involute.

The construction and properties of the metacentric curve
being a subject of geometrical reasoning, considered purely as
such, are liable neither to ambiguity nor error ; but, on what
grounds these properties are applied to measure the stability of
vessels, or to estimate their security from oversetting, when
much inclined from the upright, is not explained by M. Bou-
guer, M. Clairbois, or any other author I have had an op-
portunity of consulting : yet the opinions expressed by these
authors on the subject in question, have been adopted by many
persons as established principles ; and, being of some import-
ance in the practice, as well as theory, of naval architecture, it
cannot be thought superfluous to pay some farther attention to
them.

M. Bouguer,* having demonstrated the property of the me-
tacentre, which gives security from spontaneously oversetting,
to a vessel, whenever the centre of gravity is situated beneath
it, proceeds to observe, that his theorem, being founded on
supposing the angles of the vessel’s inclination as evanescent,
or extremely small, such as a vessel may experience in smooth
water, cannot be relied on for ascertaining the safety of ships.

* Traiie du Navire> p. 269,

the Stability of Ships.

when agitated by the winds and waves in open sea, where the
inclinations from the upright must often become considerable.
In order to extend the application of his theorem to the larger
angles of inclination, he proposes to examine whether the me-
tacentre ascends or descends as the vessel is gradually inclined.*
To effect this, the curve line E Qq (fig.ib.) is to be traced, by find-
ing the successive centres of gravity of the volumes immersed
while the vessel is inclined ; and, from this curve the metacen-
tric curve WX is to be defined : the point where the metacentric
curve meets the vertical axis in W, is the metacentre corre-
sponding to the position when the vessel floats upright and
quiescent. He observes, that if the metacentre X ascends from
its original position W, while the vessel is inclined gradually
from the perpendicular, the vessel will be secure from overset-
ting ; but will be insecure, if that point should descend while
the vessel is inclined. No demonstration of this proposition is
given, either by M. Bouguer, or by M. Clairbois, who un-
dertakes to explain the principles delivered in this chapter of
M. Bouguer's work, -f If the proposition has been suggested
by some analogies which subsist between the construction of
the lines EW, QX, and other lines similarly drawn, they will
be insufficient to establish the truth of it. The analogies are
such as the following. W being the metacentre, and E the
centre of gravity of the volume displaced, when the vessel floats
upright, WE is the radius of curvature to the curve EQ q , at
the point E ; X being also the metacentre, constructed accord-
ing to the method which has been described, when the vessel
has been inclined through an angle ASH, and Q the centre of
gravity of the corresponding volume immersed; XQ is the

* Traite duNavire, p. 271. f Clairbois sur V Architecture Navale, p. 289, et seg .

I i 2

2 44 Mr. Atwood’s Disquisition on

radius of curvature of the curve EQ at the point Q. Also, EW is
perpendicular to the water's surface AB, when the vessel floats
upright ; and XQ is perpendicular to the water s surface, when
the vessel is inclined through the angle ASH. When the vessel
floats upright, the stability is measured by the sine of inclina-
tion and the line GW jointly ; and therefore the angle of in-
clination being given, will be measured by the line GW, and
will depend in some ratio or proportion on the line EW, when
GE remains the same, or when G is made to coincide with E.

The question is, whether the stability, when the vessel is
inclined to the angle ASH, will depend in a similar degree on
the line QX ? Respecting the supposed analogy it may be re-
marked, that one condition absolutely necessary to establish it
is wanting ; namely, the centre of gravity G ought to be situ-
ated in the line XQ ; but it is considerably distant from that
line, being placed in the vertical axis of the vessel WGE. This
material difference in the conditions corresponding to the two
cases, is sufficient to destroy all inference from analogy, even
if arguments of this kind could be admitted, in geometrical
subjects, to supply the place of demonstration. It is not diffi-
cult to shew geometrically, in what position and circumstances
of the vessel the line XQ will be the correct measure of its
stability. Suppose that, by any alteration in the distribution of
the ballast or lading, the centre of gravity should be removed
from the line WGE to the line XGQ, the vessel will float per-
manently with the line XQ perpendicular to the horizon, and
the mast WE will be inclined to it at the angle — ASH.
Since XQ is the radius of curvature of the curve EQ at the
point Q, and is also perpendicular to CH, the point X will be
the true position of the metacentre, corresponding to the float-

the Stability.of Ships * 245

ing position of the vessel, when the centre of gravity is situated
out of the vertical axis in the line XQ, and Q is the centre of
gravity of the volume displaced. The measure of stability, when
the inclination is any small angle, will be the sine of that angle
and the line XG jointly ; comparing, therefore, the stability of
the vessel when the centre of gravity is situated in the line
WGE, with the stability when the centre of gravity is in the
line XGQ, the proportion of the two stabilities, at equal small
angles of inclination, will be as the line WG is to the line XG;
if the centre of gravity G should coincide with the point E in
the first case, and with the point Q in the latter case, a con-
dition often adopted by M. Bouguer, the stabilities will be in
the proportion of the lines WE to XQ, or in a triplicate ratio
of the lines BA, CH.

Such is the result of the examination proposed, from which
the only inference is, that while the centre of gravity remains
situated in the vertical axis WE, (the position it occupies in
vessels of every description,) the line XQ cannot be assumed
to measure or estimate the stability and security of a vessel at
sea, when inclined to the larger angles from the upright. M.
Clairbois, to illustrate the principles of M. Bouguer, adopts
two instances, which are the same with Case vi. (fig. 9.) and
Case vn. (fig. 12.) in these pages. In the former case, the sides
coincide with those of an isosceles wedge; the breadth BA at the
water-line being the base, and the angle BWA situated under
the water’s surface. As the vessel thus formed is gradually in-
clined from the perpendicular, he shews,* that the curve traced
by the centre of gravity of the successive volumes immersed is
an hyperbola. Of this curve he calculates the successive radii

* Clairbois, p. 291; 295,

2^6 Mr. Atwood’s Disquisition on

of curvature, which he demonstrates to increase continually
with the inclination of the vessel : he shews, that the centres of
curvature thus found, or successive metacentres, according to M.
Bouguer’s construction, ascend as the vessel is inclined; a cir-
cumstance which, according to his principle, imparts security
from oversetting. On the contrary, in the other instance, when
the sides of a vessel are inclined to an angle which is above the
water’s surface, (fig. 12.) from a similar mode of reasoning he con-
cludes, that the metacentre descends as the vessel is more and
more inclined; which, according to his proposition, would
endanger the safety of the vessel, when inclined to considerable
angles.

This determination is evidently inconsistent with the solu-
tions of Case vi. and vii. preceding, by which it appears, that
the stability acting to restore vessels thus constructed to the
upright position, under the conditions that have been stated,
will be precisely the same at all equal inclinations from the
upright, whether the sides are inclined at an angle beneath or
above the water-line ; all the other conditions being the same
in both cases.

The solution of these questions being connected with a prin-
ciple of some consequence in the practice of naval architecture,
the preceding observations have been offered with a view of
stating distinctly the opinions which are contradictory to the
solutions of Case vi. and vii. referring to the authors who have
treated on the subject, in order that a judgment may be formed
by persons conversant in naval architecture, whether the pro-
positions advanced by M. Bouguer and M. Clairbois, or the
solutions of Case vi. and Case vii. here given, may be relied
on, as founded on the genuine principles of geometry and me-
chanics ; for error must exist on one side or the other.

the Stability of Ships . 247

But, until the demonstrations of the Cases vi. and vii. are
shewn to be erroneous, and reasons are produced in support of
M. Bouguer’s propositions, which he has delivered without
any demonstration, it may be allowable to suppose that his opi-
nions are, in these particular instances, ill founded.

The same principles are extended by M. Bouguer * to ex-
press a general value of the distance between the metacentre,
and the centre of the immersed part of the ship, when inclined to

any angle : this distance he affirms to be Flu— ^3+^3- ; •f- in
which expression y and v are the parts of the total ordinate of
the water-section, (when the vessel is inclined,) at the distance
x , measured on the longer axis from the initial point ; the pro-
portion of y and v being determined by a line drawn parallel
to the axis through the centre of gravity of the section ; and p
is put for the volume immersed.

When the centre of gravity is situated in the line QX, (fig. 16.)
and the angle of inclination very small, the point of intersec-
tion of the lines CH, c h, will bisect the ordinate CH : in this
case the vessel floats permanently with the line QX vertical, and
consequently with the line WE, or plane of the masts, inclined
to the horizon at the angle ASH. But the line QX, consist-
ently with the preceding observations, cannot be applied to
measure the stability or security from oversetting of a ship,
when the centre of gravity is placed in the line WE ; that is,
in the plane of masts which divides the vessel into two parts
perfectly similar and equal; the only situation which the centre
of gravity can occupy, according to any mode of construction
hitherto practised.

* Traite du Navire, p. 273.

t No demonstration is given by M. Bouguer of this proposition.

*4$ Mr. Atwood's Disquisition on

A few remarks may be added in this place concerning a
theorem delivered by M. Bouguer,* for measuring the stabi-
lity of vessels when inclined to evanescent angles from the
upright. The theorem is this : “ When the lengths of vessels
“ are the same, the stabilities are as the cubes of the breadths."
This theorem seems at first view to stand independent of,
and not to require, any subsequent explanation : the author
immediately applies it to the discussion of some points respect-
ing the stability of vessels. If any person, relying on the au-
thor for the truth of this theorem, should only pay attention
to the proposition as it is here expressed, he would entertain
an opinion on the subject of stability which is altogether erro-
neous. M. BouGUER,-f in a subsequent page, gives a satisfac-
tory account of the limitations and restrictions under which
the theorem in question is to be understood. He observes, that
a restriction ought to be applied to the conditions of this pro-
position, in order to insure the exact correctness of it ; which
is, that the whole weight of the vessel shall be concentered in
the centre of gravity of the displaced volume ; a condition
which may be deemed amongst the most extreme cases that
can be devised, and such as is rarely known to exist. J The
vessel's centre of gravity not being supposed coincident with
the centre of the displaced volume; M. Boijguer§ gives thq
true measure of stability when the angles of inclination are

* Traite du Navire, p. 299. f Ibid. p. 299 and 300.

J In vessels of burden, the freights of which consist principally of iron, or other
metallic bodies, or blocks of stone, the vessel’s centre of gravity may be so depressed
as to coincide with, or even to be situated under, the centre of the immersed volume.
But such a disposition causes many inconveniences in the ship’s sailing ; and is never
adopted when it is possible to raise the centre of gravity to a higher position.

$ Traite du Navire, p. 300.

the Stability of Ships. 249

evanescent ; the only objection to which is, that it stands in
the author's page as being explanatory, and illustrative of a
proposition before delivered : whereas, it is in fact the real pro-
position for measuring the horizontal stability of vessels ; the
proposition it is intended to explain being a particular case of
it, and requiring a condition which scarcely ever takes place
in the practice of constructing and adjusting ships for sea.

CASE VIII.

The sides of a vessel are parallel to the masts above the
water-line, (fig. 17.) and project outward beneath it.

In the second Case, (fig. 4.) the sides project outward above
the water-line, and are parallel to the masts under it. In
Case viii. the disposition and form of the sides are the reverse of
the form according to Case 11. If, therefore, the angle of pro-
jection of the sides under the water, according to Case viii.
should be equal to the angle at which the sides project above
the water, according to Case 11. the other conditions being the
same, the stabilities* of the two vessels will be equal, at all
equal inclinations from the quiescent position. The solution
ol this case must of consequence be precisely the same with
the solution ol case ii. and need not be here repeated.

‘ ,, l.

CASE IX.

The sides of a vessel are parallel to the masts above the
water-line, (fig. 18.) and are inclined inward beneath it.

In this case, the position of the sides is the reverse of that
which is described in Case in. (fig. 6.) If, therefore, the angles
at which the sides are inclined inward, according to Case rx.

* Proposition subjoined to Case vii. page 237.

MDCCXCVIII. K k

250

Mr. Atwood’s Disquisition on

under the water-line, should be equal to the angle at which the
sides are inclined inward above the water-line, according to
Case hi. all the other conditions being the same, the stabilities
of the two vessels will be equal, at all equal inclinations from
the upright. The solution of Case ix. is therefore to be derived
from that of Case iii.

CASE X.

The sides of a vessel coincide with the surface of a cylinder,
the vertical sections being equal circles.

Let QBOAH (fig. 19.) represent a vertical section of the
vessel. The surface of the water coincides with the line BA,
when the vessel floats upright. Suppose the vessel to be in-
clined from the quiescent position, through an angle ASH, so
that the water’s surface shall intersect the vessel’s, when in-
clined, in the line CH. Bisect the line BA in D, and the line
CH in Y ; and, through the points D and Y, draw OD, FY,
perpendicular to the lines BA, CH, respectively, and meet-
ing, when produced, in the point M, which is the centre of
the circle. The angle ASH is the inclination of the vessel from
the perpendicular ; and, being the inclination of the lines BA,
CH, which are perpendicular to the lines OM, FM, respec-
tively, the inclination of the lines OM, FM, or the angle
DMF, will be equal to the angle ASH. Let E be the centre of
gravity of the area BOA, representing the volume displaced,
when the vessel floats upright, and quiescent. In the line MF,
take MQ equal to ME ; Q will be the centre of gravity of the
area CFH, representing the volume displaced when the vessel
is inclined. Let G be the centre of gravity of the vessel ; and,
through E, draw ET perpendicular to MF ; and, through G,
draw GZ perpendicular to MF, intersecting that line in the

the Stability of Ships . 25 1

point Z : GZ is the measure of the vessel’s stability. For, since
Q is the centre of gravity of the volume immersed, when the
vessel is inclined, and the line MF is drawn through it, per-
pendicular to the water’s surface CH, QM will be the direction
in which the pressure of the fluid acts, to turn the vessel round
an axis passing through G ; and GZ, being the perpendicular
distance of this line from the centre of gravity, will be the
measure of the vessel’s stability.

Let the sine of the angle of the vessel’s inclination ASH, or
OMF, be represented by the letter s to radius = 1 : by the

2DA3 BA3

properties of the circle ME == ■ „ „ A

“ r 3 X area BOA

therefore, BA be made = t, ME

12 x area BOA

if.

p

12 xarea BOA

and ET

t 3 5

12 x area BOA

The area representing the volume displaced is here con-
sidered as entirely circular : but if it should be of that form
only to the extent of the sides *AH, BC, the remaining part of
the area being of any other figure, and the whole area un-
der water should be denoted by V, the line ET will be

t3 s

area BOA

x — ~ — , or ET

t3 s

x area BOA ~ V ’ “ “ * ~ W Let GE be den0ted

by d ; then ER = ds, and RT, or the measure of the vessel’s

t 3 s

stability -j-GZ =

12V

— ds.

* Proposition and observations in pages 213, 214.

t In this expression for the measure of stability, s is the sine of the angle of the
vessel s inclination, whatever be its magnitude : this value, for the stability of vessels
which have a circular form, is the same with that which M. Bougver gives for ves-
sels of any form, when the angles of inclination are evanescent, the breadths at the
water-line being — t, and the other conditions the same ; from which circumstance,
the following remarkable conclusion is inferred : if the measure of stability should be
calculated for finite angles of inclination, by the rule M. Bouguer has given for the

Kk 2

I

252 Mr. Atwood’s Disquisition on

Let t = 1 oo, .9 = sin. ljf to rad. = 1. d = 13, V = 3600.
According to these conditions, GZ =2.63. If, therefore, the
vessel's weight should be 1000 tons, the stability will be equi-
valent to the weight of 1000 tons, acting to turn the vessel at
the distance of 2.63 from the axis passing through G, or equi-
valent to a weight of 52.5 tons, acting at a distance of 50 from
the axis.

CASE XI.

The vertical sections of a vessel are terminated by the arcs
of a conic parabola.

Let the parabola BLA (fig. 20.) represent a vertical section
of a vessel, floating with the axis DL perpendicular to BA,
which coincides with the water’s surface. G is the centre of
gravity of the vessel. Suppose a ship, so formed, to be inclined
from the upright through a given angle MOI. The breadth
BA, and depth from the water-line, DL, being given, it is re-
quired to construct the measure of the vessel’s stability.

The principal parameter being given from the conditions of
the construction, from the vertex L set off LF, equal to a fourth
part of the parameter : F is the focus of the parabola. In the
line LF, take LI to LF, as the tangent of the given angle
MOI to radius; and, in the line LI, take LX to LI, in the
same proportion of the tangent of the angle MOI to radius.
Through the point X, draw XV perpendicular to XL, inter-
secting the curve in the point V ; set off LN equal to XL :
join NV, which produce indefinitely, in the direction NVW ;

angles of inclination that are evanescent, the stability of all vessels, at equal inclina-
tions, thus calculated, whatever be their forms, would be the same as if the vertical

sections were circular ; the breadths at the water-line, position of the centres of gravity,,

■

and other elements, being the same.

the Stability of Ships. 253

NW is a tangent to the curve in the point V : through the
point V, draw VK parallel and equal to DL ; and, through the
point K, draw CH parallel to NW : let DL be divided into five
equal parts, and let LE be taken equal to three of those parts :
make VQ equal to LE ; and through Q draw Pr perpendicular
to NW ; through G draw GZ perpendicular to rP : GZ is the
measure of the vessel's stability, when inclined from the upright
through the given angle MOL The demonstration follows.
Through E, draw ET perpendicular to rP ; and, through G,
draw GR parallel to rP ; let the parameter of the curve be de-
noted by^>.

By the construction, LX : LI : : LI : LF : : tang. MOI to rad.
therefore - LX : LF : : tang.4 MOI : rad.2 and
and - LX : 4LF : : tang.2 MOI : 41'ad.2

By the properties of the curve,

wherefore

But

therefore

and

LX: XV:: XV : 4LF

LX : 4LF : : LX2 : XV2.

LX : 4LF : : tang.2 MOI : 4 rad.2
LX2: XV2: : tang.2 MOI : 4,rad.2
LX : XV : : tang. MOI : 2 rad.
or, since LX = ±XN

^XN : XV : : tang. MOI : 2 rad.
or XN: XV :: tang. MOI : rad. but, by

the construction, XN : XV : : tang. XVN : rad.

consequently tang. XVN is equal to the tangent of MOI to the
same radius; and therefore the angle XVN is equal to the
angle MOI, or the given, angle of the vessel's inclination from
the upright. Moreover, since it appears from the construction,
that the angle XVN is equal to the angle NrP, NrP is equal
to the vessel's inclination from the upright, and because the

254 Mr. Atwood's Disquisition on

line BA is parallel to XV, and the line CH parallel to NW,
by the construction, it follows, that the angle ASH is equal
to the angle XVN; wherefore the angle ASH is also equal to
the angle MOI, or the given angle of inclination from the
upright. VK being parallel to DL, and therefore a diameter of
the curve to the point V, and CH being drawn parallel to
NVW, which is a tangent to the curve in the point V, it
follows, that VK bisects the line CH in the point K ; KH
therefore will be an ordinate to the diameter VK : and, since
VK is by construction equal to DL, and DL, VK, are ab-
scissas of the segments BLA, CVH, respectively, it is known,
from the properties of the figure, that the area of the segment
BLA is equal to the area of the segment CVH ; and conse-
quently the area of the figure ASH will be equal to the area of
the figure BSC. And since, when the vessel floats upright, the
line AB coincides with the water s surface, and the area of
the segment ALB is equal to the area of the segment CVH,
it follows, that when the vessel is inclined from the perpendi-
cular, through an angle ASH, equal to the given angle MOI,
the surface of the water will intersect the vessel in the line CH.
Moreover, since LE is to LD as 3 to 5, by the construction, and
VQ is to VK in the same proportion of 3 to 5, by the proper-
ties of the figure, E is the centre of gravity of the area BLA,
and () is the centre of gravity of the area CVH, which repre-
sents the total volume displaced, when the vessel is inclined
through an angle ASH, or MOI ; and the line FQP being, by
construction, drawn perpendicular to the water’s surface CH,
will be a vertical line passing through the centre of gravity Q of
the volume displaced CVH : and GZ, drawn through the centre
of gravity G, perpendicular to this line, will be the measure of

255

the Stability of Ships.

the vessel’s stability, when inclined from the perpendicular
through the given angle MOI.

From the preceding construction and demonstration, a pro-
perty of stability is inferred, which may be expressed in the
following proposition.

If the vertical sections of a vessel are terminated by the arcs
of a conic parabola, and the sides of another vessel are parallel to
the plane of the masts, both above and beneath the water-line,
the stabilities of the two vessels will be equal at all equal in-
clinations from the upright, if the breadths at the water-line
BA, and all the other conditions, are the same in both cases.

It is thus demonstrated :

For brevity, let the angle of inclination from the upright,
or the angle ASH, be denoted by the letter S ; let BA = t3
and LD = a : rad. = 1.

From the preceding construction and demonstration, it ap-
pears that XV : XN : : i : tang. S, and by the properties
of the figure -jXN : XV : : XV : p, joining these

ratios - 1:2:: XV : p x tang. S.

Wherefore XV = £2l!in!_£

2 ’

and XL = — — tx.la"g*s — LN-

P 4 ’

also, since XV : NV : : cos. S : 1,

NV = xv — P * tang- s .

cos. S 2 X cos. S 5

and because LD = VK = a, and LE = VQ = -12-,
and the angle VQP = ASH = MOI, it follows that

yp __ 3a x^in. S . ancj tflerefore

NP = NV 4- VP — ^ x tang‘ 5 1 3a X sin. S

' 2 X COS. S * r

5

256

therefore, rN

Mr . Atwood’s Disquisition on

NP p

sin.S

2 x cos.4 S

and

Lr = rN - LN = + _!i. _

, 3a __ i> X sec.1 S . 3d

5 2 ' ‘ S ’

p x tang.8 S #

and, since LE
rE — P * sec * s __

3*

or

p x tang.4 S
4 ’

rE = 4- x sec. S “|~ 1, and, since the angle
4

ErT = S, ET = *x sin-s x sec.’ S + 1,

4

or ex = PJL^Ll x cos. S + sec. S.

4

This is the value of the line ET, when the area represent-
ing the volume immersed is terminated throughout by the

parabolic arc, the said area being = or -j~ x BA x DL ;

but, if that form should extend to the sides AH, BC, only, the
remaining part of the volume immersed being of any other
figure,* and this entire volume should be of any magnitude

V, the value of ET corresponding will be p x * jpy

X cos. S + sec. S,' or ET= - * x cos. S + sec. S. And,
since ER = d x sin. S, TR, or the measure of the ves-

sel's stability, GZ = — x cos. S -j- sec. S — d x sin. S ;

precisely the same quantity which measures the stability at
the angle of inclination S,^ when the sides are parallel to the
masts above and beneath the water-line : a coincidence not a
little remarkable, and such as would not probably have been
supposed to exist, except from the evidence of demonstration.
From this proposition it is inferred, that if the sides of a

f Case 1. page zi6.

• See pages 213, 214 .

257

the Stability of Ships,

Vessel coincide with the arcs of the conic parabola, and the
sides of another vessel coincide with the arcs of another conic
parabola, whatever be the form thereof, varying according to
the parameter, the weights of the vessels, breadths at the
water-line, and the other conditions being the same in both
cases, the stabilities of the two vessels, at all equal angles of
inclination, will be equal. If, for instance, the forms of two
vessels should be such as are represented in Tab. XII. fig. 21.
and fig. 22. the weights and other conditions being the same,
the stabilities of each of these vessels will be equal to that of a
vessel PBQFAK, the sides of which are plane surfaces, parallel
to the masts.

The propositions immediately preceding, relate to the conic
or Apollonian parabola : they have been inserted, with a view
of establishing and extending the theory of stability. It may
also be remarked, that the sides of vessels are in some instances
constructed nearly of these forms; for the same reasons, it
may be not altogether useless, to examine on what principle the
stability of vessels is to be investigated, when the forms of the
sections are parabolic curves of the higher orders, such as are
represented in fig. 23. The line cBCO is a conic or Apollonian
parabola, JBDO is a cubic, and <?BEO a biquadratic parabola.

/BFO (fig. 23.) is a parabola of 8 dimensions, and ^BGO
a parabola of 30 dimensions, which are drawn from a geome-
trical scale, in order to give a true representation of the forms
of these curves.

The general equation, determining the relation between the
abscissae and ordinates of any parabola, of the dimensions n , is
yn—pn~ 1 x x, if the ordinates are drawn perpendicular to the
axis of the curve; and y = a + px qx1 rx\ &c. -f vxn,
mdccxcviii. L 1

258 Mr. Atwood’s Disquisition on

if the ordinates are drawn parallel to the axis ; y and x signi-
fying the ordinate and corresponding abscissa ; the other let-
ters denoting constant or invariable quantities, to be determined
by the properties of the figures. In these figures, it is observ-
able, that the breadths toward the vertex O, are always greater
in the curves which are of the higher dimensions ; and, as the
dimensions are continually increased, the figure approaches
more nearly to a rectangular parallelogram,* with which it

* The radius of curvature of the conic parabola at the vertex (fig. 23.) is half the
principal parameter ; but, in all the parabolas of the higher orders, the radius of cur-
vature at the vertex is infinite. Suppose x to represent the abscissa, or distance of the
ordinate y from the vertex, measured along the axis of the curve : as x increases from
o, the radius of curvature decreases till it becomes a minimum, and then increases : a
difficulty seems to arise respecting the magnitude and variation of the radius of cur-
vature, when, the dimensions being increased sine limite, the form of the curve ap-
proaches continually, and ultimately coincides with, the rectangular parallelogram. If
the equation of the curve be yn — p* — 1 x x, where/) represents the parameter, the
radius of curvature of the curve at the extremity of an ordinate y, of which the

2» 2 2 71 2|-§

abscissa is x, will be found — p x

n*X x n +p *

n — 1

n_ * which quantity

« x w — I X/> n xt

n

is a minimum when x — p x

vature itself, or r — p x

n — 2

2 n — 2.

N

1

S

3«— 3

s: consequently, the least radius of cur-

• ; and, when n is increased sine

n— 1 2«3 — n%

h z n — 2

271 1

2TI—2,

limite , the abscissa corresponding to the least radius of curvature, or x — p*x — : — ,

v' 2 x n

\r 27

and the least radius itself, or r ~p x — — -, both of which quantities are evanescent,

shewing that if the dimensions of the parabola are increased sine limite, the curvature
at the extremity of the ordinate, when the abscissa rr o, is infinite, the radius of cur-
vature being nothing, as it ought to be, at the point H of the parallelogram BHOD,

the Stability of Ships. 259

ultimately coincides, when the dimensions are increased sine
limite. This extreme case has relation to the subject of stabi-
lity : for, whatever may be the effect of giving to the sides of
ships the forms of the several higher orders of parabolas, it is
certain, that as the dimensions of these curves are increased,
the stability will approach to that which is the consequence of
making the sides parallel to the masts ; but it has been shewn,
that when the sides coincide with the form of a conical para-

considered as a parabolic curve of infinite dimensions, the two portions of the curve
BH, HO, (fig. 23.) being inclined at a right angle, when coincident with the sides of a
rectangular parallelogram: but, since the curvature is nothing at the vertex O, the ab-
scissa being then — o, and before the abscissa has increased to any finite line, the cur-
vature at the extremity of the corresponding ordinate OH is infinite; and since the cur-
vature between the points O and H must necessarily pass through all the intermediate
gradations of magnitude, it becomes a question to define the abscissa and correspond-
ing ordinate, when the radius of curvature is a finite line : 2dly, when it becomes eva-
nescent ; and, lastly, when it is again infinitely great. By referring to the preceding
expressions for the abscissa and corresponding radius of curvature, it is found, that if

p represents the parameter, and x is made — — L— , (the number n denoting the dimen-
sions of the curve,) when n is increased sine limite, the radius of curvature will be
greater than any line that can be assigned : and such is the curvature of any portion of

the line OH, between the points O and H. 2dly, if x is =z the radius of cur-

n

vature will be = p, the ordinate approximating to equality with the line OH. 3dly,
P

if x — — , the radius of curvature will be smaller than any finite line : and, lastly,

if x — p, or any finite line, the radius of curvature will be greater than any assignable
line : which conclusions are immediately inferred from the equation expressing the

radius of curvature, or r

2« — 2 2 n — 2 '1-f'

: n +p n

P * n — 3 1 w^en the .number of di-

n x n—i x p n x x n

mensions n is increased sine limite, these successive changes in the radius of curvature
taking place while the abscissa x is increased from o to any finite magnitude.

LI 2

q6o Mr. Atwood's Disquisition on

bola*, the stability^ is the same as when the sides are plane
surfaces, parallel to the plane of the masts. It is inferred that
if the sides of a vessel are formed to coincide with a parabola
of the lowest, and the sides of another vessel with a parabolic
curve of the highest dimension, all the other conditions being
the same, the stabilities of the two vessels will be equal in
these two extreme cases.

In proceeding to ascertain the stability of vessels, the verti-
cal sections of which coincide -with any parabolic curve, the
rigid strictness of geometrical inference cannot be well pre-
served, when the oblique segments are objects of consideration,
on account of the complicated properties of the figures.]. But,
in these and similar cases, methods of approximation may be
employed, by which the stability corresponding to any given
figure of the sides may be inferred, to a degree of exactness
exceeding any that can be necessary in practice. These me-
thods of approximation are either such as are required for the
mensuration of curvilinear areas, or geometrical constructions
which exhibit the lineal measures of stability not strictly and
rigidly true, but approaching, as nearly as may be desired, to

the true and correct measures.

The methods of approximation to be used for the quadra-
ture of curvilinear spaces, are founded on Sir Isaac Newton's
discovery of a theorem, by which, from having given any

* Case xi. pages 255, 256.

f The comparative stability, in this and similar observations, is understood to im-
ply, that the vessels are inclined at equal angles of inclination from the upright, all
the other conditions (the shape of the sides excepted) being the same.

I The areas of any parabolic segments, either direct or oblique, are geometrically
quadrable, but, in the oblique segments, the positions of the centres of gravity are not
determinable generally by direct methods.

q6i

the Stability of Ships.

' \

number of points situated in the same plane, he could ascertain
the equation to the curve which would pass through them all :
and, by means of this equation, was enabled to express the
ordinate in the curve, corresponding to an abscissa of any
given length, as well as the area intercepted between any two
of the ordinates. This discovery the author himself considered
amongst his happiest inventions. Amongst the various uses of
this theorem, that of determining by approximation the areas
of curvilinear spaces is not the least considerable : for, by
this means, the fluents of fluxional quantities, not discoverable
by any known rules of direct investigation, are found, to a de-
gree of exactness fully sufficient for any practical purpose, and
with very little trouble of computation.

Mr. Stirling, in his treatise intitled 'Methodus differ entialis,
has inserted a table for measuring curvilinear spaces termi-
nated by parabolic curves, from having given 3, 5, 7, or 9 equi-
distant ordinates, and the abscissae on which they are erected.
The measures of the areas thus obtained are, under certain
conditions hereafter stated, not approximations, but geometri-
cally and strictly correct: the approximate values of curvili-
near spaces, in general, are obtained from finding the correct
areas terminated by parabolic lines which nearly coincide with
the said curves, by passing through the extremities of the same
ordinates.

The subjoined table contains Mr. Stirling's rules for ex-
pressing the areas of curvilinear spaces, from the conditions
which have been mentioned ; also additional rules for measuring
the areas which are included between the extremes of 2, 4, 6,
or 8 equidistant ordinates : the whole of this table has been re-
computed and verified.

2 6 2
if

Number of equi-
distant ordinates.

2

3

4<

5

6

Mr. Atwood’s Disquisition on

Table of Areas.

Areas.
A -p

rxR

A -f 4B
6

A + 3B
8

X

R

R

7 A 4- 32 B + 12C

~

19 A + 75 B 4- 50C
288

X

R

R

41 A + 216 B + 27 C -f 272 D „

7 8^- xR

£ 36799 A 4- 175273 B 4- 64827 C + 146461 D 0

846720 x **■

^ 989 A -f 5888 B — 928 C + 10496 D — 4540 E 0

y 28350 ~ x

In this table, the letter A denotes the sum of the first and
last ordinate of the number opposite to it in the first column :
B is the sum of the second and last but one: C is the sum of
the third and last but two, and so on. The extreme letter, sup-
pose D, (as in the rule opposite 8 ordinates,) is the sum of the
two middle ordihates, if the number of ordinates is even; or the
extreme letter, suppose D, (as in the rule opposite 7 ordinates,)
is the middle ordinate alone, if the number of ordinates is odd.
R is the entire length of the abscissa, which is always equal to
the common interval between the ordinates, multiplied by the
number of ordinates diminished by unity.

Let the area to be measured be terminated by the curve line
ABCD, &c. ( Tab. XIII. fig. 24.): A' I' is an abscissa, on which
a number of equidistant ordinates AA7, BB7, CC7, &c. are erected
at right angles. If ABCD, &c. represents a parabolic line of any
dimension, suppose ;z,the relation between the ordinates and ab-

263

the Stability of Ships.

scissse being expressed by the equation y = a -f- p x -|- q x1' -f* t. x3,
&c. -J- a xH, (in which case, the ordinates are drawn parallel to
the axis of the curve,) a measure of the area contained between
the extremes of n + 1 ordinates will be obtained with geome-
trical exactness, by computing from the rule in the table which
is opposite the number of ordinates n -f 1, supposing the table
to extend to that number : but if, as it usually happens in cases
which practically occur, that the nature of the curve is un-
known, or the conditions in other respects different from those
which are required for the mensuration of the area with perfect
correctness, it becomes a question, which particular rule in the
table should be adopted for inferring an approximate value of
the area, since an exact quadrature is not obtainable. For this
purpose, there are several reasons for preferring the rules oppo-
site the number of ordinates 2, 3, and 4 to the others, which
require a greater number of ordinates ; the common distance
between them being the same. In the first place, the rules
here pointed out are far less troublesome in the application ; a
circumstance which ought to have weight, although of less im-
portance than another consideration, which is, that the results
derived from these rules, particularly from the two latter, will
in general approximate as nearly to the true value, sometimes
more nearly, than those which are obtained by calculating
from the other more complicated theorems, unless the curve
should happen to be such as admits of being correctly measured
by any of the rules requiring a greater number of ordinates ;
a circumstance not likely to occur in practical mensurations.

Let it be proposed to measure by approximation any curvi-
linear space AA'I'IA (fig. 24.) For brevity, let the successive
ordinates AA', BB', CC', &c. be denoted by the letters a , b , c>.

Mr. Atwood's Disquisition on

&c. respectively ; also, let the common distance between the or-
dinates (hg. 23-) or A' B' = B'C' == C'D' be = r : according
to the theorem for measuring the area contained between two

ordinates, or — x R, the curve line AB is supposed to coincide

with the right line AB which joins the extremities of it ;
the space measured by this rule is the trapezium AAB'BA;
and, since A = a -|- b, and R = r, the area of the trapezium, or

T * R=s:«"+TTx ■j.

According to the rule opposite 3 ordinates, the curve line
ABC is supposed to coincide with a portion of the conic para-
bola, the axis of which is parallel to the ordinates. And since,

by this rule, the area = A -f- 4 B x in which expression
A = a -{- c, B = 6, and R = 2 r; by substituting these values,
the area AA' C' C A = a-|-4<6-f-cx-. If the curve ABC

should actually be a portion of the conic parabola, the given
ordinates being parallel to the axis of the curve, the area
AA'C'CA will be measured by this rule with exactness abso-
lutely perfect ; and, the more nearly the curve which terminates
the area approaches to the form of the conic parabola, the more
nearly will the result of calculating by this rule approximate to
the true value of the area. But it is evident, that since in this
approximation, the arc of a conic parabola is drawn through the
points A, B, C, being the same points which terminate the or-
dinates of the given curve, the difference between the parabolic
area and that which is given must, in most (except extreme)
cases, be next to an insensible quantity, when applied to prac-
tical mensuration.

In the mensuration of areas by the rule opposite 4 ordinates,

the Stability of Ships. 2%

a parabolic curve line is supposed to be drawn through the
points A, B,C, D, of the 3d dimension, such as the arc of a cubic
parabola, the ordinates of which are parallel to the axis of the
curve, and the area terminated by this curve line is assumed
to approximate to the given area AA'D'DA: by this rule, the

area = A. -j- 3 B x in which expression A = a -j- d, B = b -}- c,
and R = 3 r, which being substituted for their respective values,
the area A A' D'DA ss a -\~$b -f- 3 c d x

In order to bring these rules into a form convenient for prac-
tical use, let it be proposed to measure the area AA'G'GA (fig.
24.) intercepted between the extremes of 7 ordinates.

1st. Suppose the right lines AB, BC, CD, &c. to be assumed,
instead of the curve lines AB, BC, CD, &c. as terminations of
the space to be measured : then the area AA'G'GA will be
equal to the sum of six trapeziums ; AA'B'B, BB'C'C, CC'D'D
and so on.

The area of the trapezium AA'B'B = a -J- b x -f , by the
rule opposite 2 ordinates : by the same rule, the area of the
trapezium BB'C'C = b -f- c. x ^ ; the area of the trapezium

CC'D'D = c -J- d x f-, and so on. By adding these six sepa-
rate areas, the sum will be the area of the space AA'G'GA

= The law of con-

tinuation for a greater number of ordinates is obvious. This
rule is precisely the same with that which is given by M.
Bouguer, in his work entitled “ Trait e du Naviref * * under
a iorm somewhat different : his rule is this ; from the sum of
all the ordinates subtract ~ of the sum of the first and last ; the

* Page 1 1 2.

mdccxcviii. M m

*2 6 6 Mr. Atwood's Disquisition on

result multiplied into the common distance between the ordi-
nates will be the exact area of the figure, considered as con-
sisting of trapezia, and an approximate value of the curvilinear
area in which the said trapezia are inscribed. This rule he pro-
fesses not to be a very correct approximation, but such as may
be deemed sufficient for most practical mensurations. It must
be acknowledged, that in mensurations independent of others,
the errors arising from this rule are often not considerable,
(in many cases they are very small;) but, considering that in
naval mensurations, areas obtained by approximation are ne-
cessarily the data from which other results are to be inferred,
also by approximation, a doubt may arise whether the errors
thus accumulated may not, in some cases, become too great ;
at least it may not be improper to be provided with rules which
may be relied on, as approximating more nearly to the true
measures of areas.

Let the same area be measured by the rule opposite 3 ordi-
nates, according to which it is supposed that the curve line
ABC coincides with the arc of the conic parabola. By this

rule, the area AA'C'CA = also the area

CC'E'EC = c +V<T+ e x I; and the area EE'G'GE

= e + 4/ + g * r- : adding these three areas together, the

sum is the area AA'G'GA= a+4& + 2c-{-4d-f2<? + 4/ _|_ g

r

X

3

This rule is the same with that which Mr. Simpson has de-
monstrated in his Essays, page 109, from the properties of the
conic parabola, perhaps not noticing that it was to be found in
Mr. Stirling's table of areas.

the Stability of Ships .

267

Mr. Chapman, an eminent author on the subject of naval
architecture,* applies this theorem to naval mensurations, as a
substitute, and certainly an useful one, to the less perfect rules
which are employed for this purpose, in the works of M. Bou-
guer and other authors. The example by which Mr. Chapman
illustrates the use of this rule is the same with that which is
given in Mr. Simpson’s Essays.

This approximation to the measures of areas being applicable

* The following observation on this theorem is inserted in the Report from the
Committee of the French Royal Marine Academy, who were appointed to examine
the translation of Mr. Chapman’s Treatise on Naval Architecture, by M. Vial
de Clairbois; this report is prefixed to the French edition of Mr. Chapman’s
work.

“ Ce celebre constructed commence par donner une nouvelle methoae de calcul de
u deplacement, qui sans etre beaucoup plus longue que celle que l’on emploie com-
** munement, donne un resultat infiniment plus exact. On considere ordinairement
“ les parties curvilignes des plans de flottaisons, ou de gabarits, entre les extremites
“ des ordonnees, comme des droites; M. Chapman les regarde comme des parties

paraboliques ; et, de la nature de cette section conique, et du trapeze, il tire une
,{ expression sur laquelle il fonde un calcul assez simple,” &c.

A comparison of the results derived from this rule, and from that which is em-
ployed by M. Bouguer, does not seem to confirm the opinion of the very superior
exactness which the committee here attribute to the former rule: that it is more exact
there is no doubt, especially when the curvature is at all irregular in respect to its varia-
tion, and the results inferred are data on which other computations are to be founded ;
but, in many of the cases which occur in practical mensurations, the latter rule ap-
proximates to the required results sufficiently near the truth, as will appear by the
instances in the subsequent pages. The expression “ une nouvelle methode” can-
not be understood to mean a rule of computation newly invented, but one which
Mr. Chapman has first applied to naval mensurations. In this sense, the theorem
inserted in the 265th and 268th pages of these papers would be entitled to the appel-*
lation of c‘ a new method but it has already been shewn, that the three rules here
described, and employed in the computations which follow, are only particular cases
of the general method demonstrated in the works of Sir I. Newton, Stirling,
Simpson, and other authors.

Mm2

s68

Mr. Atwood’s Disquisition on

only when the number of given equidistant ordinates is odd,
to obtain the area when the number of ordinates is even, an-
other rule, to be employed either singly, or in conjunction with
the former, may be selected from Mr. Stirling’s table of areas.
It is that which stands opposite 4, ordinates.

Let it be proposed to measure the area AA'G'GA. Accord-
ing to this rule, the area AAl'D'DA = a-\-gb-\- 3 c -| -cl x if

also the area DD'G'GA = d -|- 3 e -|- 3/ g x ~
these two areas being added together, the sum will be the area
AA'G'GA = a -f 3 -f 3 c + 2 r/ + 3 e -f 3/ +~g x ^

When the number of given equidistant ordinates is small, these
theorems will be most conveniently used in the forms here given,*
but, when the ordinates are numerous, the trouble of arithme-
tical computation will be considerably abridged, by employing
them according to the general rules inserted underneath.

These theorems for approximating to the values of areas, may
be applied, With advantage, to the integration of fluxional quan-
tities, the fluents of which cannot be obtained by direct methods;
or, if obtained, req uiring very long and troublesome calculations. *

Suppose % to represent the abscissa, of a curve, on which the. or-

* On this principle, the rules of approximation here given are applicable to deter-
mine the positions of the centres of gravity, both of areas and solid spaces. II y is put
to represent the ordinate erected perpendicular to an abscissa, at the distance z from the
initial point thereof, the fluent of yzz (fig. 24.) will be the sum of the products ari-
sing from multiplying each ordinate into the small increment z, and also into the dis-
tance z from the initial point. And, since the area intercepted between the ordinates
A A' and y is the fluent of yz, it follows, that the distance of the centre of gravity
ef this curvilinear space from the ordinate AA', measured on the abscissa A'i', is

ftuenL? The approximate values of these fluents are obtained from the Rules n 11.

fluent yz

the Stability of Ships. 2 6g

dinates (expressed by Z, a general term or function of z), a, b , c,
d, &c. are erected at right angles, and at intervals each of which
is = r, so that when z — 0, Z = a : when z ~r, Z = b: when
2 = 2 r, Z = c, and so on. If innumerable ordinates or values
of Z be supposed drawn between each of those which are given,
at the common very small interval z, the sum of the products
arising from multiplying each of the ordinates into the incre-
ment z, that is, the fluent of Z z, will be found, by approxima-
tion, according to the three following rules ; which may be not
improperly termed, rules for approximating to the integral va-
lues of fluxional quantities. According to

rule 1.

Fluent of Z = P — — x r ;

2>

in which expression,

P == the sum of all the ordinates a + b + C + d , &c.

S 5= the sum of the first and last ordinate.
r = the common distance between the ordinates.

RULE II.

Fluent o{Zz = S~\-^P-\-^Q'x^-;
in which expression,

S = the sum of the first and last ordinate.

P = the sum of the 2d, 4th, 6th, 8th, &c. ordinate.

Q = the sum of the 3d, 3th, 7th, qth, &c. ordinate, (the last
excepted. )

r = the common distance between the ordinates.

and hi. ; and the positions of the centres of gravity are thus determined according to
the methods of computation employed in the subsequent pages. The position of the
centre of gravity in solid bodies, is determined by a similar application of these rules.

27° Mr. Atwood’s Disquisition on

RULE III,

Fluent of Z z = S -f 2 P~{-g O x ~ ;
in which expression,

S = the sum of the first and last ordinate.

P = the sum of the 4th, 7th, 10th, 13th, &c. ordinate, (the last
excepted. )

Q = the sum of the 2d, 3d, $th,£th, 8th, 9th, &c. ordinate.

r == the common distance between the ordinates.

It is to be observed, that the first of these rules approximates
to the fluent, whatever be the number of given ordinates. The
second rule only requires that the number of ordinates shall be
odd. To apply the third rule, it is necessary that the number
of ordinates given shall be some number in the progression 4,
7> 10> *3> &c- that is, the number of ordinates must be a mul-
tiple of 3 increased by unity. But, in every case, the approxi-
mate fluent may be obtained, either from the Rule 11. or the
Rule hi. or by employing both rules conjointly.

Before these theorems are applied to practical mensurations
in naval architecture, it may be satisfactory to examine, by a
few trials, to what degree of exactness they approximate to the
correct values of curvilinear spaces. This will be known, if the
area of some curve, which is exactly quadrable by other geome-
trical rules, be measured by them. Such as a parabolic figure of
which the equation is y* = p7 x, x being the abscissa coincident
with the axis, and y the corresponding ordinate perpendicular
to it.

The semi-area of this parabola (fig. 25, 2 6.) is known to

g

be xy x — ; and the curve is termed a parabola of 8 dimen-

sions.

271

the Stability of Ships.

OD is an abscissa, being a portion of the axis of this curve,
and the parameter is assumed = OD. If, therefore, an ordi-
nate BD, or DA, is drawn through the point D perpendicu-
lar to DO; the lines DB, DO, and DA, will all be equal.
Let BD be divided into 10 equal parts, * considered in this in-
stance as an abscissa, on which, at the points of division, the
several ordinates are erected perpendicular to BD, denoted in
the figure by the letters a, b, c, d, &c. If BA is assumed = 100
equal parts, DB, DO, and DA, are each = 50, and the common
interval between the ordinates = 5; the numerical values of
the successive ordinates a, b , c} &c. are expressed in the an-
nexed table.

According to the Rule 1. making
the sum of all the ordinates, or P
= 466.1341, the sum of the first and
last ordinate, or S = 50 : r = 5 = the
common distance of the ordinates.

The area BDO = P- — - x r

2

= 2205.670

correct area = — -x x 8 =2222.222

Difference •f- or error of
the approximation = 16.552

* Any line being assumed in a curve as an abscissa, lines drawn parallel to each
other, and intercepted between the abscissa and the curve, are termed ordinates. To
exemplify these rules for approximating to the areas of curvilinear spaces, it was ne-
cessary to consider the ordinates as being drawn in some cases parallel, and in others
perpendicular, to the axis.

t must not be concluded, from this instance, that the errors in measuring cur-
vilinear areas by the Rule i. will be usually so great as i6 parts in 2222. In applying
the Rule 1. to this parabolic space, the quick variation of curvature in some parts of
the curve, causes the space so measured to deviate more from the truth than would,
happen in ordinary cases, such as commonly occur in practical subjects. If, instead of

a

= 50.0000

b

= 50.0000

c

= 49-9999

d

= 49-9967

e

= 49-9672

f

= 49.8047

g

== 49.1602

h

' = 47-i 175

i

= 41-6113

k

= 28.4766

l

= 0.0000

Sum of all the

ordinates

= 466.1341

2.7 2 Mr. Atwood's Disquisitmi on

If the same area is measured by the Rule u. the process will
be as underneath :

Sum of all the ordinates - — 466.134 1

Sum of the first and last, or S = 50.0000
Sum of the 2d„ 4th, 6th, &c, or P = 225.3955

s + p = 275-3955 275-3955

Sum of the 3d, 5th, 7th, &c. (except the last) or Q = 190.7386'
and r being = 5, the area BDO == S+^Pfi-eO x~-= 2221. 765

Correct area - =2222.222

Difference or error of the approximation = .457

Let the same area be measured by the Rule in. the area
DOKI, between the two ordinates a and b, being = 5 x 50
= 250, the remaining area, from the ordinate b to the ordinate
l = o, will be obtained from the following computation :

The sum of all the ordinates = 416.1341
Sum of the first and last, reckoning

b the first, and the last / = o or S = 50.0000
Sum of the 4th, 7th, 10th, &c.. ( b being

the 1st.) or P - - = 97.0847 147.0847

Sum of the 2d, 3d, 5th, 6th, &c. from b, or Q

269.0494

the total area — DBO, (fig. 25 . ) a portion of it, which is contained between the ordi-
nates a and g , should be measured, the area, computed from either of the three rules,
would deviate very little from the truth, as appears from the following results.

Areas contained between the ordinates a and g.

computed by

Rule 1.

Rule 11.

Rule hi.

Areas

1496.74

1497.17

1497.13

Correct area
or error of
lation

1497.20

1497.20

1497.20

.46

.03

.07

If the rule in the table of areas opposite 7 ordinates should be applied to measure
the area between the ordinates a andg-, (fig. 25.) the result would be geometrically
correct.

the Stability of Ships.

And, since r = 5, the area between the ordinates

b and l — - S -J- 2 P -j- 3 Q x qj* —

Area IKOD between the ordinates a and b

area EDO - - =2221.220

Correct area EDO - - =2222.222

Difference or error of the approximation = 1.002

In applying these rules, it is necessary to observe, that if
the ordinates are drawn perpendicular to the axis of the curve,
whenever the area to be measured, or any part of it, is adjacent
to the vertex O, the area found by these rules will be the least
exact : in such cases, it will be requisite to assume an abscissa
near the vertex O, perpendicular to the axis : by erecting equi-
distant ordinates upon it, parallel to the axis, the area will be
found, with the same exactness as in the other cases, which will
appear by the following computations.

DOA (fig. 2 6.) is a semiparabola, similar and equal to DOB.

Let the line DO = 50 be divided into 10 equal parts, each
= 5 ; and, through the points of division, let the succes-
sive ordinates b, c, d, &c. be drawn perpendicular to DO:
according to the preceding observations, if the entire area DOA
should be computed by either of the three rules, the result
would be less exact than in the former cases. To obtain an
approximate value of the area, sufficiently near the truth, a por-
tion of the area adjacent to the vertex O, suppose XEO, is to
be separately computed. If OX be = 10, the line XE will
= 40.8890, which being divided into six equal parts, each of
them will = 6.815 : ^ the ordinates p , q , s, tf &c . be erected at
the points of division, perpendicular to XE :

N n

273

= 1971.220
= 250.000

MDCCXCVIII.

Mr. Atwood’s Disquisition on

From these ordinates, the area OXE
is found by the Rule n. to be

159-8390 * = 363.101.

For obtaining the area DAXE, the ordinates a, b, c} d , &c.
erected on the abscissa DX, are as expressed underneath :

The area DAXE, between the ordi-
nates a and i, is found by the Rule 11.
to be - = 1858.760

The area between the or-
dinate i and /, before found = 363 .101

* Entire area DOA =2221.861

Correct area - =2222.222

Difference or error of the
approximation - = .361

* If the area between the ordinate a and i had been computed by the Rule i. the

result would have been nearly the same.

Area between the ordinates a and i, by Rule i, is - 1857.98
Area between the ordinates i and l, before found, is 363.10

Entire area DOA - - =2221.08

Correct area - - - — 2222.22

Difference or error of the approximation - - 1.14

a

=■ 50.000

b

= 49-346

c

= 48.624

d

= 47.820

e

= 4 6-9°7

f

= 45-851

g

= 44-59°

h

= 43-ot4

i

= 40.88Q

Sum of all the

ordinates

= 4I7.O4I

k

— 37-495

l

= 0.000

Q74

p

= 10.0000

9

= 10.0000

s

= 9-9985

t

= 9.9805

u

= 7.6750

V

= 9.6100

w

= 0.0000

= 57.2640

the Stability of Ships , 275

If the total area DOA (fig. 26. ) should be measured by one
computation, suppose from Rule n. the

Area would be found - - =2176.875

Correct area - - - - = 2222.222

Difference or error of the approximation - = 45.347

If this area should be computed according to
the Rule 1. the error of the approximation will be

found 74-54

It is easily shewn, that these errors, which are far from in-
considerable, arise almost wholly from the mensuration of the
area OXE, (fig. 26.) adjacent to the vertex O. For, by measur-
ing that area, according to the Rule 11. from three ordinates
l = o, £ = 37.495, i = 40.889, the area is found to be 318.115
Whereas the correct area OXE = 10 x 40.889 x J- = 3^3*457

Difference or error from computing the area OXE
by Rule 11. - - - = 45-342

Scarcely differing from - - = 45-347

which was found to be the error from computing the entire
area DOA by this rule.

If the area between the ordinates a and k be measured by
the Rule 111. it is found to be - - = 2055.390

Area between the ordinate k and /, by the proper-
ties of the figure, is = 37.495 x 5 x ■§• - = 166.640

Entire area DOA - = 2222.030

Correct area - = 2222.222

I e f ~

Difference or error of this approximation = .192

From these computations it is evident, that the rules here
given, when employed with attention to the necessary limita-

N n 2

l

276 Mr. Atwood's Disquisition on

tions and restrictions, will approximate to the measures of
areas, to a degree of exactness fully sufficient for naval men-
surations ; and further, will be useful in determining by ap-
proximation the integral values of fluxional quantities in ge-
neral, especially those which occur in the investigation of
practical subjects.

CASE XII.

Still supposing the vertical sections of a vessel to be equal
and similar figures, let BOA (fig. 2 7.) represent one of these
sections ; the figure being either a curve of the higher dimen-
sions, or a curve not formed according to any geometrical law,
of which the lengths of the ordinates, and of any other lines
given in position, are supposed to be measurable, and given in
quantity : the angle at which the vessel is inclined from the
upright, and the other necessary conditions being known, it is
required to find, by geometrical construction, a line which
shall approximate nearly to the measure of the vessel's sta-
bility.

1st Method.

BA represents the intersection of the water's surface when
the vessel floats upright: bisect BA in the point D; and,
through D, draw the line NDM inclined to the line BA at an
angle ADM, equal to the given angle of the vessel's inclina-
tion : let the area of the figure ADM£, also the area of the
figure BDN c, be found by means of the rules which have been
described : suppose the area ADM/j to be greater than the area
BDN c} and let E represent the difference between them : from

the Stability of Ships.

2 77

the point D, in the line DA, set off a line * DS = rrr — t . :

r 7 NMxsm. ADM

a line CSH, drawn through the point S, parallel to NM,
will cut off the area ASH/jA, very nearly equal to the area
BSCcB; (fig. 27.) consequently, when the vessel is inclined
to the given angle, the water's surface will intersect the vessel
in the line CSH. (Tab. XIV. fig. 28.) Draw the lines AH, BC.
Let M and I be the centres of gravity of the triangles ASH, BSC,
respectively; through M and I, draw M/, Ik, perpendicular to
CH ; through the centre of gravity of the area A/jH, draw
h U perpendicular to SH ; and, through c, the centre of gravity of
the area BcC, draw c R perpendicular to CH : in the line /U,
take /L to LU as the area AH/j is to the area ASH£ ; and, in the
line £R, take £K to £R as the area BcC is to the area BSCc.
Let G be the centre of gravity of the vessel ; and let E be the
centre of gravity of the displaced volume when the vessel floats

* Let the area A5H£ be supposed equal to the area BSCc, (fig. 27.) and make either
of them ~ A. Let the space DMHS be denoted by M, and the space NDSC by N :
then the area ADMZ> will approximate very nearly to the quantity A -f- M, and the
area BDN to A — N. The difference of these areas will be M 4- N, which is equal to

the area NMHC = E = MN x DY ; and, consequently, DY = ; and, because

DY : DS : : sin. DSY, or ADM to radius, it will follow that DS n: — *

MNx sm. ADM

f In these small curvilinear spaces, it will be sufficient to assume the positions of the
centres of gravity by estimation, on a supposition that the curve coincides with the
arc of a common parabola ; in which case, the centre of gravity is situated at the dis-
tance of !• of the abscissa from the ordinate, or chord which joins the extremities of
the curve. The position of the abscissa is determined by drawing chords parallel to
the given chord, and by drawing a line through the points which bisect the several
chords. But, when the curvilinear spaces AHZ> are extremely small, as represented in
this figure, (fig. 28 ) no sensible difference in the result will ensue, whether the line
liU is drawn through the centre of gravity of this curvilinear space, or through any
other point which is adjacent to that centre.

278 Mr. Atwood’s Disquisition on

upright. Through the point E, draw EV parallel and equal to
KL : and, in EV, take ET to EV as the area ASH h is to the
area representing the entire volume displaced : through G,
draw GU parallel to CH ; and, through T, draw TZ perpendi-
cular to GU, intersecting the line GU in the point Z. GZ is
the measure of the vessel’s stability.

2 d Method.

Let BOA (fig. 29. ) be the given vertical section of a vessel, in-
tersected by the water’s surface BA when floating upright. G is
the vessel’s centre of gravity : E is the centre of gravity of the vo-
lume displaced in the upright position. Let the area BOA be mea-
sured by either of the three Rules, suppose Rule 1.; and through D,
the bisecting points of BA, draw NDM inclined to the line BA
in the angle ADM, equal to the given inclination of the vessel
from the upright. Let the area NOAM be measured, by erecting
equidistant ordinates on the line MN. If the area, so found, is
equal to the area BOA, the area DBN will be equal to the area
ADM. But, if they are unequal, let the difference be represented
by E,and from D, toward the largest of the areas, suppose ADM,

setoffDS= 'NM x I.adm ; and’ through s> draw CSH Pa~
rallel to NM. The area ASHZ? will approximate to equality
with the area BSCc ; and, consequently, when the vessel is in-
clined through the given angle ASH, it will be intersected by
the water’s surface in the line CH. On the line HC, let the
equidistant ordinates a , b, c, d, &c. be erected perpendicular to
CH ; and let the common interval between the ordinates be
== r. Let the measure of the area CLFK be obtained, and let
m be the centre of gravity of this area : through the point m,
draw fflP perpendicular to CH : let each of the successive

the Stability of Ships.

279

given ordinates be multiplied into its perpendicular distance
from the ordinate a. The terms resulting will be a x o, b x r,
c x 2r, d x Sr, &c-‘, let these terms be added together, and
half the sum of the first and last term being subtracted from
the amount, let the result be denoted by the letter C ; C r — area
CLFK x KP will be the sum of the products arising from mul-
tiplying each evanescent area QX into its distance QK from
the first ordinate a. In the line KH, set off a line KI*

= ----- co ah • ^ hrough the point I, draw I JT perpen-

dicular to CH ; and, through the vessel's centre of gravity G,
draw GZ perpendicular to IT. GZ is the measure of the ves-
sel's stability, when it is inclined from the upright through the
angle ASH.

In the cases which have preceded, the vertical sections of
vessels, or segments of vessels, are assumed as equal and similar
figures : whereas, in reality, the form and magnitude of the
sections are gradually changed, according as they intersect the
longer axis at a greater or less distance from the head or stern.
The solutions of the preceding cases, and the principles therein
established, may be next applied to investigate the stability of
vessels, taking into consideration the form and magnitude of
each particular section intersecting the longer axis at right
angles, and at equal distances; taking into account also, by
the methods which have been described, all the sections inter-
mediate between those which are given, that may be conceived
to intersect the axis at a very small common interval.

* Since the Rule 1. is employed in obtaining the value of the quantity C r, accord-
ing to this computation, the area COAH ought to be measured by the same rule; in
which case, the line KI will be determined nearly with the same exactness as by either
of the Rules 11. or hi.

28q

Mr. Atwood's Disquisition on

CASE XIII.

The longer axis of a vessel is supposed to be divided into a
given number of equal parts, and vertical sections to pass
through the several points of division, intersecting the axis at
right angles : the form and magnitude of each particular sec-
tion being given, with the common distance between them,
the positions of the centres of gravity of the vessel, and of the
volume displaced, and the distance of the water-section from
the keel, being known, it is required to construct the measure
of the vessel's stability, when it is inclined from the upright
through a given angle.

Let QBOAW (Tab. XV. fig. 30. ) represent any vertical section
of a vessel ; suppose it to be the greatest or principal section : BA
is the breadth of this section at the water-line, when the vessel
floats upright : let CH represent the line which coincides with
the water's surface, when the vessel is inclined from the up-
right through the given angle ASH. From the nature of the
conditions, it is sufficiently evident that the point S, in any in-
dividual section, will not be determined on the same principle
by which the position of that point was fixed according to
the former solutions ; that is, by making the area ASH equal
to the area BSC ; because, the volume immersed, and that
which is caused to emerge in consequence of the vessel's in-
clination, will not now be proportional to these areas, as
they * are on a supposition that the vertical sections are similar
and equal figures. But, in the present case, the vertical sections
being different, both in form and magnitude, the water's sur~

* See page zio.

the Stability of Ships. 281

lace, intersecting the vessel in a plane passing through the line
CH, when the vessel is inclined, will so divide the areas of the
several sections, that although the area ASH/6 may not be equal
to the area BSCc, in any of the vertical sections, yet the volume
immersed, corresponding to, and included between, the areas of
the figures ASH/j, taken from the head to the stern of the ship
shall be equal to the emerged volume which is included between
the areas BSCr, in the several sections. Suppose the breadth
of any section at the water-line* to be denoted by BA, and to
be bisected in the point D. A vertical plane passing through
the vessels longer axis and the centre of gravity G,f and divi-
ding the ship into two parts perfectly similar and equal, will
pass through the points D, in all the sections : this plane may
be termed the plane of the masts. It is easily shewn, that at
whatever distance DS, from the middle point D, the plane of
the water’s surface, passing through the lines CH, intersects
the line DA in any one section, when the vessel is inclined
through the angle ASH, it will intersect the line DA at the
same distance from the middle point D in all the other sec-
tions; that is, the distance DS will be the same in all the
sections : for, by the supposition, the vessel is inclined round
the longer axis, and consequently, the intersection of the two
planes, passing through the lines BA and CH, will be parallel

The same letters which are used to denote the several lines, in this vertical section,
must be understood to represent the lines similarly drawn in each of the other sections.
In the present instance, BA does not represent the breadth at the water’s surface, of the
principal, or any other individual vertical section, but represents generally that breadth,
in any of the sections that may be referred to.

f Represented by the point G projected on the plane BOA. The centre of gravity
B is, in like manner, here represented by projection on the same plane.
MDCCXCVIIJ. Oo

282 Mr. Atwood's Disquisition on

to the longer axis, and therefore parallel* to a line drawn
through all the points D, from one extremity of the vessel to
the other ; the several lines DS are the perpendicular distances
of these parallel lines, and are consequently all equal. In the
next place, it is requisite to determine the magnitude of the
line DS, according to the given conditions : whatever be the po-
sition of the points S, if lines CH are drawn through S, in each
of the sections, inclined at an angle to the line BA, equal to
the given angle of the vessel's inclination, the same plane will
pass through all the lines CH. It is required to ascertain at
what distance DS, from the points D, the plane CH, coinciding
with the water’s surface when the vessel is inclined, must pass,
so as to cut off a volume on the side ASH£, being the volume
immersed, which shall be equal to the volume in the Side
BSCc, which has emerged from the water, in consequence of
the vessel's inclination.

In each section, through the middle point D, draw a line
NDW, inclined to BA at an angle ADW, equal to the given
angle of the vessel’s inclination; the same plane will pass
through the lines NDW, in all the sections. By the methods
which have been described, let the area of the figure ADW h
be measured in each section ; from these equidistant areas, the
solid contents of the volume between the two planes DA-f and
DW, and the side of the vessel intercepted, may be inferred by

* The points D1 being coincident with the water’s surface, a line passing through
them must be horizontal ; and being, by the supposition, situated in the same plane
with the longer axis, must therefore be parallel to it.

f Since the same plane passes through the lines DA, drawn coincident with the
water’s surface in all the sections,- this plane may be supposed projected into the line:
DA on the plane DOA. For similar reasons, the line DW represents, the plane, which
passes through all the lines DW in all the sections*.

the Stability of Ships. 283

computing according to the Rules 11. and in.; suppose this
volume to be denoted by the letter P, and the volume contained
between the planes DB, DN, and the side of the vessel, found
by similar operations, to be denoted by the letter Q. Let the
area* of the section of the vessel passing through the lines
NDW be measured, from having given the lines NW in all
the sections from the head to the stern, and let this area be
put = R. If the volumes P and Q should be unequal, P being
the greatest, in the line DA, set off, in each section, a line DS

= R ■ sin adw . ^ a P^ane CSH be drawn passing through all

the points S, and inclined to the plane BA at the given angle of
the vessel’s inclination, the solid contents of the volume between
the planes SA, SH, and the intercepted side of the vessel, will
approximate to equality with the volume contained between the
planes SB, SC, and the intercepted side of the vessel. Since,
therefore, the water’s surface coincides with the plane BA when
the vessel is upright, when it is inclined round the longer axis,
through the given angle ASH, the water’s surface will inter-
sect the vessel in the direction of a plane passing through the
lines CH, in all the sections.

Let the solid contents of the volume immersed, or emerged,
by the inclination, be denoted by the letter A.

In the section QBOAW, let M be the centre of gravity
of the triangle ASH, and let h be the centre of gravity of the
curvilinear area AH h; also, let I be the centre of gravity of
the triangle BSC, and let c be the centre of gravity of the
curvilinear area BCc; through these points, draw the lines

* That is, the area of the section coinciding with the water’s surface, when the ves-
sel is inclined to the given angle.

O O 2

2841 Mr. Atwood's Disquisition on

Ml, b U, Ik, cR, perpendicular to CH; and, in the line IV,
take a line l L, which is to IV as the curvilinear area AH/j is
to the area ASH£. Through the points S, in all the sections,
let a line ¥f be drawn perpendicular to SH ; the same plane
will pass through all these lines. LS will be the distance of the
centre of gravity of the area ASH/j from the plane F /. The
products arising from multiplying each area ASH/j into the
distance SL, of its centre of gravity, from the plane F f, are to be
calculated in all the sections ; from which products, by gieans-
of the Rules* 1. 11. and in. the sum of the products arising from
multiplying each evanescent solid, of which the base is the
area ASH h, and the thickness a small increment of the axis,,
into the distance SL of its centre of gravity from the plane
bf, will be obtained. The sum of these products, divided by
the solid contents of the volume immersed A, will be the
distance of the centre of gravity of that volume from the ver-
tical plane Fjf. Suppose this distance to be equal to the line
SQ : let the d istance PS, of the centre of gravity of the vo-
lume emerged, or BSCc, from the plane F f, be found from
similar computations ; the line PQ will be the distance of the
centres of gravity of the volumes ASHZ>, BSCc, estimated
In the direction of the line CH, perpendicular to the plane

F/-

The solid contents of the entire volume displaced by the ship,
are to be obtained, from the. areas, either of the vertical or hori-
zontal sections..

* Whenever the Rules 1. 11. and in. are referred to, it is meant that the compu-
tation is to be made from one or more of these rules, according to the number of or=~
dinates given, or as other circumstances may direct,
f See Appendix.

the Stability of Ships: 285,

The ordinates drawn in the several sections being set down,
in regular order, the area of any horizontal section is to be
found from the corresponding series of ordinates, by means of
the Rules 11. and in, and, by the same rules, from the areas of
the horizontal section's so determined, the solid contents of the
total volume immersed are to be inferred ; some allowance be-
ing made for the irregular parts of the volume adjacent to the
head and stern, if attention to these additional volumes should
be thought necessary. That part of the volume which is con-
tained between the keel and the nearest horizontal area, is ob-
tained by first finding the area of each vertical section between
the keel and nearest ordinate : from, these areas, by means of
the Rules u. and hi, the solid contents of the volume between
the keel and nearest horizontal section will be measured, and is
to be added to the volume before found, which is contained be-
tween the two extreme horizontal sections.

Let the solid contents of the displaced, volume be denoted
by VI

From the areas of the horizontal sections, and the common in-
terval between them, the distance DE, of the centre of gravity of
the volume immersed, from the water-section, is to be obtained
by means of the Rules 11. and in.; by finding the sum: of the
products arising from multiplying each evanescent solid, of
which the base is any horizontal section, and the thickness a
small increment of the vertical axis, into that small increment,
also into its distance from the water-sec.tion : the sum of these
products, being divided by the solid contents of the volume dis-
placed, will be the distance DE,,of the water-section,, from the
centre of gravity of that volume.

The- position of the vessel’s centre of gravity G, depends.

28 6 Mr. Atwood's Disquisition on

partly on the construction and equipment of the vessel, and
partly upon the distribution of the lading and ballast, which
circumstances therefore determine the distance GE, or the dis-
tance between the centre of gravity of the vessel, and that of the
displaced volume.

These several conditions having been determined, the con-
struction of the vessel's stability will be as in the former cases.
Through the point E, draw the line EV parallel and equal to
the line PQ ; and, in EV, take ET to EV as the volume im-
mersed by the inclination is to the entire volume displaced ; or
as A to V. Through the centre of gravity G, draw GU parallel
to CH, and through the point T draw TZ perpendicular to GU.
GZ is the measure of the vessel's stability, when inclined from
the upright through the angle ASH.

The weight of the vessel and lading is found from the fol-
lowing proportion:* as i cubic foot is to V, the volume dis-
placed, so is ~ part of a ton to the vessel's weight, which will

therefore be = — ton.

35

The arithmetical operations required for ascertaining the sta-

* According to Mr. Cotes, (Hydrostatics, page 73,) the specific gravity of sea
water is = 1.03, when that of fresh water is — 1. And, since the weight of a cubic foot
of rain water is 1000 oz. or 6z{ pounds avoirdupois, it will follow, that the weight of
a cubic foot of sea water is 6z .5 x 1.03 = 64-375 pounds avoirdupois. Mr. Chapman,
in his Treatise on the Method of finding the proper Area of the Sails for Ships of the
Line, infers the weight of an English cubic foot of seawater to be 63.69 pounds avoir-
dupois. If an average between these results be taken, the weight of a cubic foot of
sea water will be very nearly 64 pounds avoirdupois ; and the weight of 35 cubic feet
of sea water will be almost exactly one ton. According to the tables published by M.
Brisson, the specific gravity of sea water is 1.0263, when that of rain water is 1. By
computing from this specific gravity, the weight of a cubic foot of sea water will be
64.14 pounds avoirdupois. 64 pounds avoirdupois is assumed as the average weight,
in the ensuing computations.

the . S lability of Ships. 287

bility of vessels, by the methods here described, are far from dif-
ficult, although they necessarily extend to some length; in
order to give an illustration of these rules, by applying them to
a particular vessel, I obtained, by favour of Messrs. Randall
and Brent, eminent constructors, a draught expressing the form
and dimensions of a large ship,* built for the service of the
East India Company. According to this draught, the vessel is
divided into 33 vertical segments, by 34 sections, intersecting
the longer axis at right angles, and at a common distance of
5 feet.-f

The lengths of the ordinates entered in the annexed table J
sufficiently define the form and magnitude of each of the 34
vertical sections ; it will not therefore be necessary to represent
their figures by separate drawings, since the constructions and
calculations founded on them, for inferring results in any one
section, are similar to those which are required in the other
sections.

The greatest or principal section, which, according to this
draught, intersects the longer axis at about bo feet from the
1st section adjacent to the head, is represented by the figure
BAO (fig. 31) : BA is the breadth at the water-line = 43.1b
feet. BA is bisected in the point D ; and DO, drawn through.
D perpendicular to BA, is the distance of the keel from the

* The ship Cuff nells.

f Mr. Brent, jun. obligingly took the trouble, at my request, of delineating each
of these sections on a large scale, and likewise of drawing and measuring the equidis-
tant ordinates necessary for calculating the areas thereof, together with such additional,
lines as are required for constructing the measure of the vessel’s stability, according to.
the principles delivered in the preceding pages.

X See Appendix.

288

Mr. Atwood's Disquisition on

water-section — 22.75 feet. The line DR = 22 feet, is divided
into 11 equal parts, and, through the points of division, 12 ordi-
nates * are drawn, parallel to the line BA, at the common dis-
tance of 2 feet.

The vessel is supposed to be inclined round the longer axis,
at an angle of 30°, and the line NDW is drawn through the
point D, inclined to BA, at an angle ADW = 30° : proceeding
according to the solution which has been given, by measuring
the line DW = 22 .6 feet, DA= 21.58 feet, the area of the

triangle ADW = 21,58 x-2— = 121.92 square feet. Also, by

mensuration, the line WA = 11.55: this line being divided
into six equal parts, of 1.925 each, if ordinates are drawn at
the points of division, perpendicular to the line WA, they are
found to be as here stated.

By computing according to the
Rule 11. from the 7 ordinates given,
of which the two extremes are = o,
the area of the curve space AW h
=. 2.95, which being added to the
area ADW= 121.92, the area of
the entire figure ADWb =124.87.
By similar calculations, the area
of the figure BDN c is found to be
= i33-6‘8*

The areas of the figures ADWZ?
and BDN c, being measured in
each of the 34 vertical sections,
are found to be as follows.

* The numerical measures of these lines are inserted in the table of ordinates ; (see
Appendix ;) the numbers are entered in the 12th vertical section. It is not necessary
to express them in the figure.

Ordinates.
Pts. of a Foot.

Numbers. Products,

a

== 0.00

1

0.00

b

= 0.15

4

0.60

c

= °-3°

2

0.60

d

= °-43

4

1.72

e

= °-38

2

0.7 6

f

= 0.23

4

0.92

8

= 0.00

1

0.00

Sum 4.60
t common interval
— .642 ; and the area
AW b = 4T x .642 = 2.95

the Stability of Ships.

Vertical

Sections

Areas of the
Figures

ADW b

Square Feet.

Areas of the
Figures

BDN c

Square Feet.

1

42.86

23.61

2

81.5s

58-92

3

IOO.80

86.80

4

1 I4.16

105.27

5

121.56

H5.7O

6

121.75

120.90

7

I23.47

I25.36

8

125.20

I29.82

9

I24.87

13^°4

10

1 24-54

132.27

11

I24.69

132-97

12

I24.87

I33.68

13

I24.87

133.68

14

I24.87

!33-68

15

124.82

133-42

16

I24.78

133.17

17

124.20

132.85

18

124.62

132-53

39

123.91

131-0 5

20

123.21

129-57

21

121.06

127.48

22

118.91

125.40

23

117.50

122.66

24

1 16.10

119-93

25

H4.OI

116.88

26

III.9I

113.83

27

I08.96

109.81

28

106.01

105.80

29

' 101.82

98.92

30

97-24.

91. 71

3i

92.41

79-95

32

86.3I

66.06

33

8l.6o

48.20

34

68.35

17.92

3767-77

3700.84

289

The vertical sections intersect the
longer axis at a common interval of 5
feet. By computing, according to the
Rule hi. from the areas of the 34 figures
ADW h here given, together with the
common interval of 5 feet, a result will
be obtained, which approximates very
nearly to the solid contents of the
volume between the planes DA, DW,
and the intercepted side of the vessel,
throughout the entire length of it.

Making, therefore, according to
Rule hi. the sum of the first and last
area - - =S= 111.21

And the sum of the 4th,

7th, 10th, 13th, &c.

area (except the 34th,) = P = 1167.58

S + P= 1278.79
Sum of all the areas - =3767.77

Sum of the 2d, 3d, 5th,

6th, &c. areas - = Q = 2488.98

Solid contents of the volume between
the planes DA, DW, and the intercept-
ed side of the vessel = S-j- 2 P -j- 3 Q

j r

x j = 18587.5 cubic feet.

From the 34 areas of the figures
BDN c, as entered in the table, by com-
puting according to the Rule hi. the
solid contents are obtained, of the vo-
lume between the planes DB, DN, and

Pp

MDCCXCVIII.

Mr. Atwood’s Disquisition on

the intercepted side of the vessel : for, observing the notation
already described, and applied to the areas BDNc.

s = 4

P= 1188.83

Q = 2470.48

The solid contents of the volume between the planes DB,
DN, and the intercepted side of the vessel,

is = S-f 2P + 3QXj = 18432
Volume between the planes
DA, DW, and the inter-
cepted side of the vessel = 18387

Difference - = 133 cubic feet.

The area of the section passing through all the lines CDW,
is found from having these lines given by mensuration, and the
common interval of 3 feet between them. This area is = 7106
square feet. The distance DY* between the plane NW and the
plane CH, which coincides with the water-section, when the
vessel is inclined 30° from the upright, is = = .022 parts

of a foot, and the distance DS-f = == .044 parts of a foot,

or little more than j- an inch : a quantity of which it is unne-
cessary to take any account in this construction. J
When, therefore, the vessel is inclined from the perpendicular,
through an angle of 30° round the longer axis, the water’s sur-
face will pass through the middle point D of the line BA, in all

* Page 283. f Ibid.

1 The form of the sides above and beneath the water line, in this vessel, causes the
points D and S almost to coincide ; but, in vessels differently formed the distance of
these points is often considerable. The present instance points out the method of com-
structing the line DS, as distinctly as if it was of greater, magnitude..

the Stability of Ships. 291

the sections ; the volume immersed, by the inclination on the
side AD W, being equal, in a practical sense, to the volume which
emerges on the side BDN. Let each or either of these volumes
be denoted by the letter A = 18509 cubic feet, being the ave-
rage value between 18432 and 18587. Through the points D, in
all the sections, draw lines F f perpendicular to the lines NW ;
the same plane passes through all the lines F/: in the next
place, the distance between the centres of gravity of the volumes
immersed, and caused to emerge, in consequence of the vessel's
inclination, estimated in the direction of a line NW, perpendicu-
lar to the plane F/, is to be obtained. To effect this, the line
~D/*, in the principal section, is found by mensuration to be
“ 13-^> /U= 7.03; and the area-j'- AW <6 = 2.95. The area
ADW£ has been found = 124.87. J Wherefore, according to

the preceding solution, the distance IL — — >J *. 7'°J. = 0 17

124,87 />

which being added to the line D/ = 13.8, the sum will be DL
= 'l3-97> anc* the product arising from multiplying this line
into the area ADW^ will be 13-97 x 124.87 = 1742. Similar
products being obtained, arising from multiplying the several
areas ADW h into the perpendicular distances of their centres
of gravity from the plane F /, also the several products arising
from multiplying the areas BDN c into the perpendicular dis-
tances of their centres of gravity from the plane F f9 in each of
the 34 vertical sections, the results will be as expressed in the
adjacent table.

The point M is the centre of gravity of the triangle ADW, as in the general
construction and solution : M/ is drawn through M, perpendicular to NW : the line
iL is found according to the method described in the same general solution,
f Page 288. j Ibid.

Mr. Atwood's Disquisition on

292

Vertical

Sections

Products on
the Side

ADH h

Products on
the Side

BDC c

1

359

*43

2

929

562

3

1269

1010

4

1522

1342

5

1683

1542

6

1699

1669

7

1723

176b

8

1747

j853

9

1 739

1873

10

173 1

1892

11

j736

1 912

12

1742

193 1

13

1742

193 1

14

1742

193 1

^5

1727

193°

ib

1713

!929

1724

1915

18

1736

1901

19

17 19

i8bb

20

1702

1832

21

lbbo

1798

22

1618

17^4

23

’591

17°S

24

1564

164b

25

1516

1584

2 6

I468

1522

27

H'5

1434

28

’363

1346

29

1290

1220

3°

1199

1088

31

1114

88l

3®

IOIO

673

33

919

4 *5

34

708

102

49908

Suppose the volume immersed, by the
inclination on the side ADW h, to be di-
vided into very thin laminas, or solids, the
bases of which are the areas of the succes-
sive figures ADH5, and the thickness a
small increment of the longer axis ; by ap-
plying the Rule in. to the products in the
adjacent table, corresponding to the figures
ADW h, we shall obtain the sum of all
the products, arising from multiplying each
of these thin solids into the perpendicular
distance of its centre of gravity from the

plane F/=S+2 P+3Q x -^ = 254367;
which sum of products being divided by
the solid contents of the said volume, or
18509, will be the distance DO, of the
centre of gravity of the volume ADW b

from the plane F / — = 13.78 > an<^>

by a similar calculation, the distance DP
from the plane F /, of the centre of gravity
of the volume on the side BDN c , caused
to emerge by the inclination, will be
= 13-54 = DR

The sum of these two lines, DQ + DP;
or PQ. will be = 27.32 feet, which is the
distance of the centres of gravity of the
volumes immersed and emerged, in con-,
sequence of the vessel's inclination, esti-.

the Stability of Ships. 293

mated in the direction of the line NW, perpendicular to the
plane F f: let the line PQ be denoted by the letter b.

The solid contents of the entire volume displaced are next
to be measured, from having first obtained the areas of the 12
horizontal sections, intersecting the vertical axis at the common
interval of 2 feet, and dividing the immersed volume into 1 1
horizontal segments. The areas of the several horizontal sec-
tions are measured by Rule hi. from having given the ordinates
drawn in the said sections, parallel to the water’s surface from
head to stern, at the common interval of 5 feet.

The mensuration of the area of the horizontal section 12,
which coincides with the water’s surface, from the ordinates,
entered in the annexed * table, is as follows ;

According to the Rule in.

s = 23-73

P = 206.4,1

Q = 444-°.5

the area of half the horizontal section
12= S+2P+3Q x 4 = 3316.3,
and the total area of the horizontal
section 12 = 6633. 6 square feet.

The areas of the 12 horizontal sec-
tions, computed by Rule in. from the
ordinates, as expressed in the table
inserted in the Appendix, are as fol-
lows..

Ordinates, in the Appendix-

Vertical

Sections

Qrdinat.cs of

the Hori-
zontal Sec-
tion 12.
Feet.

V ertica
Sections

Ordinates of
the Hori-
zontal Sec-
tion 12.
Feet.

1

IO.78

1 9

2I.48

2

16.00

20

21.32

3

18.40

21

21.22

4

19.84

22

21.05

S

20.54

23

20.82

6

20.94

24

20.61

7:

21.20

25

20.44

8

2:1 38

26

20.15

9

21.48

27

13-85

10

21.50

28

13-58

11

21.56

2 9

13-25

12

21.58

30

18.77

13

21.56

3i

l8.21

14

2 I.56

32

17-52

15

21.56

33

16.50

16

21.55

34

12-35

17

18

21-53

21.51

674.19

* See Table of

®94

Mr. Atwood’s Disquisition on

The solid contents of the volume between the
sections 12 and 2, is measured by the Rule 11. by
making the sum of the 12th and 2d areas = S
the sum of the nth, 9th, 7th, 5th, and 3d, = P
the sum of all the other areas - — Q

the common interval between the areas being
two feet.

The solid contents of the volume between
the sections 12 and 2, is S -f 4 P + 2 Q x ~
— 113791-2 cubic feet.

The contents of the volume between the sec-
tion 2 and 1, may be obtained by the Rules 11.
and in. For, by the Rule hi. the contents
between the sections 4 and 1, is found to be = 21733.8

By Rule 11. the contents between the sections

4 and 2 is - - = 17012.0

Contents between the sections 2 and 1 = 4721.8

Contents between the sections 12 and 2 = 113791.2

Water-

Lines.

Areas of the
Horizontal
Sections.

Square Feet.

12

6633.6

11

6568.4

10

6499-3

9

6324.O

8

6238.O

7

5948-8

6

5687.8

5

5353-4

4

4906.6

3

4298.6

2

1

925.0

62800.5

Contents between the sections 12 and 1 = 118513.0

To the volume thus determined, the contents of the space
between the horizontal section 1 and the keel, are to be added.
To measure this volume, the areas must be first obtained of"
the 34 vertical sections which are included between the first
ordinate in each vertical section and the keel ; which areas are
found, by the methods of mensuration already described, to be
us in the annexed table.

the Stability of Ships.

295

Verdes

Section

Areasbetween
the first Ordi-
* nate and the
s ^ Keel.
SquareFeef.

1

2

3

4

2.4

5

2.8

6

46

7

5#

8

6.6

9

7-4

10

9-3

11

10.5

12

10.5

13

10.5

34

10.5

10.5

16

10.5

17

9.8

18

9.2

39

8.5

20

8.0

2 1

6.0

22

5-3

2 3

4.8

24

43

25

3-7

2 6

3°

2 7

2.4

28

3-9

29

1.6

30

1.4

3i

1.2

32

1.1

33

1.0

34

1.0

1

’75.9

Applying the Rule hi. to these areas, and
making the common distance between them
= r = 5 feet, the solid contents between the
first horizontal section and the keel is found to
be S -p 2 P + 3 Q x -LL = 871.0 feet.

Contents between the horizontal sec-
tions 12 and 1

Contents between the first section at
the keel* -

Feet.

H8513.O

87I.O

Total contents of the volume immersed
between the first and thirty-fourth
vertical section - 119384.0

Let this volume be represented by the letter V.

By the preceding computations, the line PQ
= b was found = 27.32, and the volume
immersed by the vessel's inclination, or A =
118509. From these determinations, the line
ET is inferred; for, according to the general
theorem,

as V : A : : b : ET or

119384 18509 27.32 4.23
wherefore ET = 4.23.

The rules here employed for measuring the volume displaced,
cannot be applied to the irregular parts of the vessel, adjacent to

2,g6 Mr. Atwood’s Disquisition on

To infer the measure of stability from this value of the line ET,
it will be necessary to have given the distance GE, between the
centre of gravity of the vessel, and the centre of gravity of the
volume of water displaced. The position of this latter centre is
regulated entirely by the form and dimensions of the body under
water ; and, on this account, is to be considered as a point abso-
lutely fixed, in respect of the water-section, or other given plane.
But the position of the vessel’s centre of gravity being regulated,
partly by the construction and equipment of the vessel, and
partly by the distribution of the lading and ballast, can be as-
sumed on the ground of supposition only; unless in cases where
the position of this point has been actually ascertained. In
some vessels, the distance GE has been measured, and found
equal to about -r part of the greatest breadth air* the water-line :
without knowing what the real distance of the two centres of
gravity G and E may be, in the ship of which the dimensions
are here given as subjects of calculation, the distance GE may
be estimated (merely for the purpose of exemplifying the pre-
ceding rules) at of the breadth BA, or = 5.3 9. Con-
sequently, the inclination of the vessel from the upright being
go0, GE x sin. = 2.69 = ER : which being sub-

tracted from ET == 4.23, will leave TR or GZ = 1.53 feet,
the measure of the vessel’s stability, when inclined round the
longer axis through an angle of 30°.

The solid contents of the volume displaced being 119384
cubic feet, the weight of the vessel and contents will be equal to
that of 1 1 9384 cubic feet of sea water ; which, allowing 35 cubic
feet to each ton, will amount to 3410 tons. According to this
determination, the force of stability to turn the vessel round the
longer axis, when inclined from the upright through an angle
of 30°, is a force of 3410 tons, acting at a distance of 1.53 feet

the Stability of Ships. 297

from the axis; which force is equivalent to a weight or pressure*
of 241 tons, acting at a distance of 21.58, or half the breadth
at the water-line from the axis.

In this computation, the distance GE, between the centres of
gravity G and E, has been assumed without considering the
absolute position of these points, in respect to the water-section
or keel. But the distance DE, or OE, ought to be known, since
the point E being fixed in the same vessel, when the weight is
given, and the centre of gravity G being within certain limits
moveable, the adjustment of this centre, by means of the lading
and ballast, will be better regulated, if the position of the point
E be first ascertained: the distance DE will be found from
having given the areas of the 12 horizontal sections, and the
contents of the volume between the section 1 and the keel.

Suppose the areas of 12 horizontal sections of the vessel to
be given, as they are expressed in page 294 ; let the displaced
volume be conceived divided into laminae, or very thin solids,
of which the bases are the areas of the successive horizontal sec-
tions, and the thickness a small increment of the vertical
axis. The sum of the products arising from multiplying each
of these segments into its perpendicular distance from the sec-
tion 12, also a similar sum of products for the solid contents
adjacent to the keel, will be found from the computation
subjoined.

* This equivalent weight is not here supposed to be counterbalanced by the wind,
which probably never acts with sufficient force to keep a vessel of this weight inclined
permanently from the upright to so great an angle. But the measure of force which
acts to turn the vessel round the longer axis, when inclined to this angle of 30°, is
precisely that which is here stated, according to the given conditions.

MDCCXCVIII,

2^8

Mr. Atwood's Disquisition on

Water-
lines, or
horizontal
Sections.

Areas of the
horizontal
Sections.
Square feet.

Distances
from the
Section

12.

Products of each
area, multiplied
into its distance
from the Section
12.

Numbers for
computing
according to
the Rule n.

Products.

12

6633.6

O

00000.0

1

*

11

6568.4

2

I3I36.8

4

52547.2

IO

6499-8

4

25997-2

2

51994-4

9

6324.O

6

37944-°

4

I51776.O

8

6238.O

8

49904.0

2

99808.

7

5948.8

10

59488.0

4

237952'

6

5687.8

12

68253.6

2

136507.2

5

5353-4

14

74947-6

4

39979°-4

4

4906.6

16

78505.6

1

78505.6

Sum of the products =

1108880.8

The common interval between the sections being 2 feet, the
sum of all the products arising from multiplying each thin
horizontal segment contained between the section 12 and 4,
into its distance from the water-section, will be •§• of 1108880.8
= 739253, by the Rule 11.

Numbers for
computing
according to
Rule hi.

4

4906.6

16

78505.6

1

78505.6

3

4298.6

18

77374-8

3

232124.4

2

34.17-°

20

6834O.O

3

205020.0

1

925.0

22

20350.0

1

20350.0

536000.0

Sum of the products arising from multiplying each thin
horizontal segment between the section 4 and 1, into its distance

2 99

the Stability of Ships .

from the section 12, .= f x 536000, by the Rule 111. sa 402000
Sum of the products between section 12 and 4 = 739253

*The contents of the volume between the keel and
the ordinate 1, multiplied into the distance of its
centre of gravity from the section 12 - = 19465

Sum of the products arising from multiplying each

horizontal evanescent solid into its distance from
the water-section - = 1160718

The sum of products, thus found, divided by the entire vo-
lume displaced, will be the distance of the centre of gravity of
that volume from the water-section, or

DE =

1 160718

H9384"

= 9.7224 feet

If the distance between the centres of gravity G and E
should be assumed of the breadth BA at the water-line, or
5.39 feet, this distance being subtracted from g. 22 feet, will
be the distance of the vessel's centre of gravity beneath the
water-section, or DG = 3.83 feet.

To determine the limiting point or metacentre W, above
which if the centre of gravity G should be raised the vessel
will overset, it is only necessary to compute, by means of the

Rule 11. or hi. the value of the line EW — -F1-uen---L^A x g. ;

1 2V

where z represents any portion of the longer axis : AB = the
breadth of the water-section, at the distance z from the initial
point where the mensuration commences : z = a small incre-
ment of z : V = the solid contents of the volume displaced.

By computing the cubes of each ordinate of the water-section,
drawn at the common interval of 5 feet, as represented in the

* Equal to the sum of the products arising from multiplying each thin horizontal
segment between the section i and the keel, into its distance from the water-section.

Qq a

Soo

Mr. Atwood's Disquisition on

table* of ordinates, the total sum of these cubes is = 276573.21
Cube of the 1st ordinate, or 10.78 = 1252.73

Cube of the 34th ordinate, or 12.95 = 2171.75

Sum S = 3424.48
Sum of the cubes of the 4th, 7th,

10th, &c. 28th, and 31st = P = 88628.41

s + p = 92052.89

Sum of the cubes of the 2d, gd, 5th, 6th, f 'Sc. g2d,

33d = Q - - - = 184520.32

S + 2P + 3Q = 734242.26, this sum x 15,
will be = fluent of BA3x z = 11,013,633.9 ; and

since V= 119384.0, EW = u,°'3,6»'9 = 7.688 feet.

In this vessel, the centre of the immersed volume E is 9.72
feet beneath the water's surface; it follows, that the meta-
centre will be 2.03 feet beneath the water's surface.

The total weight of a vessel and contents is inferred from
knowing the volume of water displaced by the vessel, the solid
contents of which space have been calculated in the preceding
pages, from the areas of the 1 2 horizontal sections intersecting
the vertical axis at a common interval of 2 feet. By similar
calculations, we may determine the several weights of tonnage
which will cause the vessel to sink to any different depths, esti-
mated by the horizontal line or section which is coincident
with the water’s surface.

The solid contents of the volume included between the ho-
rizontal sections 12 and 9, is found, by the Rule in. to be 39120
cubic feet ; displacing a weight of water (allowing 35 feet to

* See Appendix.

the Stability of Ships. 301

a ton) of 1117.6 tons: the volume between the sections 11 and
9, is found, by the Rjale 11. to be ~ 25926, displacing 741 tons
of water: the volume between the sections 12 and 11 will
therefore displace a weight = 3 77 tons.

By similar computations, the following results are obtained.

From the
water-section.

Difference
of tonnage.

From

Difference
of tonnage.

12 tO

11

377 tons.

12 tO

11

377 tons-

11 to

10

374

12 tO

IO

75i

10 to

9

367

12 tO

9

1118

9 to

8

357

12 tO

8

1475

8 to

7

348

12 tO

7

1823

7 t0

6

333

12 tO

6

2156

If any one of the adjustments determining the stability of
a vessel should be altered, the several other conditions on
which that power depends, will most commonly experience
corresponding changes, the effects of which it is not easy to
estimate, without some reference to the theory of stability. If
the weight of the Cuffnells and contents should be diminished
751 tons ; or rather if another vessel, constructed in all respects
like the Cuffnells, should be loaded by 75 1 tons less weight, the
following changes will take place, by which the stability is
principally affected ; one of which is additive to, and the others
subtractive from, the stability of the vessel.

1st. The section of the water will be nearer to the keel by 4
feet ; so to coincide with the horizontal section 10 instead
of 12.

The centre of gravity of the displaced volume will be nearer to
the keel than before by 2.19 feet.

Admitting, therefore, in the first instance, the

§02

Mr. Atwood's Disquisition on

centre of gravity* of the vessel and contents
to remain at the same distance from the keel,
the distance of that centre from the centre of
the displaced volume will be increased by 2.19 feet
and this change will operate to diminish the sta-
bility.

2dly. The total weight being less, will have the
effect of diminishing the stability, in the pro-
portion of - 3410 to 2660

3dly. The breadths of the vessel at the water-
line are less than before, being the double
ordinates in the horizontal^ section 10, instead
of those in the horizontal section 12. By this
change, the stability will also be diminished.

^thly. The volume displaced being less, in con-
sequence of the diminished tonnage, in the
proportion of 2660 to 3410, the stability will
be augmented by this change; not in the
proportion of 2660 to 3410, but in a propor-
tion considerably greater; yet not sufficient to
counterbalance the effect of the alterations by
which the stability is diminished : on the

* Let the distance of the centre of gravity of any body, or system of bodies, from a
given plane, be denoted by the letter D : if weights are taken away from different
parts of the system, in such a proportion that the sum of the products arising from
multiplying each removed weight into its distance from the given plane, divided by
the sum of the removed weights, may be equal to the distance D, the distance of the
centre of gravity of the remaining weights, from the same plane, will also be the dis-
tance D : that is, the distance of the common centre of the whole system, from the
given plane, will not be affected by the removal of any portion of the weights, ac-
cording to the conditions here described,
f See Appendix.

the Stability of Ships. 303

whole, these alterations will diminish the sta-
bility of the vessel, when inclined to a given
small angle, in the proportion of about * - 100 to 67

In this estimate, the vessel's centre of gravity has
been supposed to remain at the same distance
from the keel at which it was situated in the
Cuffnells, and consequently more remote from
the centre of the displaced volume by 2.19 feet:
suppose the vessel's centre of gravity to be de-
pressed 2.19 feet, by altering the distribution
of the lading and ballast, so as to be at the
same distance from the centre of the displaced
volume, as in the CufFnells ; the effect of this
alteration will be entirely additive to the sta-
bility, which will be increased in the propor-
tion of 47 to 1 00.

An increase in the proportion of 47 to 100, com-
bined with a diminution in the proportion of
100 to 67, is, on the whole, an increase of
stability, in the proportion of 47 to 67, or about f 7 to 10

It might be difficult to depress the vessel's centre of gravity

* The subject of these observations being the relative stabilities of the two vessels,
they are supposed to be inclined from the upright to the same angle, which may be
assumed of any magnitude, either great or small: the latter supposition is here
adopted, which is well suited to the purpose of general illustration. But nothing can
be inferred from these results, respecting the stabilities, when the angles of inclination
are considerable ; which are to be obtained from computations founded on the methods
which have been described in the preceding pages.

f This proportion might have been immediately inferred from one computation
only ; but, by calculating the effects of diminishing the vessel’s weight separately, the
increase and diminution of stability, arising from the alteration of the several con-
ditions, are more distinctly expressed.

304 Mr, Atwood's Disquisition on

through so great a space as 2.19 feet; and the increase of
stability which would ensue from it, may perhaps not be ne-
cessary. If it should be required that the stability of the ves-
sel, when the weight is diminished 751 tons, shall be just equal
to that of the Cuffnells, this will be effected by adjusting the
centre of gravity lower than its original position, by only .97
parts of a foot.

These determinations relate to the vessel's stability in respect
to the longer axis. But the position of the shorter axis, round
which the ship revolves in pitching, and of the vertical axis,
round which it is caused to turn by any horizontal force not
passing through the vertical axis, will also experience some
change in consequence of diminishing the vessel's weight. For
the centre of gravity of the volume displaced, being necessarily
in the same vertical line with the vessel’s centre of gravity
when it floats quiescent, fixes the position of the latter point, in
respect to the ship’s length, when floating on an even keel.
And since the alteration of the water-section, of 4 feet in
height, causes the centre of gravity of the displaced volume to
approach nearer to the head of the vessel by about a foot,
both the shorter horizontal axis and vertical axis of the vessel
must experience the same change of position : the former al-
teration affects the motion of the vessel in pitching, and the
latter somewhat increases the action of the rudder in turning
the ship, and also affects the motion of the vessel, in turning to
and from the wind, by causes independent of the rudder.*

These observations point out the alterations of stability, in

* By altering the distance of the centre of gravity from the points of application, in
the longer axis, at which the water’s resistance and force of the wind, when not exactly
balanced, act on the vessel.

the Stability of Ships . 305

consequence of diminishing the tonnage of the vessel, without
entering into any consideration how far such changes are, on
the whole, beneficial or otherwise.

It is here necessary to observe, that the force of stability
and the measure of it, the subject of investigation in the pre-
ceding pages, is wholly independent of the waters resistance,
which co-operates with the vessel's stability only while it
is inclining, and wholly ceases as soon as the vessel has at-
tained to the greatest inclination, at which it is supposed
permanently to remain in a state pf equilibrium ; the inclining
force being exactly balanced by the force of stability. This
observation will obviate any difficulty that might possibly
occur from the principle stated in page 213; which is, that
if the shape of the zone WHFC, (fig. 1 and 2.) comprehend-
ing that portion of the sides of a vessel which may be im-
mersed under, and may emerge above, the water s surface,
should be the same in two vessels, the stability will be the same
at all equal angles from the upright, whatever shape be given
to the form of the volume immersed, which is situated beneath
the said zone, provided the vessels be in other respects similarly
constructed and adjusted : if, for instance, the keel of one ves-
sel should be very deep under the body of the vessel, the keel
of the other being of the ordinary dimensions, the deeper keel
will oppose an increased resistance to the inclination of the ves-
sel only while it is inclining, so as make it heel slower; but
will not alter the angle of permanent inclination caused by a
given force of the wind, or other uniform power ; which incli-
nation depends entirely on the stability, which has been deter-
mined in the preceding pages, and has no relation to the resist-

MDCCXCVIII, R r

306 Mr. Atwood's Disquisition on

ance of the water, which arises from the vessel's motion round
its longer axis.

The object of the preceding propositions, and inferences
founded on them, has been rather to establish general prin-
ciples, which may be of use in forming plans of construction,
than to investigate what modes of construction are the most
advantageous ; a discussion more extensive than would be con-
sistent with the subject here proposed to be considered, which
relates to the stability of vessels only.

The practice of navigation requires the co-operation of many
qualities in vessels, the laws and powers of which, considered
as acting either separately or conjointly, it is the employment
of theory to investigate. In respect to the construction of
ships, it is obvious that no one of the component qualities can
be regulated, without paying attention to all the others ; be-
cause, by increasing or diminishing any of the powers of action,
the others are commonly more or less influenced. It has been
shewn, by the propositions demonstrated in these pages, that
there are many practical methods by which the stability of
vessels, at any given angle from the upright, may be aug-
mented ; a circumstance which gives to the constructor great
choice of means for regulating this power, according to the
particular service for which the ship is designed ; for it is not
every mode that wilbbe advantageous. The several varieties of
form and adjustment by which stability is increased, may be
so unskilfully combined, that, in consequence of the ver}^
means used to obtain that essential quality, either the ship
shall not steer well, or shall drift too much to leeward, or shall
be liable to sudden and irregular motions in rolling, by which

the Stability of Ships . 307

the masts are endangered ; or those angular oscillations of the
ship shall be performed round an axis situated so much beneath
the water's surface, that the motion of rolling shall be excessive
and laborious. It is the proper use of theory, or right principle,
whencesoever derived, so to adapt the means to the end pro-
posed, that the required stability shall be imparted, without
producing inconveniences of any kind, or such only as are
unavoidable, and are the least prejudicial: the same observa-
tion applies to the other qualities of vessels. By duly com-
bining the whole, ships are constructed so as to fulfil the pur-
poses of navigation.

R r 2

308

Mr. Atwood’s Disquisition on

APPENDIX.

Note to the Investigation , Pages 232, 233.

•yAine O x sine P = tang.-s

v sec. |F x sec. S

This is investigated in the following manner :

• ' j ....

Let R be put to denote a right angle, the other notation be-
ing the same as in page 232 ;

Then the angle D WP = \ F -f P ; also DWP= R — S
wherefore F~j~ P — R — S and P = R — i F — S,

and, since 0 = 2R — F — P, it follows that 0 = R — i F -j- S ;

consequently sin. P = cos. \ F -j- S

sin. O = cos. *• F — S

and

sin. O x sin. P= cos. *

‘ F-f S x cos. \ F — S
= cos.2 \ F x cos.2 S — sin.2|- F x sin.2 S
or because cos.2 S = 1 — sin.1 S

sin. O x sin. P = cos.2 \ F x 1 — sin.2 S — sin.2 {Fx sin.2 S
or sin.Oxsin. P=cos.24-F — sin.2S

or

sec.1 \ F x sec.1 S TTT : TT

= sec.-|FXsec.»S * C0S' fF-Sln- S

sin. O X sin. P = — ■S~”C^F x^—

sec.1 |Fx sec.1 S

i -f tang.1 S — 1 + tang.1 \ F x tang.1 S

' sec.1 \ F X sec.1 S

or

. ... ^ r, 1 _ tang.1 i F x tang.1 S

sin. O x sm. P — a l F ec » s

Finally, ^sin. O x sin. P

sec.1 \ F x sec.1 S

V 1 — tang.1 J F X tang.1 S
sec. f F x sec. S

the Stability of Ships.

3°9

Note to the Investigation , Page 233.

FF. v, tang~* i F x tanS- s x sec- s

^ -/l-tang-HF X tang.*S

The investigation follows :

FP : PO : : sin. O : sin. F ; wherefore OP = FP x ; or be-

sin. O 7

sec. 4F x sin. F

cause FP *= FE x sec. i F x OP = FE x

= FE x

2 sin. | F % OP ti/a txt^ sin. 7 F

and — — PQ = FEx

V' sin.O x sin. P

v' sin.O x sin. P

; or

v'sin. O x sin.P

because ,/sin. O x sin. P+ = — ~ tanf ' * * F x ^ng-' s, PQ = FE

sec. ~ r x sec. 5

V tang. iFx sec. S

i — tang.*| F x tang. S

ButPW :WF (=FEx+v/ 1 — tang.2|- F x tang.2 S) :: sin.^-F
: sin. P; therefore PW = FE x sin. |Fx * t±5Zg

2 sin. P

and WO = PW— PQ — FE x

YE x - tang‘ * F x sec* s

V i — tang.* i F x tang.* S x sin . | F
sin. P

V' i — tang.* i F x tang.* S

or

WO = FE x sinTF— tang*|F x sin.|Fx tang.* S — tang. |F x sec. Sx sin. P

sin, P x V'i — tang.* |F x tang.* S

But sin. P = cos. S -f- \ F = cos.i F x cos. S — sin. \ F x sin. S,
which being substituted for sin. P in the value of WQ just
found, the result is

=FE y sin. IF— tang.* j-F x tang*Sx sin.|F— sin, |F-|- tang. |F x sin.|F x tang.S

cos.|Fx cos.S — sin.|F x sin.S x V' i — tang,*|F x tang.*S

or WO FF. V tang-TF* sin.|Fx tang.S— tang.*|Fx tang.* S x sin. |F

^ cos.|F x cos.S — sin.|F x sin.S x V'i— tang.*|F xtang.*S

orWQ==FExtans'2Fxsin'^Fxtans'S x rang-iF* tang-s or

^ cos.|F x cos.S — sin.f F x sin.S x i — tang.*^F xtang.*SJ

* Demonstrated in page 232.

f Page 30S.

t Page 233.

gio Mr. Atwood’s Disquisition , &c.

because cos. i F x cos. S — sin. | F x sin. S = cos. -§• F
x cos. S x 1 — tang. {F x tang. S

WO — FE X tang‘^Fx sin- 2 F X tang. S x i— tang. -|Fx tang. S

^ cos.-§F x cos. S x i — tang.fF x tang.S x V' i — tang.1 f F x tang,*S

or

WQ

v ^ ^ Q Z s'

t-tp tang.ai F x tang. S X sec. S

r E x — w~~ ~ a t r;~-=rr-
tang.1-! F x tang. 5

V l —

The work of Mr. Chapman, meritioned in page 267, is written in the Swedish
language, and has been translated into French by M. Vial de Clairbois. The
translation is intitled, Traite de la Construction des Vaisseaux, &c. par Frederic
Henri de Chapman, Premier Constructeur des Armees Navales, See.

[To face page 310,.

SHIP.

of the several vertical and horizontal Sections.

21

22

23

24

25

26

2 7

28

29

30

31

32

33

34

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

20.2

20. J5

19.85

19-75

I9.5

19-3

19.I

18.9

18.5

18.05

17-5

16.95

16.25

15-5

20.65

20.5

20, 3

20.1

I9.9

19.7

19.45

19-3

18.8

18.4

17.9

17.35

16.65

15-7

21.0

20.85

20.65

20.45

2O.25

20.0

19.7

19.4

I9.°5

18.7

I 8.2

17.6

16.9

15.8

21.15

2 1.0

20.8

20.6

2O.4

20.1

19.8

19.5

19.25

18.75

‘8-35

17.6

16.9

i5-5

21.22

2I.O5

20.82

20.61

2O.44

20.15

19.85

19.58

19.25

18.77

18.21

17.52

16.5

12.95

2I.24

2I.O75

20.82

20.61

20.43

20.12

19.8

19-52

19.15

18.62

I7.9

16.9

14.9

7-

I 2I.25

2I.O5

20.83

20.61

20.4

20.1

19-75

19.44

19.

18.4

17.4

15.65

14-5

3-4

2I.l8

21.0

20.75

20.55

2O.3

20.0

19.59

19.22

18.55

17.82

16.3

13-4

7-45

1 *95

2I.O5

20.89

20.65

20.4

20.1

1973

19-3

18.8

17.9

16.8

14.6

io-35

4-31

i-35

2I.85

| 20.68

20.4

20.15

19.85

19-35

18.8

18.1

16.85

15.05

12.0

7-i

2.7

1.05

\ 20-5

20.32

20.05

19-75

19-3

18.8

18.05

17. 1 1

15-35

12.9

9.1

4-55

1.85

.90

I9.9

1 197

19-3

19.0

18.46

I7.88

16.8

15-45

13.IO

10. 1

6.12

2.9

1.32

•7 5

f l8-9

1 18.62

18.28

17-75

I7.I

16.15

14.8

12.85

10. 1

7.05

375

1.9

1.0

.70

1 i7-25

16.9

16.40

15-7

14.8

13-4

1 1.6

9-35

6.65

4.25

2-3

1.25

.80

.65

1

14.2

13.62

12.9

119

10.8

9.2

7.2

5-3*

3-55

2.4

1.45

.98

•75

•63

7.2

I 6.4

5-9

5-i

4.2

3-35

2.5

1.8

1.4c

1 . 1 1

•9C

.8c

.62

.6a

.

il, | of a foot,
e when the vessel is loaded.

horizontal sections.

#

[To face page 310.

Dimensions of the CUFFNELLS INDIA SHIP,

The Ordinates entered in the annexed Table are the Half-breadths, expressed in Feet, of the several vertical and horizontal Sections.

VERTICAL SECTIONS.

N°

1

2

3

4

5

6

7

8

9

IO

1 1

12

13

14

15

16

17

18

*9

20

21

22

23

24

2 5

2 6

27

28

2.9

30

31

S2

33

34

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

Feet.

feet.

Feet.

Feet.

feet.

Feet.

Feet.

if)

K-5

17. t

18.9

19.85

20.4

20.55

20.65

20.55

20.5

20.5

20.5

20-5

20-5

20.5

20.5

20.5

20.5

20.5

20.4

20.3

20.2

20. J5

19.85

19-75

19.5

19.3

19.1

18.9

18.5

18.05

17-5

16.95

16.25

15-5

T5

13.0

17.0

18.9

20.0

20.55

20.75

20.9

2 1.0

21.0

210

21.0

21.0

21.0

21.0

21.0

21 O

21.0

21.0

2O.9

20.8

20.65

20-5

20.3

20.1

19.9

19.7

19.45

19-3

18.8

18.4

17.9

17-35

16.65

15-7

14

124

16.75

18.8

20.05

26.6

20-9

21 .1

21.2

21 -3

21.3

21.3

21.3

21.3

21.3

21-3

21.3

21.3

21.3

21.25

21.1

2 1.0

20.85

20.65

20.45

20.25

20.0

19.7

19.4

19.05

18.7

18.2

17.6

16.9

15.8

13

11.6

16.4

18.6

20.0

20. l6

20.105

21.15

21-3

21.4

zi-45

21.45

21.45

zi-45

21.45

21 45

21.45

21.45

2‘-45

21.4

21.25

21.15

2 I .O

20.8

20.6

20.4

20.1

19.8

19.5

19.25

18.75

*8-35

17.6

16.9

15-5

12

10.78

16.0

18.4

19.85

ZO.54

20.94

21.2

21.38

21.48

21.5

21. 56

21.56

21. 56

21.56

21.56

21-55

2I.53

21.5 I

21.48

21.31

21.22

2I.O5

20.82

20.61

20.44

zo.15

19.85

19.58

19-25

18.77

18. Zl

17.52

16.5

12.95

11

9.6

15-35

18.1

19.7

20.5

20.92

21. 2 I

21.44

21.55

21. 60

21.60

21.60

21.60

21.60

21.60

21.60

21. 60

21.52

21.50

2i-35

2I.24

2I.O75

20.82

20.61

20.43

20.12

19.8

19.52

19-15

18.62

1 7-9

16.9

14.9

7-

10

8.1

14.4

17.7

19.45

20. 3

20.8

21,15

21.38

21.52

21.60

21.60

21.60

21. 60

21.60

21.60

21.60

2 1 .60

21.55

21.50

2i-35

21.25

2I.O5

20.83

20.61

20.4

20.1

1975

I9.44

19.

18.4

17.4

15.65

i4-5

3-4

9

6.6

13.25

17.075

19.1

2O.O75

20.65

21.05

21.34

2i-45

21.5

ZI.59

21.59

2I.59

21.59

21 59

zi-59

21.55

21.52

21.44

21.3

21.18

21.0

20.75

20-55

20. 3

20.0

19.59

19.22

18.55

17.82

16.3

13-4

7-45

1.95

8

4.9

1 1 -73

16.24

18.55

I9.73

20.45

20 9

21.2

21.34

2I.47

21.51

21.51

2 1.51

2I.5I

21.5 I

21.5 I

21.50

21.4

21.3

21.2

2I.O5

20.89

20.65

20.4

20.1

19 73

19-3

18.8

17.9

16.8

14.6

10-35

4-31

1.35

7

3-3

IO. O

15.I

17.84

19 3

20.12

20. 65

2O.9

21.14

21.28

21.51

2I.5I

2I.5I

21-35

zi-35

21.32

21-3

21.2

21. 1

21.0

21. 85

20.68

20.4

20.15

19.85

19 35

18.8

18. 1

16.85

15.05

12.0

7-i

2.7

1.05

6

i*7

8.2

'3-5

16.85

18.64

*9 65

20.32

20.62

20.78

2O.93

ZO.93

21.1

21.1

21.1

21.1

21.05

21.0

2O.9

20.81

20.67

20-5

20.32

20.05

19-75

19-3

18.8

18.05

17. I I

15-35

12.9

9.1

4-55

1.85

.90

5

6.2

u-3

J5-3

17-55

18.91

I9.60

20.05

20.25

20.4

20.5

20.6

20.6

20.60

20.6

20.52

20.45

20.35

20.2 5

20.1

I9.9

19.7

19-3

19.0

18.46

17.88

16.8

15-45

13-10

IO.I

6.12

2.9

1.32

7 5

4

4.09

9.0

13.2

15.09

17.7

18.65

19-°5

1 9-3 5

19-53

I9.6

>9-75

19-75

19-75

197s

19.65

19 55

19.45

19-3

19.15

I8.9

18.62

18.28

17-75

17.1

16.15

14.8

12.85

IO.I

7.05

375

1.9

1.0

.70

3

1.8

6-35

1045

13-5

15.6

l6.8

>7-45

17.88

18.15

18.3

18.3

183

183

18.3

18 20

18.05

17-95

17-75

17.52

I7.Z5

16.9

16.40

15-7

I4.8

13-4

1 1.6

9-35

6.65

4.25

2-3

1.25

.80

.65

2

3-5

6.85

9.78

12.15

13.6

14-55

1 5"2

15 5

I5.9

15.9

15.9

15.9

15.9

15-7

i5 5

15-35

15.0

14.6

14.2

13.62

12.9

11 9

10.8

9.2

7.2

5.38

3-55

2.4

i-45

.98

•75

•63

1

z.55

3-°9

5-5

6.75

8.

9.1

9.8

IO.5

10.5

10.5

IO.5

10.5

10 3

9.8

9-2

8.5

8.

7.2

6.4

5-9

5-i

4.2

3-35

2-5

1.8

1.40

1.1 I

.90

.80

.62

.60

Common distance between the vertical sections, 5 feet. >

Common distance between the water-lines or horizontal sections, 2 feet.

Distance between the horizontal section 1, and the upper surface of the keel, I of a foot.

The ordinates of the horizontal section 12 coincide with the water’s surface when the vessel is loaded.

Tkilos. Trans. MD C CXCVIH Tab. YJll/r 3o$.

-O

B

3-

Q

l '

M \

\ Ah

wrv y

\

H

PT

A

E

C

V

D

E

o

My

?'/>{- Jr

r/iiios. ihmr. \\ i )(- cxcvin yit/i. vni/< u

. Mias. Trans. MD C C XC VJIL lab. IX p.3oS .

H|P Rl IP 'f 1 IT"- vr* ■ : - ■ !■ wt

~ ■ 2‘hilos. Tran,-, !M~D C CXC VJ H.

Thilo.?. Trans, MI) C C XC \ l 1 1 feXT. />.

Thilos. Trans. MD CCXC VET.

Tab^YX.p. 3oS.

-

-

. .

' ■ . ; ’

■

■

'

■ • V,

11

■ - :

ii ^

; i :'

•',. I ■ -

» rt

.

Jp

-

■

l ■

-

.

,

. . ■ '■ ■ , / . ■ ij " ■ • .

. I-’-, •• : ' r* ■ ' ■-

'

..

*

: / i • / . ■ • ■ ' ; /

, ■ ;

- -■ -

S/1' ' /

'

'

I

.

' *

Fhilos.Jrans.'M.D C CXC VIET. Tab. 'XXL.p.3o$.

T/iilos. Trans. MD C C XC ~VIEL Jg&XI I.p.&oS.

i1

milos.Trans.-MT> C CXC ^TH.TaT>.~SSLVp.3oS.

Thilos.lmHs.'MD C CXCVHI. Tab."sX\\./>.C>o8.

■

J^/ulus. Tran.r. ovQD C C X C V 1 1 L. Jhb X_l V.p, 3oS .

XX\.Tab?iW./>.3oS.

TZdlos.tmM.ls ID C CXC VTH. Tnb^ST. p. 3o8.

.'■■jaj i.'-c Ji-

J

C 3»

XI. Quelques Remarques d’Optique, principalement relatives d
la Refiexibilite des Rayons de la Lumiere. Par P. Prevost,
Professeur de Philosophic a Geneve , de V Academie de Berlin ,
de la Societe des Curieux de la Nature, et de la Societe Roy ale
d’ Edimbourg. Communicated by Sir Charles Blagden, Knl.
F. R. S.

Read March 22, 1798.

PREMIERE PAR TIE.

De la Refiexibilite.

§ 1. Le mot refiexibilite se prend en deux sens differens.

1. Newton [Opt. 1. 1. part 1. prop 3.) entend par la, cette
propriete d’un rayon de lumiere homogene, en vertu de la-
quelle ce rayon est rdfl^chi, s’il tombe sous un certain angle
d'incidence; et transmis, s’il tombe sous un angle plus petit:
ou, plus simplement, une disposition a etre r^flbchi, et non
transmis, a la limite qui sbpare deux milieux r^fringens. ( Opt.
1. 1. part 1. defin. 3.)

Ce philosophe pense qu'en ce sens la refiexibilite des rayons
n est pas la meme. II etablit, par des experiences, qu’il estime
concluantes, que les rayons plus rcfrangibles sont aussi plus
reflexibles. En sorte que, selon lui, toutes les ci$ Constances
etant donnees et constantes, si un rayon blanc tombe sous un
certain angle sur la surface dirimante, le rayon violet sera re~
flechi, tandis que les six autres seront encore transmis et rb-
fractes. Mais, en augmentant Tangle d'incidence, on obtiendra

3 12 M. Prevost s Optical Remarks,

successivement la reflexion de tous les rayons, depuis le violet,
qui est le plus reflexible, jusqu’au rouge, qui Test le moins.

M. Brougham [Trans. Phil, pour 1796, part I. p. 272.) ne
trouve pas concluantes les experiences par lesquelles Newton
dtablit cette proposition ; et, fondd sur une autre experience, il
etablit la proposition contraire, savoir, que tous les rayons ont
la meme disposition a etre refldchis, pourvu que Tangle d’inci-
dence soit le meme.

2. M. Brougham entend par reflexibilite, une disposition a
etre refldchi pres de la perpendiculaire a un certain d^gre :
en d’autres termes, une propriety du rayon homogene, par la-
quelle son angle de reflexion est a Tangle d’incidence en un
certain rapport, qui n’est pas celui d’dgalite, si ce n’est en
quelques cas qu’il indique.

Selon ce physicien, ce rapport varie pour chaque rayon ho-
mogene. Le rapport d’egalite a lieu pour les rayons qui con-
finent au bleu et au vert : le rapport d’inegalitd a lieu pour les
autres ; et les plus refrangibles sont le moins rdflexibles. En
sorte que, pour le rayon rouge, Tangle de reflexion est moindre,
et pour le violet, plus grand, que Tangle d’incidence.

On sait que Newton aflirme, au contraire, que Tangle de
reflexion est toujours dgal a Tangle d’incidence.

Discutons ces sentimens opposes.

§ 2. Premiere question. Les rayons homogenes different-
ils en reflexibilite au sens Newton ien ? En d’autres termes,
sous un meme angle d’incidence, arrive-t-il que le rayon violet
soit refldchi, tandis que le rouge ne Test pas; toutes choses
d’ailleurs dtant pr^cisement pareilles ?

Des deux experiences, par lesquelles Newton etablit Tinegale
reflexibilite des rayons, il suffira de rappeller celle que M.
Brougham attaque directement.

S13

chiefly on the Reflexibility of Light.

Newton fit tomber un rayon blanc perpendiculairement a
la face anterieure dfim prisme: puis, tournant le prisme sur
son axe, il observoit la reflexion qui s’opdroit a sa face poste-
rieure. II vit le violet se refldchir le premier ; puis les autres
rayons, dans 1’ordre de leurs refrangibilitds, jusqu'au rouge,
qui fut reflechi le dernier. II en conclut, que le violet est re-
flechi sous un moindre angle d’incidence que le rouge. (Ex-
pdr. g.)

(Test cette conclusion que M. Brougham attaque ; et, pour
ne point altdrer sa pensde, je vais transcrire ici ses expressions.

“ That the demonstration involves a logical error, appears
“ pretty evident. When the rays, by refraction through the
“ base of the prism used in the experiment, are separated into
“ their parts, these become divergent, the violet and red emer-
“ ging at very different angles, and these were also incident on
“ the base at different angles, from the refraction of the side
“ at which they entered ; when, therefore, the prism is moved
“ round on its axis, as described in the proposition, the base is
“ nearest the violet, from the position of the rays by refraction,
“ and meets it first ; so that the violet being reflected as soon
“ as it meets the base, it is reflected before any of the other
44 rays, not from a different disposition to be so, but merely
44 from its different refrangibility.”

Ainsi M. Brougham pense que la reflexion du rayon violet
ne precede celle du rayon rouge, que parceque la refraction
qui a lieu a la surface antdrieure, force le rayon violet a at-
teindre la surface posterieure plntdt que ne peut le faire le
rouge.

Mais il semble que Teffet est ici en sens inverse de la cause.
Ecartons d’abord un faux sens. Il est impossible que hauteur

MDCCXCVIII. S s

8*4 M. Prevost*s Optical Remarks ,

veuille dire que Toeil peut saisir Tintervalle de temps qui s’ecoule
entre 1 arrivee du rayon violet, et celle du rayon rouge, a la face
postdrieure du prisme. Maintenant, celui des deux rayons qui
decrit la route la plus courte, tombe plus pres de la perpendi-
culaire abaiss^e du point de depart ; et de cela seul on peut
conclure, qu il tombe sous un angle d’incidence plus petit.
D ou il suit, que c’est le rayon rouge qui devroit etre reflechi le
premier, et non le violet.

En effet, considerons d'abord la position du prisme au premier
moment, et telle que la r£presente la figure que Newton en a
donnee dans son Optique . Le rayon blanc FM (Tab. XVL
fig. 1.) est perpendiculaire sur AC : en ce cas, il n'est pas re-
fracte a son immergence, et suit la droite FM. A ce point,
Newton repr^sente le seul rayon violet, MN, reflechi, tandis
que tous les autres, tels que MH, MI, sont transmis et refractes;
(du moins les seuls violets sont r£fi£chis en entier.)

Cependant il est certain qu'il a fallu, pour obtenir ce pheno-
mene, chercher, en faisant tourner le prisme, a lui donner le
d£gr6 dfinclinaison qui pouvoit faire reus sir T experience i et
M. Brougham a raison d "observer, que des lors le perpendicu-
larisme du rayon, sur la face anterieure, AC, a du cesser; qifen
consequence il y a eu refraction, et que les divers rayons homo-
genes n'ont point suivi une route rectiligne, telle que FM, et
n’ont point rencontre la face posterieure, RC, sous des angles;
egaux.

Soit done maintenant A'B'C' (fig: 2.) la nouvelle position
du prisme, qu il a pris en vertu de sa rotation sur son axe ::
alors le rayon FP tombera obliquement sur A'C^, au point P;;
de sorte que la perpendiculaire PO sera du cote A /, par conse-
quent comprise dans Tangle A'PF, C’est ce qui resulte, i°. dm

chiefly on the Reflexibility of Ligh t 4 315

but que s’est propose Newton, savoir, d’augmenter Tangle d’in-
cidence sur la face posterieure; lequel angle (form 6 au point
M, dans la fig. 1. qui r£pr£sente la premiere position du prisme,)
£toit trop petit pour produire la reflexion. 20. Des expressions
precises de Newton, qui dit que le prisme ABC est tourne sur
son axe , selon le sens qu’indique Vordre des lettres A, B, C, dans
sa figure, qui, pour Tobjet que j'ai en vue, est la meme que ma

fig- 1-

Le rayon FP (fig. 2.) sera done r£fractd en s’approchant de
la perpendiculaire OP : mais le rayon le plus refrangible (le
violet) s’en approchera le plus; le moins refrangible (le rouge)
s’en approchera le moins. Ainsi les routes que suivront ces
rayons sont bien representees par les lignes PV, PR. Le rayon
violet fera done, avec la face posterieure B'C', un angle PVC',
plus grand que Tangle PRC', forme par le rouge. Or, les angles
d’incidence, aux points V, R, sont les compiemens des angles
PVC', PRC', respectivement.

II est done certain, qu’en vertu de la refraction qui s’opere a
la face anterieure, le rayon violet rencontre la posterieure sous
un angle d’incidence moindre que le rouge ; et, par consequent,
le premier est dans des circonstances plus defavorables a la re-
flexion que le second ; cependant, le premier est reflechi, tandis
que le second ne Test pas encore. On est done en droit de
conclure que, par sa nature, il est plus reflexible au sens New-

TONIEN.*

* Tout ceci s’applique egalement a la ioeme Experience de Newton, dans laquelle
il emploie deux prismes, reunis en un seul parallellpipede. Dans l’une et l’autre Ex-
perience, ( Exper . 9 et 10.) il est question d’un autre prisme, destine a rendre l’effet
plus sensible, en dispersant les rayons reflechis : il etoit inutile de parler ici de cet
accessoire.

S S 2

Si 6 M. Prevost’s Optical Remarks ,

Ainsi la consideration introduce par M. Brougham (et qui est
tres juste) fait conclure a fortiori en faveur de Bassertion New-
tonienne. On peut dire, que non seulement, a meme inci-
dence, le violet se reflechit, tandis que le rouge ne se reflechit
pas; mais meme on doit dire, que ce phenomene a lieu, quoique
Bincidence du violet soit plus defavorable a la reflexion que celle
du rouge.

Done enfin les rayons different en reflexibilitd au sens de
Newton ; et le plus refrangible est aussi le plus reflexible.

Jusqu’ici, pour rendre mon raisonnement plus simple, j’ai
laisse indetermin4 Bangle refringent, C, du prisme. Newton
le determine. Dans Y Experience g. du liv . 1. part 1. de son
Optique> il employoit un angle refringent de 450; et cependant il
dit expressdment, que les rayons entroient perpendicul air ement ;
d’ou il suit, que Bangle dhncidence, au point M, etoit aussi de
450. On seroit done fondd a croire, que des rayons tombant
sous cette incidence sur la face BC, et passant du verre a Bair,
ne sont pas tous reflechis. C’est ce qu’affirme M. Brisson, en
exposant cette meme experience ( Traite ele?n. ou Princ. de
Pbys. Paris , 1789. T. II. §. 1411.) Cependant il est bien
connu que la reflexion totale a lieu sous un angle moindre,
savoir, aux environs de 40°. On le determine meme avec Id
plus grande precision : je citerai une seule autorite. “ A ray
“ of light will not pass out of glass into air, if the angle of in-
“ cidence exceeds 40° 1 1'.” ( Lectures on Nat. and Exper. Phil, by
George Adams , London , 1794, Vol. II. p. 163.) La determina-
tion de Newton ne differe pas sensiblement de celle-ci.” “ To-
u talis reflexio turn incipit, cum angulus incidentise sit 40° 10'A
( Opt. 1. 2. p. 3. prop. 1.) Dirons-nous que, sous Bangle de 45®,
il passoit encore quelques rayons., suffisans pour rendre Bexpe-

chiefly on the Reflexibility of Light. 317

rience sensible ? Dirons-nous que le rayon FM n’^toit pas ex-
actement perpendiculaire sur la face AC ? Je pense que cette
derniere assertion est vraie ; c’est-a-dire, que dans la pre-
miere position du prisme, le rayon FM commenqoit par etre
oblique sur AC, dans le sens oppose a celui qif indique la fig. 2. ;
en sorte que Tangle APF etoit moindre que FPC. D’ou il rd-
sultoit, que les rayons les plus r^frangibles tomboient sur la face
posterieure BC, sous un angle d 'incidence plus grand, et, par la-
meme, plus favorable a la reflexion. Sous cette forme, Targu-
ment de M. Brougham reprend une nouvelle force.

Ici YOptique de Newton ne se suffit pas a elle-meme ; elle
a besoin d’un commentaire. Le meilleur sera celui que nous
offrent ses Lectiones Optica (Is. Newtoni Opuscul. Lausannce
et Geneva , 1744,- Tom. IL p. 217 — 223 ) Voici comment hau-
teur y repond a notre doute. “ Ne qua oriatur suspicio, quod
“ refractiones in superficiebus AC et AB, ad ingressum radio-
“ rum in prisma et egressum factas, possint aliquid conducere
“ ad effectus hosce producendos, observare licet, quod effectus
“ iidem producuntur, cujuscunque magnitudinis statuatur an-
“ gulus ACB *; hoc est, qusecunque sit refractio superficiei AC.
“ . Imo possis efficere, quod, cum colores partim reflectun-

“ tur .... et partim trajiciuntur .... radii perpendiculariter
“ incidant in AC, emergantque ex AB, et sic neutra super-
“ ficie refringantur, modo statuas angulum ACB esse grad.
“ 40 circiter, et iidem tamen effectus producentur.” (P. 219.)
Les auteurs les plus exacts iPont pas omis cette circonstance.
Robert Smith, apres avoir expose TExp^rience ix. de YOptique

* La texte porte ABC, par une erreur typographique rnanifeste. Ceci est d’ailleurs
indifferent a l’objet que j’ai en vue. Je ne cite pas ce qui a rapport a Eegalite requise:
entre les angles B et C, parceque cela n’influe pas sur ma recherche actuelle.

518 M. Prevost’s Optical Remarks ,

de Newton, dit (ou fait dire a Newton meme) “ Je ne me
“ suis pas apperyu ici d’aucune refraction sur les cotes, AC, AB,
“ du premier prisme, parcequela lumiere entroit presqne perpen*
(< diculairement au premier cote, et sortoit presque perpendicu-
te lairement au second, et par consequent n'en souffroit aucune,
“ ou si peu que les angles d’incidence, a la base BC, iTen etoient
“ pas sensiblement alters ; surtout, si les angles du prisme, a la

base BC, etoient chacun de 4,0° environ. Car les rayons FM
“ commencent a etre totalement reflechis lorsque Bangle CMF
“ est d’environ 50°, et, par consequent, ils formeront alors avec
“ AC un angle de 90V' ( Cours d’Optique de Robert Smith,

trad, par L. P. P. [Tezenas] Tom. I. §. 173. p. 190.)

II r^sulte de tout ceci, que lorsque Tangle refringent du
prisme est bien choisi, un rayon blanc, perpendiculaire a sa face
anterieure, AC, peut etre decompose, parcequ’il est r^flechi en
partie, et non en totality ; le rayon violet etant r^flechi, tandis
que le rouge est encore transmis.

Je remarquerai ici, que pour repeter cette experience, et la
rendre concluante, il n’est pas necessaire de circonscrire Tangle
C dans les limites qui rendent le rayon FM perpendiculaire (ou
a peu pres perpendiculaire) sur la face anterieure AC. Tous
les raisonnemens que nous avons faits ci-dessus (§.2.) seront
justes, pourvu que, dans la premiere position du prisme, Tangle
APF soit plus grand que son angle de suite FPC ; or, pour
que cette circonstance ait lieu, lorsque la reflexion s’opere au
point M, il suffit que Tangle refringent, C, soit moindre que
40°.

Cette experience variee pourroit offrir quelques resultats in-
teressans ; mais les autorites que j'ai cite ne laissent pas de
doute sur le resultat particulier que nous avions a examiner.

8'9

chiefly on the Reflexihitity of Light.

Les rayons plus refrangibles sont reflechis sous un angle
coincidence moindre. Ils sont plus reflexibles au sens New-

TONIEN.

§. 3. Mais M. Brougham etaye Topinion contraire, d’une
experience qu'il enonce ainsi :

“ I held a prism vertically, and let the spectrum of another
prism be reflected by the base of the former, so that the rays
“ had all the same angle of incidence ; then, turning round the
“ vertical prism on its axis, when one sort of rays was trans-
“ mitted or reflected, all were transmitted or reflected.”*

La discussion complette de cette experience exigeant d’assez
longs details, je me contenterai d’observer, que le plan de la face
verticale, sur lequel s’operoit la refraction, ne pouvoit point etre
ajuste de maniere a produire un meme angle d’incidence avec
tous les rayons du spectre a la fois : et, supposant meme la
chose possible, et executee un instant, la rotation du prisme
eut change cette disposition, en alterant inegalement cet angle,
pour divers rayons. On peut concevoir, en consequence, une
multitude de resultats divers ; entr'autres, on peut concevoir
que les angles dhncidence des divers rayons soient tels, que
^observation de M. Brougham se concilie avec le sentiment de
Newton, sur leur inegale reflexibilite. Mais, puisque M.
Brougham n*entre pas dans ce detail, et ne donne qu'un r£-
sultat unique, il est a prosumer qu’il n'a pas r^pete, ou du
moins varie, cette experience. Ce physicien paroit meme n’y
pas donner beaucoup d’importance, par la maniere rapide dont
il renonce.

* La meme experience, tentee par Newton, lui a donne precisement le resultat
contraire. “ Radii purpuriformes primo omnium reflectuntur, et ultimo rubriformes,’*
Left. Opt. Opuscul. Tom. II. p. 220.

g20 M. Prevost's Optical Remarks , _ ,

Je pense done, qifelle ne peut pas, quant a present, infirmer
les conclusions de Newton ; et qu’on est encore en droit d'af-
firmer, au sens de ce philosophe, que les rayons les plus rd-
frangibles sont aussi les plus reflexibles.

§. 4. Seconde Question. Les rayons homogenes different-
ils en reflexibilite au sens Broughamien ? En d’autres termes,
sous un meme angle d’incidence, le rayon rouge forme-t-il un
angle de reflexion moindre, et le violet un angle de rdflexion
plus grand, que Tangle dfincidence ?

§. 5. L'experience fondamentale de laquelle M. Brougham
deduit cette inegale reflexibilitb, au sens de sa definition, est
celle-ci.

Un cylindre brillant et poli, d’un tres petit diametre, (une
fibre metallique,) etant presente par sa convexite a un rayon
blanc, a rdflechi un spectre colore ; et, tout etant mesure ou
calcule convenablement, il a paru que les rayons qui confinent
au bleu et au vert dtoient les seuls qui fussent reflechis sous un
angle egal a Tangle d’incidence. Les rouges etoient reflechis
sous un angle moindre ; les violets sous un plus grande

Maintenant la question se reduit a savoir si cette experience
est concluante en faveur de la these de M. Brougham.

§. 6. Pour s’en assurer, il est important de rappeller un prin-
cipe pose par Newton, et admis par M. Brougham ; (p. 250.)
c'est que la force quelconque qui produit la reflexion, agitselon
une ligne perpendiculaire a la surface reflechissante.

§. 7. De ce principe il suit, que la reflexion operde par une
surface plane, doit se faire sous la loi admise jusqu’a present
par tous les opticiens. (Newton, Princip. 1. 1. prop. 96.) Et
cela est vrai, quelle que soit Tintensite de la force rbpulsive, et
aussi, quelles que soient la vitesse et Tinclinaison du rayon in-

chiefly OU the Reflexibility of Light. 321

cident; pourvu que le rayon soit r£ellement incident, et ne se
meuve pas paralieiement a la surface repulsive.

§. 8. Cette consequence, et toute la demonstration New-
ton ienne qui Fetablit, supposent que la surface agit sur le
rayon pendant toute sa route dans la sphere de son activity ; et
egalement a d’egales distances. M. Brougham n’allegue rien
contre cette hypothese, il paroit Fadmettre meme expresse-
ment; (p. 2 69.) et, en efFet, comment pourroit-on ne point
Fadmettre ?

§. 9. Ainsi, d’apres un principe qui n'est pas conteste, il
paroit que la reflexion ne peut point decomposer la lumiere
blanche, lorsque celle-ci est reflechie en totalite par une surface
plane.

Ceci est parfaitement conforme a ce qu’observe M. Brougham,
que, par aucun moyen, on ne peut reussir a operer cette decom-
position, en employantdes surfaces planes, ni des surfaces courbes
d*un rayon qui ne soit pas tres petit, et, pour ainsi dire, eva-
nouissant. On con9oit, en effet,qu'un element de surface courbe
d'un rayon plus grand, est un vrai plan, pour une particule de
lumiere. I/auteur, a la verite, explique ce phenomene d*une
autre maniere ; mais le fait, independamment de toute expli-
cation, n'en est pas moins certain et reconnu.

§. 10. Soit maintenant HHH (fig. 3.) un tres petit cylindre
ferillant et poli, (une fibre metallique;) et BRVK le cylindre,
sur meme axe, qui est la sphere d’activite de ce petit corps : Fun
et Fautre representes par leur section circulaire. (Ces deux
cercles, fort inegaux, se confondent a Fobservation. )

AB represente un rayon blanc, incident, au point B, sur le cy-
lindre, ou sur sa sphere d'activite.

Supposons les rayons homogenes inegalement repulsifs, et

MDCCXCVUI. T t

$22 M. Prevost's Optical Remarks ,

que le rouge soit plus repousse que le violet : c'est une suppo-
sition qu’admet M. Brougham. (P. 267.)

Dans cette hypothese, le rayon violet devra penetrer plus
profond^ment dans la sphere repulsive.

La route que decrit un rayon homogene, dans Tint^rieur de
la sphere d'activite repulsive, doit etre form£e de deux branches
£gales et semblables ; et Taxe doit passer par le centre de la
sphere, ou de la section du cylindre. >Cela rdsulte du principe
pos£ ci-dessus. (§. 6.)

Et une consequence immediate de cette remarque, c'est que
ce rayon homogene ressortira de la sphere d’activite, sous un
angle de reflexion egal a Tangle dfincidence.

En sorte que tous les rayons homogenes, formant en B le
meme angle dfincidence, seront tous r£fl£chis sous des angles
egaux.

Mais, puisque les uns se sont plus enfonc^s que les autres
dans la sphere d’activit^, ils sortiront done divergens; car
c*est le seul moyen de produire T£galit6 des angles de reflexion.

La fig. 3. est destin^e a exposer cet effet. Le rayon rouge,
s'enfon^ant moins dans la sphere, ou dans le cylindre d’activite
BRVK, decrit la courbe BOR, dont Taxe passe par le centre C ;
et il ressort par RG, faisant Tangle de reflexion ERG = ABD,
angle d’incidence. Le rayon violet, s'enfonyant plus, decrit la
courbe BQV, dont Taxe passe aussi par le centre C. Ce rayon
ressort par VL, et les trois angles FVL, ERG, ABD sont
£gaux.

Mais Tobservateur, voyant T arc BRV comme un point, et
sachant que les angles qu’il mesure sont la somme de Tangle
d’incidence et de Tangle de reflexion, pour les rayons de chaque
espece, sera conduit a croire, que sous une meme incidence

chiefly on the Reflexibility of Light. 323

les angles de reflexion varient: car il trouvera que la droite
AB forme, avec les droites RG, VL, des angles in^gaux ; et,
en un mot, il aura toutes les memes apparences qui se sont of-
fertes a M. Brougham. (§. 5.)

Il importe de remarquer ici, que si ce physicien affirme, que
les rayons qui confinent au bleu et au vert sont reflechis sous
un angle de reflexion egal a Tangle d’incidence, ce n’est pas
qu’il Tait reconnu,’ ou pu reconnoitre, par aucune experience
directe : mais c’est qu’il n’a eu aucune supposition plus natu-
relle a faire. Que si Ton pretendoit que tous les angles de re-
flexion sont plus petits, ou plus grands, que celui d’incidence;
ou que la limite d’egalite tombe sur quelqu’autre division
du spectre, on feroit une supposition gratuite, mais a laquelle
Tobservateur n’auroit aucun fait direct a opposer. Car il n’y a
aucun moyen concevable, par lequel, dans ces experiences, Tangle
d’incidence puisse £tre mesure directement, et separe de Tangle
de reflexion.

§. 11. Puis done, qu’en supposant (avec M. Brougham) que
le rayon rouge est plus r£pulsif que le violet, on concilie parfaite-
ment le phenomene observe avec la loi de reflexion reconnue
pour les surfaces planes, il n’y a pas de raison de smarter de
celle-ci.

Et je conclus, de tout ce qui precede, que les rayons homo-
genes ne sont point inegalement reflexibles au sens Brougha-
mien ; en d’autres termes, que la loi de reflexion admise par
Newton, est la vraie loi de la nature.

§. 12. Il resulte de la discussion precedente, que les rayons
violets se reflechissent plutot, et les rouges plus fortement.

Lors meme que ces deux effets auroient lieu dans des circon-
stances pareilles, ils ne seroient peut-etre pas inconciliables.

Tt 3

324 M. Prevost's Optical Remarks ,

On pourroit concevoir que la sphere d’activite s’^tend un peu
plus loin pour les violets que pour les rouges, mais qu'elle agit
sur ceux-ci avec plus d’intensite.

Mais il est essentiel de faire remarquer, que ces deux effets
ont lieu dans des circonstances tres diflferentes, meme opposees :
et ceci indique une exception importante a f assertion de New-
ton, sur Fin^gale reflexibilite des divers rayons homogenes.

Dans les experiences par lesquelles ce physicien fdtablit,
[Exp. g et 10.) la reflexion s'opere dans le milieu le plus
dense ;* elle se fait done par attraction. Au contraire, dans
les experiences de M. Brougham, (que j'ai exposees sommaire-
ment au §. 5.) la reflexion s’opere dans le milieu le plus rare;
e’est-a-dire, qu’elle se fait par repulsion.

Ainsi, d'une part, on voit que les rayons les plus refrangibles,
ou les plus attires dans l’acte de la transmission, sont aussi les
plus attires dans facte de la reflexion ; et, d’ autre part, on voit
que les rayons les moins refrangibles, ou les moins attires dans
la transmission, sont les plus repousses (c5est-a-dire les moins
attires,) dans facte de la reflexion.

Ceci paroit faire exception a la loi d'in£gale reflexibilite New-
ton ienne ; puisque cette inegale reflexibilite n’est prouvee par
Newton, que pour le cas ou le rayon se meut dans le milieu le
plus dense. Je ne me rappelle pas que Newton, ni aucun
opticien, jusqu a M. Brougham, ait traite f autre cas. Les
experiences de ce dernier physicien me paroissent indiquer (au

* Dans la iocmc Experience, (la seule sur laquelle on put elever un doute,) Newton
dit bien, a la verite, que la reflexion s’opere par la base commune des deux prismes,
qu’il a reunis en un seul parallelipipede. Mais il me paroit que cette reunion ne
pouvoit se faire, qu’en laissant entre les deux prismes une lame d’air, suffisante pour
produire la reflexion a la surface anterieure. {Opt. 1. 1. part. 1. prop. 3.)

chiefly on the Reflexibility of Light. 325

moins indirectement) Fin^gale inflexibility Newtonienne, pour
le cas que Newton a neglige ; je veux dire, pour celui oh le
rayon se meut dans le milieu le plus rare : et il en resulte, a
ce quil me semble, quelque penchant a croire, que cette ^flexi-
bility est en sens inverse de Tautre: ce qufll ytoit d'ailleurs
assez naturel d’attendre.

SECONDE PART I E.

Quelques Rapprochement.

§. 13. Premiere Question. Les principes qui expliquent
la ryflexion, expliquent-ils la flexion ? *

Pour parvenir a repondre a cette question, il y a deux pryii-
minaires indispensables : rappeller les principes qui ont yty
posys pour expliquer la reflexion ; et indiquer avec prycision
les lois de la flexion.

§. 14. Les principes admis ci-dessus sont, i°. la force repul-
sive agissant selon une direction perpendiculaire a la surface
reflechissante. 20. Les rayons rouges plus repulsifs que les
violets ; ou, en genyral, les rayons les moins refrangibles plus
fortement repousses que ceux qui sont plus refrangibles.

§.15. Les lois de la flexion, tres bien determinyes par M.
Brougham, peuvent (abstraction faite de certaines mesures
prydses) s'enoncer ainsi.

* M. Brougham runtime, fort a propos, flexion, ce que, jusqu’a lui, la plupart
des physiciens nommoient inflexion, (et quelques autres diffraction.) Et il distingue
la flexion en inflexion et deflexion : la premiere approchant le rayon, et la seconde
l’eloignant, du corps flecteur.

32b M. Prevost’s Optical Remarks,

i°. Le rayon le plus inflexible est aussi le plus deflexible.

20. Le rayon le plus refrangible est le moins flexible. Ainsi,
le rayon rouge est a la fois plus inflechi, et plus defldchi, que le
violet, dans les memes circonstances.

§. 16. La loi de la deflexion r£sulte bien des principes, dans
ce qui concerne ce ph£nomene seul. Les rayons rouges demon-
trent ici, comme dans la reflexion, leur plus grande force de
repulsion.

§. 17. Quant a la loi dflnflexion, elle ne r^sulte pas des prin-
cipes qui expliquent la reflexion.

On pourroit dire, que les rayons les plus rdpulsifs sont aussi les
plus attractifs. Cette proposition est admise par M. Brougham ;
mais la diverse r^frangibilite des rayons donne une indication
toute contraire.

Soit pour la deflexion, soit pour Tinflexion, on est placd dans
des circonstances pareilles a celles de Texp£rience de reflexion
par un tres petit cylindre, (§. 5.) que j*ai discut^e. (§. 10.)
Les corps flecteurs sont comparables a ce petit cylindre, Ainsi
Bangle d’incidence n'est pas donn£ imm£diatement, et on a
lieu de croire qufll est 4gal a Tangle de deflexion : il doit de
me me etre £gal a Tangle d’inflexion ; mais il doit arriver que
Tobservateur ne s’apperyoive pas de cette £galite.

Du reste, le phenomena de Tinflexion est plus compliqud,
parceque la courbe decrite par le rayon doit avoir deux re-
broussemens.

On peut entrevoir quelque explication de ses lois^ qui au pre-
mier coup d'oeil paroissent choquer des principes reconnus :
mais je m'abstiens de toute tentative pareilie.

§. 18. Seconde Question. Les principes reconnus pour la
reflexion, se concilient-ils avec ceux de la refraction ?

32 7

chiefly on the Reflexibility of Light .

Ouh Des que les rayons ont traverse la sphere repulsive, ils
entrent dans Tattractive : les rayons rouges sont, a la verity les
plus repulsifs, mais rien n’empeche que les violets ne soient
plus attractifs. On peut meme dire, que ces deux faits se lient
naturellement, et qu’on a quelque lieu de s’attendre que les
rayons moins faciles a repousser, seront plus faciles a attirer.

Or, la refraction les montre tels. Car, i°. rien de mieux prouvb
en optique theorique, que la proposition qui etablit, que la re-
fraction est produite par une attraction dirigee perpendiculaire-
ment a la surface refringente. (Newton, Princip. L 1. prop. 94,.)
20. Done aussi, les differences de refraction, et en particular la
plus grande refrangibilite des rayons violets, sont produites par
la meme cause : consequence confirmee par leur reflexibbitb
superieure dans le milieu le plus dense.

§. 19. Troisieme Q jestion. Les principes de la refraction,
peuvent-ils expliquer ceux de la flexion ?

Non. Ce qui reste inexplique, e'est la loi d’inflexion. Dans
cette loi, les rayons rouges semblent plus attractifs, tandis que
dans la refraction e'est le contraire. Shi est quelque explica-
tion qu'on entrevoie, nous y avons renonce. (§. 17.) Ainsi,
cette partie des phenomenes de flexion reste encore pour nous
comme isolee.

§. 20. <s Les rayons de la lumierb, ne sont-ils pas re-
“ flechis, refractes, et inflechis, par une seule et meme force,
“ qui se deploie diversement en diverses circonstances ?” *
Telle est la question que se faisoit Newton, au commence-
ment du siecle, et qui ne me paroit pas resolue vers sa fin.

II est vrai, qu’en bnon^ant les consequences de ses recherches,
M. Brougham conclut ainsi : “ the rays of light are reflected,

* opt. Qu. 4

328 M. Prevost’s Optical Remarks ,

61 refracted, inflected, and deflected, by one and the same power,
“ variously exerted in different circumstances.”

Cela est, sans doute, fort probable, mais non encore de-
montre.

§.21. La seule analogie nouvelle (et, sans doute, tres-impor-
tante,) qu'a saisie M. Brougham, entre ces trois classes de
phenomenes, est celle qui resulte des rapports harmoniques
entre les parties distinctes des spectres colores, produits par la
refraction, la reflexion, et la flexion.

§. 22. Le spectre par reflexion, peut-il etre calcuie exacte-
ment, d’apres les principes poses ci-dessus, (§.10.) afin de com-
parer le resultat de ce calcul a celui de F experience ? Sa division
liarmonique est un rapport saisi entre ce phenomene et celui
de la refraction, qui fortifie Fopinion, deja si probable, sur
Fidentite du principe dont ces deux phenomenes dependent.
Je n’ose aller au dela.

§. 23. Sans doute, en pesant ces considerations, on se rap-
pellera la proposition avancee par M. Brougham, savoir, que
« la reflexibilite des rayons est comme leur refrangibilite in-
« versement.” Mais il faut bien remarquer le sens de cette
assertion.

Newton a fait sortir un rayon du verre a Fair, par la face
plane d'un prisme, sous un angle dflncidence connu ; et, ayant
observe Fangle de refraction des rayons rouges, et des violets,
!1 a trouve ces angles, et leurs sinus, comme il suit*

Angle commun d'incidence 310 13' o" sinus 50
Angle de refraction du rouge 33 4 58 — — - 77

Angle de refraction du violet 34 5 2 78

M- Brougham a fait tomber un rayon sur la convexite d'une

* Opt . 1 . 1. part , 1. prop . 7.

329

chiefly on the Ueflexibility of Light,

fibre mdtallique, sous un angle cTincidence inconnu ; et, ayant
observd la reflexion, il en a conclu les angles et sinus suivans.
Angle coincidence commun - 770 20' sinus 77-i

Angle de reflexion du rouge - 75 50 77

Angle de reflexion du violet - 78 51 78

On voit bien, en effet, que les nombres 77 et 78 expriment,
de part et d'autre, des limites, d'un cote de refraction, de
Oautre de reflexion ; mais on n’en peut pas conclure la raison
inverse des quantity qu on mesure dans Tobservation de ces
deux phenomenes : la disparite des circonstances de ces deux
experiences s’y oppose. M. Brougham fait remarquer lui-meme
la difference d’incidence. Mais ee n’est pas la seule ; et il
sufRt de se rappeller, que par une meme incidence la disper-
sion des rayons colores varie selon la nature des milieux, pour
detruire toute id£e de proportion rdguliere, exprimde d'une
maniere precise et gdnerale, sans £gard a la diversity des
milieux. *-

La proposition aflirmee par M. Brougham, veut done dire
seulement, que celui des rayons qui occupe le plus de place sur
le spectre refracte, en occupe moins sur le spectre rdfldchi ; et
que Tun et Tautre offre la division harmonique. Cela suffit
bien pour indiquer quelque analogie, mais non pour fonder,
sans aucune autre preuve, Furntd du principe.

§. 24,. C’est dans le meme sens qufll faut entendre la proposi-
tion qui £tablit que “ les flexibility sont comme les ^flexibility
“ directement, et comme les refrangibilites inversement.” Et
il y a encore beaucoup plus de disparite dans les circonstances,

* Voyez entr’autres les conclusions qu; M. Robert Blair a tire de ses expe-
riences, aussi exactes que multipiiees, sur ce sujet. Trans, oftbe R. S. of Edinburgh,
Vol. III. p. 72.

MDCCXCVIII.

Uu

ggo M. Prevost's Optical Remarks ,

et dans les resultats, comrae M. Brougham a soin de le faire
remarquer. Ainsi, la division harmonique du spectre colore
fournit une analogie encore beaucoup plus foible, en faveur de
1’identite d’un principe commun, auquel ces trois phenomenes
doivent etre rapportds.

Et cependant il faut convenir que ces foibles analogies entrai-
nent la persuasion ; et que 1’ esprit ne sera point satisfait, jusqu’a
ce que, par quelque nouvel effort, la flexion soit rdunie aux phd-
nomenes de reflexion et de refraction, par Puaitd de principe.

§. 25. Notre ignorance sur la nature des forces qui produisent
ces phenomenes, en parti culier sur la nature de la force repul-
sive, et le defaut d’accord qui regne encore entre les pheno-
menes de flexion et les autres, permettent-ils d’ avoir confiance
a la cause physique, indiquee des long-temps par Newton,
reprise recemment, et mdme calcuffie, par M. Brougham; je
veux dire, la difference de masse des particules qui composent
les divers rayons ?

§. 2 6. Est-on en droit d’envisager comme une hypothese,
la thdorie Newtonienne des acces de facile et difficile trans-
mission ? Cette theorie nest que Y expression gdndralis^e d’un
fait bien observe. Si les transmissions et reflexions alternatives
ne dependent que de l’epaisseur des lames transparentes, il faut
que les rayons, ou le milieu, soient alternativement, et en tem-
puscules £gaux, dans des dispositions opposes. Des experiences
plus variees feront voir si Y epaisseur seule y influe: c est sous ce
point de vue que l’Abbe Mazeas avoit entrepris quelques expe-
riences, qui ne donnoient encore aucun rdsultat, mais qu on pou-
voit espdrer de voir suivre avec succes, [Mem. desSav. Etr. 1755*)
La theorie des lames minces transparentes, paroit avoir ete
conyue par Newton des l’age de 27 ans, et il ne 1 a public que

chiefly on the Reflexibility of Light. 33 1

35 ans plus tard : car il y faisoit allusion dans ses Legons, a
Cambridge, en 1669, et la iere Edition de son Optique est de
Fannie 1704,. ( Opusc . Tom. II. p. 27 5*) Autant il seroit ab-
surde de mettre en parallele avec la raison Tautorit6 meme la
plus respectable, autant il est juste d'exiger un examen tres
attentif d'une opinion aussi reflechie.

§. 27. Je finirai par observer, que Implication que j'ai pro-
pose, selon les principes Newtoniens, du ph^nomene observe
par M. Brougham, dans la reflexion oper£e par un cylindre
tres petit, (§.10.) ne nuit pas a Temploi que ce physicien en
fait, pour expliquer les couleurs des corps naturels. Son senti-
ment et celui de Newton, a cet 6gard, ne sont pas en contra-
diction. Il n est pas sur que les couleurs des corps naturels ne
soient produites que d'une fa$on ; mais, pour qu’elles soient
produites, il faut que la reflexion s’opere par chaque particule
des corps, sous toute sorte d’angles. Et je ne vois pas que M.
Brougham ait r£ussi a operer, sous plusieurs angles varies, la
reflexion qu’il a obtenue par ses petits cylindres. Il semble
qu’il ne parle, d’une maniere precise, que de celle ou Tangle
d'incidence etoit d* environ 770, et, par consequent, fort grand.

Uu 2

C 332 ]

XII. An Account of the Orifice in the Retina of the human Eye ,
discovered by Professor Soemmering. To which are added ,
Proofs of this Appearance being extended to the Eyes of other
Animals. By Everard Home, Esq. F. R. S.

Read April ig, 1 798.

Having had the honour of laying before this learned Society,
at different times, observations on the structure of the eye, both
in man and in other animals, I have been naturally led to avail
myself of every opportunity to investigate this subject.

My attention has been recently directed to the prosecution
of this inquiry, by a very curious discovery of an aperture in
the retina of the human eye, which we owe to Mr. Soem-
mering, an anatomist of considerable reputation, resident at
Mentz. His account of this discovery has been published on
the Continent, but I do not know that any copy of the memoir
has been brought into this country.

It was believed by Mr. Soemmering, and also by the French
anatomists, that this appearance is only to be met with in the
human eye. I have, however, been so fortunate as to discover
it in the eyes of other animals ; and the object of the present
paper is, to lay before the Society the observations I have made
upon this subject

I am indebted to my friend Sir Charles Blagden, for the
first intelligence of Mr. Soemmering’s discovery. I afterwards
received a more particular account, in a very obliging letter

333

Mr. Home's Account of an Orifice , &c.

from Mr. Maunoir, an eminent surgeon at Geneva, which
contains, I believe, the material information published by Mr.
Soemmering ; I shall therefore transcribe that part of the letter,
which is as follows.

“ The war being an obstacle to a free communication be-
“ tween England and the Continent, you are not, perhaps, ac-
“ quainted with a new discovery in the anatomy of the human
“ eye, made by a professor of Mentz, Mr. Soemmering ; permit
“ me, therefore, to say something on the subject. He was dis—
“ secting, in the bottom of a vessel filled with a transparent
“ liquid, the eyes of a young man who had been drowned, and
“ was struck on seeing, near the insertion of the optic nerve on
“ the retina, a yellow round spot, and a small hole in the mid-
“ die, through which he could see the dark choroides , (looking
“ at the surface of the retina which covers the vitreous hu-
“ mour.) He dissected other human eyes, and constantly,
" when the dissection was carefully made, found the hole of
the retina seemingly at the posterior end of the visual ra-
“ dius, nearly two lines on the temporal side of the optic nerve*
“ and the hole surrounded by the yellow zone, of above three
“ lines in diameter. The hole of the retina is not directly seen,
“ being covered with a fold of the retina itself. An anatomist
“ of Paris dissected many eyes of quadrupeds and birds, and
{e found the yellow spot and hole in no animal but the human
kind.

“ Should you think that nature has intended this hole to
“ grow large when the eye is opposed to a strong light, and
“ thereby cause a great part of the rays to fall on the choroid,
“ and vice versa , when the eye is in darkness ? And the want of
Sl such a construction in animals, is it owing to a greater power

S34f Mr. Home's Account of an Orifice

« of augmenting or diminishing the pupil, than in men ? If
“ Messrs. Mariotte and Le Cat should come to life again,
“ they would find, in that hole, the explanation of the ph^no-
“ menon of the two cards, one disappearing at a certain dis-
“ tance from one eye, &c. which may be explained by saying,
“ that where the optic nerve enters the ball, there is no cho-
“ roid, and so no vision.

“ I dissected some human eyes a short time after I had read
“ the discovery, and found the spot, the ruga concealing it, and
“ the yellow zone. The best way, I think, to see them, is to take
“ off the half posterior part of the sclerotica, then the corre-
u spondent part of the choroid ; both must be cut round the
“ insertion of the optic nerve. The retina is to remain bare and
“ untouched, sustaining alone the vitreous humour; then you
“ may see the round spot, which reaches the optic nerve, and a
“ fold of the retina, marking a diameter of the spot. Then, if
“ you press the ball a little with your finger, so as to push the
« vitreous humour rather near the bottom of the eye, the ruga
“ is unfolded, and you will see the hole perfectly round, of — of
“ a line in diameter, and its edges very thin.

“ All this can be seen on the inside of the eye, but not so
(t perfectly ; and, in that case, you must make your observa-
“ tions in water.”

Many months elapsed, after the receipt of this letter, before
I could procure an eye in a proper state for observing this aper-
ture in the retina ; but, in the course of last month, several
opportunities offered, and I saw the appearance described by
Mr. Maunoir very distinctly.

The mode I adopted for examining the retina, was that of
removing the transparent cornea; then taking away the iris,

in the Retina of the Eye. 335

and wounding the capsule of the crystalline lens, so as to
disengage the lens, without removing that part of the capsule
which adheres to the vitreous humour ; by which means, the
retina remained undisturbed, and could be accurately examined,
when a strong light was thrown into the eye.

The aperture in the retina, surrounded by a zone with a ra-
diated appearance, was distinctly seen, on the temporal side of
the insertion of the optic nerve, and about \ of an inch distant
from it, apparently a little below the posterior end of the visual
radius. The aperture itself, in this view, was very small. After
having viewed it in two different eyes, I took an opportunity of
showing it to Sir Joseph Banks and Sir Charles Blagden,
who both saw it with the same degree of distinctness.

At first, I believed it necessary to have a very fresh eye for
demonstrating this aperture, but I have since found, that it is
more readily seen in an eye two days after death ; the zone,
which is the most conspicuous part, being of a lighter colour
the first day, than it is upon the second.

I have also succeeded in preserving the posterior part of the
eye in spirits, without destroying the appearance of this aper-
ture. This preparation I am unwilling to bring to a public
meeting of the Society, since it may be liable to be injured by
being much shaken ; but I hope my having shown it to Sir
Joseph Banks and Sir Charles Blagden, will be sufficient
evidence, both to the Society and others, that such a prepa-
ration can be made.

I am induced to make this remark, by recollecting that a
celebrated anatomist of Edinburgh denied, in his last publi-
cation, that the anterior lamina of the cornea can be separated
from the others, as a continuation of the tendons of the four

g$6 Mr. Home's Account of an Orifice

straight muscles of the eye, for no other reason than because
he could not succeed in the demonstration of it ; the failure,
probably, arising from the eye not being sufficiently fresh to
admit of such a separation. Had it been mentioned in my
former paper, that the preparation, from which the engraving
was made, had been shown to this learned Society, or to any
members of it, my assertion would probably have had more
weight.

In separating the vitreous humour from the retina, I found a
greater adhesion at this particular part ; and, when the vitreous
humour was removed, the retina was pulled forward, forming
a small fold, in the centre of which was this aperture. This
doubling was sometimes produced by endeavouring to cut
through the vitreous humour, to disengage the crystalliner and
its capsule.

I have been the more particular in describing the appearance
of this aperture in the retina of the human eye, that, while I
announce this curious discovery of Mr. Soemmering to this
learned Society, I may give the most complete confirmation of
it. To have this in my power affords me a particular pleasure,
as it gives me an opportunity of doing justice to the merit of a
foreign anatomist, who deserves so highly of our art ; and who
has demonstrated to his cotemporaries, that those who labour
patiently, and follow their pursuits with ardour, may still hope to
make discoveries, in the anatomy even of those parts of the body
which are considered as the best understood ; since the human
eye, so long the favourite object of the most eminent anato-
mists and philosophers, is still but imperfectly investigated.

After having made the preceding observations upon this sin-
gular appearance in the human eye, I found, in Dr. Duncan’s

337

in the Retina of the Eye .

Annals of Medicine for 1797, an account of a publication con-
cerning it by Professor Reil, entitled, The plait, the yellow
spot, and the transparent portion of the retina of the eye.

After these are described separately, the following circum-
stances are mentioned. “ Soemmering takes this appearance
“ to be a real hole. Buzzi, on the contrary, thinks that it is
“ merely a transparent and thin portion of the retina.
“ Michaelis seems to agree with him. Reil and Meckel are
“ rather in favour of the existence of an actual hole.

“ Michaelis saw the plait more distinctly in foetuses of
w seven or eight months, than in adults ; and the transparent
“ portion lay concealed within it, but the yellow spot was
M wanting : nor is it to be observed in the eyes of newly-born
“ children. After the first year, it becomes somewhat yellow,
“ and the depth of the colour increases with the age of the
“ subject. Soemmering says that this spot is pale in children,
“ bright yellow in young people, and becomes again pale in
“ old age. Its degree of saturation seems to be intimately con-
“ nected with the state of vision : it constantly diminishes, in
“ proportion as vision is obstructed. Where one eye only is
“ diseased, in it the yellow spot is wanting, and the plait is
“ small and wrinkled ; while, in the sound one, they are rather
“ more distinct than usual.

“ Michaelis discovered no vestige of these appearances in
“ the eyes of dogs, swine, or calves.”

Professor Reil’s mode of dissecting the eye, to show the
aperture and plait, is exactly similar to that mentioned in Mr.
Maunoir’s letter.

It will appear, from the account of this orifice in the retina,
which precedes these observations of Professor Reil, that the
mdccxcviii, X x

33$ Mr. Home’s Account of an Orifice

plait so particularly mentioned is an artificial appearance, which
takes place in the dissection of the eye, and arises from the cir-
cumstance of the vitreous humour adhering more firmly to the
edge of this orifice, than to any other part of the retina ; so that
the smallest motion of the vitreous humour, in consequence of
dividing it, or removing the choroid coat, produces a plait, by
pulling forwards this portion of the retina. What is said of the
colour of the j^ellow spot, and of the difference of opinion, whe-
ther it is a, hole or a transparent portion of the retina, I shall
consider more fully in another part of this Paper.

After having ascertained the appearance of this aperture in
the human eye, and found what appeared the best mode of
seeing it, I determined to investigate this subject in the eyes of
other animals.

The monkey was the first animal which I procured for ob-
servation ; being led, from previous knowledge in comparative
anatomy, to believe that the structure of its eye must bear a
very close resemblance to that of the human subject. .

The eye was examined immediately after the death of the
animal, and was prepared in the same way that I have already
described the human eye to have been for this purpose ; so that
the concave surface of the retina appeared in its most natural
state, and the vitreous humour, being entire, kept it expanded,
and free from rugce. On the first view, nothing was to be seen
but one dark surface, surrounding the entrance of the optic
nerve. Two hours after death, the retina became sufficiently
opaque to be distinguished, and, immediately after, the orifice
was visible, appearing to be an extremely small circular aper-
ture, without any margin ; but, in half an hour more, the zone
had formed, which, when very accurately examined in a bright

339

in the Retina of the Eye .

light, had an appearance of four rays, at right angles, as ex-
pressed in the annexed plate. (See Tab. XVII. fig. 3.) Its situa-
tion, respecting the optic nerve, was precisely the same as in
the human eye. As I considered this to be a fact of some im-
portance, since it proved the aperture in the retina to be a part
of the structure of the eye, generally, and not a peculiarity in
the human eye, I requested Sir Joseph Banks, Sir Charles
Blagden, and Dr. Baillie, to examine it: to all of them it
appeared very distinct. After having shown it to those gen-
tlemen, and having an accurate drawing made of it, I preserved
that portion of the eye in spirits ; where the aperture in the
retina can still be distinctly seen, but the radiated appearance
is lost.

In the eye of a bullock, prepared in the same manner, I
looked in vain for a similar appearance : if it existed, and
bore any proportion to the size of the eye-ball, as it ap-
pears to do in the human eye and that of the monkey, it
must have been very visible. The concave surface of the
retina was examined in different lights, under a variety of cir-
cumstances, and by magnifying glasses of different powers,
but still no aperture could be discovered. I was, however,
very much struck, while looking at the optic nerve, to see
something in the vitreous humour, (in consequence of a per-
son accidentally shaking the table,) that had not been before
observed.

This proved to be a semi-transparent tube, resembling in its
coats a lymphatic vessel, rising from the retina, close to the
optic nerve, on the temporal side of its insertion, and coming
directly forwards into the vitreous humour, in which it was
lost, after being distinctly seen for ~ths of an inch of its course.

X x 2

34?o -Mr. Home's Account of an Orifce

Its appearance is accurately delineated in the annexed plate.
(Fig. 4 )

This tube is not so distinctly seen in the eye immediately
upon the animal's death, as some hours after ; and is much
more obvious in some eyes than in others. As the coats of the
tube must be nearly the same in all eyes, this difference pro-
bably arises from its contents not always having the same de-
gree of transparency.

When the eye has been kept 24 hours after the animal's
death, there is an appearance of a zone of a circular form, a
shade darker than the rest of the eye, in which the optic nerve
is included : when this zone, which is nearly ^ths of an inch
in diameter, is attentively examined, the tube I have described
is exactly in the centre of it. The tube seems to be confined
by the vitreous humour, (while that humour is entire,) and
only to move along with the central part of it ; and, in some
instances, when the vitreous humour is divided, the tube falls
down. Its attachment at the retina appears stronger than its
lateral connection with the vitreous humour; for, when I coagu-
lated the vitreous humour in spirits, and separated it from the
retina, I found the tube was left with the retina, but upon be-
ing touched was easily torn.

In the sheep's eye there is a similar tube, in exactly the same
situation, respecting the optic nerve, but much shorter, and
much less easily detected. It does not appear to be more than
-T-th of an inch in length, before it is lost in the vitreous hu-
mour. After having seen the tube distinctly in two different eyes,
and having had a drawing made of it, I looked for it in several
others, without finding it : but, examining an eye from which
the crystalline lens had not been removed, only an aperture

in the Retina of the Eye. 4 341

made into the vitreous humour, by removing a portion of the
ciliary processes along with the iris, the tube was distinctly
seen. The weight of the lens probably pulled forward the vi-
treous humour, and kept the short tube erect, in its natural
situation.

I mention this circumstance, to prevent, as much as I am
able, other anatomists from being disappointed in not finding
it ; which may readily happen, if the search be not made with
considerable attention.

In the sheep, there is no appearance of a zone surrounding
the tube.

These facts, although few in number, are sufficient to prove,
that this orifice is not peculiar to the retina of the human eye ;
and that its situation in man and in the monkey is the same : in
them, it is placed at some distance from the optic nerve ; but,
in some other animals, its situation is close to that nerve, and
it puts on the appearance' of a tube, instead of an orifice.

There is one circumstance which is curious, and which it
will require further information upon this subject to explain ;
the yellow zone, found in the human eye and that of the mon-
key, is not met with in any other animal which I have exa-
mined.

Having stated the facts, and also the opinions of other ana-
tomists, that have come to my knowledge, as well as my own
observations, upon this orifice in the retina of the human eye,
discovered by Mr. Soemmering, and having added to these,
several new facts respecting it in other animals, I shall draw
some general conclusions from the whole, with a view to show
that the conjectures which have been made, respecting its use,
are probably erroneous. I shall afterwards point out several

34s Mr. Home’s Account of an Orifice

reasons for considering it as the orifice of a lymphatic vessel
intended to carry off the vitiated parts of the vitreous humour
and crystalline lens.

In the human subject, as no examination can he made for
some considerable time after death, it is impossible to ascertain
what is the real state of this orifice in the living eye, and what
changes take place in it after death ; we only learn, that the
tinge of yellow surrounding the orifice is very slight, when the
eye is examined recently, and that the next day it becomes
much deeper.

These points appear to be satisfactorily cleared up, by the exa-
mination that was made of the monkey’s eye, as it was begun be-
fore the parts had lost the appearance belonging to them as living
parts. In that state, the retina was transparent, and no orifice
could be seen ; so that the orifice is rendered visible, by remain-
ing transparent, while the surrounding retina becomes opaque.
This appears to decide the dispute between Messrs. Soemmering
and Buzz i ; for, if this part does not undergo the change pe-
culiar to the retina, we must consider the retina as wanting
there. After the orifice is thus rendered visible, the yellow tinge
is wanting, and does not take place for several hours, and even
then is fainter than it becomes afterwards ; which appears to be
sufficient evidence, that this tinge is the effect of some change
after death, and cannot, therefore, have any effect upon vision.

The orifice has been supposed to account for a small object
becoming invisible, when placed at a certain distance from the
eye, and brought opposite a particular part of the retina. This,
however, cannot be the case, as its situation in the retina does
not correspond with the part opposed to the object, when ren-
dered invisible.

in the Retina of the Eye. 343

The orifice itself is probably too small to produce any defect
in vision, as the trunks of the blood-vessels which ramify
upon the retina cover a larger space than this orifice, for a
considerable extent, without obstructing the sight of any part
of the object.

While my observations were confined to the human eye, I
was led to consider this orifice as a lymphatic vessel, passing
from the vitreous humour through the retina, but could bring
no absolute proof of its being so. This opinion was strength-
ened by finding, that in the monkey, the orifice was only ren-
dered visible when the retina became opaque ; and it has since
been corroborated, by a distinct tube being met with in the
eyes of sheep and bullocks.

That a change must be constantly taking place in the crys-
talline and vitreous humours, to preserve to them the necessary
degree of transparency, can hardly be doubted; and that the
absorbent vessels which perform that office should have one
common trunk, which follows the course of the artery and vein,
perfectly agrees with what takes place in other parts of the body.

In the human eye, and that of the monkey, the artery is in
the centre of the optic nerve; but that would have been too cir-
cuitous a course for the lymphatic vessel to follow, and, by go-
ing through the retina, at some distance from the nerve, it can
pass out of the orbit with the blood-vessels that go through the
foramen lacernm orhitale inferius. In the bullock and sheep,
there is a plexus of vessels surrounding the optic nerve, and the

tube dips down, close by the optic nerve, probably to accom-
pany them.

From the observations made by Michaelis, of the yellow
spot not being visible in foetuses, or in infants under a year old,

su

Mr. Home's Account of an Orifice

or in eyes that are blind, also of its being brighter in young
people, and paler in old, it would appear, that it is only whe'n
the eye is capable of performing its functions, that there is any
stain communicated to the retina.

EXPLANATION OF THE PLATE (Tab. XVII.)

The drawings from which the figures are engraved were
made from preparations of the eye lying in water, with a strong
light shining upon the preparation. In making the drawings,
the principal object was, procuring a distinct view of the parts
surrounding the optic nerve; when this could be obtained, the
situation of the eye itself was not attended to.

Fig.i. A transverse section of the human eye, immediately
before the ciliary processes. The retina is viewed through the
posterior portion of the capsule of the crystalline lens.

a. The termination of the optic nerve.

b . The aperture in the retina, discovered by Professor
Soemmering.

Fig. 2. A longitudinal section of the left eye in the human
subject, to show the relative situation of the aperture in the re-
tina to the entrance of the optic nerve, and the mode in which
it appears to project, when the vitreous humour is disturbed.

a. The termination of the optic nerve.

b. The aperture in the retina.

Fig. 3. A transverse section of the eye of a large monkey,
to show the aperture in the retina : its situation is the same as
in the human eye. The zone has the appearance of a star with
four rays.

a. The entrance of the optic nerve.

b. The aperture in the retina.

Philos. TrawM D C ('XCAMU. lkb.XVlLp.344-

345

in the Retina of the Eye.

Fig. 4. A transverse section of the eye of a bullock, to show
that there is a semi-transparent tube projecting from the edge
of the entrance of the optic nerve, into the vitreous humour.
This tube is surrounded by a zone, with a distinct margin : it
is situated on the temporal side of the optic nerve.

Fig. 5. A transverse section of the eye of a sheep, to show
that there is a similar tube as in the bullock, in the same situa-
tion, but much shorter, and without the surrounding zone.

MDCCXCVIII.

C 346 3

XIII. A Description of a very unusual Formation of the human
Heart. By Mr. James Wilson, Surgeon. Communicated by
Matthew Baillie, M. D. F. R. S.

Read May 3, 1798.

Th e heart is an organ of so much importance in the animal
oeconomy, and is so immediately concerned in the support of
life, that any unusual deviation from its natural form and situa-
tion in the human body, has always been considered as a subject
of some interest by the physiologist; such deviations have,
therefore, not unfrequently been submitted to the consideration
of this and other learned Societies. Many circumstances re-
specting the circulation of the blood, and respiration, wholly
unknown to our ancestors, have lately been ascertained; but
we are not as yet arrived at a perfect knowledge of these im-
portant actions. Difficulties yet remain; more information may
still be acquired ; and the reasoning upon these subjects will
be less liable to fallacy, in proportion to the number of facts
which have been observed, and the accuracy of the observa-
tions. These are the reasons which have induced me to lay
before this Society, a description of a monstrosity in the human
heart, very singular in its nature, and which, I believe, has
not hitherto been observed or described. I have consulted the
works of those authors who were the most likely to have

Mr. Wilson's Description , &c. 347

recorded such cases, but I have not been able to meet with a
description of any which have been at all similar.

It is well known to most of the Members of this Society,
that the circulation of the blood throughout the body, and ex-
posure of it to the atmospheric air in respiration, seem, in most
animals, to be necessarily connected ; but are not equally so in
all. They are so much connected in the human subject, and in
most quadrupeds, that after birth there is a double heart ; viz.
one for the circulation of the blood throughout the body, to be
subservient to the various purposes of life and growth ; the
other for its circulation through the lungs, where it undergoes
a change which is essential to its general circulation through
the body : these two circulations, in the natural state, bear an
exact proportion to each other. Instances, however, have oc-
curred, even in the human subject, where this exact proportion
has not been preserved ; yet life has been prolonged for some
years, but in a feeble and imperfect state. In some of these
instances, the pulmonary artery has been smaller than usual,
so that much less than the natural quantity of blood was ex-
posed to the influence of the air in the lungs ; in others, the
foramen ovale has not been closed, but a considerable commu-
nication has remained between the two auricles ; and, in others,
there has been a communication between the two ventricles,
from a deficiency in the septum. The effect of all these devia-
tions is the same, upon the blood in the general circulation, viz.
that a part of the blood is not exposed to the air in the lungs ;
so that it is less pure as it circulates over the body. A more
remarkable deviation in the structure of the heart, than any to
which I have just alluded, has been lately published by Dr.
Baillie, in his Morbid Anatomy. In this heart, the aorta

Yy 2

34$ Mr. Wilson’s Description of a

arose from the right ventricle, and the pulmonary artery from
the left; the reverse of what ought, in the regular course of cir-
culation, to have taken place ; (the veins were as usual ; ) and no
communication was found between the one vessel and the other,
except through the remains of the ductus arteriosus, which was
not larger than a crow quill, and a small part of the foramen
ovale, which still continued open ; yet this child lived for two
months. In the following case of monstrous formation of the
heart, there is this very great singularity, that nature seems
to have substituted, very exactly, the circulation which takes
place in some amphibious animals, for that which is natural to
the human species.

The infant had arrived at its full time, and lived seven days
after its birth. Instead of the usual integuments, muscles, &c.
a membranous bag appeared to protrude on the upper and
fore part of the abdomen, extending from the last bone of the
sternum some way below the middle of the belly, and out-
wards, so as to be nearly circular : the navel-string seemed to
enter this membrane near its middle, and to wind superficially,
for some little way, towards the left side ; it then dipped into
the abdomen, at the place where this membrane joined the
usual coverings. Within this bag, the appearance of which
was very nearly similar to that of the chorion and amnios
which envelop the foetus at birth, but thicker in consistence,
a tumour was perceived, possessing considerable motion, from
the nature of which, no doubt was entertained that it was the
heart.

During the short period of the child’s life, it was seen and
examined by a number of professional men. Upon its death,
the tumour was carefully opened by Mr. Morell, in the pre-

349

very unusual Formation of the human Heart.

sence of Dr. Poignand; when the heart, as was previously
suspected, appeared to be situated in the epigastric region of
the abdomen, and to be imbedded, as it were, in a cavity
formed on the superior surface of the liver. In this state, the
child was sent to Dr. Baillie, by whose desire I injected the
heart, and laid its principal vessels bare, so as to bring their
uncommon distribution and course into view : a preparation of
them still remains in Dr. Baillie's possession.

A considerable part of the tendinous portion of the dia-
phragm appeared to be wanting, as likewise the lower part of
the pericardium, which is usually affixed to it. The thorax
being laid open on each side of the sternum, the two pleurae
were seen passing from that bone to the spine, and covering
the lungs, as usual. The lungs appeared perfectly natural in
colour, and nearly so in shape ; but were larger and fuller than
usual, in consequence of more room being afforded for them in
the thorax, from the peculiar situation of the heart. In the
space corresponding to the anterior mediastinum, was the thy-
mus gland, considerably longer than in other children, and
extending downwards the whole length of the sternum ; be-
hind this, was a peculiar arrangement of blood-vessels.

The heart, instead of consisting of four cavities, as in the
natural structure, consisted of a single auricle and ventricle,
which were each of them large in their size. A large arterial
trunk arose from the ventricle, and ascended into the thorax,
between the pleurae, immediately behind the thymus gland : it
soon divided into two large branches, one of which continued
to ascend, forming the aorta ; the other passed backwards, and
proved, upon examination, to be the pulmonary artery.

The aorta, having reached the common place of its curva-

350 Mr. Wilson's Description of a

ture, formed it in the same manner as it usually does ; sent
off the vessels belonging to the head and upper extremities;
descended before the vertebras, and passed into the abdomen
between the crura of the diaphragm. From the place where it
began to form the arch, it was in no respect different from the
aorta of any other infant, except that no bronchial artery was
sent to the lungs, from it or any of its ramifications.

The vessel which proved to be the pulmonary artery, almost
immediately divided into two branches ; one going to the lungs
of the left, the other to the lungs of the right side. Upon mea-
suring accurately the circumference of the aorta, where it se-
parated from the original trunk, it was found to be exactly one
inch and a quarter. Upon measuring the circumference of the
pulmonary artery, in the same manner, it was found to be
fifteen sixteenths of an inch ; so that it was five sixteenths of
an inch less than the aorta.

The vena cava inferior, having been partly surrounded by
the substance of the liver, entered the lower and back part of
the auricle. The subclavian vein of the right side crossed over
to the left of the mediastinum, where it joined the left subcla-
vian, and formed the vena cava superior. This passed down on
the left of the ascending, and before the descending, part of
the aorta ; it was then joined by a trunk formed by two large
veins, which came out of the lungs, and which were situated
immediately behind the pulmonary arteries : the union of this
trunk with the vena cava superior was continued into a large
vessel, which gradually expanded itself into the auricle. The
vena azygos ascended on the left side ; received some branches
which passed under the aorta from the right, and then entered
the upper and back part of the vena cava superior : there were

very unusual Formation of the human Heart. 351

no bronchial veins. From there being neither bronchial arteries
nor veins, it would appear that the pulmonary arteries and
veins, in addition to their usual offices, performed those of the
bronchial vessels.

The liver was not divided on its upper surface by the sus-
pensory ligament, but had a considerable cavity scooped, as it
were, out of its substance 5 which, in shape, was adapted to,
and contained, the heart : it was also, in some other particulars,
rather different from its natural shape, but not sufficiently so
to require being minutely described. The rest of the infant
was examined, but was not found to be dissimilar to any other.
These circumstances are expressed by the accompanying fi-
gures of the parts when dissected ; (Tab. XVIII.) in taking of
which, much attention was paid to render them very accurate.

It is a well ascertained fact, that the blood receives a florid
hue from the influence of the air on it in the lungs ; and this
change is supposed to be effected by the combination of a cer-
tain quantity of oxygen gas with it. In passing from the ar-
teries to the veins, in every part of the body except the lungs,
it loses the florid hue, and becomes darker : the florid blood is
that which is employed for the purposes of supporting life. In
the natural circulation, it is well known, that the whole of the
blood conveyed to, and circulating in, the pulmonary artery, is
of a dark colour ; and the whole of it, when returned by the
pulmonary veins, is florid.

It is obvious, in the case which I have described, that there
always must have been florid and dark-coloured blood mixed,
and circulating in the arteries. It would seem also, upon the
first reflection, that the quantity of dark-coloured blood would
be the greatest, in the same proportion as the capacity of the

35s Mr. Wilson’s Description of a

aorta was larger than that of the pulmonary artery. It is there-
fore necessary to recollect, that a considerable proportion of the
blood carried to the lungs was already florid or oxygenated;
and also, that the lungs in this infant were larger in propor-
tion, than in children of the same age : a smaller quantity of
blood, therefore, was to be oxygenated, and a larger surface
than usual was appropriated for this purpose. It appears also,
from experiments, (such as making a person breathe air iri
which there is a greater proportion of oxygen gas than in our
atmosphere,) that the blood can combine with more of it than
it does in natural respiration ; it therefore is not an improbable
supposition, that a larger quantity was combined here. A
small drawback must be allowed, for the quantity of oxyge-
nated blood used in the support and secretions of the lungs,
and which is usually conveyed to them by the bronchial artery;
but this quantity is too small to require more than this slight
observation of it. The blood also which passed to the lungs,
must have been again conveyed to the heart sooner, from the
shortness of its circuit ; and must have entered the heart with
a quicker or stronger current, than that blood which passed to,
and was returned from, the more remote parts of the body ;
as, in this child, the pulmonary artery and aorta were filled by
the contraction of the same ventricle. In the hearts of other
children, some time after birth, the muscular fibres of the right
side are much fewer in number than in the left.

If these circumstances are admitted as fact, viz. that the
blood circulating through the lungs of this child was combined
with a larger proportion of oxygen gas, and was returned in
a quicker and stronger current into the auricle than that re-
turned by the venee cavas, it seems reasonable to infer, that this

very unusual Formation of the human Heart. 353

blood, mixing and blending with the dark or unoxygenated
blood, would render the whole nearly as much oxygenated as
it usually is found in the left side of the heart, and in the aorta ;
therefore, that the blood circulating in the arteries of this child
would be fully equal to the support of life. Previous to birth,
this peculiarity of structure could not affect its health or growth,
as the placenta then answers the purpose which the lungs do
afterwards ; and the single ventricle seemed as equal, from its
size, to propel the blood on to the placenta, as both ventricles
in the natural state are, by means of their communication
through the ductus arteriosus *

The inference which has been drawn seems further con-
firmed, from the colour and heat of this child, during life, being
not perceptibly different from those of other children. In all
those cases of malformation of the heart where the foramen
ovale, or the ductus arteriosus, has continued open ; or where
the septum of the ventricles has been perforated, and the pul-
monary artery small, (and at the same time two ventricles,) it
has been observed, that the body had a livid colour, and, in ge-
neral, that there was a deficiency of heat.

From the particular inquiries which I made, concerning
the heat and colour of this child, of the professional gentlemen
who saw it during life, and of the nurse who attended and

* It is here not unworthy of remark, that the circulation in this child, after its
birth, was in several circumstances similar to the circulation in other children previous
to that period. A child, before birth, may be said to have a single heart ; as both the
auricles communicate together, by means of the foramen ovale ; and the pulmonary
artery communicates with the aorta, by means of the ductus arteriosus. Hence, in the
foetal heart, the blood returned from the body, which is of a dark colour, and the blood
returned from the placenta, which is florid, are poured into the same auricle ; the blood
which is sent to the placenta is therefore already in part oxygenated.

Z z

MDCCXCVIII.

354

Mr. Wilson's Description of a

dressed it, I found that the heat, so far as could be judged by
the feeling, (for it was not tried by the thermometer,) was in
no respect different from that of other children ; and that the
colour of the skin was perfectly natural, except that, on the
day on which it was born, and a short period before its death,
the lips occasionally had something of a livid appearance ; but
that this did not last any time, as they were generally pale.
This occasional lividness would happen to a child in that state,
should the heart and circulation be in no way different from
what they naturally are.

I could meet with no other remarkable circumstances, either
in the history of the mother during pregnancy, or in the child
after birth. It cried occasionally, like other children, but seemed
weak, and in pain ; it slept ; it sucked heartily, even a few
hours before its death, and had apparently healthy evacuations
of urine and feces.

Its death can be satisfactorily accounted for, from another
cause than the extraordinary formation of its heart and blood-
vessels. The membranous covering, on the fore part of the ab-
domen, did not appear to possess sufficient vascularity to retain
its life after birth ; for it immediately lost its living principle,
and became putrid and mouldy in parts. Previous to the child's
death, a process of separation had begun, between it and the
living parts to which it was connected, and a line of inflam-
mation was distinctly seen. Had this process been completed,
and the slough thrown off, the heart would have been exposed ;
but, before this, the heart itself had inflamed ; which was proved
from its being found covered with a coat of coagulable lymph
recently thrown out, and from this inflammation its death
must have arisen.

355

very unusual Formation of the human Heart.

Had the heart been covered with the usual parietes of the
abdomen, it is probable, notwithstanding its situation, that this
child might have lived in a tolerable state of health for years ;
but must constantly have been exposed to have its heart injured
by some external accident, from its not being defended by the
ribs and the sternum.

The formation and disposition of the heart and vessels, in
this child, resemble much those which are found in the frog,
and some other amphibious animals ; but this infant could not,
like them, be amphibious. Those animals are extremely tena-
cious of life, so that they live some time, even after their heart
and lungs are removed from their bodies ; and, as their circu-
lation can go on without respiration, it is therefore not won-
derful that they often live a considerable time without change
of air. Life, in the human species, depends equally on both
these actions ; for death takes place, if either of them should
stop. The circulation of the blood in this infant would have
met with no impediment, had it been immersed in water ; but,
unless respiration went on, which in that state it could not do,
the blood could undergo no change in the lungs ; and this
change is equally essential to the support of life, as the circu-
lation of the blood.

EXPLANATION OF THE FIGURES. (Tab. XVIII, )

Fig. i. represents the heart, blood-vessels, liver, &c. as they
appeared when dissected; part of the ribs, the sternum, thymus
gland, lungs, &c. having been removed.

AA. The heart, consisting of one auricle and one ventricle.

B. A large arterial trunk, arising from the ventricle.

C. The aorta, arising from this trunk.

Z Z 2

35 6 Mr. Wilson's Description , &c.

D. The pulmonary artery, arising from the same trunk,

E. The vena cava superior, descending on the left side.

F. F. The pulmonary veins, entering into the auricle with
the vena cava superior.

G. The vena azygos, ascending on the left side.

H. The diaphragm, adhering laterally to the margin of the
chest, but deficient on the fore part, towards the sternum.

II. The liver.

K. The cavity on the upper surface of the liver, in which
the heart was in part situated.

L. The membranous covering turned downwards.

M. The umbilical vein.

Fig. 2. represents the heart. The aorta and pulmonary artery
are cut off' near their origin, to shew the pulmonary veins, and
vena cava superior, entering the auricle.

A. The auricle.

B. The ventricle.

e. The trunk from which the aorta and pulmonary artery
arose.

D. A large vessel entering the auricle, and receiving the
blood from the pulmonary veins and vena cava superior,

E. The trunk formed by the pulmonary veins.

F. The vena cava superior.

G. The vena azygos.

H. The right subclavian vein.

I. The left ditto.

Thilos . Trams . MD C CXCAnnx.JT^.XV lII./?.

|.

r/iilos. /WMJIP c C-XTCATlI.Ta&.XyUI./)^ 356.

C 357 3

XIV. Account of a singular Instance of atmospherical Refraction .
In a Letter from William Latham, Esq. F. R. S. and A. S. to
the Rev. Henry Whitfeld, D. D . F. R. S. and A. S.

Read May 10, 1798.

DEAR SIR, Hastings, August i, 1 797.

On Wednesday last, July 26, about five o'clock in the after-
noon, whilst I was sitting in my dining-room at this place,
which is situated upon the Parade, close to the sea shore, nearly
fronting the south, my attention was excited by a great number
of people running down to the sea side. Upon inquiring the
reason, I was informed that the coast of France was plainly to
be distinguished with the naked eye. I immediately went
down to the shore, and was surprised to find that, even without
the assistance of a telescope, I could very plainly see the cliffs
on the opposite coast ; which, at the nearest part, are between
forty and fifty miles distant, and are not to be discerned, from
that low situation, by the aid of the best glasses. They ap-
peared to be only a few miles off, and seemed to extend for
some leagues along the coast. I pursued my walk along the
shore to the eastward, close to the water's edge, conversing
with the sailors and fishermen upon the subject. They, at first,
could not be persuaded of the reality of the appearance ; but
they soon became so thoroughly convinced, by the cliffs gra-
dually appearing more elevated, and approaching nearer, as it

35$ Mr. Latham’s Account of a

were, that they pointed out, and named to me, the different
places they had been accustomed to visit; such as, the Bay,
the Old Head or Man, the Windmill, &c. at Boulogne ; St
Vallery, and other places on the coast of Picardy ; which they
afterwards confirmed, when they viewed them through their
telescopes. Their observations were, that the places appeared
as near as if they were sailing, at a small distance, into the -
harbours.

Having indulged my curiosity upon the shore for near an
hour, during which the cliffs appeared to be at some times
more bright and near, at others more faint and at a greater
distance, but never out of sight, I went upon the eastern cliff
or hill, which is of a very considerable height, when a most
beautiful scene presented itself to my view ; for I could at once
see Dengeness, Dover cliffs, and the French coast, all along
from Calais, Boulogne, &c. to St. Vallery; and, as some of the
fishermen affirmed, as far to the westward even as Dieppe.
By the telescope, the French fishing-boats were plainly to be
seen at anchor ; and the different colours of the land upon the
heights, together with the buildings, were perfectly discernible.
This curious phenomenon continued in the highest splendour
till past eight o’clock, (although a black cloud totally ob-
scured the face of the sun for some time, ) when it gradually
vanished.

Now, Sir, as I was assured, from every inquiry I could pos-
sibly make, that so remarkable an instance of atmospherical
refraction had never been witnessed by the oldest inhabitant
of Hastings, nor by any of the numerous visitors, (it happen-
ing to be the day of the great annual fair, called Rock-fair,
which always attracts multitudes from the neighbouring places,)

singular Instance of atmospherical Refraction. 359

I thought an account of it, however trifling, would be gratify-
ing to you.

I should observe, the day was extremely hot, as you will per-
ceive by the subjoined rough journal of a small thermometer,
which was kept in the dining-room above mentioned. I had no
barometer with me, but suppose the mercury must have been
high, as that and the three preceding days were remarkably
fine and clear. To the best of my recollection, it was high
water at Hastings about two o'clock P. M. Not a breath of
wind was stirring the whole of the day ; but the small pennons
at the mast-heads of the fishing-boats in the harbour, were, in
the morning, at all points of the compass.

I am, &c.

WILLIAM LATHAM.

P. S. I forgot to mention that I was, a few days afterwards,
at Winchelsea, and at several places along the coast ; where
I was informed, the above phenomenon had been equally visi-
ble. I should also have observed, that when I was upon the
eastern hill, the cape of land called Dengeness, which extends
nearly two miles into the sea, and is about sixteen miles distant
from Hastings, in a right line, appeared as if quite close to it ;
as did the fishing-boats, and other vessels, which were sailing
between the two places: they were likewise magnified to a
great degree.

•360 Mr. Latham’s Account , &c.

State of the Thermometer at Hastings, during the Month of July, 1797.

1797.

Therm. I

Time.

Wind.

Weather.

July 1

64

10 A. M.

sw

Windy. Fair.

2

64

10

sw

Windy. Fair.

3

62

10

sw

Rain. Windy.

4

62

10

sw

Fair. Windy.

5

6l

10

sw

Rain. Windy.

6

60

10

sw

Rain. Windy.

7

6l

10

w

Rain. Windy.

8

62

10

NW

Fine.

66

5P. M.

NW

Fine.

9

66

10 A. M.

SW

Fine.

10

67

10

N afterw. SW

Fine.

11

65

10

SW

Foggy all day. -

12

63

10

sw

Fine.

13

72

10

sw

Fine.

14

76

10

w

Fine.

68

12

w

Fine.

13

72

10

w

Fine.

16

72

10

N

Fine.

78

7P. M.

E

Storm of wind. Lightning.

17

73

10 A.M.

W

Fine. .

18

70

10

W

Fine. Showers in the night.

19

67

10

wsw

Fine. Windy.

20

67

10

sw

Rain. Windy.

21

65

10

sw

Fine. Windy.

22

61

10

S

Rain.

23

65

10

s

Fine.

24

66

10

s

Fine.

25

66

10

sw

Fine.

26

68

10

sw

Fine. Dead calm all day.

76

5P. M.

sw

Fine.

27

72

10 A. M.

sw

Fine.

28

70

10

s

Fine.

29

72

10

E

Fine.

30

70

10

sw

Rain.

31

%

10

s

Fine. Windy.

C 361 3

XV. Account of a Tumour found in the Substance of the human
Placenta . By John Clarke, M. D. Communicated by the
Right Hon. Sir Joseph Banks, Bart. K. B. P. R. S.

Read May 17, 1798.

W hilst the structure and uses of any part of the animal body
remain unknown, every new fact or occurrence ought to be
recorded ; since, by this means only a more perfect knowledge
of it can be expected to be obtained.

As there are few subjects more interesting than those which
concern the functions of animals, and more especially those
processes by which they are originally formed, and afterwards
sustained, I beg leave to submit the following Paper, to the at-
tention of the Royal Society, supposing it not to be foreign to
the general views of the institution.

The exertions of the most patient industry have been hither-
to baffled, in the attempt to detect the first changes which suc-
ceed that process by which animals are propagated.

If the object of immediate pursuit has not been obtained,
much light, in the course of the investigation, has been col-
laterally thrown upon the growth and nutrition of animals, both
in the egg state of oviparous animals, and in the uterine state
of such as are viviparous.

The structure of the egg of oviparous animals serves to eluci-
date the corresponding process in the viviparous ; and, although
in many cases analogies are very inconclusive, yet in this the
mdccxcviii. 3 A

362 Dr. Clarke's Account of a Tumour

resemblance is so close, that the latter may be said to be de-
monstrated by the former.

A certain temperature , nourishment , and the application of
vital air, (or oxygen,) seem to be essential to the evolution of
the young of oviparous animals.

As the young are expelled from the mother, contained in the
cavity of the egg, at a very early period of their existence, and
as afterwards they have no connection whatsoever with her,
these are supplied by various contrivances ; and the mode of
application has been very distinctly explained, by modern in-
quirers into the structure of eggs.

Since then the same substances are to be produced, and sup-
ported, in viviparous as in oviparous animals, the conclusion is
reasonable, that similar means should be employed to attain
similar ends.

It is easy to conceive how warmth may be imparted to a
foetus situated in the uterus.

The materials for nourishment , it receives from the placenta;
but the precise manner in which they are supplied has not yet
been discovered. Of the fact there can be no doubt, because
there are many cases on record, in which there could be no
other possible way by which support could be had.*

With respect to vital air, (or oxygen,) the young of all vivi-
parous animals, whilst in the uterus, live in the same medium
as fishes, and have a structure similar to gills, for the exposure
of their blood to it : this structure is the placenta.

The heart of the foetus is adapted to this mode of life, and
in effect consists but of one auricle and one ventricle, as it is

* A case of this kind I described some time ago, which is published in the Philoso-
phical Transactions for the year 1793;

found in the human Placenta. 3 6$

found to do in fishes. The junction between the two ventricles
is attended with a great advantage, in performing the circula-
tion through the placenta ; where the length and convolution
of the umbilical vessels, in some animals, offer a great resistance
to the force of the heart, and render more exertion necessary.

In the superior aorta, the circulation is carried on by the left
ventricle alone ; as the ductus arteriosus does not join the aorta,
till after the latter has given off the carotid and subclavian
branches.

Vital air is communicated to the blood of the embryo, as it
is to the blood of fishes. This, in its passage through the
gills, is exposed to water, which is allowed by all to contain a
large proportion of vital or oxygen gas, and returns thence
fitted to answer the purposes of life.

In like manner, the blood of the mother, in the cells of the
placenta, having received the essential part of this gas from her
lungs, is applied to the capillary vessels of the umbilical arte-
ries, which receive and transmit it to the embryo ; the life of
which so entirely depends upon this communication, that an
obstruction to the circulation through the placenta, for the space
of two or three minutes, will sometimes irrecoverably destroy it.

The gills of fishes form a permanent part of their bodies ;
because they are designed to pass the whole of their lives in
the same medium. This is not the case in the embryo of vivi-
parous animals ; which, after birth, is to change its situation
for another, in which there is a direct exposure of the blood to
atmospheric air. For this reason, the placenta, whose use is
only temporary, is attached to the foetus by a slender connection,
which is soon dissolved after birth.

I have thought it necessary to introduce the foregoing obser-

3 As

3^4 Dt-. Clarke’s Account of a Tumour

vations upon the structure and functions of the placenta, in
order to shew that the principal use of it is to transmit, and
apply respectively to each other, the blood of the foetus, and
that of its mother. No other action is carried on by the ves-
sels of the foetal portion of the placenta, as far as is yet known,
than what has been described, unless so much as may be ne-
cessary for their own growth and nourishment.

The tumour which gave occasion to this Paper is, however,
an instance to prove, that these vessels are capable, like those
in other parts, of forming solid organized matter ; and that very
considerable deviations from the ordinary structure of the pla-
centa may exist, and be perfectly compatible with the life and
health of the foetus.

Previously to the birth of a healthy child, an amazing quan-
tity of liquor amnii was evacuated, which was by accident re-
ceived in a vessel, and, being afterwards measured, was found
to amount to two gallons, Winchester measure.

When the placenta came away, a hard solid body was found
in its substance. It was preserved by Mr. Mainwaring, under
whose care the case occurred, and was by him obligingly pre-
sented to me.

Fine injection was thrown into the arteries and vein of the
funis umbilicalis : when they were filled, they appeared to be
enlarged thrice beyond their natural size.

The placenta, thus prepared, was subjected to examination.
Its anterior surface was found to be covered with the amnion,
behind which lay the chorion, as usual. Some branches, both
of the arteries and veins, coming from the funis, ramified in the
common manner, forming the foetal portion of the placenta.
Others, of a very large size, not less than a swan’s quill, were

found in the human Placenta. 365

sent to the tumour ; which was situated behind the chorion,
and lay imbedded in the foetal portion of the placenta. The
general form of this tumour was oval, about four inches and
a half long, and three inches broad. The thickness of it was
about three inches. It weighed upwards of seven ounces.

Its shape resembled that of a human kidney; one edge being
nearly uniformly convex, whilst the other, where the vessels
approached it, was a little hollowed.

The general character of the surface of the tumour was con-
vexity; but in some parts of it there were slight indentations,
more particularly in the course of the large vessels.

The whole of the tumour was inclosed in a firm capsule, in
the substance of which the large vessels were contained, nearly
in the same manner as they are found in the dura mater. In
the interstices of the vessels, the capsule did not appear to be
vascular ; at least there were no vessels capable of carrying the
injected matter.

The blood-vessels, branching off from the funis to supply the
tumour, partly went over one side, and partly over the other
side of the tumour ; ramifying as they ran, till, meeting at the
convex edge of the tumour, they anastomosed very freely.
From the large trunks on the surface, small branches were
given off, penetrating into the substance, and supplying the
whole tumour with blood.

Upon making a section through the tumour, in the direction of
its length, the consistence was found to be uniform, firm, and
fleshy, very much resembling, in this respect, the kidney. The
cut surface, upon examination, had somewhat of a mottled ap-
pearance ; some parts being highly vascular, whilst others were
white and uninjected.

*66 Dr. Clarke’s Account of a Tumour

If the mere existence of such a tumour is not to be considered
as a disease, there was no appearance of any morbid tendency
in any part of it. The whole structure seemed to consist of
a regularly organized matter throughout, supplied with vessels
exclusively belonging to itself, and not passing to it from the
surrounding parts, as is generally the case in diseased masses.

They who are inclined to consider every new appearance in
the structure of parts as disease, may be disposed to include
this under that appellation.

But disease consists of such an alteration in the structure,
or functions, of a part, as occasions the natural operations of
it to be imperfectly performed, or entirely arrested. This tu-
mour appears to have produced no such effect : all the com-
mon and known functions of the placenta were performed,
notwithstanding the existence of this substance : the child
had been as well nourished, and the benefits arising from the
application of vital air or oxygen, to its blood, just as well sup-
plied, as if the tumour had not existed.

It cannot be said of this, as it might of some tumours, that it
would in time have shewn marks of a morbid tendency, so as
to have deranged the common actions of the placenta; because,
when gestation terminates, the life, and all the uses of the pla-
centa, are at an end.

I am disposed, therefore, to consider this fleshy substance,
as a solitary instance of a formative property in the vessels of
the placenta ; which they have not been hitherto generally
known to possess.*

* The placenta sometimes becomes converted into a mass of hydatids, connected to
each other by small filaments ; but this must be considered as a disease, inasmuch as
the natural structure is destroyed, and it directly interferes with the offices of the pla~

found in the human Placenta. 367

There was a remarkable circumstance attending this case,
which ought not to be lost sight of, viz. the extraordinary quan-
tity of liquor amnii, which had been contained in the ovum.
What connection there was between this and the tumour, can-
not be absolutely explained from a single instance, as there did
not seem to be any direct communication between the tumour
and the cavity of the amnion. The whole of it lay, as has been
before related, behind the chorion ; so that, between it and the
cavity of the ovum, there were two membranes interposed. In
its organization, it had all the appearance of a glandular part,
and was extremely vascular ; but, upon a very attentive exa-
mination of it, no duct could be found leading from it into the
cavity of the ovum.

Yet, although it may appear difficult to prove, that the quan-
tity of liquor amnii depended upon this substance, still, as it
so considerably exceeded that which is found in common, or
has ever been described, it is reasonably to be believed that it
did so.

The manner, however, by which the secreted fluid was con-
veyed from the tumour into the general cavity of the ovum,
must still remain unaccounted for.

centa, which no longer performs perfectly the functions for which it was designed.
Nourishment and vital air are no longer supplied properly to the foetus, which there-
fore commonly dies.

368

Dr. Clarke's Account , &c.

EXPLANATION OF THE PLATES.

(Tab. XIX.)

A view of the foetal surface of the placenta, with the arteries
and vein injected.

a. The funis umbilicalis ; the vein is exhibited to shew its
increased size.

bbb. The shaggy vessels of the chorion, forming the foetal
portion of the placenta.

c. The cavity, or cyst, in which the tumour (vide Tab. XX.)
lay.

(Tab. XX.)

An external view of the tumour, contained in the cyst formed
in the substance of the placenta, as seen in Tab. XIX.

a . A branch of the umbilical artery entering the tumour.

b. The vein returning the blood from the tumour to the um-
bilical vein.

c. A square portion of the capsule which contained the tu-
mour, turned down to shew the internal structure.

d. The substance of the tumour, seen through the opening
left by dissecting off the capsule.

JCortlavriscL- <Lb£,

$jcijur<s J 'c,

Fhilos.Tmns.'MDC C XCY T VLTab. XIX./? . 56s.

.Philos. Trans. MD C CXC *V"JIL Tab. XX,/;. 36&.

Jt{ ‘{(and, cT l.

C 369 3

XVI. On the Roots of Equations. By James Wood, B.D. Fellow
of St. Johns College , Cambridge. Communicated by the Rev.
Nevil Maskelyne, D. D, F. R. S. and Astronomer Royal .

Road May 17, 1798.

Th e great 'improvements in algebra, which modern writers
have made, are chiefly to be ascribed to Vi eta's discovery, that
“ every equation may have as many roots as it has dimensions."
This principle was at first considered as extending only to po-
sitive roots; and even when it was found that the number might*
in some cases, be made up by negative values of the unknown
quantity, these were rejected as useless. It could not, however,
long escape the penetration of the early writers on this subject,
that in many equations, neither positive nor negative values
could be discovered, which, when substituted for the unknown
quantity, would cause the whole to vanish, or answer the con-
dition of the question. In such cases, the roots were said to
be impossible, without much attention to their nature, or inquiry
whether they admit of any algebraical representation or not.
As far as the actual solution of equations was carried, viz. in
cubics and biquadratics, the imaginary roots were found to be

of this form, a + y- F ; and subsequent writers, from this
imperfect induction, concluded in general, that every equation
has as many rcots, of the form V as it has dimen-
sions. In the present state of the science, this proposition is

MDCCXCVIII. o B

37o Mr. Wood on the

of considerable importance, and its truth ought to be established
on surer grounds. The various transformations of equations,
the dimensions to which they rise in their reduction, and the
circumstances which attend their actual solution, are most
easily explained, and most clearly understood, by the help of
this principle. Mr. Euler appears to have been the first wri-
ter who undertook to give a general proof of the proposition ;
but, whatever may be thought of his reasoning in other re-
spects, as he carries it no further than to an equation of four
dimensions, and it does not appear capable of being easily ap-
plied in other cases, it gives us no insight into the subject. Dr.
Waring’s observations upon the proposition are extremely
concise ; * and, to common readers, it will still be a matter of
doubt, whether a quantity of any description whatever will, when
substituted for x in the expression xs — px1 + qx6 — . . . . + w,
cause the whole to vanish.

In the investigation of the proof here offered, it became ne-
cessary to attend to the method of finding the common measure
of two algebraical expressions ; and to observe particularly, in
what manner new values of the indeterminate quantities are
introduced; and how they may again be rejected. It ap-
pears, that these values are necessary in the division; and,
when they have been thus introduced, they enter every term
of the second remainder, from which they may be discarded.
This circumstance enables us, not only to determine the na-
ture of the roots of every equation, but also affords us a direct
and easy method of reducing any number of equations to one,
and obtaining the final equation in its lowest terms.

• Meditationes Alg. p. 272.

Roots of Equations .

S7*

PROP. I.

To find a common measure of the quantities axn - f- bxn~"
4- cxn~2 -f and Ai”-1 -f- ¥>xn~ 3 -f C.r””3

4~ D

In order to avoid fractions, multiply every term of the divi-
dend by A*, the square of the coefficient of the first term of
the divisor, and the operation will be as follows :

xn~3 + &c 2 ) a ^ Azx» — i -f- c A %xn — 2 -j. d Azxn— 3 -f &c. [a Ax -f- bA — tf B
aA'zxn-\-aBAxn — i-\-aCAxn — 2_j.aDA.27! — 34- &c.

[bA1 — flBA).r»— i-f (cA* — aCA)xn— 2_j_ (^Aa — flDA)2n-3+^.
[bA1— «BA )x»— 1 -f ( 6BA — flBa)x«— 2 -f- ( bCA — «BC)x«— 3-J-^c.

(P) (c A1 — iBA + aB2, — aCA) xn— 24-

(JA2, — 6CA-j-/zBC — aDA)x« — 3+ &c‘

Let (r A — 6B) A -j- (B*— CA ) a = a
(d'A— 6C) A + (BC — DA) a = 0
(eA — 6D) A + (BD — EA) a = y &c.
and the first remainder (P) is uxn~2 + (3 xn~3 + y x*~± -f &c.
proceed with this as a new divisor, and the next remainder ( Q)

will be (C « — B/3. a -|- /32 — ay. A) xn~~ 3 + (Da — - By .

& -|- /3y — a£ . A) a)” ^ -j- &c.

Respecting this operation we may observe :

1. That were not every term of the first dividend multiplied
by A2, that quantity would be introduced by reducing the
terms of the remainder (P) to a common denominator.

2. When P = o, Ax71-1 + Bxn~2 -\-Cxn~z + &c. is a di-
visor of A2 { axn + bxn~x + cxn~ 2 + &c.) ; and therefore it is
a divisor of axn-\- bxn~x + cxn~2 + &c. unless it be a divisor

SB 8

■ S72

Mr. Wood on the

of A*,, which is impossible ; consequently no alteration is, in
this case, made in the conclusion, by the introduction of A\

3- When P does not vanish, then every divisor of P is a
divisor of A**-1 + Bxn~2 -f Cxn~z + &c. and of Aa (ax*
+ bxn~ 1 -{- cxn~' 2 -f &c.)\ and therefore of axn -{- bxn~x
+ c xn 2 -j- &c. unless A2, = 0, in which case the remainder,
P, becomes a B (B^-2 -\-Cxn~3 + &c.)3 every divisor of which
is a divisor of Bxn~2 ~f C^-3 &c. whether it be a. divisor

of a xn -f bx -f- cxn~2 &c. or not. That is, there are two
values of the indeterminate quantity A, which, if retained, will
produce erroneous conclusions.

4. Az enters every term of the second remainder (Q), and
the two values, before introduced, may therefore be again rejected.

The coefficient of the first term of this remainder is C a — B jS"

to + — ay. A; and, by substituting for a, (2 and y, their va-

lues, and retaining only those terms in which A is not found,
and those in which it is only of one dimension, we have

C*= — &BCA+ aCB*

— a CPA

— B j3 = + 6BCA — aCB‘

+ a BDA

Co, — B/3 ^T—aC'k + a BDA
CT^BQ . « = — a* B2 CPA + az B3 DA.
p —ocy . A= a* B2 CPA.— a2 B3 DA;
therefore, those parts ofCa-— Bj0.«-f (2 z—uy. A, in which A
is of one dimension, and in which it is not found, vanish.

In the same manner it appears, that Aa enters every other
term of the remainder Q.

373

Roots of Equations.

5. If the remainder Q =s 0, then, by the second observation,
the introduction of u in the division, produces no error in the
conclusion ; and, if Q do not vanish, a will be found in every
term of the third remainder, and may there be rejected ; and
so on. Thus we obtain the conclusion, without any unneces-
sary values of A, B, C, &c, or a> b, c, &c.

6. If the highest indices of x, in the original quantities, be
equal, it will only be necessary to multiply the terms of the
dividend by A, which may be rejected after the second division.
If the difference of the highest indices of x be m , the terms of
the dividend must be multiplied by A**1, the first quotient be-
ing carried to m + 1 terms. This quantity, Am+J, will enter
every factor in. each term of the second remainder.

7. If it be necessary to continue the division, let

Ca — B/3. « -}- — ay ,A=?wA'

Da — By. a /j y ■— a L A = 71 A*

Ea — BL a -J- (3 $ — a e . A = A '

&c.

and the third remainder is (ym — + n 1 — mp . a) xn~ i

~\-($m<—l3p.m- f np — mq. a)xn~s-\ -{em — ftq.m^nq — mr.u)
xn — ® + &c. every term of which is divisible by oc. The law
of continuation is manifest.

PROP. 11.

Two roots of an equation of 2 m dimensions may be found

by the solution of an equation of m . zm — 1 dimensions.

Let xzm + px™~x + qx2m~2 -f- rx2m~ 3 + &c. = o ; and, if
possible, let v and % be so assumed that v + z, and — v + z,
may be two roots of this equation ; then,

374

Mr. Wood on the

xLnt — vtm ± 2 m z vt’n ~ 1 -f 2 m. — — -l . z1 vlm ~ % ± 2m. — - . 2CT~~3. - 3_|_ ^>Ci

/> x

•im <— 1

2 3

2?« — 2

1 + 2m— i . pzv*m-z ±2m—i . 52LJ .pzzvzn~z+ tsSc. 1.

qx™1— 1 rz

r x

%m — 3

*+■

and consequently,

2m— 2 . 2'zwa'75~3-j- Ssfc.

r v%m ■* 3 4 J

t/zm+ 2?n . — — . z2 1 t>am~ * 4 ^?c. -j- 2S ':

4 2?H — I . p z

+ ? J

— 0

+ p z*’n - 1
4 q zZm — 1 )”
+ tzL,n'-%

-j- ere.

J

and also,

2mz ~i v

+ Pi

lm ~ 1 4 2 m

-}- 2m — 1.

Sffl — 1 2W — S

2 3

'im — 2

pz~

V"m~3 4 &Ct z?nzZm~t

>

W $»

+

+

zm — 2 . qz
rJ

-f 2m — 1 . pz*m~'1,
-\-2m— 2 . qzlm~ 3 ^

42m— 3. rzzm~A'
+ &c.

Assume y = vz; and let the coefficients of the terms of the for-
mer equation be 1, b, c , d, &c. and of the latter. A, B, C, D,
&c. and the equations become

ym + by m~1 -f- cym~"* dym—3 -f- &c. = o

A ym_1 -f Bym~ 2 -f- C ym-z -f &c. = o.

These equations have a common measure of the form y=t=Z,
where Z is expressed in terms of % and known quantities ; and
this common measure may be found, by dividing, as in Prop. r.
till y is exterminated, and making the last remainder equal to
nothing.

Now, the first remainder is (c A — 6B . A -f Bs — CA) ym~*
+ (<f A — 6C . A+BC—DA) s + (TA^d . A + BD - EA)
ym— * -j- &c.; or, by substitution, aym-2~\- fiym-3 -f yy72~'* +

/

Roots of Equations . 375

i

and, in «, 2 rises to 6 dimensions ; in jS, to 8 dimensions ; and,
in 7, to 10 dimensions, &c.

The second remainder is (Ca — B/3 . a 4- (3* — uy.A.)ym-*
4- (Da — By . a [3y — A) yw-4 + &c or, by substitu-
tion, m A* ym~s -f- n AayOT— 4 -j- and, dividing by A2, the di-

mensions of % in m, are 15; in n, 17, Let tt, k, <r, r,
be the dimensions of z in the first term of the 1st, 2d, 3d, 4th,
5th, &c. remainders ; then

7 r = 6

x = 15

(> = 2K 7T -j- 4

O' — 2(0 — JC -}- 4

T = 2(T £ -f- 4

&c.

the increment of the m — 1 term of this series is 4 m 4- 1, and

therefore the m — 1 term itself is zm . m — 1 + m, or m .2 m — 1.
Now, in the m — 1 remainder, y does not appear, and, in that re-
mainder, z rises to m . 2 m — 1 dimensions ; if then, this remainder
be made equal to nothing, and a value of % determined, the last
divisor, y =■= Z, where Z is some function of z, is known ; and
this is a common measure of the two equations ym -j- by™—- 1
4- cym~z 4- &c. = o, and A ym~l 4- B yw~z -J- Cym~3 4. &c. = o ;

consequently, y ^ Z = o ; and y = =a= Z ; hence =±= V~y, or

!?, = =*=:%/ ±Z; therefore, by the solution of an equation of

m .9Lm — 1 dimensions, two roots, z =±= y^z, of the original
equation, are discovered.

Cor. 1. Since two roots of the proposed equation are z -\-v}
and z — v, xz — 2 zx zz — v* = o is a quadratic factor of

that equation.

Cor. 2. In the same manner that the two equations

Mr . Wood otz the

37s

ym + 6 y®*-1 + + &?£• — °y and Ay”2—1 + B y”*-8 -f C ym~ a -f*

6?r. — o, are reduced to one, may any two equations be re-
duced to one, and one of the unknown quantities exterminated;
also, the conclusion will be obtained in the lowest terms.

Prop. iii.

Every equation has as many roots, of the form a v/ =t= 6a,

as it has dimensions.

Case 1. Every equation of an odd number of dimensions
has, at least, one possible root ; and it may, therefore, be de-
pressed to an equation of an even number of dimensions.

Case 2. If the equation be of 2 m dimensions, and m be

an odd number, then m . 2 m — 1 is an odd number, and con-
sequently % and v* (see Prop. 11.) have possible values; there-
fore the proposed equation has a quadratic factor, jf— - 2zx
-j- z*-~ v'= o, whose coefficients are possible; that is, it has
two roots of the specified form ; and it may be reduced two
dimensions lower.

Case 3. If m be evenly odd, or ^ an odd number, then the
equation for determining 2, has either two possible roots, or
two of the form a =*= b s/ — 1, (Case 2.); and if will be of the
form r =fc= d s/ — 1 ; hence, one value of the quadratic factor
xfi— 2 % x -f- »*— if =s o, will be of this form, x*— 2a-\-zb\/ —i.x
-{- A B -j- CD v/ — 1 = o ; and another of this form,

a:1 — 2 a -T- zbs/^i. x -f AB — CD s/ — 1 = o; consequently,
x* — 4a x3 -j- (2AB -f 40*+ 4b*) x* — (4#AB -{-4b CD) x -j-AaBa
-f- C1Da=o, will be a factor of the proposed equation; and
this biquadratic may be resolved into two quadratics, whose

Roots of Equations . 377

coefficients are possible, and whose roots are therefore of the
form specified in the proposition.

In the same manner the proposition may be proved, when

y6 &c. is an odd number ; and thus it appears that it
is true in all equations. *

Cor. 1. If v *, or y, be positive, the roots of the quadratic
factor x a— qz x -f- z% — v*= o, and therefore two roots of the
proposed equation are possible. If y = o, two roots are equal ;
and if y be negative, two roots are impossible.

Cor. 2. If a possible value of % be determined, and substi-
tuted in b, c, d, &c. the original equation will have as many
pairs of possible roots as there are changes of signs in the equa-
tion ym-j- bym~l -f- cym~~2-j- &c. —o; and as many pairs of
impossible roots as there are continuations of the same sign.

# See Dr. Waking’s Med. Alg.

3c

MDCCXCVIII.

8 378 ]

XVII. General Theorems , chiefly P oris ms, in the higher Geo-
metry. By Henry Brougham, Jun. Esq. Communicated by
Sir Charles Blagden, Knt. F. R. S.

Read May 24, 1798.

The following are a few propositions that have occurred to
me, in the course of a considerable degree of attention which I
have happened to bestow upon that interesting, though diffi-
cult branch of speculative mathematics, the higher geometry.
They are all in some degree connected ; the greater part refer
to the conic hyperbola, as related to a variety of other curves.
Almost the whole are of that kind called porisms, whose nature
and origin is nozv well known ; and, if that mathematician to
whom we owe the first distinct and popular account of this
formerly mysterious, but most interesting subject,* should
chance to peruse these pages, he will find in them additional
proofs of the accuracy which. characterizes his inquiry into the
discovery of this singularly beautiful species of proposition.

Though each of the truths which I have here enunciated is
of a very general and extensive nature, yet they are all disco-
vered by the application of certain principles or properties still
more general; and are thus only cases of propositions still more
extensive. Into a detail of these, I cannot at present enter :
they compose a system of general methods* by which the dis-
covery of propositions is effected with certainty and ease ; and

* See Mr, Playfair’s Paper, in Vol. III. of the Edinburgh Trans.

Mr. Brougham's general Theorems , &c. 379

they are, very probably, in the doctrine of curve lines, what the
ancients appear to have prized so much. in plain geometry ; al-
though, unfortunately, all that remains to us of that treasure, is
the knowledge of its high value. Neither have I added the de-
monstrations, which are all purely geometrical, granting the
methods of tangents and quadratures : I have given an example, .
in the abridged synthesis, of what I consider as one of the most
intricate. It is unnecessary to apologize any farther for the
conciseness of this tract. Let it be remembered, that were
each proposition followed by its analysis and composition, and
the corollaries, scholia, limitations, and problems, immediately
suggested by it, without any trouble on the reader’s part, the
whole would form a large volume, in the style of the ancient
geometers; containing the investigation of a series of con-
nected truths, in one branch of the mathematics, all arising
from varying the combinations of certain data enumerated in
a general enunciation. *

As a collection of curious general truths, of a nature, so far
as I know, hitherto quite unknown, I am persuaded this Paper,
with all its defects, may not be unacceptable to those who feel
pleasure in contemplating the varied and beautiful relations
between abstract quantities, the wonderful and extensive ana-
logies which every step of our progress in the higher parts of
geometry opens to our view.

prop. 1. porism. (Tab. XXI. fig. 1.)

%

A conic hyperbola being given, a point may be found, such,
that every straight line drawn from it to the curve, shall cut, in

* See the celebrated account of ancient geometrical works, in the eleventh book of
Pappus Alexan drinus.

3C2

380 Mr. Brougham’s general Theorems

a given ratio, that part of a straight line passing through a
given point which is intercepted between a point in the curve
not given, but which may be found, and the ordinate to the
point where the first mentioned line meets the curve.

Let X be the point to be found, NA the line passing through
the given point N, and M any point whatever in the curve ;
join XM, ancf draw the ordinate MP; then AC is to CP in a
given ratio.

Corollary. This property suggests a very simple and accurate
method of describing a conic hyperbola, and then finding its
centre, asymptotes, and axes ; or, any of these being given, of
finding the curve, and the remaining parts.

PROP. II. PORISM.

A conic hyperbola being given, a point may be found, such,
that if from it there be drawn straight lines to all the intersec-
tions of the given curve, with an infinite number of parabolas,
or hyperbolas, of any given order whatever, lying between
straight lines, of which one passes through a given point, and
the other may he found, the straight lines so drawn, from the
point found, shall be tangents to the parabolas, or hyperbolas.

This is in fact two propositions ; there being a construction
for the case of parabolas, and another for that of hyperbolas.

PROP. in. PORISM.

If, through any point whatever of a given ellipse, a straight
line be drawn parallel to the conjugate axis, and cutting the
ellipse in another point ; and if at the first point a perpendi-
cular be drawn to the parallel, a point may be found, such, that
if from it there be drawn straight lines, to the innumerable

in the higher Geometry. 381

intersections of the ellipse with all the parabolas of orders not
given, but which may be found, lying between the lines drawn
at right angles to each other, the lines so drawn from the point
found, shall be normals to the parabolas at their intersections
with the ellipse.

PROP. IV. PORISM.

A conic hyperbola being given, if through any point thereof
a straight line be drawn parallel to the transverse axis, (and
cutting the opposite hyperbolas,) a point may be found, such,
that if from it there be drawn straight lines, to the innumerable
intersections of the given curve with all the hyperbolas of
orders to be found, lying between straight lines which may be
found, the straight lines so drawn shall be normals to the hy-
perbolas at the points of section.

Scholium. The two last propositions give an instance of the
many curious and elegant analogies between the hyperbola and
ellipse ; failing, however, when by equating the axes we change
the ellipse into a circle.

PROP. V. LOCAL THEOREM. Fig. 2.

If, from a given point A, a straight line DE moves parallel
to itself, and another CS, from a given point C, moves along
with it round C ; and a point I moves along AB, from H, the
middle point of AB, with a velocity equal to half the velocity
of DE; then, if AP be always taken a third proportional to
AS and BC, and through P, with asymptotes D'E' and AB, a
conic hyperbola be described ; also, focus I and axis AB, a
conic parabola be described, the radius vector from C to M, the

382 Mr. Brougham’s general Theorems

intersection of the two curves, moving round C, shall describe
a given ellipse.

PROP. VI. THEOREM.

A common logarithmic being given, and a point without it,
a parabola, hyperbola, and ellipse, may be described, which shall
intersect the logarithmic and each other in the same points ;
the parabola shall cut the logarithmic orthogonally ; and, if
straight lines be drawn from the given point to the common
intersections of the four curves, these lines shall be normals to
the logarithmic.

PROP. VII. PORISM.

Two points in a circle being given, (but not in one diameter,;)
another circle may be described, such, that if from any point
thereof to the given points straight lines be drawn, and a line
touching the given circle, the tangent shall be a mean pro-
portional between the lines so inflected.

Or, more generally, the square of the tangent shall have a
given ratio to the rectangle under the inflected lines,

prop. viii. poRism. Fig. 3.

Two straight lines AB, AP, (not parallel,) being given in
position, a conic parabola MN may be found, such, that if, from
any point thereof M, a perpendicular MP be drawn to the one
of the given lines nearest the curve, and this perpendicular be
produced till it meets the other line in B, and if from B a line
be drawn to a given point C, MP shall have to PB together
with CB, a given ratio.

in the higher Geometry. 383

Scholium. This is a case of a most general enunciation, which
gives rise to an infinite variety of the most curious porisms.

prop. ix. porism. Fig. 4.

A conic hyperbola being given, a point may be found, from
which if straight lines be drawn to the intersections of the
given curve with innumerable parabolas (or hyperbolas) of any
given order whatever, lying between perpendiculars which meet
in a given point, the lines so drawn shall cut, in a given ratio,
all the areas of the parabolas (or hyperbolas) contained by the
peripheries and co-ordinates to points thereof, found by the in-
numerable intersections of another conic hyperbola, which may
be found.

This comprehends, evidently, two propositions ; one for the
case of parabolas, the other for that of hyperbolas. In the for-
mer it is thus expressed with a figure.

Let EM be the given hyperbola ; BA, AC, the perpendiculars
meeting in a given point A : a point X may be found, such,
that if XM be drawn to any intersection M of EM with
any parabola AMN, of any given order whatever, and lying
between AB and AC, XM shall cut, in a given ratio, the area
AMNP, contained by AMN and AP, PN, co-ordinates to the
conic hyperbola FN, which is to be found; thus, the area ARM
shall be to the area RMNP in a given ratio.

prop. x. PORISM.

A conic hyperbola being given, a point may be found, such,
that if from it there be drawn straight lines, to the innumerable
intersections of the given curve with all the straight lines drawn
through a given point in one of the given asymptotes, the

384 Mr. Brougham’s general Theorems

first mentioned lines shall cut, in a given ratio, the areas of all
the triangles whose bases and altitudes are the co-ordinates to
a second conic hyperbola, which may be found, at the points
where it cuts the lines drawn from the given point.

PROP. XI. porism.

A conic hyperbola being given, a straight line may be found,
such, that if another move along it in a given angle, and pass
through the intersections of the curve with all the parabolas,
(or hyperbolas,) of any given order whatever, lying between
straight lines to be found, the moving line shall cut, in a given
ratio, the areas of the curves described, contained by the peri-
pheries and co-ordinates to another conic hyperbola, that may
be found, at the points where this cuts the curves described.

prop. xii. PORISM.

A conic hyperbola being given, a straight line may be found,
along which if another move in a given angle, and pass through
any point whatever of the hyperbola, and il this point of sec-
tion be joined with another that may be found, the moving line
shall cut, in a given ratio, the triangles whose bases and alti-
tudes are the co-ordinates to a conic hyperbola, which may be
found, at the points where it meets the lines drawn from the
point found.

Scholium. I proceed to give one or two examples, wherein areas
are cut in a given ratio, not by straight lines, but by curves.

PROP. xiii. porism. Fig. 5.

A conic hyperbola being given, if through any of its innu-
merable intersections with all the parabolas of any order, lying

in the higher Geometry. 385

between straight lines, whereof one is an asymptote, and the
other may be found, an hyperbola of any order be described,
(except the conic,) from a given origin in the given asymptote
perpendicular to the axis of the parabolas, the hyperbola thus
described shall cut, in a gi^en ratio, an area (of the parabolas)
which may be always found.

If from G (as origin) in AB, one of LM's asymptotes, there
be described an hyperbola IC', of any order whatever, except
the first, and passing through M, a point where LM cuts any
of the parabolas AM, of any order whatever, drawn from A
a point to be found, and lying between AB and AC, an area
ACD may be always found, (that is, for every case of AM and
IC',) which shall be constantly cut by IC', in the given ratio of
M : N ; that is, the area AMN : NMDC : : M : N.

I omit the analysis, which leads to the following construc-
tion and composition.

Construction. Let m -|- n be the order of the parabolas, and p -|- q
that of the hyperbolas. Find <p a fourth proportional to m -f- n,
q — p and m -j- 2 n ; divide GB in A, so that AR : AG : : q :
P + <p ; then find tt a fourth proportional to M -{- N, q> -\-p, and
q — p , and y a fourth proportional to q, AG, and q ■ — p\ and,
lastly, Q a fourth proportional to the parameter* of LM, 7 r and

M. If, with a parameter equal to rect-

angle r . AN, and between the asymptotes AB, AC, a conic
hyperbola be described, it shall cut the parabola in a point, the
co-ordinates to which contain an area that shall be cut by IC'
in the ratio of M : N.

Demonstration. Because AG is divided in R, so that AR : AG

* i. e. The constant rectangle or space to which AP . SM is equal.

MDCCXCVIII. 3 D

386 Mr. Brougham's general Theorems

: : q : p - f- q>, and that <p : m + n : : q~p : m + Q n, AR is

equal to AG x q

p _L m + n x g — p
* ' to -f 2 n

; and, because LM is a conic hyperbola, the

rectangle MS . RS, or MS . AP, or AP . MP + AR is equal to
the parameter, (or constant space,) therefore, this parameter
is equal to AP x MP -\- AG . q

P +

(m + n) ( q — p)
in 2 n

Again, the space ACD is equal to

m n

- of the rectangle

to -f- 2 n o

AC , CD, since AD is a parabola of the order ?n n; but (by

construction) AC . CD is equal to

m -fr- n

of <3

M+N

. T

AN:

therefore, ACD = 0

M + N
M

. T

to -j- 2 n ' M

AN, of which 0 : parameter

of LM : : nv : M, and 7r : M -{- N : : <p p : q — p\ therefore,
0 = Par- ™ x M- + q X (~ x + i,;) also, T-.q:: AG:

q — p ; consequently, ACD = N- multiplied by

(?n -f nxjr_—_p an(j diminished by — x AN x <7‘-AG‘

q-p

m 4- n Q — p , ,\ •

— ~ x + p 1 IS

to -J- 2 n A I

r • Par. LMxM + N

therefore, transposing ===== x

1 & M x q — p

equal to ACD -}- ■— x AN x » an^ Par. LM will be

equal to

ACD + x AN x — ^ ! x

M q~p q~K that is,

TL±!LX^ + P xmTn

in -j- 2 n /

M

M + N

X q — p X ACD -f- q . AN x AG

m + n
in -f- 2 n

X q -p + p

in the higher Geometry. 387

Now, it was before demonstrated, that the parameter of LM

is equal to AP x MP -{- q . AG

P jl (rf..±-'9^.~ £Lm This is therefore

r 1 m -j- 2n

M ACD + ?.ANx AG

equal to — ===== , multiplying both by

^ m 4- n t , ,

ITTTn * * ~ f + P

l^L^^.wehave^x^JxACD + ^.ANxAG

= AP x (mPx(?+^P) + ?-Ag)

From these equals take q . AG x AN, and there remains

JxACD equal to AP x PM x +p)

-f- q ■ AG x AP — AN ; or, dividing by q — f\ M ^ ^ x ACD
= AP x 2-^- 4- -i— x PM 4- -2- x AG x AP - AN. Now,

m + 2n 1 q — p 1 q — p

m + x AP x PM is equal to the area APM; therefore, the area
APM together with ~Tp x AP. PM, and ~~p x AG x AP — AN,
or APM with — x AP. PM q~ x AG x AN — AP,

q — p q — p *

or APM + -3— x AP . PM Lx rect. PT is equal to

' q — p q — p 1 M-j-N

x ACD. Now, IC'is an hyperbola of the order p\q; therefore, its
area is y-- x rect. GH . MH. But q is greater thany>; therefore,

jZTq is negative, and *-xgG^HM is the area MHKC' ; and the

area NTKC' is equal to XGTXTN; therefore, MNTH is

equal to (MHKC—NTKC'), or to JL- x GHTMH-— GT.TN?

From these equals take the common rectangle AT, and there

3 D 2

388 Mr. Brougham’s general Theorems

remains the area MPN, equal to — x AP x MP x PT;

q — p q _ p

which was before demonstrated to be, together with APM,

equal to . ACD. Wherefore MPN, together with APM,

that is, the area AMN is equal to . ACD; consequently,

AMN : ACD :: M : M -j- N; and ( dividendo ) AMN : NMDC
: : M : N. An area has therefore been found, which the hy-
perbola IC' always cuts in a given ratio.

Wherefore, a conic hyperbola being given, &c. Q. E. D .

Scholium. This proposition points out, in a very striking
manner, the connection between all parabolas and hyperbolas,
and their common connection with the conic hyperbola. The
demonstration which I have given is much abridged ; and, to
avoid circumlocution, algebraic symbols, and even ideas, have
been introduced : but, by attending to the several steps, any
one will easily perceive that it may be translated into geome-
trical language, and conducted upon purely geometrical prin-
ciples, if any numbers be substituted for m, n , p, and q ; or
if these letters be made representatives of lines, and if concise-
ness be less rigidly studied.

PROP. XIV. THEOREM.

A common logarithmic being, given, if from a given point,
as origin, a parabola (or hyperbola) of any order whatever be
described, cutting, in a given ratio, a given area of the loga-
rithmic, the point where this curve meets the logarithmic is
always situated in a conic hyperbola, which may be found.

Scholium. This proposition is, properly speaking, neither a
porism, a theorem, nor a problem. It is not a theorem, be-
cause something is left to be found, or, as Pappus expresses

in the higher Geometry. 389

it, there is a deficiency in the hypothesis : neither is it a
porism ; for the theorem, from which the deficiency distin-
guishes it, is pot local.

prop. xv. porism. Fig. 6.

A conic hyperbola being given, two points may be found,
from which if straight lines be inflected, to the innumerable
intersections of the given curve with parabolas or hyperbolas,
of any given order whatever, described between given straight
lines, and if co-ordinates be drawn to the intersections of these
curves with another conic hyperbola, which may be found, the
lines inflected shall always cut off areas that have to one
another a given ratio, from the areas contained by the co-
ordinates.

Let X and Y be the points found ; HD the given hyperbola,
FE the one to be found ; ADC one of the curves lying between
AB and AG, intersecting HD and FE: join XD, YD; then
the area AYD : XDCB in a given ratio.

prop. xvi. porism. Fig. 7.

If, between two straight lines making a right angle, an infi-
nite number of parabolas, of any order whatever, be described,
a conic parabola may be drawn, such, that if tangents be drawn
to it at its intersections with the given curves, these tangents
shall always cut, in a given ratio, the areas contained by the
given curves, the curve found, and the axis of the given curves.

Let AMN be. one of the given parabolas ; DMO the parabola
found, and TM its tangent at M ATM shall have to TMR
a given ratio.

Mr. Brougham's general Theorems

39 o

PROP. XVII. PORISM.

A parabola of any order being given, two straight lines may
be found, between which if innumerable hyperbolas of any
order be described, the areas cut off by the hyperbolas and the
given parabola at their intersections, shall be divided, in a given
ratio, by the tangents to the given curve at the intersections ;
and, conversely, if the hyperbolas be given, a parabola may be
found, &c. &c.

PROP. XVIII. PORISM.

A parabola of any order [m ?i) being given, another of
an order (m zn) may be found, such, that the rectangle
under its ordinate and a given line, shall have always a given
ratio to the area (of the given curve) whose abscissa bears to
that of the curve found a given ratio.

Example. Let m=i, n — i, and let the given ratios be those
of equality ; the proposition is this ; a conic parabola being
given, a semicubic one may be found, such, that the rectangle
under its ordinate and a given line, shall be always equal to
the area of the given conic parabola, at equal abscissae.

Scholium. A similar general proposition may be enunciated
and exemplified, with respect to hyperbolas ; and, as these are
only cases of a proposition applying to all curves whatsoever, I
shall take this opportunity of introducing a very simple, and, I
think, perfectly conclusive demonstration, of the 28th lemma,
Principia, Book I. “ that no oval can be squared .” It is well known,
that the demonstration which Sir Isaac Newton gives of this
lemma, is not a little intricate ; and, whether from this diffi-
culty, or from some real imperfection, or from a very natural

in the higher Geometry. 391

wish not to believe that the most celebrated desideratum in
geometry must for ever remain a desideratum, certain it is, that
many have been inclined to call in question the conclusiveness
of that proof.

Let AMC be any curve whatever, (fig. 8.) and D a given
line ; take in a b a part ap, having to AP a given ratio, and
erect a perpendicular p m, such, that the rectangle p m . D shall
have to the area APM a given ratio ; it is evident that m will
describe a curve amc, which can never cut the axis, unless in a.

Now, because p m is proportional to -, or to APM, p m will
always increase, ad infinitum, if AMC is infinite ; but, if AMC
stops or returns into itself, that is, if it is an oval, pm is a
maximum at b, the point of a b corresponding to B in AB ; con-
sequently, the curve amc stops short, and is irrational. There-
fore pm, its ordinate, has not a finite relation to ap, its abscissa;
But ap has a given ratio to AP; therefore pm has not a finite
relation to AP, and APM has a given ratio to p m ; therefore
it has not a finite relation to AP, that is, APM cannot be
found in finite terms of AP, or is incommensurate with AP ;
wherefore, the curve AMB cannot be squared. Now, AMB is
any oval ; wherefore no oval can be squared. By an argument
of precisely the same kind, it may proved, that the rectification ,
also, of every oval is impossible. Wherefore, &c. Q. E. D.

I shall subjoin three problems, that occurred during the con-
sideration of the foregoing propositions. The first is an example
of the application of the porisms to the solution of problems.
The second gives, besides, a new method of resolving one of
the most celebrated ever proposed, Kepler's problem; and

39 2 Mr. Brougham's general Theorems

the last presents to our view a curve before unknown, (at
least to me,) as possessing the singular property of a constant
tangent.

PROP. XIX. PROBLEM. Fig. 9.

A common logarithmic being given, to describe a conic hy-
perbola, such, that if from its intersection with the given curve
a straight line be drawn to a given point, it shall cut a given
area of the logarithmic in a given ratio. The analysis leads to
this

Construction. Let BME be the logarithmic, G its modula ; AB
the ordinate at its origin A ; let C be the given point ; ANOB
the given area ; M : N the given ratio : draw BQ parallel to AN ;
find D a fourth proportional to M, the rectangle BQ . OQ, and
M-J-N. From AD cut off a part AL, equal to AC together
with twice G ; at L, make LH perpendicular to AD, and, be-
tween the asymptotes AL, HL, with a parameter * twice
(D+^ .AB.G) describe a conic hyperbola: it is the curve
required.

PROP. XX. PROBLEM. Fig. IO.

To draw through the focus of a given ellipse, a straight line
that shall cut the area of the ellipse in a given ratio.

Construction. Let AB be the transverse axis, EF the semi-
conjugate ; E, of consequence, the centre ; C and L the foci.
On AB describe a semicircle. Divide the quadrant AK in the
given ratio in which the area is to be cut, and describe the
cycloid GMR, such, that the ordinate PM may be always a

* Or constant rectangle.

in the higher Geometry. 393

fourth proportional to the arc OO, the rectangle AB x 2 . FE, and
the line CL ; this cycloid shall cut the ellipse in M, so that, if
MC be joined, the area ACM shall be to CMB : : M : N.

Demonstration. Let AP =s x, PM=y, AC = c, AB=#,
and 2 . EF == b ; then, by the nature of the cycloid GMR,

— PM : OQ : : 2 . FE x AB : c L, and QO= AO — AQ= (by const.)

x AK — AQ; also, CL = AB — 2 . AC, since AC = LB.
Therefore. - PM : -0-^* AK— AQ : : AB x 2 . EF: AB— 2 . AC ; or
—~y : x arc . 900— arc .vers. sin. x : : a b : a — 20 ; therefore,

— y ( a — or+^ {zc—a) =ab x ( x arc • 9°° — arc.v.s.x

and, by transposition,

a b x arc . v. s. x -f- y (2/: — a) = x arc . go0. To these
equals, add 2 y (x — sc) = 0, and multiply by — 1 ; then will

ab x arc. v.s.x -f- (2^ — a)y — 2 y (x — c )■

M

M + N

xab x arc. 90“

of which the fourth parts are also equal ; therefore.

a b X arc. v. s. x . (2 x — a) y y , \ ab M

+ — ; — ~ 7 kx - 0 = — x mTn x arc • 9°

4 1 4

Now, because AFB is an ellipse, f

x ax — x\ and

b ./

ax — x ;

T (*— 0

therefore, — x 4. t^-») ts/

4 ' 4

ab M

ax — x

x m~Tm' x arc * 9°°- Multiply both numerator

and denominator of the first and last terms by a; then will

x — c ) = -

b ab . 2 x — a b / " -v ,

— x — x arc. v.s.x -j — x— v ax~x* —

a a. 1 a a 2 '

X

4 a 2 '' a

ab M

~ x m+n x arc ’ 9°°- -^ow> the fluxion of an arc whose versed

a x

which is also the

sine is x and radius—, is equal to „ , ,

2 1 zvax—x7-’

fluxion of the arc whose sine isy/— and radius unity;* wherefore,

* The semi-transver.se is supposed unity, through this demonstration.
MDCCXCVIJI. 3 E

894>

Mr. Brougham’s general Theorems

~ x ^ — x arc . sm. — -f- — - — x v ax — x J ■ — “(•£ — e)
is equal to ~ x x x arc . 90° ; and, by the quadrature

of the circle, ~x arc. sin. s/ — + x V ax — x\ is the

area whose abscissa is x; consequently, the semicircle’s area is
~ x arc . qo°. But the areas of ellipses are, to the corresponding
areas of the circles described on their transverse axes, as the

conjugate to the transverse; therefore ~ x x arc . sin. ^

r~~~ x \S ax — x2j is the area whose abscissa is x, of a
semi-ellipse whose axes are a and b; and, consequently,
—• x — x arc . 90° is the area of the semi-ellipse. Wherefore,
the area APM — (x — c) is equal to of AMFB. But
— [x — c) x AP — AC = x PC) is the triangle

CPM; consequently, APM — CPM, or ACM, is equal to — ^ ^

x AMFB; and ACM : AMFB :: M : M -]- N; or ( dividendo )
ACM : CMFB : : M : N ; and the area of the ellipse is cut in
a given ratio by the line drawn through the focus. Q. E. D.

Of this solution it may be remarked, that it does not assume
as a postulate the description of the cycloid, but gives a simple
construction of that curve, flowing from a curious property,
whereby it is related to a given circle. This cycloid too gives,
by its intersection with the ellipse, the point required, directly,
and not by a subsequent construction, as Sir I. Newton’s does.
I was induced to give the demonstration, from a conviction that
4t is a good instance of the superiority of modern over ancient

in the higher Geometry, 395

analysis ; and in itself, perhaps, no inelegant specimen of alge-
braic demonstration.

PROP. XXI. PROBLEM. Fig. 11.

To find the Curve whose Tangent is always of the same Mag-
nitude.

Analysis. Let MN be the curve required, AB the given axis,
SM a tangent at any point M, and let d be the given magni-
tude ; then, SM . q . = SP . q . + PM ,q. = d7‘; or, yz-\-d-~~
= d*3 and A. — therefore, x == f x \/ dx— yz. In order

to integrate this equation, divide y s/ d* — yr into its two parts,

d*y A — yy
y Vd* — y* and Vd* — y*'

To find the fluent of the former,

d

dzy £y

y Vdz — va y

i +

Vdz —y*

d -f V dz —y2

y Vd

2 y"

d + V dz ~yz

d x fluxion of — Z.

t" — ■=— — : therefore, the fluent of —-L-

d + Vdz-yz y VT~f

is — d x hyp. log. — Z!, and the fluent of the other part.

•yy

V d7.

= is -f- s/ dz — y2; therefore, the fluent of the aggregate

7 V^a - / is h. 1. d.±^d'~-y\ or s/d'-y 2 +

d x h. L

d + Vd* — y4

; a final equation to the curve required.

fi. £. /.

9

$g6 Mr. Brougham's general Theorems , &c.

I shall throw together, in a few corollaries, the most remark-
able things that have occurred to me concerning this curve.

Corollary 1. The subtangent of this curve is s/d*—y\

Corollary 2. In order to draw a tangent to it, from a given*
point without it ; from this point as pole, with radius equal to
d , and the curve’s axis as directrix, describe a conchoid of Nico-
medes : to its intersections with the given curve draw straight
lines from the given point ; these will touch the curve.

Corollary 3. This curve may be described (organically) by
drawing one end of a given flexible line or thread along a
straight line, whilst the other end is urged by a weight towards
the same straight line. It is, consequently, the curve of traction.
to a straight line.

Corollary 4. In order to describe this curve from its equation ;
change the one given above, by transferring the axes of its
co-ordinates : it becomes (y being == P'M and x = AP') y

~ v d7, x* fl- d x h. 1. f may be used with

ease, by changing the hyperbolic into the tabular logarithm.

Thus then, the common logarithmic has its subtangent con-
stant; the conic parabola, its subnormal; the circle, its normal ;
and the curve which I have described in this proposition, its
tangent.

Edinburgh, 1797.

B

A-

R

G-

B

B S

Philo S'. Trans. MD C CXCVJT. 7a/TXXI./j. 3 y6\

O-V

C — —

\

A

r>

mips. Trans. MI) (' ('XCVJU Ihb.'SXl.p. :\g&.

C 397 3

XVIII. Observations of the diurnal Variation of the Magnetic
Needle , in the Island of St. Helena ; with a Continuation of
the Observations at Fort Marlborough, in the Island of Su-
matra. By John Macdonald, Esq. In a Letter to the Right
Hon. Sir Joseph Banks, Bart. K. B. P. R. S.

Read May 24, 1798.

SIR,, Edinburgh, September 30, 1797'.

On my arrival in England, I had the honour of observing to
you, that I had taken some observations of the diurnal varia-
tion of the magnetic needle, in the island of St. Helena. I am
to apologize to you for having, till this period, omitted furnish-
ing you with these, and with a continuation of those formerly
taken in the island of Sumatra. The meridian was laid off’ by
means of an apparatus brought from Bencoolen ; and the re-
quisite allowance made for the alteration of the sun’s declina-
tion during the operation. The meridian-plate remains firmly
set in a pillar of teak-wood, well fixed, for the use of naviga-
tors ; who, by applying a compass-card to it, will find the va-
riation more readily, and correctly, than by amplitude or azi-
muth. A short residence at St. Helena, arising from the sud-
den departure of the fleet to which the ship I was in belonged,
has prevented the observations from being as numerous as I
eould wish. Their agreement, however, indicates that fifty-
eight observations are sufficient for affording such conclusions

»

*#

398 Mr. Macdonald's Observations of the

as philosophy may draw ; and tends to confirm some inferences
stated in a former Paper, containing similar observations taken
in the East Indies. By adding the mean of the morning and
afternoon observations, at St. Helena, and taking the half, the
general variation, in the month of November, appears to
have been if 48' 34/N west: and, by subtracting the medium
diurnal afternoon variation, from the medium diurnal morn-
ing, the vibrating variation proves to be 3' 55". It appears, that
the magnetic needle is stationary from about six o'clock in the
evening till six o'clock in the morning ; when it commences
moving, and the west variation increases, till it amounts to its
maximum, about eight o'clock ; diminishing afterwards, till it
becomes stationary. Here, the same cause seems to operate as
at Bencoolen, with a modification of effect, proportioned to the
relative situations of the southern magnetic poles, and the places
of observation. At the apartments of the Royal Society, this
species of variation is found to increase, from seven o'clock in
the morning till two o'clock in the afternoon. If the variation
is east, in the northern hemisphere in the East Indies, I con-
ceive that the diurnal variation will increase towards the after-
noon, remain some time stationary, and diminish before the
succeeding morning : if the general variation is west, in that
quarter, the reverse may be the case. The quantity of the diur-
nal variation is greater in Britain than at St. Helena, or at Ben-
coolen. This will naturally arise from this country's being
more contiguous to its affecting poles, than those islands si-
tuated near the equator. It were to be wished, that observations
were taken in as many situations as possible, similarly situated
in the opposite hemispheres, on the lines of no variation. A
greater degree of dip might be found, and conclusions might

diurnal Variation of the Magnetic Needle. 399

be deduced, that would tend considerably to illustrate this
curious and interesting subject, as yet involved in conjecture
and uncertainty. I frequently, while at Bencoolen, observed
that the needle did not retain the same level, but was some-
times depressed, and sometimes elevated, six or eight minutes.
I paid little attention to this, ascribing it to a minute alteration
in the position of the point of the socket over the pivot. I ob-
served, sometimes, a similar difference of level in the position
of the needle at St. Helena, without being able to account for it.
It may be possible, that the dip of the needle is subject to a
diurnal variation in its vertical movement. I have perused such
publications as have appeared on magnetism for some time
past : they state no theory of this obscure science, more
rational, or satisfactory, than that left us by the celebrated
Halley.

I have the honour to be, &c.

JOHN MACDONALD.

In the following observations, F. means fair; A. after;
R. rain; N. night; L. lightning; C. cloudy; S. sky; T. thun-
der; m. much; o. overcast; s. serene; W. windy; 1. little;
a, the indefinite article ; E. or W. east or west variation : a
figure, as 7, over the degree, means that the observation was
taken then, and not at the usual hour.

Observations of the diurnal Variation of the Magnetic Needle, in the Island of St. Helena.

i

Weather.

4th Observation,

•

Macdonald's Observations of the

1

• *L> u

£.S .2

T3 is O 'S

2 a- 2 * § rji §

. t: ^ . .2 -a co

0"C ui 0^ O T3 » O W O? u?

Weather.

3d Observation.

s. sunshine
s. ditto
s. ditto
s. ditto

sunsh. and show,
ditto. F.
s. C.

s. sunshine
s. ditto
a 1. C.

C.

s. sunshine

Weather,
ad Observation.

W.

R.

F. W.

R.

ditto
a 1. R.

W.

sunshine W.

R.

s. sunshine
s. ditto

C.

w. .

j

Weather.

1st Observation.

s. sunshine -
C.

R.

ditto

s. sunshine
C.

R. -
R.

C. -
ditto
a 1. W.
s. C.

W. -

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1 O O O 0 O, 0 r^oo 00 00 00 0\ O OO

r^. r^ r^AO r-^NO no no no no so r>. (v.

w

0

ON t>00 OO CN OS CO O 00 ON ON OOOnOnON

* xt- rf-

^ *0 to to 10 to s_/*\ 10 10 10 10 to to to to 10

1 l-< t-H — —

Time.

Various

Hours.

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' ^ < 5„< 2, < ^ t,

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M 00 CO K 0 00 CO totO 00 O\00 ^ 'O
• to 10 ^ t}- r^-_2 "“d"

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s -

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0 N - 0 ONCO o-«-0«NN O ■ 1~

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«C— ►“«>—< — <1— lH-( h-«

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coq\OwhN^io c^cq Os O m n co O O o >-< ’

N CO CO HK M »-( *-j « M 1-4 rm*

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X 0 0

0 & £

Continuation of the Observations of the diurnal Variation of the Magnetic Needle, at Fort Marlborough,

in Sumatra.

diurnal Variation of the Magnetic Needle . 401

3 F

MDCCXCVIII

4 02

Mr. Macdonald's Observations , & c

C 4°3 3

XIX. Cb* the Corundum Stone from Asia. By the Right Hon.

Charles Greville, F. R. S.

Read June 7, 1 798.

Analysi crystallorum, tam ejusdem quam diversae figure, multum lucis scientia.
expectat. Bergman. Opusc. de Terra Gemmarum.

Having contributed to bring into notice the mineral sub-
stance from the East Indies which is generally called Ada-
mantine Spar, I beg leave to lay before the Royal Society the
following account of its history and introduction.

About the year 17 67 or 1768, Mr. William Berry, a very
respectable man, and an eminent engraver of stone, at Edin-
burgh, received from Dr. Anderson, of Madras, a box of crys-
tals, with information of their being the material used by the
natives of India, to polish crystal, and all gems but diamonds.
Mr. Berry found that they cut agate, cornelian, &c. ; but, in his
minute engraving of figures, upon seals, &c. the superior hard-
ness of the diamond appeared preferable; and its dispatch com-
pensated for the price : the crystals were therefore laid aside,
as curiosities. Dr. Black ascertained their being different from
other stones observed in Europe, and their hardness attached
to them the name of Adamantine Spar. My friend Colonel
Cathcart sent me its native name, Corundum, from India,
with some specimens, given to him by Dr. Anderson, in 1784,
which I distributed for analysis.

3 F 2

Mr. Greville on the

4°4

When the native name was obtained, it appeared, from Dn
Woodward’s Catalogue of Foreign Fossils, published about
the year 1719* the same substance had been sent to him
from Madras, by his correspondent Mr. Bulkley.

In his first Catalogue of Foreign Fossils, p. 6. f 17. “ Nella
“ Corivindum is found ifi fields where the rice grows : it is com-
“ monly thrown up by field rats, and used, as we do emery,
“ to polish iron.”

Page 11. x. 13. “ Telia Convindum , Fort St. George, Mr.
“ Bulkley. ’Tis a talky spar, grey, with a cast of green : it
“ is used to polish rubies and diamonds.”

In Dr. Woodward’s additional Catalogue of Foreign Fossils,
published in 1725, p. 6. £*. 10. “ Nella Corivendum is found

“ by digging at the foot or bottom of hills, about 500 miles to the
“ southward of this place. They use it as emery, to clean arms,
“ &c. it serves also to grind rubies, by making it like hard
“ cement, by the help of stick-lac mixt with it. East India. Mr.
“ Bulkley.” — These, with a few others in Woodward’s Ca-
talogues, are the only instances by which any author, prior to
1768, appears to have noticed this substance.

This information being unsatisfactory, and every appearance
of the stone indicating it to be part of a stratum, I wrote re-
peatedly to friends in India, to ascertain, if possible, the situa-
tion of the rock, and, if near the sea, to send a considerable
quantity, as ballast, with a view of applying it to cut and polish
granites, porphyry, and other stones, which the high price
of cutting and polishing excluded from useful or ornamen-
tal work. But my inquiries at Madras were fruitless : by
some I was assured it came from Guzarat. From Bombay I
obtained no satisfactory information. At last, in the year

Corundum Stone from Asia. 405

1793, I obtained a satisfactory account. Sir Charles Oakley
was disposed to oblige me : he was then Governor of Madras ;
and his success is due to the activity and judgment of Mr.
Garrqw.

Mr. Garrow knew how difficult it was to avoid the causes
of my failure, from every Hindoo being occupied by the duty
of his cast ; scarcely thinking on any thing else, and, when-
ever his interest is concerned, being suspicious and reserved.
Mr. Garrow, in the first place, ascertained the cast connected
with Corundum to be the venders of glass ba?igles ; that they
used it in their business, and sold it to all other casts. This
cast of natives, at all times, had free access to every part of
Tippoo's country ; nor, until the districts about Permetty were
ceded to the English, could it be procured in any other way.
Mr. Garrow depended on his personal inspection ; the parti-
culars are contained in the following letter, communicated to
me by Sir Charles Oakley.

Sir Charles Oakley, Bart .

SIR> Tritchinopoly, 10th Nov. 1792.

“ I derived so little satisfaction from the various accounts

given me of the Corundum, from the indifference of the na-
“ tives to every subject in which they are not immediately inte-
“ rested, that I resolved to ascertain the particulars I wished
“ to know, on the spot where the stone is found. The glass-
44 men agreed in one material circumstance, that the place was
44 not far from Permetty : in other particulars they disagreed,
44 apparently with intention to mislead.

44 It is near a fortnight since I dispatched a servant I could

Mr. Greville on the

40 6

“ depend on to Permetty, with one of these people, who, on
“ his arrival there, probably through fear of his cast, said h&
“ knew no farther. My servant persevered, and informed me
“ he had found the place I wished to see.

“ I arrived at Permetty, by the route of Namcul, the 6th ;
44 and, learning that the distance to the spot was about hours
sc or 14 miles, I left Permetty in time to arrive there about
“ sunrise the next morning. At this time no person but my
4 4 servant was present, and, from a continued excavation at
44 different depths, from 6 to 16 feet, in appearance like a wa-
44 ter-course, running in length about a mile and a half east
44 and west, over the brow of a very rising ground, I saw at
44 once the place from which the stone was procured. The
44 prodigious extent that at different times appears to have been
44 dug up, with the few people employed, shews that it has been
44 a business of ages.

44 The ground through which the vein of excavation runs,
44 and of course the mineral, commands one of the finest and
44 most extensive prospects it is possible to conceive. The sur-
44 face of the ground is covered with innumerable fine alabaster
44 stones, and a variety of small shrubs, but not a tree sufficient
44 to shelter my palanquin.

44 There is not the appearance of an habitation within three
44 quarters of a mile. The nearest village is called Condrastra
44 Pollam. In this village are about 30 small thatched houses :
44 among these are 5 families, who, in descent by prescriptive
44 right, are the miners, and dig in the pits. The nearest place
44 of any consequence, in Rennell’s Map, is Caranel, on the
44 south side the Cavery. The distance of the pits from the
44 river is above 4 miles ; but the ground between prevent its

407

Corundum Stone from Asia .

u being seen in a direct line. A fine view of the river is seen
“ near Erode ; which fort, as well as Sankerdroog, are plainly
“ visible with the naked eye, as is also the Coimbitoor country,
“ south and west of the river, to an immense extent.

“ I procured, at Permetty, a cadjan from the Bramin manager
“ to the head man of the Pollam ; which, on my arrival at the
“ pits, I sent to him ; and soon after three of the miners came
“ from the Pollam, with their implements, and families follow-
“ ing with provisions. As they came up, they inquired of my
“ servant how they were to address me, having never seen an
“ European before.

“ I followed them into a pit, in the line of the excavation, above
“14 feet from the ground-level. The instrument they used
“ is a very heavy iron crow, ending in a broad point, with a
“ straight wooden handle, clampt with iron. The soil they cut
“ through is of different colours, but composed chiefly of a
“ gritty granite ; and, at the depth of seven feet, are layers of
“ a substance not unlike dried pitch, which crumbles into small
“ flakes when taken out. With considerable labour, the miners,
“ with the points of their crows, cut out several pieces of
“ the strata, of some pounds weight each ; and, when a consi-
“ derable quantity was broken off, it was carried up and crush-
“ ed to pieces, with great force, by the iron crow. Among
“ these broken lumps, the Corundum stone is found ; but in
“ many of the pieces there was none. The mode of getting it,
“ made it difficult to get any with the stratum adhering to it ;
“ this, however, after several trials I obtained very perfect, and
“ shall forward to Madras, with specimens of the strata at dif-
(C ferent depths. The stone is beyond all comparison heavier
“ than the substance which encrusts it.

Mr. Greville on the

408

“ It appears extraordinary how this stone, so concealed, should
“ under such difficulties have been sought for, and applied to
“ any purpose ; and that the knowledge of the few people who
“ dig for it, and who do so from father to son, is confined en-
“ tirely to the finding the stone. For they told me they knew
“ none of its uses, and that the labour was so hard, and their
“ gain so small, that they would, through choice, rather work
“ in the fields ; that the sale of it from the spot is confined
<c solely to the glass sellers, who vend it over the whole coun-
“ try, and who had, while I was there, above forty Parriar
“ horses, bullocks, &c. ready in the Pollam, to carry it to Tin-
“ nevelly, and the southern countries ; through which track,
“ if the stone is known in Europe, I apprehend it has found its
“ way, by means of the Dutch.

“ The people on the spot declare it is to be got in no other
“ situation or place whatever; and the stone-cutters tell me
“ they can do nothing without it. It' pays no duty, either where
“ dug up or retailed.

“ The colour of the stone is either very light brown or pur-
“ plish, in the proportion of twenty to one of the latter; but in
“ use no preference is given, and they are used equally. To an
“ indifferent person, the most striking circumstance is its great

“ weight.

“ As the spot I have been speaking of now composes a part
“ of the Company’s territories, the most minute information

“ on the subject may be acquired.

« X felt particular satisfaction in having been the first Euro-
<£ pean who was ever at the place ; and I shall be much grati-
“ lied if the account given meets with your approbation.

<£ I shall dispatch a load of the stone, in a day or two, which

Corundum Stone from Asia . 409

«■ l got at the Pollam, with the charge of it. The distance
“ from this place, by Namcul, is 84 miles.

The Charges of 50U). Weight of Corundum.

“ Nine Tritchinopoly measures of the Corundum stone weigh
“ 5olb.

Average and Cost at the Pits.*

p. f. c.

“ lj- Madras fanams per measure - - — 13 40

“ Cooley from thence to Tritchinopoly - — 28 40

“ Ditto from Tritchinopoly - - - 1 13 40

Pagodas - 2 10 40

“ The stone is delivered by measures, and paid for at the
“ Pollam, in the gold fanam.

“ I am, &c.

“ Edward Garrow.

** Nov. 15, 1792.”

This letter contains very interesting topographical observa-
tions on the mine The specimens sent were of one sort, of a
greyish colour, with a shade of green. The entire crystals, which
I selected among the broken ones, were of course few in pro-
portion ; but, with the addition of some distinct crystals, which
Col. Cathcart and Capt. Colin Macauley had sent me, have
heen sufficient to ascertain the structure and form of the crys-
tals, of which an analytical description will close this Paper. I
shall, therefore, now say nothing concerning their form, but

* The above is the prime cost. I have been informed by correspondents, who
purchased some in retail, that it was sold for about six shillings a pound, at Ma-
dras* „

3 G

MDCCXCVIII.

4io

Mr, Greville on the

proceed to give an account of the varieties of Corundum stone,
which I have obtained from India and China.

In the year 178b, Col. Cathcart sent me a small fragment
of a stratified mass from Bengal, with this label ; “ Corundum,
“ much inferior in price to that of the Coast.” It is of a purplish
hue; its fracture like compact sand-stone; and a confused
crystallization appears in all parts of the stone, by fibres of a
whiter colour, from which the light is reflected, as in feld-
spar, &c.

I have since obtained a larger lump of the stone, of the same
texture, but rather paler in its purplish hue. Sir John Mac-
gregor Murray informed me that it is called by the natives
of Bengal, Corone, and used for polishing stones, and for all
the purposes of emery.

Its specific gravity is 3,87b.

Capt. Colin Macauley procured a lump of Corundum from
a sikuldar, (a polisher, this term is most appropriate to polishers
of steel,) in whose family it had been above twenty years em-
ployed, for grinding and polishing stones or gems. The use
to which it had been so long devoted had occasioned grooves
in its surfaces, which facilitated greatly the examination of its
structure. It is about inches long, inches broad, and
above two inches thick. On one of its broad surfaces are two
oval grooves; one of them is four inches long, one broad, and

of an inch deep. On the opposite side is a shorter oval groove,
above 2-j inches long, i~ inch broad, and one inch deep. In
these grooves, the ends of the laminae of the mass reflect
the light, like the crystals. It serves as a specimen of the
simple apparatus of an Indian lapidary. Stones polished in
these grooves would be of the common India polish and form.

Corundum Stone from Asia. 411

en cabochon, which is often called tallow drop, from the French
lapidaries’ term gouite de suif, convex, oval, or circular.
A very small quantity of the Corundum powder would be
required, as the action of the powdered Corundum and gems,
on the lump of Corundum, would, as appears from the depth
of the grooves, wear away from it a supply of powder, for the
operation of polishing. It appears to be part of a larger mass,
is of a purplish colour, and of the same laminated texture as
the crystals of Corundum; it has this peculiarity, there ap-
pear cracks, branching irregularly across the laminae of the
lump, which are filled with homogeneous matter, distinguished
however by the superior purity which might be expected to
arise from the degree of filtration required for its deposition in
the fissures. Some of these cracks, which terminate on the
surface, appear to have the same crystallized arrangement which
characterizes the laminas of Corundum. The cracks not being
in any degree influenced in their direction by the laminae of the
crystallized mass, it is probable they had not been consolidated,
when they cracked ; and, from this specimen, we may expect
to find Corundum cementing masses of stone, by the same pro-
cess of stalactitical cementation by which quartz and calcedony
connect great nodules and masses of siliceous stones.

In this specimen, I consider the veins as pure Corundum,
that is, having the same specific gravity, hardness, and tex-
ture as Corundum crystals ; and I found the whole lump pos-
sessed all the qualities of Corundum, except its specific gravity,
which amounted only to 2,785; and in this property it corre-
sponded nearly with the matrix of the Corundum crystals, or
the vein in which Corundum is before stated to be found ; the
specific gravity of which is 2,768. The texture of the matrix

3 G 2

412

Mr. Greville on the

appears sometimes like adularia, and confusedly crystallized ;
often compact, like cipoline or primitive marble; sometimes
sparry, sometimes granulated, and, on the outside of the vein,
and near fissures, decomposed, and becoming opaque. In all
its states, it scratches glass, but not rock crystal, possibly from
want of adherence of its particles ; and in this it differs from the
substance of the above lump, which cuts glass and rock crystal
with great facility.

This lump, and the matrix of Corundum, appeared to possess
the same properties as Corundum, when examined by the blow-
pipe, with the different fluxes.

The matrix of Corundum having sometimes an appearance
like adularia and feldspar, I ascertained, by Mr. Hatchett's
scales, the specific gravity of adularia to be 2,558, and of feld-
spar 2,555. The Corundum, and the lighter Corundum of the
lump, cut adularia and feldspar; the latter effervesced, and
combined with soda, which the former did not.

It is therefore evident, that the matrix of Corundum, or sub-
stance of the vein, is a distinct substance from adularia and
feldspar, and nearly connected with Corundum.

The matrix or vein contains also a black substance, like
shorl, which, on closer observation, appears to be hornblende.
This substance Mr. Garrow had remarked to have the ap-
pearance of charcoal, and, on that account, he had attributed
the formation of these strata to the agency of fire. Other gen-
tlemen, from the appearance of the matrix of Corundum, have
stated it to be a calcareous vein.

Mr. Garrow observed, that there ran through the strata in
which the Corundum was found, veins of a substance like dried
pitch, apparently on their edge, which separated like a pack of

Corundum Stone f rom Asia. 413

cards. It is a brown micaceous substance, which in drying
foliates, and shews a certain degree of regular arrangement of
the component parts ; in this case, the fragments of the folia
subdivide, with some degree of regularity, into rhombs, whose
angles are 6o° and 120°: it is more smooth, and less flexible,
than pure mica.

These are all the sorts of Corundum which I procured from
India.

I now proceed to the result of my inquiries in China.

I requested Capt. Cumming, in 1786*, at that time command-
ing the Company’s ship Britannia, to take a specimen of Corun-
dum to China, to ascertain its nature, and to obtain specimens,
if possible, adhering to their matrix, and regularly crystallized.
On his arrival at Canton, he collected the information I wished,
with the good sense and zealous desire which he always exerts
for his friends. He ascertained that the stone I inquired for,
was in common use with the stone-cutters ; and he brought me
the stone, in its rude and in its pounded state, taking care to
select the most regularly crystallized pieces, and others adhe-
ring to the rock. A stone-cutter was sawing rock crystal
with a hand-saw, which he also brought to me ; it is a piece
of bamboo, slit, about 3 feet long, and i-J inch broad, thickened
at the handle by a piece of wood, rivetted with two iron pins ;
having a lump of lead tied with a thong of split rattan, steady-
ing an iron pin, on which the end of a twisted iron wire
is fastened, which, being stretched to the handle, is passed
through a hole in the bamboo, with the superabundant wire ;
a wooden peg, being pressed into the hole, keeps the bow
bent, and the wire stretched, and serves to coil the superfluous

Mr. Greville on the

4 H

wire, till, by sawing the crystal, the stretched wire is worn,
and requires to be renewed from the coil. The twisted wire
answers the purpose of a saw, and retains the powder of Co-
rundum and water, which are used in this operation. Dr. Lind
had before brought specimens similar to the above, from China.

From Sir Joseph Banks I obtained Dr. Lind's specimens,
and some in powder, which Mr. Duncan, supercargo in Chi-
na, had sent him, with the Chinese name, Pou-sa. The ma-
trix, being mixed with a red and white sparry substance and
mica, is generally called red granite ; but it appears to me of
the same nature as the matrix of Corundum from India. The
white is more fibrous, and like cyanite ; the red part of it is
compact and opaque ; other parts appear to foliate, and pure
mica is in considerable patches, and generally adheres to the
crystals. This Corundum is of a darker brown, and more
irregular on the surface than the Corundum of the Coast, and
often mixed with black iron ore, * attractable by the magnet.
It is described as the third modification of the Corundum crys-
tal, in the analytical description which follows. The chatoyant
or play of light, on these dark crystals, is very remarkable ;
some are of a bright copper colour ; others exhibit the accident
of reflection of light, which, in a polished state, gives varieties
to the cat’s eye, star-stone, sun-stone, &c. ; which, as yet, are
classed from such accident, without strict attention to their
nature, which is various, and in general has not been ascer-
tained.

* A small group, consisting of three or four octoedral crystals, presents the leasp
common variety of this kind of iron ore; the edges of the octoedra being replaced by
planes which almost cover the triangular ^planes. Rome de Vlsle. Cristallog.
Vol. IV. Plate 4. fig. 69.

Corundum Stone from Asia . 415

These are the circumstances connected with the strata worth
mentioning. The examination of Corundum on which our
present knowledge rests, is nearly that which an Indian mine-
ralogist might derive of the history of feldspar, from a lump of
Aberdeen granite, out of one or two different quarries. He
might ascertain a few modifications of the crystal of feldspar,
its fracture, and matrix ; but he would have no knowledge of
the purest or more beautiful sorts which other quarries produce,
in Scotland, at Baveno, at St. Gothard, and in Auvergne. I there-
fore think it essential to mention, that Corundum, under cir-
cumstances favourable to its crystallization, becomes glassy in
its fracture, and of various colours. I have not only observed,
in crystals of Corundum, specks of a fine ruby colour, but I
have fragments of crystals, in texture and every respect like
the colourless Corundum, of a fine red colour. It is cer-
tain that we obtain from India, Corundum which may pass
for rubies. I have sent to India some of the Corundum with
small ruby specks, which were not sufficiently distinct or large
either for measurement or analysis, in hopes of being enabled
to ascertain correctly the form of Salarn rubies found in Co-
rundum ; in the mean time, I have the Corundum of a fine red
colour. Looking over some polished rubies from India, I se-
lected one which appeared laminated like Corundum, and had
also the chatoyant or play of light on its laminae, which formed
an angle in the stone. The lapidary called it an Oriental
ruby. I altered the form of the cutting, so fortunately, that
the reflected rays formed a perfect star; a phenomenon I
had observed in the sapphire, and expected in Corundum,
but not in the octoedral ruby. The specific gravity of

Mr. Greville on the

416

this stone, being 4,166, confirmed my opinion that it is one of
the Salam rubies, so much esteemed by the natives on the Coast
or Peninsula of India, which are found in the Corundum vein.
The specific gravity of a colourless sapphire, very little less
opaque than Corundum, forming also a perfect star, was 4,000 :
that of a deep blue sapphire, and of a star-stone, 4,035 ; all
which I connect with the Corundum ; the specific gravity of a
distinct crystal of which was 3,950; of a fragment of ruby-
coloured Corundum, 3,959 ; and of a fragment of Corundum
with vitreous lustre, 3,954.

It may be objected to me, that Bergman has stated the va-
riety of specific gravity in gems to be so great, as to leave no
certain rule of judging thereby of the species. He observed,
that the topaz generally prevails in weight, being from 3,460
to 4,560; the ruby from 3,180 to 4,240 ; then the sapphire, from
3,650 to 3,940.* But in the preceding page he had said, “ Ana-
« lysi crystallorum, tarn ejusdem quam diversae figurae, multum
et lucis scientia expectat. Illse quarum antea compositionem ex-
“ plorare licuit, naturali forma per artem privatae erant/ It is
not, therefore, an hypothesis unworthy of examination which
I advance, that gems derived from the rectangled octoedra,
whose specific gravity is above 3,300 to 3,800, will be found
to be diamonds or octoedral rubies ; and these will be easily
distinguished from each other, by their lustre and hardness.
Diamonds, whether red, yellow, blue, or white, being hardest,
though their specific gravity will be less; viz. from 3,356
to 3,471, as I found among different diamonds in my collec-

* De Terra Gemmarum. Bergm. Opusc. Vol. II. p. 104.

Corundum Stone from Asia.

4*7

tion : whereas the octoedral ruby was from 3,571 to 3,625, and
inferior in hardness, not only to the diamond, but to the Co-
rundum ; the specific gravity of which, in its different appear-
ance of form and colour, I found to vary from 3,876 to 4,166;
and I suppose it to be subject to a variation from 3,300 to
4,300: after which, the jargon will come, with a specific gra-
vity of 4,600 ; easily distinguished also, by its crystallization,
from the abovementioned gems. The above specific gravities,
Mr. Hatchett very obligingly assisted me in taking, with his
accurate scales, in the temperature of 6 o°. It will not be un-
derstood that I depend entirely on the specific gravity ; on the
contrary, I connect this quality with crystallization : hardness
is the next criterion ; and analysis must separate the component
parts, and demonstrate the analogy or identity of substances,
or of compounds. The improvements of Mr. Klaproth s pro-
cess are evident, by the comparison of his first analysis, and
his last analysis, of Corundum.

In the first it consisted of

100

By the last analysis of Mr. Klaproth, the Corundum of
the Peninsula of India consisted of

Corundum earth
Siliceous earth
Iron and nickel

68 o

3 1 5°
o 50

Argillaceous earth
Siliceous earth
Oxide of iron
Loss

89 50

5 50
1 25

3 75

3H

MDCCXCVIIJ.

100

4aS

Mr. Greville on the

The Corundum of China.

Argillaceous earth

84

0

Siliceous earth

6

50

Oxide of iron

7

50

Loss

2

0

100

That the analysis of sapphire by Mr. Klaproth may be com-
pared, it is here added.

Argillaceous earth 98 50

Calx of iron - - -10

Calcareous earth - o 50

IOO

Iron ore crystallized is often mixed with the Chinese Co-
rundum, as I have before stated, and may be considered as
accidentally interposed, not combined. In the Corundum of
the Coast, the greenish colour may indicate the combination of
iron, as the blue colour does in the sapphire ; and the propor-
tion of iron in both is nearly alike.

There then is the — and of silex in Corundum, evi-
dently an integral part of the coarse Corundum crystal, and
not of the sapphire; but it will require an analysis of the vitreous
or pellucid Corundum, to decide that silex is a constituent part
of Corundum : there will then remain to account for the calca-
reous earth; and, having established its being a constituent part
of the sapphire, the small proportion of ~p, cannot be expected
to produce a very notable difference.

It is not necessary to do more than thus to hint at what
further analysis and examination of former experiments are

Corundum Stone from Asia . 419

required, to ascertain the analogy or identity of the sapphire
and Oriental ruby with Corundum.

I have before stated, that I have Corundum (which has the
same texture and fracture as the common colourless Corun-
dum) of a ruby red, and also of sapphire blue, and of sapphire
blue and white colours.

1 have sapphires, yellow and blue, white and blue, brown and
greenish, and of a purplish hue; these I should consider as
Corundum, with fracture of vitreous lustre.

Mr. Tranckell, who resides in Ceylon, and from whose
communications I derived lately much information, had, about
five years ago, a sapphire, the greater part blue, and the remain-
der of a pale ruby colour. I saw, in Rome' de i/Isle's collec-
tion, at Paris, a small gem, which was yellow, blue, and red,
in distinct spots, and he called it Oriental ruby. M. de la
Metherie, to avoid the confusion of the denomination Oriental

i

ruby with octoedral ruby, calls it a sapphire ; with more cor-
rectness, I think, the abovemention ed gems should be classed

✓

as argillaceous, under the denomination of Corundum.

I am not uninformed that Corundum is said to be found in
France. The Count de Bournon is convinced, that the spe-
cimens mentioned in Crell’s Journal, as having been found
by him in a granite in the Forez, were Corundum. M. Mor-
veau also says, he found it in Bretagne ; but the Abb£ Hauy,
in No. 28 of the Journal des Mines, asserts, that the Corundum
found in France is titanite: he does not say whether this ob-
servation extends both to the Corundum of Bretagne and that
of the Forez.

In the same manner I had observed, in the specimens
which Mr. Raspe called Jade, or a new substance, from Tiree,

3H2

42° Mr. Greville on the

on the west coast of Scotland, a great resemblance to Corun-
dum ; but, having then only had a cursory view of the sub-
stance, I am indebted to Mr. Hatchett for the examination of a
specimen of it which he had from Mr. Raspe's collection.

The Tiree stone resembles crystallized Corundum of the
Coast, in texture and colour; it is also as refractory, when exa-
mined by the blow-pipe, with different fluxes. Its specific
gravity is 3,049; consequently nearer the specific gravity of
pure Corundum than the abovementioned lump, 2,785, and the
matrix of Corundum, 2,768. The Tiree stone will scratch glass
readily, but not rock crystal ; its hardness therefore corresponds
with that of the matrix of Corundum. The substance of the
lump described in page 410, cuts glass, and rock crystal, and
the Tiree stone, readily.

It will therefore be sufficient for me to say, that there is great
probability Corundum may be found in Great Britain, and on
the continent of Europe, as well as in Asia; and the above
slight assays may show, that observations on Corundum, in its
different states of purity, may lead to accurate distinction be-
tween substances hitherto imperfectly known, and will lead to
a revision of the siliceous genus, whereby the argillaceous ge-
nus may obtain its due pre-eminence in mineralogy.

When gems, by art, or by rolling in the beds of rivers, have
been deprived of the angles of their crystals, they are unavoid-
ably subjected to uncertain external characters, which even
great practice cannot render certain; and hence the unwilling-
ness of European jewellers to deal in coloured gems. I have
some specimens of a sapphire-blue stone, India cut, very small
and pellucid; they were purchased in India, as sapphires, and
were supposed to be fluor by a lapidary in London, but are

421

Corundum Stone from Asia .

cyanite. The above could scarcely have happened, if the stones
had been of sufficient size and value to require much examina-
tion, the weight and degree of hardness being exceedingly de-
ficient. The colour, therefore, will not be a safe guide. The
diamond, whether white, blue, red, yellow, or green, can be
distinguished by its crystal, or by its specific gravity and hard-
ness, or, when it is polished, by its lustre. Other stones which
"compose the order of gems, might equally depend on their crys-
tallization, specific gravity, polish, and hardness, for a distinct -
arrangement. The near relation of argil, which Bergman gave
to this order, is daily confirmed ; and it will perhaps be to Mr.
Klaproth, more than to any other existing chemist, that we
shall owe our correct information on the subject of other gems,
as we do on the subject of Corundum.

Many of the varieties of Corundum, particularly the co-
loured and transparent sorts, with their regular crystallizations,
are yet desiderata. Many raystallized stones, from defect of
colour, lustre, &c. are of little value in the market, such as,
jargon, chrysolite, tourmaline; and an infinity of unnamed
stones of Ceylon, Pegu, Siam, &c. would be valuable to the
mineralogist, if obtained adhering to their strata, and in crys-
tals whose external form is not obliterated. I have no doubt,
when it is known how much such information will tend to
illustrate the history of the earth, and particularly that of gems,
the spirit of inquiry, so laudably afloat in British India, will
be directed to attain it.

I have not heard of any metallic veins being found in Corun-
dum, unless a stone which Alonso Barba, lib. i. c. 13. de-
scribes, should give an instance. “ The Chumpi, so called
“ from its grey colour, is a stone of the nature of emery, and

■Mr. Greville on the

422

“ contains iron ; it is of a dull lustre, difficult to work, because
“ it resists fire long. It is found at Potosi, at Chocaya, and
other places, with the minerals negrillos and rosicleres.”

Having mentioned the varieties of crystallized and amorphous
Corundum, and the miscellaneous facts relative to my collec-
tion of that substance from India and China, it might be suf-
ficient to give an icon of the crystal, and close a paper already
prolix ; but, having with satisfaction observed, within the last
years, the science of mineralogy gaining ground in Great Bri-
tain, from the knowledge acquired by several gentlemen who
have examined the mines, and formed personal acquaintance
with the most experienced and learned men on the Continent,
and also from ingenious foreigners, who have communicated
their observations on English fossils, and connected them with
the most approved systems, it may perhaps be accepted as a
sufficient apology for what follows, that I consider it as a desi-
deratum to English mineralogists, to be invited to a preference
of permanent characters, which the study of crystallization has
collected, and which promises to be a certain method of ascer-
taining the laws by which elective attraction arranges and
combines molecules of matter.

It is true, the progress of crystallography has been extremely
slow, and different nations have contributed to its present
improvement. It is rather remarkable, that the earliest trea-
tise on metallurgy, of authority, was published in Italy, by
Vanoccius Biringuccius, just before Agricola published
his treatise in 1 546 in Germany ; and the first treatise on the
structure of crystals I know, is also from Italy, by Nicolas
Steno, Prodromus Dissertationis de Solido intra Solidum na-
turaliter contento. Florentine, 1 66g, in 4to. A work of great

Corundum Stone from Asia. 423

merit. Louis Bourguet of Neufchatel, in his Lettres sur la
Formation des Sets et des Crystaux. Amst. 1729, 120, connect-
ed, by observation and measure, triangular and rhomboidal,
and cubic and pyramidal tetraedal molecules, for all different
substances. His contemporary, Maurice Antoine Capel-
ler,* attempted to deduce a system from geometrical prin-
ciples; and in this state did Linnaeus find the subject, when he
attempted to reduce the science of minerals to external cha-
racters, and crystallized bodies to salts.

None of the observations of Linnaeus will prove useless to
science ; but his system alarmed the chemists and mineralo-
gists, who rejected every other criterion than internal charac-
ter from analysis, and the system of Cronstedt was preferred
by general assent. By this means, a spirit of controversy de-
prived the chemist and lythologist of mutual assistance ; and
the general opinion was correct, on the supposition that a
mixed system of chemical and external characters would be
irreconcileable ; but it has been admitted, even by those who
most decidedly opposed Linnaeus's system, that the best sys-
tem of mineralogy should be founded on external and internal
characters combined. -f* Among the few who ventured to pro-
fess their obligations, at the same time, to Linnaeus and to
Cronstedt, was Baron Born, whose abilities and character,
in addition to his distinction as one of the counsellors of mines

• Prodromus Crystallographies, &c. and Litter a ad Scheuzerum, de Crystallorum
Generatione. Act . Nat. Cur. Vol. IV. Append, p. 9.

f Nullum itaque est dubium, quin hujusmodi methodus mixta, quae notis charao
teristicis tarn extrinsecis quam intrinsecis simul combinatis, est superstructa, proxime
ad naturalem accedens, maximam indicans symmetriam, reliquis sit praeferenda metho-
dis. J. G. rius, de Systemate Mineralogico rite condendo , §. 102.

Mr. Greville on the

424

of his Imperial Majesty, obtained his enrollment among the
Fellows of the Royal Society. He connected the intrinsic
and extrinsic characters of minerals, in the Index Fossilium,
which he published in 1772. In Sweden, Bergman’s treatise on
the forms of crystals, published in the Upsal Transactions, in
1773, was a more authoritative recommendation to the inves-
tigation of the principles of crystallization ; and it can be of
little importance for me to add, that since I have possessed
the collection of Baron Born, in 1773, I have had every con-
firmation of the same opinion. The progress of chemistry and
of crystallography, applied to mineralogy, has rendered the exa-
mination of strata, and of mines, a source of amusement as
well as instruction ; and the arrangement of interesting facts,
in the chemistry and mechanism of nature, suits my occasional
researches in geology, which, from variety of avocations and cir-
cumstances, have been very much interrupted. My acknowledg-
ment of obligation to the learned who have made this progress
in science, is the best recommendation I can give to others to
examine their works. Those whose talents and time are devoted
to the investigation of every mineral substance, can have no
respite to their labour ; minerals, in every state of their forma-
tion, perfection, and decomposition, as they occur in mines,
must have their qualities immediately ascertained, and be re-
served for profit, or thrown away on the heap. The practical
miner could not, without external characters, make any progress.
The valuable minerals are soon pointed out by assay, and their
appearance remembered. The accuracy of selection depended,
in all periods, much on the experience of the miners. It re-
mained for. Mr. Werner to give the utmost degree of accu-
racy which irregular external characters can acquire, by fixing

s

Corundum Stone from Asia .. 425

appropriate terms to all the characters which occur, and which
the senses can discriminate. In 1774, °Pei^ed his system of
external characters of minerals, and the perfection he has since
given to it, has rendered it very general. The Leskean collec-
tion, arranged after Mr. Werner’s method, has procured, in Mr.
Kirwan, a powerful support to the introduction of that system
in this country ; and we have already some other valuable pub-
lications, to recommend and introduce other favourite systems
of the Continent. It is, therefore, at this time the English
mineralogist should be invited to examine, if not to prefer, per-
manent characters, so far as the progress of crystallography has
collected them, or at least to give them a distinguished rank
among external characters of bodies.

If prejudice too long has retarded the union of intrinsic and
extrinsic characters, it has also occasioned a schism among
the advocates of crystallography.

Rome' de l’Isle, in the year 1772, published the first edi-
tion of his Essay on Crystallography, which he states to be a
supplement to Linnaeus ; and, by the assistance of a very few
friends, he was enabled to increase the number of crystals in
a degree to assume the appearance of a system. He told me,
that the accuracy of his measurement of angles of minute crys-^
tals was the acquirement of great practice, but that the Count
de Bournon, after a short practice, attained equal correctness,
and afforded him assistance, which he acknowledges in his 2d
edition to have received, particularly by the discovery of crys*
tals in Dauphine, Auvergne, Franche-Comt^, &c.

The Abb6 Hauy, an accurate and patient observer, and a
' good mathematician, considered crystallography as founded on
certain laws, reducible to demonstration by calculation. In the

MDCCXCVIII. 3 I

Mr. Greville on the

426

beginning, the differences of Bourguet and Capeller were not
more pointed than those of Rome'de i/Isle and the Abb^HAUY
but the progress of observation and calculation having demon-
strated their mutual utility, the observer and measurer of crys-
tals will now rest satisfied only when calculation confirms ac-
tual measurement. To the Abb£ Hauy is also due a late scheme
to simplify calculation, by expressing, according to algebraical
formulae, the different laws which determine the modifications
of crystals. So far as they are the result of calculation and mea-
surement, we may admit the laws of crystallization ; for, when-
ever the superposition, or subtraction, of simple or compound
molecules on a nucleus, shall, by calculation, give a series of
planes and angles, which corresponds exactly to the angles and
planes measured on natural crystals, it will amount to no more
nor less than a demonstration of the rule or arrangement of
elective attraction by figures.

These laws may be reduced to simple practice; for instance,
the Abbe Hauy, by measuring the rhombic plane of Corundum,
found its two diagonals to be as two to three ; which gives to
its acute angle 8i° 47' 10", and to its obtuse angle 98° 12' 50";
the same as martial vitriol. *

The forms of fragments in Corundum are all acute rhom-
boids. The cosine of the little angle in Corundum is ~ of the
radius ; but, in calcareous spar, the cosine is ~ of the radius ;
in shorl, -J of the radius ; in the garnet, j- ; and, in rock crys-
tal, -pf*

Thus, the application of general laws, to ascertain constant

* This result is extracted from the Journal de Physique ; but it appears, from the
Journal des Mines, No. 28, that the Abbe Hauy has since rectified this measure,
and given 86° 26' for the acute angle, and 930 34' for the obtuse angle.

e

Corundum Stone from Asia. 4^7

character, after they shall have been fully verified, may be very
simple and general. It will not require perfect crystals ; for,
when crystals separate into laminae, which subdivide into frag-
ments, and shew the form or arrangement of their molecules,
it is easy, from such fragments, to connect them with their pri-
mitive crystal, and consequently with their class'. It will be
a great step, to obtain one regular and permanent external cha-
racter. Attention to other characters will be necessary, to as-
certain the nature of the substance ; and other external charac-
ters, such as irregular fracture, colour, &c. must be resorted to,
where no permanent characters exist ; but from their nature
they are fallible, and in fact are seldom conclusive.

The progress of crystallography appearing to me of conse-
quence to the progress of mineralogy, induced me to desire the
Count de Bournon, abovementioned, one of the honourable
victims to his allegiance to his King, to describe such crystals,
in my collection, as shewed the different known modifications
of Corundum; which will develop the theory of crystalliza-
tion, so far as is consistent with the avowed object of this
Paper. The subject, I believe, has not hitherto been submit-
ted to the consideration of this Society. The translation of the
Count de Bournon's description has been carefully made to
preserve its clearness, and I hope it will be favourably received
by the Society, and make some amends for my tedious intro-
duction.

After it, I have added a table, connecting in one view the spe-
cific gravities of Corundum, & c. herein mentioned, with those
given by other authors.

3 1 2

4^8

Mr. Greville on the

An Analytical Description of the Crystalline Forms of Corundum,
from the East Indies, and from China. By the Count de Bournon*

The most usual form of Corundum is a regular hexaedral
prism ; (Tab. XXII. fig. 1.) in general, the surface of the crys-
tal is rough, with little lustre, owing to unfavourable circum-
stances under which it crystallized.

The crystals of Corundum hitherto found were not formed in
cavities, where, each crystal being insulated, its surface could
preserve that smoothness and natural brilliancy which are com-
mon to all substances that freely assume a crystalline form.
Like the crystals of feldspar which we meet with in the porphy-
roid granites, the Corundum crystals have been enveloped, at
the time of their crystallization, by the substance of the rock
which was forming, at the same time with themselves, in an
imperfect and confused crystalline mass ; and the Corundum
crystal, before it had acquired its perfect solidity, necessarily
received on its surface the impression of the different particles
of the rock which enveloped them : this naturally renders the
surface rough and dull. Crystals of feldspar found in the gra-
nitic porphyroid rocks, exhibit the same kind of appearance,
from the same cause.

The Corundum crystals are in general opaque, or at least
they have only an imperfect transparency at the edges : when
broken into thin fragments, the pieces are semi-transparent :
when held between the eye and the light, and examined with
a powerful lens, it will be perceived that their interior texture
is rendered dull by an infinite number of small flaws crossing

Corundum Stone from Asia. 429

each other, much resembling the medullary part of wood, when
it is viewed in the same manner.

The degree of transparency of the small interstices which
are between these flaws, is further evidence that this texture of
small flaws occasions opacity, which augments in proportion to
the thickness of the fragments. This kind of internal structure
has also a very strong analogy with that of feldspar in granite
and porphyry.

The endeavour to split these crystals, in a direction either
perpendicular or parallel to their axes, meets with a very con-
siderable resistance: they may indeed be broken in these direc-
tions ; but the rugged and irregular surface of the broken
parts, clearly proves that the direction in which the crystalline
laminae have been deposited one upon another, has not been
followed..

The regular hexaedral prism of these crystals, cannot there-
fore be considered as the form of the nucleus of the crystal ;
and consequently is not the primitive form of the crystals of
this substance.

If, in order to discover the direction of the crystalline lami-
nae, a variety of crystals be examined, some will hardly fail to
be met with, which, on their solid angles, formed by the junc-
tion of the sides of the prism with the planes of the extremities,
present small isosceles triangles. These are sometimes greater
and sometimes smaller, and form solid angles of 1220 34', with
the extreme planes of the crystal. They are in some instances
real faces of the crystal; but most frequently they evidently
are the effect of some violence on that part. The smoothness
and brilliancy of these small faces, in the latter case, shew that
a piece has been detached in the natural direction of the crys-

43°

Mr. Greville on the

talline lamina. It indeed much less difficult to separate a
portion of the crystal at these angles, than at any other part ;
and, in following the natural direction of the faces, with a little
patience and dexterity, all the crystalline laminae may be de-
tached, and progressively increase the size of the triangular face.

This operation, however, cannot be done indiscriminately on
all the solid angles of the crystals, but only on the alternate
ones at the same extremity, and in a contrary direction to each
other. As to the other angles, they may be broken, but it is
impossible to detach them. When, instead of the solid angles
of a hexaedral prism, small triangular planes are met with,
(which frequently happens, whether caused by violence or
otherwise,) they are always placed in the direction above-
mentioned.

If, by following this indication of nature, we continue to
detach the crystalline laminae, we shall at last cause the form
of the hexaedral prism to disappear totally, and, in place of it,
arhomboidal parallelopiped will be obtained, (fig. 2.) of which
the plane angles at the rhombs will be 86° and 940 ; the solid
angles at the summit* will measure 84° 31'; and that taken at
the re-union of the bases will be 950 29'.

We can split this parallelopiped only in a direction parallel
to its faces ; it will still consequently preserve the same form,
which is that of the nucleus of this substance, and its primitive
form.

It is, therefore, by a modification of the rhomboidal parallelo-

* For greater clearness, this rhomboidal parallelopiped may be considered as being
formed by the junction of two triedral pyramids, base to base; and the two solid angles
(each of which is formed by the re-union of three of the acute angles on the planes of
the rhomb) will then be considered as the summits of these pyramids.

Corundum Stone from Asia. 431

piped, (fig. 2.) that nature has formed the regular hexaedral
prism (fig. 1.) which this substance presents.

For, if we conceive, that in any period whatever of the in-
crease of the rhomboidal paralielopiped, a series of laminae or
crystalline plates has been deposited on all the sides of the pa-
rallelopiped ; and that these laminse have all undergone a pro-
gressive decrease of one row of crystalline molecules, at the acute
angle which tends to form the summit, and also along the sides
of the opposite acute angle, (fig. 3. and 4.) there will necessarily
result from the continuation of this superposition, to a certain
period, an hexaedral prism, terminated by two triedral pyra-
mids, placed in a contrary direction ; and their planes or faces,
which form a solid angle of 1470 26', with the sides of the
prism, will be either pentagonal, (fig. 3.) or triangular. (Fig. 4.)
They will also have, in place of a summit, an equilateral trian-
gular plane, sometimes greater and sometimes smaller.

If the superposition continues, the equilateral triangular
plane on the summit will become nonagonal, and there will
remain no other traces of the primitive planes of the rhom-
boidal paralielopiped, than small isosceles triangular planes :
(fig- 5 •) if the superposition still continues, until the last
crystalline lamina is reduced to a single molecule or point,
no appearance of the rhomboidal paralielopiped will then re-
main ; and the crystal resulting from this operation of nature
will be a regular hexaedral prism. (Fig. 1.)

In the same manner, viz. by a decrease on the lower edges
of the laminae, the primitive rhomboidal paralielopiped of cal-
careous spar passes to a regular hexaedral prism of that sub-
stance; though more frequently it does so by a decrease on the
lower angles of the laminae.

43®

Mr. Greville on the

When the laminae of the Corundum crystal have, during
their superposition on the planes of the primitive rhomboidal
parallelopiped, experienced a progressive decrease at one of
their acute angles, and along the sides of the other, at the same
time, and in the same proportion, it is easy to conceive that
the height of the hexaedral prism must be the same as that of
the rhomboidal parallelopiped, upon which it has been formed.
The height BC (fig. 1.) must therefore bear the same propor-
tion to the line AB, drawn through the middle of the two op-
posite sides of the planes on the extremities, as the whole
height EF, of the rhomboidal parallelopiped, (fig. 2.) bears to
the small diagonal GH, from one of the rhombs ; that is,
nearly as 6,45 : 5.

But, although this exact proportion appears in a very great
number of Corundum crystals, yet we meet with some whose
lengths are more or less considerable; and this is owing to
different circumstances which have existed at the time of their
crystallization. We may conceive, for instance, that if, before
the progressive decrease of the crystalline laminae, in the man-
ner abovementioned, the increase of the rhomboidal paralle-
lopiped had taken place by a superposition of laminae, in
which the rows of crystalline molecules experienced a pro-
gressive decrease along the edges of the acute angle of the
base only, (fig. 6.) and that (the sides of the prism having al-
ready acquired a certain length) the succeeding crystalline
laminae had experienced a decrease at the acute angle of the
summit, the same regular hexaedral prism would have resulted
from this process ; but the proportion between the height and
the line drawn from two of the opposite sides of the planes on
the extremities, would have been much greater than that of

433

Corundum Stone from Asia.

6, 45 : 5 ; and, consequently, this prism would have been longer
than that of the rhomboidal parallelopiped which served as its
nucleus.

On the other hand, if the increase of the rhomboidal paral-
lelopiped had taken place by a superposition of crystalline
laminae, decreasing at the acute angle of the summit, and,
some time after, decreasing also along the sides of the acute
angle of the base, (fig. 7.) the regular hexaedral prism resulting
from this process would have been shorter, in proportion to the
duration of the mode of decrease in the crystalline laminae
which were first deposited. There are some of the hexaedral
prisms, in Corundum crystals, which are so short, that they
appear no more than segments. Calcareous spar offers the
same phenomenon ; as do likewise all the substances in which
the hexaedral prism has any analogy of formation with that
which we have here described.

It happens frequently, when the superposition of the crystal-
line laminae does not go on equally on all the faces of the
rhomboidal parallelopiped, that one or two only of the solid
angles of the hexaedral prism, taken alternately, still shew, by
small isosceles triangular planes, some remains of the faces of
the parallelopiped, while the others do not shew any at all.

Mr. Greville, in his collection of this substance, has a
crystal of Corundum, upon one side of which, only two of the
planes of the rhomb have experienced an equal and perfect
superposition, while there has been but a very small number
of crystalline laminae deposited on the third plane. Conse-
quently, this crystal presents a regular hexaedral prism, one of
whose solid angles is so much truncated, that the half of the
plane of the end of the hexaedral prism disappears; (fig. 8.)
mdccxcviii. 3 K

4 34

Mr. G'reville on the

£i id this cut 01 section forms an angle of 1220 34A with the
pLne on the extremity.

It is unnecessary to observe, that the regularity of the
hexaedral prism, depends on that of the rhomboidal parallelo-
piped on which it is formed.

When, by detaching the laminae from the alternate solid
angles of the regular hexaedral prism, the planes resulting from
this operation begin to run into one another, and the crystal
begins to assume the form of the rhomboidal parallelepiped to
which it owes its origin, we frequently see the surface of these
new planes divided into an immense number of small rhombs,
formed thereon by the intersection of lines that are parallel to
the sides, which belong to the rhomboidal form of the new
faces. (Fig. 9.)

These lines are owing to the extremities of the laminae which
have been deposited on the inferior faces, corresponding with
those on which we observe them; and they serve to corro-
borate still farther, the demonstration we have given of the
formation of the regular hexaedral prism in this substance.

We frequently see small rhombs traced on the surface of
the planes, on the ends of the hexaedral prism. (Fig. 10.) This,
no doubt, is occasioned also by the intersection of the laminae,
on the planes of the primitive rhomboidal parallelopiped. But
these rhombs, formed by the re-union of lines that join in angles
of 6o° and 120°, instead of 86° and 940, (like those we have
seen traced on the faces which correspond with those of the
rhomboidal parallelopiped,) form angles of 6o° and 120°. It
would therefore be an error to consider them as indications of
the form of the elements of crystallization, as we are tempted
to do from a simple inspection of the crystal. These same lines

Corundum Stone from Asia. 435

form equilateral triangles with one another, as may be seen in
%. 10.

The cause of these small equilateral triangles, which some-
times project a little over the planes on the ends of the prism,
must now be obvious. If, during the superposition of the crys-
talline laminae on all the planes of the rhomboidal parallele-
piped, it has happened, from any cause whatever, that the la-
minae deposited on the three faces of the same summit, have
not fallen exactly on those which preceded them, or that they
have experienced some deviation, or have not had the same
decrease as all the others, at the angle of 86°, these triangles
must necessarily occur ; in the same manner it must be ob-
vious, why these small equilateral triangular projections are
frequently placed on one of the sides of the crystal.

The primitive form of the Corundum crystal is therefore a
rhomboidal parallelopiped, whose solid angle at the summit is
84° 31', and that formed by the re-union of the bases is
95 ° 29b

The crystalline laminae are rhombs of 86° and 940 : these, in
my opinion, are double crystalline molecules ; the single mole-
cules I apprehend to be isosceles triangles, of 86° at the angle
of the summit, and of 47° at those of the base *

* 1 am at present preparing a work, in which I shall, if circumstances permit me
to finish it, give the result of my observations, and my own opinion on this interesting
part of mineralogy. I shall only observe here, that although double molecules, square
and rhomboidal, are frequently formed in the process of crystallization, yet the real
form of the crystalline molecules seems to be triangular. By observing the pro-
gress of the rhomboidal parallelopiped, in its passage to the form of an hexaedral
prism, (fig. 4. and 5.) and by considering the prism terminated, it seems evident, that
the last lamina which had been deposited, after the progressive decrease in the rows
of crystalline molecules to one single molecule, must necessarily have been triangular,

3 K 2

Mr. Greville on the

436

Although the rhomboidal parallelopiped of 86° and 940 is
the primitive form of the Corundum crystal, yet it is rare to
meet with that substance under this perfect and determined
form ; and, in most mineral substances, it is more rare to meet
with their primitive crystals than their different modifications.
Amongst Mr. Greville’s numerous specimens of Corundum,
I have met with only one which has this primitive form, and it
is doubtful whether even this may not be a fragment.

The Corundum crystal presents another modification, under
which the regular hexaedral prism, instead of having three al-
ternate solid angles at each of its ends, (on which solid angles
are placed isosceles triangular planes, forming a solid angle of
1220 34b with the planes at the extremities upon which they
are inclined,) has also its angles supplied by isosceles triangu-
lar planes; but these planes, instead of 122° 34', form solid
angles of 160° 42', with the said planes on the extremities.
(See fig. 11. and 12.) These new planes, which constitute a
new modification of the primitive form of Corundum, are the
result of a different order in the decrease of the laminae; which,
in the primitive form, are deposited on the planes of its pri-
mitive rhomboid by single rows of crystalline molecules, and
increase the planes which terminate the hexaedron : whereas,
in this second modification, the decrease of molecules is by two
rows, which gives a more obtuse inclination, and forms new
planes. The surface is usually striated, parallel to the sides of
the planes which terminate this crystal ; an appearance always
announcing imperfection in the crystallization, arising either
from a change in the order of decrease or increase, or from a less
perfect union of the crystalline laminas. A section would show
gradual risings Gr steps, as appears in fig. 14. which is a section

Corundum Stone from Asia. 437

of fig. 13. in the line ADB. These striae are not to be con-
founded with those in numberless substances, as in tourmalines,
schorls, &c. which arise from the longitudinal union of num-
berless distinct crystals. The crystal resulting from this new
mode of decrease in the crystalline laminae, will represent one
or other of the varieties shewn in fig. 1 1, 12, and 13, according
to the period when such decrease has begun in the process of
the crystallization; and, if it has begun very late, the new
faces will only be small, nay almost imperceptible, isosceles
triangles, forming solid angles of 160° 42', with the planes of
the extremities of the prism, as in fig. 5. ; the measure of the
angles however must be excepted.

If this irregular mode of decrease had begun with the first
crystalline laminae which were deposited on the primitive
rhomboidal parallelopiped, the hexaedral prism resulting there-
from would have been terminated by two very obtuse triedral
pyramids, whose planes would have been rhombs ; and they
would have been placed in a contrary direction to each other,
as may be seen in fig. 12, by the dotted lines. I have not met
with this variety, but its existence may be supposed.

It happens sometimes, that the crystallization has not been
so perfect as to destroy every appearance of the faces of the
primitive rhomboidal parallelopiped; in this case, there remains,
on the solid angle of 1 120, formed by the junction of the new
faces with the edges of the prism, a small isosceles triangle,
as in fig. 13, which corresponds to those in fig. 5. of the pre-
ceding modification.

The crystals which explained the second modification, form
also a part of Mr. Greville's collection : one, in particular, is
highly worthy of notice ; it is the most perfect crystal I have

438 Mr . Greville on the

ever seen of this substance. The surface of the faces of the
prism, although rough, is infinitely less so than that of the
others, and much more brilliant. The planes on the ends have
the usual polish of crystals ; its colour is a pale red, and its
transparency may be compared to that of wax.

This substance presents a third modification, in which the
hexaedral prism diminishes in diameter, as is apparent by com-
paring the diameters of its two ends ; in some, it appears like a
regular hexaedral pyramid truncated. (Fig. 15.) The crystals
of this modification are usually irregular, and seldom admit of a
certain measure of their angles ; but, among the numerous
specimens in Mr. Greville's collection, I have been able to
ascertain, in the greater part, that the hexagonal plane at the
top forms angles of about 120°, with the planes of the pyramid;
and the hexagonal plane at the base forms angles of about 78%
with the planes of the pyramid. In other instances, the form of
the pyramid varies greatly ; in some, the angle at the upper
plane was no°, and the angle at the base about 70°; in others*
the angle at the upper plane was about ioo°, and the one at
the lower plane about 8o°.

In these three varieties, the crystalline laminae can be sepa-
rated, as in the hexagonal prism, at the three solid alternate
angles of each end, but in a contrary direction to each other.
The planes which appear when the laminae are detached re-
gularly, form solid angles of 22° 34/, with the planes of the ex-
tremity : this arrangement is analogous to that of the hexae-
dral prism. The difference of form arises from the crystalline
laminae deposited on the planes of the primitive rhomboid,
decreasing by more than one row of molecules, on the planes
of one of the triedral pyramids of the rhomboid, and by less

Corundum Stone from Asia. 439

than one row, on the planes of its other pyramid. This general
observation, on the manner in which this primitive crystal of Co-
rundum passes to the different varieties just mentioned, is the
only one I have established with any great degree of certainty
at present. Specimens with perfect crystals, whose angles may
be measured with accuracy, will probably arrive from India,
and give further demonstration, as to these and other varieties
of modifications of Corundum. We may conceive, that if, in
this modification, the crystallization had ceased before the en-
tire formation of the crystal, there would have remained small
isosceles triangular planes, on three of the alternate solid angles,
formed by the junction of the planes on the ends, with the
edges of the truncated pyramid. These isosceles triangular
planes resemble those we have seen in the first modification;
(fig. 4. and 5.) and form, in the same manner, solid angles of
1220 34', with the planes on the ends of the prism. (Fig. 16.)

Finally, if, during the formation of the crystal, in this modi-
fication, it should happen that the laminae deposited on the
three planes of the rhomboidal parallelepiped, on the side
where they undergo a greater decrease, do not undergo the
decrease of one row of molecules at the acute angle of the sum-
mit, the crystal will be a real hexaedral pyramid, (fig. 17.)
whose acute angle at the summit, measured on the sides, will
be nearly 240, in one of the varieties; 40° for the most obtuse;
and 200 for the most acute variety : the angle of their triangu-
lar planes, in the first instance, 130 41'; in the second, 220 20';
and 1 1° 28' in the third. I have not seen any perfect pyramids;
but, in manv, the hexagonal plane terminating the pyramid is
so small, that it renders its total suppression probable.

Mr. Greville on the

44°

This decrease necessarily produces a single pyramid, as
abovementioned ; nevertheless, there are instances of crystals of
Corundum, belonging to the variety where the terminal planes
make, with the planes of the pyramid, a solid angle of about
ioo°, in which, two pyramids of the same dimensions, having
their summit replaced by a small hexagonal plane, are placed
base to base.

I have also observed, among the crystals of the obtuse va-
riety abovementioned, in Mr. Greville’ s collection, an instance
of the decrease taking place by several rows, on one three-sided
pyramid of the primitive rhomboid, and by single rows on the
other. Consequently, the crystal is a short regular hexaedral
prism, terminating on one end only by an hexaedral pyramid ;
the planes of which, as well as of the prism, are alternately
broad and narrow, and almost perfect ; its apex being replaced
by a very small plane.

I shall conclude, by mentioning a variety of Corundum, de-
scribed by the Abbe Hauy, in the Journal des Mines , No. 28; in
which, the edges of the terminal planes of the hexaedral prism
are replaced by planes which form an angle of 1160 31', with
the terminal planes; but, in the numerous collection of Mr.
Greville, I have not seen this variety. One crystal had an
appearance of such planes ; but, on examination, it was clearly
accidental. The authority of the Abbe Hauy, in crystallogra-
phy, is so great, that the existence of such modification ought
not to be denied, without further examination; though I cannot
in this instance adopt it : he derives this variety, which he
calls subpyramidal, from a decrease of three rows of molecules,
at the angles of the base of the two pyramids of the primitive

Corundum Stone from Asia. 44,1

rhomboid ; and he seems to attribute the same formation to the
pyramidal variety with double pyramid, which he supposes may
exist.

The primitive crystals, and the first and second modifications
of Corundum, are from the Peninsula of India. The third
modification, or the pyramidal variety, is from China ; nothing
approaching this form being among the specimens which Mr.
Greville received from the Peninsula of India.

The preceding observations, and particularly the last men-
tioned modification of Corundum, compared with the best de-
scriptions of the sapphire, suggest the further examination of
the degree of connection, if not of identity, of these Oriental
stones.

In both, the hexaedral pyramids are usually incomplete in
their apex, and they vary in acuteness. I have stated the de-
gree in which the solid angles of the pyramid (taken as com-
plete) vary, in Corundum, to be from 20° to 40°.

Rome' de l’Isle states, that the sapphire varies from 20"
to 30°. The Abb£ Hauy ( Journal de Physique, Aug. 1793,)
mentions two varieties of the sapphire, one measuring at the so-
lid angle of the pyramid 40° 6', the other 570 24'. I never saw
a sapphire with so obtuse an angle as the last; but many,
whose angle at the top, if the pyramid had been complete,
would have been the same as that of the Corundum. Besides
the analogy between the crystals of Corundum and the sap-
phire, by the union of two hexaedral pyramids at their base, it
also exists by the measure of their angles ; and both substances
are subject to the same irregularity, sometimes appearing as
a single hexaedral pyramid, and sometimes as an hexaedral

mdccxcviii. g L

Mr. Greville on the

442

prism ; moreover, the sapphire sometimes has on its solid
angles, alternately, the same triangular planes, (fig. 5.) and
also the prominent triangles on the planes of the extremities,
(%• i°-) which often appear in the crystals of Corundum.
The Abbe Hauy, in the Journal de Physique , August, 1793,
names this variety, Orientate Enneagone , which is represented
in the annexed Plate, (fig. 18.) and says, that the small triangu-
lar planes make, with the terminal planes, an angle of 1220 18':
and, in the description of the same triangular planes in the
Corundum, (fig. 16.) it appears, that these planes are the re-
mains of the planes of the primitive rhomboid, and form, with
the terminal planes, an angle of i22°34'.

Perhaps the rhomboidal crystal, which Rome' de i/Isle had
given as one of the forms of the sapphire, should be restored
to it. He had examined it at M. Jacquemin's, jeweller to the
crown, ( Cristallographie , 1. edit. p. 221.) and he suppressed it
in his second edition, but often expressed to me his regret in
having made the alteration. I have before me a letter from
that celebrated naturalist, dated September, 1 784,* in which he
inclosed, for my opinion, a copy of a letter he had received from
Mr. Werner, with models of some crystals ; among them, two
called by him rubies ; one a rhomboid, of which the angles of
the summit are substituted by planes, (fig. 19.) the other* is
precisely the same as fig. 3, 4, and 5, of the annexed Plate.

The following is a translation of Rome' de i/Isle's words :
“The first of these rubies has exactly the same form as I have
“ represented in Plate IV. fig. 60. of my Cristallographie, viz.

* A letter to the same effect was written to M. la Met h brie, and published in
the Journal de Physique, May, 1787.

Corundum Stone from Asia. 443

■* a rhomboidal parallelopiped, truncated at each of its obtuse
“ angles, by an equilateral triangular plane.

“ You will have a correct idea of the other crystal, if you
“ suppose the crystal represented in Plate IV. fig. 87. truncated
“ at each of the summits of its pyramids, by an equilateral tri-
<c angular plane, as in the preceding modification, but deeper,
sc and in so great a degree, that the three rhombic planes of each
“ pyramid disappear, with the exception of three isosceles tri-
“ angles ; this modification differs from the first, only by the
“ hexacdral prism, and the pleeper truncature at the summits of
“ the pyramids.”

It is therefore clear, that if the primitive rhomboid of Corun-
dum decreased only at the superior angles of its laminae, it would
exhibit exactly the first of these varieties of Mr. Werner's
ruby, as in the annexed fig. 19.

As to the second variety of Mr. Werner's ruby, it is equally
clear, if in fig. 87, referred to by Rome' de l'Isle, (represented
by the annexed fig. 20.) no more of the pyramid was left than
the three small triangles 6, a , c , there would be precisely one
of the forms of Corundum before described, to which the an-
nexed figure 5 belongs.

It may perhaps be objected, that the laminae; appear to be
parallel to the terminal planes, in the sapphire, and inclined, in
the Corundum. There are crystals of Corundum, in which, very
frequently, the laminae appear parallel to the terminal plane ;

I was at first, and for some time, deceived by that appearance.
In other Corundum crystals, the laminae appear to be parallel to
the prismatic planes ; and, to conclude the instances of analogy,
the superposition of rhomboidal laminae is sometimes observable

% L 2

444

Mr. Greville on the

in Oriental rubies and sapphires. It was by this appearance, Mr,
Greville was led to try the effect of cutting the foremen-
tioned stones en cabochon ; whereby a similar effect of triple
reflection, which formed stars of six rays from a common
centre, was produced in the Oriental ruby, in the sapphire, and
in the Corundum.

It is to be lamented that Mr. Werner did not send, with
his models, the specific gravity of each of the rubies, and the
measures of their angles : we should then have had data to de*
cide, whether the rubies sent by Mr. Werner were, as I sup-
pose them to be. Oriental rubies, or sapphires ; and, with equal
certainty, whether the parallelopipedal rhomboid corresponds
precisely with that of the Corundum; by this, the perfect iden-
tity, or analogy, between the Corundum and the sapphire, would
have been no longer doubtful.

Corundum Stone from Asia.

445

Table of the specific Gravity of the Corundum, Sapphire,
Topaz, Ruby, and Diamond, on different Authorities.

Corundum.

Hatchett and Greville 2,768
H. and G. - 2,785

Klaproth - - 3,075

Klaproth - 3,710

Blumenbach - 3,808

Brisson - 3,873

Hatchett and Greville 3,876
Lichtenberg - 3,308

Gross - 3,935

Hatchett and Greville 3,350
H. and G. - 3>35 4

H and G. - 3,959

H. and G. - 3,959

H. and G. - 3,362

Klaproth - - 4,180

* Matrix of Corundum. Coast.
*Lump of Corundum. Coast.

* Corone. Bengal.

* Crystal of Corundum. Coast.

* Ditto with vitreous cross
fracture. Coast.

* Ruby-coloured. Coast.

* Chatoyant. China.

* Crystal. China.

Brisson

Bergman

Quist

Bergman

Klaproth

Bergman

Sapphire.

3,130 Brasilian;

3^50 f

3,8oof

3 >34°+

3’35° T
3>374+

probably a topaz.

Mr. Greville on the

446

Brisson

3 9.91 +

White Oriental.

Brisson

3 ’99 4+

Blumenbach

3^99 4f

Blue.

Hatchett and Greville 4,000 +

* Greyish star-stone,

Werner

4,000 +

Blumenbach

4,010 +

Hatchett and Greville 4,035-)-

Blue star-stone.

Brisson

4,076‘f

From Puy-en-Velay.

Blumenbach

4,083-)-

Crimson.

Hatchett and Greville 4,083-)-

* Pale blue crystal.

Muschenbroeck

4,090 f

Blumenbach

4,100+

Yellow.

La Metherie

4,200 +

Quist

4,200 +

Topi

2Z.

La Metherie

2,690

Siberia.

Bergman

3,460

Werner

3*464

Light blue. Brazil.

guist

3*5°°

Werner

3’52i

Eibenstocker.

Brisson

3*53*

Red. Brazil.

Brisson

S536

Dark yellow. Brazil.

Werner

3*54°

Dark yellow. Brazil.

Brisson

3*548

Oriental.

Werner

3»556

Schneckensteiner.

Brisson

3*564

Schneckensteiner.

Brisson

4,010 +

OrientaL

Bergman

4*560 +

447

Corundum Stone from Asia.

Ruby .

Bergman

3,180

Muschenbroeck

3,180

£)uist

■ 3,400

Spinel.

Blumenbach

3454

Ceylon.

Quist

■ 3,5°°

Brazil.

Brisson

3,53i

Brazil.

Klaproth

3,57°

Hatchett and Greville

3,57i

* Octoedral crystal.

H, and G.

3,625

* Made of octoedral crystal.

Blumenbach

3,^45

Blumenbach

3,760

Brisson

3,7^o

Hatchett and Greville

4,166

* Salam ruby. Star- stone.
Coast.

Quist

4,20of

Bergman

4,240+

Diamond.

Hatchett and Greville

3,35$

Perfect crystal..

Wallerius

3,400

Hatchett and Greville

3471

Aggregate crystal.

Cronstedt

3,5oo

Muschenbroeck

3,5i8

La Metherie

3,520

Brisson

3,52i

Werner

3,600

448 Mr. Greville on the Corundum Stone , &c.

The mark * distinguishes the specimens in my collection, to
which I have referred in the foregoing Paper.

The mark -f distinguishes the stones which, from their
specific gravity, I think belong to the genus of Corundum.

The generic name Corundum, I am in the habit of giving
to those sorts which have a sparry or a granulated fracture.
When Corundum has a vitreous cross fracture, I call it sap-
phire ; and distinguish its varieties by their colours, white, red,
blue, yellow, green ; and by the accidental reflection of light
from their laminae : when in one direction, I call the sap-
phire chatoyant; when the reflection is compounded of rays
which intersect each other, and appear to diverge from a com-
mon centre, I call them star-stones, as red, blue, or greyish
star-stones, or star-sapphires.

jPhilos. Trans. !M D CCXC VUL Tab.^KXILp. 44 8.

v : :v-

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Ml i” f ■ ‘ . :•••-* * Mt • f

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1

C 449 3

XX, An Inquiry concerning the chemical Properties that have
been attributed to Light. By Benjamin Count of Rumford,

F. R. S. M. R. I. A .

Read June 14, 1798.

In the Second Part of my Seventh Essay, (on the propagation
of heat in fluids,) I have mentioned the reasons which had in-
duced me to doubt of the existence of those chemical properties
in light that have been attributed to it, and to conclude, that all
those visible changes produced in bodies by exposure to the ac-
tion of the sun’s rays, are effected, not by any chemical combi-
nation of the matter of light with such bodies, but merely by
the heat which is generated, or excited, by the light that is ab-
sorbed by them.

As the decision of this question is a matter of great impor-
tance to the advancement of science, and particularly to che-
mistry, and as the subject is in many respects curious and
interesting, it has often employed my thoughts in my leisure
hours ; and I have spent much time in endeavouring to con-
trive experiments, from the unequivocal results of which the
truth might be made to appear. Though I have not been so
successful in these investigations as I could wish, yet I cannot
help flattering myself, that an account of the results of some
of my late experiments will be thought sufficiently interesting
to merit the attention of the Royal Society.

MDccxcy.111. 3 M

4>5° Count Rumford's Inquiry concerning the

Having found that gold, or silver, might be melted by the
heat (invisible to the sight) which exists in the air, at the dis-
tance of more than an inch above the point of the flame of a
wax-candle, (see my Seventh Essay, Part II. page 350.) I
was curious to know what effect this heat would produce 011
the oxides of those metals.

Experiment No. 1. Having evaporated to dryness a solution
of fine gold in aqua regia, I dissolved the residuum, in just as
much distilled water as was necessary in order that the solution
(which was of a beautiful yellow colour) might not be disposed
to crystallize ; and, wetting the middle of a piece of white
taffeta riband, inch wide, and about eight inches long, in
this solution, I held the riband, with both my hands, stretched
horizontally over the clear bright flame of a wax candle ; the
tinder side of the riband being kept at the distance of about
!-§- inch above the point of the flame. The result of this expe-
riment wras very striking. That part of the riband which
was directly over the point of the flame, began almost imme-
diately to emit steam in dense clouds ; and, in about 10 seconds,
a circular spot, about f of an inch in diameter, having become
nearly dry, a spot of a very fine purple colour, approaching to
crimson, suddenly made its appearance in the middle of it,
and, spreading rapidly on all sides, became, in one or two se-
conds more, nearly an inch in diameter.

By moving the riband, so as to bring, in their turns, all the
parts of it which had been wetted with the solution to be ex-
posed to the action of the current of hot vapour that arose from
the burning candle, all those parts which had been so wetted,
were tinged with the same beautiful purple colour.

This colour, which was uncommonly brilliant, passed quite

chemical Properties attributed to Light. 451

through the riband, and I found the stain to be perfectly in-
delible. I endeavoured to wash it out ; but nothing I applied
to it, and among other things I tried super-oxygenated marine
acid, appeared in the smallest degree to diminish its lustre.
The hue was not uniform, but varied from a light crimson to
a very deep purple, approaching to a reddish brown.

I searched, but in vain, for traces of revived gold, in its re-
guline form and colour ; but, though I could not perceive that
the riband was gilded, it had all the appearance of being co-
vered with a thin coating of the most beautiful purple enamel,
which, in the sun, had a degree of brilliancy that was some-
times quite dazzling.

Experiment No. 2. A piece of the riband which had been
wetted with the aqueous solution of the oxide, was carefully
dried in a dark closet, and was then exposed, dry, over the
flame of a burning wax candle. The part of the riband which
had been wetted with the solution, (and which on drying had
acquired a faint yellow colour,) was tinged of the same bright
purple colour as was produced in the last-mentioned experi-
ment, when the riband was exposed wet to the action of the
heat. *

Experiment No. 3. A piece of the riband which had been
wetted with the solution, and dried in the dark, was now wetted
with distilled water, and exposed wet to the action of the ascend-
ing current of hot vapour which arose from the burning candle :
the purple stain was produced as before, which extended as far

* We shall hereafter find reason to conclude, that the success of this experiment, or
the appearance of the purple tinge, was owing to the watery vapour which existed in
the hot current that ascended from the flame of the candle,

3 M 2

453 Count Rumford's Inquiry concerning the

as the riband had been wetted with the solution, but no
farther.

I afterwards varied this experiment in several ways, some-
times using paper, sometimes fine linen, and sometimes fine
cotton cloths, instead of the silk riband ; but nearly the
same tinge was produced, whatever the substance was that
was made to imbibe the aqueous solution of the metallic
oxide.

Similar experiments, and with similar results, were likewise
made with pieces of riband, fine linen, cotton, paper, &c.
wetted in an aqueous solution of nitrate of silver ; with this
difference, however, that the tinge produced by this metallic
oxide, instead of being of a deep purple, inclining to crimson,
was of a very dark orange colour, or rather of a yellowish
brown.

In order to discover whether the purple tinge, in the expe-
riments with the oxide of gold, was occasioned by the heat
communicated by the ascending current of hot vapour, or
by the light of the candle, I made the following experiment,
the result of which, I conceive to have been decisive.

Experiment No. 4. A piece of riband was wetted with the
aqueous solution of the oxide of gold, and held vertically by
the side of the clear flame of a burning wax candle, at the
distance of less than half an inch from the flame.

The riband was dried, but its colour was not in the smallest
degree changed.

When it was held a few seconds within about ■§• of an inch
of the flame, a tinge of a most beautiful crimson colour, in the
form of a narrow vertical stripe, was produced.

I

chemical Properties attributed to Light. 453

The heat which existed at that distance from the flame, on
the side of it, where this coloured stripe was produced, was suf-
ficiently intense, as I found by experiment, to melt very fine
silver wire, flatted, such as is used in making silver lace.

The objects I had in view in the following experiments, are
too evident to require any particular explanation.

Experiment No. 5. Two like pieces of riband were wetted
at the same time in the solution, and suspended, while wet, in
two thin phials, A and B, of very transparent and colourless
glass ; the mouths of the phials being left open. Both these
phials were placed in a window which fronted the south ; that
distinguished by the letter A being exposed naked to the direct
rays of a bright sun ; while B was inclosed in a cylinder of paste-
board, painted black within and without, and closed with a fit
cover, and consequently remained in perfect darkness.

In a very few minutes, the riband in the phial A began sen-
sibly to change its colour, and to take a purple hue ; and, at
the end of five hours, it had acquired a deep crimson tint
throughout.

The phial B was exposed in the window, in its dark cylin-
drical cover, three days ; but there was not the smallest appear-
ance of any change of colour in the silk.

Experiment No. 6. Two small parcels of magnesia alba, in
an impalpable powder, (about half as much in each as could be
made to lie on a shilling,) were placed in heaps, in two China
plates, A and B, and thoroughly moistened with the before-
mentioned aqueous solution of the oxide of gold. Both
plates were placed in the same window ; the moistened earth
in the plate A being exposed naked to the sun’s rays ; whUe

\

454 Count Rumford’s Inquiry concerning the

that in the plate B was exactly covered with a tea-cup, turned
upside down, which excluded all light.

The magnesia alba in the plate A, which was exposed to the
strong light of the sun, began almost immediately to change
colour, taking a faint violet hue, which by degrees became
more and more intense, and in a few hours ended in a deep
purple ; while that in the plate B, which was kept in the dark,
retained the yellowish cast it had acquired from the solution,
without the smallest appearance of change.

Experiment No. 7. A small parcel of ynagnesia alba , placed
on a china plate, having been moistened with the aqueous solu-
tion of the oxide of gold, and thoroughly dried in a dark closet,
was now exposed, in this dry state , to the action of the direct
rays of a very bright sun.

It had been exposed to this strong light above half an hour,
before its colour began to be sensibly changed ; and, at the end
of three hours, it had acquired only a very faint violet hue.

Being now thoroughly wetted with distilled water, it changed
colour very rapidly, and soon came to be of a deep purple tint,
approaching to crimson.

Experiment No. 8. A piece of white taffeta riband, which
had been wetted with the solution, and thoroughly dried in the
dark, was suspended in a clean dry phial of very fine trans-
parent glass ; and the phial, being well stopped with a dry cork,
was exposed to the strong light of a bright sun.

After the riband had been exposed, in this manner, to. the
action of the sun's direct rays about half an hour, there were
here and there some faint appearances of a change of its colour ;
but it showed no disposition to take that deep purple hue which

chemical Properties attributed to Light. 455

the riband had always acquired, when exposed to the light in
the preceding experiments.

On taking the riband out of the phial, and wetting it tho-
roughly with distilled water, and exposing it again, while thus
wetted , to the sun’s rays, it almost instantly began to change
colour, and soon became of a deep purple tint ; but, though I
examined the surface of the riband with the utmost care, and
with a good lens, both during the experiment and after it, I
could not perceive the smallest particle of revived gold , nor did
I see any vestige remaining that appeared to indicate that any
had in fact been revived.

This experiment was repeated several times, and always with
results which led me to conclude, (what indeed was reasonable
to expect,) that light has little effect in changing the colour of
metallic oxides, as long as they are in a state of crystallization.

The heat which is generated by the absorption of the rays of
light must necessarily, at the moment of its generation at least,
exist in almost infinitely small spaces ; and consequently, it is
only in bodies that are inconceivably small that it can produce
durable effects, in any degree indicative of its extreme intensity.

Perhaps the particles of the oxide of gold dissolved in water,
are of such dimensions ; and it is very remarkable, that the co-
lours produced, in some of my experiments on white ribands,
Jby means of an aqueous solution of the oxide of gold, are pre-
cisely the same as are produced from the oxide of that metal, by
enamellers, in the intense heat of their furnaces.

As the colouring substance is the same, and as the colours
produced are the same, why should we not conclude that the
effects are produced in both these cases by the same means,
that is to say, by the agency of heat ? or, in other words, and

45 6 Count Rum ford's Inquiry concerning the

to be more explicit, by exposing the oxide in a certain tempe-
rature, at which it becomes disposed to vitrify, or to undergo
a change in regard to the quantity of oxygen with which it is
combined ?

But the results of the following experiments afford still
more satisfactory information, respecting the intensity of the
heat generated in all cases where light is absorbed, and the
striking effects which, under certain circumstances, it is ca-
pable of producing.

The facility with which most of the metallic oxides are re-
duced, in the dry way, by means of charcoal, shows that, at a
certain (high) temperature, oxygen is disposed to quit those
metals, in order to form a chemical union with the charcoal,
or at least with some one of its constituent principles, if it be
a compound substance ; and hence I concluded, that gold might
be revived, in the moist way, by means of charcoal, from a solu-
tion of its oxide in water, were it possible, under such circum-
stances, to communicate to the charcoal, and to the oxide, at
the same time, a degree of heat sufficient for that purpose.

To see if this might not be done by means of light, I made,
or rather repeated, the following very interesting experiment.

Experiment No. q. Into a thin tube of very fine colourless
glass, 10 inches long, and T% of an inch in diameter, closed
hermetically at its lower end, I put as many pieces of charcoal,
about the size of large peas, as filled the tube to the height of
two inches; and, having poured on them as much of the aqueous
solution of nitro-muriate of gold as nearly covered them, ex-
posed the tube, with its contents, to the action of the direct rays
of a very bright sun.

In less than half an hour, small specks of revived gold, in all

457

chemical Properties attributed to Light.

its metallic splendour, began to make their appearance here
and there on the surface of the charcoal ; and, in six hours, the
solution, which at first was of a bright yellow colour, became
perfectly colourless , and as clear and transparent as the

PUREST WATER.

The surface of the charcoal was, in several places, nearly
covered with small particles of revived gold ; and the inside
of the glass tube, in that part where it w'as in contact with the
upper surface of the contained liquid, was most beautifully
gilded.

This gilding of the tube was very splendid, when viewed by
reflected light ; but, when the tube was placed between the
light and the eye, it appeared like a thin cloud, of a greenish
blue colour, without the smallest appearance of any metallic
splendour.

From the colour, and apparent density of this cloud, I was
induced to conclude, that the gilding on the glass was less than
one millionth part of an inch in thickness.

This interesting experiment was repeated six times, and al-
ways with nearly the same result. The gold w'as completely
revived in each of them, and the solution left perfectly colour-
less ; in most of the experiments, however, the sides of the
glass were not gilded, all the revived gold remaining attached
to the surface of the charcoal.

In two of these experiments, I made use of pieces of charcoal
which had been previously boiled several hours in a large quan-
tity of distilled water, and which were introduced wet, and hot ,
into the tube, and immediately covered by the solution, to
prevent them from imbibing any air ; and, in different experi-
ments, the solution was used of different degrees of strength.

MDCCXCVI II. 3 N

458 Count Rumford's Inquiry concerning the

I plainly perceived that the experiment succeeded best, that
is to say, that the gold was soonest revived , in those cases in
which the solution was most diluted : one of the experiments,
however, and which succeeded perfectly, was made with the
solution so much condensed, that it was nearly at the point at
which it became disposed to crystallize.*

On examining, with a good microscope, the particles of
revived gold which remained attached to the surface of the
charcoal, after it had been dried, I found them to consist of
an infinite number of small scales, separated from each other ;
not very highly polished, but possessing the true metallic
splendour, and a very deep and rich gold colour.

The gold which attached itself to the inside of the glass
tube, was in the form of a ring, about of an inch wide,
(badly defined however below,) and adhered to the glass
with so much obstinacy, as not to be removed by rincing out
the tube a great number of times with water ; it had, as has
already been observed, a very high polish, when seen by re-
flected light.

Those who enter into the spirit of these investigations, will
easily imagine how impatient I must have been, after seeing
the results of these experiments, to find out whether gold
could be revived from this aqueous solution of its oxide by
means of charcoal, without the assistance of light , and merely by
such a degree of equal heat as could be given to it in the dark.
To determine that important question, the following experi-
ment was made.

* This agrees perfectly with the results of similar experiments made by the inge-
nious and lively Mrs. Folhame. (See her Essay on Combustion, page 124.)

It was on reading her book, that I was induced to engage in these investigations ; and
it was by her experiments, that most of the foregoing experiments were suggested.

459

chemical Properties attributed to Light.

Expe?~iment No. 10. A cylindrical glass tube, of an inch
in diameter, and 10 inches long, closed hermetically at its lower
end, and containing a quantity of a diluted aqueous solution of
the oxide of gold, mixed with charcoal in broken pieces, about
the size of large peas, was put into a fit cylindrical tin case,

. which was'nicely closed with a fit cover; and the glass tube,
with its contents, so shut up in the dark, was exposed two
hours, in the temperature of 210° of Fahrenheit's scale.

On taking the glass tube out of its tin case, I found the solu-
tion perfectly colourless , and the revived gold adhering to the
surface of the charcoal.

On repeating the experiment, and using the solution nearly
saturated with the oxide, the result was precisely the same ;
the solution being found perfectly colourless, and the revived
gold adhering to the surface of the charcoal.

I own fairly, that the results of these experiments were quite
contrary to my expectations, and that I am not able to recon-
cile them with my hypothesis, respecting the causes of the re-
duction of the oxide, in the foregoing experiments ; but, what-
ever may be the fate of this, or of any other hypothesis of
mine, I hope and trust that I never shall be so weak as to feel
pain at the discovery of truth, however contrary it may be to
my expectations ; and still less, to feel a secret wish to suppress
experiments, merely because their results militate against my
speculative opinions.

It is proper 1 should observe, that the charcoal used in
this last-mentioned experiment had been boiled two hours in
distilled water, by which means its pores had been so com-
pletely filled with that fluid, that the pieces of it that

3 N 2

were

460 Count Rumford’s Inquiry concerning the

used were specifically heavier than water, and sunk in it, to the
bottom of the containing vessel.

Having been so successful in my attempts to reduce the
oxide of gold, by means of charcoal, in the moist way, I lost no
time in making similar experiments with the Oxide of silver.

Experiment No. 11. A solution of fine silver, in strong ni-
trous acid, was evaporated to dryness, and the residuum redis-
solved in distilled water.

A portion of this solution, (which was perfectly colourless,)
diluted with twice as much distilled water, was poured into a
phial containing a number of small pieces of charcoal; and the
phial, being well closed with a new cork stopple, was exposed
to the action of the sun’s rays.

In less than an hour, small specks of revived silver began
to make their appearance on the surface of the charcoal ; and,
at the end of two hours, these specks became very numerous,
and had increased so much in size, that they were distinctly vi-
sible to the naked eye, at the distance of more than three feet.
They were very white, and possessed the metallic splendour of
silver in so high a degree, that when enlightened by the sun’s
beams, their lustre was nearly equal to that of very small dia-
monds.

The phial, which was in the form of a pear, and about ij-
inch in diameter at its bulb, was very thin, and made of very
fine colourless glass ; the aqueous solution was also perfectly
transparent and colourless ; and, when the contents of the
phial were illuminated by the direct rays of a bright sun, the
contrast of the white colour of these little metallic spangles
with thd black charcoal to which they were fixed, and their

chemical Properties attributed to Light . 461

extreme brilliancy, afforded a very beautiful and interesting
sight.

As the air had been previously expelled from the charcoal,
by boiling it in distilled water, it was specifically heavier than
the aqueous solution of the metallic oxide, and consequently
remained at the bottom of the bottle.

Experime?it No. 12. A phial, as nearly as possible like that
used in the last experiment, and containing the same quantity
of diluted aqueous solution of nitrate of silver, and also of char-
coal, was inclosed in a cylindrical tin box, and exposed one hour
to the heat of boiling water, in an apparatus used for boiling
potatoes in steam, for the table.

The result of this experiment was uncommonly striking.
The surface of the charcoal was covered with a most beautiful
metallic vegetation ; small filaments of revived silver, resem-
bling fine flatted silver wire, pushing out from its surface, in all
directions !

Some of these metallic filaments were above one-tenth of an
inch in length. On agitating the contents of the phial, they
were easily detached from the surface of the charcoal, to which
they seemed to adhere but very slightly.

These experiments were repeated several times, and always
with precisely the same results.

When the oxide of gold was reduced in this way, the revived
metal appeared under the form of small scales, adhering firmly
to the surface of the charcoal. May not the difference of the
forms under which gold and silver are revived from their oxides,
in this process, be owing to the difference of the specific gravi-
ties of those metals ?

4^ 2 Count Rumford s Inquiry concerning the

The following experiments, which were first suggested by an
accident, were made with a view to investigate still farther the
causes of those effects which have been attributed to the sup-
posed chemical properties of light.

Having accidentally put away two small phials, each con-
taining a quantity of aqueous solution of the oxide of gold and
sulphuric ether, in each of which the ether had extracted the
gold completely from the solution, as was evident by the yellow
colour of the solution having been transferred to the ether, and
the solution being left colourless ; in one of the phials, which
happened to stand in a window in which there was occasionally
a strong light, (though the direct rays of the sun never fell on
it,) I found, in about three weeks, that the oxide was almost
entirely reduced ; the revived gold appearing in all its metallic
splendour, in the form of a thin pellicle, swimming on the sur-
face of the aqueous liquor in the phial, and the colour of the
ether which reposed on it having become quite faint; while no
visible change had been produced in the contents of the other
phial, which had stood in a dark corner of the room.

As these appearances induced me to suspect, or rather
strengthened the suspicions I had before conceived, that the
separation of gold from ether, under its metallic form, when a
solution of its oxide is mixed with that fluid, is always effected
by a reduction of the oxide by means of light, I made the
following experiment, with a view to the farther investigation
of that matter.

Experiment No. 13. Into a small pear-like phial, of very fine
transparent glass, I put equal quantities of an aqueous solution
of the muriatic oxide of gold and sulphuric ether ; and the phial.

chemical Properties attributed to Light . 463

which was about half filled, being closed with a good cork, well
secured in its place, was exposed to the action of the direct rays
of a bright sun.

A pellicle of revived gold, in all its metallic splendour, began
almost immediately to be formed on the surface of the aqueous
liquid, and soon covered it entirely; and, at the end of two hours,
the whole of the oxide was completely reduced, as was evident
from the appearance of the ether, which became perfectly co-
lourless.

On shaking the phial, the metallic pellicle which covered the
surface of the aqueous liquid was broken into small pieces, which
had exactly the appearance of leaf gold, possessing the true
colour, and all the metallic brilliancy, of that metal.

On suffering the phial to stand quiet, the aqueous liquor and
the ether separated, and most of the broken pieces of the thin
sheet of gold descended to the bottom of the phial : the remain-
der of them floated on the surface of the aqueous liquid ; and
the ether, as well as the aqueous liquid, appeared to be perfectly
transparent and colourless.

By the length of time which was required for the ether and
the aqueous liquid to separate, I thought I could perceive that
the ether had lost something of its fluidity; but, as this was an
event I expected, it is the more likely, on that account, that I

was deceived, when I imagined I saw proofs of its having taken
place.

On removing the cork, after the contents of the bottle had
been suffered to cool, there was no appearance of any consider-
able quantity of air, or other permanently elastic fluid, having
been either generated or absorbed, during the experiment.

Finding that the oxide of gold might be so completely and

4t>4 Count Rumford's Inquiry concerning the

so expeditiously reduced, by means of ether, I conceived it
might be possible to perform that chemical process, in the moist
way, by means of essential oils ; and this conjecture proved to
be well founded.

Experiment No. 14. Upon a quantity of a diluted aqueous
solution of nitro-muriate of gold, in a small pear-like phial,
about i|- inch in diameter at its bulb, was poured a small
quantity of etherial oil of turpentine, just as much as was suf-
ficient to cover the aqueous solution to the height of ~ of an
inch ; and the phial, being well closed with a good cork, well
secured, was exposed one hour to the heat of boiling water in
a steam-vessel.

The gold was revived, appearing in the form of a splendid
pellicle, of a bright gold colour, which floated on the surface
of the aqueous liquid. The oil of turpentine, which, at the
beginning of the experiment, was as pale and colourless as
pure water, had taken a bright yeliovv hue ; and the aqueous
fluid, on which it reposed, had entirely lost its yellow colour.

On shaking the phial, its contents were intimately mixed ;
but, on suffering it to stand quiet, the oil of turpentine soon
* separated from the aqueous liquid, retaining its bright yellow
hue, and leaving the aqueous liquid colourless.

On shaking the phial, before it had been exposed to the heat,
and mixing its contents, and then suffering it to stand quiet,
the oil of turpentine, on taking its place at the top of the
aqueous solution, was not found to have acquired any colour ;
nor was the bright gold colour of the solution found to be at
all impaired. When sulphuric ether was used, instead of the
oil of turpentine, the effect was in this respect very different.

To find out whether the oil of turpentine used in this expe-

chemical 'Properties attributed to Light. 4,6*5

riment, and which had acquired a deep yellow colour, had lost
that property by which it effected the reduction of the metallic
oxide, I now poured an additional quantity of the aqueous so-
lution of the oxide into the phial, and, shaking the phial, exposed
it, with its contents, to the heat of boiling water.

After it had been exposed to this heat about two hours, I
examined it, and found, that though a considerable quantity of
gold had been revived, yet the aqueous liquid still retained a
faint yellow colour.

The oil of turpentine had acquired a deeper and richer gold
colour, approaching to orange.

To the contents of the phial, I now added about half as much
distilled water, and, mixing the whole by shaking, I exposed the
phial again, during two hours, to the heat of boiling water;
when the remainder of the oxide was reduced, and the aqueous
liquid left perfectly colourless.

On repeating this experiment with oil of turpentine, and va-
rying it, by using a solution of the oxide of silver , (an aqueous
solution of nitrate of silver,) instead of that of gold, the result
was nearly the same : the metal was revived, and the oil of
turpentine acquired a faint greenish-yellow colour.

I also revived the oxides of gold and of silver with oil of olives ,
by a similar process, with the heat of boiling water. The oil of
olives used in these experiments lost its transparency, and be-
came deeply coloured ; that used in the reduction of the oxide
of silver, taking a very deep dirty brown colour, approaching to
black ; and that employed in reducing the oxide of gold, being
changed to a yellowish-brown, with a purple hue.

In the experiment with the oxide of silver, the inside of the
phial, in the region where the oil reposed on the aqueous solu-

mdccxcviii. 3 O

^66 Count Rumford’s Inquiry concerning the

tion, was beautifully silvered, the revived metal forming a nar-
row metallic ring, extending quite round the phial; and, in both
experiments, small detached pellicles of revived metal were vi-
sible in the oil, and adhered in several places to the inside of
the phial, forming bright spots, in which the colour of the me-
tal, and its peculiar splendour, were perfectly conspicuous.

Experiment No. 15. As carbon is one of the constituent prin-
ciples of spirit of wine, as well as of essential oils and sulphuric
ether, I thought it possible that I might succeed in the reduc-
tion of the oxide of gold, by mixing alcohol, with an aqueous
solution of nitro-muriate of gold, and exposing the mixture, in
a phial well closed, to the heat of boiling water ; but the expe-
riment did not succeed.

By pouring upon this mixture a small quantity of oil of
olives, and exposing it again to the heat of boiling water, the
gold was revived.

Is it not probable, that the reason why the oxide was not
reduced by alcohol, is the mobility of those elements, which
ought to act on each other, in order that the effect in question
may be produced ? I have no doubt but the oxide would be
reduced, could the alcohol be made to rest on the surface of the
aqueous solution, without mixing with it.

I wished to have been able to have collected and examined
the elastic fluids, which probably were formed in most of the
preceding experiments ; but my time was so much taken up
with other matters, that I had not leisure to pursue these inves-
tigations farther.

In order to see what effects would be produced by the -heat
generated at the surface of an opaque body, of a nature different
from those hitherto used in the reduction of the metallic oxides,

chemical Properties attributed to Light. 467

and one that is little disposed to form a chemical union with
oxygen, ( magnesia alba,) when, being immersed in an aqueous
solution of the oxide of gold, the rays of the sun were made
to impinge on it, I contrived the following experiment.

Experiment No. 16. I took four small thin phials, A, B, C,
and D, of very line glass, and, putting into each of them about
.five grains of dry magnesia alba, I filled the phial A, nearly full,
with a saturated aqueous solution of the oxide of gold.

I filled the phial B, in like manner, with some of the same
solution, diluted with an equal quantity of distilled water; and
the phials C and D were filled with the solution still farther
diluted.

These phials, open or without stoppers, were exposed one
whole day to the action of the direct rays of a bright sun, their
contents being often well mixed together, during that time, by
shaking.

The contents of all these phials changed colour, more or less,
but they acquired very different hues. The contents of the
phial A became of a very deep rich gold colour, approaching to
orange, the earthy sediment being throughout of the same tint.

The contents of the phial B, which were at first of a light
straw colour, first changed to a light green, and then to a
greenish blue. The phial having been suffered to stand quiet
several days, in an uninhabited room, in a retired part of the
house, the solution became nearly colourless, and the sediment
was found to be of a dirty olive colour.

The colour of the contents of the phials C and D was changed
nearly in the same manner ; and, having been suffered to stand
quiet two or three days, to settle, the solution was found to be
quite colourless, and the sediment to be deeply coloured. There

3 O 2

468 Count Rumford's Inquiry , &c.

was, however, a very remarkable difference in the hues of the
two phials ; that of the phial C being of a light greenish-blue ;
while that in the phial D was indigo, and of so deep a tint, that
it might easily have been taken for black.

These appearances were certainly very striking, and well
calculated to excite my curiosity ; but I am so much engaged
jn public business, that it is not at present in my power to pur-
sue these inquiries farther. I wish that what I have done may
induce others, who have more time to spare, to devote some
portion of their leisure to these interesting investigations.

C 469 3

XXI. Experiments to determine the Density of the Earth. By
Henry Cavendish, Esq. F.R.S. and A.S.

Read June 21, 1798.

JVIany years ago, the late Rev. John Miciiell, of this Society,
contrived a method of determining the density of the earth, by
rendering sensible the attraction of small quantities of matter ;
but, as he was engaged in other pursuits, he did not complete
the apparatus till a short time before his death, and did not
live to make any experiments with it. After his death, the
apparatus came to the Rev. Francis John Hyde Wollaston,
Jacksonian Professor at Cambridge, who, not having conveni-
ences for making experiments with it, in the manner he could
wish, was so good as to give it to me.

The. apparatus is very simple ; it consists of a wooden arm,
6 feet long, made so as to unite great strength with little
weight. This arm is suspended in an horizontal position, by
a slender wire 40 inches long, and to each extremity is hung a
leaden ball, about 2 inches in diameter; and the whole is in-
closed in a narrow wooden case, to defend it from the wind.

As no more force is required to make this arm turn round
on its centre, than what is necessary to twist the suspending
wire, it is plain, that if the wire is sufficiently slender, the most
minute force, such as the attraction of a leaden weight a few
inches ip diameter, will be sufficient to draw the arm sensibly
aside. The weights which Mr. Michell intended to use were

470 Mr. Cavendish's Experiments to determine

8 inches diameter. One of these was to be placed on one side the
case, opposite to one of the balls, and as near it as could conve-
niently be done, and the other on the other side, opposite to the
other ball, so that the attraction of both these weights would con-
spire in drawing the arm aside; and, when its position, as affected
by these weights, was ascertained, the weights were to be re-
moved to the other side of the case, so as to draw the arm the
contrary way, and the position of the arm was to be again de-
termined ; and, consequently, half the difference of these posi-
tions would shew how much the arm was drawn aside by the
attraction of the weights.

In order to determine from hence the density of the earth, it
is necessary to ascertain what force is required to draw the arm
aside through a given space. This Mr. Michell intended to
do, by putting the arm in motion, and observing the time of its
vibrations, from which it may easily be computed.*

Mr. Michell had prepared two wooden stands, on which
the leaden weights were to be supported, and pushed forwards,
till they came almost in contact with the case ; but he seems
to have intended to move them by hand.

As the force with which the balls are attracted by these

weights is excessively minute, -not more than Q- of their

weight, it is plain, that a very minute disturbing force will be
sufficient to destroy the success of the experiment; and, from
the following experiments it will appear, that the disturbing

* Mr. Coulomb has, in a variety of cases, used a contrivance, of this kind for
trying small attractions ; but Mr. Michell informed me of his intention of making
this experiment, and of the method he intended to use, before the publication of any
of Mr. Coulomb’s experiments.

47*

the Density of the Earth.

force most difficult to guard against, is that arising from the
variations of heat and cold; for, if one side of the case is warmer
than the other, the air in contact with it will be rarefied, and,
in consequence, will ascend, while that on the other side will
descend, and produce a current which will draw the arm sen-
sibly aside.*

As I was convinced of the necessity of guarding against this
source of error, I resolved to place the apparatus in a room
which should remain constantly shut, and to observe the motion
of the arm from without, by means of a telescope; and to sus-
pend the leaden weights in such manner, that I could move them
without entering into the room. This difference in the man-
ner of observing, rendered it necessary to make some alteration
in Mr. Micheli/s apparatus ; and, as there were some parts of
it which I thought not so convenient as could be wished, I
chose to make the greatest part of it afresh.

Fig. 1. (Tab. XXIII.) is a longitudinal vertical section
through the instrument, and the building in which it is placed.
ABCDDCBAEFFE, is the case ; x and x are the two balls,
which are suspended by the wires hx from the arm ghmh ,
which is itself suspended by the slender wire gl. This arm
consists of a slender deal rod bmb , strengthened by a silver

* M. Cassini, in observing the variation compass placed by him in the Observatory,
(which was constructed so as to make very minute changes of position visible, and
in which the needle was suspended by a silk thread,) found that standing near the box,
in order to observe, drew the needle sensibly aside ; which I have no doubt was caused
by this current of air. It must be observed, that his compass-box was of metal, which
transmits heat faster than wood, and also was many inches deep ; both which causes
served to increase the current of air. To diminish the effect of this current, it is by
all means advisable to make the box, in which the needle plays, nor much deeper than
h necessary to prevent the needle from striking against the top and bottom.

47s Mr- Cavendish's Experiments to determine

wire hgh ; by which means it is made strong enough to sup-
port the balls, though very light. *

The case is supported, and set horizontal, by four screws, rest-
ing on posts fixed firmly into the ground : two of them are
represented in the figure, by S and S ; the two others are not
represented, to avoid confusion. GG and GG are the end walls
of the building. W and W are the leaden weights ; which are
suspended by the copper rods RrPrR, and the wooden bar rr ,
from the centre pin P p. This pin passes through a hole in the
beam HH, perpendicularly over the centre of the instrument,
and turns round in it, being prevented from falling by the
plate p. MM is a pulley, fastened to this pin ; and Mm, a cord
wound round the pulley, and passing through the end wall;
by which the observer may turn it round, and thereby move
the weights from one situation to the other.

Fig. 2. (Tab. XXIV.) is a plan of the instrument. AAAA is the
case. SSSS, the four screws for supporting it. hh, the arm and
balls. W and W, the weights. MM, the pulley for moving them.
When the weights are in this position, both conspire in drawing
the arm in the direction h W; but, when they are removed to the
situation w and w, represented by the dotted lines, both conspire
in drawing the arm in the contrary direction hw. These weights
are prevented from striking the instrument, by pieces of wood,
which stop them as soon as they come within j of an inch of the

* Mr. Mich ell’s rod was entirely of wood, and was much stronger and stiffer
than this, though not much heavier ; but, as it had warped when it came to me, I chose
to make another, and preferred this form, partly as being easier to construct and
meeting with less resistance from the air, and partly because, from its being of a less
complicated form, I could more easily compute how much it was attracted by the
weights.

473

the Density of the Earth .

case. The pieces of wood are fastened to the wall of the build-
ing ; and I find, that the weights may strike against them with
considerable force, without sensibly shaking the instrument.

In order to determine the situation of the arm, slips of ivory
are placed within the case, as near to each end of the arm as
can be done without danger of touching it, and are divided to
eoths of an inch. Another small slip of ivory is placed at each
end of the arm, serving as a vernier, and subdividing these
divisions into 5 parts ; so that the position of the arm may be
observed with ease to tooths of an inch, and may be estimated
to less. These divisions are viewed, by means of the short tele-
scopes T and T, (fig. 1.) through slits cut in the end of the case,
and stopped with glass ; they are enlightened by the lamps L
and L, with convex glasses, placed so as to throw the light on
the divisions ; no other light being admitted into the room.

The divisions on the slips of ivory run in the direction Wzu,
(fig. 2.) so that, when the weights are placed in the positions
w and w, represented by the dotted circles, the arm is drawn
aside, in such direction as to make the index point to a higher
number on the slips of ivory; for which reason, I call this the
positive position of the weights.

FK, (fig. 1.) is a wooden rod, which, by means of an endless
screw, turns round the support to which the wire gl is fastened,
and thereby enables the observer to turn round the wire, till the
arm settles in the middle of the case, without danger of touch-
ing either side. The wire gl is fastened to its support at top,
and to the centre of the arm at bottom, by brass clips, in which
it is pinched by screws.

In these two figures, the different parts are drawn nearly in

MDCCXC VIII . g P

474 Mr- Cavendish's Experiments to determine

the proper proportion to each other, and on a scale of one to
thirteen.

Before I proceed to the account of the experiments, it will be
proper 10 say something of the manner of observing. Suppose
the arm to be at rest, and its position to be observed, let the
weights be then moved, the arm will not only be drawn aside
thereby, but it will be made to vibrate, and its vibrations will
continue a great while; so that, in order to determine how much
the arm is drawn aside, it is necessary to observe the extreme
points of the vibrations, and from thence to determine the point
which it would rest at if its motion was destroyed, or the point
of rest, as I shall call it. To do this, I observe three successive
extreme points of a vibration, and take the mean between the first
and third of these points, as the extreme point of vibration in one
direction, and then assume the mean between this and the se-
cond extreme, as the point of rest ; for, as the vibrations are
continually diminishing, it is evident, that the mean between
two extreme points will not give the true point of rest.

It may be thought more exact, to observe many extreme
points of vibration, so as to find the point of rest by different
sets of three extremes, and to take the mean result; but it must
be observed, that notwithstanding the pains taken to prevent
any disturbing force, the arm will seldom remain perfectly at
rest for an hour together ; for which reason, it is best to deter-
mine the point of rest, from observations made as soon after the
motion of the weights as possible.

The next thing to be determined is the time of vibration,
which I find in this manner : I observe the two extreme points
of a vibration, and also the times at which the arm arrives at

A

475

the Density of the Earth .

two given divisions between these extremes, taking care, as well
as I can guess, that these divisions shall be on different sides of
the middle point, and not very far from it. I then compute
the middle point of the vibration, and, by proportion, find the
time at which the arm comes to this middle point. I then, after
a number of vibrations, repeat this operation, and divide the
interval of time, between the coming of the arm to these two
middle points, by the number of vibrations, which gives the
time of one vibration. The following example will explain
what is here said more clearly.

Extreme

Points.

Division.

Time.

Point of
rest.

Time of middle
of vibration.

27,2

h.

i v

u

25

IO

23 41

n.

/

R

24

571

10

23

23

22,1

-

-

-

24 >6

27

-

-

-

24.7

22,6'

-

-

-

24-75

2b, 8

-

-

-

24,8

23

-

-

-

24*85

2 6,6

-

-

-

24*9

25

11

5 221

2 4

6 48 i

1 1

5

22

23*4

1

!

The first column contains the extreme points of the vibra-
tions. The second, the intermediate divisions. The third, the
time at which the arm came to these divisions ; and the fourth,
the point of rest, which is thus found : the mean between the
first and third extreme points is 27,1, and the mean between
this and the second extreme point is 24,6, which is the point of
rest, as found by the three first extremes. In like manner, the

3 P 2

476 Mr. Cavendish’s Experiments to determine

point of rest found by the second, third, and 4th extremes, is
24,7, and so on. The fifth column is the time at which the
arm came to the middle point of the vibration, which is thus
found: the mean between 27,2 and 22,1 is 24,65, and is the
middle point of the first vibration; and, as the arm came to 25
at ioh23/4", and to 24 at ioh 23' 57", we find, by proportion,
that it came to 24,65 at ioh 23' 23". In like manner, the arm
came to the middle of the seventh vibration at nh 5' 22"; and,
therefore, six vibrations were performed in 4 P 59", or one vibra-
tion in y' o".

To judge of the propriety of this method, we must consider
in what manner the vibration is affected by the resistance of
the air, and by the motion of the point of rest.

Let the arm, during the first vibration, move from D to B, (Tab.
XXIV. fig- 3.) and, duringthe second, from B to d; Bdbeingless
thanDB, on account of the resistance. Bisect DB in M, and B d in
m} and bisect Mm in n, and let x be any point in the vibration ;
then, if the resistance is proportional to the square of the velocity,
the whole time of a vibration is very little altered ; but, if T is
taken to the time of one vibration, as the diameter of a circle
to its semicircumference, the time of moving from B to n ex-
ceeds •§• a vibration, by nearly ; and the time of moving

from B to m falls short of j- a vibration, by as much ; and the
time of moving from B to x, in the second vibration, exceeds

that of moving from x to B, in the first, by - — /

supposing D d to be bisected in $ ; so that, if a mean is taken,
between the time of the first arrival of the arm at x and its re-
turning back to the same point, this mean will be earlier than

the true time of its coming to B, by

477

the Density of the Earth.

The effect of motion in the point of rest is, that when the
arm is moving in the same direction as the point of rest, the
time of moving from one extreme point of vibration to the
other is increased, and it is diminished when they are moving
in contrary directions ; but, if the point of rest moves uniformly,
the time of moving from one extreme to the middle point of
the vibration, will be equal to that of moving from the middle
point to the other extreme, and moreover, the time of two suc-
cessive vibrations will be very little altered ; and, therefore, the
time of moving from the middle point of one vibration to the
middle point of the next, will also be very little altered.

It appears, therefore, that on account of the resistance of
the air, the time at which the arm comes to the middle point of
the vibration, is not exactly the mean between the times of its
coming to the extreme points, which causes some inaccuracy
in my method of finding the time of a vibration. It must be
observed, however, that as the time of coming to the middle
point is before the middle of the vibration, both in the first and
last vibration, and in general is nearly equally so, the error
produced from this cause must be inconsiderable ; and, on the
whole, I see no method of finding the time of a vibration which
is liable to less objection.

The time of a vibration may be determined, either by pre-
vious trials, or it may be done at each experiment, by ascer-
taining the time of the vibrations which the arm is actually put
into by the fnotion of the weights ; but there is one advantage
in the latter method, namely, that if there should be any acci-
dental attraction, such as electricity, in the glass plates through
which the motion of the arm is seen, which should increase
the force necessary to draw the arm aside, it would also dimi-

4<78 Mr- Cavendish’s Experiments to determine

nish the time of vibration; and, consequently, the error in the
result would be much less, when the force required to draw the
arm aside was deduced from experiments made at the time,
than when it was taken from previous experiments.

Account of the Experiments .

In my first experiments, the wire by which the arm was sus-
pended was 33^ inches long, and was of copper silvered, one
foot of which weighed 2T% grains : its stiffness was such, as to
make the arm perform a vibration in about 15 minutes. I imme-
diately found, indeed, that it was not stiff enough, as the at-
traction of the weights drew the balls so much aside, as to make
them touch the sides of the case ; I, however, chose to make
some experiments with it, before I changed it.

In this trial, the rods by which the leaden weights were sus-
pended were of iron ; for, as I had taken care that there should
he nothing magnetical in the arm, it seemed of no signification
whether the rods were magnetical or not ; but, for greater se-
curity, I took off the leaden weights, and tried what effect the
rods would have by themselves. Now I find, by computation,
that the attraction of gravity of these rods on the balls, is to
that of the weights, nearly as 17 to 2300 ; so that, as the attrac-
tion of the weights appeared, by the foregoing trial, to be suffi-
cient to draw the arm aside by about 35 divisions, the attraction
of the rods alone should draw it aside about y1^ of a division ;
and, therefore, the motion of the rods from one near position to
the other, should move it about ^ of a division.

The result of the experiment was, that for the first 15
minutes after the rods were removed from one near position
to the other, very little motion was produced in the arm, and

479

the Density of the Earth.

hardly more than ought to be produced by the action of gra-
vity; but the motion then increased, so that, in about a quarter
or half an hour more, it was found to have moved or \\
division, in the same direction that it ought to have done by
the action of gravity. On returning the irons back to their
former position, the arm moved backward, in the same man-
ner that it before moved forward.

It must be observed, that the motion of the arm, in these
experiments, was hardly more than would sometimes take place
without any apparent cause ; but yet, as in three experiments
which were made with these rods, the motion was constantly
of the same kind, though differing in quantity from 4 to i £
division, there seems great reason to think that it was pro-
duced by the rods.

As this effect seemed to me to be owing to magnetism,
though it was not such as I should have expected from that
cause, I changed the iron rods for copper, and tried them as
before; the result was, that there still seemed to be some
effect of the same kind, but more irregular, so that I attributed
it to some accidental cause, and therefore hung on the leaden
weights, and proceeded with the experiments.

It must be observed, that the effect which seemed to be pro-
duced by moving the iron rods from one near position to the
other, was, at a medium, not more than one division; whereas
the effect produced by moving the weight from the midway to
the near position, was about 15 divisions; so that, if I had con-
tinued to use the iron rods, the error in the result caused there-
by, could hardly have exceeded of the whole.

480 Mr. Cavendish’s Experiments to determine

EXPERIMENT I. Aug. 5.

Weights in midway position.

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of mid.
of vibration.

Difference.

h- , „

h. , ,

, #

11,4

9 42 0

n,5

55 0

n,5

b-L

O

O

H,5

At ioh 5', weights moved to positive position.

23>4

27, 6

-

#5= mm

25,82

H>7

-

I** a

20,07

27,3

-

-

26,1

35,1

*53 SB

At nh 6', weights returned back to midway position.

5,

11

O

O

00

0 1 13

12

1 30J

18,2

GS

ea =»

12

12

16 29-1

l6 9

6,6

11

17 20/

-

-

11,92

- —

11

12

30 241

31 11 1

-

3° 45

16,3

-

-

11,72

— —

12

11

45 58 1
47 4/

-

45 5s

7.7

14 56

14 g6

15 13

Motion on moving from midway to pos.

pos. to midway
Time of one vibration

= H’3%
= 14A

= H' 55"

the Density of the Earth. 481

It must be observed, that in this experiment, the attraction
of the weights drew the arm from 1 1,5 to 25,8, so that, if no
contrivance had been used to prevent it, the momentum ac-
quired thereby would have carried it to near 40, and would,
therefore, have made the balls to strike against the case. To
prevent this, after the arm had moved near 15 divisions, I re-
turned the weights to the midway position, and let them remain
there, till the arm came nearly to the extent of its vibration, and
then again moved them to the positive position, whereby the
vibrations were so much diminished, that the balls did not
touch the sides ; and it was this which prevented my obser-
ving the first extremity of the vibration. A like method was
used, when the weights were returned to the midway position,
and in the two following experiments.

The vibrations, in moving the weights from the midway to
the positive position, were so small, that it was thought not
worth while to observe the time of the vibration. When the
weights were returned to the midway position, I determined
the time of the arm's coming to the middle point of each vibra-
tion, in order to see how nearly the times of the different vi-
brations agreed together. In great part of the following expe-
riments, I contented myself with observing the time of its
coming to the middle point of only the first and last vibration.

3 2

MDCCXCVIII.

482 Mr. Cavendish’s Experiments to determine

EXPERIMENT II. Aug. 6.

i

Weights in midway position.

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of mid.
of vibration.

Difference

1

/ u

h- , n

/ «

11

IO 4 O

11

11 O

11

17 O

11

25 O

11,

Weights moved to positive position.

%9>3

24,1

26,87

30

-

-

27,57

26,2,

-

- -

28,02

29,7

-

28,12

2 6,9

-

28,05

28,7

-

-

27,85

27,1

-

-

27,82

28,4

Weights returned to midway position.

6

18,5

12

13

1 3 5o]
4 34J

12,37

141

14 52

6,5

13

12

l8 2Q\

13 18/

11,67

18 53

14 46

15,2

11

12

S3 48}
34 51 J

» 1 1

11

33 39

13 46

13

12

45 8-1

46 22 J

—

47 25

/

the Density of the Earth .

Extreme

points.

Divisions.

Time*

Point of
rest.

Time of mid.
of vibration.

Difference.

7A

h* , n

h. , .

/ n

<Cjb

11

2 3 48^

10 75

15 25

13.6

12

5 18/

2 2 50

Motion of arm on moving weights from

midway to pos.
pos. to midway
Time of one vibration -

= 15,87
= 15,45

= 14' 4s'

Experiment iii. Aug. 7.

The weights being in the positive position, and the '

arm a little in motion.

3E5

29

-

30,12

3i

23, 1

30,02

Weights moved to midway position.

9

14

15

xo 34 18-1
35 8->

-

10 34 55

20,5

15

14

49 3i 1
3° 27/

14,8

49 39

9,2

14

15

11 5 71

6 18/

14,°7

11 4 17

322

483

♦ \

\

484 Mr. Cavendish's Experiments to determine

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of mid.
of vibration.

Difference.

h- , «

/ n

174

-

-

13>52

14 47

14

11 l8 46^

1 9 58/

-

11 19 4

10,1

-

-

13.3

-

14 27

13

33 461

33 31

14

35 26 J

15,6

Weights moved to positive position.

32

28

27

0 2 48 1

3 56 /

-

0 2 59

23,7

-

- -

27,8

31,8

-

mm mm

28,27

25,8

-

-

28,62

27

28

Of Ot

0 00

-

47 4°

3',1

Motion of the arm on moving weights from

pos.tomid. - =15,22

mid. to pos. - = 14,5

Time of one vibration, when in mid. position =14' 39"

pos. position = 14 54

These experiments are sufficient to shew, that the attraction
of the weights on the balls is very sensible, and are also suffi-
ciently regular to determine the quantity of this attraction
pretty nearly, as the extreme results do not differ from each
other by more than ~ part. But there is a circumstance in
them, the reason of which does not readily appear, namely,
that the effect of the attraction seems to increase, for half an

the Density of the Earth. 485

hour, or an hour, after the motion of the weights ; as it may be
observed, that in all three experiments, the mean position kept
increasing for that time, after moving the weights to the posi-
tive position ; and kept decreasing, after moving them from the
positive to the midway position.

The first cause which occurred to me was, that possibly there
might be a want of elasticity, either in the suspending wire, or
something it was fastened to, which might make it yield more
to a given pressure, after a long continuance of that pressure,
than it did at first.

To put this to the trial, I moved the index so much, that the
arm, if not prevented by the sides of the case, would have stood
at about 50 divisions, so that, as it could not move farther than
to 35 divisions, it was kept in a position 15 divisions distant
from that which it would naturally have assumed from the stiff-
ness of the wire; or, in other words, the wire was twisted 15
divisions. After having remained two or three hours in this
position, the index was moved back, so as to leave the arm at
liberty to assume its natural position.

It must be observed, that if a wire is twisted only a little
more than its elasticity admits of, then, instead of setting,
as it is called, or acquiring a permanent twist all at once, it
sets gradually, and, when it is left at liberty, it gradually loses
part of that set which it acquired ; so that if, in this experiment,
the wire, by having been kept twisted for two or three hours,
had gradually yielded to this pressure, or had begun to set, it
would gradually restore itself, when left at liberty, and the point
of rest would gradually move backwards ; but, though the ex-
periment was twice repeated, I could not perceive any such
effect.

The arm was next suspended by a stifter wire.

486 Mr. Cavendish’s Experiments to determine

EXPERIMENT IV. Aug. 12.
Weights in midway position.

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of mid.
of vibration.

Difference.

21,6

21,5

21,5

h- /

9 30 °

52 O

10 13 0

21,5

h- , «

/ k

Weights moved from midway to positive position.

27,2

22,1

27

22,6

26,8

23>°
2 6,6

23>4

15

22.4

15A

21.5

15>3

20,8

15.5

24,6

24,67

24>75

24,8

24,85

24*9

Weights moved to negative position

10 20 31

17

19

20

19

*9

20

20

19

18

19

19

18

19 25

20 41

}

26 45 f

27 22 /

35 1 \

48 /

40 23

41 18

}

48 3 6 \

49 24 J

54 45

55 45

}

18,73

18,52

18.35

18,22

18,1

27 31

34 28

41 5i

48 39

55 37

7^ 0

6 57

7 23

6 48

6 58

the Density of the Earth.

Weights moved to positive position.

Extreme

points.

Divisions

Time.

Point of
rest.

Time of mid
of vibration.

Difference.

h- , ,

/ a

1 ■ a

3b3

25

11 IO 25 I

11 IO 40

23

11 3/

—

-■ -

24,02

-

7 3

22

17 6 \

*7 43

3°, 6

23

26 J

J!"

-

-

24,17

-

7 1

25

24 33 1

24 44

18,4

23

25 17 /

—

— —

24,32

-

7 5

23

31 21 1

3i 49

25

32 9 J

*9>9

—

—

24,4

- -

6 59

25

23

38 39 ■>

39 31 J

-

38 48

1^,4

-

-

24,5

- -

7 6

23

43 16 1

43 34

25

46 12 /

$9,3

1

Motion of arm on moving weights from

midway to pos. - =3,1

pos. to neg. - =6,18

neg. to pos. - = 5,92

Time of one vibration in neg. position - — - f 1"

pos. position - = 7 3

I

488 Mr. Cavendish’s Experiments to determine

EXPERIMENT V. Aug. 20.

The weights being in the positive position, the arm was made
to vibrate, by moving the index.

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of mid.
of vibration.

Difference.

29,6

21,1

29,

21,6

-

h- , ,

25,2

25.17

e a

1 ®

Weights moved to negative position.

22,6

20

10 22

f )

10

23

11

19

23

go J

1^,3

-

-

-

19,27

21,9

-

-

-

19,15

1 6,5

-

-

-

19,1

21 ,5

-

-

-

19,07

16,8

-

-

19,07

21,2

-

-

-

19,07

i7,i

-

-

-

19,05

20,8

-

-

-

19,02

i7,4

-

-

-

19,05

20,6

-

-

—

19,02

-

20

11 32

16 )

11

33

53

19

33

18,97

17.5

—

MS.

—

19

41

16 }

4i

6

20

43

0 J

'

20.3

the Density of the Earth.,
Weights moved to positive position.

Extreme

points.

Divisions.

Time.

Point of
rest.

h- / ,

20,2

24

11 49 10 1

2 6

5° 19 J

29> 4

"

- -

24,95

2 6

56 15 T,

25

47 ;

20,8

-

-

24,92

28,7

-

-

24,87

21,3

-

- -

24b85

28,1

-

-

24»75

21 ,5

-

-

24,67

27,6

-

-

24,67

22

-

- -

24.7

24

0 45 48 I,

25

4 $ 43

27,2

-

- - -

24.7

25

53 11 }

24

5 4 9 J

22,4

Time of mid.
of vibration.

ii 49 37
56 44

o 46 21

53 22

Difference.

t t*

7 7

7 1

Motion of arm on moving weights from

pos. to neg. = 5,9

neg. to pos. = 3.98

Time of one vibration, when weights are in

neg. position - - = 7' 5"

pos. position - - =75

In the fourth experiment, the effect of the weights seemed to
increase on standing, in all three motions of the weights, con-
formably to what was observed with the former wire ; but, in
MDCCXCVIII. g R

49° Mr. Cavendish's Experiments to determine

the last experiment, the case was different; for though, on
moving the weights from positive to negative, the effect seemed
to increase on standing, yet, on moving them from negative to
positive, it diminished.

My next trials were, to see whether this effect was owing to
magnetism. Now, as it happened, the case in which the arm was
inclosed, was placed nearly parallel to the magnetic east and
west, and therefore,- if there was any thing magnetic in the
balls and weights, the balls would acquire polarity from the
earth ; and the weights also, after having remained some time,
either in the positive or negative position, would acquire po-
larity in the same direction, and would attract the balls ; but,
when the weights were moved to the contrary position, that
pole which before pointed to the north, would point to the south,
and would repel the ball it was approached to ; but yet, as re-
pelling one ball towards the south has the same effect on the
arm as attracting the other towards the north, this would have
no effect on the position of the arm. After some time, how-
ever, the poles of the weight would be reversed, and would
begin to attract the balls, and would therefore produce the
same kind of effect as was actually observed. ,

1 o try whether this was the case, I detached the weights**
from the upper part of the copper rods by which they were
suspended, but still retained the lower joint, namejy, that
which passed through them ; I then fixed them in tjifeir posi-
tive position, in such manner, that they could turn round on
this joint, as a vertical axis. I also made an apparatus, by which
I could turn them half way round, on these vertical axes, with-
out opening the door of the room.

Having suffered the apparatus to remain in this manner for

the Density of the Earth . y^

a day, I next morning observed the arm, and, having found it
to be stationary, turned the weights half way round on their
axes, but could not perceive any motion in the arm. Having
suffered the weights to remain in this position for about an
hour, I turned them back into their former position, but without
its having any effect on the arm. This experiment was re-
peated on two other days, with the same result.

We may oe sure, therefore, that the effect in question could
not be produced by magnetism in the weights ; for, if it was,
turning them half round on their axes, would immediately have
changed their magnetic attraction into repulsion, and have pro-
duced a motion in the arm.

As a further proof of this, I took off the leaden weights, and
in their loom placed two 10-inch magnets; the apparatus for
turning them round being left as it was, and the magnets be-
ing placed horizontal, and pointing to the balls, and with their
north poles turned to the north ; but I could not find that any
alteration was produced in the place of the arm, by turning
them half round; which not only confirms the deduction drawn
from the former experiment, but also seems to shew, that in
the experiments with the iron rods, the effect produced could
not be owing to magnetism.

The next thing which suggested itself to me was, that pos-
sibly the effect might be owing to a difference of temperature
between the weights and the case ; for it is evident, that if the
weights were much warmer than the case, they would warm
that side which was next to them, and produce a current of air,
which would make the balls approach nearer to the weights.
Though I thought it not likely that there should be sufficient
difference, between the heat of the weights and case, to have

3 R 2

492 Mr. Cavendish’s Experiments to determine

any sensible effect, and though it seemed improbable that, in
all the foregoing experiments, the weights should happen to be
warmer than the case, I resolved to examine into it, and for
this purpose removed the apparatus used in the last experi-
ments, and supported the weights by the copper rods, as before ;
and, having placed them in the midway position, I put a lamp
under each, and placed a thermometer with its ball close to
the outside of the case, near that part which one of the weights
approached to in its positive position, and in such manner
that I could distinguish the divisions by the telescope. Having
done this, I shut the door, and some time after moved the
'weights to the positive position. At first, the arm was drawn
aside only in its usual manner; but, in half an hour, the effect
was so much increased, that the arm was drawn 14 divisions
aside, instead of about three, as it would otherwise have been, and
the thermometer was raised near i°i; namely, from 6i°to 62°%.
On ^opening the door, the weights were found to be no more
heated, than just to prevent their feeling cool to my fingers.

As the effect of a difference of temperature appeared to be so
great, I bored a small hole in one of the weights, about three-
quarters of an inch deep, and inserted the ball of a small
thermometer, and then covered up the opening with cement.
Another small thermometer was placed with its ball close to
the case, and as near to that part to which the weight was ap-
proached as could be done with safety ; the thermometers
being so placed, that when the weights were in the negative
position, both could be seen through one of the telescopes, by
means of light reflected from a concave mirror.

the Density of the Earth.

493

Experiment vi. Sept. 6.
Weights in midway position.

Extreme

points.

Divisions.

Time.

Point of
rest.

Therm
in Air.

ameter
in Weight.

18,9

18,85

h >

9 43

10 3

18,85

55)5

Weights moved to negative position.

13,1

-

IO 12

-

55)5

18,4

-

l8

15,82

1.3,4

missed.

25

13.6

-

39

-

55,5

17,6

-

46

15,65

13.8

-

53

15,65

17.4

-

11 0

15,65

14,°

-

7

15,65

17,2

-

14

-

55)5

55,8

55 >8

Weights moved to positive position.

25,8

-

23

17,5

-

30

21,55

25,4

-

37

21,6

18,1

-

44

21,65

25,0

missed.

31

24,7

-

0 5

1 9)

-

12

21,77

24,4

-

19

Motion of arm on moving weights from midway to — = 3,03

494

Mr. Cavendish's Experiments to determine

Experiment vii. Sept 18.

Weights in midway position.

Extreme

Point of

Thermometer

points.

Divisions.

Time.

rest.

in Air.

inWeight.

*9>4

h •

8 30

56,7

*9>4

9 3 2

-

36,6'

Weights moved to negative position.

13^

-

40

-

-

18,8

-

47

16,23

1 3>8

-

34

Eight extreme points missed.

16,9

-

10 38

i4>3

-

11 3

13,62

1 6,6

-

12

Weights moved to positive position.

26,4

-

20

-

56,5

17,2

-

28

21,72

26,1

35

Four extreme points missed.

1 9>3

-

0 10

25*1

-

^7

22,3

19’7

-

24

Motion of arm on moving weights from midway to — = 3,15

— to = 6,1

/

the Density of the Earth.

4 95

Experiment viii. Sept. 23.
Weights in midway position.

Extreme

points.

Divisions.

Time.

Point of
rest.

Thermometer

in Air .

inWeight.

h /

W>3 9 46

ic), 2 10 45

19,2

53’1
53’ 1

Weights moved to negative position.

i3’5

-

56

-

-

18 ,6

-

11 3

16,07

13,6

—

10

Four extreme points missed.

17»4

-

44

14^

-

51

15,7

17’ 2

-

58

-

-

Weights moved to positive position.

15 ’7

-

0 1

26,7

-

8

21,42

1 8,6

-

15

-

Two extreme points missed.

2 5’9

-

36

18,1

-

43

21,9

2 5,5

—

50

Motion of arm on moving weights from midway to — - : — 3,13

— to + = 5,72.

49^ Mr. Cavendish’s Experiments to determine

In these three experiments, the effect of the weight appeared
to increase from two to five tenths of a division, on standing an
hour; and the thermometers shewed, that the weights were three
or five tenths of a degree warmer than the air close to the case.
In the two last experiments, I put a lamp into the room, over
night, in hopes of making the air warmer than the weights, hut
without effect, as the heat of the weights exceeded that of the
air more in these two experiments than in the former.

On the evening of October 1 7, the weights being placed in
the midway position, lamps were put under them, in order to
warm them ; the door was then shut, and the lamps suffered
to burn out. The next morning it was found, on moving the
weights to the negative position, that they were 7°^ warmer
than the air near the case. After they had continued an hour
in that position, they were found to have cooled i°\, so as to
be only 6° warmer than the air. They were then moved to the
positive position; and in both positions the arm was drawn aside
about four divisions more, after the weights had remained an
hour in that position, than it was at first.

May 22, 1798. The experiment was repeated in the same
manner, except that the lamps were made so as to burn only
a short time, and only two hours were suffered to elapse before
the weights were moved. The weights were now found to be
scarcely 20 warmer than the case ; and the arm was drawn aside
about two divisions more, after the weights had remained an
hour in the position they were moved to, than it was at nrst.

On May 23/ the experiment was tried in the same manner,
except that the weights were cooled by laying ice on them ;
the ice being confined in its place by tin plates, which, on

497

the Density of the Earth.

moving the weights, fell to the ground, so as not to be in the
way. On moving the weights to the negative position, they
were found to be about 8° colder than the air, and their effect
on the arm seemed now to diminish on standing, instead of in-
creasing, as it did before ; as the arm was drawn aside about
s-§- divisions less, at the end of an hour after the motion of the
weights, than it was at first.

It seems sufficiently proved, therefore, that the effect in ques-
tion is produced, as above explained, by the difference of tem-
perature between the weights and case ; for, in the 6th, 8th,
and qth experiments, in which the weights were not much
warmer than the case, their effect increased but little on stand-
ing; whereas, it increased much, when they were much warmer
than the case, and decreased much, when they were much
cooler.

It must be observed, that in this apparatus, the box in
which the balls play is pretty deep, and the balls hang near
the bottom of it, which makes the effect of the current of air
more sensible than it would otherwise be, and is a defect
which I intend to rectify in some future experiments.

Experiment ix. April 29.

Weights in positive position.

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of middle
of vibration.

34.7

h 1 a

h 1 it

35

34>65

'

mm am

34>^4

3 s

MDCCXCVIII.

498 Mr. Cavendish’s Experiments to determine

Weights moved to negative position.

Extreme

points.

Divisions.

Time.

Point of
rest.

h* 1 n

23,8 _

28

11 18 2 9j

29

58 J

33>2

-

28,52

2 9

25 271

28

57}

2 3,9

mm

- -

28,25

32

-

-

28,01

24,15

—

-

27,82

31

-

«

27.63

24,4

-

- —

27 ,55

3°>4

-

- -

27.47

28

0741

•a>

27

53 J

24,7

Motion of arm

mm

Time of mid.
of vibration.

11 l8 43

2 5 4°

o 7 26

= 6,32

Difference.

I H

Time of vibration - =6' 58"

Experiment x. May 5.
Weights in positive position.

34>5

38,5

34>4

33,97

Weights moved to negative position.

22,3

28

29

10 43 42 )
44 6 j

-

10 43 g6

33,2

-

-

27,82

-

00 !>•

5° 331
51 °J

-

5° 36

22 ,6

-

-

27,72

7

o

the Density of the Earth.

499

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of mid.
of vibration.

Difference.

h* 1 n

h* 1 a

/ 11

32,5

-

- -

27.7

23,2

-

_ -

3M 5

-

- -

37>4

23>5

-

- -

27,28

27

11 2/5 201

11 25 24

28

3°>7

-

-

27,21

- -

7 3

00

32 Ol
32 40 1

-

32 27

6 56

23>95

-

-

27,21

- -

27

28

39 19]

40 2/

-

39 23

3° >25

Motion of arm - =6, 15

Time of vibration - =6' 59"

Experiment xi. May 6.

Weights in positive position.

34.9

34. 1

-

- -

34.47

34.8

34>25

34. 49

Weights moved to negative position.

10 o 8

\

7 5

23>3

28

9 59 591

29

10 0 27 j

33.3

-

-

28,42

29

6 52

27

7 51

S 2

500 Mr. Cavendish's Experiments to determine

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of mid.
of vibration.

h* / a

28,35

h- , e

23,8

-

tarn amt

S2>5

-

-

28,3

24,4

missed.

24,8

28,17

3b3

i-

-

2Q

28

10 48 37 1
40 21i

-

10 49 8

2d, 3

-

-

2 8,2

28

2 9

56 81

56 56}

-

56 13

30,9

Motion of arm - =6,07

Time of vibration - ~ 1' 1,1

In the three foregoing experiments, the index was purposely
moved so that, before the beginning of the experiment, the
balls rested as near the sides of the case as they could, without
danger of touching it ; for it must be observed, that when the
arm is at 35, they begin to touch. In the two following expe-
riments, the index was in its usual position.

#

Experiment xii. May 9.

• Weights in negative position.

17.4

9 45

0

17.4

58

0

17.4.

10 8

0

17.4

10

0

17.4

the Density of the Earth .

Weights moved to positive position.

ixtreme

points.

Divisions.

Time.

Point of
rest.

h- , 11

28,85

24

IO 20 50 1

22

21 46 J

18,4

-

- -

23.49

28,3

-

-

23.57

1 9>3

-

-

23>6'7

27,8

-

-

23.72

20

-

-

23.8

27,4

-

- -

23>83

24

11 3 131

23

54 J

20,55

-

- -

23.87

23

9 451

24

10 28 /

27

Time of mid.
of vibration.

10 20 59

n 3 14

10 18

Motion of arm
Time of vibration

= 6,09

= 7' 3"

16

18,3

16,2

experiment xiii. May 25.
Weights in negative position.

17,2

Weights moved to positive position.

2 9>6

25

10 22 22 1

24

0 45

17*4

-

-

23,32

23

29 59}

28,9

30 23 J

-

-

23>4

24

36 58 j

18,4

23

37 24/

—

— *

23>52

10 22 56
30 3
37 7

501

5 02

Mr . Cavendish’s Experiments to determine

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of mid.
of vibration.

h* i

11

fc* , „

23

IO 44

3 1

24

31 /

IO 44 14

28,4

-

-

-

23,62

19^3

-

-

-

23T

27,8

-

-

-

23>7

24

11 5

2 6 \

23

6

1 /

11 5 31

19,9

-

-

-

23,72

23

12

12 1

24

50 J

12 35

27>3

Weights moved to negative position.

13.5

21,8

am mm

*7>75

*3*9

00 1

r-f T"i

37 34)

38 10 J

i7>67

21,1

17

18

44 26‘)

45 4 J

17,62

14.4

-

17,6

20,5

-

-

17»52

*4*7

—

- -

17*47

20

—

-

17*42

18

0 19 57 )

15

17

20 52 j

1 7*37

^9.5

17

18

2 7 15)
28 15/

37 39
44 45

o 20 24

27 30

Motion of the arm on moving weights from — to + =6,12

— to — = 5,97

+ = 7' 6"

Time of vibration at

the Density of the Earth.

experiment xiv. May 2 6. ’
Weights in negative position.

5°3

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of mid.
of vibration.

h* /

h. , „

l6,l

9 18 0

16,1

24 O

l6,l

46 O

l6,l

49 0

l6,l

Weights moved to positive position.

s7>7

23

10 0 46 1

22

1 16 J

i7>3

22

23

7 58 1

8 27 J

22,37

27,2

-

-

22,5

18,3

23

22

15 2 1
32 J

-

-

- -

22,65

26,8

-

-

22,75

-

-

22,85

26,4

23

22

43 4° 1

44 22 J

22,97

20

—

- -

23,15

26,2

22

23

49 53 )

5° 37 ;

-

1 1
8 5
15 9

43 32

50 41

Weights moved to negative position.

12,4

16

11 7 53 1

21,5

17

8 27 J

17,02

17

15 3° )

16

16 3 /

8 25

I

15 2 7'

Mr. Cavendish’s Experiments to determine

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of mid.
of vibration.

h* /

it

h* ,

12,7

-

16.9

20,7

-

-

-

16,85

13.3

-

-

-

16,82

20

-

-

-

16,72

i3>6

-

-

-

16,67

lb

11 50

33 )

11 50 58

17

5 1

13 J

*9>5

-

-

-

16,65

17

16

57 53 )
53 44 j

CO

Motion of arm by moving weights from — to + =6, 27

+ to — = 6, 13
Time of vibration at -{- = 7' 6"

— = 7 6

In the next experiment, the balls, before the motion of the
weights, were made to rest as near as possible to the sides of
the case, but on the contrary side from what they did in the
9th, 10th, and 1 ith experiments.

experiment xv. May 27.
Weights in negative position.

3>9

3S5

3$5

.

3.6 1

—

"" ""

3.61

the Density of the Earth.
Weights moved to positive position.

5°5

Extreme

points.

Divisions.

i5>4

4, 8

14,8

5:9
14,3 5
6, 8

1 3:9

7:5

10

9

9
10

10
9

11

10

10

11

13^5

Motion of the arm
Time of vibration

10 5 59
6 27

12 43

13 11

20 24
5 6

}

}

}

Point of
rest.

Time of mid.
of vibration.

48 3°

49 11

}

55 26 )

56 10 J

9:95

10,07

10,23

io?35

10,46

10,52

10,6

h.

IO 5 56

13 5

20 13

48 42

55 48

6>34
7' 7"

The two following experiments were made by Mr. Gilpin,
who was so good as to assist me on the occasion.

experiment xvi. May 28.

Weights in negative position.

22,55

8,4

-

-

15,09

21

—

-

14,9

9,2

MDCCXCVIII. 3T

Mr. Cavendish’s Experiments to determine
Weights moved to positive position.

Extreme

points.

—

Divisions.

Time.

Point of
rest.

h* , „

26,6

22

10 22 53 1

21

23 20 J

15,8

-

-

21

20

3° 7 )

21

36 j

25,8

-

-

21,05

22

37 23 1

21

55 J

16,8

-

-

2 1,11

20

44 29 ]

21

45 4

25,05

—

- -

21,11

22

51 54 1

21

52 32 j

17.57

-

-

21,2

21

59 3i )

22

11 0 13 J

24,6

-

-

21,28

22

6 24 1

21

7 9 ;

18,3

Motion of the arm

- _

Time of vibration

-

-

EXPERIMENT XVII.

May 30

Weights

in negative position

17,2

10 19 0

17,1

25 0

17,07

29 0

17,15

40 0

17.45

49 0

17,42

51 0

17,42

11 1 0

17.42

of vibration.

h.

/ //

10 23 15

30 30

37 45

45

32 20

59 34

11 6 49

6,1

7' 16"

the Density of the Earth.
Weights moved to positive position.

5°7

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of mid.
of vibration.

h* , ,

h* t t

28,8

l8,l

24

23

11 11 23 1
49 *

-

11 11 37

—

-

23,2

22

18 is -i

43 J

l8 42

27,8

23

-

-

23,12

24

25 19 1

25 4°

23

49 ;

l8,8

-

-

23,2

23

32 41 1

32 43

27.3s

24

33 13 J

-

mm IW

23,31

24

39 28 -i

39 44

4° 3 J

*9.7

-

-

23,44

23

24

46 33 }

47 11 s

-

46 46

27

—

- -

23’52

24

23

53 3s l

54 17 1

-

53 4s

20,4

—

- —

23.57

26,5

23

24

0 0 34 l

1 18 J

-

O 0 55

-

- -

2 3>55

24

7 34 1

7 5°

20,8

23

8 21 J

-

- -

2 3’59

26,25 1

23

24

H 3° )
15 24 J

-

14 58

3T2

508 Mr * Cavendish’s Eperiments to determine

Weights moved to negative position.

Extreme

points.

Divisions.

Time.

Point of
rest.

Time of midi
of vibration.

h- i „

/ a

13’3

17

18

0 3s 191

4,8 5

-

O 32 44

22,4

-

-

17 >95

« , | 5-v f

l8

17

39 4P\

40 lgJ

-

39 44

13,7

-

-

17.85

j C' v* •>

17

18

46 2 6 \

47 0 J

-

46 48

21,6

-

17,72

l8

53 43)

__

53 5°

17

54 20 J

17,6

*4

-

17

18

1 0 39)
1 20 J

-

1 0 55

20,8

-

-

17.47

l8

17 -

7 391

8 21 J

-

7 59

14’3

-

-

17.37

■ ^ • 0

17

18

14 541

15 42/

-

15 4

20,1

-

JL. Jj t

17,27

18

21 32 1

22 5

*7

22 22 J

14,6

Motion of the arm on moving weights from — to -f- = 5,78

+ to — =j 5,64

Time of vibration at » - + = 7' 2'

the Density of the Earth.

5°9

On the Method of computing the Density of the Earth from these

Experiments.

I shall first compute this, on the supposition that the arm
and copper rods have no weight, and that the weights exert
no sensible attraction, except on the nearest ball; and shall
then examine what corrections are necessary, on account of
the arm and rods, and some other small causes.

The first thing is, to find the force required to draw the
arm aside, which, as was before said, is to be determined by
the time of a vibration.

The distance of the centres of the two balls from each other
is 73>3 inches, and therefore the distance of each from the centre
of motion is 36,65, and the length of a pendulum vibrating se-
conds, in this climate, is 39,14; therefore, if the stiffness of the
wire by which the arm is suspended is such, that the force which
must be applied to each ball, in order to draw the arm aside by
the angle A, is to the weight of that ball as the arch of A to the
radius, the arm will vibrate in the same time as a pendulum

whose length is 36,65 inches, that is, in seconds ; and

therefore, if the stiffness of the wire is such as to make it vibrate
in N seconds, the force which must be applied to each ball, in
order to draw it aside by the angle A, is to the weight of the

ball as the arch of A x ^ x to ^le rac^us* But t^le ivory

scale at the end of the arm is 38,3 inches from the centre
of motion, and each division is of an inch, and therefore
subtends an angle at the centre, whose arch is — ; and there-
fore the force which must be applied to each ball, to draw fhe

510 Mr. Cavendish's Experiments to determine

arm aside by one division, is to the weight of the ball as

jdw t0 or as 8T?n^ to *•

The next thing is, to find the proportion which the attraction
of the weight on the ball bears to that of the earth thereon,
supposing the ball to be placed in the middle of the case, that
is, to be not nearer to one side than the other. When the
weights are approached to the balls, their centres are 8,85
inches from the middle line of the case ; but, through inadver-
tence, the distance, from each other, of the rods which support
these weights, was made equal to the distance of the centres of
the balls from each other, whereas it ought to have been some-
what greater. In consequence of this, the centres of the weights
are not exactly opposite to those of the balls, when they are ap-
proached together ; and the effect of the weights, in drawing the
arm aside, is less than it would otherwise have been, in the tri-
plicate ratio of ~~ to the chord of the angle whose sine is

or in the triplicate ratio of the cosine of ~ this angle to
the radius, or in the ratio of ,9779 to 1.

Each of the weights weighs 2439000 grains, and therefore
is equal in weight to 10,64 spherical feet of water; and there-
fore its attraction on a particle placed at the centre of the ball,
is to the attraction of a spherical foot of water on an equal par-

to 1. The

tide placed on its surface, as 10,64 x ,9779 x g-gj

mean diameter of the earth is 41800000 feet;* and therefore,
if the mean density of the earth is to that of water as D to one,
the attraction of the leaden weight on the ball will be to that

* In strictness, vve ought, instead of the mean diameter of the earth, to take the
diameter of that sphere whose attraction is equal to the force of gravity in this cli-
mate ; but the difference is not worth regarding.

the Density of the Earth.

511

of the earth thereon, as 10,64 x ,9779 x g-gy to 41800000 D
:: 1 to 8739000 D.

It is shewn, therefore, that the force which must be applied
to each ball, in order to draw the arm one division out of its

natural position, is of the weight of the ball ; and, if the

mean density of the earth is to that of water as D to 1, the at-
traction of the weight on the ball is of the weight of

that ball ; and therefore the attraction will be able to draw the
arm out of its natural position by 8 000p or yo68 p divisions ;

and therefore, if on moving the weights from the midway to a
near position the arm is found to move B divisions, or if it
moves 2 B divisions on moving the weights from one near po-
sition to the other, it follows that the density of the earth, or D,
. N*

^ 10683 B ’

We must now consider the corrections which must be ap-
plied to this result ; first, for the effect which the resistance of
the arm to motion has on the time of the vibration : 2d, for
the attraction of the weights on the arm : 3d, for their attrac-
tion on the farther ball : 4th, for the attraction of the copper
rods on the balls and arm : 5th, for the attraction of the case
on the balls and arm : and 6th, for the alteration of the attrac-
tion of the weights on the balls, according to the position of
the arm, and the effect which that has on the time of vibration.
None of these corrections, indeed, except the last, are of much
signification, but they ought not entirely to be neglected.

As to the first, it must be considered, that during the vibra-
tions of the arm and balls, part of the force is spent in accele-

512 Mr. Cavendish's Experiments to determine

rating the arm ; and therefore, in order to find the force re-
quired to draw them out of their natural position, we must find
the proportion which the forces spent in accelerating the arm
and balls bear to each other.

Let EDC e d c (fig. 4.) be the arm. B and b the balls. C s
the suspending wire. The arm consists of 4 parts ; first, a deal
rod D c d, 73,3 inches long ; 2d, the silver wire DC d, weigh-
ing 170 grains ; 3d, the end pieces DE and e d, to which the
ivory vernier is fastened, each of which weighs 45 grains ; and
4th, some brass work C c, at the centre. Tiie deal rod, when
dry, weighs 2320 grains, but when very damp, as it commonly
was during the experiments, weighs 2400 ; the transverse sec-
tion is of the shape represented in fig 5 ; the thickness BA, and
the dimensions of the part DE ed, being the same in all parts ;
but the breadth B b diminishes gradually, from the middle to
the ends. The area of this section is ,33 of a square inchat the
middle, and ,146 at the end ; and therefore, if any point x (fig.

4.) is taken in c d , and ~ is called x, this rod weighs —

^ ' 5 c d 0 73,3 x >23®

per inch at the middle; 2400 x— -4| at the end, and

r 73,3 X JZ38 72. 2 .228

73»3

,238

= — — 1— at x ; and therefore, as the weight of the wire is

73’3 &

~~ per inch, the deal rod and wire together may be considered

as a rod whose weight at x = per inch.

But the force required to accelerate any quantity of matter
placed at x, is proportional to xz ; that is, it is to the force re-
quired to accelerate the same quantity of matter placed at a7 as
x% to 1 ; and therefore, if c d is called /, and x is supposed to
flow, the fluxion of the force required to accelerate the deal rod

th e Density of the Earth . 513

and wire is proportional to -• ■ 1 Y-x j ~ 1 848 x-> the fluent of which,

generated while x flows from c to d, = 35°’»

so that the force required to accelerate each half of the deal rod
and wire, is the same as is required to accelerate 350 grains
placed at d.

The resistance to motion of each of the pieces de, is equal to
that of 48 grains placed at d ; as the distance of their centres of
gravity from C is 38 inches. The resistance of the brass work
at the centre may be disregarded ; and therefore the whole force
required to accelerate the arm, is the same as that required to
accelerate 398 grains placed at each of the points D and d.

Each of the balls weighs 1 126.2 grains, and they are placed at
the same distance from the centre as D and d\ and therefore, the
force required to accelerate the balls and arm together, is the
same as if each ball weighed 1 1660, and the arm had no weight;
and therefore, supposing the time of a vibration to be given, the
force required to draw the arm aside, is greater than if the arm
had no weight, in the proportion of 11660 to 11262, or of
1,0353 t0 i-

_ To find the attraction of the weights on the arm, through d
draw the vertical plane dwh perpendicular to D d, and let w
be the centre of the weight, which, though not accurately in this
plane, may, without sensible error, be considered as placed
therein, and let b be the centre of the ball; then zvb is horizontal
and = 8,85, and d b is vertical and = 5,5 ; let w a — a, w b ==

b, and let or 1-1=2:; then the attraction of the weight

on a particle of matter at x , in the direction d zv, is to its attrac-

> A.

tion on the same particle placed at b : : b3 : az + or is pro-

3U

MDCCXCVIII.

4 Mr. Cavendish’s 'Experiments to determine

portional to

b 3

a* +.«* /*\

i , and the force of that attraction to move

the arm, is proportional to -====^, and the weight of the
deal rod and wire at the point x , was before said to be
'-9°"73^48 = -■4-y--'-I-84-z per inch; and therefore, if d x flows,

the fluxion of the power to move the arm = l z x .l6f2 + ,i848^

73^3

( IP X \—z . 77 ; 63XI-

r‘3=sf=i?:m = 2: x bai + 9242; X

63 i x 821 -f 103 2TT-9245:2

b* jex 821 1032: -f --

a1 + /1 ■z1!2-

924&3*X-£+**

fla-f z*]a

ft3 ZX39S-H03 z 924 63 g

a* * *V«*+ ** *

103 b2 . 103 b 3 924 b3

7* 1

a--f /2 z2)x

The fluent of this

«2+

; which, as -J- = ,08 =

895 63 ^

fV«* +/■ *• T p a\ p~ log- -"+yT+-^-, and the force

with which the attraction of the weight, on the nearest half of
the deal rod and wire, tends to move the arm, is proportional
to this fluent generated while z flows from o to 1, that is, to
128 grains.

The force with which the attraction of the weight on the

1 3

end- piece de tends to move the arm, is proportional to 47 x
or 29 grains ; and therefore the whole power of the weight to
move the arm, by means of its attraction on the nearest part
thereof, is equal to its attraction on 157 grains placed at 6,

which is or ,0139 of its attraction on the ball.

It must be observed, that the effect of the attraction of the
weight on the whole arm is rather less than this, as its attrac-
tion on the farther half draws it the contrary way ; but, as the
attraction on this is small, in comparison of its attraction on the
hearer half, it may be disregarded.

the Density of the Earth. 515

Ihe attraction of the weight on the furthest ball, in the di-
rection bw, is to its attraction on the nearest ball ::wda: zl>D3
. : ,ooiy : i ; and therefore the effect of the attraction of the

weight on both balls, is to that of its attraction on the nearest
ball : : ,9983 : 1.

To find the attraction of the copper rod on the nearest ball,
let b and w (fig. 6. ) be the centres of the ball and weight, and
ea the perpendicular part of the copper rod, which consists of
two parts, ad and de. ad weighs 22000 grains, and is 16
inches long, and is nearly bisected by w. de weighs 41000,
and is 4 6 inches long, wb is 8,85 inches, and is perpendicular
to ew. Now, the attraction of a line ew, of uniform thickness,
on b, in the direction b w, is to that of the same quantity of
matter placed at w : : bw : eb; and therefore the attraction of

the part da equals that of 22“*** or ib3oo, placed at
and the attraction of de equals that of 41000 x - x^— 41000

dw b w

x 7T x TT’ or 25°°> placed at the same point; so that the at-
traction of the perpendicular part of the copper rod on b , is to
that of the weight thereon, as 18800 : 2439000, or as ,00771
to 1. As for the attraction of the inclined part of the rod and
wooden bar, marked Pr and rr in fig. 1, it may safely be
neglected, and so may the attraction of the whole rod on the
arm and farthest ball; and therefore the attraction of the
weight and copper rod, on the arm and both balls together, ex-
ceeds the attraction of the weight on the nearest ball, in the
pioportionof,9983 -f- ,0139 -|- ,0077 t0 one, or of 1,0199 to 1.

The next thing to be considered, is the attraction of the ma-
hogany case. Now it is evident, that when the arm stands at

3 U 2

51 6 Mr. Cavendish’s Experiments to determine

the middle division, the attractions of the opposite sides of the
case balance each other, and have no power to draw the arm
either way. When the arm is removed from this division, it is
attracted a little towards the nearest side, so that the force re-
quired to draw the arm aside is rather less than it would other-
wise bet but yet, if this force is proportional to the distance of
the arm from the middle division, it makes no error in the re-
sult ; for, though the attraction will draw the arm aside more
than it would otherwise do, yet, as the accelerating force by
which the arm is made to vibrate is diminished in the same
proportion, the square of the time of a vibration will be in-
creased in the same proportion as the space by which the arm
is drawn aside, and therefore the result will be the same as if
the case exerted no attraction ; but, if the attraction of the case
is not proportional to the distance of the arm from the middle
point, the ratio in which the accelerating force is diminished is
different in different parts of the vibration, and the square of
the time of a vibration will not be increased in the same pro-
portion as the quantity by which the arm is drawn aside, and
therefore the result will be altered thereby.

On computation, I find that the force by which the attrac-
tion draws the arm from the centre is far from being propor-
tional to the distance, but the whole force is so small as not to
be worth regarding; for, in no position of the arm does the at-
traction of the case on the balls exceed that of -Jth of a spheric
inch of water, placed at the distance of 1 inch from the centre
of the balls; and the attraction of the leaden weight equals that
of 10,6 spheric feet of water placed at 8,85 inches, or of 234
spheric inches placed at 1 inch distance; so that the attraction
of the case on the balls can in no position of the arm exceed

the Density of the Earth.

5*7

of that of the weight. The computation is given in the

117 0

Appendix.

It has been shewn, therefore, that the force required to
draw the arm aside one division, is greater than it would be if
the arm had no weight, in the ratio of 1,0353 to 1, and therefore

~ .^°3 53 0f tjie weiorht of the ball ; and moreover, the attrac-

818 N* °

tion of the weight and copper rod on the arm and both balls
together, exceeds the attraction of the weight on the nearest ball,

in the ratio of 1,0139 to 1, and therefore =

8 1 8 Na

weight of the ball; consequently D is really equal to j
x -T01 or — p—r, instead of — , as by the former

8739000 B ’ 108446’ 106836’ J

computation. It remains to be considered how much this is
affected by the position of the arm.

Suppose the weights to be approached to the balls; let W,
(fig. 7.) be the centre of one of the weights ; let M be the centre
of the nearest ball at its mean position, as when the arm is at
20 divisions ; let B be the point which it actually rests at ; and
let A be the point which it would rest at, if the weight was re-
moved; consequently, AB is the space by which it is drawn
aside by means of the attraction ; and let M /3 be the space by
which it would be drawn aside, if the attraction on it was the
same as when it is at M. But the attraction at B is greater
than af M, in the proportion of WM* : WB1 ; and therefore,

AB = M/3 x = M/3 x 1 + fjry, very nearly.

Let now the weights be moved to the contrary near position,
and let w be now the centre of the nearest weight, and b the
point of rest of the centre of the ball ; then A 6 = M/3 x 1 +

B b

Z$Ab in; -ii yT/Q 1 2lVl/i , 2MB

— valid B 6 — M/3 * 2 4- -m + ^

MW:

2M/3 X 1 +

MW

so

I

518 Mr. Cavendish’s Experiments to determine

that the whole motion B b is greater than it would be if the
attraction on the ball was the same in all places as it is at M,

in the ratio of 1 -[- to one; and, therefore, does not depend

sensibly on the place of the arm, in either position of the
weights, but only on the quantity of its motion, by moving
them.

This variation in the attraction of the weight, affects also the
time of vibration ; for, suppose the weights to be approached
to the balls, let W be the centre of the nearest weight ; let B
and A represent the same things as before ; and let x be the
centre of the ball, at any point of its vibration ; let AB repre-
sent the force with which the ball, when placed at B, is drawn
towards A by the stiffness of the wire ; then, as B is the point
of rest, the attraction of the weight thereon will also equal AB;
and, when the ball is at x, the force with which it is drawn
towards A, by the stiffness of the wire, = Ax, and that with

which it is drawn in the contrary direction, by the attraction,

WB2,

= AB x so that the actual force by which it is drawn to-
wards A = A x — = AB -f- B x — - AB x 1 -f =

Bx — — WB , very nearly. So that the actual force with

which the ball is drawn towards, the middle point of the vibra-
tion, is less than it would be if the weights were removed, in

the ratio of 1 — to one, and the square of the time of a

2ab

vibration is increased in the ratio of 1 to 1 — which differs

very little from that of 1 -j- to 1, which is the ratio in which
the motion of the arm, by moving the weights from one near
position to the other, is increased.

the Density of the Earth.

5*9

The motion of the ball answering to one division of the arm

fore, the time of vibration, and motion of the arm, must be cor-
rected as follows :

If the time of vibration is determined by an experiment in
which the weights are in the near position, and the motion of
the arm, by moving the weights from the near to the midway
position, is d divisions, the observed time must be diminished
in the subduplicate ratio of i — ^ to 1, that is, in the ratio

of 1 — ~ to 1 ; but, when it is determined by an experiment

in which the weights are in the midway position, no correc-
tion must be applied.

To correct the motion of the arm caused by moving the
weights from a near to the midway position, or the reverse,
observe how much the position of the arm differs from 20 di-
visions, when the weights are in the near position : let this be
n divisions, then, if the arm at that time is on the same side of
the division of 20 as the weight, the observed motion must be

diminished by the — part of the whole; but, otherwise, it must
be as much increased.

If the weights are moved from one near position to the other,
and the motion of the arm is 2 d divisions, the observed mo-
tion must be diminished by the part of the whole.

If the weights are moved from one near position to the other,
and the time of vibration is determined while the weights are
in one of those positions, there is no need of correcting either
the motion of the arm, or the time of vibration.

; and there-

5 20

Mr. Cavendish's Experiments to determine

CONCLUSION.

The following Table contains the Result of the Experiments .

Exper.

Mot. weight

Mot. arm

Do. corr.

Time vib.

Do. corr.

Density.

m. to 4-

14,32

13,42

/ "

-

5,5

ll

4- to m.

14U

J307

14,55

-

5,6i

„ 5

m. to -f

15.87

14,69

-

-

4,88

a l

-f tom.

15,43

14U4

14,42

-

5.07

s{

-j- to m.

15,22

13,56

14,39

-

5,26

m. to

14,5

13,28

14,54

-

5,55

m. to -j-

3,i

2,95

6,54

5,36

4-^

4- to —

6,18

-

7U

-

5,29

— to 4“

5,92

-

7.3

-

5.58

4- to —

5,9

-

7,5

-

5,65

51

— to 4-

5.98

-

7,5

-

5,57

6[

m. to —

3>°3

2,9

"l -
1 „„

-

5,53

1

— to 4-

5,9

5.7i

-

5,62

7i

m. to —

— to 4-

305

6,1

3,03

5,9

1 />4

> by
mean.

6,57

5,29

5,44

8 1

m. to —

303

3,oo

—

5.34

— to 4“

5.72

5,54

>

-■

5,79

V

9

4- to —

6, 32

-

6,5 8

-

5,1

10

-j- to —

6,15

6.59

-

5,27

n

4- to —

6, 07

-

7,i

-

5.39

12

— to 4-

6,09

-

7,3

~

542

13 |

— * to 4-

6,12

-

7,6

-

5,47

4- to —

5.97

-

7,7

-

5>6’3

V

14!

— to 4~

6,27

-

7.6

-

5.34

4_ to —

6,13

-

7,6

-

5,46

13

► — ■ to 4-

6,34

-

7,7

—

5,3

16

— to 4-

6,1

-

7,16

-

5.75

37 1

— to 4-

5.78

7,2

-

5,68

4_ to —

5>6 4

-

7.3

-

5.85

521

the Density of the Earth .

From this table it appears, that though the experiments agree
pretty well together, yet the difference between them, both in
the quantity of motion of the arm and in the time of vibration,
is greater than can proceed merely from the error of observa-
tion. As to the difference in the motion of the arm, it may
very well be accounted for, from the current of air produced
by the difference of temperature ; but, whether this can account
for the difference in the time of vibration, is doubtful. If the
current of air was regular, and of the same swiftness in all parts
of the vibration of the ball, I think it could not; but, as there
will most likely be much irregularity in the current, it may very
likely be sufficient to account for the difference.

By a mean of the experiments made with the wire first used,
the density of the earth comes out 5,48 times greater than that
of water; and by a mean of those made with the latter wire, it
comes out the same; and the extreme difference of the results of
the 23 observations made with this wire, is only ,75 ; so that
the extreme results do not differ from the mean by more than
,38, or of the whole, and therefore the density should seem
to be determined hereby, to great exactness. It, indeed, may
be objected, that as the result appears to be influenced by the
current of air, or some other cause, the laws of which we are
not well acquainted with, this cause may perhaps act always,
or commonly, in the same direction, and thereby make a con-
siderable error in the result. But yet, as the experiments were
tried in various weathers, and with considerable variety in the
difference of temperature of the weights and air, and with the
arm resting at different distances from the sides of the case,
it seems very unlikely that this cause should act so uniformly
in the same way, as to make the error of the mean result nearly

MDCCXCVIII. 3 X

522 Mr. Cavendish's Experiments to determine

equal to the difference between this and the extreme ; and,
therefore, it seems very unlikely that the density of the earth
should differ from 5,48 by so much as of the whole.

Another objection, perhaps, may be made to these experi-
ments, namely, that it is uncertain whether, in these small
distances, the force of gravity follows exactly the same law as
in greater distances. There is no reason, however, to think
that any irregularity of this kind takes place, until the bodies
come within the action of what is called the attraction of co-
hesion, and which seems to extend only to very minute dis-
tances. With a view to see whether the result could be affected
by this attraction, I made the 9th, 10th, 11th, and 15th expe-
riments, in which the balls were made to rest as close to the
sides of the case as they could ; but there is no difference to be
depended on, between the results under that circumstance, and
when the balls are placed in any other part of the case.

According to the experiments made by Dr. Maskelyne,
on the attraction of the hill Schehallien, the density of the
earth is 4^- times that of water ; which differs rather more
from the preceding determination than I should have ex-
pected. But I forbear entering into any consideration of
which determination is most to be depended on, till I have exa-
mined more carefully how much the preceding determination
is affected by irregularities whose quantity I cannot measure.

I

I

the Density of the Earth.

5*3

APPENDIX.

On the Attraction of the Mahogany Case on the Balls.

The first thing is, to find the attraction of the rectangular
plane c k (3 b (fig. 8.) on the point a , placed in the line a c per-
pendicular to this plane.

a

=a w*y and

b*

a b ,

Let a c — > a, c k — b, c b — sc, and let ^ ^

: v % then the attraction of the line b jQ on a , in the direction

bp

; and therefore, if c b flows, the fluxion of the at-

ab X «/3

traction of the plane on the point a , in the direction c b ,

b x

x

b au

— b <w

v

V az +xz x '^az-\-bz-{-xz /^az+x% ^ V'bzwz-\-az V'i-{-vz}

w b +

the variable part of the fluent of which = — log. v -j- \/ i -}- v\

and therefore the whole attraction = log. ***** x ; so

that the attraction of the plane, in the direction c b , is found
readily by logarithms, but I know no way of finding its attract
tion in the direction ac , except by an infinite series.

The two most convenient series I know, are the following :

First series. Let ~ = *r, and let A = arc whose tang, is tt,

B = A — -7T, C=B + y, D = C — ™-, See. then the attrac-

. — • 1 Bra* . 3C104 , 3 .5

tion in the direction ac—v 1 — w xA-f ~ — rTrj*“r 77^76*
&c.

3X2

524 Mr. Cavendish's Experiments to determine

For the second series, let A = arc whose tang. =~, B = A — ~ ,

B D = C — -d—> & c. then the attraction = arc . go°

3^3 5^5 u

v/T

+ v* X A

4. 3C”* _ 3-sP»6. m

2 ^ 2.4 2.4.6

It must be observed, that the first series fails when tt is
greater than unity, and the second, when it is less ; but, if b is
taken equal to the least of the two lines ck and cb} there is no
case in which one or the other of them may not be used con-
veniently.

By the help of these series, I computed the following table.

,1^2

.3714

,5H5

,6248

,7071

,7808

>8375

.9285

00

07

h

,1962

>37*4

>5* 45
,6248

,7°7i

,7808

,8575

,9285

»98>5
!• 1

,00001
,00039
,00074
OOl 10

00140

00171

00207

00244

00271

00284

00148
00277
00406'
00522
00637
00772
00910
01019
01054 1

00521

00778

01008

01245

01522

01810

02084

02135I

01 183
01525
01896
02339
02807

°3347

02002

02405

03116

03778

04368

04560

03247

039%

04867

05639

05975

05057

06310

07478

07978

08119

09931

10789

12849

14632

19612

Find in this table, with the argument — at top, and the argu-
ment ^ in the left hand column, the corresponding logarithm ;
then add together this logarithm, the logarithm of — , and the lo-
garithm of ; the sum is logarithm of the attraction.

the Density of the Earth, 525

To compute from hence the attraction of the case on the
ball, let the box DCBA, (fig. 1.) in which the ball plays, be
divided into two parts, by a vertical section, perpendicular to
the length of the case, and passing through the centre of the
ball ; and, in fig. 9, let the parallelepiped ATSDEabde be one
of these parts, ABDE being the abovementioned vertical sec-
tion ; let x be the centre of the ball, and draw the parallelogram
fin p mix parallel to B b d D, and xgrp parallel to fiB bn, and
bisect fi l in c. Now, the dimensions of the box, on the inside,
are B6= 1,75; BD = 3,6; BjG = 1,75; and/3A = 5; whence
I find, that if x c and fix are taken as in the two upper lines of
the following table, the attractions of the different parts are as
set down below.

xc

>75

.5

,25

fix

E°5

B55

Excess of attract of Ddrg above B brg

.2374

,1614

,0813

. m d rp above n b rp

.2374

,1614

,0813

— - mesp above nasp

,37°5

,2316

,1271

Sum of these

.84,53

.5744

>28.97

Excess of attract, of B b n fi above ~Ddm$

,5°°7

>327!

,l6o6

■ Aanfi above Eeml

.4677

.3079

.1525

Whole attraction of the inside surface"!
of the half box j

,1231

,06o6

,0234

It appears, therefore, that the attraction of the box on x in-
creases faster than in proportion to the distance xc.

The specific gravity of the wood used in this case is ,6 1, and
its thickness is \ of an inch ; and therefore, if the attraction of
the outside surface of the box was the same as that of the in-
side, the whole attraction of the box on the ball, when cx = ,75,

526 Mr. Cavendish’s Experiments , &c.

would be equal to 2 x ,1231 x ,61 x -J cubic inches, or ,201
spheric inches of water, placed at the distance of one inch from
the centre of the ball. In reality, it can never be so great
as this, as the attraction of the outside surface is rather less
than that of the inside ; and, moreover, the distance of x from c
can never be quite so great as ,75 of an inch, as the greatest
motion of the arm is only ij inch.

fhilos.Tmm-. MD C CXCVJ CL 7a/>.XXm./>. JxO.

t 1 .

‘ , ^

‘

■

‘

\ •• V

. •

<1 -1

jfa i>

‘ I

.

> . . ■ • ,

i i :v , .-f 1 ' > ■ ;

• 1 :

« • .

:

.

'V ■ . .

t. - - ..it -• . ■"-> *

■ • - ■ - • ■ .... 7- ... . » ' :

■ 1

' . • I

■ ,■

I

.

/ 7

,

1

- -

'

f

. ! '

«h: ■;■■■ ■■ ' ■ . "

. ■ . - ■

.

.

TMos.TmHs.3dD C CXC^V.ITI. TciljJZXW.pJzC

O

s

t //y ,, 3

o'

Af

cT

n m

E c

'ofygs;, ,5.

J B

A, D

<L a

077^

’ y 'y „ 8 .

k

a

c

,3

n

B

b

b

x

i

t

i)

r

d

m

d

1'ltitos.Tmnx. MD (TXrYm. MXXIV.a.'^'

E e.

_ A D \d n

■-'y-s- — g

6-

—

X

; t

\

j\

1 \

£ ! \

1

\

b

/

m

-

' ■

-

T -

■ .

*

’

» ,

v ■■ '■ : : , '• "

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. L

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. - . , , ' ' ''

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t 527 1

XXII. An improved Solution of a Problem in physical Astro -
nomy ; by which , swiftly converging Series are obtained, which
are useful in computing the Perturbations of the Motions of
the Earth, Mars, and Venus, by their mutual Attraction. To
which is added an Appendix, containing an easy Method of
obtaining the Sums of many slowly converging Series which
arise in taking the Fluents of binomial Surds, &c. By the
Rev. John Hellins, F.R.S. Vicar of Potter s Pury, in North-
amptonshire. In a Letter to the Rev. Nevil Maskelyne, D. D,
F. R. S. and Astronomer Royal..

Read June 28, 1798.

REVEREND SIR, Potter’s Pury, April 17, 1797.

Such is the subject of the inclosed paper, and such the repu-
tation for skill and industry, which the many valuable papers
you have communicated to the Royal Society, and your other
learned works, have justly procured to you, that it could not
with more propriety be submitted to the judgment of any other
person than yourself, even if the writer of it were a stranger
to you.

But there are circumstances which render my presenting it
to you, in some measure, a duty. I had the advantage of being,
for some years, your Assistant in the Royal Observatory at
Greenwich ; during which time, you made the important obser-
vations on the mountain Schehallien, in Scotland, which afford

£28 Mr. Hellins’s improved Solution of

an ocular demonstration of the attraction of that mountain, and
a strong argument for the general attraction of matter, a sub-
ject nearly connected with that of the following pages ; and it
was from you that I received the problem of which you will
here find an improved solution.

The diffidence with which I entered on a speculation which
had engaged the attention of such learned men as Simpson,
Euler, and De la Grange, is well known to you. Consi-
dering the great abilities of these men, and the length of time
which Euler, in particular, appears to have employed on the
subject, all that I at first expected to effect was, to facilitate
the summation of the slowly converging series by means of
which they had computed the perturbations of the motions of
the planets in their orbits, which arise from their actions on
one another, by the force of gravity ; and that this might be
done by a method which I had some time before discovered,
was evident, on inspecting their series. Here, it is probable, I
should have stopped, had not you been pleased to put into my
hands a sheet of paper, written by the late Mr. Simpson, which,
though very ingenious, was, by mistakes, which seem to have
entered in transcribing it, rendered unintelligible to some emi-
nent mathematicians who had perused it; in which state it
had remained thirty-six years. On perusing this paper, the
first thing that occurred to me was, a different method of find-
ing the fluent, from that which had been used by Mr. Simpson;
by which means, series converging by the powers of p were
obtained, while the series brought out the common way lost all
convergency by a geometrical progression, and a computation
by it was more difficult than the computation of the length of

a Problem in physical Astronomy. 529

& quadrantal arch of the circle by the series 1 + r7+rT^+
&c. Afterwards, I discovered the method of trans-

forming that series which had lost all convergency by a geo-
metrical progression, into another in which the literal powers
decrease very swiftly ; which is the improvement I now offer
to you.

In comparing the series here produced, for computing the
values of A and B in the equation ( a — by. cos. z)“”= A + B .
cos. z - j- C . cos. 2 z + D . cos. gz &c , with those which have

been published for that purpose, by Messrs. Euler and De la
Grange, it will appear, that those cases which were the most
difficult to be computed by their methods, are the most easy
by mine. For instance, if Venus's perturbation of the motion
of the Earth were to be computed, (and vice versd,) the literal
powers which have place in M. Euler's series, would be very
nearly equal to the powers of T9- ; the literal powers which have
place in M. De la Grange's series, would be nearly equal to
the powers of i- ; and, in the series now produced, the literal
powers would decrease somewhat swifter than the powers
^ sV

M. De la Grange has indeed, by a very ingenious device,
obtained a convergency in the numeral coefficients of the series
that he uses, which, for the first five terms of it, is nearly equal
to the powers of but this convergency becomes less and
less in every succeeding term, and the coefficients approach
pretty fast to a ratio of equality ; so that, to obtain the sum of
the series to six places of decimals, he proposes to compute
the first ten terms of it. The case in which those coefficients

MDCCXC VIII. 3 Y

53b JWr. Hellins's improved Solution of

have that convergency, is when n (which answers to his s,)
is = t~~~* a case which does not often happen ; however, from

the values of A and B, when n = — he derives their values

2

when n = -J, j-, &c'- another very ingenious device, worthy
of that skill for which he is justly celebrated. But, by the me-
thod now proposed, the chief part of the convergency is in the
literal powers ; and such a difference in the numeral coeffi-
cients, for a different value of n, does not take place.

For Mars's perturbation of the Earth’s motion, the literal
powers by which the three different series converge, are nearly
as follows :

M. Euler’s, ~j ^ ;

M. De la Grange’s, >by the powers of<j ;

T he series now proposed, J [_

If, indeed, the perturbation which arises from the action of
Jupiter upon the earth was to be computed, M. De la Grange’s
series would be the best that has hitherto been published for
the purpose, as the literal powers of it would, in that case, be

t . . * ,

* For obtaining nearly the different rates of convergency of the literal powers in
the three series, it will be sufficient to consider the distance of the two planets of which
the perturbations are to be computed, as ~ v' (RR-frr — zR ry.c,z), where R and r
denote their mean distances from the sun, of which R is the greater, and c, z the cosine
©f the angle of commutation. Then will M. De la Grange's series converge by the

powers of the quantity -f f- ; and, since RR -J- r rzza, and zR r—b, in our notation*

b b

and the converging quantity in M. Euler’s series is (n n) — " ifc =

. . a — b

— i-— — - ; and c c, by the powers of which the new series converges, is =: — — -r =
(RR -f r r)a 3 v «+ b

J. Ill,2- _r f r r — HLzlL. See the Memoirs of the Royal Academy of Sciences and
RR + 2Rr-[-rr (R + r)*

Belles-Lettres at Berlin, foriySi, p. Z57 ; M. Euler’s Institutions Calculi Integrals,
Vol. I. p. 186 ; and Art. 4, in what follows.

a Problem in physical Astronomy. 531

nearly equal to the powers of while the literal powers in
the new series would differ but little from those of So
that, for computing the perturbation of each of these three pla-
nets, we now have series converging so very swiftly, that the
first four terms are sufficient for the purpose.

These indeed are the perturbations of motion, arising from
the actions of the planets, which the inhabitants of this globe
have most frequent occasion to compute. And, since two of the
three are most easily calculated by the method explained in the
following pages, I am not without hopes that I have rendered
an acceptable piece of service to astronomers in general, and
more especially to those who are most intent upon improving
astronomical tables.

But it may be proper to remark, that the use of the new series
is not confined to the computations just mentioned, but may
successfully be used in computing the perturbations of the mo-
tions of other planets. For instance, in the computation of
the perturbation of Saturn’s motion by Jupiter, (and vice versa,)
the convergency of this series will be nearly by the powers of
y1— , which is a swift rate of convergency. And, for the pertur-
bation of the Georgium sidus by Saturn, (and vice versa,) the
series will converge nearly by the powers of ■§-, which is also
swiftly.

And it is further to be remarked, that in the last instance,
and indeed whenever the radii of the orbits of the two planets
differ from each other in the ratio of 2 to 1, M. De la Grange’s
series may be used with advantage, since the convergency of
the first five terms of it will then be nearly by the powers
of ; the numeral coefficients of those terms converging as

3 Ys

533 Mr. Hellins's improved Solution of

V

swiftly as the literal powers do in that case. And, when the
ratio of the two radii is greater than that of 2 to 1, his series
will converge more swiftly.

With great pleasure therefore I see, that, by one or other
of these methods, some of the longest and most difficult
calculations which formerly arose in the theory of astrono-
my, may now be exchanged for others which are short and
easy.

It is with satisfaction also, that I perceive the facility of com-
puting by the series I now present to you, is not at all less-
ened by the more general notation you have given to the
denominator of the fraction from which it is derived, at the
same time that a more accurate result is obtained than M. De
la Grange proposed. For, in the computations of which I
have been speaking, he neglected both the excentricities of the
orbits of the planets, and their inclinations to the ecliptic, as
inconsiderable : you, finding the effect of these omissions to be
greater than he imagined, have taken them in. Your other
ingenious labours on this subject will be best described by your-
self, and cannot fail of being gratefully received by all learned
astronomers.

With respect to the method by which the sums of the very
slowly converging numerical series, which occur in the sub-
sequent pages, are obtained, I need not say to you, that it is
of extensive utility, and may be successfully applied in many
cases.

I have only to request, that, if the paper here inclosed meets
with your approbation, you will communicate it to the Royal
Society. For, although I think I cannot be mistaken re-

a Problem in physical Astronomy. 533

specting the utility of the invention explained in it, }ret such

/•

is my respect for that learned body, that I am unwilling to
send them any paper of mine, on so difficult and important
a subject, till it has been examined by an able judge of the
subject.

I am,

Rev. Sir, &c..

JOHN HELLINS,

An improved Solution of a Problem in physical Astronomy, &c.

1. The perturbation of the motions of the planets in their
orbits, by the action of one upon another, is a curious pheno-
menon, which, while it affords to the philosopher a clear proof
of the general attraction of matter, produces a problem of no
small difficulty to the astronomer ; viz. to compute the quan-
tity by which a planet, so acted upon, deviates from an ellipsis
in its course round the sun : a problem which hath called forth
the skill of several of the most learned philosophers and astro-
nomers of the last and present age.

A preparatory step to the solution of this problem is, to find
a convenient expression for the reciprocal of the cube, or ra-
ther of the nth power, of the distance of any two planets. Such
an expression was first given by M. Euler, in series proceed-
ing by the cosines of the multiples, in arithmetic progression,
of the angle of commutation ; but the calculations of the first

534 Mr. Hellins’s improved Solution of

two coefficients in it were very laborious, requiring the sum-
mation of series of the common form, which converged very
slowly. Afterwards, other series were discovered by other
authors, whereby the same coefficients might be computed with
less labour ; the best of which, that I have seen, appear to be
those that were pointed out to me by Dr. Maskelyne, in-
vented by M. De la Grange, and published in the Memoirs
of the Royal Academy of Sciences at Berlin , for the year 1781.
Yet, the calculation of the two first coefficients, A and B, for
the perturbations of Mars, Venus, and the Earth, by his me-
thod, is not shorter, if it be so short as by my method, to the
investigation of which I now proceed.

PROBLEM.

2. To determine the values of A, B, C, D, &c. in the equation

TZTToZTy — * (A+B . COS. Zf-C . COS. 2£-fiD . COS. QZ, &C. )

z being the arch of a circle of which the radius is 1, and b less
than a.

First, to find the coefficient A.

3. The fluent of the right-hand side of this equation is A z
B . sin. z-\- |C . sin. zz + i D . sin. $z -f £E . sin. ^z* &c.

which evidently vanishes when z=o ; and, when 2=3* 14159,
&c. the arch of 180°, it becomes barely = A z, the sines of z,
qz, 3 z, &c. being then each = o. If, therefore, the fluent of
the first side of the equation be taken, the increase of it, while
% increases from o to 3 • 14159 &c. = tt, will be = 7r A ; and,
consequently, A will be determined.

• See M. Euler’s Instituiiones Calculi Integrals , Vol. I. p. 150.

V

a Problem in physical Astronomy. 535

a. Now, to find the fluent of 7 — r-, we have - — r-

7 (a—b.cos.z)n7 (a— b . cos. z)n

= ■ ; — 7* , x beine: put = the cosine of z; in which

expressions, while % increases from o to 3- 14159, x will de-
crease from 1 to — 1. Therefore, to obtain a more convenient
expression, put v 77 = ^^; then, while x decreases from 1 to

— 1, vv will increase from to = 1 ; and we shall have

7 a+b fl-f 0 7

the following equations :

a — bx— ( a-\-b ) vv, ( a — b x)n=. ( a-\-b )”v x = a~ ^ v % —

{a + b)zvV' _ . , a—(a + b)vv a + b-(a + b)vv a + b /<

b , l-f-X—l-\ -b = j — —(l—VV)

a— (a-{-b)vv

b—a-\-(a-\-b)vv

a-\-b 1 b — a

b —

b

b \ a 4- b

— =“-ir(vv—cc)’ cc bein£ Put = ~p- And from

these equations, the three following are easily obtained, viz.

v/(i+*) v/(i--c) = v/(1^(i— »»))

==!T~V//((1~ vv) ( vv — cc)); and,

— x (a + b)zv<v b ivw

— ■ ~ ! 1 W ■ 1 — > ■ — . ■■ ■ ■ - -

b («+,©y'((i-w) (w— cc)) v('—w)vtyv*-ccy‘

and, lastly , ^ a — bxy yqi — Vv)\/(vv — cc) (« + i) vn

_ • 1 — 2 n

(a -f b) x —7 ^—7 r ; the fluent of which may be found

v 1 ' a/ {i—vv) [yv — cc) J

when the value of n is given.

5. Now, the values of n with which astronomers are most
concerned, are J- and j-. Let, therefore, \ be written for n, and
the radical quantity ^/(i — vv) be converted into series, and
the last expression will be

'f f

53s

Mr. Hellins’s improved Solution of

=(a+b) *x_4^_rI+”4. r 2±ll’ &c \

V 1 > Viyv-cc) ^XT- 2 i 2.4 "T 246 T 2.4.6.8J

“1

:(#-(-&) 2 X

4-

2vV

\Z(vv—cc)

J f x 4- 111 4. -L 3±7. v: Mc\

* y/(vv-cc)\L^ 4 4.6 < 4.6.8 >^C'J

3-5 v* 1 3-5-7

And the fluents of these several terms, without their coeffi-
cients, are as follows :

/

/

I

f
f

These fluents, being multiplied by their proper coefficients,

and collected together, and the whole multiplied by the com-

-3.

mon factor [a 4 b) 2, the fluent sought will be

2 s/ {vv — c c)

<VV

\/{vv— cc) .

^{yv—cc)

•if

J.O 1 — - •

CC'J 5

UT »+4/(WW-<?0 _

\/{vv — cc)

— ' — Ai. * JLi . ■■ —

c

— ai 5

* vv a

v'Cv v — c c) v+e c a.

- /° .

\/{vv — c c)

2

- fa ,

V v 4

</(yv—cc) v3-f-^ccS _

\/ {vv — cc)

4

- 7 >

vv6

{vv—cc ) vs + 5ccy

- A*-

[yv—c c)

— 6

- Q 1

&C.

& C.

(a-\-b) 2 x

1

ccv

(_ + * + 7 £ + v + 4.6.8

6. We must now inquire what value this series has when
— o ; in which case, x being = 1, vv is = j^~b — cc. And

it will appear that, with this value of vv , every term of the se-
ries vanishes, so that the fluent needs no correction. If, there-
fore, we compute the value of this series when % == tt, i. e. when

3-5-7

S,&c.

z

x = — 1, and vv

a -f b
ci -|~ b

1, we shall have the value of Att,

a Problem in physical Astronomy. 537

and consequently, A will be determined. But, with this value
of v, the terms £, y, &c. lose all convergency by the geo-
metrical progression v, v3, v\ &c. and the computation of the
value of the series, by the common method, would be more
laborious than the computation of the quadrantal arch of the

circle, by the series 1 -4 — - — 1 j M — , & c. Here then

3 j 1 2.3 1 2.4.5 2.40.7

we are stopped. But, by contemplating this series, expressed
in terms of a and c, as it stands below, a very different method
of obtaining the value of it is suggested.

7. cc == a.
y g 3 v/~ ( 1 — c c)

4 4-z

3v5 __ 3-5 V (1— gg)

4.6 ' 4.6.4

3-5-7 t* 3-5.7^(i -cc)

4.6.8 4.6. 8. 6

+

3 C Co.

4-2

+

+

3-5-3 ccy{\ — cc)
4 6.4.2

3.5.7.50: — cc)

4.6.8. 6. 4

+

3-5-3^*
4.6. 4.2

+

3-S-7-5-3 cV (1— cc)
4.6. 8. 6. 4. 2

3-5-7-9 _ 3-5-7-9 \/(i— cc) , 3-5.7.9.7cg^/( i -cc) , 3-5-7-9-7-5cV(* — gQ
4.6.8.10 4.6.8.10.8 > 4.6.8.10.8.6 > 4.6.8.10.8.6.4

&iV. &c.

Here it appears, 1st. That the geometrical progression 1, cc,
c*, &c. has place in the first, second, third, &c. columns of
quantities on the right-hand side of the equation, the terms of
which, when b is nearly = a, decrease very swiftly.

2dly. That, in the diagonal line of quantities in which a en-
ters, besides this decrease of the terms, by the literal powers,
before mentioned, the numeral coefficients are so simple that
a considerable number of the terms may quickly be com-
puted.

gdly. That, if this diagonal line of quantities be taken away,
the first, second, third, &c. infinite columns of quantities which

MDCCXCVIII.

538 Mr. Hell ins's improved Solution of

remain will have the literal factors v/(i — cc ), cc</( 1 — rr)?

cc)> respectively, which are in the progression
before mentioned.

^thly. That, if the sum of the infinite series of numeral co-
efficients below the line, in each of these columns, can be ob-
tained, then the original series, which had lost all convergency
by the literal powers v, v3, v\ &c. may be transformed into
two others, in which the literal powers will be cc, c4, &c.

8. But the sums of these infinite series are attainable, and
are as follows :

3

4-

3-5

4-

3-5-7

1

3 -5-7-9

•

4.2

4.6.4

4.6.8. 6

~T

4.6.8. 10.8’

is

— T H. L. 2 =

= A;

3-5-3
4.6. 4. 2

+

35-7-5
4. 6. 8. 6. 4

+

3-5 -7-9-7
4.6.8. xo. 8. 6

+

3.5.7.9.H.9 r-

4.6.8.10.12.10.8’

is

= li + |H.L.2 =

: p;

3 * 5 * 7 • 5 * 3
4.6. 8. 6. 4. 2

+

3-5-7-9-7-5

4.6.8.10.8.6.4

+

3. 5.7.9.11. 9.7
4.6.8.10.12.10.

8.6 ’

me. is

17 + 51 H. L. 2

64

v > '

&C.

&C.

But these three sums

are

as many as are requisite, when

the

perturbation of the motion of either the Earth, Mars, or Venus,
by the attraction of any one of the other, is to be computed.

9. The sum of the coefficients in the first, second, and third

columns in Art. 7. being now obtained, take | — — a for the

' 0 0 ~ 5 cc

value of the series a (1 -f -f &c.), which will be

exact enough for the purpose, and we shall have, by Art.. 3, 5*
6, 7, and 8,

* See the Appendix.

a Problem in physical Astronomy .«

539

— s

2 \/(l — ■ CC)
CC

8 — zee

5 cc

■5rA=(a+6) 2x<j _J>

[ + A v/(l — £c) — -Cc)SvC*^/[l — CC)

and thence, by a more commodious arrangement of the terms,
and dividing both sides by 7 r.

A =

f S — zee

{a -f- b)

I X *<

8 — 5 c c

et.

“f“ V^C1 ^ “f~ (ACC -f-

10. The value of A, when n=zj., being now found, let us
next investigate the value of it when n = f ; which, for the

sake of distinction, in a use to be made of it in a subsequent
article, I denote by A'.*

By writing j. for n in the fluxionary expression obtained in

Art. 4. we have (a -f- b)k x — which bv

converting the radical quantity </(i — vv) into series, becomes

5 — ’ 4

(« + b) 5X

2 ‘VV

vv _ cc)

f~ 2 <b V ~ 4

— ( n I Z,\~ j V(vv

— (0 + 0) x<

1+&-+ ^ + ^4 + LLL* £?, ]

1 2 * 24 1 2.4.6 c 2. 4.6. 8* ^JC‘I

+

*VVm

cc) * — cc)

. -L . [3. I

L V{lJV — cc) 14 I

ysvv

4.6

, 3±7^ 4. M-9* ,

I , A a T

Now, the fluents of these terms, without their coefficients
are as follows :

\/ {vv — cc)
— 2,

/

/‘VV

\/{vv — cej

f

is = —(vv g£l 1

2 CC V3 ~

y,(pp — cc)

CCtl *

2 x/(yv — cc) m

3 C*v

V{vv — cc)
vv1

= H. L. —

c

y(vv — cc) V -f- cect

a

\/{vv — cc)

as they are exhibited in Art.

3 Z 2

£; and the rest

54° Afr. Hellins's improved Solution of

These fluents being multiplied by their proper coefficients*
and collected together, and their sum multiplied by the com-
mon factor {a + b)\ we shall have

l rfl n

+

— 5

2 */(vv — c c)

( a + b)

| 4 >/(vv — cc) | »y(vv — cc)

3 ccv* * 3 c+v

3*5 p i 3-5*7

y +

ccv
3-5-7 -9

£ &C.

4.6 w 1 4.6.8 f * 4.6.8.10

which fluent needs no correction, since, when v = c, the whole
vanishes. The value of it therefore, when v = 1, will be = AV,.
which is what we want ; and, in obtaining it, the only difficulty
is, to compute the series in which a, £, y, &c. enter, which
may be overcome in the manner shewn above in Art. 7. For
the value of this series, when v = 1, will be as follows :

3

• 4

os =

3-5

€ =

3.5 v(i_cc)

4.6

4 .6.2

3-5-7

3-5-7 \T(i—cc)

4.6.8

y —

4.6.84

+

3-5 ccco
4.6.2

3-5-7-9

4.6.8.10

3*5 *7*9-x 1
4.6.8.10.12

&C.

£

3-5*7*9 v' ( 1 — cc)

+

3-S-7-3 c<»
4.6. 8.4. 2

4.6.8. X0.6
3.57.9.11^(1.

, 3«5-7-3 C^/(I— cc)

* 4.6. 8.4.2

» 3.5.7.9.500^(1—00) ~ 3-5-7-9«5*3 cV(i— ccj

' 4.6.8. 10.6.4 ' T > Oo-

‘4.6.8.10.12.8

&c.

-cc) , 3. 57.9.1 \-7cc\/(i—cc) .

”1" a f' in 19 SI A r »

4.6.8.10.6.4,2
3.57.9.1 17.5 c4 v/(l.

■ cc)

4.6.8.10.12.8.6
&C.

4.6.8. 10. iz. 8.6. 4
&C.

Here, by attending to the same things which were observed
in Art. 7, we may easily obtain the values of as many of the
infinite columns of quantities on the right-hand side of the
equation as are wanted ; which, for the planets Mars, Venus„
the Earth, and some of the rest, are but three.

12. The value of the series, of which a is the factor, viz :

1 +

3.5 cc

+

3.57.3 c4

, — 7- , wf. I will be obtained sufficiently near

4.6.2 1 4.6.84.2’ r J

for the purpose, by this expression, ~i/cc *s some”

what more exact than the three first terms of it ; and the infi-
nite series

a Problem in physical Astronomy.

54 1

3-5
4.6. 2

3-S-7-3
4.6. 8.4.2

+

4

3-5-7
4.6. 8. 4

3-5-7-9-5

4.6.8.10.6.4

+

+

3-S-7-9 , 3-5-7-9.1 1 •

l. 6. 8. 10.6 * 4.6.8.10.12.8’

9 + — H. L. 2

l6

4

.. 3’57-9’1I-7 •

4.6.8.10.12.8.6’ ^6- ib

= ^ + TiH-L-s = f‘':

3-S-7-9-5-3 , . 3.S-7-9-U-7-5 , 3-5-7-9-1I-I3-9-7 •

I A f. « «-"> T* 8 6.4 “ 4.6.8.10.12.14.10.8,6’

4.6.8.IO.6.4.2 * 4.6.8.10.12.8.6.4

4067 , 329 rj t / *

_L 329 TT T _

~ 12288 + 777 tt'-Usj

v .

We therefore now have «+xV(i— cr)+/crv/(i— cc)

v'c* \/ (i~cc) for a near value of the infinite series — a. -f-

13. Having thus obtained a sufficiently near value of the in-
finite series which entered into the fluent, in Art. 10, we have
only to add to it the three radical terms there found, v being

put = 1, and to multiply the whole by (a+bfi, and we shall
have

~ 2a/(I— CC) 4 y/(l-cc) ^(1 —CC)

CC

3 cc 1 3c4

7rA'=(tf4-&) 1 96-23 $

| * 128— S4CC a

L+xV(#-tc)+^cv/(i-«)+/cV(i— «)

which equation being more concisely expressed, and divided
by 7 r, gives

96— 23CC

A'

it

X

128 — 84^

a

+ v/(i (~^4

# See the Appendix.

54P

Mr. Hellins's improved Solution of

Secondly, to find the Coefficient B.

14. Multiply the equation in Art. 2. by 2 cos. # = 2 x, and we

shall have— = % (A x 2 cos.# +B x 2 cos.# xcos.# + C x 2 cos.#

x cos. 2# + D x 2 cos. # x cos. 3#, &c.) ; which, because 2 cos.#
x cos. mz is = cos. (m — 1) # -f- cos. [m -f* 1) #, will be = z
(2A.cos.#-j- B( i-f-cos.2#) -j- C (cos.# -f cos. 3#) -j- D (cos. 2#

+ cos. 4 #), And, by taking the fluents, we have

/'

Ja'—bx)" = 2 A. sin.# + B#-f iB.sin. 2#-(-C(sin.#-j-isin.3#)
~f* D (i sin. 2 # -|- i sin. 4 #), &c. ; which equation, when # =
3-14159, &c.= 7T, becomes j = barely B # == B tt, the

sines of #, 2 #, 3 #, &c. being then = o.

15. Now it appears, by the notation in Art. 4, that

\/(i-xx) ( a—bx)n x ^/{yv—cc) > that X =

we therefore have, by proper substitution,

2XZ

— ZXX

z a

zvv

1

( a—bx )" — xx) ( a—bx)n b{a + b)n X ^(i— ' wj \/ {vv— cc) I

2 l V

3 — 2«

>»

b{a + b)

71 — I

(1 — vv) (vv— cc)

of which two fluxions the fluents may be found, when n has
any particular value.

16. First, let n be then the last expression in the pre-

. — 2.

ceding article becomes — — — - x — — r x

. b{a-\-b)i »/{i — vv) y/(vv — cc) b(a-i-b)z

( 1 —vvYfvv—7c) ‘ Now, the fluent of the affirmative part of this
expression is evidently = y x the fluent of the fluxion in Art. 5,

See Simpson’s Miscellaneous Tracts, lemma I. p. 76.

a Problem in physical Astronomy.

543

that is, = y At r; and the negative part, by converting ^/(i — w)
into series, will become — ~* 2--- x — — — ( 1 4- — -I- Afl 4-

b(a+b)i V^vv—cc) \ ' z ' 24 *

« C X)^

7^6 > &c.)\ the fluent of which appears, by Art. 5, to be

+ e + which will vanish

when v = c, and therefore needs no correction ; and, when
v = 1, the series, without the factor, will be as follows :

2 a =± 2 tx

g V(i—cc)

iy

3^/(1 — cc)'
'44

+

+

C C tx.

3-3 f-c ' {1 — cc)
44.2

34 £ 3 -5a/(i-

4.6 4.6.6

■ cc)

+

3.3 c4«
44-2

■ 3 54cca/(i — cc)
* 4.6.64

+

344-3cV(i — gc)
4.6.64.2

347 c 347^(1— cc) , 34.7.7CC y'(i-cc) 34774c4 ^(1— CC)

4.6.8 4.6.8. 8 “r 4.6. 8.8.6 » 4.6.8.8.64 >

8V. &?6\

Now, the sum of the infinite series
f + 54 + Hi + TXO> &c- beinS = a H . L . 2 = P,

& c. being = y H . L . 2 = <r.

3-3

4.4.2

of^ + ^ +

3477

+

347.9.9

4.6 8.8.6 T 4.6.8.10.10.8’

Of

344-3

1 34774 , 347-9-97 ,

4.6.64.2 4.6.8.8,64 * 4.6.8.10.10.8.6 "T"

^4^H.L.3 = T)6?c.

34.7.9.11.11.9

4.6.8.10.12.12.10.8’

&C.

&c.

By proceeding as above, in Art. g, a sufficiently near value
of the whole series will be obtained in this expression,

16— gcc a + n/(i — cc) (p 4- o-cc re4) ; and this, multiplied
by its proper factor, gives

£44

Mr. Hellins's improved Solution of

— z

b [a + b)k

X <

32—10 cc
16— gcc

05

- + \Z(i—cc) (p -f crc -|-tc4) for the other part
of the fluent sought. And since, by Art. 14, this fluent is
= B 7T, we have, by dividing both sides by tt,

B = ~ A

o it b (a + b)z

X

32 — 10 cc

05

16 — 9 cc

-f-v/(i' — cc ) (p + °'£:^ + Tc4)»w^c^

is its value when n = 4-

17. We are next to find the value of this coefficient, when
n = f ; which, for the sake of distinction, I denote by B7.

With this value of n , the fluxionary expression in Art. 15, be-

- •— \± “*9.

Zll

comes

zw

X

2t;»

<✓(*— w) V'(w— cc) v(i— vv) »/(yv— cc)

which being compared with the fluxions in Art. 5 and 10, it
will appear that the fluent of the former part, when v = 1, is

= ~ AV, and that the fluent of the latter part is = At r;
which fluents, taken together, are, by Art. 14, — B V. There-
fore we have B7 = ^ A' — f- A = 4- ( A7 a — A),

Thirdly, to find the Values of C, D, E, &c.

18. The values of the coefficients A and B being now found,
corresponding to the values of n ~ and we might proceed in
the same manner to find the value of C. For, if the equation
in Art. 2, be multiplied by 2 cos. 2 z, and cos. (m — ■ 2) z -f-
cos. (m -j- 2) z be written for 2 cos. qz x cos. mz, it will become

JSagfcli- = Z (2 A X cos. 2 % + B (cos. % + cos. .3 z) +

[a— &XCOS.2)” v

C (1 + cos. 42) + D (cos. 2 + cos. jz), &c.) And the sum

a Problem in physical Astronomy .

545

of the fluents on the right-hand side, when z=nr, will become
barely C z = C tt. Therefore, the fluent of the left-hand side

of the equation, when z = v, or of — -J .when-r^x,

or of 4aa— 8a (fl + &)pp.f 4(

bb(a-\-b)n

X

Z'yi/

1 — 2t»

a/(i — (yv—c c)*

when

v = 1, will be = C7r. The fluent of this fluxion, it is evident,
will consist of three parts, the first and second of which, n be-
ing = f, are obviously attainable from the values of A and B
above found in Art. g. and 16. ; and the third in series similar td
those which have been given in the former part of this paper.

It is evident also that, if n be = f, all three parts of this fluent'
are attainable from the values of the two coefficients already
found, and C' would be = — 2A'+-|-(B/a — B).

lg. And in this manner may the other coefficients, D, E,
F, &c. be determined. And since the cosines of 3 z, 4 z, &c.
are = 4 a:1— gx, 8.z4 — 8x2-f- 3, &c. respectively; and since

x — 1 > lt is evident that the numerator of the fraction

into which the fluxion in the preceding article is to be multi-
plied, will be always of this form, viz. p + qvv+rv'-^- sv*, &c.;
from which it follows, that, if the values of A', A, A, &c. cor-

responding to n, n — 1, n — 2, &c. be computed, the values of
C, D, E, F, and all the rest, may be found in terms of A', A,
A, &c. with the coefficients a and b. But, since the easiest
method, that has come to my hands, of computing the values
of C, D, E, &c. after A and B are found, is explained in M.
Euler’s Institutiones Calculi integralis, Vol. I. p. 181,* I shall

* The coefficients z, 3, and 4, after 2C sin. 3 D sin. and 4E sin. in line 8 of the
page above referred to, are wanting; and — is printed for -f- before 2 C, in line 13,
And there are press errors in many other places. It is to be regretted, that so excel-
lent a book was not more correctly printed.

MDCCXCVIII. 4 A

54$ Mr. Hellins’s unproved Solution of

not pursue this method any further ; but, having examined his
process, and corrected the errors of the press which occur in it,
now give the equations expressing the values of C, D, E, F, &c.
which were obtained by that method.

20. For the sake of brevity, let y = d ; then will the general

values of C, D, E, F, &c. be expressed by these equations :

2 n A — 2 d B
n — 2

(n -f 1) B — 4^C
n ~ 3

(n + 2) C — 6 JD
n — 4

(« 4- 3) D — 8 rfE
n — 5

&C.

where the law of continuation is very obvious. And the parti-
cular values of these letters, when n = J-, and j-, will be as

expressed in the following columns :

a

ii

a

II

II

C= 4<i B — -4 A

3 3

4“ B 6 A

— ^d B -fi 10A

D— Sl1 C 3 B

8d c 5 B

— c — i-B

5 5

3 3

i i

E=:—D — i-C

iiiD-Ac

— D — -2-C

7 7

5 5

3 3

F =— E —Ad

iilE -ad

i6^e — d

9 9

&C.

7 7

&C.

5 5

&C ,

21. The solution of the problem being now finished, it may
perhaps be satisfactory to the reader to see how the sums of
the very slowly converging numerical series, which arose in
Art. 7, ii, and i 6, were obtained; the investigations of which.

a Problem in physical Astronomy . 547

because they would have detained him too long from the im-
mediate subject of this paper, if they had been inserted in it,
are given in the following Appendix.

AN APPENDIX TO THE FOREGOING PAPER:

In which the Method of obtaining the Sums of the very slowly
converging numerical Series which are used therein, and of
many others of that Kind which arise in the Fluents of Bino-
mial Surds, is explained and illustrated ; and some Observa-
tions, tending to facilitate and abridge the Computations of
the Coefficients A and B, are added.

1. As the sums of the very slowly converging numerical
series, which arose in Art. 7, 11, and 16, of the preceding paper,
are not exhibited in any book that has come to my hands, and
as series of that kind frequently occur, I conceive that the fol-
lowing method of obtaining their sums will be acceptable to
the lovers of mathematics in general, and particularly to those
who have frequent occasion to use the sums of such series.
And, having observed, while considering the literal expressions
in the preceding paper for the values of A and B, that others,
no less accurate, might be derived from them, by which the
arithmetical operations would be facilitated and abridged, I
thought these observations might likewise be acceptable to those
who are engaged in the theory of astronomy, and have inserted
them also in this paper; which, therefore, consists of two prin-
cipal parts, the summation of the slowly converging series, and
the observations now mentioned.

4 A 2

$4>s

Mr. Hellins’s improved Solution of

I. The Summation of the slowly converging Series .

e. But, before I begin the investigation, it will be proper to

premise a few particulars, an attention to which will shorten

and facilitate the operations now to be performed.

Tint 1 *“ A'/(I ~~yy} hnn- — 1 -v'(x-yj') .. * + SU-yy)
1St illat. +'V(i_vv) beins — . + ^(,_„ x .

IS =

i + ^/(i — ;y;y)

■ is = 2 H. L

yy) i + 4/ (i -yy) ~ » + V(i-yy),

from which it follows, that H. L. of

y

i + V(i — yy)™ ‘ » + -/(*— ^j')’

sdly. That the fluxion ofH.L., + ~TJJ) is =

~~ For it is = the fluxion of — H. L. |i v/(i -yy)}

— — 21 — - x r x ; and, if both numerator and de-

vh—yy) * 1 + V(i —yy)

nominator of this expression be multiplied by l — ^(l — yy),

yl — 1 _ 'S(l—y_y)9 which is — ■

it will become

V{ *—yy)

yy

yV(i-yy)

Sdly. That the H. L. -- y*;— ^ is therefore =

-yy)

J y

2.2

+

2.4.4 1 z*4-6.6 ■ 2.4.6. 8. 8 :

4thly. That, O being put = </(i—yy), the. fluxion of

will be = + pEn)- For it; wil1 be f? ~

— — y _ (»-yy) __ -» j_ (” — *)i

— y CL y-fc-'Q^ y~‘Q, y + * y+'Q^T /-‘Q,,

y l — « 1 " — M

“ Qjy-v* t y-1 r

5thly. That, when any quantities, as

are circumscribed by a parallelogram, it denotes that a substi-

4 + — + *

\y ' yy 1

a Problem in physical Astronomy. 54$

tution for these quantities has been made in the same equation
in which it occurs, and consequently that they are no longer to
be considered as part of that equation. This I have found to
be better than cancelling, as it answers the same end without
obliteration.

We may now proceed to the summation of the series before-
mentioned, in which the utility of what has been premised will
quickly appear.

3. As it does not seem necessary to set down the operations
of computing the sums of all the series which arose in the pre-
ceding paper, I will make choice of the summation of those
which occur in Art. 11, they being the most difficult, as the
properest examples to illustrate this method.

It is well known that the expression . is = 2 y y

—5

4- 4- 1 M 2-3-5yy J_ *-3-5-&y\ j_ 2-3-5-7-9y y*

2 ' 2.4 “ 2.4.6 ~ 2. 4.6. 8 ~ 2.4.6.8.10 ’

from which equation we have —2^y - 2 y y~5 y y -3 —

^ + - W -1- Now the fluents of

the terms on the | 2 y*

first side are 1 , 1

U

f — a/(i— yy) 3^(1 —yy

4 y

+ fH.L.— 7; y-

1 4 i±y(i

-yy)

— h —

2j4 I 2 yy

-H.L.y

4 J

4IH.L. i •

4 1 + v ( 1 —yy)

on the second side, the fluents are 4. hliUl _l. 3-S-7-9/

4.6.2 ^ 4. 6. 8.4 * 4.6.8.10.6*

&c. And, to find whether these two expressions are = each
other, or have a constant difference, we may compute their
numerical values, y being put = any small simple fraction,
such as T^5, or either of which values of y is a very
convenient one for the purpose. But an easier method to dis-

550- Mr. Hellins’s improved Solution of

cover the constant quantities which lie concealed in some of
the terms on the first side, is to convert that side into series, by
the binomial theorem ; which will then be as follows :

— v'O-ouO

z /

—3 v/(i—

Ayy

— — + iy ~* + A + 3 V/ + *£*/>

— i.?-’ + 1 + 3j/+ A />

+

+ ^7 = + 1 +

-|H.L.2+TV/+Tl¥/,6fC.

2y+

4--|H.L. — T~77- 7

1 4 ! + -✓(*— ay)

The sum is = * * +*-|H.L • &, + */+

which evidently differs from the series on the second side by
the constant quantity J H.L. 2. We therefore have, by
subtracting this constant quantity from the first side,

— \/(i —yy)

2 y*

+

+

3^(1 —yy)
4 yy

i

+ *RL' T+/TZJ7)]

7

16

r

_ 3 -5yy . 3-5-7 y* , 3-5-7-9)6 Cg».
4.6.84'*' 4.6.8.10.4,*

2/*- *

which, when y becomes = 1, becomes
* * -J- H. L. 2

II

3-5

x

2

-Li 7_ _| 9

I 2 16 T 1

+

J

3-5-7

+

3-5-7-9

-, &C.

4.6.2 1 4.6. 8.4 1 4.6.8.10.6

which is the series denoted by x' in Art. 12. of the preceding
paper.

4, If the equation of fluents in the preceding Article be di-
vided by y, and if = — L. y be then taken from both sides

of it, and u be written for H.L.

— yy) 3v/(I— yy) , 3jfl

2y 5 4/ ' 4J l

1 + v{'—yy)

we shall have

3-5-7/ | 3-57-9/ ; 3-5 -7-9-”/ eg

4.6.84”** 4.6.8.10.6 ”*” 4. 6. 8. 10. 12. 8’

-L __L_ _J 1 Z 5^1

* 2_)*s * 2y 3 i6j> 16 J

And, if this equation be put into fluxions, and Q be written
for v/ C1 — yy )> f°r the sake of brevity, there will be

a Problem in physical Astronomy.

55 1

ay

+ -^r +

+

+

4 y
4

2/

-2-"}

6 f

4XV „

i

+

3 il

4 y

+

+

ay

i6yy

3

4^y

. 5
16

3«j

Ayy

3-S-7‘3yyy

y 4.6.84

-L -3-5'7‘9-5yy*
~ 4.6.8.10.6

3.5.7.9.11.7,7/ r.

4.6.8.10.12.8 ’

And this equation, more concisely expressed and divided by y,
gives

+

1 5 4-

1

3 )

y

\ 2y7 '

4 y

4/ 1

(s

3

5

5 ]

l zy

zys

i6y

l6y J

- 3uj
4 /

v 3»5-7-3i3> 1

y — 4.6.84 +

3-5-7-9-5iy

4.6.8.10.6

I 3-5-7-9-»*.7j/
”4.6.8.10.12.8

Now the fluent of the series on the second side of this equation
is found, by the methods which have been long known, to be

+

3-5‘7-9-3y4 , 3.5.7.9.11.7/ .

4.6.8.10.64 “r 4.6.8.10.12.8.6 5 &c' anc* the fluent of

3-s-7-syy

4.6.84.2

the terms on the first side will be very easily obtained, by the

following assumption, and attention to what was shewn in Art.
2. of this paper.

For the fluent of the terms on the first side of this equation,
assume

)s + “ i-Jr+g)

+ s H. L.y ; then will the fluxion of this

(~7~ +

+

/

P

b

/

+

+

+

y ' y
expression be

—6a 4 b 2 c

y7 ~y y

+ ”y“ + "Jr* +

c

y

r

~

-f- U

+ -7- +

c

II

y J

■

y

y

1

yy

+,§•))

2/lly

55® Mr. Hellins's improved Solution of

4?

y5

y, which being put = the first side o'f

the foregoing equation, there will arise as many simple equa-
tions for determining the coefficients a , b , r, &c. as there are
letters of that kind in the assumed fluent, from which their
values will easily be found. For there will be

6a = — j, from which

ft

II

_ 1

N

V*

4>b = 5a — b

0-

II

N j*vJ

r— “5

6 — 16*

II

©1

f=~*

1

II

S = -W‘

0K

*0 N

II

5-

n

v#1

If

^ — 1

32 *

to

II

O

•

The variable part, therefore, of the fluent of the first side of the
above equation is

Q

+

5

izy 4
3

J _

I 8

l6yy

I

16

12 ya 1 2y* izyy

Now, to discover the constant quantities which lie concealed in
this expression, we must proceed as above in Art. 3.

a Problem in physical Astronomy.

-5 v/ ( i — yy)

is

J_ v—6

izy° " 12

izy*

~s\/(i—yy) „

i6yy

+-h)*=

To which add the |
other terms - J

/~6+

^ — 4 -J 2 M— 2

7

x^

-Tr^4 +

+ 7IS+

— — 'V- 2

12. 2 ^

5

— wy

+

+

+

+

553

— i — i iiL.-va Afc

x 2 . 1 6 T 12.8.16/ * ^6*

■/, 6s?e.

+ T,y~6 + ■§■ y~* - j-2y-

The sum is

* +

/

12.8

+

/

12.16

5

■ +

5

16,2

16.8

3

■ +

29

8.32

67

105

y\ &c.

which exceeds the series above found, by the constant quantity
-^L. We therefore now have

19 2

Q

+

-5

12 y*
S

+

12 y
3

i6yy

I

—j— u

877

+

16

67

127° • 8 y*-

3- 5-7- 9- ”■ 7/

3-5-7 3 ^ ^ 1 3-5-7-9-57* ,

I -I /1 X in 5/i I

32yy 192 4.6. 8.4.2 ”~i 4.6.8.10.6.4

4 6X10.12.8^6'* ’ anc* w^en JV becomes = 1, Q being then

= o, and w == H. L. 2, this equation becomes

H.L. 21

-I-Jl _Lj._i_.5z (

1 12 1 8 32 192J

+

79

192

_ 3-5-73

3-5‘7-9‘5 | 3-5-7-9-»M ^

’ 4. 6. 8. 4. 2 4. 6. 8. 10. 6. 4 *"4.6. 8. io. 12. 8. 6’

which is the value of in Art. 12. of the preceding paper.

5- If the last literal equation be divided by y, and ~/'b74^'
= ~~Szy he then taken from both sides, we shall have

Q f-=T L-

^ \ X 2JI7 12 J5

+ 777?+ 3

I2J'7

1 6y

32r

+ M-I7+T5

sy

67

1923-

167
IQ5 J1 |'

512 J

5 77

16

is —

T77 + "74> &c-
4 32

■|*7 2+-~rl. See Art. 2.

4 B

MDCCXCVIII,

o<54< I^Tt. Hollins s iiYipyovcd- Solution op

1 3-5 7'9-Sy1 i 3-5‘7'9-ll-7 y5 j 3-5 7 9.1 1.13.9 y7

_L

46.8.10.6.4 '

which equation, in fluxions, gives

+ J J , y.. y

4 6. 8.10.X2. 14.10. 8 *

57

I2J8

+ T^ +

I2JC

5-6

12/

5-3
16 y*

74

1ZJ+

5.2

+ «

3

8r

+ —

1 16 y

+ 83?" +

+

57

3-5 . 3

l

12/ ’

s/5 ' 32/’-

i6yy

5

l6yy .

+

Q_

~ *> (tF +

i6j»j

67

i92yy

5

io5 'i

7l2

r^ =

83,4 l6jyjy

3-5-7-9-5-3 -W , 3.5.7.9.11.7.5^^ , 3.5.7.9.11 -13.977/

4.6.8.10.6.4 >* 4.6.8.10.12.8.6 ‘ 4.6.8. 10.12.14.10.8 9 ^ C‘

And this equation, more concisely expressed, and divided by y.
gives s.

IIJIL + J. 49 5 ] ■ f 9 1 5 )

0ji2y9~ I2y7 48js i6y3) 11 J \ 8y ‘ 1 6y*J

__ JS \5_ 9 | 7_

12^ 8^7 32jy5 ■" 192

q_ _3'57-9-,I7*5 yy1

+y(-

_ 1Q5
512^

+

3.5.7. 9.11.13.9.7^^

&C.

12 y

3-5-7-9-5 -jyy

4.6.8.10.6.4 * 4.6.8.10.12.8.6 1 4.6.8. 10. 12. 14. 10.8

Now the fluent of the fluxionary series on the second side
of the equation being obviously the series 3-5 7 9-5-3 yy

3.5.7.9.11.7.57'

4.6.8.106.4.2

4.6.8.10., z.i .6.4 + 1^8.6- ^ We are neXt to take the

fluent of the expression on the first side, and to correct it, that
it may be = this series ; which may be done as follows :

For the fluent sought, assume

Q
+

y

P

+ + T~ + 77) + “ + IF + *)

5

+ + ~zr + TT + 1 k .y, and take the fluxion

r ■ r ' r ' yy
of this expression, which will be

a Problem in physical Astronomy.

555

6b

y1

4 c

y 5

2d

y%

+

Z£_ Jj’ , 3 C

y7 ' y
+-f

+

+

r

g

+-
1 y

+ -
1

+ “ (~ +7r+^) I

y

+(

-8 P

yj • r

6 <f 4r _ zs j_ t

y7 y5 y + ;

/ g ^

jk5 y 7

Q_

— wy

4/

+

2g

’ j; and these fluxionary terms

being put — to those on the first side of the preceding equa-
tion, there will arise

8 a ~ —

12'

6 b = ja — «£.
/ 12’

from which a =

o. 1 2

b = —95.

U 2.8.12’

^ = 5b+f+fs,

c — ~75

4.8.8 ’

*d=sc + g+-l.

</— “IO>

8.8.8’

4 f=b

•>

<*|3,

II

II

e*

^ 32. ’

II

1

h— i£L
8.8.8’

*

ii

00

/>=-! S_

* 8.12’

II

to

*>

”>'2

II

s>«

4 r = #-/>

r= 0,

s— ~37
4.8.12’

/ = “los 4- /j

5J2 •

0

0

11

4 B 2

Mr. Hellins’s improved Solution of

556

Our assumed fluent, therefore, is

q I -35 __ 95 75 105

^ \ 8.123s iz.l6y6

+ ^ * -

4. 8. 83“ 8.8,833

37

+ u [tT7 + 777:, +

105

\3 2y* 1 32yy r 8.8.8/

8..*/ < 16 y6 - -p-^ *> Which may be COr-

rected in the manner shewn in the two preceding Articles, or
more expeditiously, as follows.

It is pretty evident, from the correction of the fluent in the
preceding Article, that the constant quantities which lie con-
cealed in this fluent, will appear in those terms only, (when
the radical quantity ^/(i — yy), and the logarithm u, is ex-
pressed in series,) in which the index of y is o. Thus, the con-
stant quantities will appear as below.

The 4th term of :

The gd term of ~7S^~/y)

8.8.833

is

35

X

sy*

_ *75 .

8.12

8.1638

4.12.16.16 ’

is

95

y5

_ 95

12.16

X

16 36

12.16.16 5

is

75

y*

_ . 75 .

4.8.8

X

OC

8.16.16 3

is

105

yy

1

1 M

O

v-n

a

8.8.8

X

zyy

4.16.16 9

and the terms in 'which the index of y is o, in the logarithmic
part, viz. [jyy + ~b /> ) ( ~6 + P6 y'2 +

1 27 34

are these two, - - —

4.16. i&34

27
4.i6.i6j

5

8.16 *

The sum of these six fractions is The equation of fluents

therefore is
-35 95

Q

8.123s

75

105

8 123s
35

+

I2.i636

5

16 36

16.163“"

*

8.8.8 33
37

4 -*-izyy

j + u

3

1023

4°96

+

+

105

3233 1 8.8.8

the series

a Problem in physical Astronomy.

55?

3'5'7'9-5-3 yy

3-5 7-9-1 1 -7-5 y

4.6.8.10.6.4.2 4.6.8.J0.12.8.6.4

which, when y = 1, becomes

9 , 5 , io5 \

+

3.5.7.9.11.13.97/

4.6.8.10.12.14.10.8.6

H. L. 2

+ 35

8.12
3-5*7*9*5 3

+ ^

+ “ +

32 ' 32 •

37

8.8.8

1023

4096 J

4067

12288

4^2.H.L. 2

512

4.8.12

, 3.5.7. 9. 11.7, 5 3.5.7.9.11.13.97

" i /i. 6. 8 irt.T? £6 "l

4.6.8.10.6.4.2 *T 4.6.8.10.12.8.6.4 "t“ 4.6.8.10.12.14.10.8,6

, which is

the value of / in Art. 12. of the foregoing paper.

These three examples, I conceive, are sufficient to illustrate
this method of summing the slowly converging numerical series
which arose in the solution of the problem in the preceding
paper. The three series of which the sums are now investi-
gated are, as was before observed, the most difficult to sum of
all that arose in that solution ; so that, whoever understands
what is done here, may, with great ease, compute the sums of
the rest of the series which are found there, and of many others
of this kind, which arise in the solution of problems.

II. Observations , tending to facilitate and abridge the numerical
Computations of hand B in the preceding Paper.

6. The radical factor v/(i — c c)> hi the literal expressions of
the values of A and B, may be taken away, by multiplying the

other factors by its equivalent 1 — — — — fl &Cn in con-

2 o lO

sequence of which, other expressions will be obtained, better
adapted to the purpose of numerical calculation. This will ap-
pear by the following operations.

The product of ^/(i — cc) x the other factor in the ex-
pression of the value of A, in Art. 9. of the preceding paper,
will be

55 $ Mr. Hellins's unproved Solution of

— -f -Xf-^CC-fvC4

~ + A+ fxc C -f v C*

— 1 — i^CC — \uC*
— ±cc — i-AC4

X r4

8 C

where ej, and g, are = A- i, ^ — £ a — i, and v — |a -f ,

respectively; in numbers = 0-1931472, 0*1036802, and
0*0687064, respectively. And this expression, which is evi-
dently more simple than the former, is somewhat nearer than
that to the value of the whole series, as will appear to any one
who shall compute the value of the next coefficient.

7. In like manner, the product of the two factors in the
value of A7, in Art. 13. will be

3

1

+ -fee + X' "1" p' C C + V'C 4
_1L — J— __5£l Mr

16

8.16

TF + 777 + A' + ^ cc +

v'C 4

2 5 I f 1,4

3 CC 6 2 2 ^

3 cc

5 ~ i ^ ~ 8 f— f^ + ii + k+icc+kc4.

24

1 5 4

— — CC ~~ 73 c
12 48

4

96

3

which expression also is more simple than that from which it
is derived, while the accuracy of it is not less, as is pretty evi-
dent on inspection. And, that the numerical values of b, i,
and k , are very easily attainable from the values of a7, ^7, and /,

559

a Problem in physical Astronomy.

given above in Art. 3, 4, and 5, of this paper, is very obvious.
In a subsequent Article, these values will be inserted.

8. And the product of the two factors in the value of B, in
Art. 16, may also be exchanged for a more convenient expres-
sion, by a like process.

p-{- crCC -j~ 7 C*

1 —

c c c 4

_ — &c
8 ’ '

p -j- (TCC -f- T C* 1

— ipcc — \<rc* > = p-\-lcc-\- me*, &c.

J

which expression also is more accurate than that from which
it is derived, as well as more simple. The numerical values of
/ and m, which are evidently given from those of P, 7, and r,
will be inserted a little further on, when we come to an example
of calculating the values of A and B in numbers.

9‘ The numerical calculation of the other member also, in
which « enters, may be facilitated and abridged, by the follow-
ing considerations.

If c be put for the sine of an angle, radius being t, then will
1 + s/[ i — cc) be the versed-sine of the supplement of that angle,

and I +v/(i _cc) wil1 be = tbe tangent of half that angle; from
which it follows, that the reciprocal of this quantity viz.

1 -+• vX ( 1 — c c ) . .

c ~ ’ ls ^ the co-tangent of half the angle of which the
sine is c. The common logarithm 0f l±^Ii=££l may therefore
be taken out fromTAYLOR's excellent Tables,* and quickly con-
verted into an hyperbolic logarithm, by Table XXXVII. of
Dodson's Calculator.

* These valuable Tables are computed to every second of the quadrant.

tj6o Mr. Hellins’s improved Solution of

10. An expression of this kind, $ %™cc9 w^ien c onfy va“

riable quantity, consisting of several figures, and r and s are
likewise long numbers, will be much better adapted to the use
of logarithms, when put in this form, — x r~f-c- ; because the

multiplications of r and s into cc, or additions of their loga-
rithms and taking out two numbers, are by this means ex-
changed for the addition of the constant logarithm of A : the
quotients ~ and once found, being constant numbers. Thus,
the numerical value of even 2 where r and s are single

o — j C C

figures, is more easily obtained by ~ x than by the

former expression.

11. But it will appear upon trial, that the arithmetical value
of any three terms p'^- qcc -{-r'c4, in which p', q', and r, are
constant quantities, and cc consists of five or six places of
figures, may, in general, be more easily obtained by logarithms

than the arithmetical value of — x ^ \rj~ ! ;• And, since the dif-

S (p . S 1 c c

ference of the values of these two expressions is inconsiderable
in the present case, I shall make no further use of the frac-
tional expression ; but observe, that the logarithm of q'c c, in
the other expression, being found, the logarithm of re* will be
had, by adding to it the logarithm of ^cc; for qcc x^cc — r'c\
And, since the logarithms of the numbers which stand in the

places of q' and j may be taken out and reserved for use, and
the logarithms of c c and once found, will serve for all the terms
in which these quantities occur, it will appear by an example,
that neither many logarithms, nor many numbers correspond-

a Problem in physical Astronomy. 56 1

mg to logarithms, need be taken out of other tables, in com-
puting the value of A or B.

12. It will now be proper, since the literal expressions of the
Values of A and B have been exchanged for others which are
more convenient, to bring the new equations together in one
view, and, after that, to give an example of the numerical cal-
culations by them.

It appears, by Art. 9, 10, 13, 16, and 17 of the preceding
paper, and 6, 7, and 8 of this, that

77 +e +fcc+gc '

4 cc "l-- 1- cc CC a.7.

1. A =

H (tf-J-6) z

X

a. A' =

w («-{- b) z

X <

4

3«4

+ Tc + b + icc + kdk

+

3-5

x oc -T

4- 1 4. 12

a

CC- f-

3±1L.

4.12.32

p — j- l c c — j- m c *

4. B'= y [A!a — A).

In which equations, the values of the coefficients are as follows t
^ = 0-1931472, £ = 0-0823604, P =1*3862544,

/= 0-1036802, i = 0-0551502, 7=0-3465736,

g= 0-0687064, £ = 0-0408309, m = 0-1793226.

3. B

2 a

~b~

A-

7T b (zz-f b)l

X

13. The constant numbers which will be wanted, in com*
puting the arithmetical values of A and B, are those denoted
by e, b, and p, which are given in the preceding Article ; and
the constant logarithms are the following, which are respec-
tively set down to as many places of figures as are requisite.

4C

MDCCXCVIII.

562 Mr. Hellins’s improved Solution of

L. 2 = 0*3010,300,
L- t = ~'574°>3
L-i =1796,

L. /= 1-0157,0
L. 1-821

L. y — 1-8239,087,
L.-i= 1-8750,0,

L.^ =“-6197,9,

L. i = 27415,5
L. 4- = 1-869

L. 7T = 0-4971,499*.

L-p = T8i7>
L-j= 1-6989,7,

L-^=i'75°

L. I =“.5398,0

Lm —

r7!4

14. An example, to illustrate the method of computing by
these theorems, may now be proper.

Let it be required to compute A and B by the 1st and 3d
theorems, in Art. 12. when the two planets are Venus and the
Earth.

This arithmetical work may stand as follows, in three co-
lumns, the logarithms being in the middle, and the numbers
corresponding to them on the two sides ; where a distinction
is made, which is too obvious to need any description. By this
arrangement, a frequent repetition of words, number, and lo-
garithm, will be avoided.

*

* All these constant logarithms are to be written on a slip of paper, for the sake of
expedition in the use of them.

a Problem in physical Astronomy . 563

First, for the value of A.

Numbers. Logarithms. Numbers.

Here a — 1-5236,71 1* 0-1828,913

and b = 1-4451,60 / Ar. c 1-8400,841

d — 6 = 0-0785,11
a + ^ =5 2.9688,31

a ^ b

k 4- b

2-8949,305
°’4725.855
sT-4223, 450

2

stc

1-8786,850 -

75-62841 = -~
0'i9315 = e
0-00274 =fcc

fee

cc .

g

■3-4380,4

2-244

Sum of these two logs.

5-682 “

0-00005 ^ gc*

The sum of these four terms is

75'82435

Having now found the value of the four terms ^ + e -f fee
+gc*, we must next find the value of the three logarithmic

n 5 C

terms a + — «cc + ac\ which may quickly be done as
follows :

Half the logarithm of cc is 1*211 i,725 = the sine of 90 21'
32"-28; and half this angle is 40 40' 46"-i4, the logarithmic co-
tangent of which is 1-08 69,576; and this common logarithm

* I was favoured with these numbers by Dr. Maskeuyne.

564 Mr. Hell i ns5s improved Solution of

reduced to an hyperbolic logarithm, by Table XXXVII. of
Dodson's Calculator, gives - r 2*50281 =«

* - Q,398 43

icc - 3*99638

Sum of these two logs. 2*39481 - - 0*02482 = -| acc

| -cc - - 2 "2 1 8

4‘bi3 - - 0-00041 ==-^-oiC4;

The sum of these three terms is - 2*52804 ; to which,

add the sum of the four terms above found, 75*82435,andwehave

1*8940,523 - 78-35239 = all the terms;.

<7 t(u + b)i 1-2060,283

The difference of thesel g88 . g A_

two logarithms is J * * 1 000

The value of A being now found, the computation of the
value of B will be very easy, since, of the six terms wanted,
two are already computed, and the logarithms of all the rest
are at hand. This operation may stand as follows, the loga-
rithms being still in the middle.

5%

a Problem in physical Astronomy.

1-3863°

39621

0-00916 = lee

-cc 2-136

Sum of these two logs. 4-098

0*00013 = me' *

5-00562 = 2«

*5*97.5

0-03309= \*CC

T6cc 2'17®

Sum of these two logs. 4/692

0-00049 = « c4

0-8085,343

6-43479 = {

7 *(a + b)$ 0-7334,427

DifF. of these two logs. 0-0750,916

1*18875 = A

A - 0.6880,240
a - 0.1828,913

Sum of these two logs. 0.8709,153 -

7-42874 = A a

07951,840 -

6-23999 = A# — A

T 01411,141

J

T (Aa — A) 0-9362,981

8-63571 = B.

We have now the values sought, viz. A = 4 8756, and
B = 8-6357 ; and from these may the values of C, D, E, &c. be

easily found, by the equations given in Art. 20. of the preceding
paper. I have set them down here only to five places of
figures, it being evident, from the value of a — b , that the result

cannot be depended on to more places than five, which, how-
ever, are very sufficient for the purpose.

566 Mr. Hellins’s improved Solution , &c.

I have taken notice above, in the computation of B, that, of
six terms wanted, two were ready, one of them being a con-
stant quantity, and the other computed in the preceding ope-
ration ; and it may be remarked, that, if the two terms in which
c* enters had been omitted, the difference in the result would
not have been so much as 2 in the fourth place of decimals,
which is inconsiderable. Therefore, of the six terms in this
expression, two are given, and two more only need be com-
puted in this case.

And it may be further remarked, that the term e is always
given in the expression of the value of A, and that the two
terms in which c4 enters may be omitted, as that will occasion
a difference in the result, of only 3 in the fifth place of de-
cimals, which is quite inconsiderable. Of the seven terms,
therefore, in Theorem 1. Art. 12. one is given, and four more
only need be computed, when Venus and the Earth are the two
planets of which the perturbations are to be computed.

15. My avocations calling me off from these delightful spe-
culations, I must now put an end to this paper, without men-
tioning some other observations which I have made on this
subject.

May 6, ijgj.

\

+

C 567 3

XXIII. Account of a Substance found in a Clay -pit ; and of
the Effect of the Mere of Diss, upon various Substances im-
mersed in it. By Mr. Benjamin Wiseman, of Diss, in Nor-
folk. Communicated by John Frere, Esq. F.R.S. With an
Analysts of the W iter of the said Mere. By Charles
Hatchett, Esq. F. R. S. In a Letter to the Right Hon, Sir
Joseph Banks, Bart. K. B. P , R. S.

Read April 19, 1798.

The substance I have inclosed was found near Diss, in a body
of clay, from five to eight feet below the surface of the soil.
All the pieces I observed laid nearly in a horizontal direction ;
and varied in size, from two or three ounces, to as many pounds.
The colour of the substance, when taken fresh from the clay'-pit,
was like that of chocolate ; it cuts easily, and has the striated
appearance of rotten wqt>d. The pieces were of no particular
form ; in general, they were broad and flat, but I do not re-
collect to have met with a piece that was more than two
inches in thickness : it breaks into laminae, between which are
the remains of various kinds of shells. The specific gravity of
this substance, dried in the shade, is 1 .588 ; it burns freely,
giving out a great quantity of smoke, with a strong sul-
phureous smell.

By a chemical analysis, which I cannot consider as very
accurate, one hundred grains appear to contain.

568 Mr. Wiseman’s Account of the Effect of

Of inflammable matter, including the small quantity

of water contained in the substance

*

Of mild calcareous earth -

Of iron -

Of earth, that appears to be silex - -

100

On the Effect of the Mere of Diss, upon various Substances.

Observing, several years ago, that flint stones taken out of
the Mere of Diss were incrusted with a metallic stain, I was
induced to make some experiments, in order to discover the
nature or composition of this metallic substance.

Nitrous acid readily removes it, dissolving a part, and leav-
ing a yellowish powder, which, washed and filtered, was found
to be sulphur. Vegetable fixed alkali precipitated from the ni-
trous acid a ferruginous coloured powder, which was iron.

With a view to determine what length of time was neces-
sary for the formation of this metallic stain upon flint stones,
or other substances, I inclosed in a brass wire net the follow-
ing articles : flint stones, calcareous spar, common writing
slate, a piece of common white stone ware, and likewise a
piece of black Wedgwood-pottery. After remaining in the
water from the summer of 17 92 to August, 1795, the flints and
Wedgwood-ware had acquired the metallic stain in a slight de-
gree, and the slate had assumed a rust colour ; the other sub-
stances appeared not to be at all altered. I was greatly surprised
to find the copper wire that held the net, surrounded with a
metallic coating of a considerable thickness ; it was of a deep
lead colour, and of a granulated texture. When taken from the
wire, and ground in a mortar, it had a black appearance, inter-

grains,

41-3

20.0

2.0

36.7

the Mere of Diss, upon various Substances.

spersed with very hard shining particles. The wire was evi-
dently eroded, and this substance deposited in the place of the
copper that was decomposed, somewhat similar to the decom-
position of iron in cupreous waters.

By repeated chemical analysis of this substance, one hundred
grains contain, of copper, 70; of sulphur, 16.6; of iron 13.3
grains.

I have never met with an account of the decomposition of
copper, in waters impregnated with iron, in any chemical
work; and, as iron appears to have a greater affinity to the
vitriolic acid than copper has, (as is constantly evinced in the
neighbourhood of copper mines,) it appears an anomaly in
chemistry, that I am not adept enough in the science to ac-
count for.

[The President and Council, thinking the effects of the
water of Diss Mere deserving of further inquiry, desired
Mr. Wiseman would send some of the said water, for
the purpose of examination. Mr. Wiseman accordingly
sent a quantity of the water, accompanied by the other
substances described in the following letter to the Pre-
sident ^

MDCCXCVIII.

& 7o Mr. Wiseman’s Account of the Effect of

SIR, Diss, May zg, 1798.

As the Society have expressed a wish, through Mr. Frerej
to have some of the water in which the copper wire was depo-
sited, which Mr. Frere, at my request, laid before the Society,
I have sent two gallons of the water of Diss Mere, (No. 1.)
with a small quantity of copper cuttings, (No. 2.) which laid
in the same water, a few feet from the side, and six feet in
depth, from the 7th of February, 1797, to the 20th of the pre-
sent month, May, 1798. The pieces of copper, when laid in,
weighed 3051 grains; when they were taken out, and washed
from the mud that lightly adhered to them, preserving and
weighing the scaly matter that came off, they weighed 2944.
grains, indicating a loss of 107 grains. Examining the pieces
of copper, the same evening they were taken out of the water,
I observed a number of small crystals formed upon some of
them, in the form of pyramids joined at their bases; these
crystals lost their shining appearance, by the evaporation of the
water of crystallization, in the warmth of the succeeding day.
Whether they will be preserved in a journey of nearly 100 miles,

* jk.

is perhaps doubtful. No. 3. contains two pieces of copper, on
which the crystals were most abundant. No. 4. contains a
small quantity of the substance formed upon the copper, that
came off’ in washing and in weighing it.

The town of Diss is principally situated on the NNE and E
sides of this piece of water. The land runs pretty steep on the
W and N of it, to the height of 40 or 50 feet : on the SE,
the ground comes within a few feet of the level of it. The soil
of the upper part of the town is a stiff blue clay ; that of the
lower part, to the SE, a black sand, beneath which it is a moor.

the Mere of Diss, upon various Substances. 571

The water in the higher parts of the town is good ; in the lower
parts, it is a chalybeate, of which a specimen is sent, (No. 5.)

No. 6. contains a quantity of flint stones, taken from the SE
side of the Mere, where the water is shallow ; many of which
are strongly marked with the metallic stain, which they acquire
by lying in this water a few years.

The Mere contains about eight acres, and is of various depths,
to twenty-four feet: from its situation with respect to the
town, it may naturally be supposed to contain a vast quantity
of mud, as it has leceived the silt of the streets for ages. In
summer, the water turns green ; and the vegetable matter that
swims on its surface, when exposed to the rays of the sun, af-
fords vast quantities of oxygen gas. I cannot help considering
this process as having a considerable agency in the corrosion,
and in the formation of the metallic crust upon the copper
deposited in this water. Some of this vegetable matter will be
found in the water sent to the Society.

I intend to make some further experiments with different
metallic substances, at different parts, and at various depths ;
but, as the process is slow, if in the mean time you, Sir, or any
of the members of the Society, will have the goodness to point
out any experiment you or they may wish to have made, I
shall be very glad to contribute all in my power towards the
illustration of the subject.

I have the honour to be, &c.

BENJ. WISEMAN.

The Right Hon. Sir Joseph Banks, Bart.

K. B. P. R.S.

4 D 3

,57 s Mr. Hatchett's Analysis of

/ . • *• _ X * ■} 1* * ' * ' . t ■ t _ . - K\ t - - f 2 ' • 1

£The water, and other substances described in the foregoing
letter, were delivered to Mr. Hatchett, who had been
previously requested, by the President and Council, to exa-
mine them. The result of his examination is related in
the following letter to the President. ^

Analysis of the Water of the Mere of Diss. By Charles

Hatchett, Esq.

DEAR SIR, Hammersmith, Sept. 14th, 1798.

In consequence of the request which you and the Council of '
the Royal Society have done me the honour to make, that I
would examine the water of Diss Mere, and the other substances
sent by Mr. Wiseman, I now hasten to acquaint you with the
result of my experiments.

The substances sent by Mr. Wiseman are as follows :

/ Some copper wire, with a blackish grey incrustation.

Water from Diss Mere, (marked No. i.)

Copper cuttings, covered with a blackish crust, similar to
that on the copper wire, (marked No. 2.)

Some cuttings similar to those abovementioned, (marked
No. 3.)

The paper. No. 4. contained some of the black crust, de-
tached from the cuttings.

No. 5. A quart bottle, containing some water from the
lower part of the town of Diss, and called, by Mr. W iseman,
a chalybeate water.

No. 6. Some flints, taken from the SE side of the Mere,
where the water is shallow, and having (as Mr. Wiseman
terms it) a metallic stain.

the Water of the Mere of Diss. 573

My first experiments were made on the incrustation of the
copper wire, mentioned in Mr. Wiseman's first letter.

This incrustation was easily detached from the wire, and,
being reduced to powder, was digested with nitro-muriatic acid,
in a gentle heat : a green solution was formed, and there re-
mained a residuum, of a pale yellow, which proved to be sulphur.

The solution being diluted with two parts of distilled water,
was supersaturated with pure ammoniac, by which, a few brown
flocculi of iron were precipitated. The supernatant liquor was
blue ; and, being evaporated, and redissolved by sulphuric acid,
the whole was precipitated by a plate of polished iron, in the
state of metallic copper. The component parts of this coating
were therefore copper, and a very small portion of iron com-
bined with sulphur.

I could not extend these experiments, as the whole quantity
of the coating that I was able to collect, amounted only to
three grains and an half.*

The next experiments were made on the black crust of
No. 2, 3, and 4.

This I found to be exactly the same as that formed on the
copper wire; viz. it consisted of copper combined with sulphur,
and a very small portion of iron.

* The copper wire, when the coating was removed, was perfectly flexible, and the
surface did not appear unequal or corroded : this is commonly the case under such cir-
cumstances ; for, when sulphur has combined superficially with a metal, the compound
is observed to separate easily, so as to leave the metal underneath not injured in quality,
and very lhtie, if at all, affected in appearance. Those who diminish silver coin, make
use of the following method.

They expose the coin to the fumes of burning sulphur, by which a black crust of
sulphurated silver is soon formed, which, by a slight but quick blow, comes off like a
scale, leaving the coin so little affected, that the operation may sometimes be repeated
twice or thrice, without much hazard of detection, if the coin has a bold impression.

574

Mr. Hatchett’s Analysis of

I next examined the water of Diss Mere, (No. x.) and I was
at length led on, step by step, to make a regular analysis of
the fixed ingredients.

Before I made the analysis, I examined this water with cer-
tain re-agents, and remarked the following properties.

1. The water of Diss Mere has a yellowish tinge, and the
flavour is rather saline ; but it has not any perceptible odour.

2. Prussiate of potash did not produce any effect.

3. Acetite of lead produced a slight white precipitate.

4. Nitrate of silver formed one, very copious.

5. Tincture of galls had not any effect.

6. Muriate of barytes caused a slight precipitate.

7. Ammoniac, potash, and oxalic acid, severally produced
precipitates, when added to different portions of this water.

ANALYSIS.

A. Three hundred cubic inches of the water, by a gentler
evaporation, left a pale brown scaly substance, which weighed
58 grains.

B. These 58 grains were digested in alcohol, without heat,
during 24 hours, and afforded a solution, which, by evapo-
ration, yielded muriate of lime, slightly tinged by marshy ex-
tract, 18 grains.

C. Six ounces of distilled water were then poured on the
residuum, and, with repeated stirring, remained during 24
hours. By evaporation, this afforded muriate of soda, with a
very small portion of sulphate of soda; in all, 10 grains.

D. What remained was boiled in 800 parts of distilled

water, and the solution, being evaporated, left of selenite
1.70 gr. 1 v

575

the Water of the Mere of Diss.

E. The undissolved portion now weighed 25 grains, and was
digested with diluted muriatic acid : a great part was dissolved,
with much effervescence, and, being filtrated, afforded, by am-
moniac, of alumine 1.50 gr. From this, I afterwards sepa-
rated a very minute quantity of iron, by means of prussiate of
potash.

F. Carbonate of soda was then added to the liquor, and
precipitated carbonate of lime 21 grains,

G. The insoluble residuum weighed 3.50 gr. ; and proved
to be principally carbon, (produced by decomposed vegetable
matter,) with a very small quantity of siliceous earth.

The result of this analysis was, therefore,

B. Muriate of lime - - - 18

C. Muriate of soda, with a very small portion of

sulphate of soda - - - 10

D. Selenite - - - - 1 70

E. Alumine, with a portion of iron too small to

be estimated - - *

F. Carbonate of lime - - - 21

G. Carbon, with a little siliceous earth - 3 50

£5 7o
Loss 2 30

58 o

It is worthy of notice, that the iron present was in so very
small a quantity as not to be detected by any test, till it had
been separated in conjunction with the alumine.

The water No. 5, from Mr. Wiseman's account, does not
appear to have been concerned in producing the effects which
he has observed, and the quantity was too small to be sub-
jected to a regular analysis, I noted, however, what follows.

57 6 Mr. Hatchett’s Analysis of

1. It has a very strong hepatic flavour and smelL

2 . A plate of polished silver, put into it, became black in a
few hours.

3. It became faintly bluish with prussiate of potash, aftef
standing five or six hours.

4. Tincture of galls produced a faint purple cloud.

5. Solution of acetite of lead afforded a brown precipitate.

6. Nitrate of silver produced the same.

7. Potash, and ammoniac, caused a precipitate ; but that of
the former was the most copious.

8. Oxalic acid produced a precipitate.

9. Muriate of barytes had also a slight effect.

The water No. 5. cannot, therefore, be considered as a cha-
lybeate, (the quantity of iron contained in it being scarcely per-
ceptible;) but it appears to be a water containing some hepatic
gas, together with substances similar to those contained in No. 1 .

From the above experiments it is evident, that the water
No. 1 . does not contain any of the component parts of the crust
formed on the copper wire and cuttings, although it is certain
that the incrustation took place during the immersion of those
bodies ; but, before I mention my ideas on this subject, I shall
give an account of some experiments made on the flints, No. 6.
These were coated with a yellowish shining substance, which
appeared to me to be pyrites ; and, as the flints could not have
contributed any metallic substance to form this coating, I was
enabled by their means to ascertain, whether the copper of the
crust, formed on the wire and cuttings, had been furnished by
the pieces of copper, or by any thing in the vicinity of the
water.

1. I poured nitro-muriatic acid on some of the flints, in a
matrass, so as completely to cover them.

577

the Water of the Mere of Diss ,

The coating was rapidly dissolved, with much effervescence ;
and, when the flints appeared perfectly uncoated, and in their
usual state, I decanted the liquor.

2. A yellow matter subsided, which proved to be sulphur.

3. Prussiate of potash produced Prussian blue ; and the
remaining part of the solution, being supersaturated with am-
moniac, afforded an ochraceous precipitate of iron.

The supernatant liquor did not become blue, as when cop-
per is present, nor was the smallest trace of it afforded by
evaporation.

Martial pyrites is, therefore, the only substance deposited on
bodies immersed in the water of Diss Mere ; and the copper of
the crust, formed on the wire and cuttings, was furnished by
those bodies.

It is proved by the analysis, that the water of Diss Mere does
not hold in solution any sulphur, and scarcely any iron ; it has
not, therefore, been concerned in forming the pyrites ; but it
appears to me, that the pyritical matter is formed in the mud
and filth of the Mere ; for Mr. Wiseman says in his letter, that
“ the Mere has received the silt of the streets for ages.” Now
it is a well known fact, that sulphur is continually formed, or
rather liberated, from putrefying animal and vegetable matter,
in common sewers, public ditches, houses of office, &c. &c. ;
and this most probably has been the case at Diss. Moreover,
if sulphur, thus formed, should meet with silver, copper, or
iron, it will combine with them, unless the latter should be pre-
viously oxidated.

The sulphur has therefore, in the present case, met with
iron, in, or approaching, the metallic state, and has formed py-
rites, which (whilst in a minutely divided state, or progres-

MD-CCXC VIII. 4 E

578 Mr. Hatchett's Analysis of

sively during formation,) has been deposited on bodies, such as
the flints, when in contact with the mud.

But an excess of sulphur appears to be present ; for, when
copper is put into the Mere, the sulphur readily combines with
it ; and, at the same time, a small portion of iron appears to
unite with the compound of copper and sulphur, possibly by
the mere mechanical act of precipitation.

The incrustation on the copper wire and cuttings is, in every
property, similar to that rare species of copper ore, called by the
Germans Kupfer scbwartze, ( Cuprum ochraceum nigrum; ) and
I consider it as absolutely the same. In respect to the martial
pyrites on the flints, there can be no hesitation; and, as in these
two instances, there were evident proofs of the recent formation
of ores in the humid way, I was desirous to ascertain the effect
on silver. I therefore wrote to Mr. Wiseman, to request that
he would take the trouble to make the experiment; and re-
ceived from him the following answer, accompanied by the
specimens.

“ SIR, Diss, 8th September, 1798.

“ Immediately upon the receipt of your letter, (27th July,) I
laid some silver plate, and silver wire, into the Mere ; the whole
weighed 235.6 gr. I took it out on Thursday last, (Sept. 6th)
and, after cleaning it carefully from mud and weeds, I find it
weighs 242.8 gr. ; an increase of 7.2 gr.

« The silver plate you will find much tarnished, in some parts
almost black ; the wire is in many places fairly incrusted,
which crust, upon the pressure of the fingers, comes off in thin
scales. The whole appearance of the silver strongly indicates

the W %ter of the Mere of Diss.

the presence of sulphur, which I have no doubt abounds in
every part of the Mere.

“ The peculiar smell of the mud gives me reason to sup-
pose, that a great deal of hepatic air is produced ; which, pro-
bably, uniting with the iron held in solution in the water of

the Mere, may account for the martial pyrites found on the
flints.

“ By what affinity the copper wire, laid in this water, is at-
tacked, I am not chemist enough to determine.

“ I have begun a set of experiments, with the view of pro-
ducing the same effects upon copper wire by artificial means ;
but whether I shall succeed, I am not able at present to say.

“ I am, &c.

“ BENJ. WISEMAN/*

P. S. By experiments I have lately made, I find hepatic gas
precipitates carbonate of iron in the form of a black flocculent
matter; 71 parts of which are iron, and 29 sulphur.

The silver plate I found (as Mr. Wiseman has mentioned)
much tarnished, and in many places almost black, but I could
not detach any part of it. I succeeded better with the wire, and
collected a small portion of a black scaly substance, which, as
far as the smallness of the quantity would allow it to be ascer-
tained, was sulphuret of silver ; and was similar, in every re-
spect, to the sulphurated or vitreous ore of silver, called bv the
Germans Glasertz.

4, E 2

I

580 Mr. Hatchett’s Analysis of

This effect on the silver was to be expected ; and I recollect
to have read, not many months ago, in one of the foreign jour-
nals, that Mr. Proust had examined an incrustation, of a dark
grey colour, formed in the course of a very long time, on some
silver images, in a church at (I believe) Seville. This incrus-
tation he found to be a compound of silver with sulphur, or,
in other words, vitreous silver ore.

The same principle is the cause of the tarnish which silver
plate contracts with so much ease, particularly in great cities ;
for this tarnish is principally a commencement of minerali-
zation on the surface, produced by the sulphureous and he-
patic vapours dispersed throughout the atmosphere, in such
places.

To Mr. Wiseman’s observations we are much indebted, as
they make known the recent and daily formation of martial
pyrites, and other ores, under certain circumstances. It is not
to be supposed that such effects are local, or peculiar to Diss
Mere ; on the contrary, there is reason to believe that similar
effects, on a larger scale, have been, and are now, daily pro-
duced in many places.

The pyrites in coal mines have, probably, in great measure
thus originated.

The pyritical wood also may thus have been produced ; and,
by the subsequent, loss of sulphur, and oxidation of the iron,
this pyritical wood appears to have formed the wood-like iron
ore which is found in many parts, and particularly in the mines
on the river Jenisei, in Siberia.

In short, when the extensive influence of pyrites in the mi-
neral kingdom, caused by the numerous modifications of it, in

the Water of the Mere of Diss. g8i

the way of composition and decomposition, is considered, every
thing which reflects light on its formation becomes interesting ;
and I cannot hut regard as such, the effects which Mr. Wise-
man has observed in the Mere of Diss.

With great respect, I remain, &c.

CHARLES HATCHETT.

The Right Hon. Sir Joseph Banks, Bart.

K. B. P. R. S. &c.

t 582 3

XXIV. A Catalogue of Sanscrita Manuscripts presented to the
Royal Society by Sir William and Lady Jones. By Charles
Wilkins, Esq. F. R. S.

Read June 28, 1798.

1. a. IVIaha'-bha'rata.*

A poem in eighteen books, exclusive of the part called Raghu-
vansa; the whole attributed to Crishna Hwaipdyana Vyasa ; with
copious notes by Nila-canta. This stupendous work, when per-
fect, contains upwards of one hundred thousand metrical verses.
The main subject is the history of the race of Bharata, one of
the ancient kings of India, from whom that country is said to
have derived the name of Bharata-varsha ; and more particu-
larly that of two of its collateral branches, distinguished by the
patronymics, the Cauravas and the Pauravas, (so denomi-
nated from two of their ancestors, Curu and Puru,) and of
their bloody contentions for the sovereignty of Bharata-varsha ,
the only general name by which the aborigines know the coun-
try we call India, and the Arabs and Persians Hind and Hin-
dostan. But, besides the main story, a great variety of other
subjects is treated of, by way of introduction and episode. The
part entitled Raghu-vansa, contains a distinct history of the
race of Crishna. The Mahd-bhdrata is so very popular through-
out the East, that it has been translated into most of its nu-
merous dialects ; and there is an abridgment of it in the Per-

* The Sanscrita words are spelt according to the method practised by Sir Wil-
liam Jones, in his works.

583

Mr. Wilkins's Catalogue , &c.

sian language, several copies of which are to be found in our
public libraries. The Gita , which has appeared in an English
dress, forms part of this work; but, as it contains doctrines
thought too sublime for the vulgar, it is often left out of the
text, as happens to be the case in this copy. Its place is in
the 6th book, called Bhishma-parva. This copy is written in
the character which, by way of pre-eminence, is called Dev a -
nagari. Ly J.

1. b. Ditto.

Another copy, without notes, written in the character pecu-
liar to the province of Bengal , in which the Brahmans of that
country are wont to transcribe all their Sanscrita books. Most
of the alphabets of India, though they differ very much in the
shape of their letters, agree in their number and powers, and
are capable of expressing the Sanscrita , as well as their own

particular language. This copy contains the Gita, in its proper
place. L>' J.

2. a. Rdmdyana.

The adventures of Rama, a poem in seven books, with notes,
in the Devanagari cnaracter. 1 here are several works with the
same title, but this, written by Vdlmici, is the most esteemed.
The subject of all the Ramayanas is the same : the popular
Story of Rama , surnamed Ddsarathi, supposed to be an incar-
nation of the god Vishnu , and his wonderful exploits to recover
hio beloved Sit a out of the hands of Ravana , the gigantic
tyrant of Lanca. Ly J.

2. b. Ditto.

Another copy, in the Bengal character, without notes, by
Vdlmici . Ly J.

Mr . Wilkins’s Catalogue

ss^

2. c. Ditto.

A very fine copy, in the Devanagari character, without
notes ; but unfortunately not finished, the writer having been
reduced to a state of insanity, by habitual intoxication. S. W. J.

3. a. Sri Bhagavata.

A poem in twelve books, attributed to Crishna Dwaipayana
Vyasa, the reputed author of the Mahd-bhdrata, and many other
works ; with notes by Sridbara Swami. Devanagari character.
It is to be found in most of the vulgar dialects of India, and in the
Persian language. It has also appeared, in a very imperfect
and abridged form, in French, under the title of Bagavadam ,
translated from the Tamul version. The chief subject of the
Bhagavata is the life of Crisbna ; but, being one of that spe-
cies of composition which is called Purana , it necessarily com-
prises five subjects, including that which may be considered the
chief. The Brahmans, in their books, define a Purana to be
« a poem treating of five subjects : primary creation, or creation
“ of matter in the abstract ; secondary creation, or the produc-
“ tion of the subordinate beings , both spiritual and material ;
** chronological account of their grand periods of time, called
({ Manwantaras ; genealogical rise of families, particularly of
“ those who have reigned in India ; and, lastly, a history of the
“ lives of particular families There are many copies of this
work in England. Ly J.

3. b. Ditto.

Another copy, in the Bengal character, without notes.

Ly J.

3. c. Ditto.

Another copy, on palm leaves, in the Bengal character.

S. W. J.

of Sanscrita Manuscripts. 585

4. Agni Parana.

This work, feigned to have been delivered by Agni, the god
of fire, contains a variety of subjects, and seems to have been
intended as an epitome of Hindu learning. The poem opens
with a short account of the several incarnations of Vishnu ;
particularly in the persons of Rama , whose exploits are the
theme of the Ramayana, and of Crishna , the material offspring
of Vasudeva. Then follows a history of the creation ; a tedious
dissertation on the worship of the gods, with a description of
their images, and directions for constructing and setting them
up; a concise description of the earth, and of those places
which are esteemed holy, with the forms of worship to be ob-
served at them ; a treatise on astronomy, or rather astrology ;
a variety of incantations, charms, and spells, for every occa-
sion ; computation of the periods called Manwantaras ; a de-
scription of the several religious modes of life, called A'srama ,
and the duties to be performed in each of them respectively;
rules for doing penance; feasts and fasts to be observed
throughout the year; rules for bestowing charity; a disser-
tation on the great advantages to be derived from the mystic
character OM ! with a hymn to Vasishta. The next subject
relates to the office and duties of princes ; under which head
are given rules for knowing the qualities of men and women ;
for choosing arms and ensigns of royalty; for the choice of
precious stones ; which are followed by a treatise on the art of
war, the greatest part of which is wanting in this copy. The
next head treats of worldly transactions between man and
man, in buying and selling, borrowing and lending, giving
and receiving, &c, &c. and the laws respecting them. Then
mdccxcviii. 4 F

$80

Mr. Wilkins's Catalogue

follow certain ordinances, according to the Veda , respecting
means of security from misfortunes, &c, and for the worship of
the gods. Lists of the two races of kings, called the Suryavansa ,
and the Chandravansa ; of the family of Tadu, and of Crishna;
with a short history of the twelve years war, described in the
Mahd-bharata. A treatise on the art of healing, as applicable
to man and beast, with rules for the management of elephants,
horses, and cows ; charms and spells for curing various disor-
ders ; and the mode of worshipping certain divinities. On the
letters of the Sanscrita alphabet ; on the ornaments of speech,
as applicable to prose, verse, and the drama; on the mystic
signification of the single letters of the Sanscrita alphabet ; a
grammar of the Sanscrita language, and a short vocabulary.
The work is divided into 353 short chapters, and is written in
the Bengal character. Ly. J.

5. Cdlica Parana.

A mythological history of the goddess Cali, in verse, and her
adventures under various names and characters ; a very curious
and entertaining work, including, by way of episode, several
beautiful allegories, particularly one founded upon the motions
of the moon. There seems to be something wanting at the
end. Bengal character, without notes. Ly. J.

6. a. Vdyu Parana.

This work, attributed to Vdyu the god of wind, contains,,
among a variety of other curious subjects, a very circumstan-
tial detail of the creation of all things celestial and terrestrial,
with the genealogy of the first inhabitants; a chronological
account of the grand periods called Manwantaras , Calpas , &c. ;
a description of the earth, as divided into Dwipas , Varsbas , &c.,.

of Sanscrita Manuscripts. 587

with its dimensions in Tojanas ; and also of the other planets,
and fixed stars, and their relative distances, circumferences of
orbits, &c. &c. Written in the Devanagari character. Ly. J.

6 . b. Ditto.

A duplicate in the Devanagari character. I/. J.

7- Vrihan Naradiya Parana.

This poem, feigned to have been delivered to Sanatcumara ,
by the inspired Narada , like others of the Pur anas , opens with
chaos and creation ; but it treats principally of the unity pi
God, under the title of Maha Vishnu ; arguing, that all other
gods are but emblems of his works, and the goddesses, of his
powers; and that the worshipping of either of the triad, creator,
preserver, or destroyer, is, in effect, the worshipping of him.
The book concludes with rules for the several tribes, in their
spiritual and temporal conduct through life. It is a new copy,
in the Bengal character, and, for a new copy, remarkably
correct. Ly. J.

8. Naradiya Purana.

This poem treats principally on the worship of Vishnu , as
practised by Rukmangada, one of their ancient kings. Deva-
ndgari character. S. W. J.

9* a. Bhavishyottara Purana.

The second and only remaining part. The subject is con-
fined to religious ceremonies. Devanagari character. S. W. J.

9. b. Ditto.

With an Index. Devanagari character. I/. J.

10. Gita-govinda.

A beautiful and very popular poem, by Jayadeva , upon
Crishna, and his youthful adventures. Bengal character. Ly. J.

M. a. Cumara Sambhava .

4. F 2

it

$88

Mr. Wilkins's Catalogue

An epic poem on the birth of Cartica, with notes, by Cali-
ddsa. Devandgari character. The notes are separate. Ly. J.

11. b. Ditto.

A duplicate of the text only, in the Bengal character.

Ly. J.

12. Naishadha.

The adventures of Nala; a poem, with notes. Bengal cha-
racter. Ly. J.

13. Bhatti .

A popular heroic poem, in the Bengal character. Ly. J.

14. Raghu-vansa.

The race of Crisbna, a poem by Calidas , with notes. Deva-
ndgari character. Ly. J.

15. Vrihatcatha.

Tales in verse, by Somadeva. Devandgari character. Ly. J.

16. Singh as an a.

The throne of Raja Vicramadity a ; a series of instructive
tales, supposed to have been related by thirty-two images
which ornamented it. Devandgari character. It has been
translated into Persian. Ly. J.

17. Catha Saritsagara.

A collection of tales by Somadeva. Devandgari character.

Ly. J.

18. Sue a Saptati.

The seventy tales of a parrot. Devandgari character. S.W. J.
The Persians seem to have borrowed their Tuti-nama from
this work.

19. Rasamanjari.

The analysis of love, a poem, by Bhanudatta Misra. De-
vandgari character. Ly. J.

of Sanscrita Manuscripts .

589

so. Sdntisataca.

A poem, in the Bengal character. L>\ J.

21. Arjuna Gita .

A dialogue, something in the manner of the Bbagavat Gita .
Devandgari character. L>\ J.

22. Hitopadesa.

Part of the fables translated by C. W. Written in the
Bengal character. Lv. J.

23. Brahma Nirupana .

On the nature of Brahma .. Devandgari character. Imper-
fect. 1>. J.

24. M'egbaduta ..

A poem. Bengal character. L*. J.

23. Tan Ira Sara.

On religious ceremonies, by Crishndnanda Battacharya .
Bengal character. S.W. J.

26'. Sahasra Ndma.

The thousand names of Vishnu. Devandgari character.

S.W. J.

27. Cirdtarjuniya..

A poem, in the Bengal character. £y. J.

28. Siddhanta Siromani.

A treatise on geography and astronomy, by Bhascaracharya.
Devandgari character. S.W. J.

29. Sangita Narayana.

A treatise on music and dancing. Devandgari character.

S.W.J.

30. Vrihadaranyaca.

Part of the Tajur Veda , with a gloss, by Sancara . Devand-
gari character. I>. J.

59 0

Mr. Wilkins* s Catalogue

31. Niructi , or Nairucta.

A gloss on the Veda . Devanagari character. Ly. J.

32. Aitareya.

A discourse on part of the F&fa. Devanagari character.

I A J.

33. Chandasi.

From the Sdrna Veda. Devanagari character. Ly. J.

34. Magha Tied.

A comment on some other work. Devanagari character.

Ly. J.

35. Rdjaballabha.

De materia Indorum medica; by Narayanadasa. Bengal cha-
racter. I/. J.

36. Hatha Pradipaca.

Instructions for the performance of the religious discipline
called Toga; by Swatmardma. Bengal character. Ly.J.

37. a. Manava Dharma Sastra.

The institutes of Manu, translated into English by S.W. J.
under the title of “ Institutes of Hindu Law , or the Ordinances
“ of Menu.” Devanagari character. Incorrect. I/. J.

37. b\ Ditto.

Duplicate in the Devanagari character. Very incorrect.

Ly. J.

38. Mugdha-bddha-tica.

A commentary on the Mugdha-bodha , which is a Sanscrita
grammar, peculiar to the province of Bengal, by Durga Dasa .
Bejigal character. Four vols. Ly. J.

39. Sdraswati Vyacarana.

The Sanscrita grammar called Sdraswati. (That part only
which treats of the verb.) Devanagari character. Ly. J.

of Sanscrita Manuscripts .

59*

40. Saravali.

A grammar of the Sanscrita language. Incomplete. Bengal

character. S.W. J.

41. Siddhanta Caumudi.

A grammar of the Sanscrita language, by Panini, Catya-
jana , and Patanjali ; with a duplicate of the first part, as far
as compounds. Bevanagari character. II. J.

42. a. Amara Cos a.

A vocabulary of the Sanscrita language, with a grammatical,
comment. Not perfect. Bevanagari character. 17. J.

42. b. Bitto.

The botanical chapter only, with a comment. Bevanagari

character. I/, J.

42. c. Bitto.

The whole complete. Bengal character. S. W. J.

43. Medini Cosa,

A dictionary of the Sanscrita language. Bevanagari cha-
racter. II. J.

44. Viswapracasa Cosa.

A dictionary of the Sanscrita language; by Maheswara.
Bevanagari character. \I. J.

45. Sabda Sandarbba Sindu.

A dictionary of the Sanscrita language ; by Casinatha Sar~
man. It appears from the introduction, that it was compiled
expressly for the use of S.W. J. The learned author is, at
present, head professor in the newly-established college at
Varanasi. Bevanagari character. Two vols. folio. Ly. J.

46. Venisanhara.

A drama, Sanscrita and Prdcrita , in the Bengal character..

Ly. J.

59%

Mr. Wilkins's Catalogue
4 y. Mahd Nataca .

A drama, Sanscrita and Prdcrita , in the Bengal character*

L y. J.

48. Sacuntala.

A drama, Sanscrita and Prdcrita , in the Bengal character.
This is the beautiful play which was translated into English
by S. W. J. but not the copy he used for that purpose.

Ly. J*

49. Mdlati and Madhava.

A drama, Sanscrita and Prdcrita , in the Bengal character.

Ly. J.

50. Hasyarnava.

A farce, Sanscrita and Prdcrita , in the Bengal character,

Ly. J.

51. Cautuca Sarvaswam.

A farce, Sanscrita and Prdcrita , in

the Bengal character.

Ly. J.

52. Chandrabhisheca.

A drama, Sanscrita and Prdcrita .

Bengal character.

Ly. J.

53. Ratndvali.

A drama, Sanscrita and Prdcrita. Bengal character.

Ly. J.

54. Vicramorvasi.

A drama, Sanscrita and Prdcrita. Bengal character.

Ly, J.

55. Manavicagnimitra.

A drama, Sanscrita and Prdcrita . Bengal character.

Ly. J,

I

of Sanscrita Manuscripts.

56. A catalogue of Sanscrita books, on various subjects.
Devanagari character. 17. J.

N. B. Those articles in the above catalogue, marked S.W. J.
were presented by Sir William Jones ; and those marked
17. J. by Lady Jones. A catalogue of the Persian and Arabic
MSS. presented by them, will be given in a future volume.

\

MDCCXCVIII.

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PRESENTS

RECEIVED BY THE

ROYAL SOCIETY,

From November 1797 to June 1798;

WITH THE

NAMES OF THE DONORS.

1 797- PRESENTS.

Nov. 9. Nova Acta Academiae Scientiarum Imperialis Pe-
tropolitanae. Tom. IX. et X. Petropoli, 1795 et

l>797- . _ 4®

Transactions of the Society for the Encourage-
ment of Arts, Manufactures, and Commerce,
Vol. XV. London, 1797. 8°

Catalogus Bibliothecae Historico-Naturalis Josephi
Banks, auctore J. Dryander. Tom us III. Lon-
dini, 1797. go

An authentic Account of an Embassy from the
King of Great Britain, to the Emperor of China,
by Sir G. Staunton. London, 1797. zVols.4to,
and one Volume of Plates, in folio.

Beitrage zur chemischen Kenntniss der Mineral-
korper, von M. H. Klaproth, z Band. Posen,
1797. _ 8°

Annuaire de la Republique Fran^oise, par le Bu-
reau des Longitudes, pour l’Annee 6. Paris,
PA11 5. I2mo

Papers relative to certain American Antiquities.

Philadelphia, 1796. 40

Essai politique et philosophique sur le Commerce
et la Paix, par J. B. Rougier - Labergerie.
Paris, 1 797. go

Tables of Weights and Measures, by G. Fair.
Nos. 6, 7 and 8.

Figures of non-descript Moths and Butterflies, by
T. Martyn. No. 1. 4.0

A Journal of Natural Philosophy, by W. Nichol-
son. Nos. 5, 6, 7 and 8.

16. Some Account of the Cathedral Church of Exeter.
London, 1797. fol.

4 G 2

DONORS,

The Imperial Academy
of Sciences of Peters-
burg.

The Society for the En-
couragement of Arts,
Manufactures, and
Commerce.

Right Hon. Sir Joseph
Banks, Bart. K. B.
P. R.S.

Sir George Staunton,
Bart. F. R. S.

Professor Klaproth,
F. R. S.

M. Lalande, F. R. S.

Professor Barton, of Phi-
ladelphia.

M. Rougier-Labergerie,
de l’lnstitutde France.

Mr. George Fair.

Mr. Thomas Martyn.

Mr. William Nicholson.

The Society of Anti-
quaries.

C 596 3

DONORS.

v

PRESENTS.

Dec. 7. Practical Observations on the Treatment of Stric-
tures in the Urethra, by E. Home, zd edition.
London, 1797. 8°

A Journal of Natural Philosophy, by W. Nichol-
son. No. 9.

14. The Morbid Anatomy of some of the most impor-
tant Parts of the Human Body, by M. Baillie.
2d edition. London, 1797. 8°

A Chinese Work on Medicine and Botany, in 30.
Parts.

W98- .

Jun. 11. Essai sur les Ouvrages physico-mathematiques de
Leonard da Vinci, par J. B. Venturi. Paris,
l’An 5. 4?

Recherches experimentales sur le Principe de la
Communication laterale du Mouvement dans les
Fluides, par J. B. Venturi. Paris, l’An 6. 8*

Observations on the various Systems of Canal Na-
vigation, by W. Chapman. London, 1797. 40

18. New Views of the Origin of the Tribes and Na-
tions of America, by B. S. Barton. Philadelphia,

1 791' . . . 8°

Angulorum rectseque Lines Trisectio, et consec-
taria Circuli Quadratio ; utramque detexit J. N-.
Revay. Viennae, 1797. _ 8°

Index Plantarum in Horto Botanico Academis Ha-
lensis cultarum, Hals, 1 797. 8°

A Meteorological Journal of the Year 1797, kept
in London, by W. Bent. London. 8°

Observations in Defence of a Bill lately brought
into Parliament, for erecting the Corporation of
Surgeons of London into a College, by T . Che-
valier. London, 1797. 8°

A Journal' of Natural Philosophy, by W. Nichol-
son. No. 10.

25. The Coal Viewer, and Engine Builder’s practical
Companion, hy J. Curr. Sheffield, 1797. 4°

Feb. 1 . An Essay on the comparative Advantages of verti-
cal and horizontal Windmills, by R. Beatson.
London, 1798. 8°

A Journal of Natural Philosophy, by W. Nichol-
son. No. 11.

J. F. Blumenbachii Xnstitutiones Physiologies.
Gottings, 1798. 8°

8. Annals of Medicine for the Year 1797, by A. Dun-
can, sen. and A. Duncan, jun. Vol. II. Edin-
burgh, 1798. ^ 8°

22. Connoissance des Terns pour l’Annee 7, publiee par
le Bureau des Longitudes. Paris, l’An 5. 8°

Memoire sur l’Jnterieur de l’Afrique, par J.La-
lande, Paris, PAn 3. 4°

Everard Florae, Esq.
F.R.S.

Mr. William Nicholson.

Matthew Baillie, M. D.

F.R.S.

Alexander Duncan, Esq..
F.R.S.

Professor Venturi, of
Modena.

William Chapman,, Esq.

Professor Barton, of Phi-
ladelphia.

Professor Revay, of
Gran.

Professor Curt Sprengel,
of Halle.

Mr. William Bent.

Mr. Thomas Chevalier.

Mr. William Nicholson.

Charles Hatchett, Esq.
F.R.S.

Robert Beatson, Esq.

Mr. William Nicholson.

Professor Blumenbach,
F.R.S.

Andrew Duncan, sen.
M. D. and Andrew
Duncan, jun. M. D.
M. Lalande, F. R. S.

t 597 3

PRESENTS.

DONORS.

Essai sur les Causes de la Perfection de la Sculp-
ture antique, et sur les Moyens d’y atteindre,
par M. le Chev. Louis de Gillier. Londres,
1798. 8°

March 1. Essays on the Microscope, by the late G. Adams.

2d edition. London, 1798. 40

Geometrical and Graphical Essays, by the late G.

Adams. 2d edition. London, 1 797. 8°

A Journal of Natural Philosophy, by W. Nichol-
son. Nos. xz. and 13.

8. Physiology, by E. Peart. London, 1798. 8°

15. Astronomisches Jahrbuch fur das Jahr 1799, von
J. E. Bode. Berlin, 1796. 8°

Astronomische tafeln von J. A. Koch. Berlin,
1797- . 8°

Des Revolutions de France et- de Geneve, par M.

D’lvernois. Londres, 1795. 8°

Etat des Finances et des Ressources de la Repu-

blique Fran^aise, au 1 Janvier 1796, par M.
D’lvernois. Londres, 1796. 8°

Histoire de F Administration des Finances de la Re-
publique Fran9aise pendant l’Annee 1796, par
Sir F. D’lvernois. Londres, 1 796. 8°

Tableau de F Administration de la Republique
Fran$aise pendant FAnnee 1797, par Sir F.
D’lvernois. Londres, 1798. 8°

29. Memoirs of the Literary and Philosophical Society
of Manchester, Vol. V. Part x. Manchester,
1798. 8°

April 19, A third Dissertation on Fever, by G. Fordyce,
Bart 1. London, 1798. 8®

Connoissance des Terns pour FAnnee 8, publiee par
le Bureau des Longitudes. Paris, l’An 6. 8°

A Journal of Natural Philosophy, by W. Nichol-
son. No. 14.

26. West View of the Cast-Iron Bridge over the River
Wear, at Sunderland, drawn by Robert Clark,
engraved by J. Raffield.

Hints designed to promote Beneficence, Tem-
perance, and Medical Science, Vol. I. London,
1797. 8°

Tracts relating to Natural History, by J. E. Smith.

London, X798. 8°

The Influence of metallic Tractors on the Human
Body, by B. D. Perkins. London, 1798. 8°

May 3. Antiquities of Ionia, published by the Society of
Dilettanti. Part 2. London, 1797. fol.

A Journal of Natural Philosophy, by W. Nichol-
son. No. 15.

xo. Transactions of the Royal Society of Edinburgh,
Vol. IV. Edinburgh, 1798. 40

M. Louis de Gillier.

Mr. William Jones.

Mr. William Nicholson.

E. Peart, M. D.

Mr. J. E. Bode^ F. R. Si

Sir Francis DTvernois, .
Knt.

The Literary and Philo-
sophical Society of
Manchester.

George Fordyce,- M. D.
F. R. S.

M. Lalande, F. R. S.

Mr. Williarti Nicholson.

Thomas Bowdler, Esq.
F.R.S.

John Coakley Lettsom,
M. D. F. R. S.

James Edward Smith,
M. D. F.R. S.

Benjamin Douglas Per-
kins, A. M.

The Society of Dilet-
tanti.

Mr. William Nicholson.

The Royal Society of
Edinburgh.

t 598 1

PRESENTS.

60N0R9.

17. A. Monro Disputatio inaug. de Dysphagia. Edin-
burgh 1 797. 8°

24. Astronomical Observations, made at the Royal
Observatory at Greenwich, from 1750 to 1762,
by J. Bradley. Vol. I. Oxford, 1798, fol.

Transactions of the Linnean Society. Vol. IV.
London, 1798. 40

June 7. An Account of the English Colony in New South
Wales, by D. Collins. London, 1798. 40

A Journal of Natural Philosophy, by W. Nichol-
son. No. 16.

Reflexions sur la Perfectibilite de l’Homme. 1 797.

8°

14. Athenian Letters. London, 1798. 2 Vols. 40

Museum Worsleyanum. London, 1794. 2 Vols.

fol.

Naval Sermons, by J. S. Clarke. London, 1798.

8°

Astronomisches Jahrbuch fur das Jahr 1800, von
J. E. Bode. Berlin, 1797. 8°

Dritter Supplement-band zu Dessen Astronomi-
schen jahrbiichern. Berlin, 1797. 8°

21. A Chinese Calendar for 1771.

A Book printed in the Tybet character.

Several Manuscripts in the Tybet and Mongol
Tartar characters.

The View of Hindoostan. London, 1798. 2 Vols.

1 -4°

28. Essays on the Venereal Disease, by W. Blair,
Part 1. London, 1798. 8*

Alexander Monro, jun.
M. D.

The University of Ox-
ford.

The Linnean Society.

David Collins, Esq.

Mr. William Nicholson.

The Author.

Philip Earl of Hard-
wicke, F. R. S.

Right Hon. Sir Richard
Worsley,Bart. F. R. S .

Rev. James Stanier
Clarke, F. R. S.

Mr. J.E. Bode, F.R.S.

The Rev. William Coxe,
F.R.S.

Thomas Pennant, Esq.
F. R S

Mr. William Blair.

'

INDEX

TO THE

PHILOSOPHICAL TRANSACTIONS

FOR THE YEAR 1798.

A page

A bernethy, Mr. John. Observations on the Foramina The-
besii of the heart, -

Acid lithic , remarks on, - _

Adularia , specific gravity of.

Astronomy , physical , improved solution of a problem in,

Atmospherical refraction , singular instance of,

Atwoodj George, Esq. A disquisition on the stability of ships

103

*9* 34
412

527

3 57
, 201

B

Barker, Thomas,' Esq. Abstract of a register of the barometer,
thermometer, and rain, at Lyndon, in Rutland, for the year 1 796, 1 30

Barometer, register of, at Lyndon, in Rutland, - _ 130

Beam-compass, description of one, - - _ 137

Bezoar , Oriental , remarks on, - - - 46

Bird. Mr. John. Account of some scales made by him, 137, 170, 177
Bouguer, M. Remarks on some opinions of his, - 242

Bo ur non. Count de. An analytical description of the crystalline
forms of Corundum, from the East Indies, and from China, 42S
Brougham, Henry, Jun. Esq. General Theorems, chiefly Po-
ri sms, in the higher geometry, - - - 37 8

‘ — — Remarks on his opinions re-

specting light, -

312

I

INDEX.

page

Calculi , on the composition of,

■ on one from a dog, _

• on one from a rabbit,

• on those of the horse,

Cannon , on the heat excited in boring them,
Cavendish, Henry, Esq. Experiments to determine
sity of the earth, _

Chapman, Mr . Remarks on a rule employed by him,
Clairbois , M. Remarks on some opinions of his,
Clarke, John, M. D. Account of a tumour found in
stance of the human placenta,

Clay-pit. , account of a substance found in one,
Concretions , urinary , on the composition of,

— on one from a dog, -

— on one from a rabbit,

on those of the horse,

Copper , effect of the Mere of Diss upon it,

Corundum stone , account of,

specific gravity of,

— — analysis of,

— description of its crystals,

Cube of brass, examination of one,

Cuffnells , on the dimensions of the ship so called,
Cylinder of brass, examination of one,

the den-

*5

39

42

43
81

- 469

£67

- 242
the sub-

361

567

15

- 39
42

- 43

- 5683 570

403

411,416, 445
417
428

*44> 151* *52
28 7

146, 151, 153

D

Day-labour , price of, at different periods, - - 176

Density of the earth. See Earth.

Depreciation of money, remarks on, - - - 175 •

Diamond , specific gravity of, - 416, 447

Diss Mere. See Mere of Diss.

Dog , on an urinary eoncredon from one, - - 39

E

.Earth, density of, experiments to determine, - - 469

apparatus used therein, — - 47rl

method of observing, - - - 474

. account of the experiments, - - 4 78

— method of computing, from the experiments, - 509

4

INDEX.

Earth, density of result of the experiments,

attraction of the mahogany case on the balls,

— perturbations of, method of obtaining swiftly converging

series, useful in computing them,

Equations , on their roots, _ _ „

Evelyn, Sir George Shuckburgh. An account of some en-
deavours to ascertain a standard of weight and measure.

Eye , account of an orifice in it, -

page

520

523

*

527

369

133

S32

F

Feld spar , specific gravity of,

Fluids , on the resistance of bodies moving in them,
Foot, Roman , on the length of,

Foramen Ovale of the heart , remarks on,

Foramina Thebesii of the heart , observations on,
Friction , on the heat excited by it,

412

1

169

107

103

80

G

Garrow, Mr. Edward. Letter from, concerning the Corun-
dum stone, _ _

Georgium Sidus , on the satellites of, - .

" 7 retrograde motion of its satellites,

investigation of additional satellites,

an interior satellite,

an intermediate satellite,

— an exterior satellite,

the most distant satellite, _

— on its supposed rings,

L i O '

on the flattening of its polar regions,

on the light and size of its satellites,

— - on the vanishing of its satellites,

- on the revolutions of the new satellites,

45°>

Gold , oxide of experiments on its reduction, _

Graham , Mr . George. Account of a standard of measure procured
by him,

Creville, The Right Hon. Charles. On the Corundum. stone
from Asia, «...

H

405

47

48

49

59

62

63

64

67

67

7i

7i

78

462

167

403

Harris , Mr. Account of some standard weights made by him
Hastings , account of a phenomenon seen there,

M DCCXCVI II. ^ pf

*73
3 57

INDEX.

Hatchett, Charles, Esq. An analysis of the earthy substance
from New South Wales, called Sydneia or Terra Australis , no

An analysis of the water of the Mere

ofDiss, " - - - 572

Heart , on the Foramina Thebesii of it, - _ a0g

— account of an unusual formation of it, - -

Heat, on that excited by friction, - - - 80

Helena, St. diurnal variation of the magnetic needle in that island, 397
He l l 1 ns, The Rev. John. A new method of computing the value
of a slowly converging series, of which all the terms are affirmative, 1 83

■ — An improved solution of a problem

in physical astronomy ; by which, swiftly converging series are
obtained, which are useful in computing the perturbations of the
motions of the Earth, Mars, and Venus, by their mutual attrac-
tion. To which is added an appendix, containing an easy
method of obtaining the sums of many slowly converging series
which arise in taking the fluents of binomial surds, &c. - 527

H erschel, WVlliam, LL.D. On the discovery of four addi-
tional satellites of the Georgium Sidus. The retrograde motion
of its old satellites announced ; and the cause of their disap-
pearance at certain distances from the planet explained, - 4 7

Home, Everard, Esq. An account of the orifice in the retina of
the human eye, discovered by Professor Soemmering. To
which are added, proofs of this appearance being extended to
the eyes of other animals, - - - - .332

Horse , on urinary concretions of that animal, - - 43

Hydrostatic balance, description of one, - - - 139

I

Immersion of animals, remarks on, - - 108

J

Jargon , specific gravity of, - - - 417

Jones, Sir William and Lady. A catalogue of Sanscrita ma-
1 nuscripts presented to the Royal Society by them, - 582

K

Klaproth, Mr. Analysis of Corundum, - - 417

L

Latham, William, Esq. Account of a singular instance of at-
mospherical refraction, - 3 57

INDEX.

Lecture , Baker ian, - - -

Light , on its reflexibility, - -

— - on the chemical properties attributed to it,

Liquor amnii , extraordinary quantity of,

M

Macdonald, John, Esq. Observations of the diurnal variation
of the magnetic needle, in the island of St. Helena; with a con-
tinuation of the observations at Fort Marlborough, in the
island of Sumatra, -

Manuscripts , Sanscrita , catalogue of,

Mars , perturbations of, \ method of obtaining swiftly converging
series, useful in computing them, -

Measure , endeavours to ascertain a standard of.

page

i

449

364

— account of several standards of,

— comparison of various standards of,

397

582

527

*33

167, 177

182

Mere of Diss, its effect upon various substances immersed in it, 568, 578

* analysis of its water.

Metacentre , remarks on the point so called,

Michell , The Rev. John. Method contrived by him, of determining
the density of the earth, - -

Money , on the depreciation of it, - -

Motion, remarks on, as supposed to constitute heat,

572

240

469

*75

99

N

Necessaries of life , on the prices of them at different periods, 175

Needle , magnetic , diurnal variation of, in the islands of St. Helena
and Sumatra, - _ gqy

Newton. Remarks on his opinion respecting light, - 311

O

Optical remarks , chiefly on the reflexibility of light,

Oxide, animal , experiments on,

ouric or uric , application of that name,

of gold , experiments on its reduction,

°f silver, experiments on its reduction,

P

Parabola , remarks on different ones,

Pearson, George, M. D. Experiments and observations, tending
to shew the composition and properties of urinary concretions, 15

4 H 2

3*i

29

37

45°, 462
460, 465

257

INDEX.

page

Pendulum , on its use as a standard of measure, - 133, 174

Placenta, account of a tumour found therein, - - 361

Porisms, - - - - 37S

Presents received by the Royal Society, from November, 1707,
to June, 1798, - - - - 595,

Prevost, P. Quelques remarques d’optique, principalement rela-
lives a la reflexibilite des rayons de la lumiere, - - 31 .1

Provisions, on the prices of them at different periods, - 175

Pyrites , remarks on its formation, - 580

R

Rabbit , on a calculus from one, - 42

Rain , register of, at Lyndon, in Rutland, - 1 30

Reflexibility of light, remarks on, - - - 311

Refraction , atmospherical, singular instance of, - 357

Resistance of bodies moving in fluids, experiments on, - 1

Retina of the Eye, account of an orifice therein, - 332

Roman foot, on the length of, - - 169

Roots of equations, observations on, - - 369

Ruby, specific gravity of, - - - 416, 447

R umford, Benjamin Count of. An inquiry concerning the
source of the heat which is excited by friction, - - 80

An inquiry concerning the

chemical properties that have been attributed to light, - 449

S

Saint Helena , diurnal variation of the magnetic needle in that island, 397
Sanscrita manuscripts, catalogue of, - - 582

Sapphire, specific gravity of, - - - 416,445

Satellites of the Georgium Sidus. See Georgium Sidus .

Series, slowly converging, new method of computing the value

of one, - - - - 183

... . method of obtaining the sums of many

which arise in the fluents of binomial surds, - - 547

. — — swiftly converging, method of obtaining, useful in com-
puting the perturbations of the Earth, Mars, and Venus, - 527

Ships, disquisition on their stability, - 201

Silver, effect of the Mere of Diss upon it, - - 57 3

oxide of, experiments on its reduction, - 460, 465

■ ■ • — w coin , method used by those who diminish it, - 573

INDEX.

page'

rc l& *

Soemmering , Mr. Account of an orifice in the eye, discovered by him 0Q2

™ .. ... f ' J >00

Spar, adamantine, account of,

Sphere of brass , examination of one.

Stability of Ships , disquisition on,

Standard of weight and measure , endeavours to ascertain,
Standards of measure , account of several,

comparison of various ones^

403

157

201

133

167, 177

1 82

Sterling, Mr. Remarks on his table for measuring curvilinear spaces, 261
Sumatra , diurnal variation of the magnetic needle in that island, 401
Surds , binomial , method of obtaining the sums of many slowly con-
verging series which arise in their fluents, - - 547

Sydneia , analysis of the substance so called, - - _ 11Q

T

Tarnish of silver , remarks on, _

Terra Australis, analysis of the substance so called.

Theorems , general , in the higher geometry.

Thermometer , register of, at Lyndon, in Rutland,

— at Hastings,

Tiree, account of a stone from, _

Topaz, specific gravity of, -

Troughton , Mr. Account of some instruments made by him,
Tumour , account of one in the placenta.

- 58a
110
378

1 3°

360

419

416, 446

134

361

47 1:
527

V

Variation compass, remarks on one,

V eniiSy perturbations of method of obtaining swiftly converging
series, useful in computing them,

Vince, The Rev. Samuel. The Rakerian Lecture. Experiments
upon the resistance of bodies moving in fluids^ ,

w

Water , made to boil by heat excited by friction, - _

quantity heated by the combustion of a given quantity of wax, 04

remarks on the immersion of animals in it,, - 108

weiSht of a cubic inch of it, - ’ 1 64, 1 74

Wax, quantity of heat produced by the tombustion of a given
quantity of it, - _ _ - q4

W lather of 1 796, remarks on, _ _ _

Wedgwood , Josiah, Esq. Remarks on his experiments on the Sydneia, 1 10
Weight, experiments to ascertain a standard of, - . 1n ^

INDEX.

Pagt

Weights , examination of some standard ones, - - 173

Whitehurst , Mr. John. Account of a machine made by him, 134

Wilkins, Charles, Esq. Catalogue of Sanscrita manuscripts
presented to the Royal Society by Sir William and Lady Jones, 582
W 1 lson', Mr. James. A description of a very unusual formation
of the human heart, - - - - - 346

W I SEM AN, Mr. B enjamin. Account of a substance found in a
clay-pit; and of the effect of the Mere of Diss, upon various
substances immersed in it, - - 567, 578

Wood, James, B. D. On the roots of equations, - 369

From the Press of
W. BULMER & Co.
Cleveland- Row, St. James's.

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