(1 - Σσ × (1 - _d_²) - δ²) ⁄ (1 - S_s_Σσ) = _a_²; in words thus:
_Multiply the product of the sines of the two known angles by the
square of the cosine of half the difference of the sides: add the
square of the sine of half the difference of the angles; and divide the
complement of this sum to unity, by the like complement of the product
of the four sines of the sides and angles; and the square root of the
quotient shall be the sine of half the unknown angle._
If we work by logarithms, the operation will not be very
troublesome; but the rule needs not be used, unless when a table
of the trigonometrical analogies is wanting. To supply which, the
foregoing theorems will be found sufficient, and of ready use; being
either committed to memory, or noted down on the blank leaf of the
trigonometrical tables.
_Note_, The schemes may be better, raised in card-paper, or with bent
wires and threads.
LXXI. _An Account of Two extraordinary Cases of Gall-Stones._ By James
Johnstone, _M. D. of_ Kidderminster. _Communicated by the Rev._ Charles
Lyttelton, _L. L. D. Dean of_ Exeter.
_To the Rev. Dr._ Lyttelton, _Dean of_ Exeter.
[Read Feb. 9, 1758.]
Rev. Sir,
ACcording to promise I send you a short account of the two
extraordinary cases we talked of, the last time I had the pleasure of
seeing you at Kidderminster.
The truth of the first narrated case you are already a sufficient judge
of; and if it is at all necessary to ascertain the second in like
manner, I can at any time produce the poor woman and her husband before
you, who will attest the truth of sufferings, which will not easily
escape their memory.
You are at liberty to dispose of this paper as you shall think proper.
I am,
Reverend Sir,
Your respectful and most humble Servant,
J. Johnstone.
Kidderminster, Sept. 11. 1757.
THO’ it is now pretty well known, that colicky and icteric diseases
often arise from gall-stones generated in the bilious receptacle, and
obstructing its canals; yet an example of one, of such enormous size,
voided into the _duodenum_ from the _ductus communis_, as happened in
the first of the following cases, is a very rare, if not intirely an
unexampled occurrence. It will encourage us not too easily to despair
of the expulsion of the largest _calculi_ from the gall-bladder; and
will teach us, that all violent attacks of pain about the stomach
are not owing to gout reflected upon that organ: it will make us
more cautious of giving drastic cathartics, heating and inflaming
medicines, upon such a vague presumption; and ought to dispose those,
who are trusted with the lives of their fellow-creatures, to a nicer
observation of even the minutest symptoms and circumstances, which may
occur in diseases.
The second case points out, under certain circumstances, the
practibility of extracting, by incision into the gall-bladder itself,
those _calculi_, which, from their figure, or other impediments, cannot
be voided in the natural way. The method of performing this unusual
operation, and some instances of its success, have already been made
public in the Memoires de l’Acad. de Chirurg.
1. Mrs. F----, a sedentary corpulent old lady, had been much subject
to colicky complaints, without jaundice, in the vigour of life. The
seat of the pain was chiefly under the right _hypochondrium_, as high
as the stomach. She had been tolerably free from it for at least eight
years past. December 5, 1753, about eleven o’clock in the evening, she
was suddenly seized with a violent pain, extending from that part of
the stomach lying under the right side, thro’ to her back. She compared
it to a sword driven in that direction. This pain continued not only
with unremitted violence, but even increased, till seven o’clock in the
morning: all this time she vomited and strained almost incessantly;
but after her stomach was emptied of its contents, nothing came up
besides clear slime, streaked with blood. About seven o’clock in the
morning she felt her pain fall or move lower, as she expressed it,
and from that time became remarkably easier. Soon after this change,
she became extremely sick, and vomited up, for the first time, a
prodigious quantity of greenish yellow bile. She had not before this
seizure been remarkably costive; and in her pain had a free motion
to stool with effect; but during the remainder of the (6th) day had
none, tho’ all this time emollient clysters were injected; and she
took regularly every two hours a powder of _magnes. alb. terr. fol.
tartar. tart. vitriol. ana_ ℈j. _ol. nuc. mosch. gutt._ j. with a
draught of the _succ. limon. & sal. absinth_. But in the middle of the
night, and all day (the 7th), she had an abundant discharge of loose
bilious stools. She had continued free from excessive pain since the
morning of the former day, only now and then complained of uneasiness
sometimes in one, sometimes in another, part of her bowels. About
twenty-four hours after her first seizure, she felt a great pain
striking towards the bottom of her back, and one hour after voided
the extraordinary _calculus_, of which the figure and description are
subjoined. Some time after pieces of skins were voided by stool, which
were evidently of the texture and appearance of the internal villous
coat of the intestines and gall-bladder. The above medicines were the
only ones she used, by my direction, under her painful complaint,
excepting an external fomentation, and bleeding, which the hardness and
contractedness of her pulse seemed to require. She was ordered to drink
plentifully of thin broths, and other soft diluent liquors. During the
course of her disorder she had no appearance of jaundice, nor since;
and, considering her years, enjoys at present (Sept. 1757) very good
health.
This _calculus_, as appears by the figure, was of a pyriform shape,
resembling the form of the _cystis fellea_ itself. Its surface was
quite smooth and polished, excepting towards the base, at that part
marked A, where it was scabrous, as if some other substance had lain
contiguous to it. When broken through, it was composed of concentrical
laminæ, which were alternately white and ochre-coloured. In length it
measured one inch and three tenths; its transverse section measured at
least seven tenths of an inch. It had a saponaceous smoothness, like
other gall-stones, and floated upon water. It weighed only about 126
grains.
Tho’ it be difficult to conceive, how so bulky a substance, generated
in the gall-bladder, could be conveyed along so narrow a passage as
the common biliary duct, especially considering the obliquity of its
insertion for near half an inch of length betwixt the coats of the
_duodenum_; yet there seem sufficient _data_ in the above case to
prove, that this animal stone was not formed in the alimentary tube,
but (large as it was) had come into it from the _ductus communis
choledochus_.
The shape and saponaceous smoothness, and colour of the laminæ, of this
substance, shew it was moulded in the gall-bladder, and formed from
bilious particles. The severe pain and torture, and enormous vomiting,
she underwent, for seven hours after her first seizure, argue, that it
must then be lodged in some canal much narrower and straighter than the
alimentary canal; for so soon as it dropped into that, the severe pain
in a great measure ceased.
But that straight canal, in which it was situated during those
seven hours of torture, could be no other than the _ductus communis
choledochus_; for, during this space of time, no bile was emptied
into the bowels, nor thrown up by the strongest efforts of vomiting.
But no sooner had she perceived the cause of her pain to move or drop
downwards (a sensation, which points out the precise moment the stone
must have dropped into the _duodenum_), than she began to sicken,
and instantly after vomited up a vast quantity of bilious matter;
which now, from the de-obstructed duct, began to flow freely into
the _duodenum_. The obstruction of the _ductus choledochus_ was of
too short a duration (only three hours) to occasion any observable
jaundice. And it appears by the bloody flesh-like knots, thrown up
with phlegm by vomiting, that the passage of the substance was not
effected without considerable laceration of the small bilious ducts.
And this easily accounts for the separation of the villous coat, which
afterwards appeared in this patient’s stools.
[Illustration: This coarse delineation represents the figure and true
bulk of the _calculus_; which, I believe, is still in my patient’s
custody.]
2. In February 1752. I was called to relieve a poor woman of this
place, Sarah Ewdall, aged 30 years and upwards, and the mother of
several children. She laboured under the jaundice, and complained of
a severe acute pain striking thro’ from the right _hypochondrium_
to her back, with frequent vomitings. A præternatural hardness, of
a compass not exceeding the hollow of the hand, was then plainly to
be felt at the pit of the stomach, or a little nearer to the right
_hypochondrium_. When that particular part was pressed, she complained
of great pain. The pain at this part was always increased by attempting
to lie upon the left side. She was blooded, fomented externally,
had emollient saponaceous clysters injected, and a nitrous apozem,
and pills composed of _galban. & sap. Castillens._ and soon after
recovered. She had frequent returns of the same complaint after this;
but I saw her not again till Jan. 1755, when she lay insensible in a
fit, which for several days deprived her of the use of her speech and
of all her senses, only she tossed her limbs about. About a quarter of
a year after she had recovered from this fit, Mr. Cooper of this place,
her apothecary, informed me, that from a small sore at the pit of her
stomach, which came since her last illness, she had voided several
gall-stones. Curiosity prompted me to inquire into the matter of fact
from herself. She shewed me the sore, which was now almost cicatrized.
She said, that soon after her last illness a little pimple arose upon
that part of the pit of the stomach, which had been hard ever since she
had been subject to the jaundice. This pimple broke, ran matter, and at
different times the _calculi_, which she shewed me, had come out with
the matter. Her stomach had been somewhat painful before it broke, but
was now easy. The _calculi_, which she shewed me, had the appearance
of being fragments of larger ones, and some were almost dust; tho’ she
assured me they all came from the sore in that condition. Of these
fragments I have two or three of the largest now in my custody: they
are light, swim on water, smooth like soap; are of a yellow colour, and
in some parts brown like snuff; and consist of similar concentrical
layers. The poor woman has since then been troubled with returns of
pain and jaundice, in the intervals of which her skin is perfectly
clear and white. She is still alive, and ready to attest the truth of
this narrative.
J. Johnstone.
Kidderminster, Sept. 11th, 1757.
LXXII. _A remarkable Case of Cohesions of all the Intestines_, &c.
_in a Man of about Thirty-four Years of Age, who died some time last
Summer, and afterwards fell under the Inspection of Mr._ Nicholas Jenty.
[Read Feb. 9, 1758.]
THE subject was tall, and partly emaciated. I found nothing externally
but a wound in the left side, which seemed to me to have been
degenerated into an ulcer. As I did not know the man when he was alive,
and had him two days after his decease, I cannot give an immediate
account of the cause of his death. But in opening his abdomen, I found
the epiploon adhering close to the intestines, in such a manner, that I
could not part it without tearing it. It felt rough and dry. And as I
was going to remove the intestines, to examine the mesentery, I found
them so coherent one with the other, that it was impossible for me to
divide them without laceration. Then I inflated the intestinal tube,
for the inspection of this extraordinary phænomenon; but, to my great
surprize, all the external parts of the intestines appeared smooth;
very few of the circumvolutions were seen, occasioned by the strong
lateral cohesions of their sides with each other. The substance of the
intestines was rough, and a great many pimples, as big as the head of
a pin, appeared in them, and were almost free from any moisture. It is
proper to observe, that these pimples have been taken for glands by the
late Dr. James Douglas, and others; whereas they are in reality nothing
else but the orifices of the exhaling vessels obstructed, and are not
to be met with except in morbid cases.
After having made incisions in that part of the _colon_ next to
the _rectum_, I found the _peritonæum_, or external membrane which
invests the intestines, and the _viscera_ of the _abdomen_, to be of
the thickness of a six-pence; and I fairly drew all the intestines
from their external membrane without separating their cohesions;
the _peritonæum_, or external membrane, afterwards appearing like
another set of intestines. I found a fluid in the intestines; and I
will not take upon me to say, how the peristaltic motion must have
been performed. And afterwards I parted the stomach from its external
tunic, as I had done the intestines. I found no obstruction in the
mesenteric glands; but every evolution of the mesentery firmly cohered
together. The liver also adhered closely to the diaphragm, and its
adjacent parts: and in the _vesicula fellis_ I found the bile pretty
thick, neither too green nor too yellow, but a tint between both. I
met with nothing remarkable in the other parts of the _abdomen_. In
opening the _thorax_, I found the lungs closely adhering to the ribs
laterally, and posteriorly and interiorly close to the _pericardium_.
In making an incision to open the _pericardium_, I found it so closely
adhering to the heart, that I could not avoid wounding that organ, and
with much difficulty could part it from it. I met with no fluid in the
_pericardium_. The heart was small; and in the internal side the pores
of the _pericardium_ appeared so large, that one might have insinuated
the head of a middling pin into them. They have been described by some
anatomists, who have met with cases somewhat similar to this, but
without such universal adhesions; and they have been supposed to have
been glands. The same pores likewise appeared on the heart; which, in
my opinion, are nothing but the extremities of the exhaling vessels. In
removing the heart, I found the _dorsal_, and other lymphatic glands
above the lungs, quite large, indurated, and of a dark greyish colour.
Nothing remarkable appeared in the lungs; only, that the portion of
the _pleura_, which invests the lungs, and is generally thin, was here
thick and rough; and thro’ a glass it appeared as if covered with
grains of sand; and might in several places have been easily torn from
the lungs.
The _aorta_ was pretty large; and in that part of it, which runs on the
tenth _dorsal vertebra_, I found a _cystis_, as big as an olive, full
of _pus_; and lower down, immediately before that vessel perforates the
diaphragm, I found another, something less, full of matter likewise;
both which portions I have by me. That portion of the _aorta_, where
the _cystis_ appeared, was rather thicker than the other, and osseous.
In opening the _cranium_, I found in that part of the _cerebrum_, which
lies over the _cerebellum_, a table spoonfull of _pus_, of a greenish
colour; and examining it thro’ a glass, there was an appearance of
_animalcula_ in it.
LXXIII. _Of the best Form of Geographical Maps. By the Rev._ Patrick
Murdoch, _M. A. F.R.S._
[Read Feb. 9, 1758.]
I. WHEN any portion of the earth’s surface is projected on a plane,
or transferred to it by whatever method of description, the real
dimensions, and very often the figure and position of countries, are
much altered and misrepresented. In the common projection of the
two hemispheres, the meridians and parallels of latitude do indeed
intersect at right angles, as on the globe; but the linear distances
are every-where diminished, excepting only at the extremity of the
projection: at the center they are but half their just quantity, and
thence the superficial dimensions but one-fourth part: and in less
general maps this inconvenience will always, in some degree, attend the
_stereographic_ projection.
The _orthographic_, by parallel lines, would be still less exact,
those lines falling altogether oblique on the extreme parts of the
hemisphere. It is useful, however, in describing the circum-polar
regions: and the rules of both projections, for their elegance, as well
as for their uses in astronomy, ought to be retained, and carefully
studied. As to Wright’s, or Mercator’s, nautical chart, it does not
here fall under our consideration: it is perfect in its kind; and will
always be reckoned among the chief inventions of the last age. If it
has been misunderstood, or misapplied, by geographers, they only are to
blame.
II. The particular methods of description proposed or used by
geographers are so various, that we might, on that very account,
suspect them to be faulty; but in most of their works we actually
find these two blemishes, _the linear distances visibly false_, and
_the intersections of the circles oblique_: so that a quadrilateral
rectangular space shall often be represented by an oblique-angled
rhomboid figure, whose diagonals are very far from equal; and yet,
by a strange contradiction, you shall see a fixed scale of distances
inserted in such a map.
III. The only maps I remember to have seen, in which the last of these
blemishes is removed, and the other lessened, are some of P. Schenk’s
of Amsterdam, a map of the Russian empire, the Germania Critica of the
famous Professor Meyer, and a few more[27]. In these the meridians are
straight lines converging to a point; from which, as a center, the
parallels of latitude are described: and a rule has been published for
the drawing of such maps[28]. But as that rule appears to be only an
easy and convenient approximation, it remains still to be inquired,
_What is the construction of a particular map, that shall exhibit the
superficial and linear measures in their truest proportions?_ In order
to which,
IV. Let E_l_LP, in this figure (_See_ TAB. XXI.) be the quadrant of
a meridian of a given sphere, whose center is C, and its pole P; EL,
E_l_, the latitudes of two places in that meridian, EM their middle
latitude. Draw LN, _ln_, cosines of the latitudes, the sine of the
middle latitude MF, and its cotangent MT. Then writing unity for the
radius, if in CM we take C_x_ = N_n_ ⁄ (L_l_ × MF × MT), and thro’ _x_
we draw _x_R, _xr_, equal each to half the arc L_l_, and perpendicular
to CM; the conical surface generated by the line R_r_, while the figure
revolves on the axis of the sphere, will be equal to the surface of
the zone that is to be described in the same time by the arc L_l_; as
will easily appear by comparing that conical surface with the zone, as
measured by _Archimedes_.
[Illustration: _Philos. Trans. Vol. L._ TAB. XXI. _p. 554_.
_J. Mynde sc._]
And, lastly, If from the point _t_, in which _r_R produced meets the
axis, we take the angle C_t_V in proportion to the longitude of the
proposed map, as MF the sine of the middle latitude is to radius, and
draw the parallels and meridians as in the figure, the whole space
SOQV will be the proposed part of the conical surface expanded into a
plane; in which the places may now be inserted according to their known
longitudes and latitudes.