Philosophical transactions, Vol. L. Part II. For the year 1758.

[Illustration: _Philos. Trans. Vol. L._ TAB. XX. _p. 539_.

_J. Mynde sc._]

CASE I.

_When of three given parts two stand opposite to each other, and the third stands opposite to the part required._

THEOREM I.

_The sines of the sides are proportional to the sines of angles opposite to them._

DEMONSTRATION.

Let QR (TAB. XX. _Fig._ 1.) be the base of a spherical triangle; its sides PQ, PR, whose planes cut that of the base in the diameters QC_q_, RC_r_. And if, from the angle P, the line PL is perpendicular to the plane of the base, meeting it in L, all planes drawn through PL will be perpendicular to the same, by 18. _el._ 11. Let two such planes be perpendicular likewise to the semicircles of the sides, cutting them in the straight lines PG, PH; and the plane of the base in the lines LG, LH.

Then the plane of the triangle PGL being perpendicular to the two planes, whose intersection is QGC_q_, the angles PGQ LGQ will be right angles, by 19. _el._ 11. PG likewise subtends a right angle PLG, and the angle PGL measures the inclination of the semicircle QP_q_ to the plane of the base (_def._ 6. _el._ 11.) that is (by 16 _el._ 3. and 10 _el._ 11.) it is equal to the spherical angle PQR: whence PG is to PL as the radius to the sine of PQR. The same way PL is to PH as the sine of PRQ is to the radius: and therefore, _ex æquo_. PG the sine of the side PQ is to PH the sine of PR, as the sine of PRQ is to the sine of PQR.

CASES II. _and_ III.

_When the three parts are of the same name._

And,

_When two given parts include between them a given part of a different name, the part required standing opposite to this middle part._

THEOREM II.

_Let_ S _and_ s _be the sines of two sides of a spherical triangle_, d _the sine of half the difference of the same sides_, a _the sine of half the included angle_, b _the sine of half the base; and writing unity for the radius, we have_ Ssa² + d² - b² = 0; _in which_ a _or_ b _may be made the unknown quantity, as the case requires_.

DEMONSTRATION.

Let PQR (_Fig._ 2.) be a spherical triangle, whose sides are PQ PR, the angle included QPR, the base QR, PC the semiaxis of the sphere, in which the planes of the sides intersect.

To the pole P, draw the great circle AB, cutting the sides (produced, if needful) in M and N; and thro’ Q and R, the lesser circles Q_q_, _r_R, cutting off the arcs Q_r_ _q_R equal to the difference of the sides; join MN, Q_q_, _r_R, QR, _qr_.

Then the planes of the circles described being parallel (_Theod. sphæric._ 2. 2.), and the axis PC perpendicular to them (_10. 1. of the same_), their intersections with the planes of the sides, as QT, and R_t_, will make right angles with PC; that is, QT and R_t_ are the sines (S, _s_.) of the sides PQ PR, and MC NC are whole sines. Now the isosceles triangles MCN, QT_q_, _rt_R, being manifestly similar; as also MN, the subtense of the arc which measures the angle QPR, being equal to (2_a_) twice the sine of half that angle; we shall have MN: MC∷ Q_q_: QT∷ _r_R: R_t_; or, in the notation of the theorem, Q_q_ = 2S_a_, _r_R = 2_sa_. And further, the chords Q_r_ _q_R being equal, and equally distant from the center of the sphere, as also equally inclined to the axis PC, will, if produced, meet the axis produced, in one point Z. Whence the points Q, _q_, R, _r_, are in one plane (2. _el._ 11.), and in the circumference in which that plane cuts the surface of the sphere: the quadrilateral Q_q_R_r_ is also a segment of the isosceles triangle ZQ_q_, cut off by a line parallel to its base, making the diagonals QR, _qr_, equal. And therefore, by a known property of the circle, Q_q_ × _r_R + (_q_R)² = (QR)²; which, substituting for Q_q_ and R_r_ the values found above, 2_d_ for Q_r_, 2_b_ for QR, and taking the fourth part of the whole, becomes S_sa_² + _d_² = _b_² the proposition that was to be demonstrated.

_Note_ 1. If this, or the preceding, is applied to a plane triangle, the sines of the sides become the sides themselves; the triangle being conceived to lie in the surface of a sphere greater than any that can be assigned.

_Note_ 2. If the two sides are equal, _d_ vanishing, the operation is shorter: as it likewise is when one or both sides are quadrants.

_Note_ 3. By comparing this proposition with that of the Lord Neper[26], which makes the 39th of Keill’s Trigonometry, it appears, that if AC, AM, are two arcs, then sin. (AC + AM) ⁄ 2 × sin. (AC - AM) ⁄ 2 = ((_b_ + _d_) × (_b_ - _d_) =) (sin. ½ AC + sin. ½ AM) × (sin. ½ AC - sin. ½ AM). And in the solution of Case II. the first of these products will be the most readily computed.

CASE IV.

_When the part required stands opposite to a part, which is likewise unknown_: Having from the _data_ of Case I. found a fourth part, let the sines of the given sides be S, _s_; those of the given angles Σ, σ; and the sines of half the unknown parts _a_ and _b_; and we shall have, as before, S_sa_² + _d_² - _b_² = 0; and if the equation of the supplements be (Σσα² + δ²) - β² = 0; then, because α² = 1 - _b_² = 1 - (S_sa_² + _d_²), and β² = 1 - _a_², substituting these values in the second equation, we get

THEOREM III.

(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.