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makes the 39th of Keill's Trigonometry, it appears, that if AC, AM, be two arcs, then sin. (AC + AM) × Sin. (AC — AM) = (b + d) × (b − d) = (sin. ac + sin. AM) X (sin. AC1÷3 sin. AM. And in the solution of Case 2, the first

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of these products will be the most readily computed.

Case 4. When the part required stands opposite to a part which is also unknown having from the data of Case 1 found a 4th 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, ssa2 + ď2 — b20; and if the equation of the supplements be ca2 + ♪2 — ß2 = 0; then, because a2 = 1 — f = 1 (ssa2 + d2), and g2 = 1 — a2, substituting these values in the 2d equation, we get

Theorem 3.

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a2; 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 4 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 deaf of the trigonometrical tables.

These theorems Mr. Murdoch demonstrates.

LXXI. Two extraordinary Cases of Gall-stones. By James Johnstone, M.D.* of Kidderminster. p. 543.

The calculus voided in the 1st of these cases was of a pyriform shape, resembling the form of the gall-bladder. It was smooth and polished, except at

* For the following particulars respecting the author of the above paper, we are indebted to Dr. John Johnstone, an eminent physician at Birmingham, one of Dr. James Johnstone's sons. We regret that the limited space allotted to biographical notices would not admit of our inserting more than an abstract from Dr. John Johnstone's more extended and highly interesting account of the life and writings of his late father.

James Johnstone, M.D. who practised physic for half a century with great reputation at Worcester, and distinguished himself by several ingenious works, was the 4th son of J. J. Esq. of Galabank, an ancient branch of the Johnstones of Johnstone., He was born at Annan, April 14, 1730, and in that town imbibed the rudiments of his scholastic education, under the Rev. Dr. Robert Henry, afterwards celebrated for his History of Great Britain. He studied physic at Edinburgh under the elder Monro, and the Professors Whytt, Rutherford, St. Clair, and Plummer, and there was admitted doctor in

the base. In length it measured 1 of an inch. It floated on water, and weighed about 126 grs. In the other case a number of small biliary calculi were

that faculty in June 1750, publishing an inaugural dissertation" De Aeris Factitii Imperio, in Primis Corporis Humani Viis."

His family thinking him too young, being then but in his 20th year, to settle as a practical physician immediately after taking his degree, he went to Paris, and there perfected himself in anatomy and chemistry under Ferrein and Rouelle.

In 1751 Dr. Johnstone seated himself at Kidderminster; and here he was assisted by the good offices of Dr. M'Kenzie, who had retired from practice at Worcester, and by the advice of his former masters Dr. St. Clair and Dr. Whytt. With both these eminent physicians he kept up a regular cor ́respondence, and from both he received the warmest tokens of friendship. At Kidderminster he soon became popular; even the first year of his practice, at the age of 21, he got near £100, and never afterwards had any occasion to apply to his father for money. It was here that he met with the case of gall-stones, inserted in the present vol. of these Trans.

From its low situation, and from the population being too much crowded in small habitations, malignant fevers and sore throats were often prevalent at Kidderminster, and had proved remarkably fatal; and Dr. Johnstone owed much of his reputation to his success in curing them. He gave mineral acids, Peruvian bark, and dulcified acids. He forbad bleeding. He discovered the power of mineral acids in a gaseous state to destroy putrid miasmata, and on his observations and experience he founded the method of cure of putrid fevers, which he published in 1758 in a book entitled "An historical Dissertation concerning the malignant Epidemical Fever of 1756, &c." London, printed 1758. On the discovery of the power of mineral acids in a state of gas to destroy contagions, Dr. John Johnstone has written a pamphlet* asserting and proving his father's prior claim to the discovery. This pamphlet was answered by Dr. Smyth, in his letter to Mr. Wilberforce. The reply of Dr. John Johnstone to Dr. James Carmichael Smyth, has very fairly stated and settled the point in dispute. For if the question is to be settled by dates, it is proved that Dr. James Johnstone, sen. promulgated his use of muriatic acid gas in 1758. That Guyton Morveau recommended and used muriatic acid gas for purifying the cathedral at Dijon in 1773. That Dr. James Johnstone, jun, in his Treatise on Malignant Angina, published at Worcester in 1779, recommended muriatic acid gas as the surest method to prevent the spread of contagion, and that Dr. James Carmichael Smyth professes not to have discovered the virtues of nitric acid gas till 1780, and did not publish what he professes to have discovered till 1795.

At Kidderminster, in a very early part of his practice, began Dr. Johnstone's acquaintance with the first lord Lyttelton, with whom he continued on terms of uninterrupted friendship until the death of that learned and pious nobleman. The account he has given of his last illness, under the denomination of "Case of George Lord Lyttelton" is an example of the vigour and purity of his style, and does equal honour to his head and to his heart.

In the 54th vol. of the Philosophical Transactions Dr. Johnstone published the first sketch of his opinions on the ganglions of the nerves, a subject which he afterwards pursued in the 57th and 60th vols. The publication of these papers procured their author the notice and friendship of the illustrious Haller; with whom Dr. Johnstone's correspondence began in 1764, and continued till 1775. It consists chiefly of physiological and critical observations on the doctrine of ganglions, to which he very candidly offers objections and admits of reply. In one letter, dated 25 May 1769, he says, after some prefatory observations on Dr. Johnstone's doctrine-" For any thing I know, there is but one objection (the ophthalmic ganglion), which lies entirely between nerves dedicated to * Account of the Discovery of the Power of Mineral Acids in a State of Gas to destroy Contagion. London, 1803.

voided by a woman who had long laboured under the jaundice, from a small sore at the pit of the stomach, at the part where she had felt a pain and hardness about a quarter of a year before. For a more particular account of these cases the reader is referred to this author's collected works, entitled Medical Essays and Observations.

voluntary motion. I shall look for some opportunity of showing you my just regard, and am always yours, Haller." This objection will be found very satisfactorily answered, p. 17, 5, Medical Essays and Observations. These papers were collected and enlarged, and published at Salop in 1771, under the title of Essay on the Ganglions of the Nerves. They were again published in 1795, with many valuable physiological and pathological additions, and with several other separate tracts, in one volume, under the title of "Medical Essays and Observations, with Disquisitions relating to the Nervous System."

The vol. of Medical Essays printed in 1795, contains all the tracts published by Dr. Johnstone, except his inaugural Dissertation, his Treatise on the Fever of 1756, the Life of Dr. Gregory in the Manchester Memoirs, the Life of his dearest Friend and Companion, the Rev. Dr. Job Orton, published under the article Doddridge, in the Biographia Britannica, and 2 papers in the Memoirs of the Medical Society of London on the Angina and Scarlet Fever of 1778, and on the Diseases of Needlemakers. He published besides separately a Dialogue against that most infamous of all traffics, the slave trade, and an Analysis of Walton-water near Tewkesbury, which he proved to be nearly the same in quality as the purging waters of Cheltenham. At the end of this analysis he again displayed the strong inventive tendency of his intellectual powers, by assigning the uses of the lymphatic glands.

At Kidderminster Dr. Johnstone continued in a very wide sphere of practice till August 1783, when he removed to Worcester, in consequence of the death of his eldest son Dr. James Johnstone, who was carried off by a pestilential fever caught in attending prisoners lying ill of the fever in the jail at Worcester. (See Howard on Lazarettos, p. 122.) Here Dr. J. continued to practise with an activity unabated by age and infirmities until the time of his death; which happened on the 28th of April 1802.

"In the catalogue of his private friends, he had the honour to recount the names of men eminently and deservedly distinguished for elegant taste, for classical erudition, and for deep researches into all the various branches of philosophy, which are intimately or remotely connected with the study of


"As a practical physician, Dr. Johnstone was active and humane; quick and sagacious in observing and deciding, he conformed exactly to the rule laid down by Hippocrates, and thus luminously expressed by Celsus, mederi oportere, et communia et propria intuentem. His knowledge was at once comprehensive and accurate. His application both to books and to professional duties was intense, his memory was retentive, his penetration was keen, his judgment was correct, and his elocution was ready, copious, and energetic.

"In his moral carriage, he was firm and undeviating; but his vigorous and manly mind was perhaps on some occasions too little accommodating to characters and to circumstances. In his temper he was cheerful, though sometimes hasty; in his conversation lively and instructive; in his affections warm and attached. In his domestic relations he was the best of fathers; his whole life was a sacrifice to the advantage of his children. In fine, as a public or a private man, his character has not often been surpassed; and although the memory of his personal services cannot be soon forgotten, yet has he erected a still more durable monument to his fame, in those various practical improvements of the medical art, which rank his name among the benefactors of mankind."

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LXXII. A Remarkable Case of Cohesions of all the Intestines, &c. in a Man of about 34 Years of Age, who died in the Summer 1757, and afterwards fell under the Inspection of Mr. Nicholas Jenty. p. 550.

The subject was tall, and partly emaciated. Mr. J. found nothing externally but a wound in the left side, which seemed to have been degenerated into an ulcer. As he did not know the man when he was alive, and had him 2 days after his decease, he could not give an immediate account of the cause of his death. But in opening his abdomen he found the epiploon adhering close to the intestines, in such a manner that he could not part it without tearing it. It felt rough and dry. And as he was going to remove the intestines, to examine the mesentery, he found them so coherent with one another, that it was impossible to divide them without laceration. He then inflated the intestinal tube, for the inspection of this extraordinary phenomenon ; but to his 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 number of pimples, as large as the head of a pin, appeared in them, and were almost free from any moisture. It was proper to observe, that these pimples had been taken for glands by the late Dr. J. Douglas and others; whereas they were in reality nothing else but the orifices of the exhaling vessels obstructed, and not to be met with except in morbid cases.

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After having made incisions in that part of the colon next to the rectum, he found the peritoneum, or external membrane which invests the intestines, and the viscera of the abdomen, to be of the thickness of a sixpence; and he 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. He found a fluid in the intestines; and he would not take upon him to say, how the peristaltic motion must have been performed. And afterwards he parted the stomach from its external tunic, as he had done the intestines. He 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 vesiculà fellis he found the bile pretty thick, neither too green nor too yellow, but a tint between both. He met with nothing remarkable in the other parts of the abdomen. In opening the thorax he 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, he found it so closely adhering to the heart, that he could not avoid wounding that organ, and with much difficulty could part it from it. He 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 had been described by some anatomists, who had met with cases somewhat similar to this, but without such universal adhesions; and they had been supposed to have been glands. The same pores likewise appeared on the heart; which in his opinion were nothing but the extremities of the exhaling vessels. In removing the heart he 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 through 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, he found a cystis, as large as an olive, full of pus; and lower down, immediately before that vessel perforates the diaphragm, he found another something less, full of matter likewise; both which portions he had by him. That portion of the aorta, where the cistis appeared, was rather thicker than the other, and osseous. In opening the cranium, he found in that part of the cerebrum, which lay over the cerebellum, a table spoonful of pus, of a greenish colour; and examining it through a glass, there was an appearance of animalcula in it.

LXXIII. On the best form of Geographical Maps. By the Rev. Patrick Murdoch, M.A., F.R.S. p. 553.

When any portion of the earth's surface is projected on a plane, or transferred to it by any method of description, the real dimensions, and often the figure and position of countries, are altered and misrepresented. In the common projection of the two hemispheres, the meridians and parallels of latitude 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 centre they are but half their just quantity, and thence the superficial dimensions but a 4th 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

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