Example 3.
Let the difference of longitude be 40°; but the latitudes 56° and 80°;
| And log. 2400´ + log. constant | = -1.4817217 | ||||
| Log. sin. 68° | = -1.9671659 | ||||
| S | = -1.4488876 | ||||
| T (tang. 22°) | = .4040262 | ||||
| m | = .2109980 | ||||
| Log. (T + m) | (= .6150242) | -1.7888921 | |||
| Log. (T - m) | (= .1830282) | -1.2625181 | |||
| Log. | 0.5263740 | = D | = -1.7212944 | ||
| S - D (= log. tangent 28° 6´) | = -1.7275932 | ||||
wanting of the true answer no more than 1° 4´.
And in all cases that can occur, the error of this rule will be inconsiderable.
It is not meant, however, that it ought to take place of the easier and better computation by a table of meridional parts: but it was thought proper to shew, by some examples, how safely the map itself may be depended on in the longest voyages; provided it is sufficiently large, and the necessary rumb-lines are exactly drawn[30].
LXXIV. A short Dissertation on Maps and Charts: In a Letter to the Rev. Thomas Birch, D. D. and Secret. R. S. By Mr. Wm. Mountaine, F.R.S.
London, March 21. 1758.
SIR,
Read April 6, 1758.
AMONG the several improvements made in arts and sciences by ingenious men, the construction of globes, maps, or charts, deserves a place: not only on account of the pleasure and satisfaction that arises to speculative minds, in surveying the extent and divisions of this terraqueous globe, but also for their real use and service to navigation, trade, and commerce.
Globes perhaps were first invented, as bearing the nearest semblance to the natural form of the earth and sea, with proper circles thereon described, and the several empires and kingdoms, according to their extent, latitudes, and longitudes, as far as geography and history would admit.
But tho’ these convey the most general and truest ideas of the position and situation of places; yet, as containing but a small surface, they were found not extensive enough to take in particular kingdoms or states, with their subdivisions, cities, and rivers, so as to convey an adequate and sufficient representation. Besides, they were not so portable and commodious in voyages or travels.
Maps and Charts were therefore thought of, as being most convenient for both the purposes above-mentioned; the accuracy of which depends on representing the meridians and parallels in such manner, that when places are laid thereon, according to their latitudes and longitudes, they may have such respect to each other, as they have on the globe itself; and those are either globular or rectilinear.
Globular, or curvilinear, are either general or particular.
General, are the hemispheres; for the most part constructed stereographically.
Particular, contain only some part of the terraqueous globe; and of this sort there are sundry modes of construction, which for the most part are defective, so as not to be applied with accuracy and facility to the purposes intended, in determining the courses or bearings of places, their distances, or both.
Rectilinear were therefore very early adopted, on which the meridians were described parallel to each other, and the degrees of latitude and longitude every-where equal; the rumbs were consequently right lines; and hereby it was thought, that the courses or bearings of places would be more easily determined.
But these were found also insufficient and erroneous, the meridians being parallel, which ought to converge; and no method or device used to accommodate that parallelism.
Notwithstanding the great deficiency in this plane map or chart, it was preferred, especially in nautical business; and hath its uses at this day in topographic constructions, as in bays, harbours, and very narrow zones.
However, the errors herein were sooner discovered than corrected, both by mathematicians and mariners, as by Martin Cortese, Petrus Nonius, Coigniet, and some say by Ptolemy himself.
The first step towards the improvement of this chart was made by Gerardus Mercator, who published a map about the year 1550, wherein the degrees of latitude were increased from the equator towards each pole; but upon what principles this was constructed, he did not exhibit.
About the year 1590, Mr. Edward Wright, an Englishman, discovered the true principles upon which such a chart should be constructed; and communicated the same to one Jodocus Hondius, an engraver, who, contrary to his honest faith and engagement, published the same as his own invention: This occasioned Mr. Wright, in the year 1599, to exhibit his method of construction, in his book, intitled, Correction of Errors in Navigation; in the preface of which book may be seen his charge and proof against Hondius; and also how far Mercator has any right to share in the honour due for this great improvement in geography and navigation.
Blundevill, in his Exercises, page 327, published anno 1594, gives a table of meridional parts answering to even degrees, from 1° to 80° of latitude, with the sketch of a chart constructed therefrom; but this table he acknowledged to have received from Mr. Wright, in the following words, page 326, viz. “In the mean time to reform the saide faults,” (in the plane chart) “Mercator hath in his universal chard or mappe made the spaces of the parallels of latitude to bee wider everie one than other from the equinoctial towards either of the poles, by what rule I know not, unless it be by such a table as my friend Maister Wright of Caius-college in Cambridge at my request sent me (I thank him) not long since for that purpose, which table with his consent, I have plainlie set down,” &c.
About the year 1720, a globular chart was published, said to be constructed by Mr. Henry Wilson; the errors in which were obviated by Mr. Thomas Haselden, in a letter to Dr. Halley; who at the same time exhibited a new scale, whereby distances on a given course may be measured, or laid off, at one extent of the compasses, on Wright’s projection; and was intended to render the same as easy in practice as the plane chart.
The above chart was published in opposition to Mr. Wright’s, which that author charged with imperfections and errors, and that it represented places bigger than they are upon the globe.
It is true, the surface is apparently enlarged; but the position of places, in respect to one another, are in no wise distorted; and it may be asserted, with the same parity of reason, that the lines of sines, tangents, and secants, are false, because the degrees of the circle, which are equal among themselves, are thereupon represented unequal.
Yet if a map or chart was so constructed, as to shew the situation and true extent of countries, &c. primâ facie (if I may be allowed the expression), and yet retain all the properties, uses, and simplicity, of Wright’s construction, it would be a truly great improvement; but this seems to be impossible.
The method exhibited by the Rev. Mr. Murdoch, in his paper, read before the Royal Society on the 9th of February last, shews the situation of places, and seems better calculated for determining superficial and linear measures, than any other that has occurred to me.
This Gentleman illustrates his theory with examples justly intended to point out the quantity of error, that will happen in a large extent.
For instance; Between latitudes 10° and 60° N. and containing 110 degrees difference of longitude, Mr. Murdoch computes the distance at 5594 miles; which, upon the arc of a great circle, is found to be 5477, or by other methods 5462; so that the difference is only 117, or at most 132 miles in so great an extent, and to an high latitude; and the higher the latitude the greater the error is like to be, where-ever middle latitude is concerned.
His courses also agree very nearly with computations made from the tables of meridional parts.
In example the first they are the very same:
In example the 2d they agree to half a minute:
In example the 3d they vary 1° 4´, on account of the high latitudes, which extend from 56° to 80° N.
However, I do not esteem this method so simple, easy, and concise, in the practice of navigation, as Mr. Wright’s construction, especially in determining the bearings or courses from place to place: nor will it (I presume) admit of a zone containing both north and south latitude.
Of these inconveniences Mr. Murdoch seems to be extremely well acquainted, when he expresses himself in the following very candid and ingenuous terms, viz. “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.”—And again, at the end of his nautical examples, he concludes thus, viz. “It is not meant, however, that it ought to take place of the easier and better computation by a table of meridional parts.”
I have the honour to be, with the greatest respect,
SIR,
The Royal Society’s, and
Your most obedient Servant,
William Mountaine.
Addenda to Mr. Murdoch’s Paper, Nº. LXXIII.
IF it is required “to draw a map, in which the superficies of a given zone shall be equal to the zone on the sphere, while at the same time the projection from the center is strictly geometrical;” Take Cx to CM as a geometrical mean between CM and Nn, is to the like mean between the cosine of the middle latitude, and twice the tangent of the semidifference of latitudes; and project on the conic surface generated by xt. But here the degrees of latitude towards the middle will fall short of their just quantity, and at the extremities exceed it: which hurts the eye. Artists may use either rule: or, in most cases, they need only make Cx to CM as the arc ML is to its tangent, and finish the map; either by a projection, or, as in the first method, by dividing that part of xt which is intercepted by the secants thro’ L and l, into equal degrees of latitude.
Mr. Mountaine justly observes, “that my rule does not admit of a zone containing N. and S. latitudes.” But the remedy is, to extend the lesser latitudes to an equality with the greater; that the cone may be changed into a cylinder, and the rumbs into straight lines.
LXXV. Cases of the remarkable Effects of Blisters in lessening the Quickness of the Pulse in Coughs, attended with Infarction of the Lungs and Fever: By Robert Whytt, M. D. F.R.S. Fellow of the Royal College of Physicians, and Professor of Medicine in the University of Edinburgh.
Read Feb. 16, 1758.
ONE of the most natural effects of blistering plaisters, when applied to the human body, is to quicken the pulse, and increase the force of the circulation. This effect they produce, not only by means of the pain and inflammation they raise in the parts to which they are applied, but also because the finer particles of the cantharides, which enter the blood, render it more apt to stimulate the heart and vascular system.
The apprehension, that blisters must in every case accelerate the motion of the blood, seems to have been the reason, why some eminent physicians have been unwilling to use them in feverish and inflammatory disorders, till after the force of the disease was a good deal abated, and the pulse beginning to sink. However, an attentive observation of the effects, which follow the application of blisters in those diseases, will shew, that instead of increasing, they often remarkably lessen the frequency of the pulse. This I had occasion formerly to take notice of[31], and shall now evince more fully by the following cases.
I. A widow lady, aged about 50, was seized (December 1755) with a bad cough, oppression about her stomach and breast, and a pain in her right side, tho’ not very acute. Her pulse being quick, and skin hot, some blood was taken away, which was a good deal sizy: attenuating and expectorating medicines were also prescribed. But as her complaints did not yield to these remedies, I was called on December 26th, after she had been ill about ten days; at which time her pulse beat from 96 to 100 times in a minute, but was not fuller than natural. I ordered her to lose seven or eight ounces more of blood, which, like the former, was sizy; and next day, finding no abatement of her complaints, I advised a blister to be applied, in the evening, to that part of her right side which was pained. Next morning, when the blister was removed, the pain of her side was gone, and her pulse beat only 88 times in a minute, and in two days more it came down to 78. However, after the blistered part became dry, the pulse rose in one day’s time to 96, and continued between that number and 90 for four days; after which I ordered a large blister to be put between her shoulders. When this plaister was taken off, her pulse beat under 90 times in a minute; and next day it fell to 76, and the day after to 72. The cough and other symptoms, which were relieved by the first blister, were quite cured by the second.
II. John Graham, bookbinder, in Edinburgh, aged 37, of a thin habit of body, formerly subject to coughs, and thought to be in danger of a phthisis pulmonalis, having exposed himself unwarily to cold in the night time, was, about the end of January 1756, seized with a bad cough and feverishness; for which he was blooded, and had a diaphoretic julep, a pectoral decoction, and a mixture with gum. ammoniacum and acetum scilliticum, given him by Mr. James Russell, surgeon-apothecary in this place. On the 12th of February, after he had been ill above a fortnight, I was desired to visit him. He seemed to be a good deal emaciated; his eyes were hollow, and cheeks fallen in: he was almost constantly in a sweat; coughed frequently, and spit up a great quantity of tough phlegm, somewhat resembling pus: his pulse beat from 112 to 116 times in a minute. In this condition I ordered immediately a blister to be applied between his shoulders, which lessened in some degree his cough and spitting, as well as the frequency of his pulse; but the blistered part no sooner began to heal, than he became as ill as before, and continued in this bad way nine or ten days, gradually wasting, with continued sweats, and a great spitting of a thick mucus. During this time he used tinctura rosarum, and the mixture with gum. ammon. and acet. scillit. without any sensible benefit, and had six ounces of blood taken away, which was very watery, and the crassamentum was of a lax texture. In this almost desperate condition, another blister, larger than the former, was put between his shoulders, which remarkably lessened his cough and spitting, and in two or three days reduced his pulse to 96 strokes in a minute. After this he continued to recover slowly, without the assistance of any other medicine, except the tinctura rosarum and the mixture with gum. ammon. and acet. scillit. and at present he enjoys good health.
III. Mrs.——, aged upwards of 40, who had for several years been subject to a cough and spitting in the winter months, was, in October 1756, seized with those complaints in a much greater degree than usual; to remove which, she was blooded, and got some attenuating and pectoral medicines from Mr. John Balfour, surgeon-apothecary in Leith. I was called on November 11th, after she had been ill several weeks, and found her in a very unpromising condition. She had a frequent and severe cough, with great shortness of breath and a wheezing; her lungs seemed to be quite stuffed with phlegm, of which she spit a vast quantity every day, and of such an appearance, that I was apprehensive it was, in part at least, truly purulent. When she sat up in a chair, her pulse beat above 130 times in a minute. She had a considerable thirst, and her tongue was of a deep red colour, with a beginning aphthous crust on some parts of it. She was so weak, and her pulse so feeble, that there was no place for further bleeding: a blister was therefore applied to her back, November 11th, which somewhat lowered her pulse, and lessened the shortness of breathing and quantity of phlegm in her lungs. November 16th, a second blister was laid to her side, which gave her still more sensible relief than the former, and reduced her pulse to 114 strokes in a minute. November 25th, a third blister was applied to her back; by which her cough and wheezing were rendered considerably easier, and the phlegm, which she spit up, lost its purulent appearance, became thinner, more frothy, and was much less in quantity. Her pulse beat now only 104 times in a minute. After this, her cough and spitting increasing again, she had, on the 20th of December, a fourth blister applied to her back, which, like the former, did her great service. Her stomach being extremely delicate, I scarce ordered any medicines for her all this time, except a cordial julep, with spir. volat. oleos. tincture of rhubarb as a laxative, and a julep of aqu. rosar. acet. vin. alb. and syr. balsam. of which last she took two table spoonfuls twice or thrice a day in a quarter of a pint of lintseed tea. After the fourth blister, she drank for some time a cupful of infusum amarum twice a day, and continued to recover slowly: and tho’ during the remaining part of the winter she was, as usually, a good deal troubled with a cough, yet in the spring she got free from it, and is now in her ordinary health.
IV. Christian Mcewen, aged 21, had laboured under a cough, thick spitting, pain of her breast, and pains in her sides affecting her breathing, for about a twelvemonth: and after getting, by proper remedies, in a good measure free from those complaints, her cough, from catching a fresh cold, increased to a greater degree than ever, became hard and dry, and was attended with a constant difficulty of breathing, pain in her left side, and head-ach. After having been seven or eight days in this condition, she was admitted into the Royal Infirmary, January 9th, 1757. As her pulse was small, tho’ very quick, viz. beating 130 in a minute, I thought it unnecessary to bleed her, as from former experience I did not doubt but that blistering alone would relieve her: I ordered, therefore, a large blister to be applied to her left side, where she complained of pain, and prescribed for her the following julep:
℞ Aqu. menth. simp. spirit. Minderer. ana ℥ iij. acet. scillit. ℥ i. sacchar. alb. ℥ ij. misce; cap. coch. ij. ter in die.
She was also desired to breathe frequently over the steam of hot water, and to drink lintseed tea.
January 10th. Her pulse beat only 112 times in a minute, and was somewhat fuller than on the 9th. The blister was not removed till late in the evening, and made a plentiful discharge. The cough having been so severe last night, as to keep her from sleep, I ordered her the following anodyne draught:
℞ Spirit. Minderer. ℥ ss. acet. scillit. ȝ i. syr. papav. alb. ȝ vi. misce; cap. hor. somni.
Jan. 11th. The cough easier last night; difficulty of breathing less; pulse 108 in a minute. Ordered the anodyne draught to be repeated, and the use of the julep, with acet. scillit. to be continued.
Jan. 12th. Pulse slower; cough and pain of the side easier; but still complains of a head-ach.
Jan. 13th. Pulse 94 in a minute; cough continues easier in the night, but is troublesome in the day-time.
Jan. 14th. Every way better; pulse only 80 in a minute. As her cough is still bound, ordered her, besides the medicines above-mentioned, a pectoral decoction of rad. alth. &c.
Jan. 15th. Cough and other complaints in a great measure removed; pulse 65 in a minute.
From this time her cough gave her little trouble; but on the 18th she complained of a pain in the epigastrium, with sickness at stomach, want of apetite, and a giddiness in her head, which were considerably relieved by a vomit, infusum amarum, and stomachic purges; and were almost wholly cured by the return of her menses on the 5th of February, after an interval of eight weeks.
V. A girl 21 months old, who had (December 1756) a great load of the small-pox, and not of a good kind, with a cough and obstructed breathing, was, on the seventh day from the eruption, blistered on the back; by which the pulse was lessened from 200 to 156 strokes in a minute. Next day her legs were also blistered, and the pulse thereby fell to 136. But the child’s lungs being much oppressed, and her throat being so full of pustules that she could scarce swallow any thing, she died towards the end of the ninth day.
I could add several other cases of the remarkable effects of blisters in lessening the quickness of the pulse in coughs attended with fever, pain in the side, and pituitous infarction of the lungs: but those above may be sufficient to put this matter out of doubt, as well as to remove any prejudice, that may still remain against the free use of so efficacious a remedy.
In a true peripneumony, especially where the inflammation is great, repeated bleeding is the principal remedy, and blisters early applied are not so proper. But when the peripneumony is of a mixed kind; when the lungs are not so much inflamed as loaded with a pituitous matter; when bleeding gives but little relief; when the pulse, tho’ quick, is small; when the patient is little able to bear evacuations, and the disease has continued for a considerable time; in all these cases blistering will produce remarkable good effects, and, far from increasing, will generally lessen the frequency of the pulse, and fever, more speedily than any other remedy.
On the other hand, when the fever and frequency of the pulse proceed from a true inflammation of the lungs, from large obstructions tending to suppuration, or from an open ulcer in them, blisters will be of less use, nay, sometimes will do harm, except in the last case, where they, as well as issues and setons, are often beneficial, tho’ seldom able to compleat a cure. But as in pituitous infarctions of the lungs, with cough and fever, repeated blisters applied to the back and sides are far preferable to issues or setons, so these last seem most proper in an open ulcer of the lungs. The former make a greater and more sudden derivation, and are therefore adapted to acute cases; the latter act more slowly, but for a much longer time, and are therefore best suited to chronic diseases. Further, while blisters evacuate chiefly the serous humours, issues and setons generally discharge true purulent matter, and on this account may be of greatest service in internal ulcers.
In what manner blisters may lessen the fever and frequency of the pulse attending internal inflammations, I have elsewhere endeavoured to explain[32]; and shall only add here, that in the cases above recited, where the quick pulse and feverishness proceeded more from a pituitous infarction than a true inflammation of the lungs, blisters, by relieving this organ, in some measure, of the load of humours oppressing it, would render the circulation through its vessels freer, and consequently lessen the quickness of the pulse, and other feverish symptoms.
It may not, however, be improper briefly to point out the reason, why blisters, which have been observed to be remarkably efficacious, even when early applied, in pleurisies[33], are less so in true peripneumonies. This difference, I imagine, may be accounted for from there being no immediate communication between the pulmonary vessels and those of the sides and back, to which the blisters are applied; whereas the pleura, and intercostal muscles, are furnished with blood-vessels from the intercostal arteries, which also supply the teguments of the thorax: so that while a greater flow of serous humours, and also indeed of red blood, is derived into the vessels of the external parts, to which the vesicatories are applied, the force of the fluids in the vessels of the inflamed pleura, or intercostal muscles, must be considerably lessened. Further, as the intercostal muscles and pleura are, as well as the teguments of the thorax, supplied with nerves from the true intercostals, blisters applied to the back and sides may perhaps, on this account also, have a greater effect in relieving inflammations there than in the lungs, which have nerves from the eighth pair, and from the intercostals improperly so called.
Edinburgh, May 23d, 1757.
Extract of a Letter from Dr. Whytt, Professor of Medicine in the University of Edinburgh, and F. R. S. to Dr. Pringle, F.R.S.
Edinburgh, 10 Nov. 1757.
WHAT you remark with regard to blisters being freely used by the physicians at London, in the cases mentioned in the paper I last sent you, is very just, and indeed what I knew; but altho’ their efficacy in such circumstances is now generally acknowleged both in England and Scotland, yet I do not remember that their remarkable quality in lessening the quickness of the pulse has been particularly attended to. This, therefore, I thought it might not be amiss to ascertain by a few careful observations.
I agree intirely with you, as to the use of blisters in fevers; being of opinion, that when there is no particular part obstructed or inflamed, they are of little service, and sometimes hurtful, unless perhaps towards the end, when the pulse begins to sink. Nay, in fevers, where the substance of the brain is affected, and not its membranes, I have never found any sensible benefit from blisters: and I always suspect the brain itself affected, when a fever and delirium come on without any preceding head-ach, or redness in the tunica albuginea of the eyes. This kind of fever I have met with several times, and have observed it to be generally fatal.
LXXVI. A remarkable Instance of Four rough Stones, that were discovered in an human urinary Bladder, contrary to the received Opinion; and successfully extracted by the lateral Method of Cutting for the Stone. By Mr. Joseph Warner, F. R. S. and Surgeon to Guy’s-Hospital.
Read Feb. 23, 1758.
THE favourable reception those few papers have met with from the Royal Society, which I have done myself the honour of addressing to them, encourages me to take the liberty of offering the following account to their consideration: and I am the more immediately induced to submit this paper to their perusal, as the fact hereafter related may possibly be not esteemed a matter of mere curiosity; since it is probable, that the inferences deduced from the history of the subsequent case, when attended to, may prove of the greatest consequence to the future ease and welfare of the patient, as well as be a means of preventing the operator from falling into such errors, as cannot fail of drawing an imputation upon his character, in the practice of one of the most capital and difficult undertakings in his profession.
It is a maxim laid down by the most judicious and best received writers upon operations in surgery, that when the surface of a stone, which has been extracted from the bladder, appears to be totally rough, it amounts to a proof of its having been there alone. But notwithstanding I admit it is from experience found, that the observation is in general well grounded, it may nevertheless appear, from the following case, that this rule is not unexceptionable: for which reason perhaps it may be thought right, that we should not be determined from circumstances only; but, on the contrary, that it is necessary for every surgeon to take such methods during the operation, as will enable him to judge with that degree of certainty, without which he cannot be enabled to do so.
The methods I would recommend are these: That after the extraction of a stone from the bladder, tho’ the whole of its surface be rough, the operator should nevertheless introduce the forefinger of his left or right hand thro’ the wound into the cavity of the bladder; by which means, if the subject be under twelve years of age, he will be enabled to come in contact with every internal part of the bladder with his finger: but if the subject be an adult, and of a corpulent habit of body, the finger, under these circumstances, not being found to be sufficiently long for the purpose, he must have recourse to a female catheter, or some other instrument that is nearly strait, quite smooth and polished, and of about nine or ten inches long; which will serve the purpose equally well, if of a proper form and thickness. This is the method I have made use of upon the like occasions of late years, without giving any great degree of pain to the patient, or considerably retarding the operation.
Since I have had the opportunity of making the following observation, as well as a prior observation something similar to this, where two rough stones were extracted by me a few years ago from a young man’s bladder of 15 years of age, I cannot help suspecting, that there may have been instances of one or more stones being left behind in the bladder at the time of operating, merely from the operator’s putting too great a confidence in this general rule. Which suspicion I am led into from having known people, who have undergone the operation of cutting for the stone, relapse into the like disorder in a short time after the healing of their wounds, attended with such symptoms, as have obliged them to submit to a second operation; when the stone, upon being extracted, has appeared of so considerable a size, as to make it suspicious, that this stone must probably have been of a much longer growth, than the short time betwixt the two operations could admit of. The maxim laid down to us by authors, of a smooth and polished stone in the bladder being never there alone, but always accompanied with one or more stones of the same kind, I know no exception to. But if this phænomenon should ever occur, the strict observance of that rule (delivered to us by judicious writers in surgery) of always searching the bladder under the like appearances, on presumption of one or more stones being left behind, cannot be attended with any future mischief to the patient, when carefully executed by the methods recommended above, and undoubtedly should always be strictly attended to. The smooth and polished appearances of the surfaces of human calculi are universally supposed to arise from their rubbing one against the other; which may with reason be supposed to be the case: but I confess this inference is not satisfactory to me; since it is probable, if this was the sole cause of their smoothness, the same effect would probably be always produced, when attended with the same degree of friction. But as this may be considered as a matter of mere speculation, I refer the decision of this point to those of superior abilities.