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to blame. 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.
The only maps Mr. M. remembers 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 fessor Meyer, and a few more, several of which were drawn by Senex. In these prothe meridians are straight lines converging to a point; from which, as a centre, the parallels of latitude are described: and a rule has been published for the drawing of such maps, mentioned in the preface to the small Berlin Atlas. 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,
LlX MFX MT
Let ElLP, fig. 1, pl. 8, be the quadrant of a meridian of a given sphere, its centre c, and its pole P; EL, El, the latitudes of two places in that meridian, EM their middle latitude. Draw LN, In, cosines of the latitudes, the sine of the middle latitude MF, and its cotangent MT. Then writing unity for the radius, if in Cм we take cx = and through draw x, xr, equal each to half the arc Ll, and perpendicutar to cм; the conical surface generated by the line Rr, while the figure revolves on the axis of the sphere, will be equal to the surface of the zone described in the same time by the arc Ll; as will easily appear by comparing that conical surface with the zone, as measured by Archimedes. And lastly, if from the point t, in which Rr produced meets the axis, we take the angle ctv 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 soav 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.
This construction is illustrated by a calculation in an example, having the breadth of the zone 50° lying between 10° and 60° north latitude; its longitude 110°, from 20° east of the Canaries to the centre of the western hemisphere; comprehending the western parts of Europe and Africa, the more known parts of North America, and the ocean that separates it from the old continent. And then it is remarked that a map drawn by this rule will have the following properties: 1. The intersections of the meridians and parallels will be rectangular. 2. The distances
north and south will be exact; and any meridian will serve as a scale. 3. The parallels through z and y, where the line Rr cuts the arc Ll, or any small distances of places that lie in those parallels, will be of their just quantity. At the extreme latitudes they will exceed, and in mean latitudes, from x towards z or y, they will fall short of it. But unless the zone is very broad, neither the excess nor the defect will be any where considerable. 4. The latitudes and the superficies of the map being exact, by the construction it follows, that the excesses and defects of distance, now mentioned, compensate each other; and are, in general, of the least quantity they can have in the map designed. 5. If a thread be extended on a plane, and fixed to it at its two extremities, and afterwards the plane be formed into a pyramidal or conical surface, it may be easily shown, that the thread will pass through the same points of the surface as before: and that, conversely, the shortest distance between two points in a conical surface is the right line which joins them, when that surface is expanded into a plane. Now, in the present case, the shortest distances on the conical surface will be, if not equal, always nearly equal, to the correspondent distances on the sphere; and therefore all rectilinear distances on the map, applied to the meridian as a scale, will, nearly at least, show the true distances of the places represented. 6. In maps, whose breadth exceeds not 10° or 15°, the rectilinear distances may be taken for sufficiently exact. But the above example is chosen of a greater breadth than can often be required, on purpose to show how high the errors can ever arise; and how they may, if needful, be nearly estimated and corrected, as follows: Write down, in a vacant space at the bottom of the map, a table of the errors of equidistant parallels, as from 5° to 5° of the whole latitude; and having taken the mean errors, and diminished them in the ratio of radius to the sine of the mean inclination of the line of distance to the meridian, you shall find the correction required: remembering only to distinguish the distance into its parts that lie within and without the sphere, and taking the difference of the correspondent errors, in defect and in excess.
7. The errors on the parallels increasing fast towards the north, and the line sa having at last, nearly the same direction, it is not to be wondered that the errors in our example should amount to. Greater still would happen, if we measured the distance from o to a by a straight line joining those points; for that line on the conic surface, lying every where at a greater distance from the sphere than the points o and a, must plainly be a very improper measure of the distance of their correspondent points on the sphere. And therefore, to prevent all errors of that kind, and confine the other errors in this part of our map to narrower bounds, it will be best to terminate it towards the pole by a straight line KI touching the parallel oa in the middle point K, and on the east and west by lines, as нI, parallel to the meridian through к, and meeting the tangent at FF
the middle point of the parallel sv in H. By this means too we shall gain more space than we lose, while the map takes the usual rectangular form, and the spaces GHV remain for the title, and other inscriptions.
Another, and not the least considerable, property of our map is, that it may, without sensible error, be used as a sea-chart; the rumb lines on it being logarithmic spirals to their common pole t, as is partly represented in the figure: and the arithmetical solutions thence derived will be found as accurate as is necessary in the art of sailing.
If it be required to draw a map, in which the superficies of a given zone shall be equal to the zone of the sphere, while at the same time the projection from the centre is strictly geometrical; take cr to cм, as a geometrical mean between Cм and N, 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 cм 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 at which is inter cepted by the secants through L and I, 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.
LXXIV. A short Dissertation on Maps and Charts. By Mr. Wm. Mountaine, F. R. S. P. 563.
Maps and charts are either curvilinear 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 everywhere equal; the rumbs were consequently right lines; and hereby it was thought that the courses or bearings of places would be more easily détermined. 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 has its uses at this day in topographic constructions, as in bays, harbours, and very narrow zones. However, the errors in this 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 Gerard Mercator, who published a map about the year 1550, in which the degrees of latitude were increased from the equator towards each pole; but on what principles this was constructed, he did not show.
About the year 1590, Mr. Edward Wright discovered the true principles on which such a chart should be constructed; and communicated the same to one Jodocus Hondius, an engraver, who, contrary to his engagement, published the same as his own invention: this occasioned Mr. Wright, in 1599, to show his method of construction, in his book, intitled, Correction of Errors in Navigation; in the preface of which 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.
Blundeville, in his Exercises, p. 327, published anno 1594, gives a table of meridional parts answering to even degrees, from 1o to 80° of latitude, with the sketch of a chart constructed from it: but this table he acknowledges to have received from Mr. Wright.
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. Tho. Haselden, in a letter to Dr. Halley; who at the same time exhibited a new scale, by which 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 larger than they are on the globe. It is true, the surface is apparently enlarged; but the position of places, in respect to one another, are in nowise 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 thus represented unequal. Yet if a map or chart was so constructed, as to show 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 the preceding paper, shows the situation of places, and seems better calculated for determining superficial and linear measures than any other. He 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° difference of longitude, Mr. Murdoch computes the distance at 5594 miles; which, on 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 a 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. However, this method does not appear 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 admit of a zone containing both north and south latitude.
LXXV. On the Remarkable Effects of Blisters in Lessening the Quickness of the Pulse in Coughs, attended with Infarction of the Lungs and Fever. By Rob. Whytt, M. D., F. R. S. p. 569.
One of the most natural effects of blistering plasters, 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 show, that instead of increasing, they often remarkably lessen the frequency of the pulse. This Dr. W. had occasion formerly to take notice of, in his Physiological Essays, p. 69, and now evinces more fully by the following cases.
1. A widow lady, aged about 50, was seized (Dec. 1755) with a bad cough, oppression about the stomach and breast, and a pain in her right side, though 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, Dr. W. was called on Dec. 26, after she had been ill about 10 days, at which time her pulse beat from 96 to 100 times in a minute, but was not fuller than natural. He ordered her to lose 7 or 8 oz. more of blood, which, like the former, was sizy; and next day, finding no abatement of her complaints, he advised a blister