for finding the height of the fall. He calls the breadth of the river in feet b; the mean velocity of the water in feet per fec. v; the breadth of the water-way between the obitacles c; and a the height of the fall in a fecond: from his principles 2 VV he finds 25 b 4 a с for the height of the fall. Now as this rule is to be general, whatever b and c may be, it is evident that when bc, this expreffion fhould become nothing, fince it expreffes the difference between the fall of the water, when the breadth of the river is reduced by the piers, and that of the stream, when it runs through its natural channel; 25 1: which is a plain contradiction. but in that cafe we get 21 2 If the ratio of 21 to 25 be neglected, the rule thus cor bb rected, viz. to IX " CC ขบ 4 a comes nearer to the truth than the Author's; as will appear from the example of Westminsterbridge, where b These values fubftituted into the expreffion above, gives.43 of an 994, c=820, v2, and a = 16.1:. inch, which Mr. Labellye obferved to be the real fall; whereas Mr. Robertfon's rule gives an inch and a tenth, and therefore above double to what it fhould be. The example given of the fall under London-bridge is, we conceive, out of the question; for the velocity found above bridge is occafioned by the water-way, as it is now contracted; whereas the velocity of the water before it is contracted fhould be known, according to the Author's rule: moreover it is poffible, that the fame velocity may be found above a cataract of any height; and therefore the velocity found above, when the bridge is built, cannot, we imagine, ferve to find the height of the fall under the bridge. That the example which the Author gives of London-bridge, agrees nearly with the real fall, is owing to two fuppofitions he makes, and which are not demonftrable; the one we have confidered already, and the other that the water-way is reduced from 236 feet to 196, by the piles drove round the piers, remains to be proved; as the ftarlings were never exactly measured, and befides these piles reach very little above low-water mark. Article 70. Trigonometry abridged, by Patrick Murdoch, A. M. The feveral branches of the mathematics have, within this century, been fo much cultivated and improved, that there I 4 is is fcarce any room left for new inventions; a fubject can hardly now be thought on, which Sir Ifaac Newton, his cotemporaries and followers, have not already treated of. What remains therefore for the prefent mathematicians, is to reduce their thoughts to a lefs compafs, and to render their demonftrations plainer and easier. In which application of their abilities, though an Author cannot display his genius in fo confpicuous a point of view, yet he may fhew his fagacity and judgment, by felecting and improving what is of real fervice to mankind, preferably to what is merely fpeculative; which valuable end this learned Author has chiefly in view, throughout all his performances. Moft Writers upon spherical trigonometry have reduced the cafes of right-angled triangles to fixteen, and those of oblique triangles to twelve: but they never conceived, that the rules of plain trigonometry were applicable to the fpheric! for want of which their trigonometrical works were extended to lengths which the subject by no means required; rendering that science very tedious, and much more difficult than it would have been, their principles been fewer, and no more than were neceffary. Those who are converfant with this fubject, will be furprized to fee that our Author has given all the cafes, both of plain and spheric trigonometry, in four quarto pages and a half; the whole being reduced to three theorems only, whofe demonftrations are very fhort and clear: especially if the Reader makes use of figures cut out of card-paper, fo as to raise fuch parts as fall above the plane, and are marked with the lines which are to be confidered. As the practical part of aftronomy chiefly depends on the computation of fpheric triangles, what the Author has given in this fhort paper, will be of the utmoft confequence, by leffening the labour of calculation, Article 73. Of the best form of Geographical Maps. By the fame Gentleman. In 1746, Mr. Murdoch publifhed a fmall octavo, entitled, The Elements of univerfal Perspective; wherein he has shewn, that all kind of projections may be reduced to one common principle, which he has illuftrated by feveral examples. He likewife published a thin quarto, containing tables of meridional parts, adapted to the true figure of the earth, and not to the fpheric form, as has been the cuftom: and from these tables, navigation and geography would have received confiderable improvements, had the menfurations in Peru been agreeable to what was expected from thofe made in France, and at the arctic circle. In In the prefent paper, he introduces a new construction of maps, by reprefenting a part of the globe upon a conic furface, flattened into a plane, which he conceives will reduce linear and fuperficial measures, nearer to that on the globe, than any other projections whatfoever; the reasons will be best understood by the Author's own words. When any portion of the earth's furface is projected on a plane, or transferred to it by whatever method of description, the real dimenfions, and very often the figure and pofition of countries, are much altered and mifreprefented. In the common projection of the two hemifpheres, the meridians and parallels of latitude do, indeed, interfect at right angles, as on the globe; but the linear diftances are every where diminished, excepting only at the extremity of the projection: at the center they are but half their juft quantity, and thence the fuperficial dimenfions but one fourth part: and in lefs general maps this inconveniency will always, in fome degree, attend the Stereographic projection. • The orthographic, by parallel lines, would be ftill less exact, thofe lines falling altogether oblique on the extreme parts of the hemifphere. It is useful, however, in describ❝ing the circum-polar regions: and the rules of both projections, for their elegance, as well as for their uses in aftronomy, ought to be retained, and carefully ftudied. As to 'Wright's or Mercator's nautical chart, it does not here fall ⚫ under our confideration: it is perfect in its kind.—’ After this the Author obferves, that the particular methods of projection propofed or ufed by geographers, are so various, that we might, on that very account, fufpect them to be faulty; and proceeding to fhew, upon what foundation his conftruction is to be made, he mentions the following properties. 1. The interfections of the meridians and parallels will be • rectangular. 2. The distances north and south will be exact; and any • meridian will ferve as a scale. 3. The parallels, where the line which generates the conic furface, interfects the quadrant, or any fmall distances of places that lie in thofe parallels, will be of their just quantity. At the extreme latitudes they will exceed, and in the mean latitudes, between the two foregoing interfections, they will fall fhort of it. But unless the zone is very broad, neither the excess nor the defect will be any where • confiderable. 4. The 6 4. The latitudes and the fuperficies of the map being exact, by the conftruction, it follows that the exceffes and defects of diftances now mentioned, compenfate each other; and are, in general, of the leaft quantity they can have in the map defigned. 5. If a thread is extended on a plane, and fixed to it at its two extremities, and afterwards the plane is formed into at pyramidal or conical furface, it may be cafily fhewn, that the thread will pafs through the fame points of the furface as before; and that converfely, the fhorteft diftance between two points in a conical furface is the right line which joins them, when that furface is expanded into a plane. Now, in the prefent cafe, the fhorteft diftances on the conical furface will be, if not equal, always nearly equal to the correfponding distances on the fphere: and therefore, all rectilinear ⚫ diftances on the map, applied to the meridian as a fcale, will nearly, at least, fhew the true diftances of the places re• prefented. < 6. In maps, whofe breadth exceeds not ten or fifteen degrees,' the rectilinear distances may be taken for fufficiently exact. But we have chofen our example of a greater breadth than can often be required, on purpofe to fhew how high the errors can ever arife; and how they may, if it is thought "needful, be nearly eftimated and corrected.' The Author fhews, that this construction may, without fenfible error, be applied to fea-charts; and gives feveral numerical examples, for that purpofe, which prove plainly his affertion. His reafonings are fo very obvious, that it would be needlefs to animadvert upon his conftructions: for which reason we shall close this article with a few obfervations. All the prefent maps have fcales for meafuring distances, though there is not the leaft proportion in them; which is extremely abfurd, and generally misleads people, who cannot conceive that these scales are abfolutely useless. The cafe is quite different in the paper before us: a table of corrections is propofed to be inferted in fuch maps as are very large, and which are the only ones that want them in regard to longitude; but in maps of ten or fifteen degrees, they need no corrections; and as thefe particular maps of provinces or ftates are the most ufeful with regard to the pofitions and diftances of particular places, which a common fcale of miles will fhew with as much accuracy as is neceflary. It has been obferved, that this method does not admit of a zone, containing N. and S. latitudes;-but why this objection? for if the north and fouth parts of the zone are either equal, equal, or nearly fo, the conic furface becomes, or may be made, cylindric; and when the difference is more confiderable, the center of the parallels will, it is true, be at a great distance, but yet not fo much as to become impracticable. Article 74. A short differtation on Maps and Charts. By William Mountaine, F. R. S. The author begins with fhewing, that the invention of globes, maps and charts, deferves a place among the feveral improvements, made in arts and sciences, by ingenious men: globes perhaps where firft invented, as bearing the nearest refemblance to the natural form of earth and fea; but as they contain but a small furface, maps and charts where afterwards thought of, as being more convenient for laying down the appearance, or face, of particular parts of the earth, and as being more portable for travellers. He then enters upon the defcription of the different kinds of maps, as they are divided into general and particular; in which it may be obferved, that as the difficulty naturally arofe, in reprefenting a part of a spheric furface upon a plane; different conftructions were invented, which for the moft part are so defective, as not to be applied with accuracy and facility, in determining the courfes, bearings, or distances of places. Among all the different reprefentations of a small part of the globe's furface, the rectilinear, which confiders that furface as a plane, muft have naturally occurred first to the geographers; and as the rhumbs were confequently right lines, the courses, or bearings of places could more easily be determined. It is for this reafon, that these kind of maps and charts, are ftill generally ufed to reprefent provinces and kingdoms, as likewife for fhort courfes in navigation; notwithstanding the many improvements fince suggested. The first step towards the improvement of maps, or charts, our author fays, was made by G. Mercator, who about the year 1550, published a map wherein the degrees of latitude were increased from the equator towards each pole; but upon what principles this was done, he did not explain. About the year 1590, Edward Wright, an Englishman, difcovered the true principles upon which fuch a chart fhould be co..ftructed; and in the year 1599, he exhibited his method of conftruction, in his Correction of errors in navigation; in the preface to which, may be feen how far Mercator has any right to share in the honour due to this great improvement in geography, and navigation. The |