is scarce 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 eafier. 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.

Most Writers upon fpherical trigonometry have reduced the cafes of right-angled triangles to fixteen, and thofe 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 fubject by no means required; rendering that science very tedious, and much more difficult than it would have been, had their principles been fewer, and no more than were ncceffary.

Those who are converfant with this fubject, will be furprized to fee that our Author has given all the cafes, both of plain and fpheric trigonometry, in four quarto pages and a half; the whole being reduced to three theorems only, whofe demonftrations are very fhort and clear: efpecially if the Reader makes ufe 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 utmost confequence, by leffening the labour of calculation.

Article 73. Of the best form of Geographical Maps.

fame Gentleman.

By the

In 1746, Mr. Murdoch published a small 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 publifhed a thin quarto, containing tables of meri dional parts, adapted to the true figure of the earth, and not to the fpheric form, as has been the cuftom: and from thefe 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 the prefent paper, he introduces a new conftruction of maps, by reprefenting a part of the globe upon a conic furface, flattened into a plane, which he conceives will reduce linear and superficial measures, nearer to that on the globe, than any other projections whatfoever; the reafons will be beft 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 defcription, the real dimenfions, and very often the figure and po'fition of countries, are much altered and mifrepresented. 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 distances 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 still lefs exact, thofe lines falling altogether oblique on the extreme parts of the hemifphere. It is ufeful, however, in defcribing the circum-polar regions: and the rules of both projections, for their elegance, as well as for their ufes 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 used by geographers, are fo 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 diftances north and fouth will be exact; and any ' meridian will ferve as a fcale.

3. The parallels, where the line which generates the co'nic furface, interfects the quadrant, or any small distances of places that lie in thofe parallels, will be of their juft quantity. At the extreme latitudes they will exceed, and in the mean latitudes, between the two foregoing interfec'tions, they will fall fhort of it. But unless the zone is very broad, neither the excefs nor the defect will be any where confiderable,

4. The

6

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, compensate 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 a pyramidal or conical furface, it may be eafily 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 case, the shorteft diftances on the conical furface will be, if not equal, always nearly equal to the corres 'ponding distances on the sphere: and therefore, all rectilinear diftances on the map, applied to the meridian as a scale, will C nearly, at leaft, fhew the true diftances of the places reprefented.

6. In maps, whofe breadth exceeds not ten or fifteen degrees, the rectilinear diftances may be taken for fufficiently exact. But we have chosen our example of a greater breadth than < can often be required, on purpose to fhew how high the errors can ever arife; and how they may, if it is thought needful, be nearly estimated and corrected.'

The Author fhews, that this conftruction 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 reafon we fhall clofe this article with a few observations.

All the prefent maps have scales for measuring 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 ufelefs. 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 moft ufeful with regard to the positions and diftances of particular places, which a common fcale of miles will fhew with as much accuracy as is neceffary.

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 fhort 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 several improvements, made in arts and fciences, by ingenious men : globes perhaps where firft invented, as bearing the neareft refemblance to the natural form of earth and fea; but as they contain but a fmall 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 fpheric furface upon a plane; different conftructions were invented, which for the most part are fo defective, as not to be applied with accuracy and facility, in determining the courfes, bearings, or diftances of places.

Among all the different reprefentations of a fmall part of the globe's furface, the rectilinear, which confiders that furface as a plane, must have naturally occurred first to the geographers; and as the rhumbs were confequently right lines, the courfes, or bearings of places could more eafily be determined. It is for this reafon, that thefe 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 fuggefled.

The first step towards the improvement of maps, or charts, our author fays, was made by G. Mercator, who about the year 1550, publifhed 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 conftructed; 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 fhare in the honour due to this great improvement in geography, and navigation.

The

The author obferves, that fince Mr. Wright's map made its appearance, the globular and various other conftructions were invented, but none, according to his opinion, is so convenient as Mr. Wright's for fea charts, becaufe the meridians and parallels, as likewife the rhumb lines, being reprefented by right lines, are better adapted to the capacities of moft navigators notwithstanding, that the parts towards the polcs are extended much beyond what they are on the globe. However, he thinks that a map conftructed according to Mr. Murdoch's principles, fhews the fituation of places nearer, and is better calculated for determining fuperficial and linear measures, than the former, and that his courfes alfo agree nearly with the computations made from the table of meridional parts, but he does not think it fo eafy and fimple, in the practice of navigation. To this we may add, that navigators who have been ufcd to Mercator's chart from their youth, will never approve of any other, though much better in all refpects; their attachment to the old practice being fo ftrong, that no reafoning can prevail upon them.

Article 87. An account of diftilling Water fresh from Sea-water by Wood-afhes. By Capt. William Chapman.

The captain having by accident, loft the greatest part of his stock of water, was in fear of fcarcity if the voyage fhould prove long, having neither Still nor other conveniency provided, to make fea-water fresh, by the feveral methods publifhed by authors: neceffity, however, the mother of invention, made him try experiments, as far as his circumftances would allow ; whereby he fortunately contrived a method to fupply the want: and it appears to be the moft fimple and practicable, that could be wifhed. It will not be difagreeable to the curious, to fee the account in his own words, as it may be of use to others under the fame circumstances.

I was not a ftranger to Appleby's method; I had alfo a pamphlet wrote by Dr. Butler, intituled, An eafy method of procuring fresh water at fea; and I imagined, that foap might fupply the place of capital lees, mentioned by him. I now fet myfelf at work, to contrive a ftill; and ordered an old pitch pot, that held about ten quarts, to be made clean: my carpenter, by my direction, fitted to it a cover of fir deal, about two inches thick, very clofe; fo that it was eafily made tight by luting it with pafte.. We had a hole through the cover, in which was fixed a wooden pipe, nearly perpendicular. This I call the fill-head; it was bored with an augre of one and a half-inch diameter, to

within