it having been lately insinuated, that a paper of this kind was printed in the Traniactions, Mr. Dunn also observes, that Mr. Weft has censured Mr. Wright's projection as erroneous, and given another, in which the meridian line is a scale of natural tangents from the Equinoctial to the Pole; whereas those of Mr. Wright are a scale of secants. He adds, “That both Wright and West say expressly, the Sphere being inscribed in the hollow cylinder, and the Equinoctial being fixed without swelling, while the other parts swell towards the Poles, the chart will be formed. But in this, Wright has badly expressed his own thoughts ; for his tables make it, that the Equinoctial must either swell or contract itself.”. And that Mr. Weft has therefore taken his words, but not his fense. The proposed demonstration of this tangental property, at page 58 of Mr. West's book, is no demonftration at all; there is nothing more plain, than that in order to have the Meridians at equal distances, the degrees of latitude must be enlarged to the same proportion in every part, as the circular Meridians are nearer towards the Poles, which proportion is as the Co-fine of the latitude to Radius.' This assertion is undoubtedly true; and Mr. Dunn might have added, that as in Mr. Wright's projection, the degrees of longitude are all equal, that is, the Co-line of the latitude is every where equal to the Radius, it will follow, that the degrees of latieude must be enlarged in the proportion of the Radius to the secant: for as the Co-fine of any parallel of latitude is to Radius, so is Radius to the Secant of that parallel. But if the degrees of latitude increase in the proportion of the Radius to the Secant, it follows, that the distance of any parallel of latitude from the Equator, will be equal to the sum of the Secants of all the arches contained between the Equator and that parallel ; and consequently, that the meridional line in a true sea chart, where the degrees of longitude are all equal, is nothing more than a scale formed by the addition of the natural Secants, supposed to flow with an uniform and uninterrupted motion. Hence we see the reason why the common tables of meridional parts, which are formed by the continual addition of the tabular Secants, are not strictly true; being increments of latitude formed from tables calculated to minutes only, inftead of the Secants flowing with an equal velocity. It is well known, that if a te made equal to the length of any arch, wliose radius is unity, the 24 50521 5 61 the fecant of that arch will be 1 + 2a? to at + as + 720 277 50521 as + 8064 al', &c. If therefore we multiply this se 362880 ries by a the Auxion of the arch, we shall have à + } az à + 5 61 277 50521 at à + aå as åt ato ,, &c. the 24 720 362880 Auxion of the sum of the secants, whose Auent or flowing 61 277 quantity a't i a3 + ast al + 24 5040 72576 a', &c. will be the sum of all the fecants contained 3991680 in the arch a. If therefore we put é = to the length of the arch, which we intend for the integer of the meridional parts, and multiply the above Auent być, the product will give the meridional parts of the latitude proposed. Art. 19. A Defence of Mercator's Chart against the Censure of the late Mr. Weft of Exeter. In a Letter to Charles Morton, M. D. Secretary to the Royal Society, from Wm. Mountaine, F.R.S. This ingenious Gentleman has endeavoured to thew, that the late Mr. West's objection is not well founded, and that the nautical planisphere, generally called Mercator's chart, is a true projection, from the testimony of several eminent Mathematicians. He is undoubtedly right; but we cannot help observing, that it would have been far more scientificàl, to have demonstrate ed the truth of the latter, and consequently the erroneous principles of the fo:mer. He has, however, by comparing the methods of Mr. Wright and Mr. Weft together, shewn, that they both assert the same thing, and that the latter has derived his method of construction from the former. But what Weft calls a chart, Wright calls the geometrical lineaments only, by which he obtains a rectilinear planisphere, and whence he demonstrates the principles on which his table of meridional parts is founded. After vindicating Mercator's, or rather Wright's, sea-chart, Mr. Mountaine adds, I have carefully endeavoured not to mistake the true sense and meaning of Mr. West's proposition in any part thereof; if I have not, I cannot pronounce what kind of chart may be formed from his tangent line being made the line of latitudes, or that meridian line whereupon the tangents are to determine the sections of their respective parallels : I shall only observe, that if the meridians be right lines, and parallel to each other, the rhumbs must be right lines also; but by Ec 3 by this tangential projection, these will be deflected from their true bearings, or make the angles of the courses too great, unless some expedient be devised to accommodate this error; and if the rhumbs be not right lines, such chart will then be embarrassed with more difficulttts in practice than Mr. Wright's.' Art. 29. The Difference of Longitude between the Royal Observatories of Greenwich and Paris, determined by the Observations of the Tranfits of Mercury over the Sun, in the lears 1723, 1736, 1743, and 1753. By James Short, M. A. F. R. Ś. Notwithstanding the most able Astronomers the world ever faw, have for 'near eighty years past, been constantly making observations in the royal Observatorics of Greenwich and Paris, yet it appears from Mr. Short's paper, that the difference of longitude between these two piaces, has never before been accurately determined; the Englihh Aftronomers fuppofing it to be 9 20", and the French 9 10". But neither of these are juft; for, from comparing no less than fixty-three determina. tions of the difference of longitude, deduced from the transits of Mercury over the sun, it appears that it is g' 16". Art. 31. Rules and Examples for limiting the Cafes in which the Rays of refracted Light may be reunited into a colourless Pencil. In a Letter from P. Murdoch, M. A. and F. R. S. to Robert Symmer, Esq; F.R.S. This is a very curious and useful paper ; but will not admit of any abridgment, without giving the figures with which it is clucidated. We shall therefore only obterve, that this able Mathematician has performed the task he undertook, without introducing any new principles into the science of Optics, or any dispersion of the light different from the refractions discovered by Sir Isaac Newton, near an hundred years ago. Art. 38. Ar Account of the Eclipse of the Sun, April 1, 1764. In a Letter to the Right Hon. Geo. Earl of Macclesfield, President of the Royal Society, from Mr. James Ferguson, F.R.S. After fhewing the phases of this eclipse, according to M. Meyer's tables, which make them very different from those refulting from the tables of Flamstead, Halley, and de la Caille, Mr. Ferguson makes the following fenfible remarks on the nature of eclipses in general. « If the motions of the fun and moon, were equable, any given eclipse would always return in a course of two hundred and twenty-three lunations, which would consift of 18 years, 11 days, 7 hours, 43 minutes, 20 seconds (as was observed by the alitients) or 5388 years; and would for ever do so, if at the 2 the end of each period, the sun and moon should be in conjunction either in the same node, or at the same distance from it as before. But this is not the case: for if the sun and moon are once in conjunction at eighteen degrees distance from the node, which is the greatest dittance at which the moon's shadow.can touch the earih, at the next period of 18 years, 11 days, &c. the sun and moon will be 28 minutes, 12 seconds of a degree nearer the same node than they were at the period last before. And so by falling gradually nearer and nearer the fame node every time, the moon's ihadow will pass over the center of the earth's enlightened disk, at the end of the thirty-eighth periodical return of the eclipse from the time of its firit coming in at either of the earth's poles; because the conjunction falls in the node at the end of the thirty-eighth period. • In each succeeding period the conjunctions of the fun and moon will be gradually farther and farther from the node, by the quantity of 28 minutes, 12 seconds of a degree; which will cause the moon's fhadow to pass over the disc of the earth, farther and farther on the opposite fide from the center, till it quite leaves the earth, and travels in expansion for above 12,492 years, before it can come upon the earth again at the same pole as before. « The reason of this will be plain when we consider, that 18 degrees from either of the nodes of the moon's orbit, is the greateit distance at which her shadow can touch the earth at either of its poles. And as there are 18 degrees on each side of the node, within the limits of a solar eclipse; and twice 18 make 36, these are all of the 360 degrees of the moon's orbit about either of the nodes, within which there can be an eclipse of the sum: and as these eclipses Mitt through 28. minutes 12 seconds of these 36 degrees, in every Chaldean or Plinian period, they will shift through the whole limit in 77 periods, which include 1388 years and three months. And then the periods have the remaining 324 degrees of the moon's orbit to shift through, at the rate of only 28 minutes 12 seconds of a degree in each period, before they can be near enough to the fame node again, for the moon's shadow to touch the carth ; and this cannot be gone through in less than 12,492 years : for, as 36 is to 1,388, fo is 324 to 12,492. • The eclipse April 1, 1764, fell in the open space quite clear of the earth at each return, ever since the creation till A. D. 1295, June 3, Old Stile, at 12 h. 52 min. 59 fec. p. m. when it first touched the earth at the North Pole, according to the mean (or supposed equable) mo:ions of the sun and moon; their conjunction being then 17 deg. 48 min. 27 sec. from the moon's alcending node, in the northern part of her orbit. In cach Ee4 min. 45 each period since that time, the conjunction of the sun and moon has been 28 inin. 12 sec. nearer and nearer the same node, and the moon's shadow has therefore gone more and more southerly over the earth. In the year 1962, July 8, Old Stile, at To h. 36 min. 21 sec. p.m. the same eclipse will have returned thirty-eight times; and as the conjunction will then be only 24 sec. from the node, the center of the moon's Thaa dow will fall but a little to the northward of the center of the earth's enlightened disc. At the end of the next following period, the conjunction of the sun and moon will have receded back 3 min. 27 sec. from the moon's ascending node, into the southern part of her orbit; which will cause the center of her shadow to pass a little matter south of the center of the earth's disc. After which, in every following period, the conjundion of the sun and moon will fall 28 min. 12 sec. farther and farther back from the node, and the moon's shadow will go ftill farther and farther fouthward on the earth, until A. D. 2665, September 12, Old Stile, at 23 h. 46 min. 22 sec. p.m, when the eclipse will have finished its seventy-seventh period, and will finally leave the earth at the South Pole; and cannot begin the same course over the earth again in less than 12,492 years, as above-mentioned. « And thus if the motions of the sun and moon were equable, the same eclipse would always return in eighteen Julian years, eleven days, leven hours, forty-three minutes, twenty seconds, when the last day of February in Leap-years is four times included in this period : but when it is five times included, the period is one day less, or eighteen years, ten days, seven hours, forty-three minutes, twenty seconds. • But on account of the various anomalies of the sun and moon, arising from their moving in elliptic orbs, and the effects of the fun's different attractions of the moon in different parts of her orbit, the conjunctions of the sun and moon never succeed one another at equal intervals of time; but differ sometimes no less than 14, 15, or 16 hours; and therefore, in or, der to know the true times of the returns of any eclipse, recourse must be had to long and tedious calculations. Art. 46. Problems. By Edw. Waring, M. A. and Lucasian Pro feffor of Mathematics in the University of Cambridge, F. R. S, This paper contains two subtile problems, solved in a very elegant manner; together with an useful theorem, relating ta the areas of curvilinear figures. Art. 47. Second Paper, containing the Parallax of the Sun, determired from the Obfervations of the late Transit of Venus, in which this Subject |