the fecant of that arch will be 1 + 2a2 + 1⁄2 4a++ 5 24 720 277 8064 as + 50521 362880 a1, &c. If therefore we multiply this fe ries by a the fluxion of the arch, we shall have å + 1⁄2 a2 à + 2 aloa, &c. the 5 50521 24 · a1 à + a3 å + 8064 a3 à + 362880 a2 + 5040 72576 50521 a", &c. will be the fum of all the fecants contained 3991680 in the arch a. If therefore we pute to the length of the arch, which we intend for the integer of the meridional parts, and multiply the above fluent by, the product will give the meridional parts of the latitude propofed. Art. 19. A Defence of Mercator's Chart against the Cenfure 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 fhew, that the late Mr. Weft's objection is not well founded, and that the nautical planifphere, generally called Mercator's chart, is a true projection, from the teftimony of feveral eminent Mathematicians. He is undoubtedly right; but we cannot help obferving, that it would have been far more scientifical, to have demonftrated the truth of the latter, and confequently the erroneous principles of the former. He has, however, by comparing the methods of Mr. Wright and Mr. Weft together, fhewn, that they both affert the fame thing, and that the latter has derived his method of conftruction from the former. But what Weft calls a chart, Wright calls the geometrical lineaments only, by which he obtains a rectilinear planifphere, and whence he demonftrates the principles on which his table of meridional parts is founded. After vindicating Mercator's, or rather Wright's, fea-chart, Mr. Mountaine adds, I have carefully endeavoured not to mistake the true fenfe and meaning of Mr. Weft's propofition 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 fections of their respective parallels : I fhall only obferve, that if the meridians be right lines, and parallel to each other, the rhumbs must be right lines alfo; but by this tangential projection, thefe will be deflected from their true bearings, or make the angles of the courfes too great, unlefs fome expedient be devised to accommodate this error; and if the rhumbs be not right lines, fuch chart will then be embarraffed with more difficulties in practice than Mr. Wright's.' Art. 29. The Difference of Longitude between the Royal Obfervatories of Greenwich and Paris, determined by the Obfervations of the Tranfits of Mercury over the Sun, in the Years 1723, 1736, 1743, and 1753. By James Short, M. A. F. R. Ś. Notwithstanding the most able Aftronomers the world ever faw, have for near eighty years paft, been conftantly making obfervations in the royal Obfervatorics of Greenwich and Paris, yet it appears from Mr. Short's paper, that the difference of longitude between these two places, has never before been accurately determined; the English Aftronomers fuppofing it to be 9' 20", and the French 9' 10". But neither of thefe are just; for, from comparing no lefs than fixty-three determinations of the difference of longitude, deduced from the tranfits of Mercury over the fun, it appears that it is 9' 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, Efq; 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 elucidated. We fhall therefore only obferve, that this able Mathematician has performed the tafk he undertook, without introducing any new principles into the fcience of Optics, or any difperfion of the light different from the refractions discovered by Sir Ifaac Newton, near an hundred years ago. Art. 38. An Account of the Eclipfe of the Sun, April 1, 1764. In a Letter to the Right Hon. Geo. Earl of Macclesfield, Prefident of the Royal Society, from Mr. James Ferguson, F. R. S. After fhewing the phases of this eclipfe, 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. Fergufon makes the following fentible remarks on the nature of eclipfes in general. If the motions of the fun and moon, were equable, any given eclipfe would always return in a courfe of two hundred and twenty-three lunations, which would confift of 18 years, 11 days, 7 hours, 43 minutes, 20 feconds (as was obferved by the antients) or 1388 years; and would for ever do so, if at 2 the the end of each period, the fun and moon should be in conjunction either in the fame node, or at the fame diftance from it as before. But this is not the cafe: for if the fun and moon are once in conjunction at eighteen degrees diftance from the node, which is the greatest distance at which the moon's fhadow can touch the earth, at the next period of 18 years, 11 days, &c. the fun and moon will be 28 minutes, 12 feconds of a degree nearer the fame node than they were at the period last before. And fo by falling gradually nearer and nearer the fame node every time, the moon's thadow will país over the center of the earth's enlightened disk, at the end of the thirty-eighth periodical return of the eclipfe 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 fucceeding period the conjunctions of the fun and moon will be gradually farther and farther from the node, by the quantity of 28 minutes, 12 feconds of a degree; which will cause the moon's fhadow to pass over the difc of the earth, farther and farther on the oppofite fide from the center, till it quite leaves the earth, and travels in expanfion for above 12,492 years, before it can come upon the earth again at the fame pole as before. The reason of this will be plain when we confider, that 18 degrees from either of the nodes of the moon's orbit, is the greatest distance at which her shadow can touch the earth at either of its poles. And as there are 18 degrees on each fide of the node, within the limits of a folar eclipfe; and twice 18 make 36, thefe 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 fun and as thefe eclipfes fhift through 28 minutes 12 feconds 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 fhift through, at the rate of only 28 minutes 12 feconds of a degree in each period, before they can be near enough to the fame node again, for the moon's fhadow 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 eclipfe April 1, 1764, fell in the open space quite clear of the earth at each return, ever fince 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 fuppofed equable) motions of the fun and moon; their conjunction being then 17 deg. 48 min. 27 fec. from the moon's afcending node, in the northern part of her orbit. In Ee 4 cach each period fince that time, the conjunction of the fun and moon has been 28 min. 12 fec. nearer and nearer the fame node, and the moon's fhadow has therefore gone more and more foutherly over the earth. In the year 1962, July 8, Old Stile, at 10 h. 36 min. 21 fec. p.m. the fame eclipfe will have returned thirty-eight times; and as the conjunction will then be only 24 min. 45 fec. from the node, the center of the moon's fhadow will fall but a little to the northward of the center of the 'earth's enlightened difc. At the end of the next following period, the conjunction of the fun and moon will have receded back 3 min. 27 fec. from the moon's afcending node, into the fouthern part of her orbit; which will cause the center of her fhadow to pass a little matter fouth of the center of the earth's difc. After which, in every following period, the conjunction of the fun and moon will fall 28 min. 12 fec. farther and farther back from the node, and the moon's fhadow will go ftill farther and farther fouthward on the earth, until A. Ď. 2665, September 12, Old Stile, at 23 h. 46 min. 22 fec. p.m. when the eclipfe will have finifhed its feventy-feventh period, and will finally leave the earth at the South Pole; and cannot begin the fame courfe over the earth again in less than 12,492 years, as above-mentioned. And thus if the motions of the fun and moon were equable, the fame eclipfe would always return in eighteen Julian years, eleven days, feven hours, forty-three minutes, twenty feconds, 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 lefs, or eighteen years, ten days, feven hours, forty-three minutes, twenty feconds. But on account of the various anomalies of the fun and moon, arifing 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 fun and moon never fucceed one another at equal intervals of time; but differ fometimes no less than 14, 15, or 16 hours; and therefore, in or der to know the true times of the returns of any eclipfe, recourse must be had to long and tedious calculations. Art. 46. Problems. By Edw. Waring, M. A. and Lucafian Profeffor of Mathematics in the Univerfity of Cambridge, F. R. S, This paper contains two fubtile problems, folved in a very elegant manner; together with an ufeful theorem, relating to the areas of curvilinear figures. Art. 47. Second Paper, containing the Parallax of the Sun, determined from the Obfervations of the late Tranfit of Venus, in which this Subject Subject is treated of more at length, and the quantity of the Parallax more fully afcertained. By James Short, A. M. and F. R. S. In this paper Mr. Short obferves, that there is in the Memoirs of the Royal Academy at Paris, a Memoir by Mr. Pingré, who went to the island of Rodrigues, and obferved there the tranfit of Venus; in which Memoir Mr. Pingré endeavours to fhew, that the fun's parallax, from the obfervation of the late tranfit, was 10′′, both by the obferved durations, the least distance of the centers, and by the internal contact at the egrefs; and feems to think, there must be fome mistake in Mr. Mafon's observation at the Cape of Good Hope, particularly with regard to the difference of longitude between Mr. Mafon's obfervatory and Paris; becaufe, by comparing the obfervation of Mr. Mafon at the Cape with the European obfervations, he finds the parallax of the fun to be between 8 and 9", and confequently different from the refult of his own obfervations at Rodrigues compared with the fame places. But Mr. Short has, in this paper, fhewn, beyond all doubt, both from obfervations made on this fide the Equinoctial Line, and from Mr. Pingré's own obfervations properly connected, that the fun's parallax is be-tween 8 and 9". In fhort, this elaborate paper contains the refult of all the observations made on the late tranfit of Venus, and confequently the fun's parallax is here determined to as great a degree of accuracy, as thofe obfervations will admit of. For, by taking the mean of a hundred and fixteen comparisons of the internal contacts obferved at places to the north of the Line only, the fun's parallax is 8,565. From the mean of twenty-one comparisons of the internal contacts, with that at the Cape, the fun's parallax appears to be = 8,56. The mean of twenty-one comparifons of the internal contacts with that at Rodrignes, gives the fun's parallax = 8,57. The mean of the comparisons of the total durations, fhew the fun's parallax to be = 8,61. The mean of the apparent leaft diftance of the centers, compared with that measured at Rodrigues, gives the fun's parallax = 8,56. The mean of the apparent leaft diftances of the centers, by computations from the total durations compared together, gives the fun's parallax= 8,53. The mean of these fix means, gives the fun's parallax= 8,556. And if we reject the mean arifing from the comparisons of the |