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as +





61 the fecant of that arch will be 1 + 2a +

+ 24

720 277 50521 8064

a'', &c. If therefore we multiply this fe

362889 ries by à the fluxion of the arch, we shall have a tina + 5 61 277

50521 at it a'à +

al å +

8064 720

a'o ,, &c. the

362880 fluxion of the sum of the secants, whose Auent or flowing


277 quantity a + į a3 +

at ai + al + 24

5040 50521

", &c. will be the sum of all the fecants contained in the arch a. If therefore we put e = to the length of the arch, which we intend for the integer of the meridional parts, and multiply the above Auent byė, 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 Mathematici

He is undoubtedly right; but we cannot help observing, that it would have been far more scientifical, to have demonstrate ed the truth of the latter, and consequently the erroneous principles of the former. He has, however, by comparing the methods of Mr. Wright and Mr. Weft together, thewn, that they both affert the same thing, and that the latter has derived his method of construction from the former, But what West calls a chart, Wright calls the geometrical lineaments only, by which he obtains a rcctilinear planisphere, and whence he demonstrates 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 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 forined 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 allo; but


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by this tangential projection, these will be defiected 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 difficulties 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 Transits of Mercury over the Sun, in the Years 1723, 1736, 1743, and 1753. By James Short, M. A. F.R.S.

Notwithstanding the most able Astronomers the world ever saw, have for near eighty years past, been constantly making observations in the royal Obervatories 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 Astronomers fuppofing it to be g' 20", and the French 9 10". But neither of these

are just; for, from comparing no less than fixty-three determinations of the difference of longitude, deduced from the transits of Mercury over the sun, it appears that it is 9 .6". Art. 31. Rules and Examples for limiting the Cafes in which the

Rays of refracled Light may be reunit:d into a colourless Pencil. In a Letier 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 elucidated. We shall therefore only observe, 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. An 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 shewing the phases of this eclipse, according to M. Meyer's tables, which make them very different from those resulting from the tables of Flamstead, Hailey, and de la Caille, Mr. Ferguson makes the following senlible remarks on the nature of cclipes 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 consist of 18 years, 11 days, 7 hours, 43 minutes, 20 seconds (as was observed by the antients) or 1388 years; and would for ever do so, if at


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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 distance at which the moon's fhadow can touch the earth, at the next period of 13 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 same 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 sun 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 shadow to pass over the disc of the earth, farther and farther on the opposite side 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 greatest 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 linits 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 eclipfe of the fun: and as these eclipses shift 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 earth; and this cannot be gone through in less than 12,492 years : for, as 36 is to 1,388, To 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 13, Old Stile, at 12 h. 52 min. 59 sec. D. m. wien it first touched the earth at the North Pole, according to the mean (or supposed equable) mo:ions of the fun and moon; their conjunction being then fy deg: 48 min. 27 sec. from the moon's ascending node, in the northern part of her ortit. In


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24 min.

each period since that time, the conjunction of the sun and
moon has been 28 min. 12 sec. nearer and mearer the same node,
and the moon's shadow has therefore gone more and more south-
erly over the earth. In the year 1962, July 8, Old Stile, at
10 h. 36 min. 21 sec. p.m, the same eclipse will have returned
thirty-eight times; and as the conjunction will then be only

45 lec. from the node, the center of the moon's tha-
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 pe-
riod, 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 conjunction
of the sun and moon will fall 28 min. 12 sec. farther and far-
ther back from the node, and the moon's shadow will go
ftill farther and farther fouthward on the earth, until A. Ď.
2665, September 12, Old Stile, at 23 h. 46 min. 22 sec. p.m.
when the eclipfe will have finished its feventy-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 fame eclipse would always return in eighteen Julian years,
eleven days, seven hours, forty-three minutes, twenty seconds,
when the last day of February in Leap-years is four times in-
cluded 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 feconds.

But on account of the various anomalies of the sun and
moon, arising from their moving in elliptic orbs, and the effects
of the sun's different attractions of the moon in different parts
of her orbit, the conjunctions of the sun and moon never fuc-
ceed one another at equal intervals of time; but differ some-
times 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 fubtile problems, solved in a very
elegant manner; together with an useful theorem, relating to
the areas of curvilinear figures.
Art. 47. Second Paper, containing the Parallax of the Sun, determined
on the Observations of the late Transit of Venus, in which this


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Subject is treated of more at length, and the quantity of the Parallax more fully ascertained. By James Short, A. M. and F. R.S.

In this paper Mr. Short observes, 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 observed there the transit of Venus; in which Memoir. Mr. Pingré endeavours to fhew, that the fun's parallax, from the observation of the late transit, was = 10", both by the observed durations, the least distance of the centers, and by the internal con-act at the egress; and seems to think, there must be some mistake in Mr. Mason's obfervation at the Cape of Good Hope, particularly with regard to the difference of longitude between Mr. Mason's observatory and Paris; because, by comparing the observation of Mr. Máfon at the Cape with the European observations, he finds the parallax of the fun to be between 8 and 9", and consequently different from the result of his own observations at Rodrigues compared with the same places. But Mr. Short has, in this paper, shewn, beyond all doubt, both from obfervations made on this fide the Equinoctial Line, and from Mr. Pingré's own obfervations properly connected, that the sun's parallax is between 8 and 9". In short, this elaborate paper contains the result of all the observations made on the late transit of Venus, and consequently the sun's parallax is here determined to as great a degree of accuracy, as those observations will admit of.

For, by taking the mean of a hundred and fixteen comparisons of the internal contacts observed at places to the north of the Line only, the sun's parallax is = 8,565,

From the mean of twenty-one comparisons of the internal contacts, with that at the Caps, the sun's parallax appears to be = 8,56,

The mean of twenty-one comparisons of the internal contacts with that at Rodrignès, gives the sun's parallax = 8,57,

The mean of the comparisons of the total durations, shew the fun's parallax to be = 8,61.

The mean of the apparent least distance of the centers, compared with that measured at Rodrigues, gives the sun's parallax = 8,56.

The mean of the apparent least distances of the centers, by computations from the total durations compared together, gives the sun's parallax = 8,53.

The mean of these six means, gives the sun's parallax = 8,556. And if we rejcct the mean arising from the comparisons of



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