as that of falling bodies is actually found to be.This law of gravitation Mr. Murdoch affumes as given, and makes the moon's diftance the quantity fought.

Thus writing F for the number of feet which a body falling from reft, describes, in vacuo, at the equator in one fecond, V for the verfed fine of the arc of the moon's orbit defcribed in the fame time, to the radius unity, D for the femidiameter of the equator in feet, and the ratio of the diftance of the centres of the earth and moon, to the femidiameter of the earth, that of X to 1: we have, by the general law, the moon's fall in

F

I fecond, equal to ; but the fame fall is equal to VXDXX;

whence X3

F
VXD'

in femidiameters of the equator 60.08906. And the arithmetical complement of the logarithm of the above number is the log. tang. of the moon's mean horizontal parallax at the equator, which therefore is 57'. 12", 34.

But as this distance will be fomewhat increased by the revolution of the earth and moon round their common center of gravity, this able mathematician finds, by a very curious procefs, the true distance to be 60.5883 femidiameters of the equator, and the moon's horizontal parallax 56'. 44′′, 07.

XVÍ. XVIII. XIX. XX. XXI. XXIV. XXV. XXVI. XXX. XXXI. XLV. Obfervations on the eclipfe of the fun, the first of April, 1764, made in Surry-ftreet in the Strand, by James Short, M. A.; at the houfe of Jofeph Salvador, Efq; by Dr. John Bevis; at Liverpool, by Mr. James Ferguson; at Brumpton Park, by Mr. Samuel Dunn; at Flamstead house, by Mr. Profeffor Blifs; at Thorley-hall, by Matthew Raper, Efq; at Schovezinge near Heidelberg, by Chriftian Mayer; at Chatham by Mr. Mungo Murray; and at the Jefuit's college in Rome.

As there is nothing very particular in thefe obfervations, we have placed them under one article, and fhall refer those who are defirous of perufing them, to the original, as an abridgement would be ufelefs. But muft obferve, that Dr. Bevis, Mr. Dunn, and Mr. Mayer, have alfo given in their papers, obfervations on the eclipfe of the moon, which happened on the 17th of March, 1764.

XXVII. A table of the places of the comet of 1764, difcovered at the obfervatory of the Marine at Paris, on the third of January, about eight o'clock in the evening, in the conftellation of the Dragon, concluded from its fituation obferved with regard to the ftars: By Monfieur Charles Meffier.

This table contains fixteen places of the comet deduced

from

from obfervations made from the third of January to the eleventh of February. To which Mr. Meffier has added the following elements of the theory of this comet, as deduced by M. Pingré, from his firft obfervation:

The afcending node
Inclination

Place of perihelium

3: 29°. 20. 6". 53. 54. 19.

16.

II. 48.

Logarithm of the diftance of the perihelium 9.751415. Paffage of the perihelium 12 Feb. at 10h. 29. mean time in the meridian of Paris. The motion retrograde.

XXVIII. A Supplement to Mr. Pingré's Memoir on the Parallax of the Sun in a Letter from him to the Royal Society.

Mr. Pingré, in this supplemental part to his Memoir publifhed in the preceding volume of the Philofophical Tranfactions, endeavours to fhew, from feveral obfervations made at different places, particularly thofe made by Mr. Maskelyne at St. Helena, that his own obfervations on the late tranfit of Venus, made at the island of Rodriguez, are nearer the truth than those made by Meffrs. Mafon and Dixon at the Cape of Good Hope. He has alfo corrected a pretty remarkable difference that appeared between the account of his own obfervations on the tranfit of Venus, as tranfinitted to the Royal Society, and that published in the French Memoirs.

XXIX. An Account of the Tranfit of Venus: in a Letter to Charles Morton, M. D. Sec. R. S. from Chriftian Mayer, S. J. According to the obfervations of this ingenious Aftronomer, the interior contact of the western limb of Venus, with the western limb of the fun, happened at 20h. 53. 8". true time. The moment of the egrefs, wherein the fame limb of the fun after the interior contact first appeared corniculated, 20h. 53′ 35′′. and the first outer contact at 21h. 9'. 4′′.

XXXV. Some new Properties in Conic Sections, difcovered by Edward Waring, M. A. Lucasian Profeffor of Mathematics in the University of Cambridge.

Thefe new properties are delivered in fix theorems, and are at once both curious and ufeful; but cannot be abridged, or given without the figures.

XLVI. The Defcription of a new Hygrometer, invented by James Ferguson, F. R. S.

This inftrument will certainly answer the end propofed, and point out very minute changes in the ftate of the atmosphere; but no description of it can be rendered intelligible without the copper-plate.

XLVIII. Concife Rules for computing the Effects of Refraction and Parallax in varying the apparent Distance of the Moon from the

Stan

Sun or a fixed Star; alf, an eafy Rule of Approximation for computing the Distance of the Moon from a Star, the Longitudes and Latitudes of both being given, with Demonftrations of the fame. By the Rev. Nevil Mafkelyne, A. M. Fellow of Trinity College in the University of Cambridge.

As thefe rules are very well adapted to answer the purpose intended by the Author, and, at the fame time, far more eafy than any we have yet feen on the fubject, we prefume the Reader will not be difpleafed to find them here, in the Author's own words. The demonftrations, which, in our opinion, are fufficiently concife and elegant, cannot be understood without the figures.

A rule to compute the contraction of the apparent distance of any two heavenly bodies by refraction; the zenith diftances of both, and their diftance from each other being given nearly.

Add together the tangents of half the fum, and half the difference of the zenith distances; their fum, abating 10 from the index, is the tangent of arc the first. To the tangent of arc the first, juft found, add the co-tangent of half the diftance of the ftars; the fum, abating 10 from the index, is the tangent of arc the fecond. Then add together the tangent of double the first arc, the co-fecant of double the fecond arc, and the conftant logarithm of 114" or 2,0569: the fum, abating 20 from the index, is the logarithm of the number of feconds required, by which the distance of the ftars is contracted by refraction: which therefore added to the obferved diftance gives the true diftance cleared from the effect of refraction.'

A rule to compute the contraction or augmentation of the apparent distance of the Moon from a ftar, on account of the Moon's parallax; the zenith diftances of the Moon and ftar, and their diftance from each other being given nearly.

Add together the tangents of half the fum, and half the difference of the zenith diftances of the Moon and ftar, and the cotangent of half the distance of the Moon from the star; the fum, abating 20 from the index, is the tangent of an arch, which call A. Then, if the zenith distance of the Moon is greater than that of the ftar, take the fum of the arch A, juft found, and half the distance of the Moon from the ftar; but, if the zenith distance of the Moon be less than that of the ftar, take the difference of the faid arch A and half the distance of the Moon from the ftar; and the fum, or difference called B. To the tangent of B, thus found, add the cofine of the Moon's zenith diftance, and the logarithm of the Moon's horizontal parallax, expreffed in the minutes and decimals; the fum, abating 20 from the index, is the logarithm of the effect of parallax, tendin

Q4

tending always to augment the apparent distance of the Moon from the ftar; except the zenith distance of the Moon be less than that of the ftar, and, at the fame time, the arch A be greater than half the diftance of the Moon from the star, in which cafe the effect of parallax diminishes the apparent distance of the Moon from the star.'

Remarks on the use of the two foregoing rules.

It has been remarked, after the rule for refraction above, that if the altitudes of the Moon or ftar are under 10 degrees, the zenith distances must be firft leffened by 3 times the refractions correfponding to their refpective altitudes before the effect of refraction be computed.

But in order to compute the effect of parallax from the fecond rule, the obferved diftance of the Moon from the ftar muft be first corrected by adding the effect of refraction to it found by rule the first; as muft the obferved altitudes of the Moon and ftar be alfo corrected by taking from them their respective refraction in altitude, and the corrected arches thus found muft be made ufe of in computing the parallax. Only, if the altitudes of the Moon and ftar are both 10 degrees or more, part of the calculation of rule the fecond may be faved, and arch the second, found by rule the first, taken for arch A in the fecond rule without any fenfible error. convenient to obferve the following order of computation instead In this cafe, it will be most of that before prefcribed to be used when the altitudes are under 10 degrees.

ift. Making ufe of the apparent altitudes of the Moon and ftar uncorrected, compute arches the first and second by the directions contained in the rule of refraction.

2dly. Taking arch the fecond for arch A in the rule of parallax, compute the effect of parallax according to rule the fecond.

3dly. With arches the first and second compute the effect of refraction by rule the first.

4thly, and laftly. Applying the two corrections of parallax and refraction duly, according to the rules, to the obferved distance of the moon from the ftar, you will have the true and correct distance of the Moon from the ftar, cleared both of refraction and parallax,'

A rule for computing a fecond, but fmaller correction than the firft, neceffary to be applied to the observations of the distance of the Moon from a ftar on account of parallax.

Call the principal effect of parallax, found by the preceding rule, the parallax in diftance; and find the parallax answering to the Moon's altitude. Then to the conftant logarithm 6.941 add the logarithm of the fum of the parallax in altitude and the

parallax

parallax in distance, the logarithm of the difference of the fame parallaxes, and the cotangent of the obferved distance of the Moon from the star (corrected for refraction, and the principal effect of parallax), the fum, abating 13 from the index, is the logarithm of the number of seconds required, being the fecond correction of parallax; and is always to be added to the diftance of the Moon from the ftar, first corrected for refraction, and the principal effect of parallax found above, in order to obtain the true diftance; unless the distance exceeds 90 degrees, in which cafe it is to be substracted.

• A concife rule to find the diftance of the Moon from a zodiacal ftar, very nearly; the difference of the longitudes of the Moon and ftar, and the latitudes of both being given.

To the cofine of the difference of the longitudes add the cofine of the difference of the latitudes, if both of the fame denomination, or fum; if of contrary denominations, the fum of the two logarithms, abating 10 from the index, is the cofine of the approximate diftance. This gives the true distance of the Moon from the Sun, being then nothing more than the common rule for finding the hypothenuse of a right-angled spherical triangle from the two fides given. But in the cafe of a zodiacal ftar apply the following correction to the approximate distance thus found.

To the conftant logarithm 5.3144 add the fine of the Moon's latitude, the fine of the ftar's latitude, the verfed-fine of the dif-. ference of longitude, and the cofecant of the approximate diftance; the fum of these 5 logarithms, abating 40 from the index, is the logarithm of a number of feconds, which fubftracted from the approximate distance, found before, if the latitudes of the Moon and ftar are of the fame denomination, or added thereto, if they are of different denominations, gives the true diftance of the Moon from the star.

• N. B. This rule, though only an approximation, is fo very exact, that even, if the latitude of the Moon was 5o, and that of the ftar 15°, the error would be only ro"; and if the latitude of the Moon be 5o, and that of the star 10°, the error is only 4"; and if the latitudes be lefs, will be lefs in proportion as the fquares of the fines of the latitudes decrease.' XLIX. Extract of a Letter from Mr. John Winthorp, Profeffor of Mathematics in Cambridge, New-England, to Mr. James Short.

In this Letter Mr. Winthorp has anfwered fome remarks, which it feems Mr. Short had made on his obfervation on the tranfit of Venus; and concludes with a request, that his paper may be inferted in the Philofophical Tranfactions.

L. Ok