Part 30
381. Apply one side of a Square to the Line of the Moon’s Path, and move the Square backward or forward until the other side cuts the same hour and minute both in the Path of the Place (_London_, in this Construction) and Path of the Moon; and _that_ minute, cut at the same time in both Paths, will be the precise minute of visible Conjunction of the Sun and Moon at _London_, and therefore the time of greatest obscuration, or middle of the Eclipse at _London_; which time, in this Projection, falls at _t_, 34 minutes past 10 in the Moon’s Path; and at _u_, 34 minutes past 10 in the Path of _London_. Then, upon the Point _u_ as a center, describe the Circle _zYy_ whose Radius _uy_ is equal to the Sun’s semi-diameter 16ʹ 6ʺ § 367, taken from the Scale _CA_: And upon the Point _t_ as a center, describe the Circle _Hy_ whose Radius is equal to the Moon’s semi-diameter 14ʹ 58ʺ § 367, taken from the same Scale. The Circle _zYy_ represents the Disc of the Sun as seen from the Earth, and the Circle _Hy_ the Disc of the Moon. The portion of the Sun’s Disc cut off by the Moon’s shews the Quantity of the Eclipse at the time of greatest obscuration: and if a right Line as _yz_ be drawn across the Sun’s Disc through _t_ and _u_, the minute of greatest obscuration in both Paths, and divided into 12 equal parts, it will shew what number of Digits are then eclipsed. If these two Circles do not touch one another, the Eclipse will not be visible at the given Place.
[Sidenote: It’s beginning and ending.]
382. Lastly, take the Semi-diameter of the Penumbra 31ʹ 4ʺ § 367, from the Scale _CA_ with your Compasses; and setting one foot in the Moon’s Path, to the left hand of the Axis of the Ecliptic, direct the other toward the Path of _London_; and carry this extent backwards or forwards until both Points of the Compasses fall into the same instants of time in both Paths: which will denote the time of the beginning of the Eclipse: then, do the same on the right hand of the Axis of the Ecliptic, and where both Points mark the same instants in both Paths, they will shew at what time the Eclipse ends. These trials give the Points _R_ in the Moon’s Path and _r_ in the Path of _London_, namely 9 minutes past 9 in the Morning for the beginning of the Eclipse at _London_, _April 1, 1764_: _t_ and _u_ for the middle or greatest obscuration, at 35 minutes past 10; when the Eclipse will be barely annular on the Sun’s lower-most edge, and only two thirds of a Digit left free on his upper-most edge: and for the end of the Eclipse, _S_ in the Moon’s Path and _x_ in the Path of _London_, at 4 minutes past 12 at Noon.
In this Construction it is supposed that the Equator, Tropics, Parallel of _London_, and Meridians through every 15th degree of Longitude are projected in visible Lines on the Earth’s Disc, as seen from the Sun at almost an infinite distance; that the Angle under which the Moon’s diameter is seen, during the time of the Eclipse, continues invariably the same; that the Moon’s motion is uniform, and her Path rectilineal, for that time. But all these suppositions do not exactly agree with the truth; and therefore, supposing the Elements § 367, given by the Tables to be perfectly accurate, yet the time and phases of the Eclipse deduced from it’s Construction will not answer exactly to what passeth in the Heavens; but may be two or three minutes wrong though done with the utmost care. Moreover, the Paths of all Places of considerable Latitude go nearer the center of the Disc as seen from the Moon than these Constructions make them; because the Earth’s Disc is projected as if the Earth were a perfect sphere, although it is known to be a spheroid. Consequently, the Moon’s shadow will go farther North in places of northern Latitude, and farther South in places of southern Latitude than these projections answer to. Hence we may venture to predict that this Eclipse will be more annular at _London_ (that is, the annulus will be somewhat broader on the southern Limb of the Sun) than the Diagram shews it.
383. Having shewn how to compute the times and project the phases of a Solar Eclipse, we now proceed to those of the Lunar. And it has been already mentioned § 317, that when the Full Moon is within 12 degrees of either of her Nodes, she must be eclipsed. We shall now enquire whether or no the Moon will be eclipsed _May 18, 1761, N. S._ at 32 minutes past 10 at Night. See page 193.
[Sidenote: Table IV.
Table VI.]
s ° ʹ Sun from Node at Full Moon in _March 1761_ 9 25 27 Add his distance for two Lunations, to bring it into _May_ 2 1 20 --------- And his distance at Full Moon in that month is 11 26 47
Subtract this from a Circle, or 12 Signs, and there will remain 3° 13ʹ; which is all that the Sun wants of coming round to the Ascending Node; and the Moon being then opposite to the Sun, must be just as near the Descending Node: consequently, far within the limit of an Eclipse.
384. Knowing then that the Moon will be eclipsed in _May 1761_, we must find her true distance from the Node at that time, by applying the proper Equations as taught § 363, and then find her true Latitude as taught in that article.
[Sidenote: Table IV.
Table XIII.
Table XII.]
s ° ʹ Sun’s mean distance from the Node at F. Moon in _May 1761_ 11 26 47 Add the Equation of the Node, for the Sun’s Anomaly 10^s 18° 15ʹ[85] + 6 -------- Sun’s mean distance from the Node corrected 11 26 53 Add the Equation of the Sun’s mean Place + 1 15 -------- Sun’s true distance from the Ascending Node 11 28 8 To which add 6 Signs, See § 363 6 -------- The sum is the Moon’s true distance from the same Node 5 28 8
[Sidenote: Pl. XII.]
Or the _Argument_ of her _Latitude_; which in Table XIV, gives the Moon’s true Latitude, _viz._ 9ʹ 56ʺ North Descending.
385. Having by the foregoing precepts § 355 found the true time of Opposition of the Sun and Moon in a lunar Eclipse, with the Moon’s Anomaly enter Table XV and take out her horizontal Parallax, also her true horary Motion and Semi-diameter: and likewise those of the Sun by his Anomaly, as already taught § 364 & _seq._ Then add the Sun’s horizontal Parallax, which is always 10 Seconds, to the Moon’s horizontal Parallax, and from their sum subtract the Sun’s Semi-diameter; the remainder will be the Semi-diameter of that part of the Earth’s shadow which the Moon goes through.
386. From the Sum of the Semi-diameters of the Moon and Earth’s Shadow, subtract the Moon’s Latitude; the remainder is the parts deficient. Then, as the Semi-diameter of the Moon is to 6 Digits, so are the parts deficient to the Digits eclipsed.
387. If the parts deficient be more than the Moon’s Diameter, the Eclipse will be total with continuance; if less, it will not be total; if equal, it will be total, but without continuance.
388. Now collect the Elements for projecting this Eclipse.
ʹ ʺ Moon’s horizontal Parallax 55 32 Sun’s horizontal Parallax (always) 10 The Sum of both Parallaxes 55 42 From which subtract the Sun’s Semi-diameter 15 54 Remains the Semi-diameter of the Earth’s Shadow 39 48 Semidiameter of the Moon 15 2 Sum of the two last 54 50 Moon’s Latitude subtract 9 56 Remains the parts deficient 45 0 Moon’s horary motion 30 46 Sun’s horary motion subtract 2 24 Remains the Moon’s horary motion from the Sun 28 22
[Sidenote: To project a lunar Eclipse.
Fig. III.]
389. This done, make a Scale of any convenient length as _W_, whereof each division is a minute of a degree; and take from it in your Compasses 54 Minutes 50 Seconds, the Sum of Semi-diameters of the Moon and Earth’s shadow; and with that extent as a Radius, describe that Circle _OVLG_ round _C_ as a Center.
From the same Scale take 39 Minutes 48 Seconds, the Semi-diameter of the Earth’s shadow, and therewith as a Radius, describe the Circle _UUUU_ for the Earth’s shadow, round _C_ as a Center. Subtract the Moon’s Semi-diameter from the Semi-diameter of the Shadow, and with the difference 24 Minutes 46 seconds as a Radius, taken from the Scale _W_, describe the Circle _YZ_ round the Center _C_.
Draw the right line _AB_ through the Center _C_ for the Ecliptic, and cross it at right Angles with the line _EG_ for the Axis of the Ecliptic.
Because the Moon’s Latitude in this Eclipse is North Descending, § 384, set off the Angle of her visible Path with the Ecliptic 5 Degrees 38 Minutes (Page 202.) from _E_ to _V_; and draw _VCv_ for the Axis of the Moon’s Orbit. Had the Moon’s Latitude been North Ascending, this Angle must have been set off from _E_ to _f_. _N. B._ When the Moon’s Latitude is South Ascending, the Axis of her Orbit lies the same way as when she has North Ascending Latitude; and when her Latitude is North Descending, the Axis of her Orbit lies the same way as when her Latitude is South Descending.
Take the Moon’s true Latitude 9ʹ 56ʺ in your Compasses from the Scale _W_, and set it off from _C_ to _F_ on the Axis of the Ecliptic because the Moon is north of the Ecliptic; (had she been to the South of it, her Latitude must have been set off the contrary way, as from _C_ towards _v_:) and through _F_, at right Angles to the Axis of the Moon’s Orbit _VCv_, draw the right line _LMHNO_ for the Moon’s Orbit, or her Path through the Earth’s shadow. _N. B._ When the Moon’s Latitude is North Ascending, or North Descending, she is above the Ecliptic: but when her Latitude is South Ascending, or South Descending, she is below it.
Take the Moon’s true horary motion from the Sun, _viz._ 28 Minutes 22 Seconds, from the Scale _W_ in your Compasses; and with that extent make marks in the line of the Moon’s Path _LMHNO_: then divide each of these equal spaces into 60 equal parts or minutes of time: and set the hours to them as in the Figure, in such a manner that the precise time of Full Moon, as shewn by the Tables, may fall in the Axis of the Ecliptic at _F_, where the line of the Moon Path cuts it.
Lastly, Take the Moon’s Semi-diameter 15 Minutes 2 Seconds from the Scale _W_ in your Compasses, and therewith as a Radius describe the Circles _P_, _Q_, _R_, _S_, and _T_ on the Centers _L_, _M_, _H_, _N_, and _O_; the Circles _P_ and _T_ just touching the Earth’s Shadow _UU_, but no part of them within it; the Circles _Q_ and _S_ all within it, but touching at its edges; and the Circle _R_ in the middle of the Moon’s Path through the shadow. So the Circle _P_ shall be the Moon touching the shadow at the moment the Eclipse begins; the Circle _Q_ the Moon just immersed into the shadow at the moment she is totally eclipsed; the Circle _R_ the Moon at the greatest obscuration, in the middle of the Eclipse; the Circle _S_ the Moon just beginning to be enlightened on her western limb at the end of total darkness; and the Circle _T_ the Moon quite clear of the Earth’s shadow at the moment the Eclipse ends. The moments of time marked at the points _L_, _M_, _H_, _N_ and _O_ answer to these Phenomena: and according to this small projection are as follow. The beginning of the Eclipse at 8 Hours 36 Minutes _P. M._ The total immersion at 9 Hours 42 Minutes. The middle of the Eclipse at 10 Hours 26 Minutes. The end of total darkness at 11 Hours 12 Minutes. And the end of the Eclipse at 12 Hours 18 Minutes; but the Figure is too small to admit of precision.
[Sidenote: The examination of antient Eclipses.]
390. By computing the times of New and Full Moons, together with the distance of the Sun and Moon from the Nodes; and knowing that when the Sun is within 17 Degrees of either Node at New Moon he must be eclipsed; and when the Moon is within 12 Degrees of either Node at Full she cannot escape an Eclipse; and that there can be no Eclipses without these limits; ’tis easy to examine whether the accounts of antient Eclipses recorded in history be true. I shall take the liberty to examine two of those mentioned in the foregoing catalogue, namely, that of the Moon at _Babylon_ on the 19th of _March_ in the 721st year before CHRIST; and that of the Sun at _Athens_, on the 20th of _March_, in the 424th year before CHRIST.
The time of Full Moon for the former of these Eclipses is already calculated, Page 198, and the time of New Moon for the latter, Page 196, both to the _Old Style_; so that we have nothing now to do but find the Sun’s distance from the Nodes the same way as we did the Anomalies; and if the Full Moon in _March_ 721 years before CHRIST was within 12 degrees of either Node, she was then eclipsed; and if the Sun, at the time of New Moon in _March_ 424 years before CHRIST was within 17 degrees of either Node, he must have been eclipsed at that time.
EXAMPLE I.
_To find the distance of the Sun and Moon from the Nodes, at the time of Full Moon in_ March, _the year before_ CHRIST _721, O. S._
The years 720 added to 1780 make 2500, or 25 Centuries.
Sun from Node s ° ʹ To the mean time of Full Moon in _March 1780_, Table III. 10 3 1 Add the distance for 1 Lunation [See _N. B._ Page 195, and Example III, Page 198] 1 0 40 -------- Sum 11 3 41 From which subtract the Sun’s distance from the Node for 2500 years, Table V 5 4 11 -------- Remains the Sun’s distance from the Node, _March 19_, 721 years before CHRIST 5 29 30 To which add 6 Signs for the Moon’s distance, because she was then in opposition to the Sun 6 0 0 -------- The Sum is the Moon’s dist. from the Ascend. Node 11 29 30
That is, she was within half a degree of coming round to it again; and therefore, being so near, she must have been totally, and almost centrally eclipsed.
EXAMPLE II
_To find the Suns distance from the Node at the Time of New Moon in_ March, _the year before_ CHRIST _424, O. S._
The years 423 added to 1777 make 2200, or 22 Centuries.
Sun from Node s ° ʹ At the mean time of New Moon in _March 1777_, Tab. I. 8 23 33 From which subtract the Sun’s distance from the Node for 2200 years, Table V 3 6 0 -------- Remains the Sun’s distance from the Ascending Node, _March 21_, 424 years before CHRIST 5 17 33 Which, taken from 6 Signs, the distance of the Nodes from each other 6 0 0 -------- Leaves the Sun’s distance at that time from the Descending Node, Descending _viz._ 0 12 27
Which being less than 17 degrees, shews that the Sun was then eclipsed. And as from these short Calculations we find those two antient Eclipses taken at a venture, to be truly recorded; it is natural to imagine that so are all the rest in the catalogue.
Here follow ASTRONOMICAL TABLES, for calculating the Times of NEW and FULL MOONS and ECLIPSES.