378. From the Line of Chords § [372] take the Angle of the Moon’s visible Path with the Ecliptic, viz. 5° 38ʹ § [367]: and note, that when the Moon’s Latitude is North Ascending, as in the present case, the Chord of this Angle must be set off to the left hand of the Axis of the Ecliptic CH, as from H to M, and the right line CM drawn for the Axis of the Moon’s Orbit: but when the Moon’s Latitude is North Descending, this Angle and Axis must be set to the right hand, or from H toward h. When the Moon’s Latitude South Ascending, the Axis of her Orbit lies the same way as when her Latitude is North Ascending; and when South Descending, the same way as when North Descending.
Path of the Penumbra’s center over the Earth.
379. Take the Moon’s Latitude, 40ʹ 9ʺ § [367], from the Scale CA, and set it from C to T on the Axis of the Ecliptic; and through T, at right Angles to the Axis of the Moon’s Orbit CM, draw the straight Line RTS; which is the Moon’s Path, or Line that the center of her shadow and Penumbra describes in going over the Earth’s Disc. The Point T in the Axis of the Ecliptic is the Place where the true Conjunction of the Sun and Moon falls, according to the Tables; and the Point W, in the Axis of the Moon’s Orbit, is that where the center of the Penumbra approaches nearest to the center of the Earth’s Disc, and consequently the middle of the general Eclipse.
It’s Place on the Earth’s Disc shewn for every minute of it’s Transit.
380. Take the Moon’s true Horary Motion from the Sun 27ʹ 50ʺ § [367], from the Scale CA with your Compasses (every division of the Scale being a minute of a Degree) and with that extent make marks in the Line of the Moon’s Path RTS: then divide each of these equal spaces by dots into 60 equal parts or horary minutes, and set the hours to every 60th minute, in such a manner that the dot; signifying the precise minute of New Moon by the Tables, may fall in the Point T where the Axis of the Ecliptic cuts the Line of the Moon’s Path; which, in this Eclipse, is the 25th minute past ten in the Forenoon: and then the other marks will shew the places on the Earth’s Disc where the center of the Penumbra is, at the hours and minutes denoted by them, during its transit over the Earth.
Middle of the Eclipse.
It’s Phases.
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.
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.