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out 36' and 39′, the latitude for 7°; and 41' 51", the latitude for 8°: and by making proportion be. tween these latitudes for the 42' 14" by which the Moon's distance from the node exceeds 7 degrees; her true latitude will be found to be 40 18" northascending.

6. To find the Moon's true horary motion from the Sun. With the Moon's anomaly, viz. 11'9 24' 21", enter Table XVII, and take out the Moon's horary motion; which, by making proportion in that table, will be found to be 30' 22". Then, with the Sun's anomaly, 9' 1° 26' 16", take out his horary motion 2 28" from the same table: and subtracting the latter from the former, there will remain 27 54" for the Moon's true horary motion from the Sun.

7. To find the angle of the Moon's visible path with the ccliptic. This, in the projection of eclipses, may be always rated at 5° 35', without any sensible error.

8,9. To find the semidiameters of the Sun and Moon. These are found in the same table, and by the same arguments, as their horary motions.In the present case, the Sun's anomaly gives his semidiameter 16' 6', and the Moon's anomaly gives her semidiameter 14' 57".

10. To find the semidiameter of the penumbra. Add the Moon's semidiameter to the Sun's, and their sum will be the semidiameter of the penumbra, viz. 31' 3.

Now collect these elements, that they may be found the more readily when they are wanted in the construction of this eclipse.

1. True time of new Moon in

April 1764

}

1d 10h 20' 25"

2. Semidiameter of the Earth's disc, 0 54 43 3. Sun's dist. from the nearest solst. 77 49 53 4. Sun's declination, north 5. Moon's latitude, north-ascending 0 40 18

4 49 O

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6. Moon's horary motion from the Sun O 27 54 7. Angle of the Moon's visible

path with the ecliptic

8. Sun's semidiameter

5 35 0

16 6

9. Moon's semidiameter

14 57

10. Semidiameter of the penumbra

31 3

To project an Eclipse of the Sun geometrically.

Make a scale of any convenient length, as AC, Plate XII. and divide it into as many equal parts as the Earth's Fig. I. semi-disc contains minutes of a degree; which, at the time of the eclipse in April 1764, is 54' 43'. Then, with the whole length of the scale as a radius, describe the semicircle AMB upon the centre C; which semicircle shall represent the northern half of the Earth's enlightened disc, as seen from the Sun.

Upon the centre C raise the straight line CH, perpendicular to the diameter ACB; so ACB shall be a part of the ecliptic, and CH its axis.

Being provided with a good sector, open it to the radius CA in the line of chords; and taking from thence the chord of 23 degrees in your compasses, set it off both ways from H, to g and to h, in the periphery of the semi-disc; and draw the straight line gh, in which the north pole of the disc will be always found.

When the Sun is in Aries, Taurus, Gemini, Cancer, Leo, and Virgo, the north pole of the Earth is enlightened by the Sun: but while the Sun is in the other six signs, the south pole is enlightened, and the north pole is in the dark.

And when the Sun is in Capricorn, Aquarius, Pisces, Aries, Taurus, and Gemini, the northern half of the Earth's axis C XII P lies to the right hand of the axis of the ecliptic, as seen from the Sun; and to the left hand, while the Sun is in the other six signs.

Open the sector till the radius (or distance of the two 90's) of the signs be equal to the length of Vb, and take the sine of the Sun's distance from the solstice (77° 49' 53) as nearly as you can guess, in your compasses, from the line of sines, and set off that distance from V to P in the line gb, because the Earth's axis lies to the right hand of the axis of the ecliptic in this case, the Sun being in Aries; and draw the straight line C XII P for the Earth's axis, of which P is the north pole. If the Earth's axis had lain to the left hand from the axis of the ecliptic, the distance VP would have been set off from V toward

g.

To draw the parallel of latitude of any given place, as suppose London, or the path of that place on the Earth's enlightened disc as seen from the Sun, from Sun-rise till Sun-set, take the following method.

Subtract the latitude of London, 5110 from 90° and the remainder 38 will be the co-latitude, which take in your compasses from the line of chords, making CA or CB the radius, and set it from b (where the Earth's axis meets the periphery of the disc) to VI and VI, and draw the occult or dotted line VI K VI. Then, from the points where this line meets the Earth's disc, set off the chord of the Sun's declination 4° 49' to D and F, and to E and G, and connect these points by the two occult lines FXII G and DLE.

Bisect LK XII in K, and through the point K draw the black line VI K VI. Then making CB the radius of a line of sines on the sector, take the co-latitude of London 38° from the sines in your compasses, and set it both ways from K, to VI and VI.--These hours will be just in the edge of the disc at the equinoxes, but at no other time in the whole year.

With the extent K VI,taken into your compasses, set one foot in K (in the black line below the occult one) as a centre, and with the other foot describe the semicircle VI, 7, 8, 9, 10, &c. and divide it into 12

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equal parts. Then from these points of division, draw the occult lines 7 p, 8, on, &c. parallel to the Earth's axis C XII P.

With the small extent K XII as a radius, describe the quadrantal arc XII f, and divide it into six equal parts, as XII a, ab, bc, cd, de, and ef; and through the division-points, a, b, c, d, e, draw the occult lines VII e V, VIII d IV, IX c III, X b II, and XI a I, all parallel to VI K VI, and meeting the former occult lines 7p, 8o, &c. in the points VII, VIII, IX, X, XI, V. IV, III, II, and I: which points shall mark the several situations of London on the Earth's disc, at these hours respectively, as seen from the Sun; and the elliptic curve VI VII VIII, &c. being drawn through these points shall represent the parallel of latitude, or path of London on the disc, as seen from the Sun, from its rising to its setting.

N. B. If the Sun's declination had been south, the diurnal path of London would have been on the upper side of the line VI KVI, and would have touched the line DLE in L.-It is requisite to divide the horary spaces into quarters (as some are in the figure) and, if possible, into minutes also.

Make CB, the radius of a line of chord on the sector, and taking therefrom the chord of 5° 35', the angle of the Moon's visible path with the ecliptic, set it off from H to M on the left hand of CH, the axis of the ecliptic, because the Moon's latitude is north-ascending. Then draw CM for the axis of the Moon's orbit, and bisect the angle MCH by the right line Cz.--If the Moon's latitude had been northdescending, the axis of her orbit would have been on the right hand from the axis of the ecliptic.N. B. The axis of the Moon's orbit lies the same way when her latitude is south-ascending, as when it is north-ascending; and the same way when southdescending, as when north-descending.

Take the Moon's latitude 40' 18" from the scale CA in your compasses, and set it from i to x in the

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bisecting line Cz, making ix parallel to Cy: and through x, at right-angles to the axis of the Moon's orbit CM, draw the straight line Nw x y S, for the path of the penumbra's centre over the Earth's disc. The point w in the axis of the Moon's orbit, is that where the penumbra's centre approaches nearest to the centre of the Earth's disc, and consequently is the middle of the general eclipse: the point x is that where the conjunction of the Sun and Moon falls, according to equal time by the tables; and the point is the ecliptical conjunction of the Sun and Moon. Take the Moon's true horary motion from the Sun, 27' 54", in your compasses, from the scale CA (every division of which is a minute of a degree), and with that extent make marks along the path of the penumbra's centre; and divide each space from mark to mark into sixty equal parts or horary minutes, by dots; and set the hours to every 60th minute in such a manner, that the dot signifying the instant of new Moon by the tables, may fall into the point x, half way between the axis of the Moon's orbit, and the axis of the ecliptic; and then the rest of the dots will shew the points of the Earth's disc, where the penumbra's centre is at the instants denoted by them, in its transit over the Earth.

Apply one side of a square to the line of the penumbra's path, and move the square backward and forward, until the other side of it cuts the same hour and minute (as at m and n) both in the path of London, and in the path of the penumbra's centre: and the particular minute or instant which the square cuts at the same time in both paths, shall be the instant of the visible conjunction of the Sun and Moon, or greatest obscuration of the Sun, at the place for which the construction is made, namely, London, in the present example; and this instant is at 47 minutes past X o'clock in the morning; which is 17 minutes 5 seconds later than the tabular time of true conjunction.

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