Page images
PDF
EPUB

And as they go round in 18 years, 225 days, and 8 hours, if there had been a complete number of lunations in that time, there would then be a complete and regular period of eclipses in the same time. But this never happens. However in 18 years 11d 7h 43′ 20′′, including four leap-years, or one day less, if five be included, the sun, moon, and nodes return to the same state, in which they were at the beginning of the period; save only, that the line of conjunction of the sun, moon, and nodes, will fall backwards, or to westward, 28′ 12′′ of a degree in every period. Add therefore this quantity to the time of any eclipse, and you will have the time of an eclipse of the same kind, in the next period, very nearly; or subtract it, in order to have the eclipse of the preceding period; without any trouble of calculation. This is called the Plinian Period or Chaldean Saros, as Pliny records it, to be known by the Chaldeans. But as the line of conjunction falls westward 28′ 12′′, in every period, it would pass through the whole ecliptic limits in about 1388 years, as it would shift about 36 degrees in that time; and near 12,500 years would be required to describe the remaining 324 degrees; after which time, the same period of eclipses would begin and run over again. A longer and more exact period for calculating eclipses is 557 21d 18h 30' 11"; for then, after that time, the sun, moon, and node, meet so nearly again, as to be but 11" distant.

It is however generally much safer to depend upon a calculation from astronomical tables, or, if great accuracy be not required, upon graphical constructions and delineations.

CONST

CONSTRUCTION OF SOLAR ECLIPSES.

THE design of this construction is to represent the disk of the earth, supposing it visible from the sun; for which purpose, the following elements must be collected from the tables. Suppose for the eclipse of the sun, which happened in April, 1764, as it appeared at London.

1. The true time of the conjunction, April 1a 10h 30′ 25′′; 2d, the latitude of the place 51° 30′; 3d, the semi-diameter of the earth's disk 54' 53"; 4th, the sun's distance from the nearest solstice, 77° 49′ 48"; 5th, the sun's declination north 4° 49′ 0′′; 6th, the sun's simi-diameter 16' 6"; 7th, the moon's latitude north ascending 40′ 18"; 8th, the moon's horary motion from the sun 27′ 54′′; 9th, the angle of the moon's visible path with the ecliptic 5° 35'; 10th, the moon's semi-diameter 14' 57"; 11th, the semi-diameter of the moon's penumbra 31' 3".

Take 54' 53", the semi-diameter of the earth's disk, from any large scale of equal parts, and with that radius describe the semi-circle AMB,* from the center C, to represent the ecliptic. At right angles to AB, erect CH, the axis of the ecliptic. Open the sector to the radius CA on the line of chords, and take from thence the chord of 23° 28', the obliquity of the ecliptic, which set off both ways from H to g and h, in the semi-disk, and draw the line gVh, in which the pole is always found lying to the right hand of the axis of the ecliptic, while the sun is in Capricorn, Aquaries, Pisces, Aries, Taurus and Gemini, but to the left

* Plate 17.

hand, while in the other six signs. Open the sector till the distance of the two 90°, on the line of sines, is equal to Vh, and take the sine of 77 49′ 48", the sun's distance from the nearest solstice, which set from V, the middle of h g, to P, to the right hand in this case, and draw the earth's axis CP, cutting the disk in h, from which towards VI and VI set off the chord of the co-latitude 38° 30′, making CA rad. and draw the line VI K VI; then from the points where this line cuts the disk, set off the chord of the sun's declination both ways towards E and G, and D and F, and draw the lines DE, and FG, cutting the earth's axis in L and XII, which bisect in K; then VI K, and K XII, will be the two semi-axes of the ellipsis, which will represent the path of the place of observation, London, which describe and divide into hours and minutes. Making CB radius on the line of sines, take the colatitude 38 30', from the line of sines, and set it both ways from K to the hours VI and VI in the line VI K VI, which will be just in the disk at the equinoxes, but in no other season of the year. N. B. When the sun's declination is south, the elliptic path is convex toward the pole, and vice versa.

Take the chord of 5° 35', the angle of the moon's visible path, from the line of chords, making AC radius, and set it off from H, the axis of the ecliptic, to the left hand, to M, when the moon's latitude is N. ascending, but to the right hand, when it is N. descending; draw CM for the axis of the moon's orbit, and bisect the angle HCM by the line Cxz. Take the moon's latitude from the same scale with CA, and mark the point x in the bisecting line Cxz, whose nearest distance from the line CA shall be equal to the moon's latitude, and through x draw a line

at right angles to CM, the axis of the moon's orbit, which line shall be the path of the moon's penumbra, over the disk of the earth. The point x marks the conjunction according to equal time by the tables, the point y, where this line cuts the axis of the ecliptic, is the point of the ecliptical conjunction, and the point w is that of the middle of the eclipse.

Take the moon's horary motion from the sun, 27' 54", from the same scale with AC, and with that extent make marks along the path of the moon's penumbra, each being one hour; subdivide them into 60 equal parts for minutes, and number the hours so that the instant of new moon by the tables shall fall upon the point x, half way between the axis of the ecliptic and the moon's orbit. Then apply one side of a square to the path of the penumbra, and thus move it backwards and forwards, until the other side cuts the same hour and minute, both in the path of the penumbra, and the path of the place, London, which will be, in the present example, at 10' 47'. With the semi-diameter of the sun, taken from the same scale CA, viz. 16' 16", describe a circle round 101 47' in the path of London, representing the disk of the sun; and with the semi-diameter of the moon 14′ 57′′ taken from the same scale, describe a circle round 101 47' in the path of the penumbra, to represent the disk of the moon: and the portion of the sun's disk hid by the moon's disk, will show the phases and quantity of the eclipse at that time.

Lastly, take from the same scale with AC, the semidiameter of the penumbra, and carry that extent, viz. 31' 3", backwards and forwards, along the paths of the penumbra and the place of observation, London, on both sides of the axis of the ecliptic, until both points

of the dividers fall into the same points of time in both the lines, and these times will represent the time of beginning and ending of the eclipse, at that place, according to apparent time; which, by applying the equation of time, is converted into mean time.

This construction supposes that the moon's motion is uniform and rectilineal during the whole time of the eclipse, and that the angle under which the moon's disk is seen, continues the same, which is not strictly 'true; however, it will give the times of the beginning and end within a few minutes, when executed with sufficient care.

PROJECTION OF LUNAR ECLIPSES.

THE requisites for the projection of a lunar eclipse, are these eight following; which we shall exemplify in the eclipse of May 1762, at London.

1. The true time of full moon, May, 8d 3h 50' 50": 2. The moon's horizontal parallax at that time 57′ 23": 3. The sun's semi-diameter 15′ 56′′: 4. The moon's semi-diameter 15' 38": 5. The semi-diameter of the earth's shadow at the moon, 41' 37": 6. The moon's latitude south descending, 32′ 21′′: 7. The angle of her visible path with the ecliptic, 5° 35': 3. Her true horary motion from the sun, 30' 52"; all which may be collected from the tables.

Make a scale of any convenient length, and divide it into 60 equal parts, each equal one minute. With the sum of the semi-diameters of the moon and earth's shadow, taken from this scale, 57′ 15′′, describe a semicircle ADB:* and also another with a rad. the semidiameter of the earth's shadow, 41' 37", taken from the same scale, to represent the earth's shadow. Draw two diameters at right angles to each other; one re

* Plate 18.

=

« PreviousContinue »