CALCULATION OF ECLIPSES OF THE SUN We can divide our calculations into two parts. In the first part we seek to determine if an eclipse can occur and in the second to determine its circumstances (i.e., whether or no it is visible at a given point on the earth's surface, its type [total or partial], duration, etc.) . It is most convenient to suppose the earth fixed and the sun and moon revolving around it.
We will suppose first that the observer is situated at the centre of the earth. Then the state of affairs can be represented on a "celestial sphere" as in fig. 3. In this diagram PP' is the earth's axis and the position of celestial objects is marked on the general sphere of the sky in the same way as countries are marked on an ordinary globe of the earth, looking at the celestial sphere from the "outside" as we do at the countries on a globe. Thus 0 the centre of the sphere (MEQM'E'Q') is the observer, the circle QQ' is the equator. The motion of the sun can then be de scribed thus. It moves round a great circle QQ' called the ecliptic, which is inclined at 231° nearly to the equator, once in a solar or ordinary year. The moon likewise describes once a lunar month an orbit represented by the great circle MM' which is inclined to the ecliptic at about 5° (instead of being the same circle EE' as it would be if the orbits were coplanar). The daily motion is represented by a revolution of the whole sphere and everything on it round P'P once a sidereal day. The sun can be represented by a small circle X centre on EE' about 2 ° in diameter, and the moon by a small circle Y, nearly the same size, centre on MM'. The sizes and distances of the sun and moon are such that they subtend very nearly the same angle (about 1°) at the earth, but their apparent sizes are not constant. The earth goes round the sun not in a circle but an ellipse, and the sun is further from us in summer (in the northern hemisphere) than in winter, thus its apparent size (the circle X) varies and also its speed of mo tion around EE' is faster in winter than in summer. The moon also moves around the earth in an ellipse and so its distance and speed vary so that the apparent size (i.e., the circle Y) varies. It is for these reasons that the duration and magnitude of an eclipse vary. For an eclipse to occur the sun and the moon must be so situated that the circles X and Y overlap. Clearly since they are only about 1° in radius and the circles EE' and MM' are inclined at about 5° this can only happen when the moon and sun are near one of the two points C and Z3 of intersection of EE' and MM' called the nodes of the moon's orbit. The moon's nodes revolve in the ecliptic from east to west once in about 19 years. The inter val of time between two successive passages of the sun through one of the nodes is termed an "eclipse year" and since the moon's node moves to meet the advancing sun this interval is about 18.6 days less than a tropical (or ordinary) year. Nineteen eclipse years days) are very nearly the same as 223 ordinary months days). This interval of 18 years II days (or 18 years Io days if February 29 has occurred five times) is termed the Saros, and is the simplest relation useful in predicting eclipses. After this interval the moon and sun come very nearly to the same relative position again. Thus the eclipse of 1923 (September was a repetition of that of 1905 (August 30). The coincidence of the two periods is not exact and so while there is an eclipse again after a Saros the circumstances of it will be different. If a new moon fell exactly at a node then after 18 years II days the new moon occurs before the node is reached. The dif ference between the two periods is 0.4595 days and so the moon is 28' farther west. If new moon happens within 18° of the node an eclipse of the sun may take place; if it is an ascending node the eclipse will be visible in high northern latitudes on the earth; at the next return the new moon will be 28' nearer the node and the eclipse will be visible a little south of the first position. When the new moon is within about I I ° of the node the eclipse becomes central (i.e., there is a line on the surface of the earth from any point of which the centres of the sun and moon appear coincident) and may be annular or total depending on the distance of the moon and sun from the earth. Total eclipses now occur every 18 years I I days passing south each time till the limit of I I ° is passed on the other side of the node and there remains a series of partial eclipses tailing off to the south pole. In such a series there are from 68 to 75 eclipses of which about 18 are total.
One other important relation is that 239 returns of the moon to perigee are 6585.5574 days, so after 223 lunations the moon returns not only very closely to its original position with respect to sun and node but also with respect to line of apsides so that the distance from earth to moon is nearly the same. Thus the duration is altered but little and the perturbations of the moon's orbit are almost unchanged with but little effect then on the time of the eclipse. The fraction •321I days in the period of the Saros has the effect of making each eclipse occur about iio° of longi tude further west and after three Saroses it has nearly returned to its original position but farther N. or S. Since there are two nodes the sun will come into a region where eclipse is likely twice a year, giving the two "eclipse seasons" each about one month in duration, in which at least one eclipse or possibly two small partial eclipses may happen. Generally, but not necessarily, one eclipse of the moon will occur in a season. There is possible as a maxi mum five eclipses of the sun in a year. Solar eclipses occur of tener than lunar ones, but since a solar eclipse is visible only over a very limited region of the earth and a lunar one over a whole hemisphere, lunar eclipses are more often seen at any one place. Having determined that an eclipse will take place it remains to find its circumstances at a given place on the earth's surface. For this purpose we make use of the conception of the moving shadow cones. A plane passing through the centre of the earth and per pendicular to the axis of the moon shadow is chosen as a plane of reference and termed the fundamental plane, and the co-ordinates x and y of the point in which the axis of the shadow cones cuts it are determined and also the declination d and the hour angle µ of the axis. The radii /,(penumbra) and (umbra) of the section of the shadow cones by the fundamental plane are then found, and all these quantities together with the angles of the cones and the rates of change of x, y and are given in the Ephemerides. These quantities (termed the Besselian elements) can be used to determine the circumstances at any place in the following way. Knowing the position of the observer on the earth we have his co-ordinates referred to the fundamental plane at any instant and so can find the position relative to him of the shadow cones. If he is always outside them the eclipse is invisible to him. If he be nearer the central line there will be a time when he just touches the circle formed by the penumbra. The moon is then just be ginning to touch the sun (1st contact), and if he be near enough there will also be a moment when he just touches the umbral circle and totality commences (2nd contact). The moon now touches the sun internally at one point and later he will be just on the edge of the shadow circle as it is leaving him (3rd con tact) and totality ends. The end of the eclipse is when the further rim of the penumbra reaches him and the other edge of the moon is just touching the edge of the sun externally (4th contact). Likewise the bearings of the points of contact may be found and so the position in the sky of the points of the crescents during the partial phase can be calculated. Thus it is possible to determine the form of the shadow cast on the earth's surface and the path over which it moves. In the Ephemerides maps are given showing the part covered by the shadow as it moves over the earth's sur face and the region within which the eclipse is total if anywhere (i.e., if there are any points on the earth's surface near enough to the moon to fall in the cross hatched region in fig. I), and lines are drawn joining those places where the eclipse begins and ends at certain times, e.g., sunrise or sunset. The most important in formation is the track of the line of central eclipse (the locus of the intersection of the line of centres of the sun and moon with the surface of the earth) and the northern and southern limits of the region of totality. Since the moon moves through its own di ameter in about an hour there will be about an hour between the r st and 2nd contacts and also between the 3rd and 4th. The duration of totality (the interval between the 2nd and 3rd con tacts) is greatest on the central line and may be anything up to about 71 mins. The longest duration will clearly occur when the moon is near perigee and the sun near apogee and also if the eclipse occur near mid-day on the earth's equator for then the surface of the earth is nearly perpendicular to the shadow cones and their rate of motion over it is slowest. It is possible with the aid of modern tables to predict eclipses a few years ahead with an accuracy of a few seconds. The chief difficulty lies in the uncertainty of the moon's motion.
Eclipses can of course be "predicted backwards" as well as forwards, and the calculation of ancient eclipses has been of value in historical research in fixing the dates of certain events. The main difficulty is in identifying an eclipse with certainty since often details such as the time of day of the eclipse, or even the season, are lacking. The earliest date thus accurately determined is the year 911 B.C., from an eclipse in Assyria. The eclipse men tioned in Amos viii., 9 seems to have been the one in 763 B.c. There was no eclipse visible in the neighbourhood of Samaria in 787 B.C., the date set down opposite this passage in some Bibles.