The problem may be divided into two parts. The first is to find out when an eclipse will occur, the other to determine when and where it will be visible.

For this purpose it is convenient first to consider Earth as fixed and to suppose an observer is looking out from its centre. To this observer, labeled O in the figure of the celestial sphere, the Sun and Moon appear projected on the celestial sphere. While this sphere appears to rotate daily, as measured by the positions of the stars, around the axis PP′ (Earth’s axis of rotation), the Sun’s disk, S, appears to travel slowly along the great circle EE′ (the ecliptic), making a complete revolution in one year. At the same time, the Moon’s disk, M, travels along its own great-circle path, LL′, once during a lunar month. The angular diameters of the Sun’s and the Moon’s disks, S and M, are each about 0.5° but vary slightly.

Every month, the Moon’s disk moving along its path, LL′, will overtake the more slowly moving Sun once, at the moment of the new moon. Usually the Moon’s disk will pass above or below the Sun’s disk. Overlapping of the two results in an eclipse of the Sun, which can happen only when the new moon occurs at the same time that the Sun is near the ascending node or descending node, [nodeascnd] and [nodedescd], respectively, of the Moon’s orbit. Because the nodes are 180° apart, eclipses occur in the so-called eclipse seasons, six months apart.

In the figure of the celestial sphere, the projection of Earth’s umbra is shown as a disk, U, at the distance of the Moon’s orbit. At that distance the shadow’s disk subtends an angle of about 1.4°; its centre is always opposite the Sun’s disk and travels along the ecliptic, EE′. A lunar eclipse occurs whenever the shadow’s disk overlaps the Moon’s disk; this happens only when the shadow’s disk is near one of the nodes and the Sun is near the opposite node. The Sun’s passage through the lunar nodes is thus the critical time for both solar and lunar eclipses. The plane of the Moon’s path, LL′, is not fixed, and its nodes move slowly along the ecliptic in the direction indicated by the arrows, making a complete revolution in about 19 years. The interval between two successive passages of the Sun through one of the nodes is termed an eclipse year, and, since the Moon’s node moves so as to meet the advancing Sun, this interval is about 18.6 days less than a tropical (or ordinary) year.

In the figure of the Moon’s ascending node, this region is depicted as seen from the centre of the celestial sphere and is shown much enlarged. The node is kept fixed, and the apparent motions of the Sun and the Moon (top portion of the figure) are shown relative to the node. To the observer on Earth at the centre of the sphere, the Sun’s disk will travel along the ecliptic, EE′, and the Moon’s disk along its designated path, LL′. The Sun is so distant compared with the size of Earth that, from all places on Earth’s surface, the Sun is seen nearly in the same position as it would be from the very centre. On the other hand, the Moon is relatively near, and so its projected position on the celestial sphere is different for various places of observation on Earth. In fact, it may be displaced as much as 1° from the position in which it is seen from the centre of Earth. If the radius of the Moon’s disk is enlarged by 1°, a “Moon circle,” C, is obtained that encloses all possible positions of the Moon’s disk seen from anywhere on Earth. Conversely, if any disk of the Moon’s size is placed inside this Moon circle, there is a place on Earth from which the Moon is seen in that position.

Accordingly, an eclipse of the Sun occurs somewhere on Earth whenever the Moon overtakes the Sun in such a position that the Moon circle, C, passes over the Sun’s disk; when the latter is entirely covered by the Moon circle, the eclipse will be total or annular. From the top portion of the figure of the Moon’s ascending node, it is evident that a solar eclipse will take place if a new moon occurs while the Sun moves from position S₁ to position S₄. This period is the eclipse season; it starts 19 days before the Sun passes through a lunar node and ends 19 days thereafter. There are two complete eclipse seasons, one at each node, during a calendar year. Because there is a new moon every month, at least one solar eclipse, and occasionally two, occurs during each eclipse season. A fifth solar eclipse during a calendar year is possible because part of a third eclipse season may occur at the beginning of January or at the end of December.

The bottom portion of the figure of the Moon’s ascending node illustrates the condition necessary for a lunar eclipse. If a full moon occurs within 13 days of the passage of the Sun though a lunar node—and thus of the Earth’s umbral disk, U, through the opposite node—the Moon will be eclipsed. (In the figure the umbral disk passes through the ascending node.) Most eclipse seasons, but not all, will thus also contain a lunar eclipse. When two eclipse seasons and a partial third season fall in a calendar year, there may be three lunar eclipses in that year. Eclipses of the Sun are evidently more frequent than those of the Moon. Solar eclipses, however, can be seen from only a very limited region of Earth, whereas lunar eclipses are visible from an entire hemisphere.

During a solar eclipse the shadow cones—the umbra and penumbra—of the Moon sweep across the face of Earth (see the figure of an eclipse of the Sun), while, at the same time, Earth is rotating on its axis. Within the narrow area covered by the umbra, the eclipse is total. Within the wider surrounding region covered by the penumbra, the eclipse is partial.

Astronomical ephemerides, or tables, that are published annually for the year ahead provide maps tracing the paths of the more important eclipses in considerable detail, as well as data for accurate calculation of the times of contact at any given observing location on Earth. Calculations are made some years ahead in Terrestrial Time (TT), which is defined by the orbital motion of Earth and the other planets. At the time of the eclipse, the correction is made to Universal Time (UT), which is defined by the rotation of Earth and is not rigorously uniform.

Modern computers make it possible to predict solar eclipses several years ahead with high accuracy. By means of the same calculational methods, eclipses can be “predicted backward” in time. The generation of the times and observational locations for ancient eclipses has been valuable in historical and scientific research (see below Eclipses in history).