Solar eclipse
M.V.N. Murthy, The Institute of Mathematical Sciences, Chennai 

Did you know that there was a total solar eclipse on 21 August 2017 in Madras? Well, this Madras happens to be in Oregon, USA, and not Chennai which was earlier known by the name Madras. The eclipse began on the Pacific Ocean, and was visible on land starting from 9 AM in Madras until it ended on the east coast of USA some time around 1 PM. The eclipse was visible at different times from the west coast to the east coast of the US. The last time such a total solar eclipse visible from coast to coast was actually on June 8, 1918 or nearly a hundred years ago. So this was indeed a rare event.
Something to think about: why do eclipses "travel" from west to east?
The last time a total solar eclipse happened in India was in 1995 and it was visible in North India. The next such total solar eclipse in India will occur only in the year 2034. This too will be visible only in North India. However, we are going to witness an annular solar eclipse in December 2019 passing through South India. 

The faces of the Sun and Moon
With solar eclipse we always use the qualifying words total, annular, partial, or, rarely, hybrid. What do they mean? To see this we need to look at some numbers.
The distance of the Moon and Sun from the Earth is listed in the Table. The angular diameter of both is also listed. What does this mean? Both the moon and sun appear as round discs in the sky. Angular diameter is the size of this disc as measured by a person standing on the Earth. Think of light rays entering your eyes from opposite edges of the disc. The angle between these rays is the angular diameter.
The moon is much much smaller than the sun. But because it is so much closer to the earth, the sizes of the two appear coincidentally to be almost the same. Hence the angular diameter of both is nearly the same. Since moon and Earth are in elliptical and not circular orbits, sometimes the moon is closer to the earth and its apparent size as viewed from earth varies slightly more than the apparent size of the sun. So the variation is more than that of the Sun, as the table shows.

Planar geometry?
Next we have to look at the geometry of the earth-moon-sun system. If all the three move in the same plane, we would have a solar eclipse every new-moon day when the moon is in line with the earth and the sun. However, this is not the case. The motion of earth and sun defines a plane called the ecliptic. In this plane earth makes periodic revolutions around the sun once a year. The path of the earth is almost a circle, but not quite as the numbers in the table show, it is in fact an ellipse. In an ellipse the radius is not a constant but has a minimum and a maximum. 
The path of the moon is also not quite a circle as the numbers show. However, this path is inclined at an angle, about 5 degrees, to the plane, ecliptic, of the earth's revolution around the sun. Think about Sun and Earth moving on a flat "table" and the moon bobbing up and down above and below this table as it circles the Earth. The maximum the moon is above or below the plane is about 5^o.
Therefore, the moon's orbit crosses the ecliptic, that is the plane containing the earth and sun, at two points, when it crosses below from above, and the other way around. At both these points, the moon is in line with the earth and sun. A solar eclipse occurs when this happens. That is what makes the solar eclipse a rare event. See the figure showing the favourable and unfavourable conditions for both solar and lunar eclipses. Can you figure out the conditions for lunar eclipse yourself? Also why the moon is either a full moon or new moon at eclipse?

Some more numbers
Let us look at the numbers a little more closely--the moon orbits around the earth in about 27 days. However since earth also moves away from its position during this time, it takes approximately 29.5 days to view the moon in the same position. This is called a lunar month which is the time between one full moon day to another. During this lunar month the moon crosses the ecliptic plane twice. As the earth moves around the sun, these points also move relative to the line joining the earth and the sun. Since the earth completes its orbit around the sun in a year, the point of intersection comes close to the earth and sun line approximately once in six months. That is, a new moon occurs close to this line twice a year. 

Therefore the solar eclipse, either partial or total, occurs at least twice a year in some part of the earth depending on which part of the earth is facing the sun and moon at the time moon crosses the ecliptic. As you can see the geometry is complicated, since it involves the motion of earth around the sun, motion of the moon around the earth and the inclination of the moon's orbit with respect to earth's orbit.

Types of eclipses
Partial Eclipse: The most frequent of all the solar eclipses is the Partial Eclipse. This occurs when the sun and moon are not exactly in line with the earth. Only a part of the sun is covered by the moon. It may be visible from a larger area on the earth. Because of the brightness of the sunlight, it is visible only when nearly 90 percent of the sun is covered by the face of the moon.
Annular Eclipse: The annular eclipse occurs when the earth-moon-sun system are exactly in a line but the diameter of moon as viewed from earth is slightly smaller than the diameter of the sun. During totality the sun appears like a ring of fire. This happens when the sun is closest to the earth while the moon is farthest from the earth. At its minimum, the moon covers about 91 percent of the sun's disc. But the annular eclipse occurs as long as the coverage is less than 100 percent.
Total Eclipse: The total eclipse is similar to the annular eclipse except that the moon is now the closest and the sun is at the farthest. At its maximum the coverage by moon can be 105 percent of the sun's disc. However any coverage greater than 100 percent is considered total eclipse. The most dramatic event in a total solar eclipse is when a brilliant flash of light from the last unexposed part of the sun reaches us in a "diamond ring" effect.

Umbra and Penumbra
Both total and annular eclipse are experienced in the shadow of the moon on earth when the diameters of the moon and sun coincide. This is called Umbra and is typically about 250 kms when the moon is closest but is about 100-160 kms wide for near-total to annular eclipses. However, outside this shadow a partial eclipse is visible in the region of penumbra over a much larger region.
How long does a total eclipse last? Naturally it depends on how much of the moon's diameter is covering the sun. The longest duration may be as long as 7 minutes when moon is closest while the sun is at its farthest point, but typically it lasts about a few minutes which is simply the motion of moon across the surface of the sun. 

Ancient eclipses
Has it always been like this? Actually the answer is no. When the moon was formed it was much closer to the earth than it is now. Some studies suggest that it was about 22,500 kms away when it was formed about 4.5 billion years ago and it continues to move away from earth at the rate of about 4 cm a year due to the gravitational force exerted by the Sun. So in pre-historic times, when no human lived on earth, the solar eclipses were always total! It also means that in a billion years, since the moon is moving further away, there will not be any totality either--only annular or partial eclipses. So we are at a very special moment in the history of the moon, earth and sun: when their angular diameters are just the right amounts to permit us to see spectacular eclipses.
Not only the totality, the distance to moon from earth also impacts the rotation of the earth itself. There is clear evidence to suggest the days were only 19 hours long a few billions of years ago, and it could get longer in future.

Precautions
While solar eclipse is an interesting event, care must be taken while viewing the eclipse itself. You should not view it with the naked eye of course, just as we don't see sun directly. Even when the sun is covered the intensity is so high that you can injure your eye. You should always take care to view it with sufficient dark glasses.

Eclipses and history
The eclipses of the sun have also interesting applications in history. In many mythological stories and historical events there are references to eclipses. Using present day computers we may precisely compute all the total eclipses that have occurred in the past just as we can predict future eclipses. All we need is Newton's laws of motion and the geometry of the orbits. Combined with other forms of dating, we are in a position to make historical events more precise when combined with astronomical events like eclipses.
Eclipses have also played a big role in establishing some scientific theories like Einstein's theory of general relativity where the total solar eclipse of 1919 was used to establish the bending of a ray of light which was a prediction of Einstein's theory. Many gravity related phenomena can be understood during the eclipses. Studying eclipses is indeed a fascinating field.

References:
http://www.brighthub.com/science/space/articles/5985.aspx