Aryabhatta and Eclipses

Yesterday was a solar eclipse and the usual superstitions about eclipses were doing the rounds of internet. Here is what Aryabhatta had figured out in 400-500AD about eclipses.

स्फुटशशिमासान्तेऽर्क पातासन्नो यदा प्रविशतीन्दुः |

भूच्छायाँ  पक्षान्ते तदाधिकोनं ग्रहणमध्यम् ॥

 At the end of the lunar month, the Moon, lying near a node (of the moon), enters the Sun, or, at the end of lunar fortnight, enters the Earth's Shadow, it is more or less the middle of an eclipse. (Solar eclipse in the format case and lunar eclipse in the latter case).

भूरविविवरं विभजेद् भूगुणितं तु रविभूविशेषेण|

भूच्छायादीर्घत्वं लब्धं भूगोलविष्कम्भात्  ॥

 Multiply the distance of the Sun from the Earth by the diameter of the Earth and divide the product by the difference between the diameters of the Sun and the Earth : the result is the length of the shadow of the Earth (i.e. the distance of the vertex from the Earth's shadow) from the diameter of the Earth (i.e. from the center of the Earth)

The old Hindu method of deriving this formula, called the The Lamp and Shadow method, (प्रदिपच्छ्यकर्म) is as follows.

 

In the figure above, S is the center of the Sun and E is the center of Earth. SA and EC are drawn perpendicular to SE and denote the semi-diameter of the Sun and Earth respectively. BC is parallel to SE. V is the point where SE and AC produced meet each other. If SA is considered to be a lamp post, EC is a gnomon and EV is the shadow cast by the gnomon due to the light of the lamp. Consequently EV is the length of Earth's shadow from the diameter of the earth.

The triangles CEV and ABC are similar.

Aryabhatta extensively studied phenomenon like eclipses and described various details about  it in his book. Looks like superstitious crowd is none the wiser after 1500 years.