Beruniy’s Theory of Shadows
Preface
The great encyclopedic of the Middle Ages Abu Raihon Muhammad ibn Ahmad al-Beruniy was born on the 3rd of Zulhijja month in 362 hijra year (September 4, in 973) in the capital of South Khorezm - the Kat (close to Beruniy).
He had a great capability to science from his very early years. He was taught by the outstanding scholar Abu Nasr Mansur ibn Irok, who had been famous with the pseudonym “Ptolemy” at that time. Abu Nasr ibn Irok devoted 12 of his works in the field of astrology, geometry and mathematics to Abu Rayhon Beruniy which meant respect and admitted his disciples erudition.
In 995 the Amir of Gurganj Mamun the 1st ibn Muhammad Siavushiy occupied the Kat, which was the last fortress of Afrighians’ dynasty, and declared himself the king of United Khorezm. Due to restless situation in Khorezm Beruniy had to leave home land at the age of 22. He lived in towns like Gurgan and Rai in Iran, where he became acquainted with the famous scientist Abu Mahmud Khujandiy, as well as he established relations with Ibn Sino through correspondence and exchanged opinions with him regarding some scientific problems.
In 998 after the death of Mamun the 1st Ali Mamun ascended the throne and the political situation became more stable in Khorezm. Ali ibn Mamun was a man of science, culture and education. He charged the supervisor of Beruniy Abu Nasr ibn Irok to gather the scholars in the palace and to create conditions for scientific discussions. Approximately at the end of 1003 and at the beginning of 1004 according to ibn Irok’s invitation Beruniy returned to Gurganj and his prosperious scientific period began. He managed the Mamun Academy, prepared disciples and wrote research works in various domains of science. Abu Raihon Beruniy authored about 152 scientific works, but only 30 of them have been passed to present generation.
Beruniy’s scientific activity was polyhedral with mathematics, physics, mineralogy, ethnography and history remained his main focus. His works consisting of 11 books “The Konuniy Masudiy”, “Geodesy”, “Mineralogy”, “The Monuments left by the Ancient People” (devoted to ethnography) and “Hindiston” have been used as a manual for many centuries and even presently scientists are using them in their investigations.
This brochure is a pearl of Beruniy’s activity, which concerns geometry. In this booklette together with introducing some of the Beruniy’s mathematical investigations, discussions, some commentaries are also given. Unlike his forerunners Beruniy had a capability of thinking logically, reflecting the importance of chosen problem correctly and finding out a simple form of expression for his ideas. Beruniy’s above-mentioned essential qualities are seen vividly in his works which are remained in the history as determining of the distance from the Earth to the Moon and the Sun, where he showed the attempts, vagueness and confusions of his forerunners, discussions about the unit of measurement, gave vivid explanations to them and demonstrated the character of a leader in the field of natural sciences.
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Beruniy lived in a socio-politically complicated period, which is related to the occupation of the town Kat, where Beruniy was born and lived, the removal of the capital to Gurganj, roving, destabilization, arrogance and corruption in the society. These feelings can be seen in the following Beruniy’s sentences and aphorisms:
Though some men being in a very lower level in the science, behave themselves arrogance, even they dare to insult somebody. Somehow there is richness which turns the poorness into disgrace only” (Al-Beruniy Osan al-Bokiya).
Richness can be lost but education will remain with you forever.
Who hurts harmless scholar and enjoy it, he will be punished by the Allah.
If the branches of evil are many, its source is bribery and ignorance.
Our great ancestor al-Beruniy died in 362 Hijra year, i.e. on the December 13, in 1048 at the age of 75 in the town of Ghazna. About the last days of the scholar the following words were mentioned in the book “Nomoiy Donishvoron”, which was published in 1878 in Teheran: “Beruniy had a serous illness, he was living his last days. When he regained consciousness for a moment, he could see his friend Abdulhasan Valvolijiy. Beruniy asked his friend Abdulhasan to comment on the new opinion about the heritage. Abdulhasan replied that it wasn’t appropriate moment for it. Then, looking at his friend Beruniy said, “Oh, my dear friend, every person is sure to die, but my mind is making me now to understand the importance of the problem, which you told me some years ago. So it is better to die knowing than to die not knowing,” – answered Beruniy. His friend Abdulhasan began to comment on the things, which he had asked him to explain. In some moments Beruni fell asleep forever. And this was his last talk about the science.” How a good death! It was a death worthy of great person, it was a death of a person who had spent his life profoundly, and it was a death of a scholar who had been satisfied with his activity.
Introduction
Measurements of the celestial bodies (the Earth, the Moon, the Sky), measuring the distance from the point where we stand up to them attracted attention of the most scientists from the very ancient times. The scholars from the Khorezm Mamun Academy were interested in this problem as well and first of all, its head Abu Raihon Beruniy made a great contribution to this field.
In order to determine the measures of the Earth, the Moon and the Sun, and to determine the distances from the Earth to the Moon and to the Sun, Beruniy created a theory of shades which was perfect from the mathematical point of view. The essence of this theory is that, if we put a circle with the radius equal to 
towards the Sun in some 
distance from the place where we stand, the circle’s full shade (in this point the circle covers the Sun completely) and partial shade (in this point the circle covers some parts of the Sun) fall onto the Earth. Based on the measurements of these shades Beruniy created a method of calculating the distance from the Earth to the Sun as well as the measures (size) of the Sun.
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Figure 1
Here 
is the diameter of the Sun, 
is the diameter of the circle, cover (gnomon), 
- the area of the full shade of gnomon, 
and 
- the area of the partial shade of .
In this brochure we shall see and analyse the methods created by Beruniy for measuring the radius of the Earth and the distance from the point where we stand to the object, which is far from us; also we pay attention to using them in modern practice, in making mathematical manuals. We’ll also cite Beruniy’s own sentences, some passages from his books. While reading them, the reader will for sure come to conclusion that Beruniy was and remains a great mathematician from the point of view of modern science as well.
Measuring the Distance on the Ground and the Height of a Mountain
If we are to measure the height 
of some standing object (for ex. the height of a minaret) we go to a point 
which is at some distance from the object (Figure 2), measure the angle 
using a leveling instrument and from the equation 
or 
we can determine the length of easily.
Figure 2
If we can’t get to the basis of the object 
, for example, if we are to measure the height of an object on the other side of a river or the height of the plateau, the task will become more complicated. Al-Beruniy wrote about such cases in details in his work “Gnomonics” and gave the solutions to such problems by using the Indian mathematician and astronomer Brahmagupta’s method from “Brahmassiddhta” (one of the greatest books of Brahmagupta). According to his opinion in order to measure the height of the objects with inaccessible basis, we should choose a flat place at some distance from the object (Figure 3).
Figure 3
We select a point 
on a flat site; we place a gnomon 
vertically and find out its full shade 
. In order to find out point 
, which defines the full shade of 
gnomon Beruniy stated the following: “ … then one should reach back from the point 
to such a place, where from level’s diopter 
and 
should be seen in the same landmark … as the point is on the ground you can either lie on the ground or dig a pit of depth equal to your stature, descend into the pit and look through the diopter” (Al-Beruniy Mathematical and Astronomical Treatise, “Fan”, 1987, p. 244).
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After having determined the point , we raise at point 
another gnomon 
, which is equal to gnomon and find out its shade 
in the same way as the with previous one.
From the similarities 
, 
come out the equations
