# Beruniy’s Theory of Shadows — страница 2

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. 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). 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.

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