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Beruniy’s Theory of Shadows
Предмет:  Математика, геометрия, алгебра  
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The great encyclopedic of the Middle Ages Abu Raihon Muhammad ibn Ahmad alBeruniy, who had been famous with the pseudonym "Ptolemy". Measuring the Distance on the Ground and the Height of a Mountain. The Distances between the Celestial Bodies.  
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1
Beruniy's Theory of Shadows
Preface
The great encyclopedic of the Middle Ages Abu Raihon Muhammad ibn Ahmad alBeruniy was born on the 3^{rd} 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 1^{st} 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 2→2. 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 1^{st} 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 abovementioned 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.
Beruniy lived in a sociopolitically 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” (AlBeruniy Osan alBokiya).
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 alBeruniy 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 1Here 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 MountainIf 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 2If 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. AlBeruniy 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 3We 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” (AlBeruniy 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.From the similarities, come out the equationsand with the help of these equations it is not difficult to find outBeruniy created a simple and very easy way of measuring the distances on the surface of the Earth in his book “Geodesy” (AlBeruniy, Geodesy, “Fan”, 1982, p. 167).For this he took square with the sides equal to 1, knocking nails into the points and, established long diopterical level to the point (Figure 4).Figure 4It is necessary to establish the square to the point in such a way so that the points lay on one straight line. Then we drop a stone from the point (with the Beruniy's words) and draw perpendicular.From , we shall obtain the equalities , . (1)Measuring of the EarthThe 1^{st} attempt to measure the diameter of the Globe is connected with the name of Eratosfenes (276196 BC). He determined the measurements of the Globe according to the position of the Sun in Aswan and Alexandria. According to his determination, when in Egypt the Sun is at Zenith, in Alexandria it reaches 7,2^{0} in relation to Zenith and from this, having specified, that on the globe the arch connecting Aswan and Alexandria, reaches 7,2^{0}, determines its conformity to 1/50 part of the large circumference of the Globe.So he multiplied the distance between Aswan and Alexandria by 50 and calculated the length of the Earth's large circumference. Ptolemy (II CE) also tried to calculate the measurements of the Globe and he expressed his ideas about the parameters of the Earth in his book “Geography”. But the scholars of the ancient times used “Stadiy” as a unit of measurement and as time passed by, especially during the academy “Bait ulHikma's” period (IX century), it turned out that due to the vagueness and contradictions among the units of measurement, there were many mistakes in the values of the Earth's measurements.Therefore, the chalif alMamun ibn arRashid charged the scholars of the academy “Bait ulHikma” with the task of determining the real measurements of the Globe. Observations were carried out in the Sinjor desert near Mosul and mostly the Middle Asian scholars took part in this investigation work. Under the leadership of our countryman alKhorezmiy, the scholars of “Bait ulHikma” fulfilled the task successfully.They determined that the radius of the Earth was equal to or 6406 km^{**} The modern scientist established that 1 mil = 4000 gaz = 1973,2 meters (see, Hints “Muscleman Measurements)
^{}^{)}, but in fact the actual radius of the Earth at the equator is equal to km and the Polar circle is equal to 6357 km.Describing in details the attempts of his forerunners in measuring the size of the Earth in his books “Geodesy”, “Konuniy Masudiy” Abu Raihon Beruniy (9731048) offered another new method. Beruniy wrote: “Only Greek and Indian versions of measurement of the Earth came up to us. Greek's and Indian's units of measurement were different, for example one mil which Indians used to measure the circumference of the Earth was equal to between one and eight our miles and the various measurements confused their thoughts for various scholars had different results. In each of their five “Siddihonta”s the value of the Earth circumference was different. But Greeks measured the circle of the Earth by one quantity, which was called “stadiya”. According to Galen, Eratosthenes carried out observations in Aswan and Alexandria, which are situated on the same meridian. Whenever the words in Galen's book “The Book of Provements” are combined with the words from Ptolemy's book “Entering the Art of Sphere” again the quantity will be different. Therefore, Mamun ibn arRashid charged the leaders of the science, who carried out the investigation in the Sinjor desert of Mosul to pay attention to such contradictions.If any man moves along a straight line on the Earth plane, he will move along the big circle of the Earth. But it is difficult to pass far distance along the straight line. That's why the scientists of the Mamun Academy took the pole of the Earth as a reference point (the polestar seems to be meant here). Being careful, they determined that one part of circle in 360^{0} was equal to miles.
“I myself was eager to measure the Earth and I chose a large plane land in Jurjon. But because of the inconvenient condition of the desert, the absence of the people, who could help me out, I found a high mountain with a smooth surface in the lands of India and used a different way of measuring it. From the top of the mountain I found the horizon of the Sky and Earth (Figure 5) and calculated its angle, which was equal to , measured the top of the mountain in two places and it was equal to 652 gaz, and calculated a half of a onetenth of a gaz.
Figure 5
Let the line which is perpendicular to the sphere of the Earth be the height of the mountain (Figure 6).
Figure 6
The centre of the Earth is , the line originating on the top of the mountain and going towards the horizon is , and we shall draw perpendicular to the horizon line. Consequently, we get triangle .
Its angle is a right angle and all other angles are known. Because the angle is the supplement angle of the horizon slope angle, that is, …” (alBeruniy, Konuniy Masudiy, book  5, 1973, p.p. 386387).
So according to the definition of sine, the radius of the Earth is calculated. From we get , from this
Or (2)Knowing the height of the mountain and the value of sin ^{ }^{ }Beruniy established, that the radius of the Earth is .In his book “Geodesy” Beruniy also wrote that, during Chaliph alMamun's marching to Greece (830832) he asked the mathematic scholar Abu Taiyib Sanad ibn Aly who also was with him, to ascend a mountain which stuck out of the East side of the Sea and from its top to determine the lower angle (for accuracy, during the sunset), and that when he fulfilled the task, they calculated the radius of the Earth using the lower angle and some additional angles (alBeruniy, “Geodesy”, “Fan”, 1982, p. 166).The Distances between the Celestial BodiesBeruniy writes: “Let the sun's diameter be denoted as,Figure 7
 The surface of the Earth,  gnomon object which produces a shadow, is the shadow diameter of this object on the Earth, is the centre of the shadow (Figure 7, in this drawing  full shadow,  partial shadow). If we know and , we shall obtain the distance from the Sun to the Earth and the diameter of the Sun Above mentioned quantities are distances, which can be measured being on the Earth.^{}^{)}.
Indeed, if we draw, then, and is known. Its ratio to is like the ratio of to. That is, and the triangle are known. The ratio of to is the same as the ratio of to . That is, from BZ is known and from that FZ is known (AlBeruniy, Mathematical and Astronomical Treatise, “Fan”, 1987, p. 210). According to Beruniy's proof, , from this
or
or
equalities come out. From:
or
or
equalities come out.
By these equalities we can determine easily the distance from the Earth to the Sun and the radius of the Sun: , (3)formulas are found.
Here .
If we mark the acute angles in the points and with and, by applying the theorem of sine to the formulas (3) can be changed into the following:
. (4)
Finally, by continuing the and straight lines we should find point and draw a straight line. Drawing, and using the definitions of we shall have (5)formulas, here .The formulas (3), (4), (5) are formulas of measuring the distance from the Earth to the celestial bodies the Moon and the Sun and their size. Unfortunately in practice when we use rudimentary equipment for measurement and as the Moon and the Sun are too far from us, the vales and or the angles and are almost equal to each other, and the denominators of the fractions in the formula are also almost equal to 0. Therefore, during Beruniy's period there was no possibility to use the formulas in measuring the astronomical objects. Beruniy wrote in details about the attempts in measuring the astronomical objects, about vagueness and confusion in the measurement, and although he offered theoretically simple and easy formulas for such measurements, he didn't introduce any definite figures concerning the measurements of the Moon and the Sun.But we can use the formulas suggested by Beruniy to measure the height of unapproachable objects from the surface of the Earth, which are far from us, also to measure the distance up to them. It would be just to call these formulas the formulas of Beruniy and to connect his theory of gnomon (shadows) with his name.
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