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The great encyclopedic of the Middle Ages Abu Raihon Muhammad ibn Ahmad al-Beruniy, 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.

**Ключевые слова: ***Beruniy encyclopedic Ptolemy*

**Рубрика: ***Математика*

**Предмет: ***Геометерия*

**Вид: ***реферат*

**Язык: ***английский*

**Прислал: ***Yusupov Yusufboy Ergashevich*

**Дата добавления: ***21.03.2008*

1

**Beruni****y**'**s Theory of Shadows**

**Preface**

The great encyclopedic of the Middle Ages Abu Raihon Muhammad ibn Ahmad al-Beruniy 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 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 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 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.

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.

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)

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 pole-star seems to be meant here). Being careful, they determined that one part of circle in 360^{0} was equal to *mil**es*.

“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 one-tenth 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, …” (al-Beruniy, Konuniy Masudiy, book - 5, 1973, p.p. 386-387).

So according to the definition of sine, the radius of the Earth is calculated. From we get , from this

** 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 (Al-Beruniy, 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.

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)

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 Beruni****y** and to connect his theory of gnomon (shadows) with his name.