Beruniy’s Theory of Shadows

**Beruniy’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.

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.

From the similarities_{}, _{} come out the equations

_{}

_{}

Beruniy created a simple and very easy way of measuring the distances on the surface of the Earth in his book “Geodesy” (Al-Beruniy, 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 4**

It 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)

The 1^{st}
attempt to measure the diameter of the Globe is connected with the name of Eratosfenes
(276-196 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 ul-Hikma’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 al-Mamun ibn ar-Rashid charged the scholars of the academy “Bait ul-Hikma” 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 al-Khorezmiy, the scholars of “Bait ul-Hikma” fulfilled the task successfully.

They determined that the
radius of the Earth was equal to _{} or 6406 km*^{)}, 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 (973-1048) 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 ar-Rashid 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 pole-star
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 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

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 al-Mamun’s marching to Greece (830-832) 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 (al-Beruniy, “Geodesy”, “Fan”, 1982, p. 166).

**The
Distances between the Celestial Bodies**

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[*]^{)}.

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.

_{} , _{} (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)

_{} (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.

*
The modern scientist established
that 1 *mil* = 4000 *gaz* = 1973,2 *meters* (see, Hints “Muscleman
Measurements)

[*] Above mentioned _{} quantities are distances,
which can be measured being on the Earth.

*Больше работ по теме:*

Алгебраическая проблема собственных значений

Реферат

Аппроксимация функций

Реферат

Аркфункции

Реферат

Асимптоты (определение, виды, правила нахождения)

Реферат

Афинные преобразования на плоскости

Реферат

Реферат

Аппроксимация функций

Реферат

Аркфункции

Реферат

Асимптоты (определение, виды, правила нахождения)

Реферат

Афинные преобразования на плоскости

Реферат

*Предмет: **Математика*

*Тип работы: **Реферат*

*Новости образования*

Российский государственный социальный университет реорганизует сеть своих филиалов

22 Июля 2016 16:26

Более 70 представителей крупных вузов АСЕАН подтвердили участие в форуме во Владивостоке

22 Июля 2016 12:51

Абызов: публикация первичных статсведений снизит бюрократическую нагрузку на школы

22 Июля 2016 11:53

В Чечне свыше 200 школьников сдали ЕГЭ с результатом свыше 90 баллов - министр

22 Июля 2016 05:36

Замглавы Минобрнауки РФ: задача сократить число людей с высшим образованием не стоит

21 Июля 2016 20:19

22 Июля 2016 16:26

Более 70 представителей крупных вузов АСЕАН подтвердили участие в форуме во Владивостоке

22 Июля 2016 12:51

Абызов: публикация первичных статсведений снизит бюрократическую нагрузку на школы

22 Июля 2016 11:53

В Чечне свыше 200 школьников сдали ЕГЭ с результатом свыше 90 баллов - министр

22 Июля 2016 05:36

Замглавы Минобрнауки РФ: задача сократить число людей с высшим образованием не стоит

21 Июля 2016 20:19

КОНТАКТНЫЙ EMAIL: MAIL@SKACHAT-REFERATY.RU

**Скачать реферат **
© 2018 | Пользовательское соглашение

ПРОФЕССИОНАЛЬНАЯ ПОМОЩЬ СТУДЕНТАМ