Beruniy’s Theory of Shadows
Beruniy’s Theory of Shadows
Theory of Shadows
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.
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.
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.
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.
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.
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).
We select a point on a flat site; we place a
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
and with the help of these equations it is not difficult to
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
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.
we shall obtain the equalities
, . (1)
of the Earth
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 7,20 in
relation to Zenith and from this, having specified, that on the globe the arch
connecting Aswan and Alexandria, reaches 7,20, 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
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
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
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 3600 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.
Let the line which is perpendicular to
the sphere of the Earth be the height of the mountain (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.
Distances between the Celestial Bodies
Beruniy writes: “Let the sun’s
diameter be denoted as,
- 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
equalities come out. From:
equalities come out.
By these equalities we can determine easily the distance from
the Earth to the Sun and
the radius of the Sun:
formulas are found.
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:
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
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
[*] Above mentioned quantities are distances,
which can be measured being on the Earth.