From

,

we shall obtain the equalities

, . (1)

**Measuring of the Earth**

The 1st 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 Earths 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-Hikmas” 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 Earths 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. Greeks and Indians 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 Galens book “The Book of Provements” are combined with the words from Ptolemys 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. Thats 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.*

**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

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 al-Mamuns 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**

Beruniy writes: “Let the suns 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).

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 Beruniys 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 Beruniys 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 didnt 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.