# Beruniy’s Theory of Shadows — страница 4

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-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 Beruniy 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 Sun1). 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. 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. ** The modern scientist established that 1 mil = 4000 gaz = 1973,2 meters (see, Hints “Muscleman Measurements) 1 Above mentioned quantities are distances, which can be measured being on the Earth. 1

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