History of Math. mathemetics of islam Al-Biearl devised a mothod for determining

Question

History of Math. mathemetics of islam

Al-Biearl devised a mothod for determining the radius ref the earth by sighting the horiton from the of knowaheighth. That is al Birini assumed thal oneoould a, the angle of depression from the horizontal horizon (Fig. 9.38), Show which one sights the apparel determined by the fonttula Al-Bingni performed this measurement in a particular case a mountain of heighe 6523,18 cubits. Calculate the radun of the eamh in cubits. Assuming that a cubit equals 18" convert your answer to miles and value. Comment on the efficacy of i-Binuni s procedure Al Binni's method for calculating the earth's nadiu

Explanation / Answer

The same triangle can be drawn as below

Angle C = 90 - (90-) =

cos C = r/(r+h)

=> cos = r/(r+h)   

=> 1/cos = (r+h)/r = 1+h/r

=> h/r = 1/cos -1

=> h/r = (1-cos ) / cos

=> r/h = cos / (1-cos )

=> r = h cos / (1-cos )

= 34' h = 652.318 cubits

=> r = 652.318 cos 34 / (1-cos34)

=> r = 652.318 * 20445.7 = 13337098.13 cubits

= 13337098.13 / 3520 = 3789 miles.

The current measured radius is 3959 miles. So Mr. Al-Biruni's measure was very close.

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