Math, asked by AmitJunior, 1 year ago

short note on the volume of the sphere in greek mathematic

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Answered by rajesh205
1
The diameter = dThe radius = rThe periphery = pThe area of sphere surface = sThe area of the greatest circle in the sphere = S1Volume of the sphere = vVolume of right-cone = V1The area of cone-base = S4Cone height = H4Volume of a right-cylinder = V2The area of the cylinder base = S2The cylinder height = H2The area of the side surface of a circular right-cylinder = S3Volume of the cube = V3Volume of vertical ring = V4

1. Introduction



Figure 1: A sphere circumscribed in a cylinder: the sphere has two thirds of the volume and surface area of the circumscribing cylinder.

The mathematicians of the Arabic civilization endeavoured to find a rule through which the sphere volume can be calculated. Some of them had got a cubic measure of it in comparison with the known volume of solids such as the cone, the cylinder, and so on. Likewise, they obtained a figure of the volume by finding out a relationship that links the elements of the sphere such as its surface to its radius. Consequently, the value of π played an important role in the accuracy of the cubic measure. Thus, some of the mathematicians of the Islamic tradition had the right measure, whilst others had the wrong one and proposed erroneous values.

Basically, this research is concerned with the cubic measure of the volume of the sphere in the mathematical tradition of the Arabic civilization. We begin by surveying the ancient contribution with a presentation of Archimedes' results in the Greek tradition on this issue, besides a survey of the development of cubic measure of sphere in Chinese mathematics. The fundamental points of our study discuss the following subjects. After a historical introduction on the volume of the sphere in Greek and Chinese mathematics, we present a thorough survey of the same topic in the mathematics of the Arabic-Islamic civilization from the 9th to the 17th century, especially in the works of Banu Musa, Abu ‘l-Wafa al-Buzgani, Al-Karaji, Ibn Tahir al-Baghdadi, Ibn al-Haytham, Ibn al-Yasamin, Al-Khawam al-Baghdadi, Kamal al-Din al-Farisi, Jamshid al-Kashi, and Baha' al-Din al-‘Amili. Finally, a set of conclusions is deduced.

2. Historical survey

2.1. The volume of the sphere in Greeks mathematics: Archimedes



Figure 2: Two manuscript pages of the Greek text of Archimedes' Sphere and Cylinder (Source).

The mathematical problem of the measure of the volume of the sphere is discussed by Archimedes in his known book The Sphere and Cylinder. Archimedes is considered as the best Greek scientist in the fields of mathematics and mechanical engineering. He died in 212 BCE. His scientific legacy consists in a group of influential texts presenting several important theories [2].

His book The Sphere and Cylinder [3] is composed of two parts. In the first one, Archimedes presented a number of definitions and postulates. Then he discussed the surfaces and volumes of some solids, such as the surface area of the sphere as well as its volume. In the second part, he developed some constructions and demonstrations related to the theories that he had mentioned in the first part.

Archimedes gives a rule of the volume of the sphere in comparison with the cone and cylinder. In the words of Nasir al-Din al-Tusi's edition and recension of his book The Sphere and Cylinder, Archimedes' first formula is formulated as follows:

"Each sphere is equal to four times a cone whose base is equal to the greatest circle in that sphere, and the height [of this cone] is equal to the radius of that sphere. [4]"


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