Archimedes' theorem on spheres and circular cylinders

The set of the Archimedes sphere and circular cylinder is a theorem of the solid geometry , a partial area of the geometry . It goes back to Archimedes of Syracuse (approx. 287–212 BC) and his work About Sphere and Cylinder (original title περὶ σφαίρας καὶ κυλίνδρου, Latinized De Sphaera et Cylindro ). With it, Archimedes was the first to determine the exact relationship between the volume and the surface of the sphere and the circular cylinder with the help of methods that are considered the forerunners of the methods of modern integral calculus . The theorem is therefore one of the great results of mathematics.

formulation

The ratio of the volume of a circumscribed cylinder ( ) to the volume of a sphere ( ) with a radius is ${\displaystyle V_{Z}}$ ${\displaystyle V_{K}}$ ${\displaystyle r}$ ${\displaystyle V_{Z}:V_{K}=2\cdot {\frac {3}{4}}\;\Rightarrow V_{Z}:V_{K}=3:2}$

The sentence can be stated as follows:

For a sphere, and a circular cylinder whose base a largest sphere circle of the ball and the height of the ball diameter corresponding to which are the areas of the surfaces and the volumes of the two bodies in each case in the same ratio and where: ${\displaystyle O}$ ${\displaystyle V}$

${\displaystyle O_{Kreiszylinder}:O_{Kugel}=V_{Kreiszylinder}:V_{Kugel}=3:2}$

Derivation

In Book I of On Sphere and Cylinder, Archimedes presents the above sentence as a corollary to two sentences which he previously formulated as Proposition 33 and 34 and which say the following:

Proposition 33:

For a sphere, the area of ​​the spherical surface is four times as large as the area of ​​a largest spherical circle.

Proposition 34:

For a sphere, the volume is four times as large as the volume of a circular cone , the base area of ​​which corresponds to a largest spherical circle and whose height corresponds to the spherical radius .

Related sentence

From Archimedes 'theorem about spheres and circular cylinders the following theorem results, which is sometimes also referred to as Archimedes' theorem :

The volume of a hemisphere is equal to the difference between the volumes of the surrounding circular cylinder and the circular cone contained therein of the same height and the same base area.

literature

• Archimedes: works. Translated and annotated by Arthur Czwalina . In the appendix: Circular measurement / translated by F. Rudio - Archimedes' methodology from the mechanical theorems / translated by JL Heiberg and commented by HG Zeuthen. 3rd, unchanged reprographic reprint . 3. Edition. Scientific Book Society, Darmstadt 1972, ISBN 3-534-02029-4 .
• EJ Dijksterhuis : Archimedes (translated by C. Dikshoorn) . Princeton University Press, Princeton NJ 1987, ISBN 0-691-08421-1 .
• Howard Eves: Great Moments in Mathematics (Before 1650) (=  The Dolciani Mathematical Expositions . Volume 5 ). The Mathematical Association of America , Washington 1980, ISBN 0-88385-305-1 .
• H. Fenkner: Mathematical teaching work. According to the guidelines for the curricula of the higher schools in Prussia, revised by Dr. Karl Holzmüller. Geometry. Edition A in 2 parts. I. part . 12th edition. Published by Otto Salle, Berlin 1926.
• Herbert Meschkowski : The way of thinking of great mathematicians. A path to the history of mathematics . Vieweg Verlag, Braunschweig 1990, ISBN 3-528-28179-0 .

Individual evidence

1. ↑ Dijksterhuis: p. 46
2. ↑ ^{a b} Eves: p. 85
3. ^ The mathematician Howard Eves, for example, writes in his Great Moments in Mathematics (Before 1650) , p. 88: Surely, from almost any point of view, we have here in Archimedes' work a truly GREAT MOMENT IN MATHEMATICS.
4. ^ Archimedes: Works: p. 117
5. ^ Dijksterhuis: p. 182
6. ^ Archimedes: pp. 114-117
7. ↑ Dijksterhuis: pp. 180-181
8. ↑ Meschkowski: p. 33
9. ↑ Fenkner / Holzmüller: p. 347