圆柱容球 (公元前 225)

圆柱内切球体是由古希腊著名数学家及科学家阿基米德创立，圆柱容球可得出圆柱体表面积，并且还可以同时将球体表面积巧妙得出。

历史

古希腊著名的数学家阿基米德（Archimedes）是史上杰出的数学家。按照他生前遗愿，他墓碑刻了“圆柱容球”的几何图形（义即刻出一个二维的模型图）。

阿基米德是城防者，他挡住许多罗马军队的攻击，但最后在叙拉古城庆祝节日时罗马军队还是乘防守疏忽之机闯入。

当罗马军队冲进城时，玛尔凯路曾下令不要杀害这位伟大的物理学家。可是那时，阿基米德正在他的实验室里画他的图形。士兵冲进后，脚踏声惊扰了他。这种惊扰，使他惊醒过来，愤怒喊道：“喂！你弄坏了我的图画，赶快跑开些！”结果，他的愤慨激怒了罗马兵，阿基米德便死于刀下。

规律

圆柱容球时，球直径与圆柱的高和底直径相等。设圆柱底半径为r，圆柱体积即为V柱=πr²×2r=2πr³。阿基米德发现并证明了球体积公式是V球=

r³，所以V球=⅔V柱。即当圆柱容球时，球体积是圆柱体积的⅔。

此外，阿基米德还发现，圆柱容球时，球表面积正好也是圆柱表面积的⅔。

On the Sphere and Cylinder (Greek: Περὶ σφαίρας καὶ κυλίνδρου; Perì sphaíras kaì kulíndrou) is a treatise that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so.

Contents

The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder. Let be the radius of the sphere and cylinder, and be the height of the cylinder, with the assumption that the cylinder is a right cylinder—the side is perpendicular to both caps. In his work, Archimedes showed that the surface area of a cylinder is equal to:

and that the volume of the same is:

When the inscribing cylinder's height is equal to its diameter, so that the sphere touches the cylinder at the top and bottom, can be substituted:

On the sphere, he showed that the surface area is four times the area of its great circle. In modern terms, this means that the surface area is equal to:

and that the volume of the contained ball is two-thirds the volume of a circumscribed cylinder, meaning that the volume is

Thus showing that both the volume and the surface area of the sphere were two-thirds that of the cylinder. This implies the area of the sphere is equal to the area of the cylinder minus its caps. This result would eventually lead to the Lambert cylindrical equal-area projection, a way of mapping the world that accurately represents areas. Archimedes was particularly proud of this latter result, and asked for a sketch of a sphere inscribed in a cylinder to be inscribed on his tombstone. Later, Roman philosopher Marcus Tullius Cicero rediscovered the tomb, which had been overgrown by surrounding vegetation.

The argument Archimedes used to prove the formula for the volume of a ball was rather involved in its geometry, and many modern textbooks have a simplified version using the concept of a limit, which did not exist in Archimedes' time. Archimedes used an inscribed half-polygon in a semicircle, then rotated both to create a conglomerate of frustums in a sphere, of which he then determined the volume.

It seems that this is not the original method Archimedes used to derive this result, but the best formal argument available to him in the Greek mathematical tradition. His original method likely involved a clever use of levers. A palimpsest stolen from the Greek Orthodox Church in the early 20th century, which reappeared at auction in 1998, contained many of Archimedes works, including The Method of Mechanical Theorems, in which he describes a method to determine volumes which involves balances, centers of mass and infinitesimal slices.

Hat-box theorem

The proof of the surface-area result also shows that any zone of the sphere between two parallel planes has the same area as the portion of the circumscribed cylinder's lateral surface between the same planes; the projection of the sphere onto the cylinder is therefore area-preserving. This is known as the hat-box theorem. It is the basis of the Lambert cylindrical equal-area projection. In quantum mechanics it implies that uniformly distributed pure states of a qubit on the Bloch sphere yield uniformly distributed measurement probabilities.

Recent discoveries

A missing page of the manuscript was recovered in 2026. While one side of the page has visible drawings, the other is completely covered by later additions of biblical paintings.