''On the Sphere and Cylinder'' ( el, Περὶ σφαίρας καὶ κυλίνδρου) is a work that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...

and the volume of the contained ball and the analogous values for a

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...

, and was the first to do so.

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

r

be the radius of the sphere and cylinder, and

h

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: :

A_C = 2 \pi r^2 + 2 \pi r h = 2 \pi r ( r + h ).\,

and that the volume of the same is: :

V_C = \pi r^2 h. \,

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: :

A_S = 4\pi r^2.\,

The result for the volume of the contained ball stated that it is two-thirds the volume of a

circumscribe In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

, meaning that the volume is :

V_S = \frac\pi r^3.

When the inscribing cylinder is tight and has a height

h = 2r

, so that the sphere touches the cylinder at the top and bottom, he showed 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 (as it was allegedly inscribed on his tombstone discovered by

Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the esta ...

), and so he asked for a sketch of a sphere inscribed in a cylinder to be inscribed on his grave. Later,

Roman Roman or Romans most often refers to: *Rome, the capital city of Italy *Ancient Rome, Roman civilization from 8th century BC to 5th century AD *Roman people, the people of ancient Rome *'' Epistle to the Romans'', shortened to ''Romans'', a lette ...

philosopher

Marcus Tullius Cicero Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the esta ...

discovered 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

frustum In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are ...

s 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 In textual studies, a palimpsest () is a manuscript page, either from a scroll or a book, from which the text has been scraped or washed off so that the page can be reused for another document. Parchment was made of lamb, calf, or kid skin an ...

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 ''The Method of Mechanical Theorems'' ( el, Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as ''The Method'', is one of the major surviving works of the ancient Greek polymath Ar ...

, in which he describes a method to determine volumes which involves balances, centers of mass and infinitesimal slices.

Notes

References

* * *S. H. Gould, The Method of Archimedes, The American Mathematical Monthly. Vol. 62, No. 7 (Aug. - Sep., 1955), pp. 473–476 * Lucio Lombardo Radice, ''La matematica da Pitagora a Newton'', Roma,

Editori Riuniti Editori Riuniti is an Italian publishing house based in Rome that publishes books and magazines on the history of socialism, socialist thought, physics and mathematics theory, and the history of Central and Eastern Europe and the Balkans. Histor ...

, 1971. * Attilio Frajese, ''Opere di Archimede'', Torino, U.T.E.T., 1974. {{DEFAULTSORT:On The Sphere And Cylinder Ancient Greek mathematical works Euclidean geometry Works by Archimedes 225 BC

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Notes

References