Pappus's centroid theorem

In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.

The theorems are attributed to Pappus of Alexandria^[a] and Paul Guldin.^[b]

The first theorem

The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by its geometric centroid.

A=sd.\,

For example, the surface area of the torus with minor radius r and major radius R is

A=(2\pi r)(2\pi R)=4\pi ^{2}Rr.\,

The second theorem

The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by its geometric centroid.

V=Ad.\,

For example, the volume of the torus with minor radius r and major radius R is

V=(\pi r^{2})(2\pi R)=2\pi ^{2}Rr^{2}.\,

This special case was derived by Johannes Kepler using infinitesimals.^[c]

Generalizations

The theorem can be generalized for arbitrary curves and shapes, under appropriate conditions.^[5]

Footnotes

^ See:^[1]

They who look at these things are hardly exalted, as were the ancients and all who wrote the finer things. When I see everyone occupied with the rudiments of mathematics and of the material for inquiries that nature sets before us, I am ashamed; I for one have proved things that are much more valuable and offer much application. In order not to end my discourse declaiming this with empty hands, I will give this for the benefit of the readers:

The ratio of solids of complete revolution is compounded of (that) of the revolved figures and (that) of the straight lines similarly drawn to the axes from the centers of gravity in them; that of (solids of) incomplete (revolution) from (that) of the revolved figures and (that) of the arcs that the centers of gravity in them describe, where the (ratio) of these arcs is, of course, (compounded) of (that) of the (lines) drawn and (that) of the angles of revolution that their extremities contain, if these (lines) are also at (right angles) to the axes. These propositions, which are practically a single one, contain many theorems of all kinds, for curves and surfaces and solids, all at once and by one proof, things not yet and things already demonstrated, such as those in the twelfth book of the First Elements.

— Pappus, Collection, Book VII, ¶41‒42
^ "Quantitas rotanda in viam rotationis ducta, producit Potestatem Rotundam uno gradu altiorem, Potestate sive Quantitate rotata."^[2] That is: "A quantity in rotation, multiplied by its circular trajectory, creates a circular power of higher degree, power, or quantity in rotation." ^[3]
^ Theorem XVIII of Kepler's Nova Stereometria Doliorum Vinariorum (1615):^[4] "Omnis annulus sectionis circularis vel ellipticae est aequalis cylindro, cujus altitudo aequat longitudinem circumferentiae, quam centrum figurae circumductae descripsit, basis vero eadem est cum sectione annuli." Translation:^[3] "Any ring whose cross-section is circular or elliptic is equal to a cylinder whose height equals the length of the circumference covered by the center of the figure during its circular movement, and whose base is equal to the section of the ring."

References

^ Pappus of Alexandria (1986) [c. 320]. Jones, Alexander (ed.). Book 7 of the Collection. New York: Springer-Verlag. doi:10.1007/978-1-4612-4908-5. ISBN 978-1-4612-4908-5.
^ Guldin, Paul (1640). "De centro gravitatis trium specierum quanitatis continuae". Vol. 2. Vienna: Gelbhaar, Cosmerovius. p. 147. Retrieved 2016-08-04.
^ ^a ^b Radelet-de Grave, Patricia (2015-05-19). "Kepler, Cavalieri, Guldin. Polemics with the departed". In Jullien, Vincent (ed.). Seventeenth-Century Indivisibles Revisited. Science Networks. Historical Studies. Vol. 49. Basel: Birkhäuser. p. 68. doi:10.1007/978-3-319-00131-9. ISBN 978-3-3190-0131-9. ISSN 1421-6329. Retrieved 2016-08-04.
^ Kepler, Johannes (1870) [1615]. "Nova Stereometria Doliorum Vinariorum". In Frisch, Christian (ed.). Joannis Kepleri astronomi opera omnia. Vol. 4. Frankfurt: Heyder and Zimmer. p. 582. Retrieved 2016-08-04.
^ Goodman, A. W.; Goodman, G. "Generalizations of the Theorems of Pappus". JSTOR. The American Mathematical Monthly. Retrieved 2014-06-28.

External links

Weisstein, Eric W. "Pappus's Centroid Theorem". MathWorld.

[2] See:^[1]

They who look at these things are hardly exalted, as were the ancients and all who wrote the finer things. When I see everyone occupied with the rudiments of mathematics and of the material for inquiries that nature sets before us, I am ashamed; I for one have proved things that are much more valuable and offer much application. In order not to end my discourse declaiming this with empty hands, I will give this for the benefit of the readers:

The ratio of solids of complete revolution is compounded of (that) of the revolved figures and (that) of the straight lines similarly drawn to the axes from the centers of gravity in them; that of (solids of) incomplete (revolution) from (that) of the revolved figures and (that) of the arcs that the centers of gravity in them describe, where the (ratio) of these arcs is, of course, (compounded) of (that) of the (lines) drawn and (that) of the angles of revolution that their extremities contain, if these (lines) are also at (right angles) to the axes. These propositions, which are practically a single one, contain many theorems of all kinds, for curves and surfaces and solids, all at once and by one proof, things not yet and things already demonstrated, such as those in the twelfth book of the First Elements.

— Pappus, Collection, Book VII, ¶41‒42

[5] "Quantitas rotanda in viam rotationis ducta, producit Potestatem Rotundam uno gradu altiorem, Potestate sive Quantitate rotata."^[2] That is: "A quantity in rotation, multiplied by its circular trajectory, creates a circular power of higher degree, power, or quantity in rotation." ^[3]

[7] Theorem XVIII of Kepler's Nova Stereometria Doliorum Vinariorum (1615):^[4] "Omnis annulus sectionis circularis vel ellipticae est aequalis cylindro, cujus altitudo aequat longitudinem circumferentiae, quam centrum figurae circumductae descripsit, basis vero eadem est cum sectione annuli." Translation:^[3] "Any ring whose cross-section is circular or elliptic is equal to a cylinder whose height equals the length of the circumference covered by the center of the figure during its circular movement, and whose base is equal to the section of the ring."

[1] Pappus of Alexandria (1986) [c. 320]. Jones, Alexander (ed.). Book 7 of the Collection. New York: Springer-Verlag. doi:10.1007/978-1-4612-4908-5. ISBN 978-1-4612-4908-5.

[3] Guldin, Paul (1640). "De centro gravitatis trium specierum quanitatis continuae". Vol. 2. Vienna: Gelbhaar, Cosmerovius. p. 147. Retrieved 2016-08-04.

[RdG2015-4] Radelet-de Grave, Patricia (2015-05-19). "Kepler, Cavalieri, Guldin. Polemics with the departed". In Jullien, Vincent (ed.). Seventeenth-Century Indivisibles Revisited. Science Networks. Historical Studies. Vol. 49. Basel: Birkhäuser. p. 68. doi:10.1007/978-3-319-00131-9. ISBN 978-3-3190-0131-9. ISSN 1421-6329. Retrieved 2016-08-04.

[6] Kepler, Johannes (1870) [1615]. "Nova Stereometria Doliorum Vinariorum". In Frisch, Christian (ed.). Joannis Kepleri astronomi opera omnia. Vol. 4. Frankfurt: Heyder and Zimmer. p. 582. Retrieved 2016-08-04.

[generalizations-8] Goodman, A. W.; Goodman, G. "Generalizations of the Theorems of Pappus". JSTOR. The American Mathematical Monthly. Retrieved 2014-06-28.

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