Editing Pappus's centroid theorem (section)

{{Short description|Results on the surface areas and volumes of surfaces and solids of revolution}}
[[File:Pappus centroid theorem areas.gif|thumb|right|400px|The theorem applied to an open cylinder, cone and a sphere to obtain their surface areas. The centroids are at a distance ''a'' (in red) from the axis of rotation.]]

In mathematics, '''Pappus's centroid theorem''' (also known as  the '''Guldinus theorem''', '''Pappus–Guldinus theorem''' or '''Pappus's theorem''') is either of two related [[theorem]]s dealing with the [[surface area]]s and [[volume]]s of [[surface of revolution|surface]]s and [[solid of revolution|solid]]s of revolution.

The theorems are attributed to [[Pappus of Alexandria]]{{efn|See:<ref>{{cite book| author=Pappus of Alexandria|author-link=Pappus of Alexandria|editor-last=Jones| editor-first=Alexander| year=1986| title=Book 7 of the ''Collection''| volume=8|location=New York | publisher=Springer-Verlag |isbn=978-1-4612-4908-5 |doi=10.1007/978-1-4612-4908-5 |orig-year=c. 320|series=Sources in the History of Mathematics and Physical Sciences}}</ref>
{{Quote|
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:<br>

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''.
 |author=Pappus
 |source=''Collection'', Book VII, ¶41‒42}}
}} and [[Paul Guldin]].{{efn|"Quantitas rotanda in viam rotationis ducta, producit Potestatem Rotundam uno gradu altiorem, Potestate sive Quantitate rotata."<ref>{{cite book|author-last=Guldin|author-first=Paul|author-link=Paul Guldin|title=De centro gravitatis trium specierum quanitatis continuae| volume=2| pages=147| year=1640 |publisher=Gelbhaar, Cosmerovius | location=Vienna | url=https://books.google.com/books?id=CNaI61CYc94C&pg=PA147|access-date=2016-08-04}}</ref>
That is: "A quantity in rotation, multiplied by its circular trajectory, creates a circular power of higher degree, power, or quantity in rotation."<ref name="RdG2015">{{cite book|author-last=Radelet-de Grave|author-first=Patricia |editor-last=Jullien|editor-first=Vincent |title=Seventeenth-Century Indivisibles Revisited|chapter=Kepler, Cavalieri, Guldin. Polemics with the departed |pages=68 |publisher=Birkhäuser |series=Science Networks. Historical Studies |volume=49 |isbn=978-3-3190-0131-9 |chapter-url=https://books.google.com/books?id=8Vt1CQAAQBAJ&pg=PA68 |date=2015-05-19 |doi=10.1007/978-3-319-00131-9 |issn=1421-6329 |location=Basel |hdl=2117/28047 | access-date=2016-08-04}}</ref> }} Pappus's statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640.<ref>{{cite journal |last1=Bulmer-Thomas |first1=Ivor |title=Guldin's Theorem--Or Pappus's? |journal=Isis |date=1984 |volume=75 |issue=2 |pages=348–352 |doi=10.1086/353487 |jstor=231832 |url=https://www.jstor.org/stable/231832 |issn=0021-1753}}</ref>