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Cavalieri’s principle

Two stacks of British coins with the same volume, illustrating

Cavalieri’s principle in three dimensions 

In   geometry,   Cavalieri’s principle, a modern imple-mentation of the method of indivisibles, named afterBonaventura Cavalieri, is as follows:[1]

•   2-dimensional case: Suppose two regions in aplane are included between two parallel lines in thatplane. If every line parallel to these two lines inter-sects both regions in line segments of equal length,

then the two regions have equal areas.•   3-dimensional case: Suppose two regions in three-

space (solids) are included between two parallelplanes. If every plane parallel to these two planes in-tersects both regions in cross-sections of equal area,then the two regions have equal volumes.

Today Cavalieri’s principle is seen as an early step to-wards   integral calculus, and while it is used in someforms, such as its generalization in Fubini’s theorem, re-sults using Cavalieri’s principle can often be shown moredirectly via integration. In the other direction, Cavalieri’s

principle grew out of the ancient Greek  method of ex-haustion, which used limits but did not use infinitesimals.

1 History

Cavalieri’s principle was originally called the method ofindivisibles, the name it was known by in   RenaissanceEurope.   Archimedes was able to find the volume of asphere given the volumes of a cone and cylinder usinga method resembling Cavalieri’s principle. In the 5thcentury AD, Zu Chongzhi and his son  Zu Gengzhi  es-

tablished a similar method to find a sphere’s volume.[2]The transition from Cavalieri’s indivisibles to EvangelistaTorricelli's and John Wallis's infinitesimals was a major

Bonaventura Cavalieri  , the mathematician the principle is named 

after.

advance in the history of the  calculus. The indivisibleswere entities of codimension 1, so that a plane figure wasthought as made out of an infinity of 1-dimensional lines.Meanwhile, infinitesimals were entities of the same di-mension as the figure they make up; thus, a plane figurewould be made out of “parallelograms” of infinitesimalwidth. Applying the formula for the sum of an arithmeticprogression, Wallis computed the area of a triangle bypartitioning it into infinitesimal parallelograms of width

1/∞.

2 Examples

2.1 Spheres

If one knows that the volume of a   cone   is1

3 (base  ×  height)   , then one can use Cavalieri’s

principle to derive the fact that the volume of a sphere is4

3πr

3 , where  r  is the radius.

That is done as follows: Consider a sphere of radius   rand a cylinder of radius   r   and height   r   . Within thecylinder is the cone whose apex is at the center of the

1

https://en.wikipedia.org/wiki/Spherehttps://en.wikipedia.org/wiki/Cone_(geometry)https://en.wikipedia.org/wiki/Codimensionhttps://en.wikipedia.org/wiki/Integral_calculushttps://en.wikipedia.org/wiki/Bonaventura_Cavalierihttps://en.wikipedia.org/wiki/Infinitesimalhttps://en.wikipedia.org/wiki/John_Wallishttps://en.wikipedia.org/wiki/Evangelista_Torricellihttps://en.wikipedia.org/wiki/Evangelista_Torricellihttps://en.wikipedia.org/wiki/Zu_Gengzhihttps://en.wikipedia.org/wiki/Zu_Chongzhihttps://en.wikipedia.org/wiki/Archimedeshttps://en.wikipedia.org/wiki/Renaissance_Europehttps://en.wikipedia.org/wiki/Renaissance_Europehttps://en.wikipedia.org/wiki/Infinitesimalhttps://en.wikipedia.org/wiki/Method_of_exhaustionhttps://en.wikipedia.org/wiki/Method_of_exhaustionhttps://en.wikipedia.org/wiki/Fubini%2527s_theoremhttps://en.wikipedia.org/wiki/Integral_calculushttps://en.wikipedia.org/wiki/Cross_section_(geometry)https://en.wikipedia.org/wiki/Bonaventura_Cavalierihttps://en.wikipedia.org/wiki/Geometry

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2   2 EXAMPLES 

yy

y

r 

The disk-shaped cross-section of the sphere has the same area as 

the ring-shaped cross-section of that part of the cylinder that lies 

outside the cone.

sphere and whose base is the base of the cylinder. Bythe Pythagorean theorem, the plane located  y  units abovethe “equator” intersects the sphere in a circle of areaπr2− y

2

 . The area of the plane’s intersection withthe part of the cylinder that is  outside of the cone is also

πr2− y

2 . The aforementioned volume of the cone is

1

3 of the volume of the cylinder, thus the volume  outside

of the cone is   23

 the volume of the cylinder. Therefore thevolume of the upper half of the sphere is   2

3 of the volume

of the cylinder. The volume of the cylinder is

base  ×  height =  πr2 ·  r  =  πr3

(“Base” is in units of area; “height” is in units of distance.Area × distance = volume.)

Therefore the volume of the upper half-sphere is2

3

πr

3

and that of the whole sphere is4

3

πr

3 .

2.2 Cones and pyramids

The fact that the volume of any pyramid, regardless ofthe shape of the base, whether circular as in the case of acone, or square as in the case of the Egyptian pyramids,or any other shape, is (1/3) × base × height, can be estab-

lished by Cavalieri’s principle if one knows only that it istrue in one case. One may initially establish it in a sin-gle case by partitioning the interior of a triangular prisminto three pyramidal components of equal volumes. Onemay show the equality of those three volumes by meansof Cavalieri’s principle.

In fact, Cavalieri’s principle or similar infinitesimal ar-gument is   necessary   to compute the volume of conesand even pyramids, which is essentially the content ofHilbert’s third problem – polyhedral pyramids and conescannot be cut and rearranged into a standard shape,and instead must be compared by infinite (infinitesimal)

means. The ancient Greeks used various precursor tech-niques such as Archimedes’s mechanical arguments ormethod of exhaustion to compute these volumes.

2.3 The napkin ring problem

Main article: Napkin ring problemIn what is called the napkin ring problem, one shows by

h

If a hole of height  h  is drilled straight through the center of a

sphere, the volume of the remaining band does not depend on the

size of the sphere. For a larger sphere, the band will be thinner 

but longer.

Cavalieri’s principle that when a hole of length h is drilledstraight through the center of a sphere, the volume of theremaining material surprisingly does not depend on thesize of the sphere. The cross-section of the remainingring is a plane annulus, whose area is the difference be-tween the areas of two circles. By the Pythagorean theo-rem, the area of one of the two circles is  π times r 2 − y 2,where  r  is the sphere’s radius and  y  is the distance fromthe plane of the equator to the cutting plane, and that ofthe other is  π  times  r  2 − (h/2) 2. When these are sub-

tracted, the r 2 cancels; hence the lack of dependence ofthe bottom-line answer upon r .

2.4 Cycloids

The horizontal cross-section of the region bounded by two cy-

cloidal arcs traced by a point on the same circle rolling in one

case clockwise on the line below it, and in the other counterclock-

wise on the line above it, has thesame length as thecorresponding

horizontal cross-section of the circle.

N. Reed has shown[3] how to find the area bounded bya cycloid  by using Cavalieri’s principle. A circle of ra-

dius r  can roll in a clockwise direction upon a line belowit, or in a counterclockwise direction upon a line aboveit. A point on the circle thereby traces out two cycloids.

https://en.wikipedia.org/wiki/Cycloidhttps://en.wikipedia.org/wiki/Napkin_ring_problemhttps://en.wikipedia.org/wiki/Napkin_ring_problemhttps://en.wikipedia.org/wiki/Method_of_exhaustionhttps://en.wikipedia.org/wiki/Hilbert%2527s_third_problemhttps://en.wikipedia.org/wiki/Pyramid_(geometry)https://en.wikipedia.org/wiki/Pythagorean_theorem

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3

When the circle has rolled any particular distance, the an-gle through which it would have turned clockwise and thatthrough which it would have turned counterclockwise arethe same. The two points tracing the cycloids are there-fore at equal heights. The line through them is thereforehorizontal(i.e. parallel to thetwo lines on which the circle

rolls). Consequently each horizontal cross-section of thecircle has the same length as the corresponding horizon-tal cross-section of the region bounded by the two arcs ofcyloids. By Cavalieri’s principle, the circle therefore hasthe same area as that region.

It is a short step from there to the conclusion that the areaunder a single whole cycloidal arch is three times the areaof the circle. Which then means that the area of a rect-angle bounding one half of a single cycloidal arch is twotimes the area of the circle, the area of a rectangle bound-ing a single whole cycloidal arch is four times the areaof the circle, and the rectangularly-bounded area above a

single whole cycloidal arch is exactly equal to the area ofthe circle.

3 See also

•   Fubini’s theorem(Cavalieri’s principle is a particularcase of Fubini’s theorem)

4 References

[1] Howard Eves, “Two Surprising Theorems on Cavalieri

Congruence”,  The College Mathematics Journal , volume

22, number 2, March, 1991), pages 118–124

[2] Zill, Dennis G.; Wright, Scott; Wright, Warren S. (2009).

Calculus: Early Transcendentals  (3 ed.). Jones & Bartlett

Learning. p. xxvii.   ISBN 0-7637-5995-3.   Extract of

page 27

[3] N. Reed, “Elementary proof of the area under a cycloid”,

Mathematical Gazette, volume 70, number 454, Decem-

ber, 1986, pages 290–291

5 External links

•   Weisstein, Eric W.,   “Cavalieri’s Principle”,MathWorld .

•   (German) Prinzip von Cavalieri

•   Cavalieri Integration

http://researchspace.csir.co.za/dspace/bitstream/10204/5267/1/Grobler5_2011.pdfhttp://www.dividano.de/prinzip-von-cavalieri.htmlhttps://en.wikipedia.org/wiki/MathWorldhttp://mathworld.wolfram.com/CavalierisPrinciple.htmlhttps://en.wikipedia.org/wiki/Eric_W._Weissteinhttps://en.wikipedia.org/wiki/Mathematical_Gazettehttps://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27https://en.wikipedia.org/wiki/Special:BookSources/0-7637-5995-3https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://books.google.com/books?id=R3Hk4Uhb1Z0Chttps://en.wikipedia.org/wiki/Fubini%2527s_theorem

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4   6 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 

6 Text and image sources, contributors, and licenses

6.1 Text

•   Cavalieri’s principle   Source:    https://en.wikipedia.org/wiki/Cavalieri’{}s_principle?oldid=711849241   Contributors:    Toby Bartels,Michael Hardy, William M. Connolley, Ciampix, Jason Quinn, Histrion, Bender235, Btyner, DVdm, Siddhant, Pb30, Nbarth, Thenub314,TheEditrix2, David Eppstein, CommonsDelinker, Chiswick Chap, Alexbot, Five-toed-sloth, Franklin.vp, DumZiBoT, Addbot,  ,.דני ל

Luckas-bot, Yobot, , Materialscientist, ArthurBot, Xqbot, GrouchoBot, Point-set topologist, Imveracious, Orloffsr, Tkuvho, RandomDS-devel, The.megapode, WikitanvirBot, 478jjjz, Thecheesykid, GeorgeBarnick, ClueBot NG, Vinícius Machado Vogt, Widr, Rurik theVarangian, CsDix, Matrix1919 and Anonymous: 21

6.2 Images

•   File:Bonaventura_Cavalieri.jpeg Source:  https://upload.wikimedia.org/wikipedia/commons/4/49/Bonaventura_Cavalieri.jpeg License: Public domain Contributors:  ?  Original artist:  ?

•   File:Cavalieri’{}s_Principle_in_Coins.JPG Source:  https://upload.wikimedia.org/wikipedia/commons/5/5a/Cavalieri%27s_Principle_in_Coins.JPG License:  CC BY-SA 3.0 Contributors:  Own work Original artist:  Chiswick Chap

•   File:Folded_cycloid.svg Source:  https://upload.wikimedia.org/wikipedia/commons/f/ff/Folded_cycloid.svg License:   Public domain Con-tributors:  Transferred from en.wikipedia Original artist:  Michael Hardy at en.wikipedia

•   File:Sphere_bands.svg  Source:   https://upload.wikimedia.org/wikipedia/commons/2/27/Sphere_bands.svg License:   Public domain  Con-

tributors:  Own work (Original caption: “I created this with xfig.”) Original artist:  Michael Hardy at en.wikipedia•   File:Sphere_cavalieri.svg Source:   https://upload.wikimedia.org/wikipedia/commons/5/58/Sphere_cavalieri.svg License:  Public domain

Contributors:  Transferred from en.wikipedia Original artist:  Michael Hardy at en.wikipedia

6.3 Content license

•   Creative Commons Attribution-Share Alike 3.0

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