In geometry , Cavalieri's principle , a modern implementation of the method of indivisibles , named after Bonaventura Cavalieri , is as follows:
31-471: [REDACTED] Look up indivisible in Wiktionary, the free dictionary. Indivisible may refer to: Mathematics [ edit ] Method of indivisibles, the historical name of what is now known as Cavalieri's principle Absence of divisibility (ring theory) Arts [ edit ] Films [ edit ] Indivisible (2016 film) ,
62-593: A 2016 Italian film Indivisible (2018 film) , a 2018 American film Music [ edit ] Indivisible, album by Lungfish (band) 1997 "Indivisible", song by Marie-Mai "Indivisible", song by Pillar from Fireproof (Pillar album) "Indivisible", song by Hatebreed "Indivisible", song by Lungfish (band) "Indivisible", song by The Dirtbombs "Indivisible", song by Plankeye "Indivisible", song by Crüxshadows "Indivisible", song by Betty Wright "Indivisible", song by Yellowjackets Other media [ edit ] Indivisible ,
93-467: A novel by Fanny Howe (2003) Indivisible (video game) Other uses [ edit ] French ship Indivisible (1799) Indivisible movement , a progressive movement initiated as a reaction to the election of Donald Trump as US President in 2016 Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Indivisible . If an internal link led you here, you may wish to change
124-467: A novel by Fanny Howe (2003) Indivisible (video game) Other uses [ edit ] French ship Indivisible (1799) Indivisible movement , a progressive movement initiated as a reaction to the election of Donald Trump as US President in 2016 Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Indivisible . If an internal link led you here, you may wish to change
155-464: A plane figure was thought as made out of an infinite number of 1-dimensional lines. Meanwhile, infinitesimals were entities of the same dimension as the figure they make up; thus, a plane figure would be made out of "parallelograms" of infinitesimal width. Applying the formula for the sum of an arithmetic progression, Wallis computed the area of a triangle by partitioning it into infinitesimal parallelograms of width 1/∞. N. Reed has shown how to find
186-539: A standard shape, and instead must be compared by infinite (infinitesimal) means. The ancient Greeks used various precursor techniques such as Archimedes's mechanical arguments or method of exhaustion to compute these volumes. Consider a cylinder of radius r {\displaystyle r} and height h {\displaystyle h} , circumscribing a paraboloid y = h ( x r ) 2 {\displaystyle y=h\left({\frac {x}{r}}\right)^{2}} whose apex
217-457: Is at the center of the bottom base of the cylinder and whose base is the top base of the cylinder. Also consider the paraboloid y = h − h ( x r ) 2 {\displaystyle y=h-h\left({\frac {x}{r}}\right)^{2}} , with equal dimensions but with its apex and base flipped. For every height 0 ≤ y ≤ h {\displaystyle 0\leq y\leq h} ,
248-485: Is seen as an early step towards integral calculus , and while it is used in some forms, such as its generalization in Fubini's theorem and layer cake representation , results using Cavalieri's principle can often be shown more directly via integration. In the other direction, Cavalieri's principle grew out of the ancient Greek method of exhaustion , which used limits but did not use infinitesimals . Cavalieri's principle
279-471: Is the radius. That is done as follows: Consider a sphere of radius r {\displaystyle r} and a cylinder of radius r {\displaystyle r} and height r {\displaystyle r} . Within the cylinder is the cone whose apex is at the center of one base of the cylinder and whose base is the other base of the cylinder. By the Pythagorean theorem ,
310-412: Is the sphere's radius and y {\displaystyle y} is the distance from the plane of the equator to the cutting plane, and that of the other is π × ( r 2 − ( h 2 ) 2 ) {\textstyle \pi \times \left(r^{2}-\left({\frac {h}{2}}\right)^{2}\right)} . When these are subtracted,
341-409: The 5th century AD, Zu Chongzhi and his son Zu Gengzhi established a similar method to find a sphere's volume. Neither of the approaches, however, were known in early modern Europe. The transition from Cavalieri's indivisibles to Evangelista Torricelli 's and John Wallis 's infinitesimals was a major advance in the history of calculus . The indivisibles were entities of codimension 1, so that
SECTION 10
#1732764643835372-426: The area bounded by a cycloid by using Cavalieri's principle. A circle of radius r can roll in a clockwise direction upon a line below it, or in a counterclockwise direction upon a line above it. A point on the circle thereby traces out two cycloids. When the circle has rolled any particular distance, the angle through which it would have turned clockwise and that through which it would have turned counterclockwise are
403-425: The area of the circle, and so, the area bounded by the arch is three times the area of the circle. The fact that the volume of any pyramid , regardless of the shape of the base, including cones (circular base), is (1/3) × base × height, can be established by Cavalieri's principle if one knows only that it is true in one case. One may initially establish it in a single case by partitioning
434-413: The area of the intersection of that plane with the part of the cylinder that is "outside" of the cone; thus, applying Cavalieri's principle, it could be said that the volume of the half sphere equals the volume of the part of the cylinder that is "outside" the cone. The aforementioned volume of the cone is 1 3 {\textstyle {\frac {1}{3}}} of the volume of the cylinder, thus
465-643: The centre of a sphere where the remaining band has height h {\displaystyle h} , the volume of the remaining material surprisingly does not depend on the size of the sphere. The cross-section of the remaining ring is a plane annulus, whose area is the difference between the areas of two circles. By the Pythagorean theorem, the area of one of the two circles is π × ( r 2 − y 2 ) {\displaystyle \pi \times (r^{2}-y^{2})} , where r {\displaystyle r}
496-410: The cylinder part outside the inscribed paraboloid. Therefore, the volume of the flipped paraboloid is equal to the volume of the cylinder part outside the inscribed paraboloid. In other words, the volume of the paraboloid is π 2 r 2 h {\textstyle {\frac {\pi }{2}}r^{2}h} , half the volume of its circumscribing cylinder. If one knows that
527-512: The disk-shaped cross-sectional area π ( 1 − y h r ) 2 {\displaystyle \pi \left({\sqrt {1-{\frac {y}{h}}}}\,r\right)^{2}} of the flipped paraboloid is equal to the ring-shaped cross-sectional area π r 2 − π ( y h r ) 2 {\displaystyle \pi r^{2}-\pi \left({\sqrt {\frac {y}{h}}}\,r\right)^{2}} of
558-909: The free dictionary. Indivisible may refer to: Mathematics [ edit ] Method of indivisibles, the historical name of what is now known as Cavalieri's principle Absence of divisibility (ring theory) Arts [ edit ] Films [ edit ] Indivisible (2016 film) , a 2016 Italian film Indivisible (2018 film) , a 2018 American film Music [ edit ] Indivisible, album by Lungfish (band) 1997 "Indivisible", song by Marie-Mai "Indivisible", song by Pillar from Fireproof (Pillar album) "Indivisible", song by Hatebreed "Indivisible", song by Lungfish (band) "Indivisible", song by The Dirtbombs "Indivisible", song by Plankeye "Indivisible", song by Crüxshadows "Indivisible", song by Betty Wright "Indivisible", song by Yellowjackets Other media [ edit ] Indivisible ,
589-428: The interior of a triangular prism into three pyramidal components of equal volumes. One may show the equality of those three volumes by means of Cavalieri's principle. In fact, Cavalieri's principle or similar infinitesimal argument is necessary to compute the volume of cones and even pyramids, which is essentially the content of Hilbert's third problem – polyhedral pyramids and cones cannot be cut and rearranged into
620-460: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Indivisible&oldid=1081625691 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages indivisible [REDACTED] Look up indivisible in Wiktionary,
651-402: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Indivisible&oldid=1081625691 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Cavalieri%27s principle Today Cavalieri's principle
SECTION 20
#1732764643835682-408: The other half of the rectangle with it. The new rectangle, of area twice that of the circle, consists of the "lens" region between two cycloids, whose area was calculated above to be the same as that of the circle, and the two regions that formed the region above the cycloid arch in the original rectangle. Thus, the area bounded by a rectangle above a single complete arch of the cycloid has area equal to
713-418: The plane located y {\displaystyle y} units above the "equator" intersects the sphere in a circle of radius r 2 − y 2 {\textstyle {\sqrt {r^{2}-y^{2}}}} and area π ( r 2 − y 2 ) {\displaystyle \pi \left(r^{2}-y^{2}\right)} . The area of
744-421: The plane's intersection with the part of the cylinder that is outside of the cone is also π ( r 2 − y 2 ) {\displaystyle \pi \left(r^{2}-y^{2}\right)} . As can be seen, the area of the circle defined by the intersection with the sphere of a horizontal plane located at any height y {\displaystyle y} equals
775-451: The principle, in his publications he denied that the continuum was composed of indivisibles in an effort to avoid the associated paradoxes and religious controversies, and he did not use it to find previously unknown results. In the 3rd century BC, Archimedes , using a method resembling Cavalieri's principle, was able to find the volume of a sphere given the volumes of a cone and cylinder in his work The Method of Mechanical Theorems . In
806-400: The same area as that region. Consider the rectangle bounding a single cycloid arch. From the definition of a cycloid, it has width 2π r and height 2 r , so its area is four times the area of the circle. Calculate the area within this rectangle that lies above the cycloid arch by bisecting the rectangle at the midpoint where the arch meets the rectangle, rotate one piece by 180° and overlay
837-408: The same. The two points tracing the cycloids are therefore at equal heights. The line through them is therefore horizontal (i.e. parallel to the two lines on which the circle rolls). Consequently each horizontal cross-section of the circle has the same length as the corresponding horizontal cross-section of the region bounded by the two arcs of cycloids. By Cavalieri's principle, the circle therefore has
868-428: The volume outside of the cone is 2 3 {\textstyle {\frac {2}{3}}} the volume of the cylinder. Therefore the volume of the upper half of the sphere is 2 3 {\textstyle {\frac {2}{3}}} of the volume of the cylinder. The volume of the cylinder is ("Base" is in units of area ; "height" is in units of distance . Area × distance = volume .) Therefore
899-445: The volume of a cone is 1 3 ( base × height ) {\textstyle {\frac {1}{3}}\left({\text{base}}\times {\text{height}}\right)} , then one can use Cavalieri's principle to derive the fact that the volume of a sphere is 4 3 π r 3 {\textstyle {\frac {4}{3}}\pi r^{3}} , where r {\displaystyle r}
930-416: The volume of the upper half-sphere is 2 3 π r 3 {\textstyle {\frac {2}{3}}\pi r^{3}} and that of the whole sphere is 4 3 π r 3 {\textstyle {\frac {4}{3}}\pi r^{3}} . In what is called the napkin ring problem , one shows by Cavalieri's principle that when a hole is drilled straight through
961-501: Was originally called the method of indivisibles, the name it was known by in Renaissance Europe . Cavalieri developed a complete theory of indivisibles, elaborated in his Geometria indivisibilibus continuorum nova quadam ratione promota ( Geometry, advanced in a new way by the indivisibles of the continua , 1635) and his Exercitationes geometricae sex ( Six geometrical exercises , 1647). While Cavalieri's work established