Bonaventura Cavalieri (1598-1647)

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Bonaventura Cavalieri (1598-1647)

• Joined the religious order Jesuati in Milan in
  1615 while he was still a boy.
• In 1616 he transferred to the Jesuati monastery
  in Pisa. After meeting Galileo, considered
  himself a disciple of his.
• In 1629 Cavalieri was appointed to the chair of
  mathematics at Bologna
• Cavalieri's theory of indivisibles, presented in
  his Geometria indivisibilis continuorum nova of
  1635. After criticism Cavalieri improved his
  exposition publishing Exercitationes geometricae
  sex which became the main source for 17th Century
  mathematicians.
• Galileo wrote
• few, if any, since Archimedes, have delved as
  far
• and as deep into the science of geometry.

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Bonaventura Cavalieri (1598-1647)

• Cavalieris principle
• If two plane figures cut by a set of parallel
  lines intersect, on each of these straight lines,
  equal chords, the two figures are equivalent
  i.e. of equivalent area.
• Cavalieris triangle paradox

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Bonaventura Cavalieri (1598-1647)

• Cavalieri thinks of a triangle or any plane
  figure as being made up of all lines (a.l.) of
  the figure. The area is then given by a.l..
• When he writes all squares (a.s.) of a plane
  figure he means the all the squares obtained by
  producing a square from each line of the plane
  figure.
• When he writes all cubes (a.c.) of a plane figure
  he means the all the cubes obtained by producing
  a cube from each line of the plane figure.
• We should think of the word all as a kind of
  integration sign.

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Bonaventura Cavalieri (1598-1647)

• Proposition 21 ac(AD)4ac(ACF)4ac(FDC).
• Notice that together with means plus and
  with means that the collection of indivisibles
  is over the product. With abuse of notation we
  will write just the product. Going sentence by
  sentence and giving comments in , denotes
  together with and juxtaposition denotes with
  
• ac(AD)ac(ACF)ac(FDC)3al(ACF)as(FDC)3as(ACF)al
  (FDC). (NE)3(NHHE)3NE3HE33NH(HE)23(NH)2HE.
  
• ac(AD)al(AD)as(AD) ac(AD)/al(AD)as(FDC)al(AD)
  as(AD)/al(AD)as(FDC) as(AD)/as(FDC)3 For
  this one needs that each line of AD has the same
  length and thus the term al(AD) yields a constant
  factor. Also as(AD)/as(FDC)3 from the
  quadrature of the parabola.
• ac(AD)3al(AD)as(FDC), al(AD)as(FDC)al(ACF)as(FDC
  )al(FDC)as(FDC), al(FDC)as(FDC)ac(FDC).
  NE(HE)2(NHHE)(HE)2NH(HE)2(HE)3.
• ac(AD)3 (ac(FDC)al(ACF)as(FDC))
• ac(AD)ac(ACF)ac(FDC)3 al(FDC)as(ACF)3al(ACF)as
  (FDC).
• 3al(ACF)as(FDC) 3al(ACF)as(FDC)
• Cryptic Subtract 6 from RHSs of 4 and 5.
• ac(ACF)ac(FDC)3as(ACF)al(FDC)3ac(FDC).
• ac(ACF)ac(FDC)2ac(ACF), since ac(ACF)ac(FDC).
  The triangles are congruent.
• 3al(ACF)as(FDC)3as(ACF)al(FDC)
  ac(ACF)ac(FDC)ac(AD)4ac(FDC)4ac(FAC). From
  1,8,9,11.
• al(ACF)as(FDC)as(ACF)al(FDC), by symmetry.
• 3al(ACF)as(FDC)3as(ACF)al(FDC) This is not
  needed but clarifies the proof. Each summand of 1
  contributes the same.

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Bonaventura Cavalieri (1598-1647)

• Proposition 21 ac(AD)4ac(ACF)4ac(FDC)
• ac(AD)ac(ACF)ac(FDC)3al(ACF)as(FDC)3as(ACF)al(
  FDC)
• ac(AD)/al(AD)as(FDC)al(AD)as(AD)/al(AD)as(FAD)
• as(AD)/as(FAD)3
• Now al(AD)al(ACF)al(FDC) and al(FDC)as(FDC)ac(F
  DC) so
• ac(AD)3al(AD)as(FDC)3al(ACF)as(FDC)3ac(FDC)
  
• By symmetry
• ac(ACF)ac(FDC)
• al(ACF)as(FDC)as(ACF)al(FDC)
• So (1) ac(AD)2ac(FDC)6al(ACF)as(FDC)
• (2) ac(AD)3ac(FDC)3al(ACF)as(FDC)
• thus 3al(ACF)as(FDC)ac(FDC)
• and ac(AD)2(ac(FDC)3al(ACF)as(FDC))4ac(FDC)

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Bonaventura Cavalieri (1598-1647)
Effectively Cavalieri integrates of xy3 from 0
to 1 without the fundamental theorem by
decomposing the square 0,1X0,1 into the two
triangles above and below xy, and using that
Cavalieri does slightly more than this, since he
integrates uv3 between 0 and 1 in any linear
coordinates system. It is enough to consider
xaubv, ycv. (why?)
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Bonaventura Cavalieri (1598-1647)

• Proposition 23. Any parallelogram has twice the
  area of its first diagonal space, three times the
  area of its second diagonal space, four times the
  area of its third diagonal space etc.
• Rephrased In any (not necessarily Cartesian)
  linear coordinate system the area under the curve
  xyn between a and b is 1/n1 times the area of
  the parallelogram 0,ax0,b.

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Bonaventura Cavalieri (1598-1647)

• The link between proposition 23 and 21 is given
  by noting that for third diagonal space al are
  given by the FI.
• Now EFFIDA3AF3
• all DA3ac(AC)
• all AF3 ac(ABC)ac(CDA),
• all EF al(AC)area(AC)
• all FIal(3rd diagonal space)area(3rd diagonal
  space).
• So the area of the parallelogram is to the area
  of the third diagonal space as ac(AC)ac(ABC)1/4
  (by proposition 21).