• Archimedes showed that the three corresponding slices would always balance, and so the three solids are in balance.
  
  

• The cone and sphere at A balance 4 cylinders at C.

• Using d₁W₁ = d₂W₂ (the law of the lever), Archimedes reasoned that

1 x (cone volume + sphere volume) = 1/2 x (4 cylinder volumes)

• Archimedes already knew the volume of the cylinder and the cone, so he could finally conclude that

sphere volume = 4/3 pi r³

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