Statement
- Plane: if every line parallel to a given line cuts two figures in segments of equal length, the figures have equal area.
- Space: if every plane parallel to a given plane cuts two solids in sections of equal area, the solids have equal volume.
A stack of coins pushed into a slant keeps the same volume, so an oblique prism or cylinder has the same volume as a right one with equal base and height.
History
The Italian mathematician Bonaventura Cavalieri set out the method of indivisibles in Geometria indivisibilibus (1635), treating figures as made of infinitely many lines or planes. Similar ideas appear in Archimedes and in the 5th-century Chinese mathematician Zu Geng. The method fed into the calculus of Newton and Leibniz.
Volume of a sphere
Compare a hemisphere of radius r with a cylinder of radius and height r from which a cone has been removed. At height h both have cross-sectional area π(r²-h²), so the hemisphere's volume is πr³-πr³/3=2πr³/3 and the sphere's is 4πr³/3.
Link to integration
Today volume is written as V=∫A(x)dx, so equal cross-sectional area functions give equal volumes, which justifies the principle rigorously.
References
- Boyer, C. B., & Merzbach, U. C. (2011). A History of Mathematics (3rd ed.). Wiley.
- Eves, H. (1991). Two surprising theorems on Cavalieri congruence. The College Mathematics Journal, 22(2), 118-124.
- Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage.