Measurements of volume

Summary: Volume has been measured in numerous ways throughout history, with calculus playing an integral role.

Volume is the amount of space that is occupied by an object. In other words, volume is a three-dimensional analogue of the area. Volume is important in construction, engineering, and physics. Early on, weight was easier to measure than volume, especially because crops and other real-life objects often had irregular shapes. One early volume calculation can be found in an ancient Egyptian mathematical work called the “Moscow Papyrus,” named for the country where it resided in the twentieth century. It dates back to almost 2000 b.c.e., and its author is unknown.

One problem provides a method for calculating units of volume of a truncated pyramid. However, mathematicians developed a variety of methods to calculate volume, including via the displacement of water, the method of exhaustion, and connections to determinant and integration methods. In twenty-first-century mathematics classrooms, students investigate volume relationships and formulas. Students calculate the volume of geometric objects like cylinders, cones, and spheres. The volume of a cylinder (πr²h), is obtained by multiplying the area of the base, πr² where r is the radius of the given circle, by the height (h). The volume of a cone is one-third that amount, and the volume of a sphere is

However, these formulas took a long time to develop and could only be approximated before the development of calculus and a fuller understanding of π. Students also compare quantities of water or sand-filled relational geosolids and examine volume integrations and the volume interpretation of determinants.

Early Methods

The method of exhaustion has long been used to estimate volumes. Democritus of Abdera is noted as the first to state that the volume of a cone is one-third that of a cylinder of the same height and radius and that the volume of a pyramid is one-third of the corresponding prism. Eudoxus of Cnidus developed the method of exhaustion that uses what would now be referred to as “limits” of sums of well-known areas or volumes. He justified Democritus’ relationships and explored other areas and volumes. Some of Eudoxus’s work appears in Euclid of Alexandria’s Elements. In ancient China, volume calculations were published in the Nine Chapters on the Mathematical Art. In his commentary of 263 c.e., Liu Hui calculated the volume of figures like a tetrahedron and the frustum of a cone. The volume of the sphere was challenging and he noted: “Let us leave the problem to whoever can tell the truth.” Archimedes of Syracuse researched the volume of various figures, including surfaces of revolution. He showed that the volume of a cylinder equals the sum of the volume of a cone of the same height and the volume of a sphere of the same height. Here, the height can be expressed as twice the radius. Archimedes is reported to have considered this as one of his greatest achievements because a related inscription appeared on his tombstone. In twenty-first-century classrooms, students understand Archimedes’ statement by pouring sand or water from a cone and a sphere into a cylinder to fill it up. They also investigate the related formulas. Archimedes also reportedly noticed that water displacement could be used to measure volume while famously expressing: “Eureka!”

Using Calculus and Integration

Computing volume by integration allowed for the calculation of the volume of irregular objects. In 1615, Johannes Kepler published Nova Stereometria Doliorum Vinarorum (New Solid Geometry of a Wine Barrel). He apparently became interested in the volume of casks on his wedding day. Methods of integration and volume calculations developed along with calculus before the related analysis was well understood. Cavalieri’s principle is named for seventeenth-century mathematician Bonaventura Cavalieri. Cavalieri combined the method of exhaustion with Kepler’s work and computed volumes by comparing cross-sectional areas. This method predates the analysis that was needed to put it on a sound footing, and Cavalieri was criticized for his ideas. Some results seemed counterintuitive and provided additional fodder for critics. For example, the surface of revolution obtained from revolving the region under 1/x between 1 and infinity has finite volume. In the seventeenth century, mathematician Thomas Hobbes is noted as having remarked about this result: “To understand this for sense it is not required that a man should be a geometrician or a logician, but that he should be mad.”

Mathematicians eventually developed the analysis rigorously. The Riemann integral is named for nineteenth-century mathematician Bernhard Riemann. Measuring the area below a graph of the function is accomplished by dividing the region under the graph into extremely small rectangles and adding these rectangles up. Roughly, the volume of a region in space would be computed with a similar idea. The given space would be divided into small rectangular boxes. Each piece would have the volume (dx×dy×dz), and the volume of the whole space would be computed by a triple integral. However, this method supposes that one understands the functions that make up the surface. Many mathematical theories about approximation of the boundary surface have been developed for a long time, and they have played an important role since they are indeed extremely useful in actual computation.

Other Methods

Another method of computing volume that is explored in linear algebra and physics classes is by the determinant. In a 1773 paper on mechanics, Joseph Lagrange calculated the volume of a tetrahedron in terms of the locations of the coordinates. In modern terms one would recognize the connection of the expression to a determinant calculation of a 3-by-3 matrix.

Integration formulas such as Green’s theorem and the divergence theorem, which are studied in a multivariable calculus course, connect volume to other calculations. Mathematician and physicist George Green worked on vector calculus integral theorems, and Green’s theorem is named after him. Green’s theorem relates surface and volume integrals. Mathematician Carl Friedrich Gauss contributed to the geometry of surfaces as well as the divergence theorem. The divergence theory relates the volume integral of the divergence inside a surface to the flux of a vector field on the closed surface. A well-studied question related to volume measurements dates back to ancient Greece. Archimedes and Zenodorus examined the sphere as the surface that would enclose a given volume with the least amount of surface area. Mathematician Hermann Schwarz proved that the sphere maximized volume with minimal surface area in 1884. In the twentieth century, mathematicians solved the Double Bubble Problem, showing that a standard configuration is the most efficient way to enclose two regions.

Bibliography

Anastassiou, George A., and Karl-Georg Steffens. The History of Approximation Theory: From Euler to Bernstein. Boston: Birkhäuser, 2010.

Hirshfeld, Alan. Eureka Man: The Life and Legacy of Archimedes. New York: Walker, 2009.

Lawn, Richard E., and Elizabeth Prichard. Measurement of Volume. Cambridge, England: Royal Society of Chemistry, 2003.