horocycle in the appropriate Q quadrant. For example, in thermodynamics the isothermal process explicitly follows the hyperbolic path and work can be interpreted as a hyperbolic angle change. Similarly, a given mass M of gas with changing volume will have variable density δ = M / V, and the ideal gas law may be written P = k T δ so that an isobaric process traces a hyperbola in the quadrant of absolute temperature and gas density. For hyperbolic coordinates in the theory of relativity see the History section. Statistical applications Comparative study of population density in the quadrant begins with selecting a reference nation, region, or urban area whose population and area are taken as the point (1,1). Analysis of the elected representation of regions in a representative democracy begins with selection of a standard for comparison: a particular represented group, whose magnitude and slate magnitude (of representatives) stands at (1,1) in the quadrant. Economic applications There are many natural applications of hyperbolic coordinates in economics: Analysis of currency exchange rate fluctuation:The unit currency sets . The price currency corresponds to . For we find , a positive hyperbolic angle. For a fluctuation take a new price Then the change in u is: Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent measure. The quantity is the length of the left-right shift in the hyperbolic motion view of the currency fluctuation. Analysis of inflation or deflation of prices of a basket of consumer goods. Quantification of change in marketshare in duopoly. Corporate stock splits versus stock buy-back. History The geometric mean is an ancient concept, but hyperbolic angle was developed in this configuration by Gregoire de Saint-Vincent. He was attempting to perform quadrature with respect to the rectangular hyperbola y = 1/x. That challenge was a standing open problem since Archimedes performed the quadrature of the parabola. The curve passes through (1,1) where it is opposite the origin in a unit square. The other points on the curve can be viewed as rectangles having the same area as this square. Such a rectangle may be obtained by applying a squeeze mapping to the square. Another way to view these mappings is via hyperbolic sectors. Starting from (1,1) the hyperbolic sector of unit area ends at (e, 1/e), where e is 2.71828…, according to the development of Leonhard Euler in Introduction to the Analysis of the