At large distances from the source (compared to the size of the source), this power is distributed over larger and larger spherical surfaces as the distance from the source increases. Since the surface area of a sphere of radius "r" is "A" = 4"πr" 2, the intensity "I" (power per unit area) of radiation at distance "r" is  The energy or intensity decreases (divided by 4) as the distance "r" is doubled; measured in dB it would decrease by 6.02 dB per doubling of distance. In acoustics, the sound pressure of a spherical wavefront radiating from a point source decreases by 50% as the distance "r" is doubled; measured in dB, the decrease is still 6.02 dB, since dB represents an intensity ratio. The pressure ratio (as opposed to power ratio) is not inverse-square, but is inverse-proportional (inverse distance law):  The same is true for the component of particle velocity formula_6 that is in-phase with the instantaneous sound pressure formula_7:  In the near field is a quadrature component of the particle velocity that is 90° out of phase with the sound pressure and does not contribute to the time-averaged energy or the intensity of the sound. The sound intensity is the product of the RMS sound pressure and the "in-phase" component of the RMS particle velocity, both of which are inverse-proportional. Accordingly, the intensity follows an inverse-square behaviour:  For an irrotational vector field in three-dimensional space, the inverse-square law corresponds to the property that the divergence is zero outside the source. This can be generalized to higher dimensions. Generally, for an irrotational vector field in "n"-dimensional Euclidean space, the intensity "I" of the vector field falls off with the distance "r" following the inverse ("n" − 1)th power law  given that the space outside the source is divergence free. John Dumbleton of the 14th-century Oxford Calculators, was one of the first to express functional relationships in graphical form. He gave a proof of the mean speed theorem stating that "the latitude of a uniformly difform movement corresponds to the degree of the midpoint" and used this method to study the quantitative decrease in intensity of illumination in his "Summa logicæ et philosophiæ naturalis" (ca. 1349), stating that it was not linearly proportional to the distance, but was unable to expose the Inverse-square law.