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The square–cube law was first mentioned in Two New Sciences (1638). The square–cube law (or cube–square law) is a mathematical principle, applied in a variety of scientific fields, which describes the relationship between the volume and the surface area as a shape's size increases or decreases.
At low , is asymptotically proportional to , which means that the drag is linearly proportional to the speed, i.e. the drag force on a small sphere moving through a viscous fluid is given by the Stokes Law: = At high , is more or less constant, but drag will vary as the square of the speed varies.
The divergence of a vector field which is the resultant of radial inverse-square law fields with respect to one or more sources is proportional to the strength of the local sources, and hence zero outside sources. Newton's law of universal gravitation follows an inverse-square law, as do the effects of electric, light, sound, and radiation ...
The ratio between the volumes of similar figures is equal to the cube of the ratio of corresponding lengths of those figures (for example, when the edge of a cube or the radius of a sphere is multiplied by three, its volume is multiplied by 27 — i.e. by three cubed). Galileo's square–cube law concerns similar solids.
Because of the necessary phase change, the expander cycle is thrust limited by the square–cube law. When a bell-shaped nozzle is scaled, the nozzle surface area with which to heat the fuel increases as the square of the radius, but the volume of fuel to be heated increases as the cube of the radius.
Isometric scaling is governed by the square–cube law. An organism which doubles in length isometrically will find that the surface area available to it will increase fourfold, while its volume and mass will increase by a factor of eight. This can present problems for organisms.
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Square–cube law (ratio of surface area to volume) A 3/2-power law can be found in the plate characteristic curves of triodes . The inverse-square laws of Newtonian gravity and electrostatics , as evidenced by the gravitational potential and Electrostatic potential , respectively.