Windmills were developed to a high degree between their initial European use in the twelfth century and the twentieth century when mathematical analysis was first applied to optimise the design of their sails. It led immediately to the concept of twisted sails in which the weather angle decreased with radius. Various theories were tested experimentally by Smeaton, who found twisted sails to be better than common sails but less good than the Dutch sails then normally used. A century passed before Rankine applied Newton's laws of motion to derive their power and efficiency (power coefficient). Annular windmills were developed in both Britain and the United States where they were extensively tested by Perry, who found them as efficient as Smeaton's sails. In the twentieth century, Lanchester, Betz, and Joukowski independently derived an upper limit for windmill power coefficient and this, combined with aerodynamic theory and streamlined aerofoils led to the development of the modern wind turbine.
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