Period After Hero and Before Galileo

The surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length 5 of C and the distance d traveled by its geometric centroid. 

For example, when a rectangular of sides r and I is rotated by 360° about its one side of length, a right cylinder of length I and radius r is generated. The area of the rectangular is rl and its centroid travels a distance of 2tt(r/2). Hence, by the second theorem of the Pappus, the volume of the cylinder is nr x rl = ni l. 

Leonardo da Vinci (1452-1519) stated the two basic laws of friction (http // www.phy.davidson.edu/fachome/dmb/PY430/Friction/history.html)  

He also stated that every fiictional body has a resistance equal to one quarter of its weight. In other words, he suggested a Coulomb coefficient of friction of 0.25 for the bodies. This value is high considering most of the sliding bodies today however, in Leonardo da Vinci s time, the tribological characteristics of the bodies were definitely inferior in comparison to present age. Leonardo da Vinci did not publish his theories on friction and he never got credit for it. 

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