By J.H. Jeans, M.A., F.R.S
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This quantity comprises the papers of the eleventh Symposium of the AG STAB (German Aerospace Aerodynamics Association). during this organization these scientists and engineers from universities, research-establishments and are concerned, who're doing learn and undertaking paintings in numerical and experimental fluid mechanics and aerodynamics for aerospace and different purposes.
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For the two points we' ll use to calculate the slope we choose x 1 = x and x 2 = x + h. We are using h as an alternative symbol for Ax, as was done in Chapter . The corresponding values of y will be y 1 = y(x) and Y2 = y(x + h). ---'--- Ax = (x 2xh + h + h) - x 2 h = 2x + h. 3) Because h has canceled out of the denominator, we can now safely let h shrink to zero, obtaining the slope at any particular point along the parabola: slope = 2x. This result may look familiar, and indeed it is: we have j ust calculated the derivative y'(x) of the function y(x) = x 2 with respe~t to x [just as we calculated the derivative of the d istance function s(t) = ct 2 with respect to t in Chapter and found s' (t) = 2ct].
One of these resembled Leonardo's formulation. He said that the distance in successive time intervals is proportional to the odd numbers I , 3, 5, . . In other words, if a body falls one unit of distance in one unit of time; then in the second unit of time it falls three units of distance, five in the third, and so on. 2 Laws interrelating the time, position, and speed of falling bodies, as proposed by Albert of Saxony, Oresme, Leonardo da Vinci, and Galileo. We've now encountered four different attempts to describe falling motion: speed is proportional to distance or to time; distance per time interval is proportional to the integers or to the odd numbers.
Now we let h shrink to 0 and see what happens to this equation. dt, and x 2 becomes t 2 • ln other words we have shown that -dA = r2 • dt the derivative of A(t) is t 2 . Therefore A(t) must be an antiderivative of t 2 • But we know all antiderivatives of t 2 have the form ~ t 3 + C for some constant C, so A(t) = ~t3 + C. But when t = 0 the area is 0 and hence C = 0, so we find A(t) = ~t 3 . We have obtained Archimedes' qiUadrature formula using antidifferentiation! The reason this discovery is important is not that Newton and Leibniz solved the quadrature problem for a parabolic segment.
An Elementary Treatise on Theoretical Mechanics by J.H. Jeans, M.A., F.R.S