By Petr Beckmann
The heritage of pi, says the writer, even though a small a part of the background of arithmetic, is however a replicate of the background of guy. Petr Beckmann holds up this replicate, giving the historical past of the days whilst pi made development -- and likewise while it didn't, simply because technology was once being stifled by means of militarism or spiritual fanaticism.
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Extra info for A History of Pi
Let us suppose that u does represent some wave propagating in a region having a boundary curve x,(z), yo(z). For this incident wave, the objective is to find the reflected wave, with u for both the incident and reflected waves being solutions of the Fig. 8. The two ray directions at a point on the initial curve. 22 1 First-Order Partial Differential Equations eikonal equation on the same side of the boundary curve. On physical grounds, one can argue that the phase of the incident and reflected waves should be the same at the boundary.
We introduce tiR, and aT by setting pol = cos UI. POR = por cos UR > =n cos aT. 7) qOT = n sin aT. Here aI is known. 3) is no longer equal to unity since uTis a solution on the right of the boundary curve. 11) to describe the direction numbers for the boundary curve and conclude as in that discussion that aR = 2e - cII. 4, the incident ray direction and the reflected ray direction make equal angles with the tangent to the boundary curve. 7) as n cos(8 - U T ) = cos(8 - El). 9) This equation is Snell's law, and the wave we are calling the transmitted wave is actually the refracted wave.
4). If 4 satisfies the eikonal equation, what effect will discontinuities in U,, have on the solution? 8 Let U be a solution of Uxx + uyy+ (w”c2)U Assume a solution of the form = 0. 5 Propagation of Wave Fronts 27 with A being a power series in inverse powers of o whose coefficients are functions of x and y. Substitute this form into the equation for U. 4). 9 Suppose that a set of base characteristics is given by x =fh4, y = do,4. Then we solve for n in the first equation and substitute into the second to obtain y as a function of x and 7 .
A History of Pi by Petr Beckmann