By Mourrain B., Pavone J.-P.
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Extra info for Subdivision methods for solving polynomial equations
Note that the points A' and B' of the curve I with abscissas —d/ k and + d/ik correspond to the points A" and B" of the curve 2 with the abscissas — d and + d (Fig. 51). By virtue of the similarity of the arcs AB and AUBH it follows that the length ofA"B" is equal to (21//k)k = = 2l,, that is. to the given length of the chain. This is the advantage of 43 the curve 2 over the initial curve 1. Its drawback, however, lies in the fact that the curve I passes through the points of suspension A and B, and the curve 2 may not pass through them.
The Catenary Only half a century after the publication of Galilei’s book the elder of the Bemoulli brothers, Jacob, found in a purely theoretical way an exact formula giving the shape of a suspended chain. Not publishing his findings he challenged other mathematicians to do what he had done. That was in 1690. lakob, Johann Bernoulli. All of them used in the solution of the problem the laws of mechanics and the powerful technique of the recently developed mathematical analysis, the derivative and integral.
In looking for the answer we shall depend on the fact mentioned above that all catenaries are similar. Suppose l'>l. Then the chain will hang along some arc ACB beneath the arc ACB (F ig. 49). We shall show that the required equation of the catenary to which the arc ACB belongs can be found in two steps. First we have to transform the curve I. y = (1 /2) (ex + e"‘), into the curve 2, y=(k/2)(e’”‘+e"‘”‘). The latter cn be obtained from curve I by the similarity transformation with the point O as centre and k as the ratio of magnification (k is positive).
Subdivision methods for solving polynomial equations by Mourrain B., Pavone J.-P.