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By Arutyunov A.V., Jacimovic V.

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Since I am not in a position to illuminate this remarkable personality with a measure of thoroughness, I must refer the refer to two books on Kepler. 1 In his book Predtecenskii describes with rare beauty Kepler's moral eminence. I n his recent book Belyi describes in detail Kepler's scientific progress and his genius. Belyi's book also includes an ex­ tensive bibliography. It seems to have been Kepler's fate to be spared no trial. He endured poverty, privation, sickness, the death of loved ones, upheavals, and exile.

We will show that for any other point =f. the time of traversal of the path is greater than the time of traversal of the path To this end we erect perpendiculars to the line at and respectively. Let be the point of intersection of and the perpendicular at . We draw a line through parallel to and denote its points of intersection and R , respectively. 2 we see that the angles and are respectively equal to o: 1 and Tr/2 - o:2 , respectively. 3 What underlies Fermat's principle is the assumption that light is propa­ gated along certain lines.

Fermat himself obtained the required result in a far more elaborate way. It is natural to ask: Why did Fermat not use his algorithm? The answer is very simple: Fermat could apply his method to polynomials-and here he actually antic­ ipated the notion of a derivative-but he did not know how to apply it to radical expressions. That is why the deduction of Snel's law using derivatives was first accomplished by Leibniz, who introduced this concept in the very same work of 1 684 in which he laid the foundations of the grandiose edifice of mathematical analysis.

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2-Normal Processes in Controlled Dynamical Systems by Arutyunov A.V., Jacimovic V.


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