By C. Edward Sandifer

Sandifer has been learning Euler for many years and is likely one of the global s prime specialists on his paintings. This quantity is the second one selection of Sandifer s How Euler Did It columns. every one is a jewel of ancient and mathematical exposition. The sum overall of years of labor and research of the main prolific mathematician of heritage, this quantity will go away you marveling at Euler's shrewdpermanent inventiveness and Sandifer's fabulous skill to explicate and positioned all of it in context.

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**Extra resources for How Euler Did Even More**

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However, the Summarium, as well as the title I chose for the column, advertised an infinite product of secants. We haven’t seen such an infinite product, nor have we seen anything of that spiral illustration that caught my eye in the first place. It’s time to see what we can do about that. Euler begins a new problem. Problem. Taking ϕ to be any arc of a circle of radius 1, to find the sum of the infinite series tan ϕ + 1 1 1 1 1 1 1 1 tan ϕ + tan ϕ + tan ϕ + tan ϕ + etc. 2 2 4 4 8 8 16 16 To solve this problem, Euler brings back the figure from his first problem.

Continue constructing rectangles, each with 1/4 the area of the previous one, and each with its corner on the ray. It is easy to see that the sum of the bases of these rectangles, ab + bc + cd + de + · · · converges to some length, call it ax, but it is not so easy to see how ax is related to ab, or how this has anything to do with the circumference of a circle. Descartes, in the style of his times, doesn’t tell us. Euler, though, sets out to prove it, and he shares the details of his proof with us.

236–252. org 34, 35 Did Euler Prove Cramer’s Rule? 37 [Burckhardt 1983] Burckhardt, J. , Fellman, E. , Leonhard Euler 1707– ¨ zu Leben und Werk, Birkh¨auser, Basel, 1983. 32 1783: Betrage [Cramer 1750] Cramer, Gabriel, Introduction a l’analyse des lignes courbes alg´ebriques, Geneva: Cramer fr`eres, 1750. 33, 34, 36 [Gua de Malves 1740] Gua de Malves, J. P. de, Usages de l’analyse de Descartes pour d´ecouvrir, sans le secours du Calcul Differentiel, les Propri´et´es, ou affctations principales des lingnes g´eometriques de tous les ordres, Paris: Briasson, 1740.