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1. De Linea in quam Flexile se pondere proprio curvat ejeuque usu insignia adinveniendi quotcunque medias proportionales & Logarithmos. - 2. De Solutionibus Problematis Catenarii vel Funicularis in Actis A. 1691 aliisque a Dn. I.B. propositis. 1-2:.

1691 · 41859 — Leipzig Grosse & Gleditsch 1691. 4to. Contemp. full vellum. Faint handwritten title on spine. a small stamp on titlepage. In: "Acta Eruditorum Anno MDCLXXXXI". 85906 pp. and 13 of 15 folded engraved plates. The 2 first plates lacks but they do not belong to the papers listed.Leibniz' papers: pp.277-281 a. 1 plate pp. 435-439. Johann Bernoulli: pp. 274-276 a. 1 plate. Huygens: pp. 281-282. - Jacob Bernoulli: pp. 282-290 a. 1 plate. <br/><br/><em>All papers first apperance. All 5 of extreme importence in the development of the Calculus. Leibniz' 2 papers on the catenary curve paper 1-2 offered here was written at the instigation of Jacques Bernoulli. Following the example of Blaise Pascal who had initiated in 1658 a contest for the construction of the cycloid Leibniz also provoked the geometers of his time by challenging them to submit at the fixed date of mid-1691 their geometric method for the construction of the catenary curve. Leibniz later provided the answer followed by Johann Bernoulli and Huygens.'These two papers are a historical account of the origin of the study of this transcendental curve and at the same time the first physical-geometric construction showing the species-relationship between the catenary and the logarithmic curves as two companion curves; one arithmetic the other geometric. All of the differentials of the catenary curve are arithmetic means of corresponding differentials of the logarithmic curve; and all of the differentials of the logarithmic curve are geometric means of the catenary.'"The Catenary is the form of a hanging fully flexible rope or chain the name comes from "catena" which means 'chain' suspended on two points. The interest in this curve originated with Galileo who thought that is was a parabola. Young Christiaan Huygens proved in 1646 that this cannot be the case. What the actual form was remained an open question till 1691 when Leibniz Johann Bernoulli and the then much older Huygens sent solutions to the problem to the "Acta" Jakob Bernoulli 1690 Johann Bernoulli 1691 Huygens 1691 and Leibniz 1691 - these 4 1691-papers offered here - in which the previous year Jakob Bernoulli had challenged mathematicians to solve it. As published the solutions did not reveal the methods but through later publications of manuscripts these methods have been known. Huygens applied with great paper 4 virtuosity the by then classical methods of 17th century infinitesimal mathematics and he needed all his ingenuity to reach a satisfactory solution. Leibniz the papers 1-2 and Bernoulli paper 3 applying the new Calculus found the solutions in a much direct way. In fact the catenary was a test-case between the old and the new style in the study of curves and only because the champion of the old style was a giant like Huygens the test-case can formally be considered as ending in a draw." Grattan-Guiness in "From the Calculus to Set Theory 1630-1910.".The paper by JACOB BERNOULLI no. 5 offered here is a milestone papers as it marks the invention of the "SYSTEM OF POLAR COORDINATES" with points located by reference to a fixed point and a line through that point. Although newton had earlier also devised such a coordinate system in 1671 his work was not known so that the credit for the discovery generally goes to Bernoulli. Parkinson Breakthroughs 1691.Further papers contained in this volume of Acta Eruditorum:DENYS PAPIN: Mecanicorum de Viribus Motricibus sententia asserta a D. Papino adversius C.G.G. L. Leibniz objectiones. pp. 6-13. The plate lacks. - and Dion. Papini Observationes quaedam circa materias ad Hydraulicam spectantes. Pp. 208-213 a. 1 plate. This importent paper is part of the LEIBNIZ-PAPIN-CONTROVERSY.JACOB BERNOULLI: Specimen Calculi Differentialis in dimensione Parabolæ helicoidis ubi de flexuris curvarum in genere carundem evolutionibus. Pp. 13-22. The plate lacks. - and J.B. Demonstratio Centri Oscillationis ex Natura Vectis reperta occassione eorum quæ super hac materia in Historia Literaria Roterodamensi recensentur articulo.Pp.317-321.LEIBNIZ: O.V.E. Additio ad Schediasma de Medii Resistentia publicatum in Actis mensis Febr. 1889. Pp. 177-178. and O.V.E. Quadratura Arithmetica Communis Sectionum Conicarum quæ centrum babent.Pp. 178-182 a. 1 plate.TSCHIRNHAUS: Singularia Effecta Vitri Caustici bipedalis quod omnia magno sumtu hactenus constructa specula ustoria virtute superat per D.T. Pp. 517-520 </em> hardcover