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:.

Leipzig, Grosse & Gleditsch, 1691. 4to. Contemp. full vellum. Faint handwritten title on spine. a small stamp on titlepage. In: "Acta Eruditorum Anno MDCLXXXXI". (8),590,(6) 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. 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, rep.…