## “Supplementum defectus geometria Cartesianae circa inventionem locorum” [Joh. Bernoulli, pp. 264-269] WITH [Two other papers] WITH Notatiuncula ad acta decemb. 1695 [Leibniz, pp. 45-47] in Acta Eruditorum Anno M DC XCVI, 1696

Leipzig: Grosse & Gleditsch. 1st Edition. FIRST EDITION, FIRST PRINTING OF JOHANN BERNOULLI’S FAMOUS BRACHISTOCHRONE PROBLEM (sometimes called the Brachistochrone challenge). The Brachistochrone (Greek for ‘shortest’ and ‘time) problem itself was one of the earliest examples of a class of problem now known as the calculus of variations. By the 19th century, thanks to Lagrange among others, all fundamental laws of nature could be formulated in terms of the calculus of variations – a kind of universal language of physics.

Bernoulli issued his mathematical challenge in the context of the argument between Newton and Leibniz over who first invented the calculus, the mathematical study of change. While Newton, save for a small notice, made something of a secret of his discovery of fluxions, Leibniz had published his calculus in 1684. By 1695, he and his student, Johann Bernoulli, had developed the calculus into a magnificent tool for solving an array of problems. Bernoulli and Leibniz, however, were curious about how much Newton really knew. Together they devised a test, the Brachistochrone problem, confident that only a mathematician who knew the calculus could solve it. Following the custom of the time, they published it.

The problem read as follows: "Let two points A and B be given in a vertical plane. To find the curve that a point M, moving on a path AMB , must follow such that, starting from A, it reaches B in the shortest time under its own gravity" (ibid). Johann added that this curve is not a straight line, but a curve well known to geometers.

Wanting to lure Newton, Bernoulli introduced his challenge as follows: “I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise” (Bernoulli, Acta 1696, p. 264).

The solution to the Brachistochrone problem was a segment of a cycloid and it was solved by Leibniz, Newton (absent a proof), von Tschirnhaus, l’Hôpital and both Bernoullis. "Among the solutions, the two by the Bernoulli brothers were of particular importance. Johann had ingeniously reformulated the challenge as a problem belonging to optics, i.e., the bending of a light ray through a transparent nonuniform media, and applied Fermat’s principle of least time. He thus demonstrated the fundamental parallel between geometric optics and point mechanics, which would lead to the work of W.R. Hamilton in the 1830’s. The solution by Jacob also contains a general principle, namely, that a curve which constitutes a maximum or minimum as a whole must also possess this property in the infinitesimal - the fundamental principle of the calculus of variations" (DSB).

Two of the solutions would be of significant import: one, Johann’s own, because it would present the first example of the optical-mechanical analogy, and his brother Jakob’s because it represented an important early step in the calculus of variations in that he realized that the problem itself was of a new type, the variable being a function (upon which, not incidentally, Lagrangian mechanics is based).

ALSO PRESENT ARE OTHER PAPERS OF SIGNIFICANCE: Four by Jakob Bernoulli: “Observatiuncula ad ea quaenupero mense novembri de Dimensionibus Curvarum leguntur” WITH “Constructio Generalis omnium Curvarum transcendentium ope simplicioris Tractoriae et Logarithmicae”, WITH “Problema Beaunianum universalius conceptum”, WITH “Complanatio Superficierum Conoidicarum et Sphaeroidicarum” AND two more by Johann Bernoulli “Demonstratio Analyticea et Syntetica fuae Constructionis Curvae Beaunianae” WITH “Tetragonismus universalis Figurarum Curvilinearum per Construitionem Geometricam continuo appropinquantem” AND two by Tschirnhaus “Intimatio singularis novaeque emendationis Artis Vitriariae” WITH “Responsio ad Observationes Dnn. Bernoulliorum, quae in Act. Erud. Mense Junio continentur” WITH “Additio ad Intimationem de emendatione artis vitriariae”. Item #974

CONDITION & DETAILS: Leipzig: Grosse & Gleditsch. Octavo (slightly trimmed by a conservator at the top margin; no loss whatsoever and if one didn’t know it wouldn’t be evident). [4], 604 [inclusive of errata], 2. 9 copperplates. The only institutional evidence is a small, unreadable stamp on the front flyleaf. Tightly and solidly bound in half-calf over marbled paper boards; some of the calf at the spine chipped and has been professionally re-glued. The boards are scuffed and rubbed (see photo); internally, light toning; largely bright and very clean. Very good condition.

Price: \$1,500.00