Très rare premier état à la date de 1830. Edition originale de la première, et seule partie parue, de l'ultime oeuvre mathématiques de Fourier publiée posthume par son ami Navier. "In constrast with the famous work on heat diffusion, Fourier's interest in the theory of equations is remarkably little know. Yet it has a much longer personal history, for it began in his sixteenth year when he discovered a new proof of Descarte's rule of sign and was just as much in progress at the time of his death [...]. Fourier's proof was based on multiplying f(x) by (x + p), thus creating a new polynomial which contained one more sign in its sequence and one more positive (or negative) root, according as p was less (or greater) than zero, and showing that the number of preservations (or variations) in the new sequence was not inscreased relative to the old sequence. Hence the number of variations (or preservations) is increased by at least one, and the theorem follows. The details of the proof may be seen in any textbook dealing with the rule, for Fourier's youthful achievement quickly became the standard proof, even if its authorship appears to be viertually unknown[...]". "Fourier appears to have proved his own theorem while in his teens and he sent a paper to the Academy in 1789. However, it disappeared in the thrumoil of the year in Paris, and the pressure of administrative and other scientific work delayed publication of the resultats untiel the late 1810's. Then he became involved in a priority row with Ferdinand Budan de Bois-Laurent, a part-time mathematician who had previously published similar but inferior result. At the time of his death, Fourier was trying to prepare thse and many other result for a book to be called Analyse des équations déterminées ; he had almost finished only the first two of its seven "livres". His friend Navier edited it for publication in 1831 [sic], inserting an introduction to establish from attested documents (including the delayed 1789 paper) Fourier's priority on results which had by then become famous. Perhaps Fourier was aware that he would not live to finish the work, for he wrote a synopsis of the complete book which also appeared in this edition. The synopsis indicated his wide interests in the subject, of which the most important not yet mentioned were various means of distinguishing between real and imaginary roots, refinements of the Newton-Raphson method of approximating to the root of an equation, extensions to Daniel Bernoulli's rule for the limiting value of the ratio of successive terms of a recurrent series, and the method of solution and applications of linear inequalities. Fourier's remarkable understanding of the last subject makes him the great anticipator of linear programming." On trouve à la suite, deux extraits d'articles de Fourier tirés des Mémoires de l'Académie des Sciences portant sur le sujet de la théorie des équations : -Sur la distinction des racines imaginaires, et sur l'application des théorèmes d'analyse algébrique aux équations transcendantes qui dépendent de la théorie de la chaleur (Mémoires de l'Académie royale des sciences de l'Institut de France, tome VII, Paris, Didot, 1827, pages 605 à 624) ; -Remarques générales sur l'application des principes de l'analyse algébrique (lues à l'Académie des Sciences le 9 mars 1829 et publiées dans les Mémoires de l'Académie Royale des Sciences de l'Institut de France, tome X, Paris, Didot, 1831 ; pages 119 à 146). Bel exemplaire, à toute marge, portant l'ex-libris imprimé du bibliophile Henri Viellard et l'estampille, annulée, de l'Institut Catholique de Paris. DSB, V, p. 93-99.