Avignon, Libraires Associés, 1767. Small 8vo. Cont. hcalf. Richly gilt back. Back lightly rubbed. VII,(9),247 pp. Title with small rubberstamp and a little browned, lower right corner neathly repaired, no loss. Internally fine with a few brownspots. Printed on good paper. First edition. A work, treating the history of the Mendicant Friars, although issued anonymously, is attributed to d'Alembert. - Barbier II:684.

Berlin, C.F. Voss, 1779. Small 8vo. Unbound, but stitched. 96 pp. A few marginal brownspots, otherwise fine. First edition.

(Berlin, Haude et Spener, 1771). 4to. No wrappers as issued in "Memoires de l'Academie Royale des Sciences et Belles Lettres". tome XXV, pp. 265.-284. First appearance of one of D'Alembert's importent papers on the mathematical theory of acromatic lenses and abberation.

(Berlin, Haude et Spener, 1770). 4to. No wrappers, as issued in "Mémoires de l'Academie Royale des Sciences et Belles-Lettres", L'Année 1763, tome XIX, , pp. 235-254 + pp. 255-266 + pp. 267-291. Lambert's paper: pp. 292-310. First printing of all 4 papers. D'Alemberts letters to Lagrange deals with the mathematical treatment of a problem that taxed the minds of the major mathematicians of the day, discussing Bernouilli's and Eulers solutions on the vibrating string. D'Alembert had introduced the wave-equation in physics for the first time in in 1747.<br><br>Lambert made importent contributions to the theory of equations.

D'ALEMBERT, (JEAN le ROND). - [THE FIRST USE OF PARTIAL DIFFERENTIAL EQUATIONS IN MATHEMATICAL PHYSICS]

Reference : 31091

Paris, David l'aine, 1747. 4to. Cont. full calf, raised bands. Rebacked in old style. Inner hinges strenghtened. Corners restored. Engraved title-vignette and 1 large engraved vignette in the text. (8),XXVIII,194,138 pp. and 2 engraved folded plates (all). First 4 and last 6 leaves waterstained in margins. Occational marginal dampstaining. First edition, issued in the same year in both Paris and Berlin, but only the Paris-edition also has the Latin text, which was translated into French by d'Alembert himself.<br><br>The work is highly important, as it is the first work at all in which the general use of partial differential equations in mathematical physics appeared. D'Alembert discusses the mathematical theory of vibrations of cords and hereby he was led to partial differential equations which he applied to the "Theory of Winds" and laid the base of a scientific meteorology. He rejected the conception of Edmund Halley that the general circulation of the atmosphere is significantly controlled by the distribution of solar heating, and applies a mathematical theory, based on Newton's law of gravitation, thus explaining the winds by means of the gravitational forces from the sun and moon. - D'Alembert's name survives in the mathematics of today, the "Dalabertian" for wave equation, D'alembert's paradox in hydrodynamics etc.

(Berlin, Haude et Spener, 1767). 4to. Without wrappers as issued in "Memoires de l'Academie des Sciences et Belles Lettres", tome XXI, pp. 381-413. First edition, in the periodical form. In this paper D'Alembert comments on the previous solutions - by Huygens, Fontaine, Euler, Lagrange - to the tautochrone problem, the problem of finding the curve for an object sliding without friction in uniform gravity to its lowest point and in its shortest time.- Withbound a paper by M.J.A. Euler: Recherches des Forces dont les Corps célestes sont sollicités qu'ils ne sont pas spherique. Berlin, Acad. Royale, tome XXI, 1767.

D'ALEMBERT, JEAN LE ROND. - D'ALEMBERT'S THEOREM - THE FUNDAMENTAL THEOREM OF ALGEBRA.

Reference : 46603

Berlin, Haude et Spener, 1848-52. 4to. No wrappers as extracted from "Mémoires de l'Academie Royale des Sciences et Belles-Lettres", tome II (1846), tome IV, tome VI a. tome VI. Pp. 182-224, pp. 249-291, pp. (361-) 378, pp. 413-416 and 1 folded engraved plate. First apperance of d'Alembert's 3 importent papers on the Calculus of Integration, a branch of mathematical science which is greatly indepted to him. He here gives the proof of THE FUNDAMENTAL THEOREM OF ALGEBRA, called d'Alembert's theorem, and later corrected by Gauss (1799).<br><br>The theorem is based on these three assumptions:<br>Every polynomial with real coefficients which is of odd order has a real root. (This is a corollary of the intermediate value theorem. <br>Every second order polynomial with complex coefficients has two complex roots. <br>For every polynomial p with real coefficients, there exists a field E in which the polynomial may be factored into linear terms.<br><br>Also with an importent paper by Leonhard Euler "Mémoire sur l'Effet de la Propagation successive de la Lumiere dans l'Apparition tant des Planetes que des Cometes" (Memoir on the effect of the successive propogation of light in the appeareance of both comets and planets). Pp. 141-181 and 2 folded engraved plates. - The paper is founded on Euler's theory of light as waves and not as particles. It is from the same year as his fundamental work on light as waves: "Nova Theoria" - Enestroem E 104.

D'ALEMBERT, LEONHARD EULER, DANIEL BERNOULLI. - THE CONTROVERSY ABOUT VIBRATING STRINGS. - [FIRST APPEARANCE OF THE WAVE EQUATION.]

Reference : 38975

(Berlin, Haude et Spener, 1749-67). 4to. Bound together in one very nice recent marbled paper binding with gilt leather title-label to front board. All 9 without wrappers as extracted from "Memoires de L'Academie Royale des Sciences et Belles-Lettres", Tome III pp. 214-219 a. 1 engraved plate, Tome III pp. 220-249 a. 2 engraved plates, Tome VI pp. 355-360, Tome IV pp. 67-85 a. 1 engraved plate., Tome XXI pp. 307-335 a. 3 engraved plates, Tome IX pp. 196-222 a. 1 engraved plate, Tome IX pp. 173-195 a. 2 engraved plates, Tome IX, pp. 147-172 a. 1 engraved plate, Tome XXI pp. 281-306 a. 1 engraved plate. All 9 papers in first editions, in the periodical form. In D'Alemberts paper the WAVE-EQUATION appeared for the first time in print, and thus he was the first person to solve the mathematical equation for a vibrating string, and in the same connection it was the first success with partial differential equations. This paper opened the series of papers which are offered here. The three authors came to different conclusions on the nature of an "arbitrary" function and its expansion in trigonometric functions, a controversy brought to a conclusion only in the 19th century by Fourier, Cauchy, Dirichlet and Riemann. <br>"Nowadays this is the starting point for SUPERSTRINGS, which some people call "THE THEORY OF EVERYTHING". D'Alembert could not have guessed the importance his discovery would one day have. He was interested in strings because he was interested in music. He was a friend of the composer Rameau, and once wrote a book explaining Rameau's theory of harmony. So his broad interest in both the arts and sciences led D'Alembert to an important insight. Perhaps the entire universe is made of vibrating strings obeying an equation D'Alembert was first alerted to by the sound of a harpiscord." (Andrew Crumey).<br><br>The wave equation is an important partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. It arises in such fields as acoustics, electromagnetics, and fluid dynamics.<br>D'Alembert thought that he had given the general solution of the problem, but Euler pointed out that there is no physical reason to require that the initial position of the string is given by a single function - D'Alembert had operated with this arbitrary function (the initial condition) and one other arbitrary function, the one that gives the shape of the travelling wave - different parts of the string could very well be described by different formulas as long as they fitted together smoothly. Moreover, the travelling wave solution could be extended to his situation. The point is that for Euler and D'Alembert every function had a graph represented a single function. Euler argued that any graph - even if not given by a function - should be admitted as a possible initial position of the string. D'Alembert did not accept Euler's physical reasoning (Euler's first paper here offered). In 1755 Daniel Bernoulli joined the argument with his paper (Bernoulli's first paper offered here). He found another form of solution for the vibrating string, using "standard waves". A standing wave is a motion of the string in which there are fixed "nodes" which are stationary; between the nodes each segment of the string moves up and down in unison. The "principal mode" is the one without nodes, the "second harmonic" is the name given to the motion with a single node at the mid-point. The "third harmonic" has two equally spaced-nodes, and so on - each "harmonic" thus corresponds to a pure tone of music. <br>"Just as D'Alembert had rejected Euler's reasoning, now Euler rejected Bernoulli's. First of all, as Bernoulli acknowledged, Euler himself had already found the standing wave solution in one special case. Euler's objection was to claim that the standing-wave solution was general - applicable to all motions of the string" (Davis & Hersh). D'Alembert also attacked Bernoulli's solution (see his article in the Encyclopedie).<br>Euler's method was more useful than D'Alemberts, but, as it turned out, Bernoulli was closer to the truth. The later use by Fourier of sines and cosines on heat flow was very similar to Bernoulli's method of studying the vibrations. "The debate about the Vibrating Strings raged throughout the 1760s and 1770s. Even Laplace entered the fray in 1779, and sided with d'Alembert. D'Alembert continued in a series of booklets, entitled Opuscules, which began to appear in 1768". (Morris Kline).

Berne et Lausanne, Sociétés Typographiques, 1778. Large 8vo. Contemp. hcalf. Raised bands, title-and tomelabels with gilt letterint. Small nicks to leather at lower compertment. Corners bumped. Some scratches to coverpaper. Engraved portrait of D'Alembert as frontispiece. (4),CIV,784 pp. and 4 folded tables. Internally clean and fine, printed on good paper.