Page:Encyclopædia Britannica, Ninth Edition, v. 16.djvu/767

Rh result so quickly that the commandant at first refused to receive it the time necessary for the work had not been taken ; but upon examination the value of the discovery was recognized, and the method was adopted. And Monge, continuing his researches, arrived at that general method of the application of geometry to the arts of construc tion which is now called descriptive geometry. But such was the system in France before the Revolution that the officers instructed in the method were strictly forbidden to communicate it even to those engaged in other branches of the public service ; and it was not until many years af ter- Avards that an account of it was published. The method consists, as is well known, in the use of the two halves of a sheet of paper to represent say the planes of xy and xz at right angles to each other, and the consequent repre sentation of points, lines, and figures in space by means of their plan and elevation, placed in a determinate relative position. In 1768 Monge became professor of mathematics, and in 1771 professor of physics, at Mezieres; in 1778 he married Madame Horbon, a young widow whom he had previously defended in a very spirited manner from an unfounded charge; in 1780 he was appointed to a chair of hydraulics at the Lyceum in Paris (held by him together with his appointments at Mezieres), and was received as a member of the Academy ; his intimate friendship with Berthollet began at this time. In 1783, quitting Mezieres, he was, on the death of Bezout, appointed examiner of naval candidates. Although pressed by the minister to prepare for them a complete course of mathematics, he declined to do so, on the ground that it would deprive Madame Bezout of her only income, arising from the sale of the works of her late husband ; he wrote, however (1786), his Traite elementaire de la Statique. Monge contributed (1770-1790) to the Memoirs of the Academy of Turin, the Memoires ties Savants Strangers of the Academy of Paris, the Memoires of the same Academy, and the Annales de Chimie, various mathematical and physical papers. Among these may be noticed the memoir &quot; Sur la theorie des deblais et des remblais &quot; (Mem. de VAcad. de Paris, 1781), which, while giving a remarkably elegant investigation in regard to the problem of earth work referred to in the title, establishes in connexion with it his capital discovery of the curves of curvature of a surface. Euler, in his paper on curvature in the Berlin Memoirs for 1760, had considered, not the normals of the surface, but the normals of the plane sections through a particular normal, so that the question of the intersection of successive normals of the surface had never presented itself to him. Monge s memoir just referred to gives the ordinary differential equation of the curves of curvature, and establishes the general theory in a very satisfactory manner ; but the application to the interesting particular case of the ellipsoid was first made by him in a later paper in 1795. A memoir in the volume for 1783 relates to the production of water by the combustion of hydrogen ; but Monge s results in this matter had been anticipated by Watts and Cavendish. In 1792, on the creation by the Legislative Assembly of an executive council, Monge accepted the office of minister of the marine, but retained it only until April 1793. When the Committee of Public Safety made an appeal to the savants to assist in producing the materiel required for the defence of the republic, he applied him self wholly to these operations, and distinguished himself by his indefatigable activity therein ; he wrote at this time his Description de I art de fabriquer les canons, and his Avis aux ouvriers en fer sur la fabrication de Vacier. He took a very active part in the measures for the establishment of the Normal School (which existed only during the first four months of the year 1795), and of the School for Public Works, afterwards the Polytechnic School, and was at each of them professor for descriptive geometry; his methods in that science were first pub lished in the form in which the shorthand writers took down his lessons given at the Normal School in 1795, and again in 1798-99. In 1796 Monge was sent into Italy with Berthollet and some artists to receive the pictures and statues levied from several Italian towns, and made there the acquaintance of General Bonaparte. Two years afterwards he was sent to Rome on a political mission, which terminated in the establishment, under Massena, of the shortlived Roman republic ; and he thence joined the expedition to Egypt, taking part with his friend Berthollet as well in various operations of the war as in the scientific labours of the Egyptian Institute of Sciences and Arts ; they accompanied Bonaparte to Syria, and returned with him in 1798 to France. Monge was appointed president of the Egyptian commission, and he resumed his connexion with the Polytechnic School. His later mathematical papers are published (1794-1816) in the Journal and the Correspondence of the Polytechnic School. On the forma tion of the Senate he was appointed a member of that body, with an ample provision and the title of count of Pelusium ; but on the fall of Napoleon he was deprived of all his honours, and even excluded from the list of mem bers of the reconstituted Institute. He died at Paris on the 28th July 1818. For further information see B. Brisson, Notice hislorique sur Gaspard Monge ; Dupin, Essai historique sur les services et les tra- vaux scicntifiqucs de Gaspard Monge, Paris, 1819, which contains (pp. 162-166) a list of Monge s memoirs and works ; and the bio graphy by Arago (CEuvres, t. ii., 1854). Monge s various mathematical papers are to a considerable extent reproduced in the Application de V Analyse a la Geometric, 4th edition (last revised by the author), Paris, 1819 the pure text of this is reproduced in the 5th edition (revue, corrigee et annotee par M. Liouville), Paris, 1850, which contains also Gauss s Memoir, &quot; Disquisitiones generales circa superficies curvas,&quot; and some valu able notes by the editor. The other principal separate works are Traite elementaire de la Statique, 8 edition, conformee a la precc- dcnte, par M. Hachette, et suivie d une Note etc., par M. Cauchy, Paris, 1846; and the Geometric Descriptive (originating, as mentioned above, in the lessons given at the Normal School). The 4th edition, published shortly after the author s death, seems to have been sub stantially the same as the 7th (Geometric Descriptive par O. Monge, suivie d une theorie des Ombres et dc la Perspective, extraite dcspapiers de I auteur, par M. Brisson, Paris, 1847). (A. CA.) MONGHYR, or MUNGIE, a district in the lieutenant- governorship of Bengal, lying between 24 22 and 25 49 N. lat., and 85 40 and 86 52 E. long., is bounded on the N. by Darbhangah and Bhagalpur, on the E. by Bhagal- pur, on the S. by the Santal Pargands and Haz.iribagh, and on the W. by Gaya, Patna, and Darbhangah, with an area of 3922 square miles. The Ganges divides the district into two portions. The northern, intersected by the Burl Gandak and Tiljuga, two important tributaries of the Ganges, is always liable to inundation during the rainy season, and is a rich, flat, wheat and rice country, support ing a large population. A considerable area, immedi ately bordering the banks of the great rivers, is devoted to permanent pasture. Immense quantities of buffaloes are sent every hot season to graze on these marshy prairies ; and the ghi, or clarified butter, made from their milk forms an important article of export to Calcutta. To the south of the Ganges the country is dry, much less fertile, and broken up by fragmentary ridges. The soil consists of quartz, mixed in varying proportions with mica. Ranges of hills intersect this part of the district, and in the extreme south form conical peaks, densely covered with jungle, but of no great height. Irrigation is necessary throughout the section lying on the south of the Ganges. In 1872 the population of Monghyr was 1,812,986 (males, 897,074; females 915,912): Hindus, 1,613,546 ; Mohammedans,