Place of birth: Beaune, Burgundy, France
Areas of activity: mathematics, mechanics, technology
Gaspard Monge Comte de Péluse (Gaspard Monge, comte de Péluse, 1746, Beaune, Burgundy, France - July 28, 1818, Paris) - French mathematician, geometer, engineer, statesman. The founder of descriptive geometry. His research in the fields of physics, chemistry, optics, metrology and practical mechanics is known.
Gaspard Monge was born in the small town of Beaune in eastern France into the family of a local merchant. His parents were Jacques Monge and Jeanne Rousseau. Gaspard was the eldest of five children, to whom his father, despite the low origins and relative poverty of the family, tried to provide the best education that a person from a lower class could afford. Gaspard's brother Louis became a professor of mathematics and astronomy, another brother Jean also became a professor of mathematics, hydrography and navigation. Gaspard Monge received his initial education at the city school of the Oratorian Order. After graduating in 1762 as the best student, he entered the College of Lyon, which also belonged to the Oratorians. Soon Gaspard was entrusted with teaching physics there.
Already at the age of 14, the boy invented a fire pump, which had an original design and thoughtfulness of all the details. In the summer of 1764, Monge drew up a remarkably accurate plan of his hometown of Beaune. To draw up this plan, the young self-taught geometer used measuring and drawing instruments of his own making and invention. The plan was so successful that one abbot used it for his small historical work. Now this plan, like an expensive relic, is kept in one of the city libraries of Beaune.
While studying in Lyon, Gaspard received an offer to join the order and remain as a college teacher, however, instead, having shown great abilities in mathematics, drawing and drawing, he managed to enter the Mezières School of Military Engineers, but (due to his origin) only as an auxiliary non-commissioned officer department and without pay. However, advances in the exact sciences and original solution one of the important tasks of fortification (the placement of fortifications depending on the location of the enemy’s artillery) allowed him in 1769 to become an assistant teacher of mathematics, and then physics, and with a decent salary of 1800 livres per year.
In 1769, at the age of 23, Monge took the position of professor of mathematics, and in 1770 - professor of physics, at the School of Military Engineers, and, in addition, taught classes in stone cutting. Starting with the task of accurately cutting stones according to given sketches in relation to architecture and fortification, Monge came to the creation of methods that he later generalized in a new science - descriptive geometry. Gaspard based his science on the rectangular projection of a spatial figure onto two mutually perpendicular planes (horizontal and vertical) and an original method of depicting it on a plane (method of diagrams). At the Military Engineering School, where Monge taught, a new department of descriptive geometry was organized. Monge was made the head of this department.
Considering the possibility of using the methods of descriptive geometry for military purposes in the construction of fortresses and all other military structures, Monge was forbidden to print anything about his discovery for fear that foreigners would use it and thereby deprive France of military superiority over others. Descriptive geometry was declared a military secret. The leadership of the Mézières school did not allow open publication of Monge's works until 1799.
In 1777, Monge married the young widow of a foundry owner, Maria Catherine Huart (Orboni). The marriage was happy and lasted until the end of Monge's life. Finding himself the owner of a workshop, he mastered foundry, became interested in metallurgy, and became seriously involved in physics and chemistry.
Monge taught at the Mézières school for 20 years. There they taught geometry, physics, fortification, and construction with an emphasis on practical exercises. This school became the prototype of the future famous Polytechnic School. In addition to the basics of descriptive geometry, Monge also developed other mathematical methods, including the theory of developments, calculus of variations and others. Several reports that he made with great success at meetings of the Paris Academy of Sciences, and the recommendations of academicians d'Alembert, Condorcet and Bossu ensured Monge's election in 1772 to one of the twenty corresponding members of the French Academy of Sciences, and in 1780 he was already elected academician. Monge moved to Paris, retaining his position at the Mézières school. In addition, he began teaching hydrodynamics and hydrography at the Paris Nautical School, and subsequently took the position of examiner of maritime schools. However, working and living alternately for six months in Paris and Mézières over time became very tiring for him and did not suit the leadership of the Mézières school. In 1783, Monge stopped teaching at school and in 1784 finally moved to Paris.
Elected to academician, Monge, in addition to research on mathematical analysis, published in the “Memoirs” of the Academy, worked together with Berthollet and Vandermonde to study various states of iron, carried out experiments on capillarity, made observations on optical phenomena, and worked on constructing a theory of main meteorological phenomena. Independently of Lavoisier and Cavendish, he discovered that water is a compound of hydrogen and oxygen. In 1781, Monge published Memoire sur la théorie des deblais et des remblais (Memoire sur la théorie des deblais et des remblais), in 1786-1788. prepared a textbook on practical mechanics and machine theory “Treatise on statics for maritime colleges” (Traité élémentaire de statique, á l ́usage des colléges de la marine). This course was reprinted eight times, the last one in 1846, and was repeatedly translated into other languages, including Russian (Elementary Foundations of Statics).
Monge welcomed the French Revolution, which proclaimed social justice and equality. He experienced firsthand how difficult it is for a member of the lower class to get a good education and gain a position in society. Unlike many fellow citizens who left the country, Monge continued his scientific and teaching activities, participated in meetings of the Academy of Sciences, and willingly and conscientiously carried out assignments new government. In May 1790, together with academicians Borda, d'Alembert, Condorcet, Coulomb, Lagrange, Laplace, he was appointed by the National Assembly to the commission to establish a new, uniform for the entire country, metric system of measures and weights, based on the decimal system, to replace the old measures, various in every province.
Monge organized 12 schools in the ports of France to train specialist hydrographers. In August 1792, taking into account his commitment to the ideals of the Revolution and his knowledge of maritime disciplines, the Legislative Assembly appointed him Minister of the Navy to the new government, the Provisional Executive Council.
The fleet entrusted to Monge was in dire condition: there were not enough officers and sailors, ammunition and food. France had already suffered several defeats at sea, and soon she had to enter into a war with England. Despite the meager state treasury, Monge managed to partially replenish the empty arsenals and begin building the necessary fortifications on the banks. During his six-month tenure as President of the Council, he had to make two important political decisions - he signed the verdict on the execution of Louis XVI and the declaration of war with England. However, he did not have the necessary administrative and military experience, he was burdened by ministerial work and resigned in April 1793, continuing to work in the name of the Revolution.
The Committee of Public Safety instructed Monge to organize the production of gunpowder, steel, the casting of cannons and the manufacture of guns. His talent as a scientist, versatile knowledge and amazing performance allowed him to successfully as soon as possible cope with all assigned tasks. To obtain the saltpeter necessary for the production of gunpowder, Monge found and popularly described methods for extracting it from the ground in barns and cellars. He organized new foundries and developed methods for smelting steel, changed the technology for making guns and organized their production of up to 1000 pieces per day in Paris alone. Without receiving any remuneration for his work, Monge often left for work early in the morning and returned late at night, eating only bread, since there was a shortage of food in the country, and he did not consider it possible to stand out among the starving workers. However, even this did not save him from periodic accusations of disloyalty to the authorities, so that one day he was forced to hide from persecution for two months. Since 1794, Monge no longer took a direct part in the affairs of public administration, and completely devoted himself to scientific and teaching activities.
In 1794, Monge published a manual for the production of cannons (Description de l'art de fabriquer les canons) and began organizing the Central School of Public Works, which was supposed to replace the Academies and Universities abolished by the decrees of the Convention in 1793. According to the plan, this was to be a new type of higher school with a three-year course to train engineers and scientists in a wide range of civilian and military specialties on a solid scientific basis. On September 1, 1795, the school was renamed the Ecole Polytechnique.
In January 1795, the so-called Higher Normal School was organized, intended for four-month training of professional personnel (mainly teachers). Together with Monge, classes were taught by Berthollet, Laplace, Lagrange and others. For students of the first intake of the School, Monge prepared and taught a course in descriptive geometry, a recording of which was published in the Proceedings of the Normal School (1795). At the same time, Monge created another of his main works - Application of analysis to geometry (L "application de l" analysis la gometrie, 1795), where, in addition to discoveries in differential geometry, a geometric interpretation of partial differential equations was given. This direction was continued in the works of such mathematicians as K. Gauss, J. Steiner and J. Plücker. In October 1795, the Convention formed an association of renewed academies, called the French Institute (later the National Institute of Science and Art). It was assumed that the Institute would become a scientific institution consisting of three classes (departments): physical and mathematical sciences, moral and political sciences, literature and fine arts. Monge was among the most active organizers and then teachers of these scientific institutions.
In May 1796, the Directory instructed Monge and Bertollet to take part in the commission for the selection of monuments of art and science in the regions of Italy conquered by the army of the Republic for indemnity. Monge fulfilled the order, delivering paintings by Raphael, Michelangelo, Titian, Veronese and other works of art to Paris, as well as scientific exhibits and instruments for the Polytechnic School. While in Italy, he met and became friends with the young General Bonaparte, whose devotion largely determined Monge's future life. Returning from Italy, on October 1, 1797, he made a speech before the Directory about the victories of the French army with threats against the English government, but, at the same time, with calls to preserve the nation that gave Newton to the world.
In February 1798, Monge was again sent to Italy as part of a commission to clarify the events taking place in Rome. On March 20, a Republic was proclaimed there and papal power was overthrown. Monge, however, did not stay in Rome for long - together with Berthollet, Fourier, Malus and other academicians, he participated in the Egyptian campaign of Bonaparte, who counted heavily on the help of scientists in building roads, canals, dams, drawing maps, organizing the production of gunpowder, guns and guns, as well as in the creation of new scientific institutions similar to French ones in the conquered territories. On August 29, 1798, in Cairo, members of this expedition and some military personnel, among whom Bonaparte himself belonged, established the Egyptian Institute of Sciences and Arts, modeled after the French one and electing Monge as its president for the first trimester, Bonaparte as vice-president, and Fourier as permanent secretary. .
Monge continued his scientific work, published in the scientific and literary collection “Egyptian Decades” (“Décade Égyptienne”) published by the Institute. In it, for the first time, his report was published with a simple explanation of the phenomenon of the mirage that frightened the soldiers in the desert (Memoire sur le phenomene doptique connu sous le nom de mirage). At times, Monge had to remember his short military past - in October 1798 he led the defense of the Institute against the rebellious Cairo population, and in 1799 he participated in Bonaparte’s unsuccessful campaign in Syria. Having received information about the difficult situation in France, on August 18, 1799, Bonaparte, accompanied by Monge and Berthollet, secretly left Cairo and after a difficult and dangerous two-month journey they reached Paris.
Having concentrated all the power in his hands, Bonaparte appointed Monge a senator for life; at the Polytechnic School he teaches courses in the application of algebra and analysis to geometry, draws up a charter and work plan for the school. In August 1803, Monge was appointed vice-president of the Senate, and in September - senator of Liege with instructions to organize the production of cannons there. Devotion to the new government and services to the Empire were rewarded - he received the highest degree of the Legion of Honor, in 1806 he was appointed president of the Senate for another one-year term, a year later he received the title of Count of Pelus and 100,000 francs to purchase an estate. However, his health soon began to fail, and his arm was temporarily lost. Monge ceases teaching at the École Polytechnique, but continues his scientific work and advises on proposed technical projects. So, in 1805, the emperor instructed him to study the possibility of building a canal from the Ourcq River to supply Paris with water.
Events of 1812-1814 ended with the defeat of France and the exile of Bonaparte. Monge remained committed to the Empire and was still on Bonaparte's side throughout the Hundred Days. After the restoration of Bourbon power, Monge was stripped of his titles, awards and pension, and expelled (though only for a year) from the Ecole Polytechnique. By government order in 1816, he and Carnot were excluded from the reformed new way Institute and were replaced by Cauchy and Breguet. As one of the “regicides,” Monge could expect more serious reprisals. From all these blows of fate, completed by the exile of his son-in-law Echasserio, as a former member of the Convention, Monge received several apoplectic attacks and soon died. He was buried in the Père Lachaise cemetery. Monge's wife survived him by 24 years.
The creation of “Descriptive Geometry”, the treatise of which was published only in 1799 under the title “Géométrie descriptive”, served as the beginning and basis of work that allowed the new Europe to master geometric knowledge Ancient Greece; work on the theory of surfaces, in addition to its immediate significance, led to the clarification of the important principle of continuity and to the disclosure of the meaning of that extensive uncertainty that arises when integrating equations with partial derivatives, arbitrary constants, and even more so with the appearance of arbitrary functions.
Monge's other, less significant contributions to science include the theory of polar planes as applied to second-order surfaces; discovery of circular sections of hyperboloids and hyperbolic paraboloid; the discovery of a twofold method of forming the surfaces of the same bodies using a straight line; creating the first idea of the lines of curvature of surfaces; the establishment of the beginnings of the theory of reciprocal polars, later developed by Poncelet, the proof of the theorem that the locus of the vertex of a trihedral angle with right plane angles described near a second-order surface is a ball, and, finally, the theory of constructing orthogonal projections of three-dimensional objects on a plane, called Monge diagram (Monge Project).
Numerous memoirs of Monge were published in the works of the Paris and Turin academies, published in the Journaux de l'Ecole Polytechnique et de l'Ecole Normale, in the Dictionnaire de Physique, Diderot and d'Alembert's "Methodical Encyclopedia", in the Annales de Chimie " and in the "Décade Egyptienne", published separately: "Dictionnaire de Physique" (1793-1822), compiled with the collaboration of Cassini, "Avis aux ouvriers en fer sur la fabrication de l'acier" (1794), compiled together with Berthollet, and others. In the book “Gaspard Monge. Collection of articles for the 200th anniversary of his birth” contains a bibliography of Monge’s works (72 titles) and a list of publications about his life and work (73 titles).
The name of Gaspard Monge is included in the list of the 72 greatest scientists of France, placed on the first floor Eiffel Tower(N 54).
In Gaspard Monge's hometown, Beaune, on the square named after him in 1849, a monument was erected in his honor.
Named after him:
Navy building.
A street in Paris (Rue Monge), running along the former buildings of the Ecole Polytechnique, as well as a square in the 5th Parisian arrondissement and the Place Monge metro station located on it.
Street in Dijon.
Primary school in Lille.
Educational institutions (lyceums of general and technological education or colleges) in the following cities: Beaune, Chambery, Charleville-Mézières, Saint-Joire, Savigny-sur-Orge, Nantes, Knutanger.
Research Institute of Electronics and Informatics Gaspard Monge - IGM (Institut d "Electronique et d" Informatique Gaspard-Monge) in Marne-la-Vallée, a suburb of Paris.
Monge G. Memoire sur la theory des deblais et des remblais. - Paris, 1781.
Monge G. Traité élémentaire de statique, á l ́usage des colléges de la marine. - Paris, 1788. - 227 p.
Monge G. Description de l'art de fabriquer les canons. - Paris, 1794.
Monge G. Géométrie descriptive. - Paris, 1799. - 132 p.
Monge G. Memoire sur le phenomene doptique connu sous le nom de mirage//Décade Egyptienne. - Caire, 1799. - V. 1. - R. 37-46.
Monge G. Initial foundations of statics or equilibrium of solid bodies for navigating schools. - St. Petersburg, 1803. - 151 p.
Monge G. The art of casting cannons. - St. Petersburg, 1804.
Monge G. Application de l’Algèbre à la Géométrie. - Paris, 1805.
Monge G. Application de l’Analyse à la Géomètrie. - Paris, 1807.
Monge G. Initial foundations of statics. - St. Petersburg, 1825. - 208 p.
Monge Gaspard. Application of analysis to geometry / Ed. M. Ya. Vygodsky. M.-L.: ONTI, 1936. - 699 p.
Monge Gaspard. Descriptive geometry / Ed. prof. D.I. Kargina. - M.: Publishing house. USSR Academy of Sciences, 1947. - 292 p.
Literature
Arago F. Biographies of famous astronomers, physicists and geometers. - St. Petersburg, 1859. - T. 1. - P. 499-589.
Launay Louis de. Monge fondateur de l'École polytechnique. - Paris, 1933. - 380 p.
Staroselskaya-Nikitina O. Essays on the history of science and technology during the French bourgeois revolution of 1789-1794. - M.-L., 1946. - 274 p.
Gaspard Monge. Collection of articles for the 200th anniversary of his birth / Rep. ed. V.I. Smirnov. - L.: Ed. USSR Academy of Sciences, 1947. - 85 p. - 5,000 copies.
Kargin D.I. Gaspard Monge and his “Descriptive Geometry” // Gaspard Monge. Descriptive geometry. - M.: Publishing house. USSR Academy of Sciences, 1947. - pp. 245-257.
Kargin D.I. Gaspard Monge is the creator of descriptive geometry. 1746-1818. To the 200th anniversary of his birth // Nature, - 1947. - No. 2. - P. 65-73.
Lukomskaya A.M. List of works and literature about the life and work of Gaspard Monge // Gaspard Monge. Descriptive geometry. - M.: Publishing house. USSR Academy of Sciences, 1947. - pp. 258-270.
Vavilov S.I. Science and technology during the French Revolution / Collected works. - M.: USSR Academy of Sciences, 1956. - T. 3. P. 176-190. - 3,000 copies.
Bogolyubov A.N. Gaspard Monge / Ed. acad. I. I. Artobolevsky. - M.: Nauka, 1978. - 184 p. - 30,000 copies.
Demyanov V.P. Geometry and Marseillaise. About the French mathematician and revolutionary G. Monge / Rep. ed. V. I. Smirnov. - M.: Knowledge, 1986. - 252 p.
Borodin A.I., Bugai A.S. Outstanding mathematicians. – Kyiv: Radyanska School, 1987.
He received his primary education at the city school of Bon. Teaching at this school focused almost exclusively on ancient languages; he had to study the physical and mathematical sciences, to which Monge had a special attraction, without outside help.
At the age of 16, Monge drew up a remarkably accurate plan of his hometown of Beaune. The methods and instruments necessary for its compilation for measuring angles and drawing lines were invented by the compiler himself.
Having entered the additional department for training engineer conductors at the Mezières School of Military Engineers, Monge soon advanced from among his comrades. The direct and easy solution he gave to the problem of defiling the fortifications gave reason to the school authorities to appoint him as a mathematics tutor.
From the same time, Monge's scientific activity began, the first result of which was the creation of “Descriptive Geometry” - this most important of his scientific works. The reluctance to give foreigners the opportunity to enjoy the fruits of the inventions of the French genius prompted the head of the Mézières school to prohibit Monge from making his discovery public. Monge's other major work was research on the theory of surfaces, set out in a series of memoirs that he presented at the Paris and Turin Academies.
In 1768, Monge was appointed professor of mathematics; in addition, in 1771 (after the death of Abbot Nollet), the department of physics was also transferred to Monge. It should be noted that he carried out the decomposition of water in 1783, although this work was done after the homogeneous work of Henry Cavendish, but before receiving information about this latter and therefore constitutes the inalienable property of Monge.
In 1780, Monge was appointed teacher of hydraulics at a school established in the Louvre, with the obligation to live in Mézières and in Paris for six months each. In the same year, Monge was elected to the Academy. Monge had to leave Mézières completely in 1783.
Elected to academician, Monge, in addition to research on higher analysis, set out in a number of excellent memoirs in the publications of the Academy, was engaged, together with Berthollet and Vandermont, in the study of various states of iron, carried out experiments on capillarity, made observations on optical phenomena, worked, albeit unsuccessfully, on By constructing a theory of the main meteorological phenomena, he finally significantly improved practical mechanics. In this latter he showed that all complex machines, no matter how complex they may be, are reduced to a very small number of constituent elements; gave tables explaining the replacement of one movement by another, caused by the connection between parts of the machine; showed more advantageous ways to use the forces of water, air and steam during work. The composition of his famous “Traité de statique” (P., 1788) dates back to the same time. Great french revolution found an ardent supporter in Monge. During this era, he was first appointed as a member of the commission for establishing a new system of weights and measures, and in 1792 he took the post of Minister of Navy, which he remained until April 10, 1793.
Despite the stinginess of the state treasury, Monge's energy managed to partially replenish the depleted arsenals and begin the construction of the necessary buildings on the banks. Even more important was that Monge indicated and popularly expounded methods of extracting the saltpeter necessary for making gunpowder from the ground in barns, cellars and cemeteries, and that he set up many cannon foundries, edged weapons factories and for making guns. From his instructions to workers, his famous work on artillery technology “L’Art de fabriquer les canons” (1794) was subsequently compiled.
Not receiving any remuneration from the bankrupt state for all his work, he reached such poverty that he had to live on bread alone, and the accusation based on the denunciation of the gatekeeper forced him to flee. A quick change of direction very soon, however, allowed him to return to Paris. From that time on, he no longer took a direct part in public administration affairs and devoted himself entirely to scientific and teaching activities.
At the Normal School, established after 9 Thermidor, he for the first time introduced a course of descriptive geometry into the curriculum, the notes of which, compiled by students, quickly spread.
The turning point of Monge's pedagogical activity was the work on organizing teaching and implementing it in practice at the famous Polytechnic School, founded at the end of 1794. Upon the closure of the academies in 1793 and the establishment a year later of the National Institute that replaced them, in the development of the charter of which Monge took a significant part, he was among the first 48 members of the new scientific institution who were appointed by the government.
Sent to Italy in 1796 to receive paintings and statues included in the war indemnity, he met and became friends with Napoleon Bonaparte. In 1798, the government entrusted him, together with two other persons, with the difficult task of establishing, on the basis of the French constitution of the 3rd year of the Roman Republic, which was supposed to replace the temporal power of the popes, destroyed by French troops. However, Monge and his comrades could not triumph over the difficulties of the task entrusted to them.
Napoleon, preparing for a campaign against Egypt, invited him and Bertholla to assemble a scientific expedition, which was supposed to accompany the army setting out on the campaign and had the goal of studying the conquered countries and spreading enlightenment in them. A significant part of this expedition consisted of people belonging to the Polytechnic School. On August 29, 1798, in Cairo, from the members of this expedition and some military personnel, among whom Napoleon himself belonged, the Egyptian Institute was formed, modeled after the French one and electing Monge as its president.
The works of the members of the new institute were published in the Décade Egyptienne, published by it, which was published at ten-day intervals. In it, Monge's memoir about the mirage appeared for the first time. During the Empire, he was appointed senator and received the title of Count of Peluse and the highest degree of the Legion of Honor. At his petitions, more or less significant sums were squeezed out of the emperor’s personal funds in the form of benefits to various individuals, and once the emperor sent him the sum of 100,000 francs. Little by little, Monge's beliefs changed, turning from republican to imperialist.
After the fall of the Empire and the restoration of the Bourbons, Monge lost everything he had received under the Empire and even the academic chair he had occupied before the revolution. By order of the government in 1816, he and Carnot were excluded from the institute, which had been transformed in a new way, and were replaced by Cauchy and Breguet. From all these disasters, completed by the exile of his son-in-law Echasserio, as a former member of the convention, Monge became mentally ill and soon died.
Scientific activities
The creation of “Descriptive Geometry”, the treatise of which appeared only in 1799 under the title “Géométrie descriptive”, served as the beginning and foundation of works that allowed the new Europe to master the geometric direction of Ancient Greece; work on the theory of surfaces, in addition to its immediate significance, led to the clarification of the important principle of continuity and to the discovery of the meaning of that wide uncertainty that is generated when integrating partial differential equations, arbitrary constants, and even more so the appearance of arbitrary functions.
The principle of continuity, as it appears in Monge, can be stated as follows. Any property of a figure that expresses relations of position and is justified in an innumerable number of continuously interconnected cases can be extended to all figures of the same kind, even if it can only be proven under the assumption that constructions that are feasible only within certain limits can actually be produced. This property exists even in cases where, due to the complete disappearance of some intermediate quantities necessary for proof, the proposed constructions cannot be carried out in reality.
Among the smaller contributions to science, one should point out Monge's theory of polar planes to second-order surfaces; on the discovery of circular sections of hyperboloids and a hyperbolic paraboloid; to the discovery of a twofold method of forming the surfaces of the same bodies using a straight line; to create the first idea about the lines of curvature of surfaces; to establish the first foundations of the theory of reciprocal polars, later developed by Poncelet, and, finally, to prove the theorem that the locus of the vertex of a trihedral angle with right plane angles, described about a second-order surface, is a ball.
The Monge method uses the method of rectangular projections or the method of orthogonal projection of a geometric image (point, line, plane, surface) onto two mutually perpendicular and mutually connected projection planes with rays perpendicular to these projection planes, this is the essence of the Monge method:
Rice. 18 Monge method: H - horizontal projection plane; V - frontal plane of projection; W - profile projection plane.
The lines of intersection of the projection planes are called the projection axis or coordinate axis:
A` - projection of point A onto plane H (horizontal projection of point A);
A" - projection of point A onto plane V (frontal projection of point A);
A" is the projection of point A onto the plane W (profile projection of point A).
Projection methods using one-picture drawings allow solving a direct problem (i.e., using a given original to construct its projection). However, the inverse problem (i.e., reproducing the original using a projection) is definitely impossible to solve. This problem allows for countless solutions, because Each point Ab of the projection plane b can be considered a projection of any point of the projecting ray SAb passing through Ab.
Thus, the considered one-picture drawings do not have the property of reversibility.
To obtain reversible one-picture drawings, they are supplemented with the necessary data.
There are various ways such an addition. For example, drawings with numerical marks.
The method is that, along with the projection of point A1, the height of the point is specified, i.e. its distance from the projection plane. The scale is also set.
This method is used in construction, architecture, geodesy, etc. However, it is not universal for creating drawings of complex spatial forms.
Rice. 19
In 1798, the French geometer-engineer Gaspard Monge, summarizing the theoretical knowledge and experience accumulated by that time, for the first time gave a scientific justification for the general method of constructing images, proposing to consider a flat drawing consisting of two projections as the result of combining with a plane two mutually connected mutually perpendicular planes projections.
This is where the principle of constructing drawings originates, called the Monge Method, by which it was said above that the projection of a point does not determine the position of the point in space, and in order to establish this position, having a projection of the point, additional conditions are required. For example, a rectangular projection of a point on a horizontal projection plane is given and the distance of this point from the plane is indicated by a numerical mark; the projection plane is taken to be the “zero level plane”, and the numerical mark is considered positive if a point in space is above the zero level plane, and negative if the point is below this plane.
The method of projections with numerical marks ") is based on this.
In the following presentation, the position of points in space will be determined by their rectangular projections on two or more projection planes.
In Fig. 20 shows two mutually perpendicular planes. Let us take them as projection planes. One of them, designated by the letter k1, is located horizontally; the other, designated by the letter R2, is vertical. This plane is called the frontal plane of projections, pl. I is called the horizontal plane of projections. The projection planes Kj and R2 form the system Kj, R2.
The line of intersection of the projection planes is called the projection axis. The projection axis divides each of the planes I! and i2 on the half-plane. For this axis we will use the notation l or the notation in the form of a fraction r2/rj. Of the four dihedral angles formed by projection planes, the first is considered to be the one whose faces are shown in Fig. 9 have the designations I! and I2.
In Fig. Figure 10 shows the construction of projections of a certain point A in the system R15 R2. Drawing perpendiculars from A to itj and r2, we obtain projections of point A: horizontal, designated A", and frontal, designated A".
The projecting lines, respectively perpendicular to l and r2, define a plane perpendicular to the planes and to the axis of the projections. This plane, at the intersection with I and I2, forms two mutually perpendicular straight lines A"AX and A"AX, intersecting at point Ax on the projection axis. Consequently, the projections of a certain point are located on straight lines, perpendicular to the axis of projections and intersecting this axis at the same point.
The method of projections with numerical marks is not included in the program of the course presented. Those interested are referred to books on descriptive geometry for construction and architectural specialties.
If the projections A" and A" of a certain point A are given (Fig. 21), then, by drawing perpendiculars through A" to the area TCj and through A" to the area. l2 - we get a certain point at the intersection of these perpendiculars. So, two projections of a point completely determine its position in space relative to a given system of projection planes.
Turning pl. Kj around the axis of projections at an angle of 90° (as shown in Fig. 22), we obtain one plane - the plane of the drawing; projections A" and A" will be located on the same perpendicular to the projection axis - on the communication line. As a result of the indicated combination of planes I and L2, a drawing is obtained known as diagram ") (Monge diagram). This is a drawing in system 2 (or in the system of two rectangular projections).
Having switched to the diagram, we have lost the spatial picture of the location of the projection planes and points. But, as we will see later, the diagram ensures the accuracy and measurability of images with significant simplicity of construction. To imagine a spatial picture from it requires the work of imagination.
Since, in the presence of a projection axis, the position of point A relative to the projection planes Tij and n2 is established, the segment A"AX expresses the distance of point A from the projection plane l2, and the segment A"AX - the distance of point A from the projection plane n^ It can also be determined the distance of point A from the projection axis. It is expressed by the hypotenuse of a triangle constructed along the legs A"AX and A"AX (Fig. 23): plotting on the diagram the segment AA, equal to A"AX, perpendicular to A"AX, we obtain the hypotenuse AAX, expressing the required distance.
Attention should be paid to the need to draw a line of connection between the projections of a point: only with this line interconnecting the projections is it possible to establish the position of the point they define.
Let us agree in what follows to call Monge's diagrams, as well as projection drawings, which are based on Monge's method (see § 3), in one word - drawing and to understand this only in the indicated sense. In other cases where the word “drawing” is used, it will be accompanied by an appropriate definition (perspective drawing, axonometric drawing, etc.).
Yorige (French) - drawing, project. Sometimes instead of “epure” they write and pronounce “epure”, which corresponds not to the pronunciation of the word eurige, but to the feminine gender of this word in French.
During the Directory, he became close to Napoleon, took part in his campaign in Egypt and the founding of the Egyptian Institute in Cairo (1798); was elevated to count.
Monge Gaspard (10.5.1746-28.7.1818) - French geometer and public figure, Member of the Paris Academy of Sciences (1780). Creator of descriptive geometry, one of the organizers of the Ecole Polytechnique in Paris and its long-term director. Born in Bon Cote d'0r. Graduated from the School of Military Engineers in Mézières. From 1768 he was a professor of mathematics, and from 1771 also a professor of physics at this school. From 1780 he taught hydraulics at the Louvre School (Paris). He was engaged in mathematical analysis, chemistry, meteorology, practical mechanics. During the French bourgeois revolution, he worked on the commission to establish a new system of weights and measures, then he was the Minister of Naval Affairs and the organizer of national defense. During the Directory, he became close to Napoleon, took part in his campaign in Egypt and his founding in Cairo. Egyptian Institute (1798); was elevated to rank. Received worldwide recognition by creating (in the 70s) modern methods projection drawing and its basis - descriptive geometry. Monge's main work on these issues is “Descriptive Geometry”; published in 1799 He also made important discoveries in differential geometry. Monge's first works on surface equations were published in 1770 and 1773. In 1795 and 1801, Monge's works on finite and differential equations of various surfaces were published. In 1804 the book “Application of Analysis in Geometry” was published. In it, Monge considered cylindrical and conical surfaces formed by the movement of a horizontal line passing through a fixed vertical line, surfaces of “channels,” surfaces in which the lines of greatest slope everywhere form a constant angle with the horizontal plane; transfer surfaces, etc. As an appendix to the book, Monge gave his theory of integration of 1st order partial differential equations and his solution to the problem of string vibration. For each type of surface, I first derived a differential equation, then a finite equation. The first one denoted the letters p and q for the partial derivatives of z with respect to x and y, and the letters r, s and t for the 2nd order derivatives.
Gaspard Monge
After successfully graduating from school, its management recommended Gaspard Monge for further studies at the College of the Holy Trinity in Lyon. He was accepted there and soon became a physics teacher there (at the age of 16), occupying this position until 1764. To receive special education at the age of 18, Monge entered the Military Engineering School in Mézières, but he was accepted not into the officer class, since he did not have a noble origin, but into the department that trained craftsmen and producers of work. There, students mastered the basics of algebra, geometry, drawing, and also made all kinds of models of buildings and fortifications. At the Mézières school, Monge quickly became one of the first students. Having a good mathematical background, he could easily and originally solve the most complex problems.
After graduation, Monge was retained at the Mézières School as a teacher: first as an assistant in the mathematics department under Professor Charles Bossu(1730–1814), and then as an assistant in the department of physics with professor ZhanaAntoine Nollet(1700–1770). In 1770, after Nollet's death and Bossu's transfer to another job, Monge became the head of both of these departments at once. In addition to physics and mathematics, he also taught a course in chemistry, as well as the theory of perspective and shadows. It was during the Mézières period of his life that Monge began to develop the ideas of descriptive geometry and found numerous applications for them, in particular, for calculating the relief of fortifications.
The school students of that period were very fond of their young professor. He was not handsome, he spoke quickly and not always clearly, but he was very kind and never regretted his personal time for anyone. Often during classes he approached some gape listener with the words: “My friend, I will repeat from the moment from which you stopped understanding me.”
Professor Monge knew how to convey his passion for science to others; there were no slackers or laggards among his students. He didn't care at all about his career.
In 1777 he married, and three years later he became a teacher of hydraulics at the Louvre School in Paris. During these years, he was actively involved in issues of mathematical analysis, chemistry, meteorology, and practical mechanics. For achievements in these areas, the Paris Academy of Sciences in 1780 elected 34-year-old Monge as a full member.
Participation in the meetings of the Academy required the young scientist to remain permanently in Paris, so he was allowed to stay there for six months a year. During Monge's absence, his younger brother Louis lectured at the Mézières school. Monge(1748–1827), also a professor of mathematics.
When the French Revolution began, Monge became its ardent supporter. These years were filled with extremely active social and practical activities for him. At first he worked on the commission to establish a new system of weights and measures, then he became one of the organizers of national defense and the French military industry. This happened under the following circumstances. On August 10, 1792, after the deposition of King Louis XVI, Monge was elected to the provisional government, where he received the portfolio of Minister of the Navy. After the creation of the National Convention, which finally abolished the royal power, in September of the same year he retained his post as Minister of the Republic, responsible for the navy. This appointment of a scientist far from the problems of the navy can be explained as follows: after the revolution, all the specialists and aristocrats in the Admiralty fled, and what was needed was simply a person devoted to the nation, an authoritative and honest person.
Monge always sought to apply his adored mathematics to any field in which fate threw him. He was an encyclopedist, like any scientist of that time, and, having become an examiner of midshipmen, he did not do any leniency to future naval officers. However, the fleet at that time was not the government’s highest priority. France needed ammunition much more. Under the king, the brilliant Lavoisier dealt with this issue, but the revolutionaries executed him, thereby exposing the most important front, and without gunpowder, their guns and cannons became like sticks that were useless in a real battle.
And so Monge took up the production of gunpowder. Together with Claude-Louis Berthollet, he figured out how and where to mine saltpeter in France. The result was amazing: if before 1789 France consumed no more than a million pounds of saltpeter per year, through the efforts of Monge and his employees, 12 million pounds of it were produced in ten months!
But getting the components is not a solution to the problem. Powder mills, the number of which was very limited, did not have time to process all this. Then Monge suggested putting copper balls in ordinary barrels. These “mills in miniature” could be placed in any yard, and through his efforts France turned into a huge gunpowder factory. Of course, without general popular enthusiasm this enormous work could not have been completed, but even without Monge’s brilliant head nothing would have happened.
Guns at that time were made of cast iron and bronze. Cast iron cannons were easier to cast, but they were much heavier. As a rule, they were used in the navy or in fortresses. Monge increased the number of iron cannon factories from four to thirty. Instead of 900 guns per year, 30 thousand were cast. Through the efforts of Monge, the number of copper cannon factories increased from two to fifteen. They began to produce seven thousand guns. For this purpose, church bells were used as a source of copper. True, the composition of bell copper was not suitable for the production of cannons, but Monge attracted chemists and found new ways to separate copper from tin. Previously, clay molds of tools were needed for production. Monge suggested casting cannons in sand. The first cannon obtained in this way was tested on the Champ de Mars, and all of Paris applauded the successful results. During the day, Monge did not leave his workshops; at night he wrote the manual “On the Art of Cannon.” Everything that did not relate specifically to issues of defense and armament of the army seemed unimportant.
Monge bravely endured hunger and cold. In general, he ate mostly bread, allowing people to make fun of him. For example, the following joke is known: “Monge began to live in luxury; Now he eats radishes!”
One day Madame Monge found out that a denunciation had been written against her husband and Berthollet. She ran to Bertolla, but the great chemist only muttered thoughtfully: “It is very possible that we will be convicted and taken to the guillotine, but this will not happen earlier than in eight days.”
Why in eight days and what would happen in eight days, Madame Monge did not understand, but it was obvious that the scientist was worried about something completely different at that time. Monge himself, in response to his wife’s crying, said: “The most important thing is that my foundries work wonderfully.”
In 1794, together with Berthollet, Monge became the founder and first professor of the Ecole Polytechnique, one of the best higher educational institutions in France (he lectured here for more than ten years). This contribution of Monge to science is difficult to overestimate: as a result of his fruitful organizational and teaching activities, the Polytechnic School quickly became a center for the general scientific training of highly qualified specialists; all the major engineers and mathematicians of France in the 19th century either graduated from this school or were its teachers.
Back to scientific activity, Monge devoted himself to descriptive geometry. This is now the name of an engineering discipline consisting of a set of algorithms for studying the properties of spatial geometric objects and based on the representation of these objects using two independent projections. Simply put, it is a science that studies spatial figures by projecting them onto planes.
However, Monge's main works on this section were published only in 1799, since for many years the French government kept this discipline secret, qualifying it as a military secret. It is known that Monge created his significant work “Application of Analysis to Geometry” in 1795. This work was a textbook analytical geometry, in which special emphasis was placed on differential equations.
Within the walls of the Polytechnic School, Monge managed to ensure that descriptive geometry and geometry in general became the central, defining subjects of the curriculum. He was able to present the most complex issues with amazing clarity and clarity.
During the years of the Directory, Monge became close to Napoleon and it was thanks to him that he achieved great ranks and fame. Napoleon, as you know, never promoted idlers to high positions. And for Monge, even then he was an example of a statesman and commander. Napoleon and Monge became especially close in 1796 in Italy, where the latter was sent by the Directory with instructions to select the most outstanding works of science and art for the museums and repositories of Paris.
When Napoleon signed peace with the Austrians in 1797, Monge was sent from Milan to Paris to transmit this document to the Directory with a view to ratifying it. At the same time, Napoleon wrote about Monge like this:
“Citizen Monge is famous for his knowledge and his patriotism. By his behavior in Italy, he achieved that the French were respected. He deserves my friendship."
In 1797, Monge facilitated Napoleon's entry into the Institut de France (National Institute of Sciences and Arts), created by the Convention to replace the "bourgeois" Academy of Sciences, abolished in 1793.
When Monge returned from Italy to Paris in October 1797, he was already aware of Napoleon’s desire to “join science” and immediately began to “cook public opinion" Another academician devoted to Napoleon, Claude-Louis Berthollet, helped him in this. A convenient opportunity turned up very opportunely: a vacancy appeared in the ranks of academicians. But two more people claimed it, and they were much more famous in science than General Bonaparte. The first was Jacques Dillon(1760–1807) - engineer who built the first iron bridge in France, the second was an 84-year-old engineer MarkRene Montalembert(1713–1799), author of an eleven-volume work on fortification.
The secret vote took place on December 25, 1797: 305 votes were cast for Napoleon, 166 votes for Dillon, and 123 votes for Montalembert. As we see, the faithful Monge and Berthollet did not disappoint: they chose Napoleon, who had no scientific works or other merits except victories on the battlefields. After this, it was written in the newspapers that General Bonaparte was elected as an academician, “ amazing person, a philosopher who stood at the head of the army."
When Napoleon began planning his Egyptian expedition, he, without a moment’s hesitation, invited Monge and Berthollet to his “team”. They happily agreed.
About 150 scientists and specialists representing more than fifteen different professions were invited to participate in the expedition.
Historian Jean Tulard provides the following data:
“The trip was attended by 21 mathematicians, 3 astronomers, 17 civil engineers, 13 naturalists and mining engineers, selected by Monge and Berthollet, the same number of geographers, 3 chemists, gunpowder and saltpeter specialists, 4 architects, 8 draftsmen, 10 mechanics, 1 sculptor, 15 translators, 10 writers, 22 typesetters."
The list of names of scientists who traveled with Napoleon to Egypt is impressive. It was led by Monge and Berthollet. Under their command were mathematicians JeanBatisteJoseph Fourier(1768–1830) and Louis Costaz(1767–1842), chemists Hippolyte ColletDecotille(1773–1815) and JacquesPierre Champy(1744–1816), naturalist EtienneGeoffroy SaintIler(1772–1844), astronomers NicolasAntoine Nouet(1740–1811) and PierreJoseph de Beauchamp(1752–1801), geologist Deoda de Dolomieu(1750–1801), artists Dominique VivantDenon (1747–1825), HenriJoseph Redouté(1766–1852) and Andre Dutertre (1753–1842).
And many luminaries of French science, by the way, refused. The number of “refuseniks” included, for example, an engineer and mathematician Gaspard de Prony(1755–1839), chemist AntoineFrancois Fourcroix(1755–1809), naturalists GeorgesLeopold Cuvier(1769–1832) and Frederic Cuvier (1773–1838).
Of course, everyone had their own reasons for this. “My calculation,” Georges-Leopold Cuvier explained his refusal, “is this: I am now in the center of science, among the most remarkable collections, and I am confident that here in Paris I will make much more important discoveries than by participating in even the most fruitful journey.”
Already in Cairo, Monge became one of the founders of the Institute of Egypt.
The Egyptian Institute was a very important research institution, consisting of four departments: mathematics, physics, political economy, literature and the arts. Napoleon himself became the vice-president of the Institute, and Monge became the president. The opening of this “academy” was very solemn, and at the same time Napoleon declared that “the triumph over ignorance is the greatest of triumphs, and the successes of his weapons are the successes of enlightenment.”
In Egypt, Monge effectively became Napoleon's right hand. They spent a lot of time in scientific discussions, traveling together to Suez to see traces of the ancient canal that once connected the Nile with the Red Sea.
