What is the monge method. Meaning of monge, gaspar in the collier dictionary

If information about the distance of a point relative to the projection plane is given not with the help of a numerical mark, but with the help of the second projection of the point, built on the second projection plane, then the drawing is called two-picture or complex. The basic principles for constructing such drawings are set forth by G. Monge.

The method set forth by Monge - the method of orthogonal projection, and two projections are taken on two mutually perpendicular projection planes - providing expressiveness, accuracy and measurability of images of objects on a plane, has been and remains the main method for drawing up technical drawings.

The model of three projection planes is shown in the figure. The third plane, perpendicular to both P1 and P2, is denoted by the letter P3 and is called the profile plane. The projections of points on this plane are denoted by capital letters or numbers with index 3. The projection planes, intersecting in pairs, define three axes 0x, 0y and 0z, which can be considered as a system of Cartesian coordinates in space with the origin at point 0. Three projection planes divide the space into eight trihedral angles - octants. As before, we will assume that the viewer viewing the object is in the first octant. To obtain a diagram, the points in the system of three projection planes of the P1 and P3 planes are rotated until they coincide with the P2 plane. When designating axes on a diagram, negative semiaxes are usually not indicated. If only the image of the object itself is significant, and not its position relative to the projection planes, then the axes on the diagram are not shown. Coordinates are numbers that correspond to a point to determine its position in space or on a surface. In three-dimensional space, the position of a point is set using rectangular Cartesian coordinates x, y, and z (abscissa, ordinate, and applicate).

Lecture 7, SRSP-7

2. The location of the straight line relative to the projection planes.

3. Mutual arrangement of a point and a line, two lines.

Projection straight

To determine the position of a straight line in space, there are the following methods: 1. Two points (A and B). Consider two points in space A and B (Fig.). You can draw a straight line through these points. learn the segment. In order to find the projections of this segment on the projection plane, it is necessary to find the projections of points A and B and connect them with a straight line. Each of the segment projections on the projection plane is smaller than the segment itself:<; <; <.

2. Two planes (a; b). This method of setting is determined by the fact that two non-parallel planes intersect in space in a straight line (this method is discussed in detail in the course of elementary geometry).

3. Point and angles of inclination to the projection planes. Knowing the coordinates of a point belonging to the line and its angles of inclination to the projection planes, you can find the position of the line in space.

AT Depending on the position of the straight line in relation to the projection planes, it can occupy both general and particular positions. 1. A straight line that is not parallel to any projection plane is called a straight line in general position (Fig.).

2. Straight lines parallel to the projection planes occupy a particular position in space and are called level lines. Depending on which projection plane the given line is parallel to, there are:

2.1. Direct projections parallel to the horizontal plane are called horizontal or contour lines (Fig.).

2.2. Direct projections parallel to the frontal plane are called frontal or frontals (Fig.).

2.3. Direct projections parallel to the profile plane are called profile projections (Fig.).

3. Straight lines perpendicular to the projection planes are called projecting. A line perpendicular to one projection plane is parallel to the other two. Depending on which projection plane the investigated line is perpendicular to, there are:

3.1. Frontally projecting straight line - AB (Fig.).

3.2. Profile projecting straight line - AB (Fig.).

Gaspar Monge, graph de Peluz(French Gaspard Monge, comte de Pluse; 1746, Beaune, Burgundy, France - July 28, 1818, Paris) - French mathematician, geometer, statesman, Minister of Marine of France.

Biography

From student to academic

Gaspard Monge was born on May 10, 1746 in the small town of Beaune in eastern France (within the modern department of Côte d'Or) into the family of a local merchant. He was the eldest of five children, to whom his father, despite the low origin and relative poverty of the family, tried to provide the best education available at that time for people from the humble class. His second son, Louis, became a professor of mathematics and astronomy, the youngest - Jean - also a professor of mathematics, hydrography and navigation. Gaspard received his initial education at the city school of the Oratorians. After graduating in 1762 as the best student, he entered the college of Lyon, also owned by the Oratorians. Soon Gaspard was entrusted with teaching physics there.

In the summer of 1764, Monge drew up a plan of his native city of Beaune, remarkably accurate. The necessary methods and instruments for measuring angles and drawing lines were invented by the compiler himself. While studying in Lyon, he received an offer to join the order and remain a college teacher, however, instead, having shown great abilities in mathematics, drafting and drawing, he managed to enter the Mézieres School of Military Engineers, but (due to origin) only as an auxiliary non-commissioned officer officer department and without a paycheck. Nevertheless, success in the exact sciences and an original solution to one of the important problems of fortification (the placement of fortifications depending on the location of enemy artillery) allowed him in 1769 to become an assistant (teaching assistant) in mathematics, and then in physics, and already with a decent salary at 1800 livres a year.

In 1770, at the age of 24, Monge held the position of professor at the same time in two departments - mathematics and physics, and, in addition, conducts classes in cutting stones. 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, the creator of which he is rightfully considered. Considering the possibility of using the methods of descriptive geometry for military purposes in the construction of fortifications, the leadership of the Mézières school did not allow open publication until 1799 (the lectures were recorded verbatim in 1795).

In 1777, Monge married the young widow of the foundry owner Marie-Catherine Huart, after her first husband, Orboni (Marie-Catherine Huart, 1747–1846). The marriage was happy and lasted until the end of Monge's life, they had two daughters (the third died in infancy). Once the owner of the workshop, Monge mastered the foundry business, was fond of metallurgy, and was seriously engaged in physics and chemistry.

Monge taught at the Mezieres school for 20 years. They taught geometry, physics, fortification, construction, with an emphasis on practical exercises. This school became the prototype of the famous Polytechnic School in the future. In addition to the basics of descriptive geometry, Monge developed other mathematical methods, including the theory of sweeps, the calculus of variations, and others. Several reports, with great success, made by him at the meetings of the Paris Academy of Sciences, and the recommendations of Academicians d'Alembert, Condorcet and Bossu ensured Monge in 1772 the election of twenty "associs" members of the Academy ("attached", that is, corresponding members of the Academy), and in 1780 he was already elected an academician. Monge moves to Paris, retaining his position at the Mézières school. In addition, he teaches hydrodynamics and hydrography at the Paris Maritime School, and subsequently holds the position of examiner of maritime schools. However, the alternation of work and residence for six months in Paris and Mezieres eventually became very tiring for him and did not suit the leadership of the Mezieres school. In 1783, Monge stopped teaching at school and in 1784 finally moved to Paris.

Place of Birth: Beaune, Burgundy, France

Activities: mathematics, mechanics, technology

Gaspard Monge Count de Peluse (Gaspard Monge, comte de Péluse, 1746, Beaune, Burgundy, France - July 28, 1818, Paris) - French mathematician, geometer, engineer, statesman. Ancestor of descriptive geometry. Known for his research in the field of physics, chemistry, optics, metrology and practical mechanics.
Gaspard Monge was born in the small town of Beaune in eastern France in the family of a local merchant. His parents were Jacques Monge and Jeanne Rousseau. Gaspard was the eldest of five children, whom the father, despite the low origin and relative poverty of the family, tried to provide the best education that a person from the humble 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 Oratory order. After graduating in 1762 as the best student, he entered the college of Lyon, also owned by the Oratorians. Soon Gaspard was entrusted with the teaching of physics there.
Already at the age of 14, the boy invented a fire pump, which had an original design and thoughtfulness of all details. In the summer of 1764, Monge drew up a plan of his native city of Beaune, remarkably accurate. To draw up this plan, the young self-taught geometer used measuring and drawing instruments of his own manufacture and invention. The plan was so successful that one abbot used it for his little history essay. 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 a college teacher, however, instead, having shown great abilities in mathematics, drafting and drawing, he managed to enter the Mézières School of Military Engineers, but (due to his origin) only for an auxiliary non-commissioned officer department and without a paycheck. Nevertheless, success in the exact sciences and an original solution to one of the important problems of fortification (the placement of fortifications depending on the location of enemy artillery) allowed him to become an assistant teacher of mathematics in 1769, and then of physics, and already with a decent salary of 1800 livres. in year.
In 1769, at the age of 23, Monge held the position of professor of mathematics, and in 1770 - professor of physics, at the School of Military Engineers, and, in addition, conducts 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 on two mutually perpendicular planes (horizontal and vertical) and the original way of its representation on a plane (diagram method). At the Military Engineering School, where Monge taught, a new department of descriptive geometry was organized. Monge was made head of this department.
Given the possibility of using the methods of descriptive geometry for military purposes in the construction of fortresses and all other military installations, 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 the owner of the foundry, Maria Catherine Huar (Orboni). The marriage was happy and lasted until the end of Monge's life. Once the owner of the workshop, he mastered foundry, became interested in metallurgy, seriously engaged in physics and chemistry.
Monge taught at the Mezieres school for 20 years. They taught geometry, physics, fortification, construction, with an emphasis on practical exercises. This school became the prototype of the famous Polytechnic School in the future. In addition to the basics of descriptive geometry, Monge developed other mathematical methods, including the theory of sweeps, the calculus of variations, and others. Several reports, with great success, made by him at the meetings of the Paris Academy of Sciences, and the recommendations of Academicians d'Alembert, Condorcet and Bossu ensured Monge's election in 1772 to the twenty corresponding members of the French Academy of Sciences, and in 1780 he was already elected an academician. Monge moved to Paris, retaining his position at the Mézières school. In addition, he began to teach hydrodynamics and hydrography at the Paris Maritime School, and later took the position of examiner of maritime schools. However, work and living for six months alternately in Paris and Mézières eventually 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 he finally moved to Paris.
Elected to the academician, Monge, in addition to research on mathematical analysis, published in the "Memoirs" of the Academy, studied together with Berthollet and Vandermonde the study of various states of iron, made experiments on capillarity, made observations on optical phenomena, worked on building a theory of the main meteorological phenomena. Independently of Lavoisier and Cavendish, he discovered that water is a combination of hydrogen and oxygen. In 1781, Monge published Memoire sur la theorie des deblais et des remblais, in 1786-1788. prepared a textbook on practical mechanics and machine theory "Treatise on statics for marine colleges" (Traité élémentaire de statique, á l ́usage des colléges de la marine). This course was reprinted eight times, the last - in 1846, and was repeatedly translated into other languages, including Russian (Basic foundations of statics).
Monge welcomed the French Revolution, which proclaimed social justice and equality. He experienced first hand how hard it is for a representative of the lower class to get a good education and take 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, willingly and conscientiously carried out the instructions of the new government. In May 1790, together with Academicians Borda, d'Alembert, Condorcet, Coulomb, Lagrange, Laplace, he was appointed by the National Assembly to a commission to establish a new, common for the whole country, metric system of measures and weights based on the decimal system, instead of the old measures, various in every province.
Monge organized 12 schools in the ports of France to train hydrographers. In August 1792, taking into account his commitment to the ideals of the Revolution and knowledge of maritime disciplines, the Legislative Assembly appoints him Minister of Marine in the new government - the Provisional Executive Council.
The fleet entrusted to Monge was in a difficult condition: there were not enough officers and sailors, ammunition and food. France had already suffered several defeats at sea, and soon she was to go to war with England. Despite the scarcity of the state treasury, Monge managed to partly replenish the empty arsenals and begin building the necessary fortifications on the banks. During the semi-annual performance of the duties of President of the Council, he had to make two major political decisions - he put his signature under 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 already in April 1793 resigned, 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 capacity for work allowed him to successfully cope with all the tasks in the shortest possible time. To obtain the saltpeter necessary for the production of gunpowder, Monge found and popularly outlined 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 a day in Paris alone. Not receiving any remuneration for his work, Monge often went to work early in the morning and returned late at night, eating only bread, because there was not enough 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 once he was forced to hide from persecution for two months. Since 1794, Monge no longer took a direct part in the affairs of state administration, but completely devoted himself to scientific and teaching activities.
In 1794, Monge published a guide to the production of cannons (Description de l'art de fabriquer les canons) and set about organizing the Central School of Public Works, which was supposed to replace the Academies and Universities abolished by decrees of the Convention in 1793. According to the plan, it was supposed to be a new type of higher school with a three-year education to train engineers and scientists on a solid scientific basis in a number of civilian and military specialties. On September 1, 1795, the school was renamed the Polytechnic School.
In January 1795, the so-called Higher Normal School was organized, intended for a four-month training of professional personnel (mainly teachers). Together with Monge, classes were conducted by Berthollet, Laplace, Lagrange and others. For the students of the first set of the School, Monge prepared and read a course in descriptive geometry, a record of which was published in the Proceedings of the Normal School (1795). At the same time, Monge created another of his main works - the Application of Analysis to Geometry (L "application de l" analyze la gometrie, 1795), where, in addition to discoveries in differential geometry, a geometric interpretation of partial differential equations is 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 later teachers of these scientific institutions.
In May 1796, the Directory instructed Monge and Berthollet to take part in the selection committee for the indemnity of monuments of art and science in the regions of Italy conquered by the army of the Republic. Monge fulfilled the order by delivering to Paris paintings by Raphael, Michelangelo, Titian, Veronese and other works of art, as well as scientific exhibits and instruments for the Polytechnic School. During his stay in Italy, he met and became friends with the young General Bonaparte, whose devotion largely determined the future life of Monge. Returning from Italy, on October 1, 1797, he delivered a speech to the Directory about the victories of the French army with threats against the British 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, the Republic was proclaimed there, the 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 on the help of scientists in building roads, canals, dams, compiling maps, organizing the production of gunpowder, guns and cannons, as well as in the creation of new scientific institutions in the conquered territories, similar to the French ones. On August 29, 1798, in Cairo, members of this expedition and some military men, including Bonaparte himself, established the Egyptian Institute of Sciences and Arts, modeled on the French Institute and elected Monge as its president for the first trimester, Bonaparte's vice-president, Fourier's indispensable 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 a mirage that frightened 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, 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 power in his hands, Bonaparte appointed Monge a senator for life, at the Polytechnic School he reads 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 Order of the Legion of Honor, in 1806 he was appointed President of the Senate for another year, a year later he received the title of Count of Peluza and 100,000 francs to buy the estate. However, soon his health began to fail, his hand was taken away for a while. Monge stops teaching at the Polytechnic School, but continues his scientific work and advises on proposed technical projects. So, in 1805, the emperor instructs him to study the possibility of building a canal from the Urk River to supply Paris with water.
Events of 1812-1814 ended with the defeat of France and the exile of Bonaparte. Monge remained a supporter of the Empire and throughout the Hundred Days was still on the side of Bonaparte. After the restoration of power of the Bourbons, Monge was deprived of titles, awards and pensions, expelled (albeit only for a year) from the Polytechnic School. In 1816, by government decree, he and Carnot were expelled from the Institute, which had been reorganized in a new way, and were replaced by Cauchy and Breguet. As one of the "regicides", Monge could expect more serious repressions. 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 strokes and soon died. He was buried in the Père Lachaise cemetery. Monge's wife survived him by 24 years.
The creation of the "Descriptive Geometry", a treatise of which was published only in 1799 under the title "Géométrie descriptive", served as the beginning and basis of the work that allowed the new Europe to master the geometric knowledge of Ancient Greece; works on the theory of surfaces, in addition to their immediate significance, led to the elucidation of the important principle of continuity and to the disclosure of the meaning of the vast uncertainty that arises when integrating equations with partial derivatives, arbitrary constants, and even more so with the appearance of arbitrary functions.
Of the other, less significant contributions of Monge to science, one should name 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 way of forming the surfaces of the same bodies with the help of a straight line; creation of the first idea about the lines of curvature of surfaces; the establishment of the principles of the theory of mutual polars, developed later by Poncelet, the proof of the theorem that the locus of the vertex of a trihedral angle with right planar 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 diagram Monge (Project Monge).
Numerous memoirs of Monge were published in the works of the Paris and Turin academies, were published in the Journaux de l'Ecole Polytechnique et de l'Ecole Normale, in the Dictionnaire de Physique, Diderot and d'Alembert's Methodological 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, etc. In the book “Gaspard Monge. Collection of Articles for the 200th Anniversary of the Birth” contains a bibliography of Monge's works (72 items) and a list of publications about his life and work (73 items).
The name of Gaspard Monge is included in the list of the 72 greatest scientists of France, placed on the first floor of the Eiffel Tower (N 54).
In the hometown of Gaspard Monge, Beaune, on the square named after him in 1849, a monument was erected in his honor.

Named after him:
The building of the Navy.
A street in Paris (Rue Monge), running along the former buildings of the Polytechnic School, as well as a square in the 5th Parisian district and the metro station Place Monge 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, Chambéry, Charleville-Mezieret, Saint-Joiret, Savigny-sur-Orge, Nantes, Knutange.
Research Institute of Electronics and Informatics Gaspard Monge - IGM (Institut d "Electronique et d" Informatique Gaspard-Monge) in Marne-la-Vallee, a suburb of Paris.

Monge G. Memoire sur la theorie 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. Geometric descriptive. - Paris, 1799. - 132 p.
Monge G. Memoire sur le phenomene doptique connu sous le nom de mirage//Decade Egyptienne. - Caire, 1799. - V. 1. - R. 37-46.
Monge G. Initial foundations of statics or equilibrium of rigid bodies for navigation 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.
Monzh 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.: Ed. Academy of Sciences of the USSR, 1947. - 292 p.

Literature

Arago F. Biographies of famous astronomers, physicists and geometers. - St. Petersburg, 1859. - T. 1. - S. 499-589.
Launay Louis de. Monge fondateur de lÉcole polytechnique. - Paris, 1933. - 380 p.
Staroselskaya-Nikitina O. Essays on the history of science and technology in the period of the French bourgeois revolution 1789-1794. - M.-L., 1946. - 274 p.
Gaspar Monge. Collection of articles for the 200th anniversary of the birth / Ed. ed. IN AND. Smirnov. - L .: Ed. Academy of Sciences of the USSR, 1947. - 85 p. - 5,000 copies.
Kargin D.I. Gaspard Monge and his "Descriptive Geometry" / / Gaspard Monge. Descriptive geometry. - M.: Ed. Academy of Sciences of the USSR, 1947. - S. 245-257.
Kargin D.I. Gaspard Monge is the creator of descriptive geometry. 1746-1818. To the 200th anniversary of the birth // Priroda, - 1947. - No. 2. - P. 65-73.
Lukomskaya A.M. List of works and literature on the life and work of Gaspard Monge / / Gaspard Monge. Descriptive geometry. - M.: Ed. Academy of Sciences of the USSR, 1947. - S. 258-270.
Vavilov S.I. Science and technology during the French Revolution / Collected Works. - M.: AN SSSR, 1956. - T. 3. S. 176-190. - 3,000 copies.
Bogolyubov A.N. Gaspar 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. Monzhe / Otv. ed. V. I. SMIRNOV - M.: Knowledge, 1986. - 252 p.
Borodin A.I., Bugai A.S. Outstanding mathematicians. - Kyiv: Radyansk school, 1987.

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 the graph.

Monge Gaspard (May 10, 1746-July 28, 1818) - French geometer and public figure, Member of the Paris Academy of Sciences (1780). Creator of descriptive geometry, one of the organizers of the Polytechnic School in Paris and its director for many years. Born in Bon Côte d "0r. He graduated from the School of Military Engineers in Mezieres. From 1768 he was a professor of mathematics, from 1771 he was 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 measures and weights, then he was the minister of the sea and the organizer of national defense.During the Directory, he became close to Napoleon, took part in his campaign in Egypt and founding in Cairo Egyptian Institute (1798); was elevated to the graph. Received worldwide recognition, creating (in the 70s) modern methods of 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 papers on the equations of surfaces were published in 1770 and 1773. In 1795 and 1801 Monge's work on finite and differential equations of various surfaces. 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; translation surfaces, etc. As an appendix to the book, Monge gave his theory of integration of equations with partial derivatives of the 1st order and his solution to the problem of string vibrations. For each of the types of surfaces, he first derived a differential, then a final equation. The first denoted by letters p and q partial derivatives of z with respect to x and y, and by letters r, s and t - derivatives of the 2nd order.

Information and methods of construction, conditioned by the need for flat images of spatial forms, have been accumulating gradually since ancient times. For a long period, flat images were performed mainly as visual images. With the development of technology, the question of applying a method that ensures the accuracy and readability of images, i.e., the ability to accurately determine the location of each point of the image relative to other points or planes, and by simple methods to determine the size of line segments and figures, has become of paramount importance. The gradually accumulated separate rules and techniques for constructing such images were brought into the system and developed in the work of the French scientist Monge, published in 1799 under the title "Géometrie déscriptive".

Gaspard Monge (1746-1818) went down in history as a major French geometer of the late 18th and early 19th centuries, engineer, public figure and statesman during the revolution of 1789-1794. and the reign of Napoleon I, one of the founders of the famous Ecole Polytechnique in Paris, a participant in the work on the introduction of the metric system of measures and weights. As one of the ministers in the revolutionary government of France, Monge did a lot to protect her from foreign intervention and to win the revolutionary troops. Monge did not immediately get the opportunity to publish his work outlining the method he developed. Given the great practical importance of this method for making drawings of objects of military importance, and not wanting the Monge method to become known outside the borders of France, her government banned the publication of the book. Only at the end of the XVIII century this prohibition was lifted. After the restoration of the Bourbons, Gaspard Monge was persecuted, forced into hiding and ended his life in poverty. Monge's method - parallel projection method (moreover, rectangular projections are taken on two mutually perpendicular projection planes)- providing expressiveness, accuracy and readability of images of objects on a plane, has been and remains the main method of drawing up technical drawings.

Word rectangular often replaced by the word orthogonal, formed from the words of the ancient Greek language, denoting "straight" and "angle". In the following, the term orthogonal projections will be used to designate a system of rectangular projections on mutually perpendicular planes.

This course focuses on rectangular projections. In the case of using parallel oblique projections, this will be specified each time.

Descriptive geometry (NG) has been the subject of teaching in our country since 1810, when classes in descriptive geometry, along with other disciplines of the curriculum, began at the newly founded Institute of the Corps of Railway Engineers. This was due to its ever-increasing practical importance.

Yakov Alexandrovich Sevastyanov (1796-1849), who graduated from this institute in 1814, taught at the Institute of the Corps of Railway Engineers 1), whose name is associated with the appearance in Russia of the first works on modern times. , first translated from French, and then the first original work entitled "Fundamentals of Descriptive Geometry" (1821), mainly devoted to the presentation of the method of orthogonal projections.

1) Now the Leningrad Institute of Railway Engineers. Academician V. N. Obraztsov.

Ya. A. Sevastyanov gave lectures in Russian, although teaching in those years was generally conducted in French. Thus, Ya. A. Sevastyanov laid the foundation for teaching and establishing terminology in n. g. in their native language. Even during the life of Ya. A. Sevastyanov n. city ​​was included in the curricula of a number of civilian and military educational institutions.

A major trace in the development of n. In the 19th century, Nikolai Ivanovich Makarov (1824-1904), who taught this subject at the St. Petersburg Institute of Technology, and Valerian Ivanovich Kurdyumov (1853-1904), who, being a professor at the St. Petersburg Institute of Railway Engineers in the department of building art, left at this institute, the course of N. d. In his teaching practice, V. I. Kurdyumov gives numerous examples of the application of n. to solving engineering problems.

The activities and works of V. I. Kurdyumov, as it were, ended almost a century of development of AD. city ​​and its teaching in Russia. During this period, the greatest attention was paid to the organization of teaching, the creation of works intended to serve as textbooks, the development of improved techniques and methods for solving a number of problems. These were essential and necessary moments in the development of teaching n. G.; however, its scientific development lagged behind the achievements in the field of method of presentation of the subject. Only in the works of V. I. Kurdyumov did the theory receive a more vivid reflection. Meanwhile, in some foreign countries in the 19th century A.D. city ​​has already received significant scientific development. Obviously, in order to eliminate the backlog and for the further development of the scientific content, n. it was necessary to expand its theoretical basis and turn to research work.

This can be seen in the works and activities of Evgraf Stepanovich Fedorov (1853-1919), the famous Russian scientist, crystallograph geometer, and Nikolai Alekseevich Rynin (1877-1942), who already in the last years before the Great October Socialist Revolution turned to the development of descriptive geometry as science. To date, descriptive geometry as a science has received significant development in the works of Soviet scientists N.A. Glagolev (1888-1945), A.I. Dobryakov (1895-1947), D.D. Y. Gromov (1884-1963), S. M. Kolotov (1885-1965), N. F. Chetverukhin (1891-1974), I. I. Kotov (1909-1976) and many others.

Questions for Chapter I

1. How is the central projection of a point constructed?
2. When is the central projection of a straight line a point?
3. What is the method of projection called parallel?
4. How is a parallel projection of a straight line constructed?
5. Can the parallel projection of a straight line be a point?
6. If a point belongs to a given line, how are their projections mutually arranged?
7. In which case, in a parallel projection, a straight line segment is projected to its full size?
8. What is the "Monge method"?
9. What is the meaning of the word "orthogonal"?