The Polytechnic School of France
Contents
| Page. | |
| Polytechnic School at Paris, | 11 |
| Subjects of Instruction as Prescribed for Admission in 1850, | 13 |
| Preparatory Course in the Lycees, | 49 |
| History, Management, Conditions of Admission, Course of Study, Examinational System, and Results, | 55 |
| I. Foundation and History, | 55 |
| Out growth of the Necessities of the Public Service in 1794, | 56 |
| High Scientific Ability of its first Teachers, | 58 |
| Peculiar Method of Scientific Teaching, | 59 |
| Characteristic features of the Répétitorial System, | 59 |
| The Casernement, or Barrack Residence of the Pupils, | 60 |
| Permanent Organization in 1809, | 60 |
| Commission of 1850, | 62 |
| II. Outline of the Plan, Objects, and Management, | 63 |
| Public Services provided for in its General Scientific Course, | 63 |
| Admission by Competition in an Open Examination, | 63 |
| Annual Charge for Board and Instruction, | 64 |
| Exhibitions, (or bourses, demi-bourses,) and Outfits (trousseaux,) | 64 |
| Length of Course of Study, | 64 |
| Number of Professors and Teachers, besides its Military Staff, | 64 |
| Military Establishment, | 65 |
| Civil Establishment, | 65 |
| General Control and Supervision, | 65 |
| 1. Board of Administration, | 65 |
| 2. Board of Discipline, | 65 |
| 3. Board of Instruction, | 65 |
| 4. Board of Improvement, | 66 |
| III. Conditions and Examinations for Admission, | 66 |
| Who may be Candidates for Admission, | 66 |
| Subjects of Entrance Examination, | 66 |
| Preliminary Examination, | 67 |
| Written Examination, | 67 |
| Oral Examination, | 68 |
| Scale of Merit, and Latitude in Amount of Credit given, | 68 |
| Reports of Examiners to Minister of War, | 69 |
| Co-efficients of Influence, varying with the Study and Mode of Examination, | 69 |
| Decision of Jury on all the Documents of each Candidate, | 70 |
| Final Action of the Minister of War, | 70 |
| IV. School Buildings, Course and Method of Study, | 70 |
| Situation, Number, and Purposes of Buildings, | 70 |
| Daily Routine of Exercises, | 72 |
| Method of Teaching and Study, | 73 |
| Professorial and Répétitorial, | 74 |
| Interrogations, Général, | 74 |
| Interrogations, Particulieres by the Répétiteurs, | 74 |
| One Répétiteur to every eight Pupils, | 74 |
| System of Credits for every Lecture, every Interrogation, and Exercise, | 75 |
| Final Admission to Public Service, depends on daily and hourly fidelity, | 76 |
| Division of First Year’s Work into three portions, | 76 |
| First portion—Analysis and Descriptive Geometry, | 76 |
| Second portion—Mechanics, Geodesy, Physics, &c., | 76 |
| Third portion—General Private Study, | 76 |
| Number and Subjects of Lectures in Second Course, | 78 |
| V. Examinational System, | 78 |
| Ordinary Examinations, | 78 |
| 1. By Professors on their own Lectures, both Written and Oral, | 78 |
| 2. By Examiners on the Manipulations of the Pupils, | 78 |
| 3. By Répétiteurs every ten or fourteen days, | 78 |
| 4. By Professors and Répétiteurs at the close of each Course, | 79 |
| First Annual Examination, | 79 |
| Table—Co-efficient of Influence in Second Division of First Year’s Course, | 79 |
| Specimen of Credits gained by one Student in First Year’s Course, | 80 |
| Persons excluded from the Second Year’s Course, | 81 |
| Second Annual or Great Final Examination, | 81 |
| Conducted by the same Examiners as the First, | 81 |
| Oral, and extends over the whole Two Years’ Course, | 81 |
| Results based on each Day’s Study’s, Year’s, and Examination’s results, | 82 |
| Tables—Co-efficients of Influence in Final Classification, &c., | 82 |
| Order in which the Public Services are Selected, | 83 |
| VI. General Remarks on Character and Results of the Polytechnic School, | 84 |
| Appendix, | 88 |
| Public Services Beside the Army Supplied by This School, | 88 |
| 1. Gunpowder and Saltpetre, | 88 |
| 2. Navy, | 88 |
| 3. Marine Artillery and Foundries, | 88 |
| 4. Naval Architects. School of Application at L’Orient, | 88 |
| 5. Hydrographers, | 88 |
| 6. Roads and Bridges. School of Application at Paris, | 89 |
| 7. Mining Engineers. School of Mines at Paris and St. Etienne, | 89 |
| 8. Tobacco Department | 90 |
| 9. Telegraphs, | 90 |
| Programmes of Internal Instruction During the Two Years of Study, | 91 |
| 1. Analysis, | 91 |
| First Year—Calculus, Differential, | 91 |
| Calculus, Integral, | 93 |
| Second Year—Calculus, Integral, (continuation,) | 94 |
| 2. Descriptive Geometry and Stereotomy, | 97 |
| First Year—Descriptive Geometry, Geometrical Drawing, | 97 |
| Second Year—Stereotomy: Wood-work, | 103 |
| Masonry, | 103 |
| 3. Mechanics and Machines, | 104 |
| First Year—Kinematics, | 105 |
| Equilibrium of Forces, | 105 |
| Second Year—Dynamics, | 112 |
| Hydrostatics, | 115 |
| Hydraulics, | 115 |
| Machines in Motion, | 116 |
| 4. Physics, | 116 |
| First Year—General Properties of Bodies, Hydrostatics, Hydrodynamics, | 117 |
| Heat, | 119 |
| Statical Electricity, | 123 |
| Second Year—Dynamical Electricity, | 124 |
| Acoustics, | 125 |
| Optics, | 126 |
| 5. Manipulations in Physics, | 129 |
| First and Second Year, | 130 |
| Distribution of Time, | 131 |
The Polytechnic School at Paris8
I. Foundation and History
The origin of the Ecole Polytechnique dates from a period of disorder and distress in the history of France which might seem alien to all intellectual pursuits, if we did not remember that the general stimulus of a revolutionary period often acts powerfully upon thought and education. It is, perhaps, even more than the Institute, the chief scientific creation of the first French Revolution. It was during the government of the committee of public safety, when Carnot, as war minister, was gradually driving back the invading armies, and reorganizing victory out of defeat and confusion, that the first steps were taken for its establishment. A law, dating the 1st Ventose, year II., the 12th of March 1794, created a “Commission des Travaux Publics,” charged with the duty of establishing a regular system for carrying on public works; and this commission ultimately founded a central school for public works, and drew up a plan for the competitive examination of candidates for admission to the service. It was intended at first to give a complete education for some of the public services, but it was soon changed into a preparatory school, to be succeeded by special schools of application. This was the Ecole Polytechnique.
The school and its plan were both owing to an immediate and pressing want. It was to be partly military and partly civil. Military, as well as civil education had been destroyed by the revolutionists. The committee of public safety had, indeed, formed a provisional school for engineers at Metz, to supply the immediate wants of the army on the frontier, and at this school young men were hastily taught the elements of fortification, and were sent direct to the troops, to learn as they best could, the practice of their art. “But such a method,” says the report accompanying the law which founded the school, “does not form engineers in any true sense of the term, and can only be justified by the emergency of the time. The young men should be recalled to the new school to complete their studies.” Indeed no one knew better than Carnot, to use the language of the report, “that patriotism and courage can not always supply the want of knowledge;” and in the critical campaigns of 1793–4, he must often have felt the need of the institution which he was then contributing to set on foot. Such was the immediate motive for the creation of this school. At first, it only included the engineers amongst its pupils. But the artillery were added within a year.
We must not, however, omit to notice its civil character, the combination of which with its military object forms its peculiar feature, and has greatly contributed to its reputation. Amongst its founders were men, who though ardent revolutionists, were thirsting for the restoration of schools and learning, which for a time had been totally extinguished. The chief of these, besides Carnot, were Monge and Fourcroy, Berthollet and Lagrange. Of Carnot and Lagrange, one amongst the first of war ministers, the other one of the greatest of mathematicians, we need not say more. Berthollet, a man of science and practical skill, first suggested the school; Monge, the founder of Descriptive Geometry, a favorite savant of Napoleon though a zealous republican, united to real genius that passion for teaching and for his pupils, which makes the beau idéal of the founder of a school; and Fourcroy was a man of equal practical tact and science, who at the time had great influence with the convention, and was afterwards intrusted by Napoleon with much of the reorganization of education in France.
When the school first started there was scarcely another of any description in the country. For nearly three years the revolution had destroyed every kind of teaching. The attack upon the old schools, in France, as elsewhere, chiefly in the hands of the clergy, had been begun by a famous report of Talleyrand’s, presented to the legislative assembly in 1791, which recommended to suppress all the existing academies within Paris and the provinces, and to replace them by an entirely new system of national education through the country. In this plan a considerable number of military schools were proposed, where boys were to be educated from a very early age. When the violent revolutionists were in power, they adopted the destructive part of Talleyrand’s suggestions without the other. All schools, from the university downwards, were destroyed; the large exhibitions or Bourses, numbering nearly 40,000, were confiscated or plundered by individuals, and even the military schools and those for the public works (which were absolutely necessary for the very roads and the defense of the country) were suppressed or disorganized. The school of engineers at Mézières (an excellent one, where Monge had been a professor,) and that of the artillery at La Fère, were both broken up, whilst the murder of Lavoisier, and the well known saying in respect to it, that “the Republic had no need of chemists,” gave currency to a belief, which Fourcroy expressed in proposing the Polytechnic, “that the late conspirators had formed a deliberate plan to destroy the arts and sciences, and to establish their tyranny on the ruins of human reason.”
Thus it was on the ruin of all the old teaching, that the new institution was erected; a truly revolutionary school, as its founders delighted to call it, using the term as it was then commonly used, as a synonym for all that was excellent. And then for the first time avowing the principle of public competition, its founders, Monge and Fourcroy, began their work with an energy and enthusiasm which they seem to have left as a traditional inheritance to their school. It is curious to see the difficulties which the bankruptcy of the country threw in their way, and the vigor with which, assisted by the summary powers of the republican government, they overcame them. They begged the old Palais Bourbon for their building; were supplied with pictures from the Louvre; the fortunate capture of an English ship gave them some uncut diamonds for their first experiments; presents of military instruments were sent from the arsenals of Havre; and even the hospitals contributed some chemical substances. In fine, having set their school in motion, the government and its professors worked at it with such zeal and effect, that within five months after their project was announced, they had held their first entrance examination, open to the competition of all France, and started with three hundred and seventy-nine pupils.
The account of one of these first pupils, who is among the most distinguished still surviving ornaments of the Polytechnic, will convey a far better idea of the spirit of the young institution than could be given by a more lengthy description. M. Biot described to us vividly the zeal of the earliest teachers, and the thirst for knowledge which, repressed for awhile by the horrors of the period, burst forth with fresh ardor amongst the French youth of the time. Many of them, he said, like himself, had been carried away by the enthusiasm of the revolution, and had entered the army. “My father had sent me,” he added, “to a mercantile house, and indeed I never felt any great vocation to be a soldier, but Que voulez vous? les Prussiens etaient en Champagne.” He joined the army, served two years under Dumouriez, and returned to Paris in the reign of terror, “to see from his lodgings in the Rue St. Honore the very generals who had led us to victory, Custine and Biron, carried by in the carts to the guillotine. “Imagine what it was when we heard that Robespierre was dead, and that we might return safely to study after all this misery, and then to have for our teachers La Place, Lagrange, and Monge. We felt like men brought to life again after suffocation. Lagrange said, modestly, “Let me teach them arithmetic.” Monge was more like our father than our teacher; he would come to us in the evening, and assist us in our work till midnight, and when he explained a difficulty to one of our chefs de brigade, it ran like an electric spark through the party.” The pupils were not then, he told us, as they have since been, shut up in barracks, they were left free, but there was no idleness or dissipation amongst them. They were united in zealous work and in good camaraderie, and any one known as a bad character was avoided. This account may be a little tinged by enthusiastic recollections, but it agreed almost entirely with that of M. de Barante, who bore similar testimony to the early devotion of the pupils, and the unique excellence of the teaching of Monge.
We are not, however, writing a history of this school, and must confine ourselves to such points as directly illustrate its system of teaching and its organization. These may be roughly enumerated in the following order:
1. Its early history is completed by the law of its organization, given it by La Place in his short ministry of the interior. This occurred in the last month of 1799, a memorable era in French history, for it was immediately after the revolution of the 18th of Brumaire, when Napoleon overthrew the Directory and made himself First Consul. One of his earliest acts was to sign the charter of his great civil and military school. This charter or decree deserves some attention, because it is always referred to as the law of the foundation of the school. It determined the composition of the two councils of instruction and improvement, the bodies to which the direction of the school was to be, and still is, intrusted; some of its marked peculiarities in the mode and subject of teaching. It is important to notice each of the two points.
The direction of the school was at first almost entirely in the hands of its professors, who formed what is still called its Council of Instruction. Each of them presided over the school alternately for one month, a plan copied from the revolutionary government of the Convention. In the course of a few years, however, another body was added, which has now the real management of the school. This is called the “Council of Improvement” (Conseil de perfectionnement,) and a part of its business is to see that the studies form a good preparation for those of the more special schools (écoles d’application) for the civil and military service. It consists of eminent men belonging to the various public departments supplied by the school, and some of the professors. It has had, as far as we could judge, an useful influence; first, as a body not liable to be prejudiced in its proposals by the feelings of the school, and yet interested in its welfare and understanding it; secondly, as having shown much skill in the difficult task of making the theoretical teaching of the Polytechnic a good introduction to the practical studies of the public service; thirdly, as being sufficiently influential, from the character of its members, to shield the school from occasional ill-judged interference. It should be added that hardly any year has passed without the Council making a full report on the studies of the school, with particular reference to their bearing on the Special Schools of Application.
The method of scientific teaching has been peculiar from the beginning. It is the most energetic form of what may be called the repetitorial system, a method of teaching almost peculiar to France, and which may be described as a very able combination of professional and tutorial teaching. The object of the répétiteur, or private tutor, is to second every lecture of the professor, to explain and fix it by ocular demonstration, explanations, or examination. This was a peculiarity in the scheme of Monge and Foureroy. The latter said, in the first programme, “Our pupils must not only learn, they must at once carry out their theory. We must distribute them into small rooms, where they shall practice the plans of descriptive geometry, which the professors have just shown them in their public lectures. And in the same manner they must go over in practice (répéteront) in separate laboratories the principal operations of chemistry.” To carry out this system the twenty best pupils, of whom M. Biot was one, were selected as répétiteurs soon after the school had started. Since then the vacancies have always been filled by young but competent men, aspiring themselves to become in turn professors. They form a class of teachers more like the highest style of private tutors in our universities, or what are called in Germany Privat-docenten, than any other body—with this difference, that they do not give their own lectures, but breaking up the professor’s large class into small classes of five and six pupils, examine these in his lecture. The success of this attempt we shall describe hereafter.
2. A change may be noticed which was effected very early by the Council of Improvement—the union of pupils for artillery and engineers in a single school of application. The first report in December 1800, speaks of the identity in extent and character of the studies required for these two services; and in conformity with its recommendation, the law of the 3rd of October 1802, (12th Vendémiaire, XI.) dissolved the separate artillery school at Châlons, and established the united school for both arms in the form which it still retains at Metz.
3. In 1805 a curious change was made, and one very characteristic of the school. The pupils have always been somewhat turbulent, and generally on the side of opposition. In the earliest times they were constantly charged with incivisme, and the aristocracy was said to have “taken refuge within its walls.” In fact, one of its earliest and of its few great literary pupils, M. de Barante, confirmed this statement, adding, as a reason, that the school gave for a while the only good instruction in France. It was in consequence of some of these changes that the pupils who had hitherto lived in their own private houses or lodgings in Paris, were collected in the school building. This “casernement” said to be immediately owing to a burst of anger of Napoleon, naturally tended to give the school a more military character; but it was regarded as an unfortunate change by its chief scientific friends. “Ah! ma pauvre école!” M. Biot told us he had exclaimed, when he saw their knapsacks on their beds. He felt, he said, that the enthusiasm of free study was gone, and that now they would chiefly work by routine and compulsion.
4. The year 1809 may be called the epoch at which the school attained its final character. By this time the functions, both of boards and teachers, were accurately fixed, some alterations in the studies had taken place, and the plan of a final examination had been drawn up, according to which the pupils were to obtain their choice of the branch of the public service they preferred. In fact, the school may be said to have preserved ever since the form it then assumed, under a variety of governments and through various revolutions, in most of which, indeed, its pupils have borne some share; and one of which, the restoration of 1816, was attended with its temporary dissolution.
Thus, during the first years after its foundation the Polytechnic grew and flourished in the general dearth of public teaching, being indeed not merely the only great school, but, until the Institute was founded, the only scientific body in France. Working on its first idea of high professorial lectures, practically applied and explained by répétiteurs, its success in its own purely scientific line was, and has continued to be, astonishing. Out of its sixteen earliest professors, ten still retain an European name. Lagrange, Monge, Fourcroy, La Place, Guyton de Morveau were connected with it. Malus, Hauy, Biot, Poisson, and De Barante, were among its earliest pupils. Arago, Cauchy, Cavaignac, Lamoricière, with many more modern names, came later. All the great engineers and artillerymen of the empire belonged to it, and the long pages in its calendar of distinguished men are the measure of its influence on the civil and military services of France. In fact its pupils, at a time of enormous demands, supplied all the scientific offices of the army, and directed all the chief public works, fortresses, arsenals, the improvement of cities, the great lines of roads, shipbuilding, mining—carried out, in a word, most of the great improvements of Napoleon. He knew the value of his school, “the hen” as he called it, “that laid him golden eggs”—and perhaps its young pupils were not improved by the excessive official patronage bestowed by him upon “the envy of Europe,” “the first school in the world.” It can not, however, be matter of surprise, that its vigor and success should have caused Frenchmen, even those who criticise its influence severely, to regard it with pride as an institution unrivaled for scientific purposes.
It is not necessary to give any detailed account of the later history of the school, but we must remark that disputes have frequently arisen with regard to the best mode of harmonizing its teaching with that of the special schools of application to which it conducts. These disputes have been no doubt increased by the union of a civil and military object in the same school. The scientific teaching desirable for some of the higher civil professions has appeared of doubtful advantage to those destined for the more practical work of war. There has been always a desire on the one side to qualify pure mathematics by application, a strong feeling on the other that mathematical study sharpens the mind most keenly for some of the practical pursuits of after life. We should add, perhaps, that there has been some protest in France (though little heard among the scientific men who have been the chief directors of the school) against the esprit faux, the exclusive pursuit of mathematics to the utter neglect of literature, and the indifference to moral and historical studies. Some one or other of these complaints any one who studies the literature, the pamphlets, and history of the school will find often reproduced in the letters of war ministers, of artillery and engineer officers commanding the school of application at Metz, or of committees from the similar schools for the mines and the roads and bridges. The last of these occasions illustrates the present position of the school.
On the 5th of June 1850, the legislative assembly appointed a mixed commission of military men and civilians, who were charged to revise all the programs of instruction, and to recommend all needful changes in the studies of the pupils, both those preparatory to entrance9 and those actually pursued in the school. The commission was composed as follows:—
M. Thenard, Member of the Academy of Sciences, and of the Board of Improvement of the Polytechnic School, President.
Le Verrier, Member of the Academy of Sciences and of the Legislative Assembly, Reporter.
Noizet, General of Brigade of Engineers.
Poncelet, General of Brigade of Engineers, Commandant of the Polytechnic School, Member of the Academy of Sciences.
Piobert, General of Brigade of Artillery, Member of the Academy of Sciences.
Mathieu, Rear Admiral.
Duhamel, Member of the Academy of Sciences, Director of Studies at the Polytechnic School.
Mary, Divisional Inspector of Roads and Bridges.
Morin, Colonel of Artillery, Member of the Academy of Sciences.
Regnault, Engineer of Mines, Member of the Academy of Sciences.
Olivier, Professor at the Conservatoire des Arts et Metiers.
Debacq, Secretary for Military Schools at the Ministry of War, Secretary.
A chronic dispute which has gone on from the very first year of the school’s existence, between the exclusive study of abstract mathematics on the one hand, and their early practical application on the other, was brought to a head (though it has scarcely been set at rest) by this commission. All the alterations effected have been in the direction of eliminating a portion of the pure mathematics, and of reducing abstract study to the limits within which it was believed to be most directly applicable to practice. The results, however, are still a subject of vehement dispute, in which most of the old scientific pupils of the Polytechnic, and many of what may be styled its most practical members, the officers of the artillery and engineers, are ranged on the side of “early and deep scientific study versus early practical applications.” It is, indeed, a question which touches the military pupils nearly, since it is in their case particularly that the proposed abstract studies of the Polytechnic might be thought of the most doubtful advantage. We do not try to solve the problem here, though the facts elsewhere stated will afford some materials for judgment. We incline to the opinion of those who think that the ancient genius loci, the traditional teaching of the school, will be too strong for legislative interference, and that, in spite of recent enactments, abstract science and analysis will reign in the lecture-rooms and halls of study of the Polytechnic, now as in the days of Monge.
II. An Outline of the Management and of the Establishment of the School, etc.
The Polytechnic, as we have said, is a preparatory and general scientific school; its studies are not exclusively adapted for any one of the departments to which at the close of its course the scholars will find themselves assigned; and on quitting it they have, before entering on the actual discharge of their duties of whatever kind, to pass through a further term of teaching in some one of the schools of application specially devoted to particular professions.
The public services for which it thus gives a general preparation are the following:
Military: Under the Minister at War.
Artillery (Artillerie de terre.)
Engineers (Génie.)
The Staff Corps (Corps d’Etat Major.)
The Department of Powder and Saltpetre (Poudres et Salpétres.)
Under the Minister of Marine.
Navy, (Marine.)
Marine Artillery (Artillerie de mer.)
Naval Architects (Génie maritime.)
The Hydrographical Department (Corps des Ingénieurs Hydrographes.)
Civil: Under the Minister of Public Works.
The Department of Roads and Bridges (Ponts-et-chaussées.)
The Department of Mines (Mines.)
Under the Minister of the Interior.
The Telegraph Department (Lignes Télégraphiques.)
Under the minister of Finance.
The Tobacco Department (Administration des Tabacs.)
To these may be added at any time, by a decree on the part of the government, any other departments, the duties of which appear to require an extensive knowledge of mathematics, physics, or chemistry.
Admission to the school is, and has been since its first commencement in 1794, obtained by competition in a general examination, held yearly, and open to all. Every French youth, between the age of sixteen and twenty, (or if in the army up to the age of twenty-five,) may offer himself as a candidate.
A board of examiners passes through France once every year, and examines all who present themselves, that have complied with the conditions, which are fully detailed in the decree given in the appendix. It commences at Paris.
A list of such of the candidates as are found eligible for admittance to the Polytechnic is drawn up from the proceedings of the board, and submitted to the minister at war; the number of places likely to be vacant has already been determined, and the minister fixes the number of admissions accordingly. The candidates admitted are invariably taken in the order of merit.
The annual charge for board and instruction is 40l. (1,000 fr.,) payable in advance in four installments. In addition there is the cost of outfit, varying from 20l. to 24l. Exhibitions, however, for the discharge of the whole or of one-half of the expense (bourses and demi-bourses,) are awarded by the state in favor of all the successful candidates, whose parents can prove themselves to be too poor to maintain their children in the school. Outfits and half outfits (trousseaux) and demi-trousseaux) are also granted in these cases, on the entrance of the student into the school; and the number of these boursiers and demi-boursiers amounts at the present time to one-third of the whole.
The course of study is completed in two years. On its successful termination which is preceded by a final examination, the students are distributed into the different services, the choice being offered them in the order of their merit, and laid down in the classified list drawn up after the examination. If it so happen that the number of places or the services which can be offered is not sufficient for the number of qualified students, those at the bottom of the list are offered service in the infantry or cavalry, and those who do not enter the public service, are supplied with certificates of having passed successfully through the school. Students who have been admitted into the school from the army, are obliged to re-enter the army.
All others, as has been said, have the right of choosing, according to their position on the list, the service which they prefer, so far, that is, as the number of vacancies in that service will allow; or they may if they please decline to enter the public service at all.
Such is a general outline of the plan and object of the school. We may add that, besides its military staff, it employs no less than thirty-nine professors and teachers; that it has four boards of management, and that ten scientific men unconnected with the school, and amongst the most distinguished in France, conduct its examinations. The magnitude of this establishment for teaching may be estimated by the fact, that the number of pupils rarely exceeds three hundred and fifty, and is often much less.
A fuller enumeration of these bodies will complete our present sketch.
I. The military establishment consists of:—
The Commandant, a General Officer, usually of the Artillery or the Engineers, at present a General of Artillery.
A Second in Command, a Colonel or Lieutenant-Colonel, chosen from former pupils of the school; at present a Colonel of Engineers.
Three Captains of Artillery and Three Captains of Engineers, as Inspectors of Studies, chosen also from former pupils of the school.
Six Adjutants (adjoints,) non-commissioned officers, usually such as have been recommended for promotion.
II. The civil establishment consists of:—
1. A Director of Studies, who has generally been a civilian, but is at present a Lieutenant-Colonel of Engineers.
2. Fifteen Professors, viz.:—Two of Mathematical Analysis. Two of Mechanics and Machinery. One of Descriptive Geometry. Two of Physics. Two of Chemistry. One of Military Art and Fortification. One of Geodesy. One of Architecture. One of French Composition. One of German. One of Drawing. Of these one is an officer of the Staff, another of the Artillery, and a third of the Navy; two are Engineers in Chief of the Roads and Bridges; nine are civilians, of whom two are Members of the Academy of Sciences.
3. Three Drawing Masters for Landscape and Figure Drawing; one for Machine Drawing, and one for Topographical Drawing.
4. Nineteen Assistant and Extra Assistant Teachers, (répétiteurs and répétiteurs adjoints) whose name and functions are both peculiar.
5. Five Examiners for Admission, consisting at present of one Colonel of Artillery, as President, and four civilians.
6. Five Examiners of Students (civilians,) four of them belonging to the Academy of Sciences.
7. There is also a separate Department for the ordinary Management of Administration of the affairs of the school, the charge of the fabric and of the library and museums; and a Medical Staff.
III. The general control or supervision of the school is vested, under the war department, in four great boards of councils, viz.:—
1. A board of administration, composed of the commandant, the second in command, the director of studies, two professors, two captains, and two members of the administrative staff. This board has the superintendence of all the financial business and all the minutiae of the internal administration of the school.
2. A board of discipline, consisting of the second in command, the director, two professors, three captains (of the school,) and two captains of the army, chosen from former pupils. The duty of this board is to decide upon cases of misconduct.
3. A board of instruction, whose members are, the commandant, the second in command, the director, the examiners of students, and the professors; and whose chief duty is to make recommendations relating to ameliorations in the studies, the programmes of admission and of instruction in the school, to—
4. A board of improvement, charged with the general control of the studies, formed of—
The Commandant, as President.
The Second in Command.
The Director of Studies.
Two Delegates from the Department of Public Works.
One Delegate from the Naval Department.
One Delegate from the Home Department.
Three Delegates from the War Department.
Two Delegates from the Academy of Sciences.
Two Examiners of Students.
Three Professors of the School.
III. Conditions and Examinations for Admission
The entrance examination is held yearly in August; the most important conditions for admission to it are always inserted in the Moniteur early in the year, and are—
1st. All candidates must be bachelors of science.
2nd. All candidates (unless they have served in the army) must have been as much as sixteen and not more than twenty years old on the 1st of January preceding.
3rd. Privates and non-commissioned officers of the army must be above twenty and under twenty-five years of age; must have served two years, and have certificates of good conduct.
4th. Candidates who propose to claim pecuniary assistance (a bourse or demi-bourse) must present formal proofs of their need of it.
The subjects of the entrance examination are the following:—
Arithmetic, including Vulgar and Decimal Fractions, Weights and Measures, Involution and Evolution; Simple Interest.
Geometry of Planes and Solids; application of Geometry to Surveying; Properties of Spherical Triangles.
Algebra, including Quadratic Equations with one unknown quantity, Series and Progressions in general; Binomial Theorem and its applications; Logarithms and their use; on Derived Functions; on the Theory of Equations; on Differences; application of the Theory of Differences to the Numerical Solution of Equations.
Plane and Spherical Trigonometry; Solution of Triangles; application of Trigonometry to Surveying.
Analytical Geometry, including Geometry of two dimensions; Co-ordinates; Equations of the first and second degree, with two variables; Tangents and Asymptotes; on the Ellipse, Hyperbola, and Parabola; Polar Co-ordinates; Curved Lines in general.
Geometry of three dimensions, including the Theory of Projections; Co-ordinates; the Right Line and Plane; Surfaces of the second degree; Conical and Cylindrical Surfaces.
Descriptive Geometry; Problems relative to a Point, Right Line and Plane; Tangent Planes; Inter of Surfaces.
Mechanics; on the Movement of a Point considered geometrically; on the Effect of Forces applied to points and bodies at rest and moving; on the Mechanical Powers.
Natural Philosophy, including the Equilibrium of Liquids and Gasses; Heat; Electricity; Magnetism; Galvanism; Electro-magnetism and Light; Cosmography.
Chemistry, the Elements; French; German; Drawing, and (optionally) Latin.
This examination is partly written and partly oral. It is not public, but conducted in the following manner:—
Five examiners are appointed by the minister of war to examine the candidates at Paris, and at the several towns named for the purpose throughout France.
Two of these examiners conduct what may be called a preliminary examination (du premier degré,) and the other three a second examination (du second degré.) The preliminary examiners precede by a few days in their journey through France those who conduct the second examination. The written compositions come before either.
The preliminary examination (du premier degré) is made solely for the purpose of ascertaining whether the candidates possess sufficient knowledge to warrant their being admitted to the second examination; and the second examination serves, in conjunction with the written compositions, for their classification in the order of merit.
Prior to the examination, each candidate is called upon to give in certain written sheets containing calculations, sketches, plans and drawings, executed by him at school during the year, certified and dated by the professor under whom he has studied. Care is taken to ascertain whether these are the pupils’ own work, and any deception in this matter, if discovered, excludes at once from the competition of the school.
This done, the candidates are required to reply in writing to written or printed questions, and to write out French and German exercises; great care being taken to prevent copying. This written examination occupies about twenty-four hours during three and a half separate days, as shown in the following table. It usually takes place in the presence of certain official authorities, the examiners not being present.
| First Sitting. | |
| Hours. | |
| Arithmetic, | 1 |
| Geometry, | 1 |
| Latin, | 1 |
| 3 | |
| Second Sitting. | |
| Algebra, | 1 |
| History, geography, and French, | 3 |
| 4 | |
| Third Sitting. | |
| Descriptive geometry, and diagram, or sketch, | 4 |
| Fourth Sitting. | |
| Mechanics, | 1 |
| Physics, chemistry, and cosmography, | 2 |
| 3 | |
| Fifth Sitting. | |
| Applied analysis, | 1½ |
| German exercise, | 1½ |
| 3 | |
| Sixth Sitting. | |
| Solution of a triangle by logarithms | 3 |
| Seventh Sitting. | |
| Drawing | 4 |
| Total | 24 |
Next each candidate is examined orally for three-quarters of an hour, on two successive days, by each of the two examiners separately, and each examiner makes a note of the admissibility or non-admissibility of the candidate.
At the close of this oral examination, the notes relating to the various candidates are compared, and if the examiners differ as to the admissibility of any candidate, he is recalled, further orally examined, and his written exercises carefully referred to, both examiners being present. A final decision is then made.
The preliminary examiners then supply the others with a list of the candidates who are entitled to be admitted to the second oral examination. On this occasion each candidate is separately examined for one hour and a half by each examiner, but care is taken that in all the principal subjects of study the candidate is examined by at least two out of the three examiners.
Each examiner records his opinion of the merits of every candidate in replying, orally and in writing, by awarding him a credit varying between O and 20, the highest number indicating a very superior result.
This scale of merit is employed to express the value of the oral replies, written answers, or drawings. It has the following signification, and appears to be generally in use in the French military schools:—
| 20 | denotes perfect. |
| 19 18 | denotes very good. |
| 17 16 15 | denotes good. |
| 14 13 12 | denotes passable. |
| 11 10 9 | denotes middling. |
| 8 7 6 | denotes bad. |
| 5 4 3 | denotes very bad. |
| 2 1 | denotes almost nothing. |
| 0 | denotes nothing |
Considerable latitude is granted to the examiner engaged in deciding upon the amount of credit to be allowed to the student, for the manner in which he replies to the various questions. He is expected to bear in mind the temperament of the candidate, his confidence or timidity, as well as the difficulty of the questions, when judging of the quality of the reply, more value being given for an imperfect answer to a difficult question than for a more perfect reply to an easy one.
The reports of the examiners, together with the various documents belonging to each candidate, are sent from each town to the minister at war, who transmits them to the commandant of the Polytechnic School to make out a classified list.
Very different value of course is attached to the importance of some of the subjects, when compared with others; and the measure of the importance is represented in French examinations by what are termed co-efficients of influence, varying for the several subjects of study and kind of examination. The particular co-efficients of influence for each subject in these written and oral examinations, are as follows:—
| Co-efficients of Influence. | |||
| Oral examination— | analytical mathematics, | 20 | |
| Oral examination— | geometrical ditto, | 14 | 52 |
| Oral examination— | physics and mechanics, | 16 | |
| Oral examination— | German language, | 2 | |
| Written compositions on | mathematical subjects, | 5 | |
| Written compositions on | descriptive geometry, drawing, and description, | 5 | 34 |
| Written compositions on | logarithmic calculations of a triangle, | 2 | |
| Written compositions on | mechanics, | 2 | |
| Written compositions on | physics or chemistry, | 4 | |
| German exercise, | 1 | ||
| French composition, | 5 | ||
| Latin translation, | 5 | ||
| Copy of a drawing, | 5 | ||
| Total, | 86 | ||
In order to make out the above mentioned classified list, the respective credits awarded by the examiners to each candidate are multiplied by the co-efficients representing the weight or importance attached to each subject; and the sum of their products furnishes a numerical result, representing the degree of merit of each candidate.
A comparison of these numerical results is then made, and a general list of all the candidates is arranged in order of merit.
This list, and the whole of the documents from which it has been drawn up, are then submitted to a jury composed of the Commandant of the School.
The Second in Command.
The Director of Studies.
Two Members of the Board of Improvement.
The Five Examiners.
It is the special business of this jury carefully to scrutinize the whole of the candidates’ documents, drawings, &c., and they further take care that a failure in any one branch of study is duly noted, as such failure is a sufficient reason for the exclusion of the candidate from the general list.
As soon as this general list has been thoroughly verified, it is submitted to the minister of war, who is empowered to add one-tenth to the number actually required for the public services; and thus it may happen that one-tenth of the pupils may annually be disappointed.
IV. The School Buildings and the Course and Method of Study
A brief description of the buildings may be a suitable introduction to an account of the studies that are pursued, and the life that is led in them.
The Polytechnic School stands near the Pantheon, and consists of two main buildings, one for the official rooms and the residence of the commandant and director of studies, the other, and larger one, for the pupils. Detached buildings contain the chemical lecture room and laboratory, the laboratory of natural philosophy, the library, fencing and billiard rooms.
The basement floor of the larger building contains the kitchen and refectories. On the first floor, are the two amphitheaters or great lecture rooms, assigned respectively to the pupils of the two years or divisions, in which the ordinary lectures are given. The rooms are large and well arranged; the seats fixed, the students’ names attached to them. The students are admitted by doors behind the upper tier of seats; at the foot of all is a platform for the professor, with a blackboard facing his audience, and with sufficient room for a pupil to stand and work questions beside him. Room also is provided for one of the captains, inspectors of studies, whose duty it is to be present, for the director of studies, whose occasional presence is expected, and for the assistant teachers or répétiteurs, who in the first year of their appointment are called upon to attend the course upon which they will have to give their subsequent questions and explanations. On this floor are also the museums, or repositories of models, instruments, machines, &c., needed for use in the amphitheaters, or elsewhere. The museum provided for the lecturer on Physics (or Natural Philosophy) appeared in particular to be well supplied.
The whole of the second floor is taken up with what are called the salles d’interrogation, a long series of small cabinets or studies, plainly furnished with six or eight stools and a table, devoted to the interrogations particulières, which will presently be described.
The third floor contains the halls of study, salles d’étude, or studying rooms, in which the greater part of the student’s time during the day is passed—where he studies, draws, keeps his papers and instruments, writes his exercises, and prepares his lectures. These are small chambers, containing eight or, exceptionally, eleven occupants. A double desk runs down the middle from the window to the door, with a little shelf and drawers for each student. There is a blackboard for the common use, and various objects are furnished through the senior student, the sergeant, a selected pupil, more advanced than the rest, who is placed in charge of the room, and is responsible for whatever is handed in for the use of the students. He collects the exercises, and generally gives a great deal of assistance to the less proficient. “When I was sergeant,” said an old pupil, “I was always at the board.” The spirit of camaraderie, said to exist so strongly among the Polytechnic students, displays itself in this particular form very beneficially. Young men of all classes work heartily and zealously together in the salles d’étude, and no feeling of rivalry prevents them from assisting one another. The sergeant does not, however, appear to exercise any authority in the way of keeping discipline.
These chambers for study are arranged on each side of a long corridor which runs through the whole length of the building, those of the juniors being separated from those of the seniors by a central chamber or compartment, the cabinet de service, where the officers charged with the discipline are posted, and from hence pass up and down the corridor, looking in through the glass doors and seeing that no interruption to order takes place.
The fourth story is that of the dormitories, airy rooms, with twelve beds in each. These rooms are arranged as below, along the two sides of a corridor, and divided in the same manner into the senior and junior side. A non-commissioned officer is lodged at each end of the corridor to see that order is kept.
Such is the building into which at the beginning of November the successful candidates from the Lycées and the Ecoles préparatoires are introduced, in age resembling the pupils whom the highest classes of English public schools send annually to the universities, and in number equal perhaps to the new under-graduates at one of the largest colleges at Cambridge. There is not, however, in other points much that is common, least of all in the methods and habits of study we are about to describe. This will be best understood by a summary of a day’s work.
The students are summoned to rise at half-past five, have to answer the roll-call at six, from six to eight are to occupy themselves in study, and at eight they go to breakfast. On any morning except Wednesday, at half-past eight, we should find the whole of the new admission assembled in an amphitheater, permanent seats in which are assigned to them by lot, and thus placed they receive a lecture from a professor, rough notes of which they are expected to take while it goes on. The first half hour of the hour and a half assigned to each lecture is occupied with questions put by the professor relating to the previous lecture. A name is drawn by lot, the student on whom the lot falls is called up to the blackboard at which the professor stands, and is required to work a problem and answer questions. The lecture concluded, the pupils are conducted to the salles d’étude, which have just been described, where they are to study. Here for one hour they devote themselves to completing and writing out in full the notes of the lecture they have just heard. The professor and his assistants, the répétiteurs, are expected to follow and make a circuit through the corridors, to give an opportunity to ask for information on any difficult points in the lecture. A lithographed summary of the substance of the lecture, extending perhaps to two octavo pages, is also furnished to each studying room for the use of its pupils.
The lecture, as we have said, commences at half-past eight o’clock; it lasts an hour and a half; the hour of writing up the notes brings us to eleven. The young men are now relieved by a change of occupation, and employ themselves (still in their places in the rooms of study) at drawing. A certain number, detached from the rest, are sent to the physical and chemical laboratories. The rotation is such as to admit each student once a month to two or three hours’ work at a furnace for chemistry, and once in two months to make experiments in electricity, or other similar subjects. In this way, either at their drawing or in the laboratories, they spend three hours, and at two o’clock go to their dinner in the refectories below, and after dinner are free to amuse themselves in the court-yard, the library, the fencing and the billiard rooms, till five. At five they return to the studying rooms, and for two hours, on Mondays and Fridays, they may employ themselves on any work they please (étude libre;) on Tuesday there is a lecture in French literature, and on Thursday in German; at seven o’clock they commence a lesson, which lasts till nine, in landscape and figure drawing, or they do exercises in French writing or in German; at nine they go down to supper; at half-past nine they have to answer to a roll-call in their bedrooms, and at ten all the lights are put out.
Wednesday is a half-holiday, and the pupils are allowed to leave the school after two o’clock, and be absent till ten at night. The morning is occupied either in study, at the pleasure of the students, or in set exercises till eleven, when there is a lecture of one hour and a half, followed, as usual, by an hour of special study on the subject of the lecture. On Sunday they are allowed to be absent almost the whole day till ten P.M. There is no chapel, and apparently no common religious observance of any kind in the school.
Such is a general sketch of the ordinary employment of the day; a couple of hours of preparatory study before breakfast, a lecture on the differential calculus, on descriptive geometry, on chemistry, or natural philosophy, followed by an hour’s work at notes; scientific drawing till dinner; recreation; and general study, or some lighter lecture in the evening. Were we merely to count the hours, we should find a result of eleven or eleven and a half hours of work for every day but Wednesday, and of seven and a half hours for that day. It is to be presumed, however, that though absolute idleness, sleeping, or reading any book not authorized for purposes of study, is strictly prohibited, and when detected, punished, nevertheless the strain on the attention during the hours of drawing and the lectures of the evening is by no means extreme. Landscape and figure drawing, the lecture in French literature, and probably that in German, may fairly be regarded as something like recreation. Such, at least, was the account given us of the lectures on literary subjects, and it agrees with the indifference to literature which marks the school. Of wholesome out-of-door recreation, there certainly seems to be a considerable want. There is nothing either of the English love of games, or of the skillful athletic gymnastics of the German schools.
The method of teaching is peculiar. The plan by which a vast number of students are collected as auditors of professorial lectures is one pursued in many academical institutions, at the Scotch universities, and in Germany. Large classes attend the lectures in Greek, in Latin, and in mathematics at Glasgow; they listen to the professor’s explanations, take notes, are occasionally questioned, and do all the harder work in their private lodgings. Such a system of course deserves in the fullest sense the epithet of voluntary; a diligent student may make much of it; but there is nothing to compel an idle one to give any attention.
It seems to have been one especial object pursued in the Polytechnic to give to this plan of instruction, so lax in itself, the utmost possible stringency, and to accumulate upon it every attainable subsidiary appliance, every available safeguard against idleness. Questions are expressly put vivâ voce by the professor before his lecture; there is a subsequent hour of study devoted to the subject; there is the opportunity for explanation to individual students; the exaction of notes written out in full form; the professor also gives exercises to the students to write during their hours of general study, which he examines, and marks; general vivâ voce examinations (interrogations générales,) conducted by the professors and répétiteurs, follow the termination of each course of lectures; and lastly, one of the most important and peculiar parts of the method, we have what are called the interrogations particulières. After every five or six lectures in each subject, each student is called up for special questioning by one of the répétiteurs. The rooms in which these continual examinations are held have been described. They occupy one entire story of the building; each holds about six or eight pupils, with the répétiteurs. Every evening, except Wednesday, they are filled with these little classes, and busy with these close and personal questionings. A brief notice, at the utmost of twenty-four hours, is served upon the students who are thus to be called up. Generally, after they have had a certain number of lectures, they may expect that their time is at hand, but the precise hour of the summons can not be counted upon. The scheme is continually varied, and it defies, we are told, the efforts of the ablest young analysts to detect the law which it follows.
It will be seen at once that such a system, where, though nominally professorial, so little is left to the student’s own voluntary action, where the ordinary study and reading, as it is called in our English universities (here such an expression is unknown) is subjected to such unceasing superintendence and surveillance, and to so much careful assistance, requires an immense staff of teachers. At the Polytechnic, for a maximum of 350 pupils, a body of fifteen professors and twenty-four répétiteurs, are employed, all solely in actual instruction, and in no way burdened with any part of the charge of the discipline or the finance, or even with the great yearly examinations for the passage from the first to the second division, and for the entrance to the public services.
With a provision of one instructor to every eight students, it is probable that in England we should avoid any system of large classes, from the fear of the inferior pupils being unable to keep pace with the more advanced. We should have numerous small classes, and should endeavor, above all things, to obtain the advantage of equality of attainment in the pupils composing them.
The French, on the other hand, make it their first object to secure one able principal teacher in each subject, a professor whom they burden with very few lectures. And to meet the educational difficulty thus created, to keep the whole large class of listeners up to the prescribed point, they call in this numerous and busily employed corps of assistants to repeat, to go over the professor’s work afresh, to whip in, as it were, the stragglers and hurry up the loiterers. Certainly, one would think, a difficult task with a class of 170 freshmen in such work as the integral and differential calculus. It is one, however, in which they are aided by a stimulus which evidently acts most powerfully on the students of the Polytechnic School. During the two years of their stay, the prospect of their final admission to the public services can rarely be absent from the thoughts even of the least energetic and forethinking of the young men. Upon their place in the last class list will depend their fortune for life. A high position will secure them not only reputation, but the certainty of lucrative employment; will put it in their power to select which service they please, and in whichever they choose will secure them favorable notice. Let it be remembered that fifty-three of these one hundred and seventy are free scholars, born of parents too poor to pay 40l. a year for their instruction; to whom industry must be at all times a necessity, and industry during their two years at the Polytechnic the best conceivable expenditure, the most certainly remunerative investment of their pains and labor. The place on the final class list is obviously the prize for which this race of two years’ length has to be run. What is it determines that place? Not by any means a final struggle before the winning-post, but steady effort and diligence from first to last throughout the course. For the order of the class list is not solely determined by success in the examination after which it is drawn up, but by the result of previous trials and previous work during the whole stay at the school.
For, during the whole time, every written exercise set by the professor, every drawing, the result of every interrogation particulière by the répétiteurs, and of each general interrogation by the professors and répétiteurs, is carefully marked, and a credit placed according to the name of the student and reserved for his benefit, in the last general account. The marks obtained in the examination which closes the first year of study form a large element in this last calculation. It had been found that the work of the first year was often neglected: the evil was quickly remedied by this expedient. The student, it would seem, must feel that he is gaining or losing in his banking account, so to call it, by every day’s work; every portion of his studies will tell directly for or against him in the final competition, upon which so much depends.
Such is the powerful mechanism by which the French nation forces out of the mass of boys attending their ordinary schools the talent and the science which they need for their civil and military services. The efforts made for admission to this great scientific school of the public services, the struggle for the first places at the exit from it, must be more than enough, it is thought, to establish the habits of hard work, to accumulate the information and attainment, and almost to create the ability which the nation requires for the general good.
We may now follow the student through his course of two years’ study. The first year’s work may be mainly divided into three portions of unequal length; two of them of about four months each (with an additional fortnight of private study and examination,) are mainly given to hard lecturing, whilst the third portion of two months is devoted to private study and to the examinations.
In accordance with this arrangement of the year, the four hardest subjects are thus distributed. Analysis and descriptive geometry, the staple work of the school—its Latin, as M. de Barante called it—come in the first four months; there is then a pause for private study and a general examination in these two subjects (interrogations générales as distinct from the interrogations particulières of the répétiteurs.) This brings us to the middle of March. Analysis and geometry are then laid aside for the rest of the year, and for the next portion of four months the pupils work at mechanics and geodesy, private study and a general examination completing this course also. Important lectures on physics and chemistry run on during both these periods, and are similarly closed by private study and a general examination. The less telling evening classes of French literature and German end at the beginning of June, and landscape and figure drawing only last half the year. It may be observed also, that, as a general rule, there is on each day one, and only one, really difficult lecture. This is immediately preceded and followed by private study, but then comes something lighter, as a relief, such as drawing or work in the laboratories.
The chief feature in the third portion of the year is the complete break in the lectures for general private study (étude libre,) a month or six weeks before the closing examination at the end of the year. The immediate prospect of this prevents any undue relaxing of the work; and it is curious to observe here how private efforts and enforced system are combined together, for even the private efforts are thus systematized and directed. The closing examination of the first year begins on the 1st and ends on the 25th of September.
The total number of lectures in each branch of study, with the dates when they respectively commence and finish, and the period when the general examinations (interrogations générales) take place, are exhibited in the following tables, and we should add that the interval between the close of each course and the commencement of the chief yearly examination is devoted to free study.
TABLE FOR THE SECOND OR LOWER DIVISION, FOLLOWING THE FIRST YEAR’S COURSE OF STUDY.
NL No. of Lectures
E Annual Examination
| Subject of Study. | NL | Course of Lectures | General Examinations Interrogations Générales. | E | ||
|---|---|---|---|---|---|---|
| Commenced. | Finished. | Commenced. | Finished. | |||
| Analysis | 48 | 3rd Nov. | 25 Feb. | 13th March | 18th March | * |
| Mechanics & Machines | 40 | 21st March | 29th June | 24th July | 2nd August | |
| Descriptive Geometry | 38 | 3rd Nov. | 3rd March | 13th March | 18th March | |
| Physics | 34 | 2nd “ | 28th June | 10th July | 19th July | |
| Chemistry | 38 | 5th “ | 17th “ | 10th “ | 19th “ | |
| Geodesy | 35 | 20th March | 30th “ | 24th “ | 2nd August | |
| French Literature | 30 | 8th Nov. | 6th “ | |||
| German | 30 | 2nd “ | 15th “ | |||
| Figure and Landscape Drawing | 50 | 4th “ | 28th April | |||
| Total | 343 | |||||
* Begins on the 1st Sept., and ends on the 25th Sept.
The work of the second year is almost identical in its general plan with that of the first. A continuation of analysis with mechanics in place of descriptive geometry is the work of the first four months, then comes the private study and the interrogations générales, and then again, from the middle of March to the middle of July, work of a more professional character, stereotomy, the art of war and topography, forms the natural completion of the pupil’s studies. Chemistry and physics follow the same course as during the first year, and terminate with the private study and the general examination at the beginning of August. The evening lectures in French literature and German end about the middle of June, and those in figure and landscape drawing at the beginning of May. The last portion is again given to private study and the great Final Examination.
TABLE FOR THE FIRST OR UPPER DIVISION, FOLLOWING THE SECOND YEAR’S COURSE OF STUDY.
NL No. of Lectures
E Annual Examination
| Subject of Study. | NL | Course of Lectures | General Examinations Interrogations Générales. | E | ||
|---|---|---|---|---|---|---|
| Commenced. | Finished. | Commenced. | Finished. | |||
| Analysis | 32 | 11th Nov. | 3rd March | 13th March | 18th March | * |
| Mechanics and Machines | 42 | 10th “ | 2nd “ | 13th “ | 18th “ | |
| Stereotomy | 32 | 20th Mar. | 26th June | 10th July. | 19th July. | |
| Physics | 36 | 12 Nov. | 29th “ | 24th “ | 2nd Aug. | |
| Chemistry | 38 | 14th “ | 28th “ | 24th “ | 2nd “ | |
| Architecture and Construction | 40 | 10th “ | 8th “ | |||
| Military Art and Fortification | 20 | 21st Mar. | 21st “ | 10th “ | 19th July. | |
| Topography | 10 | 3rd Jan. | 21st “ | |||
| French literature | 30 | 11th Nov. | 9th “ | |||
| German | 30 | 14th “ | 19th “ | |||
| Figure and Landscape Drawing | 48 | 12th “ | 2nd May. | |||
| Total | 358 | |||||
* Begins on the 10th Sept. and ends on the 10th Oct.
V. The Examinations, Particularly That of the First Year and the Final One
We have now brought the pupil nearly to the end of his career, but must previously say a few words about his examinations, the chief epochs which mark his progress, and the last of which fixes his position almost for life. For this purpose it is necessary to recapitulate briefly what has been said in different places of the whole examinatorial system of the Polytechnic School.
1. All the professors require the students in their studying rooms, to answer questions in writing on the courses as they go through them: a different question is given to each student, and every third question is of such a nature as to involve a numerical example in the reply.
These questions are given in the proportion of one to about every four lectures, and the replies after being examined by the professor or répétiteur, are indorsed with a credit, varying from 0 to 20, and the paper is then given back to the student, to be produced at the close of the year.
2. Credits are assigned to the students for their ordinary manipulations in chemistry and physics, during the first year; and at the close of each year, for their manipulations, in chemistry alone, before the examiners.
3. The répétiteurs examine, (in the interrogations particulières,) every ten or fourteen days, from six to eight students during a sitting of two hours, on the subject of study lectured on since the previous examination of the same kind. All these students must continue present, and at the close the répétiteur assigns to each a previous examination of the same kind. All these students must continue present, and at the close the répétiteur assigns to each a credit entirely dependent on the manner in which each has replied. The professors and captains inspectors are occasionally present at these examinations, which are discontinued at certain periods according to the instructions of the director of studies.
4. At different intervals of time, from a fortnight to a month, as may happen, after the close of the course in each branch of study, general examinations (interrogations générales) are made by the professors and répétiteurs. From four to six students are examined together for at least two hours, and at the conclusion the professor makes known to the director of studies the credit he has granted to each student for the manner in which he has passed his examination.
Such may be called the minor or ordinary examinations. But there is an annual closing examination at the end of each year, which we will now describe. The first year’s annual examination commences on the 1st and ends on the 25th September. It is carried on by special examiners, (a different set from those who conduct the entrance examinations,) and not by the professors. These give to every student a credit between 0 and 20 in each branch of study, according to the manner in which he replies.
The following table shows the co-efficients of influence allowed to the different studies of the first year, subdivided also among the particular classes of examination to which the student has been subjected. The component parts of the co-efficients as well as the co-efficients themselves, slightly vary from year to year, dependent on the number of examinations:—
TABLE I.—FIRST YEAR’S COURSE OF STUDIES: SECOND DIVISION.
TC Total Co-efficients. (repeated)
WA Written Answers to Professors’ Questions.
ER Examinations by Répétiteurs. (Int. Part.)
GE General Examinations. (Int. Gen.)
Man. Manipulations.
O Ordinary.
Ex At Examination.
SN Sheets of notes on descriptive Geometry.
GD Graphical Representations and Drawing.
1st First Annual Examination.
| Nature of Study. | TC | Co-efficient of Influence awarded to | TC | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| WA | ER | GE | Man. | SN | GD | 1st | ||||
| O | Ex | |||||||||
| Analysis, | 56 | 9 | 10 | 9 | .. | .. | .. | .. | 28 | 56 |
| Mechanics, | 60 | 7 | 9 | 8 | .. | .. | .. | 14 | 22 | 60 |
| Descriptive Geometry, | 48 | .. | 7 | 7 | .. | .. | 4 | 12 | 18 | 48 |
| Geodesy, | 39 | 6 | 5 | 7 | .. | .. | .. | 3 | 18 | 39 |
| Physics, | 45 | 6 | 9 | 7 | 2 | .. | .. | .. | 21 | 45 |
| Chemistry, | 45 | 5 | 9 | 7 | 4 | 2 | .. | .. | 18 | 45 |
| French Literature, | 12 | 12 | .. | .. | .. | .. | .. | .. | .. | 12 |
| German Language, | 10 | 2 | 3 | .. | .. | .. | .. | .. | 5 | 10 |
| Drawing, | 10 | .. | .. | .. | .. | .. | .. | 10 | .. | 10 |
| Shading & Tinting Plans, | .. | .. | .. | .. | .. | .. | .. | .. | .. | 3 |
At the conclusion of this examination the director of studies prepares a statement for each student, exhibiting the credits he has obtained at each of the preceding examinations in each subject, multiplied by the co-efficient of influence, and the sum of the products represents the numerical account of the student’s credit in each branch of study.
As the process is somewhat intricate, we append the following example, to show the nature of the calculation performed, in order to ascertain the amount of credits due to each student:—
REPORT OF THE CREDITS GAINED IN THE FIRST YEAR’S COURSE OF STUDY BY M. N., STUDENT AT THE POLYTECHNIC SCHOOL.
CI Co-efficient of Influence (twice)
Cr Credit obtained by the Student.
SP Sum of Products.
MC Mean Credit in each Subject of the Course.
| Subject of Examination | CI | Nature of Examination or Proof. | Cr | CI | Product | SP | MC |
|---|---|---|---|---|---|---|---|
| Analysis, | 56 | Written answers to Professors’ questions | 17.16 | 9 | 154.44 | ||
| Examinations by répétiteurs (interrogations particulières) | 15.47 | 10 | 154.70 | 845.81 | 15.00 | ||
| General Examination (interrogations générales) | 13.71 | 9 | 123.39 | ||||
| Annual Examination | 14.75 | 28 | 413.28 | ||||
| Mechanics | 60 | Written answers to Professors’ questions | 13.45 | 7 | 94.15 | ||
| Examinations by répétiteurs | 12.72 | 9 | 114.48 | 664.13 | 11.07 | ||
| General Examination | 11.37 | 8 | 90.96 | ||||
| Graphical representations and drawing | 5.61 | 14 | 78.54 | ||||
| Annual Examination | 13.00 | 22 | 286.00 | ||||
| Descriptive Geometry | 48 | Examinations by répétiteurs | 17.15 | 7 | 120.05 | ||
| General Examination | 11.72 | 7 | 82.04 | 633.15 | 13.19 | ||
| Sheets of notes | 12.45 | 4 | 49.80 | ||||
| Graphical representation and drawing | 11.88 | 12 | 142.76 | ||||
| Annual Examination | 13.25 | 18 | 238.50 | ||||
| Geodesy | 39 | Written answers to Professors’ questions | 9.16 | 6 | 54.96 | ||
| Examinations by répétiteurs | 7.85 | 5 | 39.25 | 229.01 | 5.87 | ||
| General Examination | 5.74 | 7 | 40.18 | ||||
| Graphical representation and drawing | 4.36 | 3 | 13.08 | ||||
| Annual Examination | 4.53 | 1 | 81.54 | ||||
| Physics | 45 | Written answers to Professors’ questions | 2.76 | 6 | 13.56 | ||
| Examinations by répétiteurs | 3.54 | 9 | 31.86 | 112.21 | 2.49 | ||
| General Examination | 5.74 | 7 | 40.18 | ||||
| Ordinary manipulation | 1.55 | 2 | 3.10 | ||||
| Annual Examination | 1.84 | 21 | 38.84 | ||||
| Chemistry | 45 | Written answers to Professors’ questions | 2.46 | 5 | 12.30 | ||
| Examinations by répétiteurs | 3.25 | 9 | 29.95 | 131.16 | 2.91 | ||
| General Examination | 2.47 | 7 | 17.29 | ||||
| Ordinary manipulation | 2.26 | 4 | 9.04 | ||||
| Manipulation at Exam | 1.58 | 2 | 3.16 | ||||
| Annual Examination | 3.34 | 18 | 60.12 | ||||
| French Literature | 12 | Written answers to Professors’ questions | 5.46 | 12 | 67.68 | 67.68 | 5.64 |
| German Language | 10 | Written answers to Professors’ questions | 6.57 | 2 | 13.14 | ||
| Examinations by répétiteurs | 4.86 | 3 | 14.58 | 55.92 | 5.59 | ||
| Annual Examination | 5.64 | 5 | 28.20 | ||||
| Drawing | 10 | Graphical representation and drawing | 4.36 | 10 | 43.60 | 43.60 | 4.36 |
| Shading and Tinting Plans | 3 | Graphical representation and drawing | 3.86 | 3 | 11.58 | 11.58 | 3.86 |
| Sum | 10)70.07 | ||||||
| General Mean Credit, = | (7.00) | ||||||
It is important to remark that any student whose mean credit, given in the eighth column of the preceding table, in any branch of study does not exceed three, or whose general mean credit for the whole of the studies being the arithmetical mean of all the values recorded in the eighth column, and given at the bottom in the example, does not exceed six, is considered to possess an insufficient amount of instruction to warrant his being permitted to pass into the first division for the second year’s course. He is accordingly excluded from the school, unless he has been prevented from pursuing his studies by illness, in which case, when the facts are thoroughly established, he will be allowed a second year’s study in the second division, comprising the first year’s course of study.
We now pass to the second annual or great final examination for admission to the public services, remarking only that in the interrogations générales of the second year the principal subjects of both years are included.
The final examinations for admission into the public service commence about the 10th September, and last about one month. They are conducted by the same examiners who examined at the close of the first year. These are five in number, and appointed by the minister of war. One of these takes analysis; a second, mechanics; a third, descriptive geometry and geodesy; the fourth, physics; and the fifth, chemistry.
The examination in military art and topography is conducted by a captain of engineers specially appointed for the purpose; and in the same manner the examination in German is carried on by a professor, usually a civilian, specially but not permanently appointed.
The questions are oral, and extend over the whole course of study pursued during the two years. Each student is taken separately for one hour and a quarter on different days by each of the five examiners; each examiner examines about eight students daily.
A table, very similar to that already given, is prepared under the superintendence of the Director of studies for every student, to ascertain the numerical amount of his credits in each branch of study, the co-efficients of influences for the particular subject of study and nature of examination being extracted from a table similar to that in page 80, and when these tables have all been completed, a general list of all the students is made out, arranged in the order of their merits.
Formerly, conduct was permitted to exercise some slight influence on their position, but that is no longer the case.
The same regulations exist, as regards the minimum amount of credit that will entitle the students to enter into the public service, as have already been stated above in reference to the passage from the first to the second year’s course of study.
TABLE II. SECOND YEAR’S COURSE OF STUDY: FIRST DIVISION.
RP Result of previous Year’s Examination.
WA Written Answers to Professors’ Questions.
ER Examinations by Répétiteurs. (Int. Part.)
GE General Examinations. (Int. Gen.)
Man. Manipulations.
O Ordinary.
Ex At Examination.
SN Sketches and Notes in Architecture
GD Graphical Representations and Drawing
EA Examination in Architecture
2d 2d Annual or Final Examination.
TC Total Co-Efficients.
| TC | Co-efficient of Influence awarded to | TC | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| RP | WA | ER | GE | Man. | SN | GD | 2d | ||||
| O | Ex | ||||||||||
| Analysis, | 28 | 8 | 10 | 9 | .. | .. | .. | .. | .. | 26 | 81 |
| Mechanics, | 25 | 8 | 12 | 9 | .. | .. | .. | 10 | .. | 28 | 92 |
| Descriptive Geometry, | 36 | .. | .. | .. | .. | .. | .. | .. | .. | .. | 36 |
| Geodesy, | .. | 6 | 5 | 7 | .. | .. | .. | 1 | .. | 18 | 37 |
| Physics, | 23 | 5 | 10 | 8 | .. | .. | .. | .. | .. | 22 | 68 |
| Chemistry, | 20 | 5 | 10 | 8 | 4 | 2 | .. | .. | .. | 19 | 68 |
| Architecture, | .. | .. | .. | .. | .. | .. | 12 | 14 | 10 | .. | 36 |
| Military Art and Tophography, | .. | .. | 3 | 5 | .. | .. | .. | 9 | .. | 8 | 25 |
| French Literature, | 6 | 12 | .. | .. | .. | .. | .. | .. | .. | .. | 18 |
| German, | 5 | 2 | 3 | .. | .. | .. | .. | .. | .. | .. | 15 |
| Drawing, | 5 | .. | .. | .. | .. | .. | .. | 10 | .. | .. | 15 |
| Shading and Tinting, | 2 | .. | .. | .. | .. | .. | .. | 3 | .. | .. | 5 |
From the preceding tables and explanations, it will be apparent that, as the whole of the students for each year are compelled to follow precisely the same course of study, the system of professorial instruction, combined with the constant tutelage and supervision exercised by the répétiteurs, and the examinations (interrogations particulières) of the répétiteurs, at short intervals of time, have for their principal object the keeping alive in the minds of the students the information which has been communicated to them. As a stimulus to continuous and unceasing exertion, it will be seen by an inspection of the tables of the co-efficients of influence, that the manner in which the students acquit themselves from day to day, and from week to week, is made an element, and a very important one, in determining their final position in the list arranged according to merit, exceeding as it does in most instances the influence exerted on their classification by their final examination at the close of each year. This principle thus recognizes not only their knowledge at the end of each year, but also the manner in which they have proved it to the professors and répétiteurs in the course of the year; and with reference to the second year’s study, the final result of the first year’s classification exercises an influence amounting to about one-third of the whole, in the final classification at the end of the second year.
It follows also, that as the examinations at the end of each year are made by examiners, otherwise unconnected with the school, and not by the professors belonging to it, the positions of the students in the classified list is partly dependent on the judgment of the professors with whom they are constantly in communication, and partly on the public examiners, whom they meet only in the examination rooms.10
The examiners of the students are not frequently changed, and practically the same may be said of the examiners for admission.
PR By Professors and Répétiteurs.
Ex By Examiners.
Y1 By the results of the first Year’s Examination.
CL In the Classified List at the end of 2nd year.
| Subjects of Study. | Per-centage of influence exercised on the position of the Students. | ||||||
|---|---|---|---|---|---|---|---|
| During the 1st Year. | During the 2nd Year. | CL | |||||
| PR | Ex | Y1 | PR | Ex | PR | Ex | |
| Analysis, | 50.0 | 50.0 | 34.5 | 32.5 | 33.0 | 49.75 | 50.25 |
| Mechanics, | 63.2 | 36.7 | 27.2 | 42.4 | 30.4 | 59.6 | 40.40 |
| Descriptive Geometry, | 62.5 | 37.5 | 100.0 | 0.0 | 0.0 | 62.5 | 37.5 |
| Geodesy, | 53.8 | 46.2 | 0.0 | 51.4 | 48.6 | * 51.4 | 48.6 |
| Physics, | 53.3 | 46.7 | 33.8 | 33.8 | 32.4 | 51.8 | 48.2 |
| Chemistry, | 60.0 | 40.0 | 29.4 | 43.2 | 27.4 | 60.8 | 39.2 |
| Architecture, | .. | .. | 0.0 | 100.0 | 0.0 | 100.0 | 100.0 |
| Military Art and Topography, | .. | .. | 0.0 | 68.0 | 32.0 | 68.0 | 32.0 |
| French Literature, | 100.0 | 0.0 | 33.3 | 66.7 | 0.0 | 100.0 | 0.0 |
| German Language, | 100.0 | 0.0 | 33.3 | 33.3 | 33.4 | 66.7 | 33.3 |
| Drawing | 100.0 | 0.0 | 33.3 | 66.7 | 0.0 | 100.0 | 0.0 |
| Shading and Tinting Plans, | 100.0 | 0.0 | 40.0 | 60.0 | 0.0 | 100.0 | 0.0 |
| * | |||||||
* When taught in the 2nd year
The students at the head of the list have generally since the wars of the first Empire entered into the civil rather than into the military services, the former being much better remunerated.
The services are usually selected by preference, nearly in the following order:—
| The Roads and Bridges (Ponts et chaussées) and Mines (Mines,) | very nearly on an equality. |
| Powder and Saltpetre (Poudres et Salpêtres.) | |
| Naval Architects (Génie maritime.) | |
| Engineers (Génie militaire.) | |
| The Artillery (Artillerie de terre.) and the Staff Corps (Etat Major,) | very nearly on an equality. |
| The Hydographical Corps (Ingénieurs Hydrographes.) | |
| Tobacco Department (Administration des Tabacs.) | |
| Telegraph Department (Lignes Télégraphiques.) | |
| Navy (Marine.) | |
| Marine Artillery (Artillerie de mer.) | |
Such, at least, is the result of a comparison of the selections made by the students during eight different years.
This preference of the civil to the military services has been the subject of frequent complaints on the part of the military authorities to the minister of war.
No steps have, however, been taken by the French government to prevent the free choice of a profession being granted to the most successful students.
We have now followed the student at the Polytechnic to the end of his school career. He is then to pass to his particular School of Application, in which (as the name implies) he is taught to apply his science to practice. It is difficult to state precisely the amount of such science which the highest pupils may be thought to possess on leaving; the best idea of it will be gained by reference to the programmes of the most important of the lectures. It is also needless to dwell again on the main features of the school—the emulation called forth, the minute method, the great prizes offered for sustained labor. We must, however, make some remarks on these points before concluding our account, so far as they bear on the subject of military education.
VI. General Remarks
1. Keeping out of sight for the moment some defects both in the principles and details of the education of this school, the method of teaching adopted seems to us excellent, and worthy of careful study. In this remark we allude principally to the skillful combination of two methods which have been generally thought incompatible; for it unites the well-prepared lecture of a German professor, with the close personal questioning of a first-rate English school or college lecture. But besides this, its whole system is admirably adapted for the class of pupils it educates.
These pupils are generally not of the wealthy classes; they are able, and struggling for a position in life. On all these grounds their own assistance in the work may be calculated upon. Yet they are not left to themselves to make the most of their professors’ lectures. The aid of répétiteurs, even more valuable in its constant “prudent interrogations,” than in the explanations afforded, is joined to the stimulus given by marking every step of proficiency, and by making all tell on the last general account. But though the routine and method of the school are so elaborate, play is given to the individual freedom of the pupils in their private work, and this is managed so skillfully that the private work is made immediately to precede the final examination, on which mainly depends the pupil’s place for life. Thus from first to last they are carried on by their system without being cramped by it; every circumstance favorable to study is made the most of; rigorous habit, mental readiness, the power of working with others, and the power of working for themselves, the ambition for immediate and permanent success, all the objects and all the methods which students ever have in view, support and stimulate those of the Polytechnic in their two years’ career.
2. The mainspring, however, of the school’s energy is the competition amongst the pupils themselves, and this could hardly exist without the great prizes offered to the successful. This advantage, added to the general impulse of the early days of the Empire, has no doubt powerfully contributed to the great position of the school. It has made it a kind of university of the élite mathematicians, and as in England young men look to the prizes of the universities, and the professions to which they lead, as their best opening in life, so in France, ever since the first revolution, the corresponding class has inclined to the active and chiefly military career which is offered by the great competitive school of the country.
3. A preparatory school of this remarkable character can not but exercise a very powerful influence over those three-fourths of its pupils who leave it to enter the army. The obvious question is whether the attempt is not made to teach more than is either necessary or desirable for military purposes, and to this suspicion may be added the fact that the civil prizes being more in request than the military, many of those who enter the army do so in the first instance reluctantly, and that the pupils at the bottom of the list appear to be often such marked failures as to imply either great superficiality or premature exhaustion.
4. In studying the Polytechnic School we have had these points constantly brought before us, and feeling the difficulty of discussing them fully, we beg to invite attention to the evidence sent us in reply to some questions which we addressed on the subject to some distinguished scientific officers and civilians connected with the school. We will give briefly the result of our own inquiries.
5. The complaint of General Paixhans has been quoted. He urges that a considerable proportion of the army pupils are mere queues de promotion, and quite insufficient to form le corps et surtout la tête of troops d’élite.
Other not inconsistent complaints we heard ourselves, of the mental exhaustion and the excessively abstract tendencies of many of the military pupils of the school.
6. Such are the complaints. There is certainly reason to think that, with regard to the twenty or thirty lowest pupils on the list, those of General Paixhans are well founded. These are the breaks down, and we are at first surprised that, entering as they must do,11 with high attainments, they should fall so low as the marks in the tables (with which we are most liberally supplied) prove to be the case.
At the same time, we believe that no teaching ever has provided or will provide against many failures out of one hundred and seventy pupils, even among those who promised well at first: and if the standard of the majority of pupils is high at the Polytechnique, and the point reached by the first few very high, it is no reproach that the descent amongst the last few should be very rapid.
With regard to the assertion, that the teaching is excessive and leads too much to abstract pursuits for soldiers, it may be partially true. Perhaps the general passion for science has led to an overstrained teaching for the army, even for its scientific corps; and yet would it be allowed by officers of the highest scientific ability, either in the French or the English army, that less science is required for the greatest emergencies of military than for those of civil engineering, or for the theory of projectiles than for working the department of saltpetre?
It may, however, be true that an attempt is made at the Polytechnic to exact from all attainments which can only be reached by a few.
7. With this deduction, we must express our opinion strongly in favor of the influence of the Polytechnic on the French army. We admit that in some instances pupils who have failed in their attempt at civil prizes enter the army unwillingly, but they are generally soon penetrated with its esprit de corps, and they carry into it talent which it would not otherwise have obtained. Cases of overwork no doubt occur, as in the early training for every profession, but (following the evidence we have received) we have no reason to think them so numerous as to balance the advantage of vigorous, thoughtful study directed early towards a profession which, however practical, is eminently benefited by it. “It can not be said,” was the verdict of one well fitted to express an opinion, “that there is too much science in the French army.”
8. Assuming, however, the value of the scientific results produced in the French army by the Polytechnic, it by no means follows that a similar institution would be desirable in another country. Without much discussion it may be safely said that the whole history and nature of the institution—the offspring of a national passion for system and of revolutionary excitement—make it thoroughly peculiar to France.
9. Some obvious defects must be noticed. The curious rule of forbidding the use of all books whatever is a very exaggerated attempt to make the pupil to rely entirely on the professors and répétiteurs. The exclusive practice of oral examination also seems to us a defect. Certainly every examination should give a pupil an opportunity of showing such valuable qualities as readiness and power of expression; but an examination solely oral appears to us an uncertain test of depth or accuracy of knowledge; and however impartial or practiced an examiner may be, it is impossible that questions put orally can present exactly the same amount of difficulty, and so be equally fair, to the several competitors.
At the same time, although in all great competing examinations the chief part of the work (in our opinion) should be written, the constant oral cross-questioning of the minor examinations at the Polytechnic, appeared to be one of the most stimulating and effective parts of their system,
10. A more serious objection than any we have named lies against the exclusive use of mathematical and scientific training, to the neglect of all other, as almost the only instrument of education. The spirit of the school, as shown especially by its entrance examinations, is opposed to any literary study. This is a peculiar evil in forming characters for a liberal profession like the army. Such a plan may indeed produce striking results, if the sole object is to create distinguished mathematicians, though even then the acuteness in one direction is often accompanied by an unbalanced and extravagant judgment in another. But a great school should form the whole and not merely a part of the man; and as doing this, as strengthening the whole mind, instead of forcing on one or two of its faculties—as giving, in a word, what is justly called a liberal education—we are persuaded that the system of cultivating the taste for historical and other similar studies, as well as for mere science, is based on a sounder principle than that which has produced the brilliant results of the Polytechnic.
11. It may be added, in connection with the above remark, that as the entrance examination at the Polytechnic influences extensively the teaching of the great French schools, and is itself almost solely mathematical, it tends to diffuse a narrow and exclusive pursuit of science, which is very alien from the spirit of English teaching.
12. We may sum up our remarks on the Polytechnic School thus:—
Regarded simply as a great Mathematical and Scientific School, its results in producing eminent men of science have been extraordinary. It has been the great (and a truly great) Mathematical University of France.
Regarded again as a Preparatory School for the public works, it has given a very high scientific education to civil engineers, whose scientific education in other countries (and amongst ourselves) is believed to be much slighter and more accidental.
Regarded as a school for the scientific corps of the army, its peculiar mode of uniting in one course of competition candidates for civil and military services, has probably raised scientific thought to a higher point in the French than in any other army.
Regarded as a system of teaching, the method it pursues in developing the talents of its pupils appears to us the best we have ever studied.
It is in its studies and some of its main principles that the example of the Polytechnic School may be of most value. In forming or improving any military school, we can not shut our eyes to the successful working at the Polytechnic of the principle, which it was the first of all schools to initiate, the making great public prizes the reward and stimulus of the pupil’s exertions. We may observe how the state has here encouraged talent by bestowing so largely assistance upon all successful, but poor pupils, during their school career. We may derive some lessons from its method of teaching, though the attempt to imitate it might be unwise. Meanwhile, without emulating the long established scientific prestige of the Polytechnic, we have probably amongst ourselves abundant materials for a military scientific education, at least as sound as that given at this great School.
Note
In addition to the Schools of Application for Artillery and Engineers at Metz, and of Infantry and Cavalry at St. Cyr, of which a pretty full account will be given, the following Public Services are supplied by the Polytechnic School.
Gunpowder and Saltpetre.—(Poudres et Salpêtres.)
In France the manufacture of gunpowder is solely in the hands of the Government. The pupils of the Polytechnic who enter the gunpowder and saltpetre service, are sent in succession to different powder-mills and saltpetre refineries, so as to gain a thorough acquaintance with all the details of the manufacture.
On first entering the service they are named élèves des poudres. They afterwards rise successively to the rank of assistant-commissary, commissary of the third, of the second, and of the first class.
Navy.—(Marine.)
A small number of the pupils of the Polytechnic enter the Navy. They receive the rank of élève de première classe, from the date of their admission.
They are sent to the ports to serve afloat. After two years’ service they may be promoted to the rank of enseigne de vaisseau, on passing the necessary examinations, on the same terms precisely as the élèves de premiere classe of the Naval School.
Marine Artillery.—(Artillerie de la Marine.)
The French marine artillery differs from the English corps of the same name, in not serving afloat. Its duties are confined to the ports and to the colonies. It is governed by the same rules and ordinances as the artillery of the army.
The foundries of La Villeneuve, Rochefort, Ruelle, Névers, and Saint Gervais are under its direction.
The officers of the marine artillery are liable to be sent on board ship to study naval gunnery, so as to be in a position to report upon alterations or improvements in this science.
Naval Architects.—(Génie Maritime.)
The naval architects are charged with the construction and repair of vessels of war, and with the manufacture of all the machinery required in the ports and dockyards. The factories of Indret and La Chaussade are under their direction.
The pupils of the Polytechnic enter the corps of naval architects with the rank of élève du Génie Maritime. They are sent to the School of Application of Naval Architects at L’Orient. After two years’ instruction they undergo an examination, and, if successful, they are promoted to the rank of sub-architect of the third class, so far as vacancies admit. They may be advanced to the second class after a service of two years.
Hydrographers.—(Ingénieurs Hydrographes.)
The hydrographers are stationed at Paris. They are sent to the coast to make surveys, and the time so spent reckons as a campaign in determining their pension. On their return to Paris they are employed in the construction of maps and charts.
The hydrographers have the same rank and advantage as the naval architects.
On leaving the Polytechnic, the pupils enter the corps of hydrographers with the rank of élève hydrographe. After two years’ service, and one season employed on the coast, they become sub-hydrographers without further examination.
Roads and Bridges.—Government Civil Engineers.—(Fonts et Chaussées.)
The Polytechnic furnishes exclusively the pupils for the Government Civil Engineer Corps. On leaving the Polytechnic, the pupils enter the School of Application in Paris. The course of instruction here extends over a period of three years. It commences each year on the first of November, and lasts till the 1st of April. After the final examination, the pupils are arranged according to the results of the examination and the amount of work performed.
The pupils enter the college with the rank of élève de troisième classe. They rise successively to the second and to the first class, on making the requisite progress in their studies.
From the 1st of May to the 1st of November the élèves of the second and the third class are sent on duty into the provinces. The élèves of the first class who have completed their three years’ course of instruction, are employed in the duties of ordinary engineers, or are detached on special missions. In about three years after quitting the college, they may be appointed ordinary engineers of the second class.
The engineers of the Ponts et Chaussées prepare the projects and plans, and direct the execution of the works for the construction, preservation, and repair of high roads, and of the bridges and other structures connected with these roads, with navigable rivers, canals, seaports, lighthouses, &c. They are charged with the superintendence of railways, of works for draining marshes, and operations affecting water-courses; they report upon applications to erect factories driven by water. Under certain circumstances, they share with the Mining Engineers the duty of inspecting steam-engines.
Permission is not unfrequently granted to the engineers of the Ponts et Chaussées to accept private employment. They receive leave of absence for a certain time, retaining their rank and place in their corps, but without pay.
Mining Engineers.—(Mines.)
The Mining School of Application is organized almost exactly on the same plan as that of the Ponts et Chaussées: like the latter, it is in Paris.
The course of instruction, which lasts three years, consists of lectures, drawing, chemical manipulation and analysis, visits to manufactories, geological excursions, and the preparation of projects for mines and machines. Journeys are made by the pupils, during the second half of the last two years of the course, into the mineral districts of France or foreign countries for the purpose of studying the practical details of mining. These journeys last one hundred days at least. The pupils are required to examine carefully the railroads and the geological features of the countries they pass through, and to keep a journal of facts and observations. In the final examination, marks are given for every part of their work.
The mining engineers, when stationed in the departments, are charged to see that the laws and ordinances relating to mines, quarries, and factories are properly observed, and to encourage, either directly or by their advice, the extension of all branches of industry connected with the extraction and treatment of minerals.
One of their principal duties is the superintendence of mines and quarries, in the three-fold regard of safety of the workmen, preservation of the soil, and economical extraction of the minerals.
They exercise a special control over all machines designed for the production of steam, and over railways, as far as regards the metal and fuel.
The instructors in the School of Application in Paris, and in the School of Mines at St. Etienne, are exclusively taken from the members of the corps.
Like the engineers of the Ponts et Chaussées, the mining engineers obtain permission to undertake private employment.
Tobacco Department.—(Administration des Tabacs.)
The pupils who enter the tobacco service, commence, on quitting the Polytechnic, with the rank of élève de 2e classe. They study, in the manufactory at Paris, chemistry, physics, and mechanics, as applied to the preparation of tobacco. They make themselves acquainted at the same time with the details of the manufacture and with the accounts and correspondence.
They are generally promoted to the rank of élevè de 1re classe in two years. They rise afterwards successively to the rank of sub-inspector, inspector, and director.
After completing their instruction at the manufactory of Paris, the élevès are sent to tobacco manufactories in other parts of France.
Promotion in the tobacco service does not follow altogether by seniority. Knowledge of the manufacture and attention to their duties are much considered, as the interests of the treasury are involved in the good management of the service.
Telegraphs.—(Lignes Telégraphiques.)
On entering the telegraphic service the pupils of the Polytechnic receive the rank of élevè inspecteur.
They pass the first year at the central office. During the six winter months they study, under two professors, the composition of signals, and the regulations which insure their correctness and dispatch, the working of telegraphs and the manner of repairing them, the theory of the mode of tracing lines and of determining the height of the towers, electro-magnetism and its application to the electric telegraph. During the summer months they make tours of inspection. They assist in the execution of works, and practice leveling and the laying down of lines.
At the end of the year the élevès inspecteurs undergo an examination, and, if there are vacancies, are appointed provisional inspectors. After a year in this rank they may be appointed inspectors either in France or Algeria.
Each inspector has charge of a district containing from twelve to fifteen stations. He is obliged to make a tour of inspection once a month of at least ten days’ duration.
After a certain number of years’ service the inspector rises to the rank of director. Besides their other duties, the directors exercise a general superintendence over the inspectors.
Programmes of the Principal Courses of Instruction of the Imperial Polytechnic School During the Two Years of Study
I. ANALYSIS.—FIRST YEAR.
DIFFERENTIAL CALCULUS.
Lessons 1–9. Derivatives and Differentials of Functions of a Single Variable.
Indication of the original problems which led geometers to the discovery of the infinitesimal calculus.
Use of infinitesimals; condition, subject to which, two infinitely small quantities may be substituted for one another. Indication in simple cases of the advantage of such substitution.
On the different orders of infinitely small quantities. Infinitely small quantities of a certain order may be neglected in respect of those of an inferior order. The infinitely small increment of a function is in general of the same order as the corresponding increment of the variable, that is to say, their ratio has a finite limit.
Definitions of the derivative and differential of a function of a single variable. Tangents and normals to plane curves, whose equation in linear or polar coordinates is given.
A function is increasing or decreasing, according as its derivative is positive or negative. If the derivative is zero for all values of the variable, the function is constant. Concavity and convexity of curves; points of inflection.
Principle of function of functions. Differentiation of inverse functions.
Differentials of the sums, products, quotients, and powers of functions, whose differentials are known. General theorem for the differentiation of functions composed of several functions.
Differentials of exponential and logarithmic functions.
Differentials of direct and inverse circular functions.
Differentiation of implicit functions.
Tangents to curves of double curvature. Normal plane.
Differential of the area and arc of a plane curve, in terms of rectilinear and polar co-ordinates.
Differential of the arc of a curve of double curvature.
Applications to the cycloid, the spiral of Archimedes, the logarithmic spiral, the curve whose normal, sub-normal, or tangent, is constant; the curve whose normal passes through a fixed point; the curve whose arc is proportional to the angle which it subtends at a given point.
Derivatives and differentials of different orders of functions of one variable. Notation adopted.
Remarks upon the singular points of plane curves.
Lessons 10–13. Derivatives and Differentials of Functions of Several Variables.
Partial derivatives and differentials of functions of several variables. The order in which two or any number of differentiations is effected does not influence the result.
Total differentials. Symbolical formula for representing the total differential of the nth order of a function of several independent variables.
Total differentials of different orders of a function; several dependent variables. Case where these variables are linear functions of the independent variables.
The infinitesimal increment of a function of several variables may in general be regarded as a linear function of the increments assigned to the variables. Exceptional cases.
Tangent and normal planes to curved surfaces.
Lessons 14–18. Analytical Applications of the Differential Calculus.
Development of F(x + h,) according to ascending powers of h. Limits within which the remainder is confined on stopping at any assigned power of h.
Development of F(x,) according to powers of x or x - a; a being a quantity arbitrarily assumed. Application to the functions sin(x,) cos x, ax, (1 + xm) and log.(1 + x.) Numerical applications. Representation of cos x and sin x by imaginary exponential quantities.
Developments of cosm x and sinm x in terms of sines and curves of multiples of x.
Development of F(x + h, y + k,) according to powers of h and k. Development of F(x, y) according to powers of x and y. Expression for the remainder. Theorem on homogeneous functions.
Maxima and minima of functions of a single variable; of functions of several variables, whether independent or connected by given equations. How to discriminate between maxima and minima values in the case of one and two independent variables.
True values of functions, which upon a particular supposition assume one or another of the forms
0⁄0, ∞⁄∞, ∞ + 0, 00, 4∞
Lessons 19–23. Geometrical Applications. Curvature of Plane Curves.
Definition of the curvature of a plane curve at any point. Circle of curvature. Center of curvature. This center is the point where two infinitely near normals meet.
Radius of curvature with rectilinear and polar co-ordinates. Change of the independent variable.
Contacts of different orders of plane curves. Osculating curves of a given kind. Osculating straight line. Osculating circle. It is identical with the circle of curvature.
Application of the method of infinitesimals to the determination of the radius of curvature of certain curves geometrically defined. Ellipse, cycloid, epicycloid, &c.
Evolutes of plane curves. Value of the arc of the evolute. Equation to the involute of a curve. Application to the circle. Evolutes considered as envelops. On envelops in general. Application to caustics.
Lessons 24–27. Geometrical Applications continued. Curvature of Lines of Double Curvature and of Surfaces.
Osculating plane of a curve of double curvature. It may be considered as passing through three points infinitely near to one another, or as drawn through a tangent parallel to the tangent infinitely near to the former. Center and radius of curvature of a curve of double curvature. Osculating circle. Application to the helix.
Radii of curvature of normal s of a surface. Maximum and minimum radii. Relations between these and that of any, normal or oblique.
Use of the indicatrix for the demonstration of the preceding results. Conjugate tangents. Definition of the lines of curvature. Lines of curvature of certain simple surfaces. Surface of revolution. Developable surfaces. Differential equation of lines of curvature in general.
Lesson 28. Cylindrical, Conical, Conoidal surfaces, and Surfaces of Revolution.
Equations of these surfaces in finite terms. Differential equations of the same deduced from their characteristic geometrical properties.
INTEGRAL CALCULUS.
Lessons 29–34. Integration of Functions of a Single Variable.
Object of the integral calculus. There always exists a function which has a given function for its derivative.
Indefinite integrals. Definite integrals. Notation. Integration by separation, by substitution, by parts.
Integration of rational differentials, integer or fractional, in the several cases which may present themselves. Integration of the algebraical differentials, which contain a radical of the second degree of the form √c+bx+ax2. Different transformations which render the differential rational. Reduction of the radical to one of the forms
√x2+x2, √a2-x2, √x2-a2.
Integration of the algebraical differentials which contain two radicals of the form
√a+x, √b+x,
or any number of monomials affected with fractional indices. Application to the expressions
| xmdx | dx | xmdx | ||
| √1-x2 | , | xm √1-x2 | , | √ax-x |
Integration of the differentials
F(log x) dx⁄x, F sin-1x dx⁄√1-x2, x(log xn)dx, xm eax dx, (sin-1xm)dx.
Integration of the differentials eax sin bxdx and eax cos bxdx.
Integration of (sin xm.)(cos xn) dx.
Integration by series. Application to the expression
dx
√ax-x2 √1-bx
Application of integration by series to the development of functions, the development of whose derivatives is given: tan-1x, sin-1x, log(1 + x.)
Lessons 35–38. Geometrical Applications.
Quadrature of certain curves. Circle, hyperbola, cycloid, logarithmic spiral, &c.
Rectification of curves by rectilinear or polar co-ordinates. Examples. Numerical applications.
Cubic content of solids of revolution. Quadrature of their surfaces.
Cubic content of solids in general, with rectilinear or polar co-ordinates. Numerical applications.
Quadrature of any curved surfaces expressed by rectangular co-ordinates. Application to the sphere.
Lessons 39–42. Mechanical Applications.
General formula for the determination of the center of gravity of solids, curved or plane surfaces, and arcs of curves. Various applications.
Guldin’s theorem.
Volume of the truncated cylinder.
General formula which represent the components of the attraction of a body upon a material point, upon the supposition that the action upon each element varies inversely as the square of the distance. Attraction of a spherical shell on an external or internal point.
Definition of moments of inertia. How to calculate the moment of inertia of a body in relation to a straight line, when the moment in relation to a parallel straight line is known. How to represent the moments of inertia of a body relative to the straight lines which pass through a given point by means of the radii vectores of an ellipsoid. What is meant by the principal axes of inertia.
Determination of the principal moments of inertia of certain homogeneous bodies, sphere, ellipsoid, prism, &c.
Lessons 43–45. Calculus of Differences.
Calculation of differences of different orders of a function of one variable by means of values of the function corresponding to equidistant values of the variable.
Expression for any one of the values of the function by means of the first, and its differences. Numerical applications; construction of tables representing a function whose differences beyond a certain order may be neglected. Application to the theory of interpolation. Formulæ for approximation by quadratures. Numerical exercises relative to the area of equilateral hyperbola or the calculation of a logarithm.
Lessons 46–48. Revision.
General reflections on the subjects contained in the preceding course.
ANALYSIS.—SECOND YEAR.
CONTINUATION OF THE INTEGRAL CALCULUS.
Lessons 1–2. Definite Integrals.
Differentiation of a definite integral with respect to a parameter in it, which is made to vary. Geometrical demonstration of the formula. Integration under the sign of integration. Application to the determination of certain definite integrals.
Determination of the integrals ∫ sinax⁄x dx, and ∫ cosbxsinax⁄x dx, between the limits 0 and x. Remarkable discontinuity which these integrals present.
Determination of ∫e-x²dx and ∫e-x²cos mx dx between the limits 0 and ∞.
Lesson 3. Integration of Differentials containing several Variables.
Condition that an expression of the form M dx + N dy in which M and N are given functions of x and y may be an exact differential of two independent variables x and y. When this condition is satisfied, to find the function.
Extension of this theory to the case of three variables.
Lessons 4–6. Integration of Differential Equations of the First Order.
Differential equations of the first order with two variables. Problem in geometry to which these equations correspond. What is meant by their integral. This integral always exists, and its expression contains an arbitrary constant.
Integration of the equation M dx + N dy = 0 when its first member is an exact differential. Whatever the functions M and N may be there always exists a factor µ, such that µ (M dx + N dy) is an exact differential.
Integration of homogeneous equations. Their general integral represents a system of similar curves. The equation (a + b x + c y) dx + (a’ + b’ x + c’ y) dy = c, may be rendered homogeneous. Particular case where the method fails. How the integration may be effected in such case.
Integration of the linear equation of the first order dy⁄dx + P y = Q, where P and Q denote functions of x. Examples.
Remarks on the integration of equations of the first order which contain a higher power than the first of dy⁄dx. Case in which it may be resolved in respect of dy⁄dx. Case in which it may be resolved in respect of x or y.
Integrations of the equation y = x dy⁄dx + φ (dy⁄dx). Its general integral represents a system of straight lines. A particular solution represents the envelop of this system.
Solution of various problems in geometry which lead to differential equations of the first order.
Lessons 7–8. Integration of Differential Equations of Orders superior to the First.
The general integral of an equation of the m order contains m arbitrary constants.
(The demonstration is made to depend on the consideration of infinitely small quantities.)
Integration of the equation dmy⁄dxm = φ (x.)
Integration of the equation d2y⁄dx2 = φ (y, dy⁄dx).
How this is reduced to an equation of the first order. Solution of various problems in geometry which conduct to differential equations of the second order.
Lessons 9–10. On Linear Equations.
When a linear equation of the mth order contains no term independent of the unknown function and its derivatives, the sum of any number whatever of particular integrals multiplied by arbitrary constants is also an integral. From this the conclusion is drawn that the general integral of this equation is deducible from the knowledge of m particular integrals.
Application to linear equations with constant co-efficients. Their integration is made to depend on the resolution of an algebraical equation. Case where this equation has imaginary roots. Case where it has equal roots. The general integral of a linear equation of any order, which contains a term independent of the function, may be reduced by the aid of quadratures to the integration of the same equation with this term omitted.
Lesson 11. Simultaneous Equations.
General considerations on the integration of simultaneous equations. It may be made to depend on the integrations of a single differential equation. Integration of a system of two simultaneous linear equations of the first order.
Lesson 12. Integrations of Equations by Series.
Development of the unknown function of the variable x according to the powers of x-a. In certain cases only a particular integral is obtained. If the equation is linear, the general integral may be deduced from it by the variation of constants.
Lessons 13–16. Partial Differential Equations.
Elimination of the arbitrary functions which enter into an equation by means of partial derivatives. Integration of an equation of partial differences with two independent variables, in the case where it is linear in respect to the derivatives of the unknown function. The general integral contains an arbitrary function.
Indication of the geometrical problem, of which the partial differential equation expresses analytically the enunciation. Integration of the partial differential equations to cylindrical, conical, conoidal surfaces of revolution. Determination of the arbitrary functions.
Integration of the equation d2u⁄dy2 = a2d2u⁄dx2. The general integral contains two arbitrary functions. Determination of these functions.
Lessons 17–23. Applications to Mechanics.
Equation to the catenary.
Vertical motion of a heavy particle, taking into account the variation of gravity according to the distance from the center of the earth. Vertical motion of a heavy point in a resisting medium, the resistance being supposed proportional to the square of the velocity.
Motion of a heavy point compelled to remain in a circle or cycloid. Simple pendulum. Indication of the analytical problem to which we are led in investigating the motion of a free point.
Motion of projectiles in a vacuum. Calculation of the longitudinal and transversal vibrations of cords. Longitudinal vibrations of elastic rods. Vibration of gases in cylindical tubes.
Lessons 24–26. Applications to Astronomy.
Calculation of the force which attracts the planets, deduced from Kepler’s laws. Numerical data of the question.
Calculation of the relative motion of two points attracting one another, according to the inverse square of the distance.
Determination of the masses of the earth and of the planets accompanied by satellites. Numerical applications.
Lessons 27–30.
Elements of the calculus of probabilities and social arithmetic.
General principles of the calculus of chances. Simple probability, compound probability, partial probability, total probability. Repeated trials. Enunciation of Bernouilli’s theorem (without proof.)
Mathematical expectation. Applications to various cases, and especially to lotteries.
Tables of population and mortality. Mean life annuities, life interests, assurances, &c.
Lessons 31–32. Revision.
General reflections on the subjects comprised in the course.
II. DESCRIPTIVE GEOMETRY AND STEREOTOMY.
General Arrangements.
The pupils take in the lecture-room notes and sketches upon sheets, which are presented to the professor and the “répétiteurs” at each interrogation. The care with which these notes are taken is determined by “marks,” of which account is taken in arranging the pupils in order of merit.
The plans are made according to programmes, of which the conditions are different for different pupils. The drawings are in general accompanied with decimal scales, expressing a simple ratio to the meter. They carry inscriptions written conformably to the admitted models, and are, when necessary, accompanied with verbal descriptions.
In the graphic exercises of the first part of the course, the principal object is to familiarize the pupils with the different kinds of geometrical drawing, such as elevations and shaded s, oblique projections and various kinds of perspective. The pupils are also accustomed to different constructions useful in stereotomy.
The subjects for graphic exercises in stereotomy are taken from roofs, vaults, and staircases. Skew and oblique arches are the subject of detailed plans.
FIRST YEAR.
DESCRIPTIVE GEOMETRY.—GEOMETRICAL DRAWING.
Lessons 1–3. Revision and Completion of the Subjects of Descriptive Geometry comprised in the Programme for Admission into the School.
Object of geometrical drawing. Methods of projection. Representation of points, lines, planes, cones, cylinders, and surfaces of revolution. Construction of tangent planes to surfaces, of curves, of inter of surfaces, of their tangents and their assymplotes.
Osculating plane of a curve of double curvature. A curve in general cuts its osculating plane.
When the generating line of a cylinder or a cone becomes a tangent to the directrix, the cylinder or cone in general has an edge of regression along this generating line. The osculating plane of the directrix at the point of contact touches the surface along this edge.
Projections of curves of double curvature; infinite branches and their assymplotes, inflections, nodes, cusps, &c.
Change of planes of projection.
Reduction of scale; transposition.
Advantage and employment of curves of error; their irrelevant solutions.
Lessons 4–6. Modes of Representation for the Complete Definition of Objects.
Representation by plans, s, and elevation.
Projection by the method of contours. Representation of a point, a line, and a plane; questions relative to the straight line and plane. Representation of cones and cylinders; tangent planes to these surfaces.
Lessons 7–11. Modes of Representation which are not enough in themselves to define objects completely.
Isometrical and other kinds of perspective.
Oblique projections.
Conical perspective: vanishing points; scales of perspective; method of squares; perspective of curved lines; diverse applications. Choice of the point of sight. Rules for putting an elevation in perspective. Rule for determining the point of sight of a given picture, and for passing from the perspective to the plan as far as that is possible. Perspective of reflected images. Notions on panoramas.
Lessons 12–13. Representations with Shadows.
General observations on envelops and characteristics.
A developable surface is the envelop of the position of a movable plane; it is composed of two sheets which meet. It may be considered as generated by a straight line, which moves so as to remain always a tangent to a fixed curve.
Theory of shade and shadow, of the penumbra, of the brilliant point, of curves of equal intensity, of bright and dark edges.
Atmospheric light: direction of the principal atmospheric ray. Notions on the degradation of tints; construction of curves of equal tint.
Influence of light reflected by neighboring bodies.
Received convention in geometrical drawing on the direction of the luminous ray, &c.
Perspective of shadows.
Lessons 14–15. Construction of Lines of Shadows and of Perspective of Surfaces.
Use of circumscribed cones and cylinders, and of the normal parallel to a given straight line.
General method of construction of lines of shadow and of perspective of surfaces by plane s and auxiliary cylindrical or conical surfaces.
Construction of lines of shadow and perspective of a surface of revolution.
The curve of contact of a cone circumscribed about a surface of the second degree is a plane curve. Its plane is parallel to the diametral plane, conjugate to the diameter passing through the summit of the cone. The curve of contact of a cylinder circumscribed about a surface of the second degree is a plane curve, and situated in the diametral plane conjugate to the diameter parallel to the axis of the cylinder.
The plane parallel s of a surface of the second degree are similar curves. The locus of their centers is the diameter conjugate to that one of the secant planes which passes through the center of the surface.
General study of surfaces with reference to the geometrical constructions to which their use gives rise.
Lesson 16. Complementary Notions on Developable Surfaces.
Development of a developable surface; construction of transformed curves and their tangents. Developable surface; an envelop of the osculating planes of a curve. The osculating plane of a curve at a given point may be constructed by considering it as the edge of regression of a developable surface; this construction presents some uncertainty in practice. Notions on the helix and the developable helicoid.
Approximate development of a segment of an undevelopable surface.
Lessons 17–18. Hyperbolic Paraboloid.
Double mode of generation of the paraboloid by straight lines; plane-directers; tangent planes, vertex, axis, principal planes; representation of this surface. Construction of the tangent plane parallel to a given plane. Construction of plane s and of curves of contact, of cones, and circumscribed cylinders.
Scalene paraboloid. Isosceles paraboloid.
Identity of the paraboloid with one of the five surfaces of the second degree studied in analytical geometry.
Re-statement without demonstration of the properties of this surface found by analysis, principally as regards its generation by the conic s.
Lessons 19–20. General Properties of Warped or Ruled Surfaces.
Principal modes of generation of warped surfaces. When two warped surfaces touch in three points of a common generatrix, they touch each other in every point of this straight line. Every plane passing through a generatrix touches the surface at one point in this line. The tangent plane at infinity is the plane-directer to all the paraboloids of “raccordement.”
Construction of the tangent planes and curves of contact of circumscribed cones and cylinders. When two infinitely near generatrices of a warped surface are in the same plane, all the curves of contact of the circumscribed cones and cylinders pass through their point of concourse.
The normals to a warped surface along a generatrix form an isosceles paraboloid. The name of central point of a generatrix is given to the point where it is met by the straight line upon which is measured its shortest distance from the adjoining generatrix. The locus of these points forms the line of striction of the surface. The vertex of the normal paraboloid along a generating line is situated at the central point. If the point of contact of a plane touching a warped surface moves along a generatrix, beginning from the central point, the tangent of the angle which the tangent plane makes with its primitive position is proportional to the length described by the point of contact. The tangent plane at the central point is perpendicular to the tangent plane at infinity upon the same generatrix. Construction of the line of striction by aid of this property.
Lessons 21–22. Ruled Surfaces with plane-divecters Conoids.
The plane-directer of the surface is also so to all the paraboloids of “raccordement.” Construction of the tangent planes and curves of contact of the circumscribed cones and cylinders.
The line of striction of the surface is its curve of contact with a circumscribed cylinder perpendicular to the directer-plane. Determination of the nature of the plane s.
The lines of striction of the scalene paraboloid are parabolas; those of the isosceles paraboloid are straight lines.
Construction of the tangent plane parallel to a given plane.
Conoid: discussion of the curves of contact of the circumscribed cones and cylinders.
Right conoid. Conoid whose inter with a torus of the same height, whose axis is its rectilinear directrix, has for its projection upon the directer-plane two arcs of Archimedes’ spiral. Construction of the tangents to this curve of inter.
Lessons 23–25. Ruled Surfaces which have not a Directer-Plane. Hyperboloid. Surface of the “biais passe.”
Directer-cone: its advantages for constructing the tangent plane parallel to a given plane, and for determining the nature of the plane s. The tangent planes to the points of the surface, situated at infinity, are respectively parallel to the tangent plane of the directer-cone. Developable surface which is the envelope of these tangent planes at infinity. Construction of a paraboloid of raccordement to a ruled surface defined by two directrices and a directrix cone.
Hyperboloid; double mode of generation by straight lines; center; assymptotic cone.
Scalene hyperboloid; hyperboloid of revolution. Identity of the hyperboloid with one of the five surfaces of the second degree studied in analytical geometry.
Re-statement without demonstration of the properties of this surface, found by analysis, principally as to what regards the axis, the vertices, the principal planes, and the generation by conic s.
Hyperboloid of raccordement to a ruled surface along a generatrix; all their centers are in the same plane. Transformation of a hyperboloid of raccordement.
Surface of the biais passé. Construction of a hyperboloid of raccordement; its transformation into a paraboloid.
Construction of the tangent plane at a given point.
Lessons 26–28. Curvature of Surfaces. Lines of Curvature.
Re-statement without proof of the formula of Euler given in the course of analysis.
There exists an infinity of surfaces of the second degree, which at one of their vertices osculate any surface whatever at a given point.
In the tangent plane, at a point of a surface, there exists a conic, whose diameters are proportional to the square roots of the radii of curvature of the normal s to which they are tangents. This curve is called the indicatrix. It is defined in form and position, but not in magnitude. The normal s tangential to the axes of the indicatrix are called the principal s.
The indicatrix an ellipse; convex surfaces; umbilici; line of spherical curvatures.
The indicatrix a hyperbola; surfaces with opposite curvatures.
The assymplotes of the indicatrix have a contact of the second order with the surface, and of the first order with the of the surface by its tangent plane.
A ruled surface has contrary curvatures at every point. The second assymplotes of the indicatrices of all the points of the same generatrix form a hyperboloid, if the surface has not directer-plane,—a paraboloid, if it have one.
Curvature of developable surfaces.
There exists upon every surface two systems of orthogonal lines, such that every straight line subject to move by gliding over either of them, and remaining normal to the surface, will engender a developable surface. These lines are called lines of curvature.
The two lines of curvature which cross at a point, are tangents to the principal s of the surface at that point.
Remarks upon the lines of curvature of developable surfaces, and surfaces of revolution.
Determination of the radii of curvature, and assymplotes of the indicatrix at a point of a surface of revolution.
Lessons 29–30. Division of Curves of Apparent Contour, and of Separation of Light and Shadow into Real and Virtual Parts.
When a cone is circumscribed about a surface, at any point whatever of the curve of contact, the tangent to this curve and the generatrix of the cone are parallel to two conjugate diameters of the indicatrix.
Surfaces, as they are considered in shadows, envelop opaque bodies, and the curve of contact of a circumscribed cone, only forms a separation of light and shadow, for a luminous point at the summit of the cone, when the generatrices of this cone are exterior. This line is thus sometimes real and sometimes virtual.
Upon a convex surface, the curve of separation of light and shade is either all real or all virtual. Upon a surface with contrary curvatures, this curve presents generally a succession of real and virtual parts: the curve of shadow cast from the surface upon itself presents a like succession. These curves meet tangentially, and the transition from the real to the virtual parts upon one and the other, take place at their points of contact in such a way that the real part of the curve of shadow continues the real part of the curve of separation of light and shade. The circumscribed cones have edges of regression along the generatrices, which correspond to the points of transition.
The lines of visible contour present analogous circumstances.
General method of determining the position of the transition points. Special method for a surface of revolution.
Lessons 31–34. Ruled Helicoidal Surfaces.
Surface of the thread of the triangular screw; generation, representation, s by planes and conical cylinders.
Construction of the tangent plane at a given point, or parallel to a given plane. The axis is the line of striction.
Construction of lines of shadow and perspective: their infinite branches, their assymplotes. Determination of the osculating hyperboloid along a generatrix.
Representation and shading of the screw with a triangular thread and its nut.
Surface of the thread of the square screw; generation, s by planes and conical cylinders; tangent planes; curve of contact of a circumscribed cone.
The curve of contact of a circumscribed cylinder is a helix whose step is half that of the surface. Determination of the osculating paraboloid. At any point whatever of the surface, the absolute lengths of the radii of curvature are equal.
Representation and shading of the screw with a square thread, and of its nut.
Observations on the general ruled helicoidal surface, and on the surface of intrados of the winding staircase.
Lesson 35. Different Helicoidal Surfaces.
Saint-Giles screw, worm-shaped screw and helicoidal surfaces to any generatrix. Every tangent to the meridian generatrix describes a screw surface with triangular thread, which is circumscribed about the surface, along a helix, and may be used to resolve the problems of tangent planes, circumscribed cylinders, &c.
Helicoid of the open screw, its generation, tangent planes.
Lessons 36–37. Topographical Surfaces.
Approximate representation of a surface by the figured horizontal projections of a series of equidistant horizontal s. This method of representation is especially adapted to topographical surfaces, that is to say, surfaces which a vertical line can only meet in one point.
Lines of greatest slope. Trace of a line of equal slope between two given points.
Inter of a plane and a surface, of two surfaces, of a straight line and a surface.
Tangent planes, cones, and cylinders circumscribed about topographical surfaces.
Use of a topographical surface to replace a table of double-entry when the function of two variables, which it represents, is continuous. It is often possible, by a suitable anamorphosis, to make an advantageous transformation in the curves of level.
Lesson 38. Revision.
Review of the different methods of geometrical drawing. Advantages and disadvantages of each.
Comparison of the different kinds of surfaces, résumé of their general properties.
Object, method, and spirit of descriptive geometry.
SECOND YEAR.
STEREOTOMY.—WOOD-WORK.
Lessons 1–4. Generalities.
Notions on the mode of action of forces in carpentry. Resistance of a piece of wood to a longitudinal effort and to a transversal effort. Distinction between resistance to flexure and resistance to rupture. Beams.
Advantages of the triangular system, St. Andrew’s cross.
Lessons 5–8. Roofs.
Ordinary composition of roofs.
Distribution of pressures in the different parts of a girded roof.
Design of the different parts of roofs, &c., &c.
Lessons 9–10. Staircases.
MASONRY.
Lessons 11–12. Generalities.
Notions on the settlement of vaulted roofs. Principal forms of vaults, en berceau, &c., &c.
Distribution of the pressures, &c.
Division of the intrados. Nature of the surfaces at the joints, &c., &c.
Lessons 13–15. Berceaux and descentes.
Lessons 16–22. Skew Arches.
Study of the general problem of skew arches.
First solution. Straight arches en échelon.
Second solution: Orthogonal appareil. True and principal properties of the orthogonal trajectories of the parallel s of an elliptical or circular cylinder. Right conoid, having for directrices the axis of the circular cylinder and an orthogonal trajectory. The inter of this conoid by a cylinder about the same axis is an orthogonal trajectory for a series of parallel s.
Third solution: helicoidal. Determination of the angular elevation at which the surfaces of the beds become normal to the head planes; construction in the orthogonal and helicoidal appareil of the curves of junction upon the heads, and the angles which they form with the curves of intrados. Cutting of the stones in these different constructions. Broken helicoidal appareil, for very long skew arches.
Helicoidal trompes at the angles of straight arches; voussures or widenings, which it is necessary to substitute near the heads at the intrados of an arch with a considerable skew; case where the skew is not the same for the two heads. Orthogonal trajectories of the converging s of a cylinder.
Lessons 23–25. Conical Intrados—Intrados of Revolution.
Skew trompe in the angle. Suggestions on the general problem of conical skew vaulted roofs.
Spherical domes, &c.
Lessons 26–27. Intrados, a Ruled Surface.
Winding staircases, &c., &c.
Lesson 28. Helicodial Intrados.
Staircase on the Saint-Giles screw.
Lessons 29–31. Composite Vaulted Roofs.
Various descriptions of vaults.
Suggestions on vaulted roofs with polygonal edges and with ogival edges.
Lesson 32. Revision.
Spirit and method of stereotomy.
Degree of exactness necessary. Approximate solutions. Case where it is proper to employ calculation in aid of graphical constructions.
Review and comparison of different appareils.
MECHANICS AND MACHINES.
GENERAL ARRANGEMENTS.
The pupils execute during the two years of study:—
1. Various drawings or plans of models in relief representing the essential and internal organs of machines, such as articulations of connecting rods, winch-handles and fly-wheels, grease-boxes, eccentrics worked by cams or circles giving motion to rods; the play of slides, &c.; cylinders of steam-engines, condenser, pistons, and various suckers; Archimedes’ screw, and other parts of machines.
The sketches of the plan drawings are traced by hand and figured. The drawings in their finished state are washed and colored according to the table of conventional tints; they all carry a scale suitably divided.
2. A drawing of wheel-work by the method of development, and tracing the curves of teeth by arcs of circles from which they are developed. This drawing represents, of the natural size, or on any other scale of size considered suitable to show the nature of the partial actions only, a small number of teeth either in development or projection; the entire wheel-work is represented by the usual method of projection, where in drawings on a small scale the teeth are replaced by truncated pyramids with a trapezoidal base.
3. Finally, numerical exercises concerning the loss of work due to the proejudicial resistances in various machines, the gauging of holes, orifices, &c.
Models in relief or drawings on a large scale, of the machines or elements of the machines mentioned in the course, assist in explaining the lessons. They are brought back, as often as found necessary, under the eyes of the students. When possible, lithographic sketches of the machines, or the elements of the machines, which ought to enter into the course, are distributed among the pupils.
The pupils, divided into s, pay their first visit to the engine factories towards the end of their first year of study; they make one or more additional visits at the end of the second year.
FIRST YEAR.
PART I. KINEMATICS.—PRELIMINARY ELEMENTARY MOVEMENTS OF INVARIABLE POINTS AND SYSTEMS.
Lessons 1–2.
Object of kinematics, under the geometrical and experimental point of view. Its principal divisions.
Re-statement of the notions relative to the motion of a point, its geometrical representation, and more especially the determination of its velocity.
Simultaneous Velocities of a Point and the Increments of its Velocities.
Ratio of the elementary displacement and the velocity of a point to the displacement, and velocity of its projection upon a straight line or plane. Use of infinitesimals to determine these ratios. Example:—Oscillatory motion of the projection upon a fixed axis of a point moving uniformly upon the circumference of a circle.
Analogous considerations for polar co-ordinates. Relations of the velocity of a point, of its velocity of revolution and its angular velocity about a fixed pole; of its velocity in the direction of the radius vector; of the velocity of increase of the area which this radius describes.
Simple Motions of Solids, or Rigid Systems.
1. Motion of rectilinear or curvilinear translation; simultaneous displacements, and velocities of its different points.
2. Motion of rotation about a fixed axis; relation of the velocities of different points to the angular velocity.
Geometrical notions and theorems relative to the instantaneous center of rotation of a body of invariable figure and movable in one plane, or to the instantaneous axis of rotation of a rigid system situated in space, and movable parallel to a fixed plane. Relation of the velocities of different points to their common angular velocity. Use of the instantaneous center of rotation for tracing tangents; examples—and amongst others—that of the plane curve described by a point in a straight line of given length, whose extremities slide upon two fixed lines. Rolling of a curve upon another fixed curve in a plane. Descartes’ theorems upon the inter of the normals at the successive points of contact: cycloids, epicycloids, involutes, and evolutes. Extension of the preceding motions to the instantaneous axis of rotation of a rigid system movable about a fixed point.
COMPOSITION OF MOTIONS.
Lessons 3–6. Composition of the Velocities of a Point.
Polygon of velocities. Example of movements observed relatively to the earth. Particular cases; composition of velocities taken along three axes; composition of the velocity of a point round a fixed pole, and its velocity along the radius vector. Method of Roberval for tracing tangents.
Composition of the Simple Motions of a Solid System.
Composition of any number of translatory displacements of a solid. Composition of two rotations about two intersecting axes. Composition of any number of rotations about axes cutting one another at the same point; parallelopiped and polygon of rotations. Composition of two simultaneous rotations about parallel axes; case where the rotations are equal and of opposite kinds. Decomposition of a rotation about an axis into an equal rotation about any axis whatever parallel to the first, and a translation perpendicular to the direction of this axis. Direct and geometrical decomposition of the most general motions of a body into a rotation about, and a translation along, an axis called the instantaneous axis. Composition of any two motions whatever. Every movement of an invariable system is at each instant of time decomposable into three movements of rotation, and three movements of translation with respect to three axes, which are neither parallel nor lying in the same plane, but otherwise arbitrarily chosen.
Relative or Apparent Motions.
Relative motion of two points whose absolute motions are given graphically à priori. Trajectory of the relative motions, relative velocities, and displacements upon curves or upon the direction of the mutual distance of the two points; use of the parallelogram to determine its amount. Relative motion of a point in motion in respect of a body turning about a fixed axis; relative motion of two bodies which turn about parallel or converging axes, and in general of two rigid bodies or systems impelled by any motions whatever. How this problem is immediately reduced to that of the composition of given motions.
The most general continued motion of an invariable figure in a plane is an epicycloidal motion, in which the instantaneous center describes a curve fixed in relation to absolute space, and traces relatively to the proposed figure a movable curve, which is rigidly connected with that figure and draws it along with it in its motion of rolling upon the other fixed curve. Case of space or spherical figures.
ON THE ACCELERATED MOTION OF A POINT.
Lessons 7–9. Accelerated Rectilinear Motion.
Re-statement of the motions acquired relatively to the acceleration in the variable rectilinear motion of a point. Brief indication of the solution of six problems arising out of the investigation of the laws of the motion in terms of the space, time, velocity, and accelerating force. For the most part these solutions may be brought to depend on exact or approximate quadratures. Numerical exercises.
Accelerated Curvilinear Motions.
Re-statement of the notions acquired relative to the composition of accelerating forces; the resulting acceleration, the normal and tangential acceleration animating a point in motion on a curve. The total acceleration of a point upon an axis or plane is the projection upon this axis or plane of the acceleration of the moving body in space. In uniform curvilinear motion the total or resultant acceleration becomes normal to the curve. Particular case of the circle; value of the normal acceleration in terms of the velocity of revolution or the angular velocity of the radius vector. Case of any curve whatever; geometrical expression of the total or resultant acceleration.
Accelerated Compound and Relative Motions.
Geometrical investigation of the simple and compound accelerations arising out of the hypothesis in which the motion of any system of points whatever is referred to another system of invariable form, but also in motion. Geometrical and elementary explanations of the results obtained by means of the transformation of co-ordinates.
Examples or Exercises chosen from among the following Questions:—
Projection of circular and uniform motion upon a fixed straight line or plane; motion of a circle which rolls uniformly on a straight line; comparison of the motions of the planets relatively to each other, treating them as circular and uniform: comparison of the accelerating force on the moon with that of bodies which fall to the earth.
GEOMETRICAL THEORY AND APPLICATION OF MECHANISMS OR CONTRIVANCES FOR THE TRANSFORMATION OF MOTION.
Lessons 10–19.
Succinct notions on the classification of elementary motions and organs for transmission of motion in machines after Monge and Hachette, Lanz and Bétancourt.
The most essential details upon this subject are set forth in the following order, and made clear by outline drawings previously distributed among the pupils.
Organs fitted to regulate the direction of the circular or rectilinear motion of certain pieces.
Axle; trunnions, gudgeons; pivots and bearings; couplings of axes; adjustment of wheels and of their arms. Joints with hinges, &c.; sheaves and pulleys; chains, ropes, and straps; means of securing them to the necks. Grooves and tongue-pieces. Eyelet-holes sliding along rectilinear or curvilinear rods. Advantages and disadvantages of these different systems of guides under the point of view of accuracy.
Rapid indication of some of their applications to drawbridges and to the movable frames or wagons of saw-works and railways.
Transmission at a Distance of Rectilinear Motion in a determinate Direction and Ratio.
Inclined plane or wedge guiding a vertical rod. Wedge applied to presses. Rods, winch-handles, &c. Disposition of drums or pulleys in the same plane or in different planes; geometrical problem on this subject. Fixed and movable pulleys. Blocks to pulleys. Simple and differential wheel and axle moved by cords. Transmission through a liquid. Ratios of velocities in these different organs.
Direct Transformation of circular progressive motion into progressive and intermittent rectilinear motion.
Rod conducted between guides: 1o, by the simple contact of a wheel; 2o, by cross-straps or chains; 3o, by a projecting cam; 4o, by means of a helicoidal groove set upon the cylindrical axis of the wheel. To-and-fro movement, and heart-shaped or continuous cam, waves, and eccentrics. Simple screw and nut. Left and right handed screws; differential screw of Prony, called the micrometric screw. Ratio of the velocities in these different organs.
The example of the cam and pile-driver will be particularly insisted upon; 1o, in the case where this cam and the extremity of the rod have any continuous form given by a simple geometrical drawing; 2o, in the case where this form is defined geometrically by the condition, that the velocity is to be transmitted in an invariable ratio, as takes place for cams in the form of epicycloids or involutes of circles.
Transformation of a circular progressive motion into another similar to the first.
1o, by contact of cylinders or cones, the two axes being situated in the same plane; 2o, by straps, cords, or endless chains, the axes being in the same situation; 3o, by cams, teeth, and grooves, at very slight intervals; 4o, by the Dutch or universal joint. Case, where the axes are not situated in the same plane; use of an intermediate axis with beveled wheels or a train of pulleys; idea of White or Hooke’s joint in its improved form. Endless screw specially employed in the case of two axes at right angles to one another. Combinations or groupings of wheels. Idea of differential wheels. Relations of velocities in the most important of these systems of transmission.
Transformation of circular progressive Motion into rectilinear or alternating circular motion.
Ordinary circular eccentric. Eccentrics with closed waves or cams. Examples and graphical exercises in the class-rooms relative to the alternate action of the traveling frames of saw-mills, of the slides or entrance valves of steam-engines. Cams for working hammers and bellows.
Transformation of alternating circular motion into alternating rectilinear motion, or into intermittent and progressive circular motion.
Pump rods with or without circular sectors, &c. Examples taken from large exhausting pumps, fire-engines, and common pumps. Suggestions as to the best arrangement of the parts. Lagarousse’s lever, &c. Application of the principle relative to the instantaneous center of rotation to give the relations of the velocities in certain simple cases.
Transformation of alternating circular or rectilinear motions into progressive circular motion.
The knife-grinder’s treadle. System of great machines worked with connecting rods, fly-wheel, &c. Watt’s parallelogram, and the simplest modifications of it for steamboats, for instance. The most favorable proportions for avoiding the deviation of piston-rods. Simplification of parts in the modern steam-engines of Maudsley, Cavé, &c. Variable ratios of the velocities.
Of organs for effecting a sudden change of motion.
Suspendors or moderators, &c. Dead wheels and pulleys, &c. Mechanisms for stretching cords or straps, and make them change pulleys during the motion. Brakes to windmills, carriages, &c., &c. Case where the axes are rendered movable. Means for changing the directions and velocity of the motions. Coupled and alternate pulleys; alternate cones; castors moving by friction and rotation upon a plate or turning-cone; eccentric and orrery wheels. Means of changing the motion suddenly and by intervals; wheels with a detent pile-drivers; Dobo’s escapement for diminishing the shock, &c.
Geometrical Drawing of Wheel-work.
General condition which the teeth of toothed wheels must satisfy. Consequence resulting from this for the determination of the form of the teeth of one of two wheels, when the form of the teeth of the other wheel is given.
Cylindrical action of toothed wheels or toothed wheels with parallel axes. External engagement of the teeth; internal engagement. Particular systems of toothed wheels; lantern wheels, flange wheels, involutes of circles. Reciprocity of action; case where the action can not be rendered reciprocal. Pothook action. Details as to the form and dimensions given in practice to the teeth and the spaces which separate them.
Conical action of toothed wheels, or toothed wheels with converging axes. Practical approximate method of reducing the construction of a conical to that of a cylindrical engagement of toothed wheels.
Means of observation and apparatus proper for discovering experimentally the law of any given movement.
Simple methods practiced by Galileo and Coulomb in their experiments relative to the inclined plane and the motion of bodies sliding down it. Various means of observing and discovering the law of the translatory and rotatory motion of a body according as the motion is slow or rapid. Determination of the angular velocity, &c. The counter in machines. Apparatus of Mattei and Grobert for assigning the initial velocity of projectiles (musket balls.) Colonel Beaufoy’s pendulum apparatus. Chronometrical apparatus for continuous indications by means of a pencil. Eytelwein’s apparatus with bands, and its simplest modifications. Apparatus with cylinders or revolving disks. Use of the tuning-fork for measuring with precision very small fractions of time.
(The principal sorts of the apparatus above described are made to act under the eyes of the pupils.)
PART II.—EQUILIBRIUM OF FORCES APPLIED TO MATERIAL SYSTEMS.
Lesson 21.
Résumé of the notions acquired upon the subject of forces, and their effects on material points.
Principle of inertia, notion of force, of its direction, of its intensity. Principle of the equality of action and reaction. What is meant by the force of inertia? Principle of the independence and composition of the effects of forces. Forces proportional to the acceleration which they produce on the same body. Composition of forces. Relation between the accelerating force, the pressure, and the mass. Definition of the work done by a force. The work done by the resultant is equal to the sum of the works done by the components. Moment of a force in relation to an axis deduced from the consideration of the work of the force applied to a point turning about a fixed line. The moment of the resultant of several forces applied to a point is equal to the sum of the moments of the components. Corresponding propositions of geometry.
Lessons 22–25.
Succinct Notions upon the Constitution of Solid Bodies.
Every body or system of bodies may be regarded as a combination of material points isolated or at a distance, subject to equal and opposite mutual actions. Interior and exterior forces. Example of two molecules subject to their reciprocal actions alternately, attractive and repulsive, when the forces applied draw them out of their position of natural equilibrium. Different degrees of natural solidity, stability, or elasticity; they can only be appreciated by experience.
Equilibrium of any Systems whatever of Material Points.
General theorem of the virtual work of forces applied to any system whatever of material points. It is applicable to every finite portion of the system, provided regard be had to the actions exercised by the molecules exterior to the part under consideration. Determination of the sum of the virtual works of the equal and reciprocal actions of two material points. Demonstration of the six general equations of equilibrium of any system whatever. They comprise implicitly every equation deduced from a virtual movement compatible with the pre-supposed solidification of the system.
Theorem on the virtual work in the case of systems where one supposes ideal connections, such as the invariability of the distance of certain points of the system from one another, and the condition that certain of them are to remain upon curves either fixed or moving without friction.
Equilibrium of Solid Bodies.
The six general equations of equilibrium are sufficient as conditions of the equilibrium of a solid body. Theory of moments and couples.
APPLICATIONS.
Lessons 26–29. Equilibrium of Heavy Systems.
Recapitulation of some indispensable notions for the experimental determination of the center of gravity of solids when the law of their densities is unknown. Re-statement of the theorem relative to the work done by gravity upon a system of bodies connected or otherwise. In machines supposed without friction submitted, with the exception of their supports, to the action of gravity alone, the positions of stable or unstable equilibrium correspond to the highest or lowest points of the curve which would be described by the center of gravity of the system when made to move. Influence of defect of centering in its wheels, upon the equilibrium of a machine. Case where the center of gravity always remaining at the same height the equilibrium is neutral. Examples relative to the most simple drawbridges, &c.
Equilibrium of Jointed Systems.
Equilibrium of the funicular polygon deduced from direct geometrical considerations: Varignon’s theorem giving the law of the tensions by another polygon whose sides are parallel and proportional to the forces acting upon the vertices of the funicular polygon. Case of suspension bridges; investigation of the curve which defines the boundary of the suspension chain; tensions at the extremities.
Equilibrium of systems of jointed rigid bodies without friction. Determination of the pressure upon the supports and the mutual actions at the joints.
Equilibrium and stability of solid bodies submitted to the action of stretching or compressing forces.
Permanent resistance and limiting resistance of prisms to longitudinal extension and compression. Equilibrium and stability of a heavy solid placed upon a horizontal plane and submitted to the action of forces which tend to overset it. Resultant pressure and mean pressure; hypothetical distribution of the elements of the pressure on the base of support. Conditions of stability, regard being had to the limit of resistance of solid materials, co-efficient of stability deduced from it.
PART III.—ON THE WORK DONE BY FORCES IN MACHINES.
Lessons 30–39. General Notions.
Principle of work in the motion of a material point. Extension of this principle to the case of any material system whatever in motion. Considerations relative to mechanical work in various operations, such as the lifting of weights, sawing, planing, &c. It is the true measure of the productive activity of forces in industrial works. It may always be calculated either rigorously or approximately when the mathematical or experimental law which connects the force with the spaces described is given. Uniform work, periodical work, mean work, for the unit of time. Horse-power unit. Examples and various exercises, such as the calculation of the work corresponding to the elasticity of gases on the hypothesis of Mariotte’s law, the elongation of a metallic prism, &c.
Dynamometrical Apparatus.
Dynamometer of traction by a band or rotating disc or register. Dynamometer of rotation with simple spring, with band or register. Dynamometer of rotation with multiple springs and with register for the axles of powerful machines. Improved indicator of Watt.
(These pieces of apparatus are made to act under the eyes of the pupils.)
Work of Animal Prime Movers upon Machines.
Results of experience as to the values of the daily work which animal motors can supply under different circumstances without exceeding the fatigue which sleep and nourishment are capable of repairing.
Theory of the Transmission of Work in Machines.
Principal resistance. Secondary resistances. Two manners in which bodies perform the duty of motors. Ratio of work done to work expended always inferior to unity. Different parts of machines; receiver; organs of transmission; tools as machines.
Calculation of the Work due to the passive resistances in machines.
Résumé of the notions previously acquired on friction. Application to the inclined plane, to the printing-press, to guides or grooves, to the screw with a square thread; different cases of uniform motion being impossible under the action of forces of given directions. Friction of trunnions, pivots, eccentrics, and insertions of winch-handles. Prony’s dynamometrical brake; conditions of its application. Resistance to rolling; its laws according to experiment. Use of rollers and friction-wheels; their practical inconveniences.
Mixed friction of toothed wheels; the Dobo escapement: friction of the teeth in the endless screw.
Stiffness and friction of cords. Results of experience. Friction of cords and straps running round drums. Different applications; brakes; transmission by cords, endless straps, or chains.
Examples and exercises; effects of passive resistances in the capstan, the crane, pulleys, &c.
Lesson 40. Revision.
SECOND YEAR.
PART I.—DYNAMICS.— DYNAMICS OF A MATERIAL POINT.
Lessons 1–2. Completion of the Notions acquired on this Subject.
Differential equations of the motion of a material point submitted to the continued action of one or more forces. The acceleration of the projection of a point upon any axis or plane is due to the projection of the forces on this axis or plane. The acceleration along the trajectory is due to the tangential force. Relation of the curvature to the centripetal force. Introduction of the force of inertia into the preceding enunciations.
The increase of the quantity of motion projected upon an axis or taken along the trajectory is equal to the impulsion of the projected resultant, or to that of the tangential force. The total impulsion of a force is got by methods of calculation and of experiment analogous to those which relate to work. The increase of the moment of the quantity of motion in relation to any axis is equal to the total moment of the impulsions of the forces during the same interval of time; direct geometrical demonstration of this theorem. In decomposing the velocity of the moving body into a velocity in the plane passing through the axis of the moments, and a velocity of revolution perpendicular to this plane, we may replace the moment of the quantity of motion in space by the quantity of motion of revolution. Particular case known under the name of the principle of areas.
Extension of the preceding theorems to the case of relative motions. Apparent forces which must be combined with the real ones that the relative motion of a point may be assimilated to an absolute motion. Particular case of relative equilibrium. Influence of the motion of the earth upon the accelerating force of gravity.
DYNAMICS OF ANY MATERIAL SYSTEMS.
Lessons 3–8.
Principle or general rule which reduces questions in dynamics to questions in equilibrium by the addition of the forces of inertia to the forces which really act on the system. Equation of virtual work which expresses this equilibrium; it comprises in general the external and internal forces.
General Theorems.
These theorems, four in number, are founded upon the principle of the equality of action and reaction applied to internal forces. They may be deduced from the preceding rule, but the three last are obtained more simply by extending to a system of material points analogous theorems established for isolated material points.
General theorem of the motion of the center of gravity of a system. Particular case called principle of the conservation of the motion of the center of gravity.
General theorem on the quantities of motion and impulsions of exterior forces projected on any axis.
General theorems of moments of quantities of motion and impulsions of exterior forces, projected on any axis whatever.
General theorems of the moments of quantities of motion and impulsions of exterior forces about any axis. Analogy of these two theorems with the equations of the equilibrium of a solid, in which the forces are replaced by impulsions and quantities of motion.
Composition of impulsions, of quantities of motion, or the areas which represent them. All the equations which can be obtained by the application of the two theorems relative to quantities of motion and impulsions, reduce themselves to six distinct equations. Particular case called principle of the conservation of areas. Fixed plane of the resulting moment of the quantities of motion called plane of maximum areas.
General theorem of work and vis viva. Part which appertains to the interior forces in this theorem. Particular case called principle of the conservation of vires vivæ, where the sum of the elements of work done by the exterior and interior forces is the differential of a function of the co-ordinates of different points of the system. Application of the theorem of work to the stability of the equilibrium of heavy systems.
Extension of the preceding theorems to the case of relative motions. Particular case of relative equilibrium. Motion of any material system relative to axes always passing through the center of gravity, and moving parallel to themselves. Invariable plane of Laplace. Relation between the absolute vis viva of a material system, and that which would be due to its motion, referred to the system of movable axes above indicated.
Examples and Applications.
The following examples, amongst others, to be taken as applications or subjects of exercises relative to the general principles which precede.
Walking. Recoil of guns. Eolypile. Flight of rockets.
Pressure of fluid veins, resistance of mediums, &c. Direct collision of bodies more or less hard, elastic, or penetrable. Exchange of quantities of motion. Loss of vis viva under different hypotheses. Influence of vibrations and permanent molecular displacements.
Pile driving; advantage of large rammers. Comparison of effects of the shocks and of simple pressures due to the weight of the construction. Oblique collision, and ricochet. Data furnished by experiment.
Oscillations of a vertical elastic prism suspended to a fixed point, and loaded with a weight, neglecting the inertia, and the weight of the material parts of this prism. Case of a sudden blow. What is meant by the “resistance vive” of a prism to rupture? Results of experiments.
Work developed by powder upon projectiles, estimated according to the vis viva which it impresses on them, as well as upon the gun and the gases upon hypothesis of a mean velocity.
SPECIAL DYNAMICS OF SOLID BODIES.
Lessons 9–12. Simple Rotation of an invariable Solid about its Axis.
In applying to this case the first general rule of dynamics, the theorem of the moments of the quantities of motion, and the theorem of work, we are led to the notion of the moment of inertia; explanation of the origin of this name. The angular acceleration is equal to the sum of the moments of the exterior forces divided by the moment of inertia about the axis of rotation. Sum of the moments of the quantities of motion relative to this axis. Vis viva of a solid simply turning about an axis. What is meant by radius of gyration?
Remind of the geometrical properties of moments of inertia, of the ellipsoid which represents them, of the principal axes at any point, of those which are referred to the center of gravity.
Pressure which a rotating body exercises on its supports. Reduction of the centrifugal and tangential forces of inertia to a force which is the force of inertia of the entire mass accumulated at the center of gravity, and a couple.
Particular case where the forces of inertia have a single resultant; different examples. Center of percussion. Compound pendulum; length of the corresponding simple pendulum. Center of oscillation; reciprocity of the centers or axes of suspension and oscillation. Pressure upon the axis. Influence of the medium; experience proves that the resistance, varying with the velocity, changes the extent of the oscillations, but does not sensibly affect the time. Experimental determination of the center of oscillation and the moment of inertia about an axis.
Motion of an invariable Solid subject to certain Forces.
General notions on this subject. Motion of the center of gravity; motion of rotation about this point.
Lessons 13–19. Various Applications.
Motion of a homogeneous sphere or cylinder rolling upon an inclined plane, taking friction into account.
Motion of a pulley with its axis horizontal, solicited by two weights suspended vertically to a thread or fine string passing round the neck of the pulley, the axle of which rests upon movable wheels. Atwood’s machine serving to demonstrate the laws of the communication of motion.
Motion of a horizontal wheel and axle acted on by a weight suspended vertically to a cord rolled round the axle, or upon a drum with the same axis, and presenting an eccentric mass. To take account of the variable friction of the bearings, and the stiffness of the cord, with recourse, if necessary, to approximation by quadratures. Oscillations of the torsion balance.
Balistic pendulum. Condition that there may be no shock on the axis. Experimental determination of the direction in which the percussion should take place.
Theory of Huyghen’s conical pendulum considered as a regulator of machinery. How to take account of the inertia and friction of the jointed rods, as well as of the force necessary to move the regulating lever, &c.; appreciation of the degree of sensibility of the ball apparatus with a given uniform velocity.
Windlass with fly-wheel. Dynamical properties of the fly-wheel. Reduced formulæ for a crank with single or double action. Advantages and disadvantages of eccentric masses. Tendency of the tangential forces of inertia to break the arms. Numerical examples and computations.
Mutual action of rotating bodies connected by straps or toothed wheels in varying motion.
The wedge and punching-press. Stamping screw or lever used in coining, cams, lifting a pile or a hammer. To take account of the friction during the blow, and afterwards to estimate the loss of vis viva in cases which admit of it.
PART II.—SPECIAL MECHANICS OF FLUIDS.—HYDROSTATICS.
Lessons 20–22.
Principle of the equality of pressure in all directions. Propagation of the pressures from the surface to the interior of a fluid, and upon the sides of the vessel. Equations of equilibrium for any set of forces. Pressure exerted in the containing orifices. Measure of the pressure upon a plain portion of surface inclined or vertical (sluice-gate, embankments, &c.) Center of push or pressure. Pressure against the surfaces of a cylindrical tube. Effect, and resistance to oppose to the pressure. Manometer and piezometer. Equilibrium of a body plunged in a heavy fluid or floating at its surface. Stability of floating bodies. Metacenter. Laws of the pressure in the different atmospheric strata.
HYDRAULICS.
Lessons 23–27. Flow of Fluids through small Orifices.
Study of the phenomena which accompany this flow in the case of a thin envelop and a liquid kept at a constant level. Conditions of this constancy in the level, and the permanence of the motion in general. Motion of the lines of fluid; form; contraction; reversal and discontinuity of liquid veins. Fundamental formulæ for liquids and gases based upon the principle of vis viva, and Bernoulli’s hypothesis of parallel s or Borda’s of contiguous threads. Torricelli’s theorem relative to small orifices. What is called the theoretical expenditure, effective expenditure, and co-efficient of geometrical contraction. Co-efficient deduced from the effective expenditure. Its variations with the volume of the fluid contents, and the form of the inner surfaces of the reservoir. Results of the experiments of Michelotti, Borda, Bossut, &c. Phenomenon of adjutages. Venturi’s experiments; influence of atmospheric pressure; loss of vis viva; reduction of the velocity and augmentation of the expenditure. Results of experience relative to the co-efficient of expenditure, the form and range of the parabolic jets, showing the initial vis viva, and the loss of vis viva.
Large orifices.—Sluice holes and floodgates; reservoirs or open orifices; expenditure; practical formulæ and results of experiment. Influence of the proximity of the sides and the walls. Arrangement to avoid the effects of contraction or the losses of vis viva.
Flow through conducting Pipes and Open Canals.
Practical formulæ relative to the case of uniform s of great length. Measure of the pressures at different points of a conduit-pipe. Expression for the losses of effect due to corners and obstructions. Flow of gases. Principal methods of measuring the volume consumed adopted in practice. Floats. Pitot’s tube. Woltman’s mill. Register mill in air or gas. Waste in such instruments. Modulus and scale for water-supply.
PART III.—DIFFERENT MACHINES CONSIDERED IN THE STATE OF MOTION.
Lesson 28. General Considerations. Résumé of the Notions acquired on this Subject.
Equation of vis viva, and transmission of work in machines, account being taken of the different causes of power and resistance. Physical constitution of machines; receiver, communicators, and operator. Influence of the weights, of frictions, of shocks, and any changes in the vis viva. Parts with continuous or uniform motion, with alternating or oscillating motion. Laws of the motion on starting from rest, and when the stationary condition is established. The positions to which the maximum and minimum of the vis viva correspond are those in which there is equilibrium between all the forces, exclusive of the forces of inertia. Advantage of uniform or periodic motion. General methods for regulating the motion; symmetrical distribution of the masses and strains; flys and various regulators. Brakes and moderators; their inconveniences. Object and real advantages of machines.
Lessons 27–35. Hydraulic Wheels.
Vertical wheels with float-boards, with curved ladles, and with spouts. Figure of the surface of the fluid in these latter. Horizontal wheels working by float-boards, buckets, and reaction. Turbines. Description, play, and useful effects compared according to the results of experiment. Vertical wheels of windmills and steamboats. Screw propeller.
Windmills.
Description. Result of Coulomb’s observations.
On the principal kinds of Pumps.
Special organs of pumps. Valves and pistons, force pump, sucking pump; limit to the rise of the water. Sucking and force pump. Dynamical effects. Indication as to the losses of vis viva and the waste in different pumps. Explanation of the hydraulic ram. Air vessel. Fire pumps. Double action pumps.
Various Hydraulic Machines.
Hydraulic press. Water engine. Exhausting machines; norias; under and overshot wheels; Archimedes’ screw, construction and experimental data.
Lessons 36–39. Steam Engines.
Succinct description of the principal kinds of steam-engine with or without detent. Effects and advantages of the detent. Condenser. Air Pump. Furnace and feeding-pump.
Variable detent. Formulæ and experimental results.
Lessons 40–42. Revision.
Reflections on the totality of the subjects of the course.
IV. PHYSICS.—FIRST YEAR.
GENERAL PROPERTIES OF BODIES.—HYDROSTATICS.—HYDRODYNAMICS.
Lessons 1–5. Preliminary Notions.
Definitions of physics. Phenomena. Physical laws. Experiments are designed to make them spring out of the phenomena. Method of induction. Physical theories; different character of the experimental and mathematical methods.
General Properties of Bodies.
Extension. Measure of lengths. Vernier. Cathetometer. Micrometer screw. Spherometer. Dividing engine.
Divisibility. Porosity. Ideas generally received on the molecular constitution of bodies. These conceptions, which are purely hypothetical, must not be confounded with physical laws. Elasticity. Mobility. Inertia. Forces; their equilibrium, their effects, their numerical estimation.
Weight or Gravity.
Direction of gravity. Plumb-line. Relation between the direction of gravity and the surface of still water.
Weight. Center of gravity.
Experimental study of the motion produced by weight. In vacuum, all bodies fall with the same velocity. Disturbing influence of the air. Inclined plane of Galileo. Atwood’s machine. To prove by experiment; 1o the law of the spaces described; 2o the law of velocities. Morin’s self-registering apparatus with revolving cylinder.
Law of the independence of the effect produced by a force upon a body, and the motion anteriorily acquired by this body. Law of the independence of the effects of forces which act simultaneously upon the same body. Experimental demonstration and generalization of these laws. Law of the equality of action and reaction.
Mass. Acceleration. For equal masses the forces are as the accelerations which they produce. Relation between the force, mass, and acceleration. Collision.
General laws of uniformly accelerated motion. Formulæ.
Pendulum. Law of the isochronism of small oscillations and law of the lengths deduced from observation.
Method of coincidences or beats. Use of the pendulum as the measure of time. Simple pendulum; formulæ. Compound pendulum: the laws of the oscillations of a compound pendulum are the same as the laws of the oscillations of a simple pendulum whose length may be calculated.
Determination by means of the pendulum of the acceleration produced by gravity. This acceleration is independent of the nature of the body.
Remark that the formulæ for the motion of oscillation apply to the comparison of forces of any kind, that may be regarded as constant and parallel to themselves in all positions of the oscillating body.
Identity of gravity and universal attraction.
Measure of weights. Balance. Conditions to be attended to in making it. Absolute sensibility; proportional sensibility. Method of double weighing. Details of the precautions necessary in order to obtain an exact weight.
Different States of Bodies. Hydrostatics.
Solids. Cohesion. Transmission of external pressures.
Elasticity. The true laws of elasticity are unknown. Empirical laws in certain simple cases, and for a very small action. Elasticity of compression, extension, torsion. Experimental determination of the co-efficients of elasticity. Limits of elasticity. Limits of tenacity.
Ductility. Temper. Cold hammering. Annealing.
Liquids. Fluidity. Viscosity. Physical laws which form the basis of hydrostatics:—1o the transmission of external pressures is equal in all directions; 2o the pressure exercised in the interior of a liquid upon an element of a surface is normal to that element, and independent (as to amount) of its direction. These principles are demonstrated by the experimental verification of the consequences drawn from them.
Application to heavy liquids. Free surface, and surface de niveau. Pressure upon the parts of the containing vessel, and upon the bottom in particular; hydrostatic paradox; verificatory experiments. Haldat’s apparatus. Hydrostatic press.
Application to immersed or floating bodies (principle of Archimedes;) verificatory experiments. (In treating of the equilibrium of floating bodies, the conditions of stability are not gone into.)
Superposed liquids.
Communicating vessels. Water level. Spirit level; its use in instruments.
Densities of solids and liquids. Anemometers.
Compressibility of liquids. Piezometer. Correction due to the compressibility of the solid envelop.
Gas. Expansibility. Other properties common to liquids and gases. Principle of the equal transmission of pressures in all directions. Weight of gases. Pressure due to weight (principle of Archimedes.) Weight of body in air and in vacuum. Aerostation.
Superposed liquids and gases.
Communicating vessels. Barometer.
Detailed construction of barometer. Barometers of Fortin, Gay-Lussac, Bunten. Indication of the corrections necessary.
Mariotte’s law. Regnault’s experiments.
Manometer with atmospheric air—with compressed air. Bourdon’s manometer.
Law of the mixture of gases.
Air pump. Condensing pump.
Primary Notions of Hydrodynamics.
Toricelli’s principle. Mariotte’s vessel and syphon. Uniform flow of liquids. The same of gases.
Molecular Phenomena.
Cohesion of liquids. Adhesion of liquids to solids. Capillary phenomena. Apparent attractions and repulsions of floating bodies.
Adhesion of drops.
Molecular actions intervene as disturbing forces in the phenomena of the equilibrium and motion of liquids.
HEAT.
EFFECTS OF HEAT ON BODIES.
Lessons 6–9. Generalities.
General effects. Arbitrary choice of one of these effects to define the thermometric condition of a body. Conventional adoption of a thermometer. Definition of temperature.
Dilating Effects.
Definition of the co-efficients of linear, superficial, and cubic dilatation. Approximate relation between the numerical values of these three co-efficients. The value of the co-efficient of dilatation depends upon the thermometric substance and the temperature selected as the zero point. It becomes nearly independent of the zero point when the co-efficient is very small.
Relation between volume, density, and temperature. Linear dilatation of solid bodies. Ramsden’s instrument. Cubical dilatation of liquids. Dulong and Petit’s experiments on mercury. Discussion. Regnault’s experiments.
Cubical dilatation of solids and of other liquids when that of mercury is given.
Relations between the volume, density, and elasticity of a gas, and its temperature.
Cubical dilatation of gases. Experiments of Gay-Lussac, Rudberg, and M. Regnault. Advantage of varying the methods of experimenting in these delicate researches.
Methods based upon the changes of volume under a constant pressure, and upon the changes of pressure for a constant volume.
The disagreement of these two methods is due to deviations from the law of Mariotte.
The constancy of the co-efficients of dilatation previously defined is only approximately true.
Necessity of employing two different co-efficients of dilatation according as consideration is being had to the variations of volume to a given pressure, or of pressure to a given volume.
Empirical formulæ for the dilatation of liquids.
Graphical constructions.
Lesson 10. Thermometers.
Construction of thermometers. Mercurial thermometer. Details of construction. Fixed points. Different scales; their relation. Arbitrary scales. Change which takes place in the zero point. Different precautions to be observed in using the mercurial thermometer.
General want of comparability of mercurial thermometers with tubes of different material.
Air thermometers. They are comparable with one another within the limits of the errors of experiment, whatever the nature of the tube employed. This property entitles the air thermometer to a preference for all accurate measures. Comparison of the air and mercurial thermometers.
THERMOSCOPE, DIFFERENTIAL THERMOMETER, PYROMETERS, BREGUET’S THERMOMETER.
Lessons 11–13. Changes of State produced by Heat.
Exposition of the phenomena which accompany the liquefaction of solids and the solidification of liquids. Constancy of the temperature whilst the phenomenon is going on.
Sudden melting and freezing. Persistance of the liquid state beneath the melting point.
Influence of pressure.
Exposition of the phenomena which accompany the conversion of liquids or solids into vapor, and the inverse passage from the gaseous to the liquid or solid state. Constancy of the temperatures whilst the phenomenon is going on.
Influence of pressure.
Phenomena of ebullition in free space. Augmentation of the temperature and pressure in a confined space. Papin’s digester.
Properties of vapors in spaces and in gases. Saturated vapors. Their tension does not depend upon the space which they occupy, but only upon their temperature.
Effects of a diminution or increase of pressure without change of temperature; the same without change of pressure. Effects of lowering the temperature in a limited region of space occupied by vapor.
Tension of a saturated vapor at the boiling point of its liquid.
Measure of the tensions of the vapor of water. Experiments of Dalton, Gay-Lussac, Dulong, and Arago, and of M. Regnault.
Tables of the tensions of steam. Empirical formulæ. Graphical constructions.
It is assumed that non-saturated vapors are subject to the same laws as gases.
APPLICATIONS. CORRECTION OF THE BOILING-POINT IN THE CONSTRUCTION OF THERMOMETERS. BAROMETRICAL THERMOMETERS.
Lessons 14–16. Various Applications of the Laws previously established.
A phenomenon can not always be separated from the accessory phenomena which concur with it in producing the final result. Necessity of corrections to render complex results comparable inter se.
Density of solids when regard is had to the temperature and weight of the gases displaced by them.
Precautions to be attended to in the experiments. Empirical formulæ for the density of liquids. Maximum density of water. The temperature corresponding to the maximum must be determined graphically, or by interpolation.
Corrections for measures of capacity, for barometric measures.
The uncertainty of the corrections can not, in any considerable degree, affect the densities of solids and liquids.
Density of gases. Biot and Arago’s experiments. Special difficulties of the question. The uncertainty of the corrections may sensibly affect the results. Regnault’s method.
The same method may be applied to the determination of the co-efficient of dilatation for gases.
Density of vapors. Definition founded on the hypothetical application of the same laws to gases and vapors. Formulæ. Experimental method of Gay-Lussac and of Dumas. Corrections. Comparison of the two methods. Necessity of conducting the experiments at a distance from the saturation point. Latour’s experiments. Relations between the weight and volume of a gas, and its temperatures; between the weight and volume of a gas mixed with vapors, and its temperature. Various problems.
Hygrometry. Chemical hygrometry. Hygrometry by the dew-point. Psychrometry.
PROPAGATION OF HEAT.
Lessons 17–18. Propagation at a Distance.
Rapid propagation of heat at a distance, in vacuum, in gases, in certain liquid or solid mediums. Experiments which establish this.
Rays of heat. Velocity of propagation. Intensity of heat received at a distance. Intensity of heat received or emitted obliquely. Emitting power, power of absorption, reflection, diffusion. The emitting and absorbing power are expressible by the same number in terms of their proper units respectively.
Analysis of calorific radiations by absorption. Different effects of deathermanous or thermochroic medium. Different influences of increasing thicknesses of the combination of different mediums. Radiations proceeding from different sources, various effects of different mediums on these radiations.
The calorific radiations emanating from different sources, have all the characters of differently colored heterogeneous rays of light.
THEORY OF RADIATION AND OF THE DYNAMICAL EQUILIBRIUM OF TEMPERATURES. APPARENT REFLECTION OF COLD.
Lesson 19. Law of Cooling.
Definition of the rate of cooling. Many causes may conspire in the cooling of a body.
Cooling in space. Newton’s law only an approximation. Experimental investigation of the true law. Method to be followed in this investigation. The velocity of cooling is not a datum directly observable. It must be deduced provisionally from an empirical relation between the temperature and the time. Preliminary experiments. Course of the definitive experiments. Elementary experimental laws.
Hypothetical form of the function which expresses the velocity of cooling. To determine by means of the preceding experimental laws the unknown form of the function which expresses the law of radiation. Relation between the temperatures and the times. This relation only contains data immediately observable, and may be verified à posteriori.
The contents which enter into the preceding relation depend upon thermometric constants and the nature of the radiating surface.
The contact of a gas modifies the law of cooling.
Lessons 20–21. Propagation by Contact.
Slow propagation of heat in the interior of bodies, in solids, liquids, and gases. Confirmatory experiments. Hypothesis of partial radiation. Theoretical law resulting from this hypothesis upon the decrease of temperatures in a solid limited by two indefinite parallel planes maintained at constant temperatures. Determination of the co-efficient of conductibility by the experimental realization of these conditions. This experiment determines a numerical value of the co-efficients; it is not of a nature to serve as a check upon the theoretical principles. Enunciation of the law resulting from the same theoretical principles upon the decrease of temperatures in a thin bar heated at one end.
CALORIMETRY.
Lessons 22–23. Specific Heats.
Comparison of the quantities of heat. The quantities of heat are not proportioned to the temperatures. Definitions of the unity of heat. General method of mixtures to estimate the quantities of heat. Experimental precautions and corrections.
Application of the general method of mixtures. Specific heats of solids and liquids. Law of the specific heat of atoms. Heat absorbed by expansion, restored by the compression of bodies. Experiments on gases. Specific heats of gases under constant pressure. Measure of specific heats of gases under constant pressure. Special difficulties of the question. Succinct indication of one of the methods. Specific heats to a constant volume.
Lesson 24. Latent Heat.
Component heat of liquids absorbed into the latent state during fusion, restored to the free state during solidification.
Influence of the viscous state. Latent heat of ice. Ice calorimeter; its defects.
Component heat of vapors, absorbed into the latent state during vaporization, restored to the free state during condensation. Measure of the latent heat of vapors. Regnault’s experiments.
Empirical laws on the latent heat of vaporization.
Applications of Calorimetry.
Means of producing heat or cold; 1, by changes in density; 2, by changes of state. Freezing mixtures. Vaporization of liquids. Condensation of vapors.
Steam-boilers. Warming by hot air and hot water. Various problems. Sensations produced by a jet of vapor.
Different physical and chemical sources of heat; percussion, friction, chemical combinations, animal heat, natural heat of the globe, solar heat, &c. It will be remarked that mechanical work may become a source of heat, and heat a source of mechanical work.
STATICAL ELECTRICITY.—MAGNETISM.—STATICAL ELECTRICITY.
Lessons 25–27.
General phenomena. Distinction of bodies into conductors and non-conductors. Distinction of electricity into two kinds. Separation of the two electricities by friction. Hypothesis of electric fluids. Effects of vacuum of gases and vapors of points. Electrical attractions and repulsions. Electrization by influence. Case where the influenced body is already electrized. Sparks; power of points. Electrization by influence preceding the motion of light bodies.
Electroscopes.
Electrical machines of Van-Marum, Nairne, Armstrong.
Condenser. Accumulation of electricity upon its surface. Leyden jar. Batteries. Electrical discharges. Effects of electricity.
Condensing electroscope. Electrophorus.
Velocity of statical electricity.
Atmospherical electricity. Phenomena observed with a serene sky. Electricity of clouds. Storms. Lightning. Thunder. Effects of thunder. Return-shock. Lightning conductor.
Different sources of statical electricity.
MAGNETISM.
Lessons 28–30.
Natural magnets. Action upon iron and steel. Artificial magnets. The attractive action appears as if it were concentrated about the extremities of magnetic bars. First idea of poles.
Direction of a magnetized bar under the earth’s action. Reciprocal action of the poles of two magnets. Names given to the poles.
Phenomena of influence. Action of a magnet upon a bar of soft iron; upon a bar of steel. Coercive force. Effects of the rupture of a magnetized bar. Theoretical ideas on the constitution of magnets. More precise definition of the poles.
Action of the earth upon a magnet. The earth may be considered as a magnet. Its action may be destroyed by means of a magnet suitably placed. Astatic needles. The magnetic action of the earth is equivalent to a couple. Three constants define the couple of terrestrial action. Declination. Inclination. Intensity. Measure of the declination; of the inclination.
Magnetic metals. Influence of hammering, tempering, &c. Methods of magnetizing. Saturation. Loss of magnetism. Influence of heat. Magnetic lines. Armatures.
Magnetization by the earth’s influence. Means of determining the magnetic state of a body.
Measure of Magnetism and Electricity.
Lessons 31–32.
Coulomb’s balance. Distribution of magnetism on a magnetized bar; distribution of electricity at the surface of isolated conductors. Comparative discussion of the conditions of the two problems and the methods of experiment.
Laws of the magnetic attractions and repulsions. Law of electric attractions and repulsions. Comparative discussion of the conditions of the two problems, and the methods of experiment.
Determination of the law of magnetic attractions and repulsions by the method of oscillations.
Comparison of the magnetic intensity at different points of the earth’s surface.
Lessons 33–34. Revision.
Considerations on the totality of the subjects of the course.
SECOND YEAR.
DYNAMICAL ELECTRICITY.—GALVANISM.
Lessons 1–2.
Chemical sources of electricity. Experimental proofs. Arrangement devised by Volta to accumulate, at least in part, at the extremities of a heterogeneous conductor the electricity developed by chemical actions.
Pile. Tension at the two isolated extremities; at one single isolated extremity; at the two extremities reunited by a conductor. Continuous current of electricity. Poles. Direction of the current, &c.
Various modifications of the pile of Volta. Woollaston’s pile, Münch’s pile, &c. Dry piles; their application to the electroscope.
Principal effects of electricity in motion, and means of making the currents perceptible. Experiment of Oersted. Galvanoscopes.
Currents produced by heat in heterogeneous circuits. Thermo-electric piles. Thermometric graduation of thermo-electric piles.
Currents produced by the sources of statical electricity.
PROPERTIES OF CURRENTS.
Lesson 3. 1. Chemical Actions.
Definitions. Phenomena of decomposition and transference. Reaction of the elements transferred upon electrodes of different kinds.
Principles of electrotyping.
Causes of the variation of the current in ordinary piles; means of remedying this; Daniell’s pile. Bunsen’s pile.
Lessons 4–8. 2. Mechanical Properties.
Reciprocal actions of rectilinear or sinuous currents parallel or inclined. Reaction of a current on itself.
Reciprocal actions of helices or solenoids. Continuous rotation of currents by their mutual action; by reaction. Analogy of magnets and solenoids. Electro-dynamical theory of magnetism. Action of magnets upon currents and solenoids. Action of currents upon magnets. Experiments of Biot and Savart. Continual rotation of a current by a magnet; of a magnet by a magnet.
Action of the earth upon currents; it acts as a rectilinear current directed from east to west, perpendicularly to the magnetic meridian.
Continual rotation of a current by the action of the earth.
Astatic conductors.
Lessons 9–10. 3. Magnetic Properties.
Action of an interposed conductor upon iron filings.
Electro-magnets. Magnetization temporary or permanent. Principles of the electric telegraph. Electrometers. Reference to diamagnetic phenomena.
4. Electro-motive Properties.
Phenomena of induction by currents, by magnets. Phenomena of magnetism in motion. Induction of a current upon itself.
Induction of different orders.
Interrupted currents. Clarke’s machine.
Lesson 11. 5. Calorific Properties.
Influence of the nature of the interposed conductor; of its; of the intensity of the current. Unequal temperatures at the different junctions of a heterogeneous circuit.
6. Luminous Properties.
Incandescence of solid conductors. Spectrum of the electric light. Voltaic arc. Transfer of ponderable matter. Action of the magnet upon the Voltaic arc.
7. Physiological Action of Currents.
Some words on this subject. Muscles and nerves. Actions of discontinuous currents. Reotomic contrivances.
Reometry.
Compass of sines, of tangents. Experimental graduation of galvanometers.
The dynamical intensity of a current diminishes when the length of a current increases. Reostat.
Laws of the dynamical intensity of a current in a homogeneous circuit. Reduced length and resistance of a circuit. Specific co-efficients of resistance. Laws of the dynamic intensity of a current in a heterogeneous circuit.
The intensity of currents is in the inverse ratio of the total reduced length, and proportional to the sum of the electromotive forces. Formula of the pile. Discussion of the case of hydro-electric piles—thermo-electric piles. Conditions for the construction of a pile, with reference to the effects to be produced. Conditions for the construction of a galvanometer with reference to its intended application.
Laws of secondary currents in the simplest cases. The chemical intensity of a current is proportional to its dynamical intensity.
ACOUSTICS.
Lessons 12–15.
Noise, sound, quality of the sound, pitch, intensity, timbre. A state of vibration in a solid, liquid, or gaseous body is accompanied with the production of sound.
The pitch depends on the number of vibrations. Unison. Instruments for counting the vibrations:—1st. Graphic method. 2nd. Toothed wheels. 3rd. Lever. Feeling of concord. Musical scale. Gamut. Limit of appreciable sounds.
Study of vibrating motions in solids. Vibrating cords. Vibrations transversal, longitudinal. Experimental laws. Sonometer.
Spontaneous division of a cord into segments. Fundamental sounds. Harmonic sounds.
Straight and curved rods. Transversal and longitudinal vibrations. Experimental laws. Division into segments. Nodes. Ventral segments. Membranes.
Plane and curved plates. The vibrations divide them into “concamerations.” Nodal lines. Harmonic sounds.
Study of the vibrations in liquids and in gases.
Theoretical ideas upon the propagation of a vibratory motion in indefinite elastic media, on an indefinite cylindrical tube. Waves of condensation of dilatation. Progressive nodes and ventral divisions. Laws of the intensities of sound. Direct measure of the velocity of the propagation of sound in water. Measure of the velocity of the propagation of sound in air. Formulæ without demonstration. Comparison of the formulæ with experiment.
Sonorous waves reflected in an indefinite medium.
Fixed nodes and ventral divisions. Sonorous waves reflected in closed and open tubes. Fixed nodes and ventral divisions; the vibratory state and density thereat.
Series of sounds afforded by the same tube. Effect of holes.
Sonorous reflected waves in rods. Series of sounds afforded by the same rod vibrating longitudinally. Indirect measure of the velocity of sound in gases, liquids, and solids.
Experiments on the communication of vibrating motion in heterogeneous mediums, on the general direction of the vibrating motion communicated.
Intensification of sounds. Interferences. Beats. Different stringed and wind instruments. Means of setting them in vibration.
A few words on the organs of voice and hearing. Incompleteness of our knowledge on this subject.
OPTICS.
Lessons 16–17. Propagation of Light.
Propagation of light in a straight line. Rays of light. Geometrical theory of shadows. Velocity of light. Rœmer’s observations. Laws of intensity of light. Photometers of Bouguer, Rumford. Intensity of oblique rays. Comparison of illuminating powers. Total brightness. Intrinsic brightness.
Reflection.
Reflection of light: its laws. Experimental demonstration. Images formed by one or more plane mirrors. To ascertain if a looking-glass has its two faces parallel.
Spherical mirrors. Foci, formulæ. Discussion. Images by reflection. Measure of the radius of a spherical mirror.
Definition of caustics by reflection. Definition of the two spherical aberrations in mirrors.
Woollaston’s goniometer.
Lesson 18. Refraction.
Refraction of light in homogeneous mediums. Descartes’ law. Experimental demonstration for solids and liquids.
Inverse return of the rays. Successive refractions. Indices of transmission in terms of the principal indices. Consequences of Descartes’ law. Total reflection. Manner of observing it.
Irregular refractions. Mirage.
Refraction is always accompanied with the accessory phenomenon of dispersion.
Geometrical consequences of the law of refraction. Focus of a plane surface. Focus of a medium bounded by two parallel plane surfaces; by two plane surfaces inclined in the form of a prism.
Foci of a spherical surface; of a medium limited by two spherical surfaces. Lenses.
Formula for lenses. Discussion. Varieties of lenses. Optic center. Images. Measure of the focal distance of lenses.
Definition of caustics by refraction. Definition of the two spherical aberrations of a lens.
Lessons 19–20. Dispersion.
Unequal refrangibility of the differently colored rays which compose white light. Analysis of heterogeneous light by the prisms. Newton’s method. Solar spectrum. Homogeneity of the different colors. Second refraction of a homogeneous pencil. Experiment with crossed prisms. Precautions to be attended to in the experiments. The spectrum, obtained by Newton’s method, differs from the spectrum produced at the focus of a lens placed between the prism and the picture, according to the method of Fraunhofer. Reasons of the comparative purity of this latter spectrum. Fraunhofer’s lines. Different spectra of different sources of heterogeneous light. Marginal iridescence of a large pencil of natural light traversing a prism. Dispersion of light by lenses. Iridescence of focal images. Recomposition of light, by means of a prism at the focus of a spherical mirror or a lens, by the rapid rotation of a plane mirror, by the rotation of a disk with party-colored sectors. Compound colors.
Chemical and calorific radiations accompany luminous radiations.
Analysis of light by absorption. Characteristic action of transparent colored mediums upon different sorts of compound light. Different influences of increasing thickness. Effects of differently colored mediums upon heterogeneous light. Effects of differently colored mediums upon homogeneous rays separated by the prism.
Lesson 21. Measure of the Indices of Refraction.
Determination of the indices of refraction.
1. In solids. Measure of the refracting angles. Minimum of deviation. Measure of the corresponding deviation. Use of Fraunhofer’s lines.
2. In liquids.
3. In gases. Special difficulties of the question. Experimental method. Biot’s and Arago’s experiments.
Any power whatever of the index of refraction diminished by unit is sensibly proportional to the density of the gas. Method of Dulong founded on this remark.
Lessons 22–23. Application of the preceding Laws.
Rainbow. Different orders of bow.
Achromatism.
Achromatic prisms. Diasperometer achromatism of lenses; how to verify it. Definition of secondary spectra: their nature gives the means of recognizing, whether flint or crown glass predominates, in an imperfectly achromatic lens.
Instruments essentially consisting of an achromatic lens. Magic lantern; megascope; solar microscope; camera obscura; collimators.
Vision.
Summary description of the principal optical parts of the eye. They act like the lens of a camera obscura to form an image upon the retina. Distinct vision; optometers; short sight; long sight; spectacles.
Binocular vision; perspective peculiar to each eye; estimation of distances; sensation of solidity; stereoscope; estimation of magnitudes.
PERSISTENCE OF IMPRESSIONS; DIVERS EXPERIMENTS.
Lessons 24–26. Optical Instruments.
Camera lucida. A lens is necessary to reduce to the same apparent distance the two objects seen simultaneously. Instruments to assist the sight; simple microscope; the magnifying power; distinctness; field; advantage of a diaphragm; it modifies the field and the brightness variously according to its position.
Woollaston’s double glass; its advantages.
General principle of compound dioptrical instruments.
Compound microscope; experimental measure of its magnifying power, by means of the diaphragm, by means of the camera lucida.
Astronomical telescope; object glass; simple eye-glass. Necessity for a diaphragm; its place; the wires, their place; optic axis of a telescope. Parallax of the threads of the wires; magnifying power of the object-glass; of the eye-glass; field of view of a telescope.
Optic ring; different methods of measuring the magnifying power.
Distinctness of a telescope; night-glass.
Different distances of drawing out the eye-glass for short-sighted and long-sighted observers.
Different sorts of eye-pieces; positive eye-pieces; ordinary double eye-piece of the astronomical telescope. Ramsden’s eye-piece; treble eye-piece of the terrestrial telescope. Negative eye-pieces; simple eye-piece of Galileo. Compound ditto of Huyghens; advantages and disadvantages of these different combinations; general principle of catadioptrical instruments.
Lessons 27–29. Double Refraction.
Crystallized mediums do not all act upon light like homogeneous mediums.
Double refraction of Iceland spar: the extraordinary image turns round the ordinary image. The ordinary and extraordinary rays cross at the interior of the crystal.
Huyghens’ construction; measure of the ordinary and extraordinary indices of refraction; attractive and repulsive crystals; a ray falling perpendicularly does not always bifurcate in a camera with parallel faces, nor in a prism. Definition of uniaxial and biaxial crystals.
The dispersion of the ordinary ray differs from that of the extraordinary ray.
The two rays are unequally absorbed in many colored mediums. Tourmaline.
Doubly-refracting prisms; their construction. Use of doubly-refracting prisms to measure apparent diameters, &c.
Lessons 30–31. Polarization.
Successive refractions in doubly-refracting prisms. Special properties of the two rays emerging from the first doubly refracting crystal. Polarization by double refraction.
Reflection from transparent media polarizes the light partially or wholly according to the incidence. Brewster’s law. Reflection of polarized light from a transparent medium.
Simple refraction partially polarizes the light. Many successive refractions polarize it almost totally. Piles of glasses.
Different methods to obtain a ray of polarized light, 1st, by reflection; 2nd, by simple refraction; 3rd, by double refraction, by eliminating one of the refracted pencils;—by a screen,—by total reflection, Nicol’s prism, by absoption, tourmaline.
Distinctive characters of light completely or partially polarized.
Lessons 32–34. Theory of Undulations.
Hypothesis of luminous undulations.
Vibratory state of a simple ray of homogeneous light. Vibratory state at the inter of two simple rays of homogeneous light intersecting at a very small angle.
Experimental proofs in support of this hypothesis:
1st. Experiment with interferences, fringes. Their breadth is different for different colors; they give the various colors of the prism in white light. The alternately bright and dark sheets are hyperboloids of revolution. The measure of the fringes give the means of estimating the lengths of the undulations corresponding to different colors.
2nd. Colored rings of Newton, observed by reflection, by refraction. Law of the diameters; these vary in absolute length for different colors. Variously colored rings with white light. Reflected rings with a white spot at the center.
The theory of the undulations does not apply merely to theses phenomena. Explication of the laws of reflection and refraction. Definition of polarization in the system of waves. Elementary application of double refraction and the polarization which accompanies it in uniaxial crystals when the face of the crystal is parallel to the axis, and the plane of incidence normal or parallel to this axis.
Chemical and Calorific Radiations.
Chemical and calorific radiations are subject, like luminous radiations, to the laws of reflection, refraction, dispersion, double refraction, polarization, interferences.
Lessons 35–36. Revision.
Considerations on the totality of the subjects of the course.
MANIPULATIONS IN PHYSICS.
The practical exercises which constitute the subject of this programme will be performed in part by the pupils under the direction of the professors and répétiteurs, in part by the professors and répétiteurs, with the coöperation of the pupils.
FIRST YEAR.
Use of various instruments, designed for measuring lengths. Experiments on weight with Atwood’s machine, the inclined plane, Morin’s apparatus, and the pendulum.
Some experiments on elasticity.
Various verifications of the principles of hydrostatics and hydrodynamics.
Construction of aerometers.
Construction of a barometer, of a manometer. Various verifications of the law of Mariotte.
Various experiments with the air-pump.
Determination the density of solids or liquids by different methods.
Construction of a thermometer.
Experiments on the dilatation of liquids and solids by means of the ordinary thermometer and by means of the statical thermometer.
Experiments upon the dilatation of air by various methods.
Experiments upon the tension of vapors by different methods.
Determination of the density of vapors and gases by various methods.
Leading experiments on calorific radiation.
Experiments on cooling.
Determination of specific heats, heats of fusion, heats at which bodies pass into vapor.
Cooling mixtures.
Use of the chemical hygrometer, the wet bulb hygrometer.
Rehearsal of the leading experiments on magnetism.
To magnetize a needle, to reverse its poles.
Rehearsal of the principal experiments of statical electricity.
Experiments verificatory of the laws of electricity and magnetism.
Use of compasses.
SECOND YEAR.
Experiments upon the chemical actions of poles.
Leading experiments in electro-dynamics.
Leading experiments upon the magnetic properties of currents.
Experiments on induction.
Experiments on the calorific and luminous actions of currents.
Quantitative experiments on the laws of currents.
Experiments on the propagation of sound; on the vibrations of rods of plane or curved plates, membranes, sonorous tubes.
Experiments on mirrors, plane or curved.
Experiments on lenses. Experiments on the decomposition of light by the prism—by absorption. Measures of the indices of the refraction of solids. Use of the magnifying glass and microscope; measure of the magnifying power. Use of different telescopes, with and without corrections. Measure of the magnifying power. Experiments on double refraction and polarization. Experiments on interferences and colored rings.
ORGANIZATION AND CONDITION IN 1869.
The organization of the school, which is fixed by a Decree dated Nov. 30th, 1863, is of a military character. There is a staff of military officers in addition to, and quite separate from, the staff employed in the duties of instruction. The pupils wear uniform, which, however, is more civil than military in appearance. They are formed into four companies which together constitute a battalion; and, although they are not actually subject to the penal code of the army, the discipline maintained and the punishments inflicted are entirety military in character.
The military establishment remains exactly as it was in 1856, and consists of:
The Commandant, a General Officer, usually of the Artillery or the Engineers, at present a General of Artillery.
A Second Commandant, a colonel or lieutenant-colonel, chosen from among the former pupils of the school; at present a colonel of Engineers.
Three captains of Artillery and three captains of Engineers, as inspectors of studies, chosen also from former pupils of the school.
Six adjutants (adjudants), non-commissioned officers, usually such as have been recommended for promotion.
Slight changes have been made in the civil establishment; it now consists of:—
1. A Director of Studies, at present a colonel of Engineers.
2. Seventeen professors,12 (two additional professors for history) seventeen Répétiteurs and assistant Répétiteurs, and five drawing masters. Of the 17 professors, two are at present officers of Engineers, and one an officer of Artillery; the remainder are civilians, of whom three are members of the Academy of Sciences.
3. Five examiners for admission, and five for conducting the examinations at the school. All of these at present are civilians.
4. An administrative staff consisting of a treasurer, librarian, &c.; and a medical staff.
The general control or supervision of the school is vested, under the War Department, in four great boards or councils, viz.:—
1. A Board of Administration, composed of the Commandant, the Second Commandant, the Director of Studies, two professors, two captains of the military staff, and two members of the administrative staff. This board has the superintendence of all the financial business, and all the minutiæ of the internal administration of the school.
2. A Board of Discipline, consisting of the Second Commandant, the Director of Studies, three captains of the Military Staff, and one major of the army, selected from former pupils of the school.13 The duty of this board is to decide upon cases of misconduct.
3. A Board of Instruction, whose members are, the Commandant, the Second Commandant, the Director of Studies, the Examiners of Students, the Professors, and two captains of the Military Staff; and whose chief duty is to make recommendations relating to ameliorations in the studies and the programmes of admission and of instruction in the school to—
4. A Board of Improvement (Conseil de Perfectionnement), charged with the general control of the studies, and formed of:—
The Commandant, president,
The Second Commandant,
The Director of Studies,
Two delegates from the Naval Department,
Two delegates from the Department of Public Works,
One delegate from the Home or Finance Department,
Three delegates from the War Department,
Two members of the Academy of Sciences,
Two examiners of students,
Three professors of the school.
The delegates from the public departments are appointed by the respective ministers; the members of the Academy, the examiners, and the professors are selected by the Minister of War. The real management of the school, so far as the course of instruction is concerned, is in the hands of the Conseil de Perfectionnement; it will be seen that of the 18 members composing it more than half are entirely independent of the school, and are men of eminence in the various public services for which the instruction at the Polytechnic is preparatory. One of the chief duties of the Council is to see that the studies form a good preparation for those of the more special schools (Ecoles d’ Application) for the civil and military services; and the eminent character of its members gives great weight to the recommendations they make to the Minister of War.
The annual expenses of the school, as extracted from the Budget for 1869, are as follows:—
| Francs. | ||
|---|---|---|
| Pay of staff, professors, &c., | 331,850 | |
| Instruction, maintenance, examination of candidates, clothing, books, &c., | 321,073 | |
| Francs. | ||
| Outfits for 30 new pupils at 600 francs each | 18,000 | |
| Allowances (premières mises) to 25 exhibitioners on admission to the military services at 750 fr. each | 18,750 | |
| 36,750 | ||
| Maintenance and repair of buildings, | 30,000 | |
| Total sum charged in the schools estimate, | 719,673 | |
| Add regimental pay of 28 officers and non-commissioned officers employed at the school, | 85,515 | |
| Total expenditure | 805,188 | |
| Deduct repayments from pupils, | 237,000 | |
| Cost to the State, | 568,188 | |
| Or about 22,720l. | ||
The chief changes that have been made in regard to the course of instruction since 1856, may be summarized as follows:
1. The more elementary portions of chemistry and physics which are required in the entrance examination, but which were formerly repeated at the school, have been omitted. The course of instruction in these subjects is now confined to the more advanced portions which do not enter into the entrance examination.
2. The mathematical courses have in some points been slightly curtailed, and the number of lectures in French literature and German have been diminished. By the modifications thus made in the programmes, it has been found possible to shorten the whole course of study and to increase the length of the vacations.
3. The subject of “Military Art,” which formerly entered into the final examination is no longer taken into consideration in determining the order of merit of the pupils. In this respect the course of instruction may be said to have even less of a military character than formerly. Topographical drawing is the single military subject which has any influence on the final classification of the pupils, and this only to a very slight extent.
4. History has been introduced as a subject of instruction. This change was made in 1862. The course comprises general history, both ancient and modern, but more especially the history of France in modern times. The introduction of this subject appears to have arisen partly from a feeling that an acquaintance with history was a necessary element of a liberal education, and partly from a wish to meet, to some extent, an objection often made to the Polytechnic course of instruction, that it was too deficient in studies of a literary character. History, however, like military art, is evidently still regarded as a subject of only secondary importance and has no influence on the final classification.
5. A diminution has been made in the number of examinations during the course, by the suppression of one of the half-yearly examinations by the professors (interrogations générales, as distinct from the interrogations particulières) in each year. Further reference will be made to this point when speaking of the examinations at the school.
6. The importance of written exercises in determining the respective merits of the pupils has been decreased, apparently from the difficulty of establishing a security that such compositions were the unaided work of the individual.
The following table shows the present course of instruction during the two years, and the alterations which have been made in the number of lectures in each subject since 1856:—
| Subject.—First Year’s Course. | Lectures in—1868. | 1856. | |
|---|---|---|---|
| Analysis | Differential calculus, | 25 | 28 |
| Integral calculus, | 18 | 20 | |
| Descriptive geometry and geometrical drawing, | 32 | 38 | |
| Mechanics and machinery, | 40 | 40 | |
| Physics, comprising heat and electricity, | 30 | 34 | |
| Chemistry:—The metals, | 30 | 38 | |
| Astronomy and geodesy, | 30 | 35 | |
| French composition and literature, | 25 | 30 | |
| History, | 25 | 0* | |
| German, | 25 | 30 | |
| Figure and landscape drawing, | 48 | 50 | |
| Second Year’s Course. | |||
| Analysis:—Integral calculus, | 32 | 32 | |
| Stereotomy:—Geometrical drawing of constructions in timber and masonry, | 28 | 32 | |
| Mechanics:—Dynamics, hydrostatics, and machinery, | 40 | 42 | |
| Physics:—Acoustics, optics, and heat, | 30 | 36 | |
| Chemistry:—Continuation of the metals and organic chemistry, | 30 | 38 | |
| Architecture and buildings, construction of roads, canals, and railways, | 40 | 40 | |
| French composition and literature, | 25 | 30 | |
| History, | 25 | 0* | |
| German, | 25 | 30 | |
| Military art, | 20 | 20 | |
| Topography, | 2 | 10 | |
| Figure and landscape drawing, | 48 | 48 | |
In connection with several of the courses, such as descriptive geometry, stereotomy, machinery, and architecture, much drawing is done by the pupils; hand sketches are taken of the diagrams shown in the lecture-room, and finished drawings are afterwards executed in the salles d’étude. In addition to this, 30 attendances of two or three hours each, distributed over the two years, are especially devoted to drawing more elaborate plans and elevations of architectural constructions and machinery. The practical applications of the theoretical instruction are limited to manipulations in the laboratory in connection with the course of lectures on chemistry and physics. Towards the close of the second year the pupils are also taken to visit some of the large manufacturing establishments in Paris, in order to gain a practical acquaintance with machinery.
All the subjects taught at the school are obligatory, but history and military art, as already stated, have no influence in determining the order of merit of the pupils in the final result.
The only instruction in practical military exercises, which is compulsory upon all, is that in drill. The pupils are exercised under arms in company drill, and are also occasionally drilled as a battalion; but very little importance is attached to this point—the only really military portion of their training. Drill goes on only for about three months in each year during the spring and summer, and even during this brief period only takes place about twice a week. By the regulations of the school the pupils should be exercised in musketry practice, but although they are armed with the Chassepot rifle this regulation is never carried out. Instruction is given in fencing and gymnastics, but attendance at both is voluntary, and scarcely more than half the pupils take advantage of it. Neither riding nor swimming are taught at the school.
The school year commences about the 1st of November, and terminates about the first of August. Some seven months of the year are given up to lectures and the ordinary routine of study; about two months are occupied with the annual examinations and private preparation for them; the remaining three months—August, September, and October—are the vacation. In addition to this long vacation, from eight to twelve days are allowed after the periodical examination, which takes place near the end of February, at the close of the first portion of each year’s study.
One peculiarity in the arrangements of the school is that the subjects of each year’s course are not all studied simultaneously. The lectures in the main subjects of instruction—those which, as a rule, present the most difficulty—are divided into courses which continue only during a certain portion of each year. Thus in the junior division, analysis and descriptive geometry are the mathematical subjects studied during the first three months, or three months and a half. The course in them is then concluded; an examination by the professors (interrogation générale) is held in these subjects, and they are laid aside for the remainder of the year, though they enter into the examination at the close of the year. Their place is then taken by a course of lectures in mechanics and geodesy. Similarly in the second year, analysis and mechanics are the subjects of the first course of lectures, at the termination of which there is an examination; and for the remainder of the year no further lectures in them are given, stereotomy and military art taking their place.
The subjects involving as a rule less difficulty—such as history, French literature, German, and drawing—are spread over the whole year, forming generally the evenings’ occupation.
Notes
* Introduced in 1862.