Chapter VI: Part I: Military System and Schools in France (5)

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 2^{e} 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 1^{re} 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 _n_^{th} 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_, _a^{x}_, (1 + _x^{m}_) and log.(1 + _x_.) Numerical applications. Representation of cos _x_ and sin _x_ by imaginary exponential quantities.

Developments of cos^{m} _x_ and sin^{m} _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, 0^0, 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 sections of a surface. Maximum and minimum radii. Relations between these and that of any section, 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+ax^{2}_). Different transformations which render the differential rational. Reduction of the radical to one of the forms

√(x^{2}+x^{2}), √(a^{2}-x^{2}), √(x^{2}-a^{2}).

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

x^{m} dx dx x^{m} dx
---------- , ---------------- , --------
√(1-x^{2}) x^{m} √(1-x^{2}) √(ax-x)

Integration of the differentials

dx dx
F(log x)-- , F sin^{-1}x ---------- ,
x √(1-x^{2})

x(log x^{n})dx, x^{m} e^{ax}dx, (sin^{-1}x^{m})dx.

Integration of the differentials e^{ax} sin _bxdx_ and e^{ax} cos _bxdx_.

Integration of (sin x^{m}.)(cos x^{n}) _dx_.

Integration by series. Application to the expression

dx
-------------------
√(ax-x^{2}) √(1-bx)

Application of integration by series to the development of functions, the development of whose derivatives is given: tan^{-1}_x_, sin^{-1}_x_, 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 ∫{(sin _ax_)/_x_}_dx_, and ∫{(cos _bx_ sin _ax_)/_x_}_dx_, between the limits _0_ and _x_. Remarkable discontinuity which these integrals present.

Determination of ∫e^{-_x_^{2}}_dx_ and ∫e^{-_x_^{2}}cos _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(_d y_/_d x_). 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 _d^{m}y_/_dx^{m}_ = φ(_x_.)

Integration of the equation _d^{2}y_/_dx^{2}_ = 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 _m_^{th} 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 _d^{2}u/dy^{2} = a^{2}d^{2}u/dx^{2}_. 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 sections, 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 intersection 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, sections, 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 sections 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 sections 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 sections 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 sections.

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 sections.

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 intersection 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 intersection.

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 sections. 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 sections.

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 section, whose diameters are proportional to the square roots of the radii of curvature of the normal sections to which they are tangents. This curve is called the indicatrix. It is defined in form and position, but not in magnitude. The normal sections tangential to the axes of the indicatrix are called the principal sections.

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 section 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 sections 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, sections 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, sections 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 sections. 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.

Intersection 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 sections of an elliptical or circular cylinder. Right conoid, having for directrices the axis of the circular cylinder and an orthogonal trajectory. The intersection of this conoid by a cylinder about the same axis is an orthogonal trajectory for a series of parallel sections.

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 sections 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 sections, 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._