PART I MILITARY SYSTEM AND SCHOOLS IN FRANCE. (continued 2)
Reading edition
PROGRAMME OF ANALYTICAL GEOMETRY.
1. GEOMETRY OF TWO DIMENSIONS.
Rectilinear co-ordinates.—Position of a point on a plane.
Representation of geometric loci by equations.
Homogeneity of equations and of formulas.—Construction of algebraic expressions.
Transformation of rectilinear co-ordinates.
Construction of equations of the first degree.—Problems on the right line.
Construction of equations of the second degree.—Division of the curves which they represent into three classes.—Reduction of the equation to its simplest form by the change of co-ordinates.5
Problem of tangents.—The coefficient of inclination of the tangent to the curve, to the axis of the abscissas, is equal to the derivative of the ordinate with respect to the abscissa.
Of the ellipse.
Centre and axes.—The squares of the ordinates perpendicular to one of the axes are to each other as the products of the corresponding segments formed on that axis.
The ordinates perpendicular to the major axis are to the corresponding ordinates of the circle described on that axis as a diameter, in the constant ratio of the minor axis to the major.—Construction of the curve by points, by means of this property.
Foci; eccentricity of the ellipse.—The sum of the radii vectors drawn to any point of the ellipse is constant and equal to the major axis.—Description of the ellipse by means of this property.
Directrices.—The distance from each point of the ellipse to one of the foci, and to the directrix adjacent to that focus, are to each other as the eccentricity is to the major axis.
Equations of the tangent and of the normal at any point of the ellipse.6—The point in which the tangent meets one of the axes prolonged is independent of the length of the other axis.—Construction of the tangent at any point of the ellipse by means of this property.
The radii vectores, drawn from the foci to any point of the ellipse, make equal angles with the tangent at that point or the same side of it.—The normal bisects the angle made by the radii vectores with each other.—This property may serve to draw a tangent to the ellipse through a point on the curve, or through a point exterior to it.
The diameters of the ellipse are right lines passing through the centre of the curve.—The chords which a diameter bisects are parallel to the tangent drawn through the extremity of that diameter.—Supplementary chords. By means of them a tangent to the ellipse can be drawn through a given point on that curve or parallel to a given right line.
Conjugate diameters.—Two conjugate diameters are always parallel to supplementary chords, and reciprocally.—Limit of the angle of two conjugate diameters.—An ellipse always contains two equal conjugate diameters.—The sum of the squares of two conjugate diameters is constant.—The area of the parallelogram constructed on two conjugate diameters is constant.—To construct an ellipse, knowing two conjugate diameters and the angle which they make with each other.
Expression of the area of an ellipse in function of its axes.
Of the hyperbola.
Centre and axes.—Ratio of the squares of the ordinates perpendicular to the transverse axes.
Of foci and of directrices; of the tangent and of the normal; of diameters and of supplementary chords.—Properties of these points and of these lines, analogous to those which they possess in the ellipse.
Asymptotes of the hyperbola.—The asymptotes coincide with the diagonals of the parallelogram formed on any two conjugate diameters.—The portions of a secant comprised between the hyperbola and its asymptotes are equal.—Application to the tangent and to its construction.
The rectangle of the parts of a secant, comprised between a point of the curve and the asymptotes, is equal to the square of half of the diameter to which the secant is parallel.
Form of the equation of the hyperbola referred to its asymptotes.
Of the parabola.
Axis of the parabola.—Ratio of the squares of the ordinates perpendicular to the axis.
Focus and directrix of the parabola.—Every point of the curve is equally distant from the focus and from the directrix.—Construction of the parabola.
The parabola may be considered as an ellipse, in which the major axis is indefinitely increased while the distance from one focus to the adjacent summit remains constant.
Equations of the tangent and of the normal.—Sub-tangent and sub-normal. They furnish means of drawing a tangent at any point of the curve.
The tangent makes equal angles with the axis and with the radius vector drawn to the point of contact.—To draw, by means of this property, a tangent to the parabola, 1o through a point on the curve; 2o through an exterior point.
All the diameters of the parabola are right lines parallel to the axis, and reciprocally.—The chords which a diameter bisects are parallel to the tangent drawn at the extremity of that diameter.
Expression of the area of a parabolic segment.
Polar co-ordinates.—To pass from a system of rectilinear and rectangular co-ordinates to a system of polar co-ordinates, and reciprocally.
Polar equations of the three curves of the second order, the pole being situated at a focus, and the angles being reckoned from the axis which passes through that focus.
Summary discussion of some transcendental curves.—Determination of the tangent at one of their points.
Construction of the real roots of equations of any form with one unknown quantity.—Investigation of the inters of two curves of the second degree.—Numerical applications of these formulas.
2. GEOMETRY OF THREE DIMENSIONS.
The sum of the projections of several consecutive right lines upon an axis is equal to the projection of the resulting line.—The sum of the projections of a right line on three rectangular axes is equal to the square of the right line.—The sum of the squares of the cosines of the angles which a right line makes with three rectangular right lines is equal to unity.
The projection of a plane area on a plane is equal to the product of that area by the cosine of the angle of the two planes.
Representation of a point by its co-ordinates.—Equations of lines and of surfaces.
Transformation of rectilinear co-ordinates.
Of the right line and of the plane.
Equations of the right line.—Equation of the plane.
To find the equations of a right line, 1o which passes through two given points, 2o which passes through a given point and which is parallel to a given line.
To determine the point of inter of two right lines whose equations are known.
To pass a plane, 1o through three given points; 2o through a given point and parallel to a given plane; 3o through a point and through a given right line.
Knowing the equations of two planes, to find the projections of their inter.
To find the inter of a right line and of a plane, their equations being known.
Knowing the co-ordinates of two points, to find their distance.
From a given point to let fall a perpendicular on a plane; to find the foot and the length of that perpendicular (rectangular co-ordinates).
Through a given point to pass a plane perpendicular to a given right line (rectangular co-ordinates).
Through a given point, to pass a perpendicular to a given right line; to determine the foot and the length of that perpendicular (rectangular co-ordinates).
Knowing the equations of a right line, to determine the angles which that line makes with the axes of the co-ordinates (rectangular co-ordinates).
To find the angle of two right lines whose equations are known (rectangular co-ordinates).
Knowing the equation of a plane, to find the angles which it makes with the co-ordinate planes (rectangular co-ordinates).
To determine the angle of two planes (rectangular co-ordinates).
To find the angle of a right line and of a plane (rectangular co-ordinates).
Surfaces of the second degree.
They are divided into two classes; one class having a centre, the other not having any. Co-ordinates of the centre.
Of diametric planes.
Simplification of the general equation of the second degree by the transformation of co-ordinates.
The simplest equations of the ellipsoid, of the hyperboloid of one sheet and of two sheets, of the elliptical and the hyperbolic paraboloid, of cones and of cylinders of the second order.
Nature of the plane s of surfaces of the second order.—Plane s of the cone, and of the right cylinder with circular base.—Anti-parallel of the oblique cone with circular base.
Cone asymptote to an hyperboloid.
Right-lined s of the hyperboloid of one sheet.—Through each point of a hyperboloid of one sheet two right lines can be drawn, whence result two systems of right-lined generatrices of the hyperboloid.—Two right lines taken in the same system do not meet, and two right lines of different systems always meet.—All the right lines situated on the hyperboloid being transported to the centre, remaining parallel to themselves, coincide with the surface of the asymptote cone.—Three right lines of the same system are never parallel to the same plane.—The hyperboloid of one sheet may be generated by a right line which moves along three fixed right lines, not parallel to the same plane; and, reciprocally, when a right line slides on three fixed lines, not parallel to the same plane, it generates a hyperboloid of one sheet.
Right-lined s of the hyperbolic paraboloid.—Through each point of the surface of the hyperbolic paraboloid two right lines may be traced, whence results the generation of the paraboloid by two systems of right lines.—Two right lines of the same system do not meet, but two right lines of different systems always meet.—All the right lines of the same system are parallel to the same plane.—The hyperbolic paraboloid may be generated by the movement of a right line which slides on three fixed right lines which are parallel to the same plane; or by a right line which slides on two fixed right lines, itself remaining always parallel to a given plane. Reciprocally, every surface resulting from one of these two modes of generation is a hyperbolic paraboloid.
General equations of conical surfaces and of cylindrical surfaces.
VI. DESCRIPTIVE GEOMETRY.
The general methods of Descriptive Geometry,—their uses in Stone-cutting and Carpentry, in Linear Perspective, and in the determination of the Shadows of bodies,—constitute one of the most fruitful branches of the applications of mathematics. The course has always been given at the Polytechnic School with particular care, according to the plans traced by the illustrious Monge, but no part of the subject has heretofore been required for admission. The time given to it in the school, being however complained of on all sides as insufficient for its great extent and important applications, the general methods of Descriptive Geometry will henceforth be retrenched from the internal course, and be required of all candidates for admission.
As to the programme itself, it is needless to say any thing, for it was established by Monge, and the extent which he gave to it, as well as the methods which he had created, have thus far been maintained. We merely suppress the construction of the shortest distance between two right-lines, which presents a disagreeable and useless complication.
Candidates will have to present to the examiner a collection of their graphical constructions (épures) of all the questions of the programme, signed by their teacher. They are farther required to make free-hand sketches of five of their épures.
PROGRAMME OF DESCRIPTIVE GEOMETRY.
Problems relating to the point, to the straight line, and to the plane.7
Through a point given in space, to pass a right line parallel to a given right line, and to find the length of a part of that right line.
Through a given point, to pass a plane parallel to a given plane.
To construct the plane which passes through three points given in space.
Two planes being given, to find the projections of their inter.
A right line and a plane being given, to find the projections of the point in which the right line meets the plane.
Through a given point, to pass a perpendicular to a given plane, and to construct the projections of the point of meeting of the right line and of the plane.
Through a given point, to pass a right line perpendicular to a given right line, and to construct the projections of the point of meeting of the two right lines.
A plane being given, to find the angles which it forms with the planes of projection.
Two planes being given, to construct the angle which they form between them.
Two right lines which cut each other being given, to construct the angle which they form between them.
To construct the angle formed by a right line and by a plane given in position in space.
Problems relating to tangent planes.
To draw a plane tangent to a cylindrical surface or to a conical surface, 1o through a point taken on the surface; 2o through a point taken out of the surface; 3o parallel to a given right line.
Through a point taken on a surface of revolution, whose meridian is known, to pass a plane tangent to that surface.
Problems relating to the inter of surfaces.
To construct the made, on the surface of a right and vertical cylinder, by a plane perpendicular to one of the planes of projection.—To draw the tangent to the curve of inter.—To make the development of the cylindrical surface, and to refer to it the curve of inter, and also the tangent.
To construct the inter of a right cone by a plane perpendicular to one of the planes of projection. Development and tangent.
To construct the right of an oblique cylinder.—To draw the tangent to the curve of inter. To make the development of the cylindrical surface, and to refer to it the curve which served as its base, and also its tangents.
To construct the inter of a surface of revolution by a plane, and the tangents to the curve of inter.—To resolve this question, when the generating line is a right line which does not meet the axis.
To construct the inter of two cylindrical surfaces, and the tangents to that curve.
To construct the inter of two oblique cones, and the tangents to that curve.
To construct the inter of two surfaces of revolution whose axes meet.
VII. OTHER REQUIREMENTS.
The preceding six heads complete the outline of the elementary course of mathematical instruction which it was the object of this article to present; but a few more lines may well be given to a mere enumeration of the other requirements for admission to the school.
Mechanics comes next. The programme is arranged under these heads: Simple motion and compound motion; Inertia; Forces applied to a free material point; Work of forces applied to a movable point; Forces applied to a solid body; Machines.
Physics comprises these topics: General properties of bodies; Hydrostatics and hydraulics; Densities of solids and liquids; Properties of gases; Heat; Steam; Electricity; Magnetism; Acoustics; Light.
Chemistry treats of Oxygen; Hydrogen; Combinations of hydrogen with oxygen; Azote or nitrogen; Combinations of azote with oxygen; Combination of azote with hydrogen, or ammonia; Sulphur; Chlorine; Phosphorus; Carbon.
Cosmography describes the Stars; the Earth; the Sun; the Moon; the Planets; Comets; the Tides.
History and Geography treat of Europe from the Roman Empire to the accession of Louis XVI.
German must be known sufficiently for it to be translated, spoken a little, and written in its own characters.
Drawing, besides the épures of descriptive geometry, must have been acquired sufficiently for copying an academic study, and shading in pencil and in India ink.
Will not our readers agree with M. Coriolis, that “There are very few learned mathematicians who could answer perfectly well at an examination for admission to the Polytechnic School”?
SCHOOLS OF PREPARATION FOR THE POLYTECHNIC SCHOOL
There are strictly speaking no Junior Military Schools preparatory to the Polytechnic School, or to the Special Military School at St. Cyr. These schools are recruited in general from the Lycées and other schools for secondary instruction, upon which they exert a most powerful influence. Until 1852 there was no special provision made in the courses of instruction in the Lycées for the mathematical preparation required for admission into the Polytechnic, and the Bachelor’s degree in science could not be obtained without being able to meet the requirements in Latin, rhetoric, and logic for graduation in the arts, which was necessary to the profession of law, medicine, and theology. In consequence, young men who prepared to be candidates for the preliminary examinations at the Polytechnic and the St. Cyr, left the Lycées before graduation in order to acquire more geometry and less literature in the private schools, or under private tuition.
A new arrangement, popularly called the Bifurcation, was introduced by the Decrees of the 10th of April, 1852; and has now come into operation. The conditions demanded for the degree in science were adapted to the requirements of the Military Schools; and in return for this concession it is henceforth to be exacted from candidates for the Military Schools. The diploma of arts is no longer required before the diploma of science can be given. The instruction, which in the upper classes of the Lycées had hitherto been mainly preparatory for the former, takes henceforth at a certain point (called that of Bifurcation) two different routes, conducting separately, the one to the baccalaureate of arts, the other to that of science. The whole system of teaching has accordingly been altered. Boys wanting to study algebra are no longer carried through a long course of Latin; mathematics are raised to an equality with literature; and thus military pupils—pupils desirous of admission at the Polytechnic and St. Cyr, may henceforth, it is hoped, obtain in the Lycées all the preparation which they had latterly sought elsewhere.
Under this new system the usual course for a boy seems to be the following:—
He enters the Lycée, in the Elementary Classes; or, a little later, in the Grammar Classes, where he learns Latin and begins Greek. At the age of about fourteen, he is called upon to pass an examination for admission into the Upper Division, and here, in accordance with the new regulations, he makes his choice for mathematics or for literature, the studies henceforth being divided, one course leading to the bachelorship of science, the other to that of arts.
In either case he has before him three yearly courses, three classes—the Third, the Second, and what is called the Rhetoric. At the close of this, or after passing, if he pleases, another year in what is called the Logic, he may go up for his bachelor’s degree. The boy who wants to go to St. Cyr or the Polytechnic chooses, of course, the mathematical division leading to the diploma he will want, that of a bachelor of science. He accordingly begins algebra, goes on to trigonometry, to conic s, and to mechanics, and through corresponding stages in natural philosophy, and the like. If he chooses to spend a fourth year in the Logic, he will be chiefly employed in going over his subjects again. He may take his bachelor’s degree at any time after finishing his third year; and he may, if he pleases, having taken that, remain during a fifth or even a sixth year, in the class of Special Mathematics.
If he be intended for St. Cyr, he may very well leave at the end of his year in Rhetoric, taking of course his degree. One year in the course of Special Mathematics will be required before he can have a chance for the Polytechnic. Usually the number of students admitted at the latter, who have not passed more than one year in the mathematiques spéciales is very small. Very probably the young aspirant would try at the end of his first year in this class, and would learn by practice to do better at the end of the second.
The following are the studies of the mathematical of the upper division as laid down by the ordinance of 30th August, 1854.
The Third Class (Troisième,) at fourteen years old.
Arithmetic and first notions of Algebra. Plane Geometry and its applications. First notions of Chemistry and Physics. General notions of Natural History; Principles of classification. Linear and imitative Drawing.
The Second Class (Seconde,) at fifteen years old.
Algebra; Geometry, figures in space, recapitulation; Applications of Geometry, notions of the geometrical representations of bodies by projections; Rectilineal Trigonometry; Chemistry; Physics; and Drawing.
The Rhetoric, at sixteen years old.
Exercises in Arithmetic and Algebra; Geometry; notions on some common curves; and general recapitulation; Applications of Geometry; notions of leveling and its processes; recapitulation of Trigonometry; Cosmography; Mechanics; Chemistry concluded and reviewed; Zoölogy and Animal Physiology; Botany and Vegetable Physiology; Geology; Drawing. (The pupil may now be ready for the Degree and for St. Cyr.)
The Logic, at seventeen years old.
Six lessons a week are employed in preparation for the bachelorship of science, and in a methodical recapitulation of the courses of the three preceding years according to the state of the pupil’s knowledge.
Two lessons a week are allowed for reviewing the literary instruction; evening lessons in Latin, French, English, and German, and in History and Geography, having been given through the whole previous time.
The Special Mathematics, at eighteen and nineteen years old.
Five lessons a week are devoted to these studies; in the other lessons the pupils join those of the Logic class for reviewing all their previous subjects, whether for the bachelorship in science or for competition for admission at the Ecole Normale or the Polytechnic.
It will only be necessary to add a few sentences in explanation of the methods pursued in the upper classes of the Lycées. The classes are large—from 80 to above 100; the lessons strictly professorial lectures, with occasional questions, as at the Polytechnic itself. In large establishments the class is divided, and two professors are employed, giving two parallel courses on the same subject. To correct and fortify this general teaching, we find, corresponding to the interrogations of the Polytechnic, what are here called conferences. The members of the large class are examined first of all in small detachments of five or six by their own professors once a week; and, secondly, a matter of yet greater importance, by the professor who is conducting the parallel course, and by professors who are engaged for this purpose from other Lycées and preparatory schools, and from among the répétiteurs of the Polytechnic and the Ecole Normale themselves. It appeared by the table of the examinations of this latter kind which had been passed by the pupils of the class of Special Mathematics at the Lycée St. Louis, that the first pupil on the list had in the interval between the opening of the school and the date of our visit (February 16th) gone through as many as twenty-four.
The assistants, who bear the name of répétiteurs at the Lycées, do not correspond in any sense to those whom we shall hereafter notice at the Ecole Polytechnique. They are in the Lycées mere superintendents in the salles d’étude, who attend to order and discipline, who give some slight occasional help to the pupils, and may be employed in certain cases, where the parents wish for it, in giving private tuition to the less proficient. The system of salles d’étude appears to prevail universally; the number of the pupils placed in each probably varying greatly. At the Polytechnic we found eight or ten pupils in each; at St. Cyr as many as 200. The number considered most desirable at the Lycée of St. Louis was stated to be thirty.
It thus appears that in France not only do private establishments succeed in giving preparation for the military schools, but that even in the first-class public schools, which educate for the learned professions, it has been considered possible to conduct a series of military or science classes by the side of the usual literary or arts classes. The common upper schools are not, as they used to be, and as with us they are, Grammar schools, they are also Science schools. In every Lycée there is, so to say, a sort of elementary polytechnic department, giving a kind of instruction which will be useful to the future soldier, and at the same time to others, to those who may have to do with mines, manufactures, or any description of civil engineering. There is thus no occasion for Junior Military Schools in France, for all the schools of this class are more or less of a military character in their studies.
The conditions of admission to the examination for the degree of Bachelor of Science are simply, sixteen years of age, and the payment of fees amounting to about 200 fr. (10l.) Examinations are held three times a year by the Faculties at Paris, Besançon, Bordeaux, Caen, Clermont, Dijon, Grenoble, Lille, Lyons, Marseilles, Montpellier, Nancy, Poitiers, Rennes, Strasburg, and Toulouse, and once a year at Ajaccio, Algiers, and nineteen other towns. There is a written examination of six hours, and a viva voce examination of an hour and a quarter. It is, of course, only a pass examination, and is said to be much less difficult than the competitive examination for admission to St. Cyr.—Report of English Commissioners, 1856.