Geometry in space.

The plane and the right line. Conditions in order that a right line may be perpendicnlar to a plane. Properties of the perpendicular and the oblique lines drawn from the same point to a plane. Parallelism of right lines and of planes.

Dihedral angles. Generation of dihedral angles by the rotation of a plane about a right line. Dihedral right angle. Measurement of dihedral angles. Properties of planes perpendicular to each other.

Trihedral angles. Cases of equality and of symmetry. Properties of the supplementary trihedral angle. Limit of the sum of the faces of a convex polyhedral angle. Limits of the sum of the dihedral angles of a trihedral angle. Analogies and differences between trihedral angles and rectilinear triangles.

Polyhedrons. The prism, parallelopiped, cube, pyramid. Plane and parallel sections of the prism and the pyramid

Measurement of volumes. Volume of the parallelopiped, the prism, the pyramid, the frustum of a pyramid, and the frustum of a triangular prism.

Symmetry in polyhedrons. Plane of symmetry. Center of symmetry.* Comparison of the faces, the dihedral angles, the homologous polyhedral angles of two symmetrical polyhedrons. Equivalence of their volumes.

Similar polyhedrons. Cases of similitude of two triangular pyramids. Ratio of the volumes of two similar polyhedrons, similarly placed.

The round bodies.

Right cylinder with circular base. Measure of the lateral surface and of the volume; extension to right cylinders with any base whatever.

Right cone with circular base. Sections parallel to the base. Convex surface and volume of the cone, and of the frustum of a cone.

Sphere. Plane sections; great circles; small circles. Poles of a circle. Given a sphere, to find its radius by a plane construction.

Tangent plane. Angle of two arcs of a great circle.

Spherical triangles. Analogy with trihedral angles. Measure of the surface generated by a line turning regularly about an axis drawn in its plane and through its center. Area of the zone of the entire sphere. Exercises.

Measure of the volume generated by a triangle turning about an axis drawn in its plane, and through one of its angles. Application to a regular polygonal sector turning about an axis drawn in its plane, and through its center. Volume of a spherical sector, of the entire sphere, of a spherical segment. Exercises. Approximate volume of a solid bounded by any surface whatever.

Properties and definitions of certain curves.

Definition of the ellipse by the properties of the foci. Drawing of the curve by determining points, and by a continuous motion. Axes. Vertices. Radius vector. General definition of a tangent to a curve.

The radii drawn from the foci to any point of the ellipse make equal angles with the tangent at this point. To draw a tangent to the ellipse to a point taken on the curve, through an exterior point. Normal of the ellipse.

Definition of the parabola by means of the property of the foci, and by its directrix. Tracing of the curve through points, and with a continuous motion. Axis. Vertex. Radius vector.

The tangent makes equal angles with the line parallel to the axis, and the radius vector, drawn through the point of contact. To draw a tangent to a parabola *The study of symmetry with reference to a point, reduces itself to that of symmetry with reference to a plane, by rotating one of the two figures through an angle of 1800, about an axis perpendicular to the plane and passing through the center of symmetry.

through a point taken on its curve, and through an exterior point. Normal. Subnormal.

Relation between the square of an ordinate perpendicular to the axis, and the distance of this ordinate from the vertex.

Definition of the helix, considered as resulting from the unrolling of the plane of a right-angled triangle upon a right cylinder with circular base. Distance between the turns of the helix. The tangent to the helix makes a constant angle with the side of the cylinder. To construct the projection of the helix and of the tangent upon a plane perpendicular to the base of the cylinder.

PLANE TRIGONOMETRY.

Class of elementary mathematics.

Trigonometric lines. Relations between the trigonometric lines of the angle. Expressions of the sine and the cosine as a function of the tangent. Formulas relating to the sine, cosine, and tangents of the sum and the difference of two ares. Expressions of sin 2a, cos 2a, and tang 2a. Given cos a or sin a, to calculate sin ja and cos a.

To adapt the sum of two trigonometric lines to calculations by logarithms; e. g., the sine, cosine, and tangent. Construction of trigonometric tables. Use of tables. Relation between the angles and the sides of a right triangle or of any triangle whatever. Resolution of right triangles. Resolution of any triangles whatever in the four cases that may present themselves. To determine the area of a triangle as a function of given parts.

Application of trigonometry to the different questions presented in surveying; distance of an inaccessible point. Measurement of heights. Three point problem.

DESCRIPTIVE GEOMETRY.

Class of elementary mathematics.

Inadequacy of ordinary drawing for the representation of solid bodies. Usefulness of a geometric method, which, by graphic operations executed on one and the same plane, determines exactly the form and the position of a figure of three dimensions. Projection of a point, of a right line, of any line, on a plane. Plane of projection. Traces of a plane. True length of a line that joins two points given by their projections. Angles of a right line with the planes of projection. Representation of a plane by its traces. Angles made by a plane with the planes of projection. Method of revolutions.

Exercises.

Intersection of two planes. Intersection of a right line and a plane. Distance of a point from a plane. Distance of a point from a right line. Angle of two right lines; of a right line and a plane; of two planes.

Projections of a prism, a pyramid, a cylinder, a cone with circular base, executed from models.

Plane sections of polyhedrons.

Method of plans cotés.

STATICS.*

Class of elementary mathematics.

Forces. Condition of equality of two forces. Their numerical representation. Comparison of forces with weights by means of the dynamometer. Translation of the point of application of a force to any point in its direction, supposing it to be inva riably connected with the first point.

Composition of two forces applied at the same point. Theorem of the moments with reference to a point taken in the plane of the forces. Composition of any number of forces applied at the same point. Conditions of equilibrium. Composition of

* After 1880.

two parallel forces. Couples. Composition of any number of parallel forces. Centers of parallel forces. Centers of gravity; their determination in certain simple cases, as the triangle and pyramid. Composition of a system of forces applied to a solid body; their reduction to two forces, one of which acts at any given point. General conditions of equilibrium.

Simple machines.

The lever: General condition of equilibrium; relation between the power and the weight. The balance: Ordinary balance, Roman balance, scales of commerce. The pulley: Equilibrium of the fixed pulley; of the movable pulley. Combinations of pulleys. The capstan or windlass: Equilibrium; relation between the power and the weight. The inclined plane: Equilibrium of a body placed upon an inclined plane.

Elements of kinematics and dynamics.

Uniform rectilinear motion; velocity. Varying rectilinear motion; mean velocity; velocity at any instant. Rectilinear motion uniformly varied; acceleration. Acceleration at any instant in a varying rectilinear motion. Composition of two simultaneous rectilinear motions, varying uniformly or otherwise. Uniform motion of rotation about a fixed axis; angular velocity.

Law of inertia.

Law of relative motion: The inference that a constant force, acting upon a material point which starts from rest, or which has an initial velocity in the same direction as the force, imparts to it a uniformly varied motion. Converse proposition. Two constant forces are proportional to the accelerations that they produce in acting separately upon the same material point which starts from rest, or which has an initial velocity in the same direction as the force.

Mass: Its measure by means of weight.

Work of forces.

The work of a constant force applied to a point whose displacement is rectilinear. Unit of work. To show that, in simple machines, in a state of uniform motion and acted upon only by a power and a resistance, the motive power is equal to the resistance. Influence of passive resistance. In practice, the motive power always exceeds the useful resistance.

3.-ACADEMIC YEAR.

The course at the Naval School is two years. The academic session on board the Borda begins on the 1st of October and lasts nearly ten months. There are three terms or trimesters, of three months each. The regular programme of studies extends only over the first eight months, June being the mois de pioche, which is occupied in studying for the coming examinations and in special exercises. The examinations take up nearly the whole of the tenth month, July.

After the close of the annual examinations, the students of the second class are embarked on board the Bougainville, a small vessel attached to the school. This a screw steamer, with engines of 120 horse-power, built specially for the school. In this they take a short practice cruise, during which they visit various points on the neighboring coast of France, including Cherbourg on the one hand and Lorient on the other. Sometimes the cruise extends as far as Ferrol, in Spain. During the cruise, the pupils are stationed with the men, and work the ship, receiving at

the same time instruction of a practical character in seamanship, navigation, and steam-engineering, and performing exercises in drawing. For the first few weeks, the first class also takes part in the cruise, in the same or another vessel; and during this time, the practice ships maneuver in and about the roads of Brest; and when the second class goes to sea, the members of the first class, which has now been graduated, are sent home for a six weeks' vacation. The Bougainville returns to Brest early in September, and the second class men have leave for a month. On the 1st of October, they return for the second year course in the Borda, while the graduates, now for the first time admitted with a specific rank to the Navy, are embarked on board the Flore, for a more extended cruise. They are called aspirants de deuxième classe, a term nearly equivalent to cadets, and they are still undergoing instruction, though instruction of a more distinctly practical character.

During the course on board the Borda, the young men under instruction are known simply as pupils, élèves. There are two classes or divisions, the first and second, being those respectively in their second and first year. The average number in a class is between 40 and 50. In June, 1878, there were 53 of the anciens, or first class, and 44 of the second. Each class is divided into two sections (escouades) for purposes of administration, and each of the four sections is intrusted to the particular attention of a lieutenant. Each section is again divided into three subsections (séries) for the interrogations. Each subsection numbers about eight members.

4.-COURSE OF INSTRUCTION.

The methods of instruction adopted in the Naval School of Brest are similar, in a general way, to those of other French schools and colleges, and resemble nothing in America. Recitations in the ordinary sense do not exist, and text-books are almost unknown. A few books of reference are used, including the nautical almanac, a work on physics (by Almeida), English and French grammars, and the admirable series of manuals published under the authority of the Ministry of Marine.* These books are not, however, used as text-books. The main feature of the system of instruction is the cours, or lecture. This is delivered in a more or less formal manner in the amphitheater, and the students take notes, which are inspected from time to time by the instructors. They also receive, after the lecture, a full summary of the points treated; a summary so

*This series includes

1. Manual of the seaman-gunner (matelot canonnier).

2. Manual of the topman (gabier).

3. Manual of the small-arm man (marin fusilier).

4. Manual of the coast-pilot (pilote côtier).

5. Manual of the helmsman (matelot timonier).

6. Manual of torpedoes (défenses-sous-marines).

7. Manual of fencing and gymnastics.

8. Manual of miscellaneous exercises (diving, &c.).

full that it amounts pretty nearly to the lecture itself. It is reproduced on lithographic sheets (feuilles autographiées) similar to those in use at the Polytechnic and elsewhere, and contains all the necessary drawings, diagrams, and tables. The time allowed for the cours is from an hour to an hour and a quarter. It is followed by a short recess, and by a period of study from one to two hours long, under the supervision of the same or a subordinate instructor, but always on the same branch of study. Thus a lecture in mechanics is followed by study in mechanics; a lecture in naval architecture by study in naval architecture, and so on. During the study period, assistance and explanations are given; various exercises are performed, in some cases an abstract of the previ ous lecture being made; and the interrogations are held. The latter are a peculiar feature of the system, and form the complement of the cours. They are conducted orally, and are something between a recitation and a monthly examination. They are of two kinds, particular and general; and each student has an interrogation of both kinds in each term. The particular interrogations take place at irregular intervals, during the study period, and are held without previous notice; the captain designating each day the students who are to be interrogated. They cover the ground gone over since the beginning of the term. The general interrogations are held at the end of each term on the studies of the whole term. They are longer and fuller than the others, and they have a greater weight in determining the term-mark, counting double in the two first terms, and three times as much in the third term. The professor may call upon students to answer questions at other times, during the hours of lecture, for example, but this is not done to any very great

extent.

The general interrogation of the third term covers the course for the whole year, but it is independent of the final annual examination. This, of course, has the same scope as the last interrogation, and the latter seems to be conducted chiefly with a view to preparing the students to pass it. The examination is conducted by an outside board appointed for the purpose, of which the Préfet Maritime is president, and the prineipal working member in scientific subjects is a professor of hydrography. It is thorough and searching in character, and upon it chiefly depends the advancement or discharge of students of doubtful capacity. The essential point, however, to be noticed about it is that it is conducted independently of the school, and thus acts as check upon in

structors.

The marking system is exceedingly simple. The maximum is, as usual, 20. The term marks, as has been stated, are ascertained by combining the marks for the two interrogations, giving the second double weight, and, in the case of third term, triple weight. The term-marks are combined with equal weight to determine the school-mark for the year; and the latter, combined with the mark for the annual examination, given by the examining board, fixes the mark in each subject for the year. The