17. Through a point taken upon the hyperbolic paraboloid, to draw a plane tangent to that surface.

Ninth construction.-18. Through a point taken in a regular surface, to draw a plane tangent to this surface ;-application, 1, to the conoid;-2, to the helicoid, [surface of the thread of a screw.]

Of the intersection of surfaces.

Tenth construction.-To construct the section of a right cylinder, through a plane perpendicular to one of the planes of projection. To draw the tangent to the curve of intersection, to make the development of the cylindrical surface, and to refer to it the curve of intersection as well as its tangent.

Eleventh construction.-To construct the intersection of a cone; 1, right; 2, oblique, through a plane perpendicular to one of the planes of projection; development and tangent.

Twelfth construction.-To construct the intersection of a surface of revolution through a plane, and the tangent to the curve of intersection; application, 1, to the ellipsoid; 2, to the sheet hyperboloid.

Thirteenth construction.-To construct the right section of a cylinder, to draw the tangent, and to construct the development. Fourteenth construction.-To construct the intersection,-1, of two cylindrical surfaces,-2, of two conical surfaces,-3, of a cylindrical surface and a conical surface,-4, of two surfaces of revolution whose axes meet,-5, of a conical surface and a surface of revolution.

Application to the solution of certain problems.

Fifteenth construction.-Four points being given in space, to find a fifth which may be at equal distances from each of them, that is, to circumscribe a sphere about a pyramid.-Four planes being given, to find a point which may be at equal distances from each of them, or to find a sphere tangent to four given planes.

Sixteenth construction.—In a solid angle of three faces, there exist three rectilinear angles formed by the edges among themselves, and three diedral angles formed by the mutual inclinations of the faces. Of these nine angles, three being given, to find the others.

Seventeenth construction.-To draw to a helix traced upon a right cylinder a tangent parallel to a given plane.-Through a given point upon a spherical epicycloid, to draw a tangent to this curve.

Eighteenth construction.-Through a given right line to draw a plane tangent to a sphere.

Nineteenth construction.-Through a given right line to draw a plane tangent to a surface of revolution;-1, to the ellipsoid ;2, to the sheet hyperboloid.

Twentieth construction.-To construct a right line resting upon four given right lines.

Applications of descriptive geometry.

1. To Linear Perspective.-General notions upon perspective and upon the apparent outlines of bodies. The pupils will execute various constructions.

2. To Shades.-General notions upon shades. The pupils will execute numerous constructions of shade.

Wash. The pupils will wash for effect, under the direction of the professor of design, some of the drawings in shade which they shall have executed.

3. To Stone-cutting.-1. Construction. Curve slope, in bevel, presenting a cylindrical scoop.-2. Curve slope in a round tower, in bevel, and presenting a spherical scoop.-3. Turned slope, or cow's horn (two solutions).-4. Back vault of Marseilles.5. Long gothic-arch vault, and cloister-arch vault.-6. Right descent.-7. Sloping descent (two solutions).-8. Trumpet slope in the angle.-9. Ridge-arch vault in a round turn.-10. Winding stair-case, open for light, and with solid newels.

4. To Carpentry.-1. Construction. Right ridge;-2. Oblique ridge;-3. Inclined span ;-4. Winding staircase. The pupils will have to trace a certain number of combinations pointed out by the professor.

5. To Machinery.-Tracing of gearings and of curves described by moveable points in machines, &c. &c.

6. To Astronomy.-Projection of solar dials.

Course of Geometry of three dimensions.

The professor will give, during the course of descriptive geometry, numerous lessons, in which he will determine, by geometry, the principal properties of conic sections. and of surfaces of the second order, and those of transcendental curves, and of surfaces of superior orders, which find their application in the

arts.

Course of Topography.

Elementary notions of Topography.

The pupils will execute in the field, surveys with the plain table, the compass, the surveyor's square, and will besides be exercised in levelling,

At the commencement of each lesson, the professor will question a certain number of the pupils; the adjunct professor will moreover have two interrogatories a week. At the end of each year the pupils will undergo a general examination.

Course of Natural Philosophy.

By M. Péclet-Former pupil of the Normal School, Master of Conferences in the Preparatory School, Ex-Professor of Natu

ral Philosophy at Marseilles, &c. M. Caladon, Adjunct Professor.

The course of Natural Philosophy is divided into two parts, corresponding to the two years of studies. The first part includes the physical properties of solids, liquids, and gases, at rest or in motion; the second, those of the imponderable fluids. The first part will comprehend all the facts of rational mechanics, which are important in applications. The most of them will only be enunciated, since the mathematical knowledge possessed by the pupils will be insufficient to enable them to comprehend their demonstration. But the manner of applying them will be enlarged upon. In this part of the course much development will be given to the theory of the resistance of solid bodies at rest or in motion, in liquids and gases, and to that of friction. In the second part, the theory of heat will principally be enlarged upon.

Although this course of natural philosophy is chiefly directed towards useful applications, the plan comprehends all classes of physical phenomena; but a great number will be examined only in a very succinct manner. This arrangement has the advantage of uniting among themselves the different parts of natural philosophy, of facilitating their study, and of imprinting the facts on the memory; advantages which would not be attained by treating merely of the physical phenomena which have useful applications to the arts. Besides, if the pupils ought not to study certain theories, they ought to know of their existence, in order that they may labour under no illusion in relation to the difference which exists between the part of the science which they have studied, and the science considered in all its develop

ments.

First Year.

General properties of bodies.-Extension, measure of extension; system of measurement; guaging; different instruments; impenetrability; divisibility; atoms; mobility; different kinds of forces; time; units of time; instruments for the measure of time; velocity; inertia; measure of forces; composition of forces which act upon the same point; equilibrium of a material point; motion of a material point, free or required to rest upon a curve or a surface.

Permanent forces which act upon bodies.-Gravitation; weight; general phenomena; centre of gravity; law of weight both above and below the surface of the earth; law of the fall of a body at a small distance from the earth; intensity of weight at the surface of the earth; apparatus designed to measure the bulk of bodies, balance, steelyard, bent lever balance.

Molecular attraction.
Repulsive force of heat.

Of solid bodies.-Porosity; density, different methods of determining it; phenomena which result from the greater or less stability of the equilibrium between the particles of solid bodies. Elasticity; ductility; frangibility; resistance to the stroke; resistance to pressure; resistance to traction; resistance to being worn or used; structure of solid bodies; equilibrium of solid bodies; motion of a free solid body; motion of a solid body around a point, a right line, or upon a surface; friction; laws; tables; shock of bodies, solid, soft, and elastic; use of solid bodies to transmit motions.

Of liquid bodies. Porosity; density; areometer of Baume, of Cartier; alcoometer, volumeter; areometer giving the densities; phenomena which result from the greater or less stabillity of equilibrium between the particles of liquid bodies; compressibility; elasticity; viscosity; cohesion; equilibrium of liquid bodies; principles upon which the laws of the equilibrium of liquids are founded; equilibrium of liquids subjected to whatever forces, equilibrium of liquids enclosed in vessels of large dimensions; lateral pressure; equilibrium of liquids in capillary spaces; equilibrium of floating bodies; motion of liquids; motion in open canals; motion in conduit pipes; flowing through orifices in very thin partitions, through pipes; shock of liquids against solid bodies; motion of floating bodies; use of liquids for transmitting and modifying forces; use of liquids as moving forces.

Of gaseous bodies.Atmosphere; general properties of the gases; measure of the elastic force of the gases; barometers; law of Mariote; density of the gases; bodies floating in gas; motions of gaseous bodies; motions in conduit tubes; flowing through orifices in their partitions, and through pipes; shock of the gases against solid bodies; machines and apparatus, the play of which is founded upon the properties of the air; use of the gases in transmitting and modifying forces; of the wind as a moving force; of air, considered as the vehicle of sound.

Second Year.

Of heat.-Sensible caloric; radiated caloric; laws of radiation; influence of surfaces; equilibrium of temperature by exchange; propagation of heat through bodies; laws of the diminution and increase of heat; latent caloric; expansion of bodies; vapor; hygrometry; specific caloric; phenomena developed in the changes of state of bodies; measure of temperatures; sources of heat; sources of cold.

Fuel.-Combustibles employed for fuel; calorific powers in weight and volume; radiating powers; motion of hot air in conduit pipes; laws of the motions in having no regard to friction; laws deduced from observation; influence of superior and infe

rior partitions; of the curvature of pipes; of lateral openings; effect produced by the meeting of currents.

Of chimneys.-Elements which determine their dimensions; law of the cooling of the air in chimneys; determination of the minimum diameter of a chimney; the most favorable dispositions for augmenting the draft; construction of shop and house chimneys; influence of the state of the atmosphere upon the draft of chimneys; apparatus designed to secure the draft of chimneys from the influence of the winds.

Steam Boilers, apparatus of supply, apparatus for safety, furnaces, hearths, grates, pipes of circulation, chimneys, registers, form of the different apparatus in use.

Simple distillation, by an open fire, in the open air, with pressure, by steam; rapid distillation; distillation with analysis of vapors; double distillation.

Spontaneous evaporation; by a current of forced air; by fire in open vessels; by hot air; by steam and by baths of hot oil; double apparatus; evaporation in the open air, spontaneous or by heat.

Drying in free air; by hot air; by contact with steam pipes; by a current of air put in motion and dried; drying of pulverised matters; drying in the open air.

Heating by the gases; ventilation of habitations; warming by combustion direct; internal warming by stoves; warming of the air by radiation; pipe stoves; hot air calorifers; steam calorifers; hot water calorifers; comparison of the different modes of warming.

Heating of liquids; direct heating; heating by steam.
Heating of solid bodies.

Cooling of bodies; ice houses; frigorific mixtures; ventilation by cold air.

Health apparatus for unhealthy factories.

Light.-Transmission; reflection; refraction; dispersion; coloring of light in passing through thin lamina; inflection; double refraction; polarization; luminous meteors; vision; optical instruments; construction of lenses and mirrors.

Lighting-Combustibles employed in lighting; examination of flame; lighting by solid materials; lighting by liquid materials; lamps, different forms of tubes, different arrangements of reservoirs of supply; hydrostatic lamps, mechanic lamps; lighting by gas; comparison of the different modes of lighting; apparatus for modifying lights; glasses ground or colored; reflectors; lens apparatus; lighthouses; flint and steel.

Electricity-General phenomena; inequality of the conducting power of bodies; electricity developed by rubbing two bodies together is of a different nature in each of them; law of electrical attractions and repulsions; causes of the dispersion of

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