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, 10 which passes through two given points, 20 which passes through a given point and which is parallel to a given line.

To determine the point of intersection of two right lines whose equations are known. To pass a plane, 10 through three given points; 20 through a given point and parallel to a given plane; 30 through a point and through a given right line. Knowing the equations of two planes, to find the projections of their intersection. To find the intersection 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 coordinates).

Knowing the equation of a plane, to find the angles which it makes with the coordinate 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 Co-ordinates of the centre.

any.

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 sections of surfaces of the second order.-Plane sections of the cone, and of the right cylinder with circular base.-Anti-parallel section of the oblique cone with circular base.

Cone asymptote to an hyperboloid.

Right-lined sections 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 sections 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 Stonecutting 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.*

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 intersection."

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, 10 through a point taken on the surface; 20 through a point taken out of the surface; 8° parallef 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.

The method of the change of the planes of projection will be used for the resolution of these problems.

Problems relating to the intersection of surfaces.

To construct the section 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 intersection. To make the development of the cylindrical surface, and to refer to it the curve of intersection, and also the tangent.

To construct the intersection of a right cone by a plane perpendicular to one of the planes of projection. Development and tangent.

To construct the right section of an oblique cylinder.-To draw the tangent to the curve of intersection. 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 intersection of a surface of revolution by a plane, and the tangents to the curve of intersection.-To resolve this question, when the generating line is a right line which does not meet the axis.

To construct the intersection of two cylindrical surfaces, and the tangents to that

curve.

To construct the intersection of two oblique cones, and the tangents to that curve.
To construct the intersection 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"!

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BY 8. G. HOWE.

THE importance of the study of the ancient Greek language, has been set forth in this Journal. Valuable hints and suggestions upon the subject are to be found in Prof. Lewis' articles in the preceding volume.

Our object now is to show how new interest and importance may be given to the study of the language and literature of the old Greeks, by connecting it with the study of the language and the literature of the modern Greeks. For which, Prof. Felton has given increased facilities by the publication of a volume of Selections from their best writers.*

In our utilitarian age and country, there is a growing prejudice against the study of the Greek and Latin, partly because a knowledge of those languages is difficult to be attained; partly because some regard it as a sort of aristocratic accomplishment; partly because others think that the time spent in attaining it might be better spent in something else; but mainly because very few know anything about the matter. The popular misnomer of dead language as applied to Greek, is proof of this. The language is not dead, and probably never will be. The Greeks of to-day can read Homer more easily than we can read Chaucer; and they can read Xenophon about as easily as we read Spencer. But suppose they could not. Suppose, indeed, there were no living Greeks, would the old Greek language be dead? Not in any just sense. We have hundreds of aspirants for immortality in every branch of literature, and they have their respective thousands of admirers, who believe their reputations will win in the race against time; but we will back old Homer against any living poet, Aristotle against any philosopher, Socrates against any moralist, Demosthenes against any orator; we will give to their living rival two thousand years the start, and feel sure that they will be beaten, and left out of sight in ages, when the names and work of the ancients will be as fresh and green as they are now.

But taking it for granted that an intimate knowledge of the Greek will always be sought by those who aspire to high scholar

* Selections from modern Greek writers in Prose and Poetry, with notes by C. C. Felton, LL. D. Eliot Professor of Greek in Harvard University.

No. 5.-VOL. II, No. 1.]-13.

ship, and that the study of it will not be banished from our hig seminaries, we would urge a few considerations in favor of having the language taught in such a manner that it will be in no sense a dead language; to wit, in connection with the spoken and written language of several millions of living men.

This can be done without much additional study, and when done may become very useful by opening to the student the living language of the most active and intelligent people of the East; a people who have their universities, their gymnasia, and their common schools, their periodicals, and their newspapers; and who are fast building up a literature which shows them to be worthy desendants of their illustrious ancestors.

The political revolution which the Greeks recently effected so completely, is not the only one which they have attempted. They have aimed also at effecting an equally remarkable revolution in their national language, by driving out all foreign words and phrases which their conquerors, especially the Italians and Turks had left; by correcting grammatical corruptions, and by bringing it back as nearly as possible to its old condition of most beautiful, flexible, and expressive language yet contrived by man.

This attempt was certainly as remarkable as it was bold. We know no other instance in history where a people, or race, has consciously and purposely undertaken such a task. Such changes in language are usually made slowly and unconsciously; but the Greeks went to work earnestly, purposely, and almost unanimously. It was not merely the work of scholars; they could have done nothing alone; but the people, who clung to the memory of their high descent, who always persisted in calling their boys Pericles, and Socrates, and Leonidas, and their girls Aspasia, and Helena, and Penelope,the common people seconded the scholars in the high attempt, and set about discarding what they understood to be foreign words, and using native ones with an eagerness which would have seemed puerile and useless, if its purpose had not been so good, and its success so remarkable.

This redemption of a language is such an extraordinary thing, that it is worth notice.

The natural brotherhood of man is shown in the tendency to a common form of speech which is manifested as soon as social relations are established. So surely as men of different nations come into relation with each other, even if it be relations of war, so surely do they begin to form a common language. Their leaders bring them together as enemies, but they soon form relations as friends.