being alone exempt from these difficulties, we should seek, as far as possible, to resolve all questions by means of it. Thus, let us suppose that we know the hypothenuse and one of the sides of a right-angled triangle, the direct determination of the included angle will be given by a cosine, which will be wanting in exactness if the hypothenuse of the triangle does not differ much from the given side. In that case we should begin by calculating the third side, and then use it with the first side to determine the desired angle by means of its tangent. When two sides of a triangle and the included angle are given, the tangent of the half difference of the desired angles may be calculated with advantage; but we may also separately determine the tangent of each of them. When the three sides of a triangle are given, the best formula for calculating an angle, and the only one never at fault, is that which gives the tangent of half of it.

The surveying for plans, taught in the course of Geometry, employing only graphical methods of calculation, did not need any more accurate instruments than the chain and the graphometer; but now that trigonometry furnishes more accurate methods of calculation, the measurements on the ground require more precision. Hence the requirement for the pupil to measure carefully a base, to use telescopes, verniers, etc., and to make the necessary calculations, the ground being still considered as plane. But as these slow and laborious methods can be employed for only the principal points of the survey, the more expeditious means of the plane-table and compass will be used for the details.

In spherical trigonometry, all that will be needed in geodesy should be learned before admission to the school, so that the subject will not need to be again taken up. We have specially inscribed in the programme the relations between the angles and sides of a right-angled triangle, which must be known by the students; they are those which occur in practice. In tracing the course to be pursued in the resolution of the three cases of any triangles, we have indicated that which is in fact employed in the applications, and which is the most convenient. As to the rest, ambiguous cases never occur in practice, and therefore we should take care not to speak of them to learners.

In surveying, spherical trigonometry will now allow us to consider cases in which the signals are not all in the same plane, and to operate on uneven ground, obtain its projection on the plane of the horizon, and at the same time determine differences of level.

It may be remarked that Descriptive Geometry might supply the place of spherical trigonometry by a graphical construction, but the degree of exactitude of the differences of level thus obtained would be insufficient.

PROGRAMME OF TRIGONOMETRY.

1. PLANE TRIGONOMETRY.

Trigonometrical lines.-Their ratios to the radius are alone considered.--Relations of the trigonometric lines of the same angle.-Expressions of the sine and of the cosine in functions of the tangent.

Knowing the sines and the cosines of two arcs a and b, to find the sine and the cosine of their sum and of their difference.-To find the tangent of the sum or of the difference of two arcs, knowing the tangents of those arcs.

Expressions for sin. 2a and sin. 3 a; cos. 2a and cos. 3a; tang. 2a and tang. 8 a. Knowing sin. a or cos. a, to calculate sin. ta and cos. ta.

Knowing tang. a, to calculate tang. a.

Knowing sin. a, to calculate sin. a.-Knowing cos. a, to calculate cos. §a.

Use of the formula cos.p+cos. q = 2 cos. † ( p + q) cos. ↓ (p-q), to render logarithms applicable to the sum of two trigonometrical lines, sines or cosines.-To render logarithms applicable to the sum of two tangents.

Construction of the trigonometric tables.

Use in detail of the tables of Callet.-Appreciation, by the proportional parts, of the degree of exactness in the calculation of the angles.-Superiority of the tangent formulas.

Resolution of triangles.

Relations between the angles and the sides of a right-angled triangle, or of any triangle whatever.-When the three angles of a triangle are given, these relations determine only the ratios of the sides.

Resolution of right-angled triangles.-Of the case in which the hypothenuse and a side nearly equal to it are given.

Knowing a side and two angles of any triangle, to find the other parts, and also the surface of the triangle.

Knowing two sides a and b of a triangle and the included angle C, to find the other parts and also the surface of the triangle.—The tang. (A-B) may be determined; or tang. A and tang. B directly.

Knowing the three sides a, b, c, to find the angles and the surface of the triangle.Employment of the formula which gives tang. I A.

Application to surveying for plans.

Measurement of bases with rods.

Measurement of angles.-Description and use of the circle.-Use of the telescope to render the line of sight more precise.-Division of the circle.-Verniers. Measurement and calculation of a system of triangles.-Reduction of angles to the centres of stations.

How to connect the secondary points to the principal system.-Use of the plane table and of the compass.

2. SPHERICAL TRIGONOMETRY.

Fundamental relations (cos, a cos. b cos. c + sin. b sin. c cos. A) between the sides and the angles of a spherical triangle.

To deduce thence the relations sin. A: sin. B=sin. a: sin. b; cot. a sin.b-cot. A sin. C=cos. bcos. C, and by the consideration of the supplementary triangle cos. Acos. B cos. C+ sin. B sin. C cos. a.

Right-angled triangles.-Formulas cos. a=cos. b cos. c; sin. b= sin. a sin. B; tang. c=tang. a cos. B, and tang. b=sin. c tang. B.

In a right-angled triangle the three sides are less than 90°, or else two of the sides are greater than 90°, and the third is less. An angle and the side opposite to it are both less than 90°, or both greater.

Resolution of any triangles whatever:

10 Having given their three sides a, b, c, or their three angles A, B, C.-Formulas tang. a, and tang. A, calculable by logarithms:

20 Having given two sides and the included angle, or two angles and the included side.-Formulas of Delambre:

80 Having given two sides and an angle opposite to one of them, or two angles and a side opposite to one of them. Employment of an auxiliary angle to render the formulas calculable by logarithms.

Applications.-Survey of a mountainous country.-Reduction of the base and of the angles to the horizon.-Determination of differences of level.

Knowing the latitude and the longitude of two points on the surface of the earth, to find the distance of those points.

V. ANALYTICAL GEOMETRY.

The important property of homogeneity must be given with clearness. and simplicity.

The transformation of co-ordinates must receive some numerical applications, which are indispensable to make the student clearly see the meaning of the formulas.

The determination of tangents will be effected in the most general manner by means of the derivatives of the various functions, which we inserted in the programme of algebra. After having shown that this determination depends on the calculation of the derivative of the ordinate with respect to the abscissa, this will be used to simplify the investigation of the tangent to curves of the second degree and to curves whose equations contain transcendental functions. The discussion of these, formerly pursued by laborious indirect methods, will now become easy; and as curves with transcendental equations are frequently encountered, it will be well to exercise students in their discussion.

The properties of foci and of the directrices of curves of the second degree will be established directly, for each of the three curves, by means of the simplest equations of these curves, and without any consideration of the analytical properties of foci, with respect to the general equation of the second degree. With even greater reason will we dispense with examining whether curves of higher degree have foci, a question whose meaning even is not well defined.

We retained in algebra the elimination between two equations of the second degree with two unknown quantities, a problem which corresponds to the purely analytical investigation of the co-ordinates of the points of intersection of two curves of the second degree. The final equation is in general of the fourth degree, but we may sometimes dispense with calculating that equation. A graphical construction of the curves, carefully made, will in fact be sufficient to make known, approximately, the co-ordinates of each of the points of intersection; and when we shall have thus obtained an approximate solution, we will often be able to give it all the numerical rigor desirable, by successive approximations, deduced from the equations. These considerations will be extended to the investigation of the real roots of equations of any form whatever with one unknown quantity.

Analytical geometry of three dimensions was formerly entirely taught within the Polytechnic school, none of it being reserved for the course of admission. For some years past, however, candidates were required to know the equations of the right line in space, the equation of the plane, the solution of the problems which relate to it and the transfor

mation of co-ordinates. But the consideration of surfaces of the second order was reserved for the interior teaching. We think it well to place this also among the studies to be mastered before admission, in accordance with the general principle now sought to be realized, of classing with them that double instruction which does not exact a previous knowledge of the differential calculus.

We have not, however, inserted here all the properties of surfaces of the second order, but have retained only those which it is indispensable to know and to retain. The transformation of rectilinear co-ordinates, for example, must be executed with simplicity, and the teacher must restrict himself to giving his pupils a succinct explanation of the course to be pursued; this will suffice to them for the very rare cases in which they may happen to have need of them. No questions will be asked relating to the general considerations, which require very complicated theoretical discussions, and especially that of the general reduction of the equation of the second degree with three variables. We have omitted from the problems relating to the right line and to the plane, the determination of the shortest distance of two right lines.

The properties of surfaces of the second order will be deduced from the equations of those surfaces, taken directly in the simplest forms. Among these properties, we place in the first rank, for their valuable applications, those of the surfaces which can be generated by the movement of a right line.

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.*

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.

*The students will apply these reductions to a numerical equation of the second degree, and will determine the situation of the new axes with respect to the original axes, by means of trigonometrical tables. They will show to the examiner the complete calculations of this reduction and the trace of the two systems of axes and of the curves.

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.*-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 trans

verse 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, 10 through a point on the curve; 20 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 intersections of two curves of the second degree.-Numerical applications of these formulas.

They will be deduced from the property, previously demonstrated, of the derivative of the ordinate with respect to the abscissa.