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.2_a_ and sin.3_a_; cos.2_a_ and cos. 3_a_; tang.2_a_ and tang.3_a_. Knowing sin._a_ or cos._a_, to calculate sin.½_a_ and cos.½_a_. Knowing tang._a_, to calculate tang.½_a_. Knowing sin._a_, to calculate sin.⅓_a_--Knowing cos._a_, to calculate cos.⅓_a_. Use of the formula cos._p_+cos._q_ = 2cos.½(_p + q_)cos.½(_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.