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.½(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.½A. _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._b_ cos.C, and by the consideration of the supplementary triangle cos.A = -cos.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: 1º Having given their three sides _a_, _b_, _c_, or their three angles A, B, C.--Formulas tang.½_a_ and tang.1/2A, calculable by logarithms: 2º Having given two sides and the included angle, or two angles and the included side.--Formulas of Delambre: 3º 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.