1o Having given their three sides a, b, c, or their three angles A, B, C.—Formulas tang.½a 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.
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.