From any point on a hemisphere, two arcs of great circles can always be drawn which shall be perpendicular to the circumference of the hemisphere, and they will in general be unequal. Now, it may be proved, by a course of reasoning analogous to that employed in Book I., Proposition XV. : 1o. That the shorter of the two arcs is the shortest arc that can be drawn from the given point to the circumference. 2o. That two oblique arcs drawn from the same point, to points of the circumference at equal distances from the foot of the perpendicular, are equal: 3o. That of two oblique arcs, that is the longer which meets the circumference at the greater distance from the foot of the perpendicular. This property of the sphere is used in the discussion of triangles in spherical trigonometry. INTRODUCTION TO TRIGONOMETRY. LOGARITHMS. 1. THE LOGARITHM of a number is the exponent of the power to which it is necessary to raise a fixed number, to produce the given number. The fixed number is called the base of the system. Any positive number, except 1, may be taken as the base of a system. In the common system, the base is 10. 2. If we denote any positive number by n, and the corresponding exponent of 10, by x, we shall have the exponential equation, In this equation, x is, by definition, the logarithm of n, which may be expressed thus, 3. From the definition of a logarithm, it follows that, the logarithm of any power of 10 is equal to the exponent of that power: hence the formula, If a number is an exact power of 10, its logarithm is a whole number. |