Functions

Functions, in Mathematics, are quantities whose magnitudes depend on the magnitudes of other quantities. The surface of a sphere depends on its radius; hence it is said that the surface is a function of the radius. The connection is given by a simple type of algebraic equation, s = 4irr2; where s is the surface required, ir is the ratio of circumference to diameter of any circle (q.v.), and r is the given radius. The connection here shown between s and r being purely algebraical, the function is termed algebraical. Again, the cosine of an angle is a function of the angle itself; this is a simple case of trigonometrical functions. The logarithm (q.v.) of a number is a function of that number; this is logarithmic. In fact, there are functions of various types, and dependent on all kinds of variable quantities. To take one case of a complex function depending on more than one variable, the temperature of a point in a cubical block of metal, raised to a white heat and then allowed to cool in air, will be a function of the distances of the point from the faces of the cube, of the original temperature conditions of the block when first it began to cool, of the temperature of the air surrounding the block, and of the time which has elapsed since the experiment began. Numerous problems in physics introduce functions such as this, and demand very refined mathematical skill for their complete solution.