FUNCTION. (1) In biology, the action proper to tissues, organs or groups of organs in plant and animal life. The function of respira tion is the joint action of lungs and skin; diges tion is a very compound function, to which organs and groups of organs contribute. The actions are capable of being grouped in subordi nation to three leading phenomena of every liv ing thing — namely, sustentation, reproduction and relation. To the first belong digestion and all the other functions which contribute to the vegetative life; the processes of the second are, as examples of cell transformation, so far iden tical with those of the other two, but the results are different; the cell changes of the nervous system which regulates the relations of living things, are again identical with those of the other two sets of phenomena. Functional dis eases are those due to organs perfect in struc ture but not performing their functions prop erly; as opposed to organic or structural dis eases, due to defect of structure.
Organs often have more than one function —a primary, or that for which it is principally intended, and a secondary, some subsidiary pur pose which it performs. It sometimes happens that important changes take place in the course of the evolution of a type, or the development of an individual, whereby the primary function disappears and some secondary use becomes pre-eminent or exclusive. Thus "a brilliant speculation' says Carpenter, "has indicated pairs of tracheal gills on the meso- and meta thorax as the possible origin of insect wings. The primeval insects forsook, so it is thought, the water for the landi and the plates, becom ing useless for breathing, were enlarged and finally changed into organs of flight." Another strong and familiar example is the case so often presented among crustaceans where the mouth parts are largely structures ("foot-jaws) orig inally ambulatory, but now entirely devoted to the seizing and mastication of food. Change of function results in change of structure. Con sult Darwin, 'Origin of Species' (6th ed., Lon don 1882) ; Dohni, A., 'Der Ursprung der Wir belthiere und das Princip des Functionwechsels' (Leipzig 1875) ; Marshall, A. M., 'Biological Lectures and Addresses' (London 1894) ; Saint George Mivart, 'Genesis of Species' (New York 1871). See also FUNCTIONALISM.
(2) In mathematics, one quantity is said to be a function of another, or of several others, when its value depends on those of the latter. Thus the area of a triangle is a function of its three sides, and is a function of a, b, c, and x. Functions receive distinctive names according to the nature of the depend ence above referred to. Thus thefunction
above written is said to be an algebraical func tion of x, since y is obtainable from x by the performance of a limited and definite number of algebraical operations. Log x, sin x az, on the other hand, are said to be transcendental func tions of x, and for obvious reasons receive the distinctive names of logarithmic, trigonomet rical and exponential functions. The term function in its mathematical sense was due to Leibnitz (1692), but in its present sense was first defined by Johann Bernoulli (1718). La grange first used the term "theory of functions" in has (Theorie des functions analytiques) (Paris 1797). The object of the theory is the study of functions of one or more variables, in which either the variables or the coefficients, or both, are complex numbers. Lagrange (1772, 1797, 1806) may be said to have been the real founder of this general theory but others before him — Newton, Leibnitz, Bernoulli, Clairaut (1734), D'Alembert (1747), and Euler (1753) —had already worked in its direction. Landen (1775) is usually credited with found ing the theory of elliptic functions, though this theory had been suggested by Jakob Bernoulli (1691) ; Maclaurin (1742), and D'Alembert (1746). The real development of the theory, however, is due to Legendre, who after great labor produced his (Traite des fonctions ellip tiques et des integrales Euleriennes) (1825-28). Abel, Jacobi and Cayley also contributed much to this theory. The present form of the gen eral theory of functions is based largely on the works of Cauchy, Riemann and Weierstrass. For a list of the special functions consult Muller, 'Mathematische Terminologie,' in 'Bib liotheca Mathematica> (Leipzig 1901), a work in which are given about 200 functions. Brill and Noether, in 'Die Entwickelung der Theoric der algebraischen Functionen in alterer und neuerer Zeit,' in Vahresbericht der deutschen Mathematiker Vereingung' (Vol. II, Berlin 1894), gave a valuable history of the develop ment of functions. Consult also Forsyth, 'Theory of Functions' (Cambridge 1893) ; Harkness and Morley, 'Theory of Functions' (New York 1893) ; and Merriman and Wood ward, 'Higher Mathematics' (New York 1896), in all of which will be found the historical de velopment, bibliography and full discussion of the theory of functions. See also articles in this encyclopedia: COMPLEX VARIABLE THEORY OF THE FUNCTIONS OF A; REAL VARIABLE, THE ORY OF THE FUNCTIONS OF A; MATHEMATICS; TatcoNourrint; etc., to which extended bibliog raphies are appended.