pure mathematics. Numerous new fields were opened up,
and have been diligently explored by many mathematicians.
Skew-determinants were studied by Cayley; axisymmetric-determinants by
Jacobi, V. A. Lebesque, Sylvester and O. Hesse, and centro-symmetric
determinants by W. R. F. Scott and G. Zehfuss. Continuants have been
discussed by Sylvester; alternants by Cauchy, Jacobi, N. Trudi, H.
Nagelbach and G. Garbieri; circulants by E. Catalan, W. Spottiswoode
and J. W. L. Glaisher, and Wronskians by E. B. Christoffel and G.
Frobenius. Determinants composed of binomial coefficients have been
studied by V. von Zeipel; the expression of definite integrals as
determinants by A. Tissot and A. Enneper, and the expression of
continued fractions as determinants by Jacobi, V. Nachreiner, S.
Guenther and E. Fuerstenau. (See T. Muir, _Theory of Determinants_,
1906).

[1] The expression, a linear function, is here used in its narrowest
sense, a linear function without constant term; what is meant is that
the determinant is in regard to the elements a, a', a", ... of any
column or line thereof, a function of the form Aa + A'a' + A"a" + ...
without any term independent of a, a', a" ...

[2] The reason is the connexion with the corresponding theorem for the
multiplication of two matrices.

DETERMINISM (Lat. _determinare_, to prescribe or limit), in ethics, the
name given to the theory that all moral choice, so called, is the
determined or necessary result of psychological and other conditions. It
is opposed to the various doctrines of Free-Will, known as voluntarism,
libertarianism, indeterminism, and is from the ethical standpoint more
or less akin to necessitarianism and fatalism. There are various degrees
of determinism. It may be held that every action is causally connected
not only externally with the sum of the agent's environment, but also
internally with his motives and impulses. In other words, if we could
know exactly all these conditions, we should be able to forecast with
mathematical certainty the course which the agent would pursue. In this
theory the agent cannot be held responsible for his action in any sense.
It is the extreme antithesis of Indeterminism or Indifferentism, the
doctrine that a man is absolutely free to choose between alternative
courses (the _liberum arbitrium indifferentiae_). Since, however, the
evidence of ordinary consciousness almost always goes to prove that the
individual