small ones in Maine and in Northampton county, Pennsylvania, U.S.A. Large
crystals have also been found in Japan. In its occurrence in basic rather
than in acid eruptive rocks, axinite differs from the boro-silicate
tourmaline, which is usually found in granite. The specific gravity is
3.28. The hardness of 6 1/2-7, combined with the colour and transparency,
renders axinite applicable for use as a gemstone, the Dauphine crystals
being occasionally cut for this purpose.

(L. J. S.)

AXIOM (Gr. [Greek: axioma]), a general proposition or principle accepted as
self-evident, either absolutely or within a particular sphere of thought.
Each special science has its own axioms (cf. the Aristotelian [Greek:
archai], "first principles") which, however, are sometimes susceptible of
proof in another wider science. The Greek word was probably confined by
Plato to mathematical axioms, but Aristotle (_Anal. Post._ i. 2) gave it
also the wider significance of the ultimate principles of thought which are
behind all special sciences (_e.g._ the principle of contradiction). These
are apprehended solely by the mind, which may, however, be led to them by
an inductive process. After Aristotle, the term was used by the Stoics and
the school of Ramus for a proposition simply, and Bacon (_Nov. Organ._ i.
7) used it of any general proposition. The word was reintroduced in modern
philosophy probably by Rene Descartes (or by his followers) who, in the
search for a definite self-evident principle as the basis of a new
philosophy, naturally turned to the familiar science of mathematics. The
axiom of Cartesianism is, therefore, the _Cogito ergo sum_. Kant still
further narrowed the meaning to include only self-evident (intuitive)
synthetic propositions, _i.e._ of space and time. The nature of axiomatic
certainty is part of the fundamental problem of logic and metaphysics.
Those who deny the possibility of all non-empirical knowledge naturally
hold that every axiom is ultimately based on observation. For the Euclidian
axioms see GEOMETRY.

AXIS (Lat. for "axle"), a word having the same meaning as axle, and also
used with many extensions of this primary meaning. It denotes the imaginary
line about which a body or system of bodies rotates, or a line about which
a body or action is symmetrically disposed. In geometry, and in geometrical
crystallography, the term denotes a line which serves to aid the
orientation of a figure. In anatomy, it is, among othe