Thus every quaternion may be written in the form q = Sq+Vq, where either Sq or Vq may separately vanish; so that ordinary algebraic quantities (or **scalars**, as we shall call them) and pure vectors may each be regarded as special cases of quaternions.

A n are **scalars**, and in particular applications may be restricted to real or complex numerical values.

A n) in which only its scalar coefficients occur; in fact, the special units only serve, in the algebra proper, as umbrae or regulators of certain operations on **scalars** (see Number).

This idea finds fuller expression in the algebra of matrices, as to which it must suffice to say that a matrix is a symbol consisting of a rectangular array of **scalars**, and that matrices may be combined by a rule of addition which obeys the usual laws, and a rule of multiplication which is distributive and associative, but not, in general, commutative.

Combebiac's tri-quaternions, which require the addition of quasi-**scalars**, independent of one another and of true **scalars**, and analogous to true **scalars**.

To fix a weighted point and a weighted plane in Euclidean space we require 8 **scalars**, and not the 12 **scalars** of a tri-quaternion.

Let n, w be two quasi-**scalars** such that r t e =n, con = w, nw =w 2 = o.

The axis of the member xQ+x'Q' of the second-order complex Q, Q' (where Q=nq+wr, Q'=nq'+wr' and x, x' are **scalars**) is parallel to a fixed plane and intersects a fixed transversal, viz.

Thus, in place of his general tri-quaternion we might deal with products of an odd number of point-plane-**scalars** (of form, uq+wr) which are themselves point-plane-**scalars**; and products of an even number which are octonions; the quotient of two point-plane-**scalars** would be an octonion, of two octonions an octonion, of an octonion by a point-plane-scalar or the inverse a point-plane-scalar.