Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/173

 our formulæ to space of four or more dimensions. It will not be claimed that the notions of quaternions will apply to such a space, except indeed in such a limited and artificial manner as to rob them of their value as a system of geometrical algebra. But vectors exist in such a space, and there must be a vector analysis for such a space. The notions of geometrical addition and the scalar product are evidently applicable to such a space. As we cannot define the direction of a vector in space of four or more dimensions by the condition of perpendicularity to two given vectors, the definition of $$\text{V}\alpha \beta$$, as given above, will not apply totidem verbis to space of four or more dimensions. But a little change in the definition, which would make no essential difierence in three dimensions, would enable us to apply the idea at once to space of any number of dimensions.

These considerations are of a somewhat a priori nature. It may be more convincing to consider the use actually made of the quaternion as an instrument for the expression of spatial relations. The principal use seems to be the derivation of the functions expressed by $$\text{S}\alpha\beta$$ and $$\text{V}\alpha\beta.$$ Each of these expressions is regarded by quatemionic writers as representing two distinct operations; first, the formation of the product $$\alpha\beta,$$ which is the quaternion, and then the taking out of this quaternion the scalar or the vector part, as the case may be, this second process being represented by the selective symbol, $$\text{S}$$ or $$\text{V.}$$ This is, I suppose, the natural development of the subject in a treatise on quaternions, where the chosen subject seems to require that we should commence with the idea of a quaternion, or get there as soon as possible, and then develop everything from that particular point of view. In a system of vector analysis, in which the principle of development is not thus predetermined, it seems to me contrary to good method that the more simple and elementary notions should be defined by means of those which are less so.

The quaternion affords a convenient notation for rotations. The notation $$q ( \,\, ) q^{-1},$$ where $$q$$ is a quaternion and the operand is to be written in the parenthesis, produces on all possible vectors just such changes as a (finite) rotation of a solid body. Rotations may also be represented, in a manner which seems to leave nothing to be desired, by linear vector functions. Doubtless each method has advantages in certain cases, or for certain purposes. But since nothing is more simple than the definition of a linear vector function, while the definition of a quaternion is far from simple, and since in any case linear vector functions must be treated in a system of vector analysis, capacity for representing rotations does not seem to me sufficient to entitle the quaternion to a place among the fundamental and necessary notions of a vector analysis. Another use of the quaternionic idea is associated with the symbol $$\nabla.$$