It is chiefly through the labors of Gibbs and Heaviside that an analysis has been perfected which not only does away with the unnecessary complexity and artificiality of other analyses but offers a strictly natural and therefore as direct and simple a substitute as possible, and, at the same time in no wise is at variance, but runs paralel to them. THIS NEW, yet old method is VECTOR ANALYSIS; it COMBINES within itself most of the advantages of both Quaternions and of Cartesian Analysis. The adoption of Vector Analysis is urged on the grounds of naturalness, simplicity and directness; with it the true meaning of processes and results is brought out as clearly as possible, and desirable abbreviation is obtained. It is admitted, that to a straight and clear thinker, almost any notation or mathematical method suffices, and to such a one, changes in notation or method may appear hardly worth while. He has already attained one of the results which, perforce, follow the intelligent assimilation of a vector method of thinking. To him there is left but the attainment of a simple notation which is the logical accompaniment of clear thought.