However great the skill and ingenuity shown by the pioneer in solving problems by Quaternions, there was always left the thought to the unbiased student that a lack of parallelism existed between the old and the new methods of treatment. Such a lack undoubtedly does exist, but it is only during the last few years that a method has been evolved which avoids this fatal defect. 

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.  

A few examples of vector concentration are to be found in the exercises of the last chapter of this book. But the sole use of vector notation, without the insightand clear conceptions which should obtain at the same time, is without any value whatsoever, vitiates the vector point of view, and is contrary to the spirit of it.