An Elementary Treatise on Quaternions #224729

Sometimes, indeed, this rule is most absurdly violated, for it is usual to take cos2a' as equal to (cos a)2, while cos — 1x Is not equal to (cos No such incongruities appear in Quaternions; but what is true of operators and functions in other methods, that they are not generally commutative, IS in Quaternions true in the multipli cation of (vector) coordinates. It will be observed by those who are acquainted with the Cal culus that I have, in many cases, not given the shortest or simplest proof of an important proposition. This has been done with the view of including, in moderate compass, as great a variety of methods as possible. With the same object I have endeavoured to supply, by means of the Examples appended to each Chapter, hints (which will not be lost to the intelligent student) of farther develop ments of the Calculus. Many of these are due to Hamilton, who, in spite of his great originality, was one of the most excellent examiners any University can boast of. It must always be remembered (that Cartesian methods are mere particular cases of Quaternions, where most of the distinctive fea tures have disappeared; and that when, in the treatment of any particular question, scalars have to be adopted, the Quaternion solution becomes identical with the Cartesian one. Nothing there fore is ever lost, though much is generally gained, by employing Quaternions in preference to ordinary methods. In fact, even when Quaternions degrade to scalars, they give the solution of the most general statement of the problem they are applied to, quite inde pendent of any limitations as to choice of particular coordinate axes.