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To The first edition of this work, published in 1867, the following was prefixed :

The present work was commenced in 1859, while I was a Professor of Mathematics, and far more ready at Quaternion analysis than I can now pretend to be. Had it been then completed I should have bad means of testing its teaching capabilities, and of improving it, before publication, where found deficient in that respect.

The duties of another Chair, and Sir W. Hamilton's wish that my volume should not appear till after the publication of his Elements, interrupted my already extensive preparations. I had worked out nearly all the examples of Analytical Geometry in Todhunter's Collection, and I had made various physical applications of the Calculus, especially to Crystallography, to Geometrical Optics, and to the Induction of Currents, in addition to those on Kinematics, Electrodynamics, Fresnel's Wave Surface, &c., which are reprinted in the present work from the Quarterly Mathematical Journal and the Proceedings of the Royal Society of Edinburgh.

'Sir W. Hamilton, when I saw him but a few days before his death, urged me to prepare my work as soon as possible, his being almost ready for publication. He then expressed, more strongly perhaps than he had ever done before, his profound conviction of the importance of Quaternions to the progress of physical science ; and his desire that a really elementary treatise on the subject should soon be published.

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