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The Theory of Quaternions is due to Sir William Rowan Hamilton, Royal Astronomer of Ireland, who presented his first paper on the subject to the Royal Irish Academy in 1843. His Lectures on Quaternions were published in 1853, and his Elements, in 1866, shortly after his death. The Elements of Quaternions by Tait is the accepted text-book for advanced students.The following development of the theory is prepared for average students with a thorough knowledge of the elements of algebra and geometry, and is believed to be a simple and elementary treatment founded directly upon the fundamental ideas of the subject. This theory is applied in the more advanced examples to develop the principal formulas of trigonometry and solid analytical geometry, and the general properties and classification of surfaces of second order.In the endeavour to bring out the number idea of Quaternions, and at the same time retain the established nomenclature of the analysis, I have found it necessary to abandon the term vector for a directed length. I adopt instead Clifford's suggestive name of step, leaving to vector the sole meaning of right quaternion. This brings out clearly the relations of this number and line, and emphasizes the fact that Quaternions is a natural extension of our fundamental ideas of number, that is subject to ordinary principles of geometric representation, rather than an artificial species of geometrical algebra.The physical conceptions and the breadth of idea that the subject of Quaternions will develop are, of themselves, sufficient reward for its study. At the same time, the power, directness, and simplicity of its analysis cannot fail to prove useful in all physical and geometrical investigations, to those who have thoroughly grasped its principles....

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