of Hamilton's, as well as his so-called 'Elements of Quaternions,' is frequently unpleasant in style, besides being obscure and difficult of interpretation.
Hamilton's method involved a remarkable extension of science. He showed that the 'impossible quantities' which so frequently occur in analysis admit of easy interpretation by a natural extension of the symbol's meaning. The so-called imaginary or unreal factor really denoted an operation to be performed on the line or surface in question, the operation of rotation. If we multiply a line by (–1) the result is the same as if the line were turned through 180° in its plane, and hence if multiplied by (–1)½ the line will be turned through 90°. On that discovery of the operational character of 'imaginary' factors and expressions was based the whole science of quaternions. Warren in 1828, Peacock (see Algebra, vol. ii. chap, xxxi.), De Morgan in his 'Double Algebra,' and others had clearly discussed the interpretation of (–1)½. The notion of motion, virtual transference and rotation, was now combined with the application of algebra to geometry, and while the word 'add' represented motion forward and backward, the word 'multiply' was specialised to represent circular motion. Hamilton freed the science from the limitations of ages, and by his new adaptation of symbols dealt with lines in all possible planes, quite irrespective of any such restricting axes of reference as were necessary to the Cartesian system. To bring any line in space to complete coincidence with any other line may be called finding its quaternion: so named from the four numbers or elements occurring in the geometrical question of comparing two lines in space, viz. their mutual angle, the two conditions determining their plane and their relative length.
This new algebra accordingly could express the relations of space directionally as well as quantitatively, and recommended itself as a powerful organ in solid geometry, dynamical questions involving rotation, spherical conics or surfaces of the second order, besides innumerable applications in physical and astronomical problems, crystallography, electrical dynamics, wherever, in short, there occurs motion or implied translation in tridimensional space, or where the notion of polarity is involved.
In spite of the undoubted power of this 'algebra of pure space' and its trenchant disposal of many classes of physical and geometrical problems, the method has not attracted much attention, except among a few advanced mathematicians. Professor Kelland for several years showed the application of the method to elementary geometry, conics, and some central surfaces of the second-order; but at present none of our universities appear to encourage the study, partly from lack of time to deal adequately with the highest physical applications of mathematical work. There are great difficulties from the use of familiar terms in an extended sense which is frequently difficult of interpretation geometrically. As a whole the method is pronounced by most mathematicians to beneither easy nor attractive, the interpretation being hazy or metaphysical and seldom clear and precise.
As a professor of astronomy Hamilton was not successful, especially in the practical part of his duties, partly perhaps from want of previous training in instrumental and technical work. Some of his professorial lectures, however, were admired for their fluent ornate style, frequently rising into eloquence. From the knowledge of languages which he acquired in youth he was able to read Latin, Greek, German, and Arabic for relaxation, and was frequently seen reading Plato and Kant. He had excellent taste in poetical composition, and wrote many sonnets and other poems. He corresponded with Wordsworth, Coleridge, and Southey, and lived on terms of intimacy with Miss Edgeworth and Mrs. Hemans. He had also an extensive correspondence with Professor De Morgan from 1841 till 1865, the year of his death. A mere 'selection' of the letters occupies 390 pages of the concluding volume of the Rev. R. P. Graves's 'Life of Hamilton.' From his genial and candid disposition and the simplicity of his manners, Hamilton was esteemed both by young and old, not only by those in his home circle, but by all with whom he came in contact.
The second great literary work of Hamilton, 'The Elements of Quaternions,' was published posthumously, edited by his son, William Edwin Hamilton, C.E., in 1866. Besides the previous four years spent in accumulating the material of the 'Elements of Quarternions,' the last two years of the author's life were incessantly occupied in the work of revision, selection, and compression. So devoted indeed was his attention that he is supposed to have seriously injured his health, which had already been affected by a gouty illness, and even his brain-power. Latterly there were also epileptic symptoms. He died on 2 Sept. 1865. The pension of 200l. which he had received since he was knighted was afterwards continued to his widow.
A list of Hamilton's papers, memoirs, and posthumous publications is given in the Rev.
