540 



guage, must have been the result of a world of reading and toilsome 

 antiquarian research. In 1830 he supplied by his treatise on Algebra 

 one of the greatest deficiencies in our whole circle of mathematical 

 reading, that, namely, of a sound elementary work on that subject 

 based on truly philosophical principles, and explaining the true gist 

 and nature of symbolical reasoning, in its relation to ordinary arith- 

 metic and the science of concrete numerical magnitude, and pointing 

 out (on the principle of the * Permanence of equivalent forms') the 

 origin and the solution of many of those difficulties which were 

 usually slurred over by the student, in a way little conducive to the 

 formation of clear logical habits of thought. In this remarkable 

 work, the ideas propounded by Buee, Argand, Mourey and Warren, 

 respecting the geometrical interpretation of imaginary symbols, were 

 for the first time presented to the student in an elementary treatise 

 as part and parcel of the general subject, and as intimately interwoven 

 in the very texture of the algebraic methods ; thus preparing them 

 to understand and appretiate those more abstruse and powerful 

 systems of imaginary representation subsequently developed in the 

 double and triple algebra of Professor De Morgan and the quater- 

 nions of Sir William Hamilton. A report which he presented to the 

 British Association in 1834, "On the recent progress of certain 

 branches of Analysis," afforded him the occasion of still further 

 maturing his views of the subject; and finally, in 1842 and 1845, 

 he published in two successive volumes a more elaborate and com- 

 plete treatise, in which the purely arithmetical or technical view 

 of algebra is presented quite separately from the purely symbolic 

 or formal one, and which leaves little to desire in respect of meta- 

 physical completeness, and nothing in that of lucid exposition. 

 The position which he then held in the University, as Lowndean 

 Professor of Mathematics (to which office he was elected in 1837), 

 identifies this work with the University in which it was produced 

 as a contribution to scientific literature of which it may well be 

 proud. 



In this, his capacity of Lowndes Professor, he at first gave a series 

 of lectures on practical and theoretical astronomy ; and when, by mu- 

 tual arrangement with the Plumian Professor, these lectures, belong- 

 ing more properly to the department of the latter, were given by 

 that officer, he delivered a course on geometry, and for three sue- 



