Obituary Notice of Mr. Horner. 459 



Which is against the hypothesis (4). 



6. As at step (3), it may be shown that Q m = Q u is im- 

 possible, ifb^a. 



(C.) Hence the series of remainders Q p Q 2 , Q 3 , 



(Q a = ) q 9 recur perpetually in the same order, and are all 



distinct from each other, and from the terms of series (B). 



7. Should any term of (A) yet remain, a continuation of 



c 

 this process (4) will exhaust them. For, that — is integral 



may be ascertained by reflecting that every unanticipated as- 

 sumption, such as q, entails a> terms such as series (C), all un- 

 anticipated. 



8. But since — -II- is integral, and ~ m is also integral, 

 _ £ is integral ; that is _ .. q.e.d. 



#* # Since this paper was received we were pained to learn 

 that its distinguished author is no more. We have been 

 favoured by a mathematical friend with the following notice 

 of his decease and his character. " Mr. Horner died on the 

 21st of Sept. after a week of extreme suffering, arising from a 

 complication of asthma and ossification of the heart, aged 

 fifty-one. His health had been for many years in a very pre- 

 carious state, and his sufferings (as in this class of disorders 

 they almost always are) often very great ; yet his equanimity 

 and firmness of character were fully equal to cope with pain 

 and bodily decay; and his end was such as a Christian's should 

 be — calm, peaceful, happy. 



" As a mathematician Mr. Horner was undoubtedly amongst 

 the first men of our time. His method of continuous approxi- 

 mation to the roots of equations of all kinds, but which has 

 hitherto been little applied except to algebraical equations, is 

 probably the most important invention with which algebra has 

 been enriched during the last two centuries. The Philoso- 

 phical Magazine contains a considerable number of his papers 

 on mathematical and physical subjects, which are of course 

 well known to its readers, and which, therefore, we need not 

 specify or eulogise. Some alterations in this paper were sent 

 off' only a few days before his lamented decease. 



" Mr. Horner was not a mathematician only, he was a man 

 of the most refined taste in literature, both classical and ge- 

 neral. He has left behind him many papers on these sub- 

 jects, on biblical criticism, and on other subjects, as well as on 



3 N 2 



