traced upon the Surface of the Sphere. 427 



NOTE E, p. 361. 



I may here notice, that the error of Mr HOWARD, in assigning the nature of the 

 curve referred to in this note, was pointed out by Mr LOWRY in the old series of the 

 Mathematical Repository, vol. i. p. 419. The same gentleman also in that (now ex- 

 tremely scarce) work gave a considerable number of new and very curious theorems 

 in spherical geometry and spherical trigonometry, as well as in the successive vo- 

 lumes of the new series of the Repository. He appears, indeed, to be almost the only 

 English geometer who has given much attention to these subjects, or made any ad- 

 ditions of consequence to our knowledge on the subject. 



I have no wish, however, to disparage the work of Mr HOWARD, for it manifests 

 much ingenuity in several parts, though it bears all the marks of its hasty composi- 

 tion ; and that ill-fated mathematician has also the additional merit of directing Mr 

 LOWRY'S attention to that class of inquiries. But setting aside the general merit of 

 his work, the beauty of several of his propositions, and the mathematical difficulties 

 which were inherent in the method of investigation he employed, it may be predicted, 

 that HOWARD'S name will be remembered as long as spherical geometry is culti- 

 vated, for that beautiful theorem which he discovered, that the triangles upon equal 

 bases and between equal parallel circles, being equal. It is, indeed, one of the most 

 beautiful properties yet known in spherical geometry, and one, perhaps, yielding to 

 no other in the richness of its consequences. HOWARD, too, was the first systematic 

 writer on the subject amongst us. 



I had in that note mentioned all the writings that had then come to my know- 

 ledge on the subject of spherical co-ordinates. I have now to add, that Professor 

 GUDERMANN of Cleve published a little work on the same subject in 1830, under 

 the title of Grundriss der Sphdrik Analytischen. This work, though I have so long 

 delayed the completion of my own paper, with a hope to avail myself of the advan- 

 tage of comparison with the work of an independent inquirer, I have never been able 

 to procure ; and, indeed, since I have tried every method that offered the least 

 prospect of success, without avail, I scarcely now can expect to obtain it. 



The learned German Professor has also published some articles on the same and 

 collateral subjects in CRELLE'S Berlin Journal fur die reine und angzvandte Mathe- 

 matik. My attention was first called to this circumstance by a learned friend, who 

 pointed out a notice in the Bulletin des Sciences Mathematiques de FERUSSAC, in 

 which a very brief notice of one of GUDERMANN'S papers was inserted. Upon re- 

 ferring to CHELLE for the original paper, I found it impossible to ascertain with cer- 

 tainty the nature of GUDERMANN'S processes, which rested upon theorems and me- 

 thods laid down in his treatise, and which were only referred to in the said paper. 

 All his subsequent writings that have reference to loci, are similarly incapable of in- 

 VOL. XII. PART II. 3 I 



