Mr. T! S. Davies on Geometry and Geometers. 543 



then already developed ; just as Euclid gives the demonstrations 

 of the problems in his lirst four and his sixth books. The num- 

 ber of problems, however, which are now required, would render 

 it next to impossible to write such a book ; and even if it were 

 written, it would be of little value to the class for whom it was 

 designed. The other is, to give the problems according to any 

 convenient arrangement for easy reference, with the best con- 

 structions that could be devised, and with varieties of construc- 

 tion to meet different practical exigences; and thus omit all 

 considerations respecting the different principles on which the 

 operations were based, in reference to a logical system of theo- 

 retical geometry. The proofs may be added or not — making it 

 the rule of science or the " rule of thumb "—as the author and 

 his publisher may deem best ; though there is little doubt as to 

 the course that would in most cases be pursued. The " cheap " 

 and the " easy " would prevail. But even such a book has never 

 been produced that is not perfectly contemptible as a system of 

 rules. For the most part the problems are those of Euclid, and 

 constructed in Euchd's manner; though for merely practical 

 pm-poses, there are few of them which cannot be effected m a far 

 better manner. Euclid was bound to certain processes in his 

 constructions by the theoretical truths which he had developed 

 anterior to his arriving at them; and he only introduced them 

 at the places where they occur to subserve a futm-c (generally 

 the next) portion of his theoretic development. It formed no 

 part of his plan to compose a treatise on geometry expressly for 

 the use of the architect, the engineer, and the artisan ; and his 

 plan ought to form no exemplar for those who undertake such 

 works. The truth however is, that such works are and have 

 been generally undertaken by men in all respects incompetent to 

 the task. They have been men alike unacquainted with geometiy 

 and of the arts to which it is capable of being applied. Very 

 rarely indeed is even a moderate acquaintance with the best geo- 

 metry found combined with an intimate knowledge of the pro- 

 blems that perpetually embarrass the "practical man;" and such 

 a work as is really required is rather a dream of the possible 

 future than a hope of the actual present. 



Mr. Swale appears to have been expert m the solution ot 

 " Diophantine Problems," as several of apparent difficulty occur 

 in these MSS. Probably it was a taste acquired by intercourse 

 with his friends Whitley and Ryley; as none, either of these 

 papers or his published writings, indicate that algebraical specu- 

 lations in general interested liini greatly. 1 am, however too 

 little acquainted with the literature of this subject to be able to 

 judge of the value- of these; solutions as compared with others; 

 or whether any of them have been elsewhere discussed, or are 



