Notices respecting Neixi Books. 73 



the employment of analytical artifices by which many of the results 

 might have been obtained more concisely. In every case he has endea- 

 voured to point out the form in which the student would be expected 

 to present the solution to the examiner." 1'he geometrical papers are 

 also evidently solved on the same principle ; and we feel bound to 

 say that under such conditions, a better set of solutions could not 

 have been produced. I'hey are everywhere marked by a ])erfect 

 knowledge of the broad principles of the science, and by an address in 

 the management of his symbols, which is almost peculiar to Mr. Gaskin. 

 We are not offering any opinion as to whether Mr. Gaskin'splan be the 

 best that could have been adopted ; but we merely affirm that, having 

 selected a plan, he has fulfilled his objects with consummate ability. 

 The gem of Mr. Gaskin's two volumes, however, in the eyes of a 

 mathematician, will be the three "appendices" to the geometrical 

 problem.s. The first and third of these are upon a problem which 

 has received much attention both from English and foreign geome- 

 ters, in one form or other : — " In a given conic section to inscribe a 

 polygon all whose sides (produced if necessary) shall pass through 

 given points." Much as this problem has been discussed (for a pretty 

 complete list of these discussions, we must refer to Mr. Potts's "Ap- 

 pendix," p. 98), we believe that Mr. Gaskin has not been anticipated 

 in his method by any one whatever ; and certainly of all the attempts 

 to solve this difficult problem by algebraic methods, his process is 

 the most direct and elegant. We are glad to see such models of in- 

 vestigation laid before the men ; and especially by an author whose 

 name carries that weight in the University which his does. 



To the second "Appendix" we must award less praise. It is 

 mainly composed of a discussion of the equation of the second degree 

 between ,r and y. It is complete, but complex and cumbrous : and 

 is is inferior in all respects but completeness, to many others that we 

 could easily point out ; whilst even in respect to its one good quaUty, 

 it is not superior to several of them. We have not room, however, 

 to particularise. 



The remaining part of this Appendix is devoted to the deduction of 

 a considerable number of properties of the conic sections, references 

 for which are made to different papers set either in the Senate House 

 or in the various Colleges. In these solutions a mixed method is 

 pursued. Certain cardinal propositions are established by coordinate 

 methods ; and from these, by geometrical considerations, the otiiers 

 are deduced. Great difference of opinion exists among geometers as 

 to the i)ropriety of this mode of investigation ; and we shall not offer 

 ourselves either as dictators or umpires on the question. It is suffi- 

 cient to say, tliat granting these cardinal properties (as Pascal's Hex- 

 agram, for instance) to have been established in a manner to satisfy 

 any given geometer, he will find the deductions from them to have 

 been effected with great brevity, clearness and elegance. 



On the whole, then, we cannot but congratulate the University 

 on tiie a]ipcarance of the works of Messrs. Potts and Gaskin ; and 

 we trust they will be speedily followed by others as valuable, either 

 by those authors themselves, or by other men possessing an equal 

 amount of learning and soundness of judgement. 



