366 Reviews, and Notices respecting New Books. 



tion of the triangle defined by the conditions of the question. The 

 third part contains a summary of the principles of Navigation and 

 Nautical Astronomy. We think that sufficient is here taught to 

 answer the purposes of most men who enter upon the study ; and 

 it is certainly taught in a way to enable the student to see the 

 reason of the complicated rules that occur in the practice of navi- 

 gation — a property belonging to few books that have been published 

 on the subject. 



The fourth part consists of a series of miscellaneous trigonome- 

 trical inquiries, such as are commonly to be found in the most re- 

 cent continental and English works on trigonometry, and they are 

 all conducted in the most simple and general manner. Amongst 

 them we see some formulae from De Gua for the spherical excess, 

 which we think have not been before given in any English work. 

 The chapter on the " Relations between the corresponding varia- 

 tions of the parts of a Triangle" is taken from Professor Airy's 

 Trigonometry, in the E?icyclopcedia Metropolitans which forms 

 the best treatise on this branch of the subject that we are ac- 

 quainted with. 



The remaining part of the work consists of three supplementary 

 chapters, containing some original speculations upon Spherical 

 Geometry and Trigonometry, from the pen of Mr. Davies, which, 

 together with other numerous productions of that accomplished 

 mathematician, are well deserving the attention of geometers, as well 

 for the novel views they afford as for the striking results they furnish 

 in what may be termed spherical analysis, — a subject to which 

 he has been long, and we believe, successfully, devoting his atten- 

 tion. We regret that we cannot find room for a complete analysis 

 of these researches, and that we must content ourselves with a very 

 brief account of the several chapters. 



The first of these is devoted chiefly to the demonstration of cer- 

 tain properties of spherical triangles, having remarkable analogies 

 to those of plane triangles. The second chapter is employed in 

 establishing the properties of associated triangles, a term which we 

 must explain, and we shall do so by giving a short account of the 

 whole of the terminology which Mr. Davies has employed in this 

 supplement 



When a spherical triangle ABC is formed, we may, by pro- 

 ducing its sides till they meet, 



two and two, form three others, c^ ""-"*«^e (l^ ^?f 



AB'C,C A' B, B Q A. These f^fC^X^^^\^>^/ 



four triangles form what Mr. \\^/ yxv y\ \t 



Davies calls an associated system \\. /^\/ 



of triangles. The triangle ABC \. j /S 



(not necessarily the central one) rtn;' / 



is called the fundamental triangle \ ^trtf 



of the associated system, and the V \ j 



other three the supplemental ^«5/ 



ones. When a new triangle, a be, C *V? 



whose poles are ABC, is de- \ 



scribed, this, it is well known, is 



denominated the polar triangle ; 



