312 Notices respecting New Books. 



I remark as to the analytical theory, that, taking the origin at 

 the intersection of the perpendiculars, and for the coordinates 

 of the summits (a v J3J,... (« 5 , £ 5 ) respectively, then we have 



« ] «4 + /3 1 /54 = a 2 a 5 + /3 2 ^ 5 = a 3 a, + ^ 3 /3 1 = a 4 a 2 + ^ 4 ^ 2 

 = a 5 ag + /3 5 /3 3 , =-y 2 , 

 where 7 2 is the above-mentioned product, or y is the radius of 

 the sphere. 



Cambridge, September 14, 1871. 



XXXVII. Notices respecting New Books. 



Text-books of Science. Elements of Plane and Solid Geometry. By 

 H. W. Watson, M.A., some time Fellow of Trinity College, Cam- 

 bridge. Small 8vo, pp. 285. London: Longmans, 1871. 



TT is scarcely possible to take an elementary treatise on geometry 

 -*- in hand without comparing it with Euclid's Elements ; and we 

 shall most readily convey to our readers a notion of the contents of 

 this volume by stating that it covers very nearly the same ground as 

 a good school edition of Euclid ; that is to say, it contains, in sub- 

 stance, the first six books and the first twenty-one propositions of 

 the eleventh book, with such additional propositions and remarks as 

 are commonly given in the form of notes. On the other hand, the 

 demonstrations of the propositions are in most cases not the same as 

 Euclid's, and the order in which they are arranged is materially dif- 

 ferent. The work, in fact, is not in any sense an edition of Euclid, 

 but a distinct and independent treatise on geometry, every page of 

 which shows that a great deal of care and thought have been be- 

 stowed on its composition. It would not be easy in a short notice 

 to mention the various points of difference, which in many cases 

 extend to minute details ; but the following may be particularized : — 

 The propositions relating to areas, corresponding to Euclid I. 35-48 

 and II., form a separate book ; the problems are kept distinct from 

 the theorems, and form separate sections ; the subject of proportion 

 is treated algebraically ; Loci are introduced at the end of the first 

 book ; in several parts Limits are employed ; and all the books are 

 subdivided into sections : e. g. the book on Planes and Lines in Space 

 (a subject which is treated very elaborately) is subdivided into four 

 sections, under the head of Miscellaneous Propositions, Perpendi- 

 culars and Obliques to Planes, Dihedral Angles, and Polyhedral 

 Angles. Exercises, about 200 in all, are added to most of the 

 sections. 



The first section of the first book is on triangles ; and the starting- 

 point is an axiom equivalent to the familiar statement that " a straight 

 line is the shortest distance between its extreme points." In Mr. 

 Watson's hands, however, the axiom takes the following form: — "The 

 length of the straight line joining any two points is less than the 

 length of any broken line whatever joining the same two points " 



