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  XLI. 
  Notices 
  respecting 
  New 
  Boohs, 
  

  

  An 
  Introduction 
  to 
  the 
  Theory 
  of 
  Infinite 
  Sei'ies. 
  By 
  T. 
  J. 
  I' 
  A. 
  

   Beomwich. 
  Macmillan 
  & 
  Co. 
  1908. 
  

  

  A 
  Course 
  of 
  Pure 
  Mathematics. 
  By 
  Gr. 
  H. 
  Hardy. 
  Cambridge 
  

   University 
  Press. 
  1908. 
  

  

  Plane 
  Geometry 
  for 
  Advanced 
  Students. 
  — 
  Part 
  I. 
  By 
  Clemei^t 
  

   V. 
  DuKELL. 
  Macmillan 
  & 
  Co. 
  London 
  : 
  1909. 
  

  

  ^HE 
  authors 
  of 
  these 
  three 
  books 
  deserve 
  the 
  thanks 
  of 
  teachers 
  

   -^ 
  and 
  of 
  serious 
  students 
  for 
  placing 
  in 
  their 
  hands 
  well-planned 
  

   and 
  clearly 
  written 
  treatises 
  on 
  certain 
  important 
  parts 
  of 
  pure 
  

   mathematics. 
  Of 
  the 
  three 
  the 
  Geometry 
  is 
  the 
  most 
  elementary, 
  

   and 
  may 
  be 
  studied 
  with 
  great 
  profit 
  by 
  any 
  intelligent 
  student 
  

   familiar 
  with 
  the 
  equivalent 
  of 
  Euclid's 
  first 
  six 
  books. 
  With 
  the 
  

   exception 
  of 
  a 
  few 
  obvious 
  sphere 
  and 
  plane 
  problems 
  in 
  inversion 
  

   and 
  a 
  short 
  paragraph 
  on 
  solid 
  geometry, 
  the 
  geometry 
  is 
  throughout 
  

   plane 
  and 
  deals 
  with 
  the 
  properties 
  of 
  triangles, 
  plane 
  quadri- 
  

   laterals, 
  and 
  circles. 
  The 
  demonstrated 
  properties 
  are 
  arranged 
  

   under 
  93 
  Theorems, 
  while 
  the 
  numerous 
  examples 
  to 
  each 
  chapter 
  

   contain 
  a 
  large 
  number 
  of 
  other 
  interesting 
  theorems 
  which 
  appa- 
  

   rently 
  do 
  not 
  attain 
  to 
  the 
  status 
  of 
  the 
  ninety-three. 
  The 
  diagrams 
  

   are 
  excellent 
  and 
  add 
  much 
  to 
  the 
  beauty 
  of 
  the 
  page. 
  Occasionally 
  

   the 
  author 
  indicates 
  instructive 
  statical 
  interpretations 
  of 
  the 
  

   geometrical 
  theorems. 
  One 
  definition 
  only 
  have 
  we 
  noted 
  as 
  

   being 
  incomplete. 
  On 
  page 
  68 
  we 
  read 
  that 
  "Any 
  quantity 
  which 
  

   can 
  be 
  completely 
  represented 
  by 
  a 
  straight 
  line 
  of 
  appropriate 
  

   length, 
  drawn 
  in 
  an 
  appropriate 
  direction, 
  is 
  called 
  a 
  vector." 
  

   A 
  finite 
  rotation 
  can 
  be 
  so 
  represented, 
  and 
  yet 
  it 
  is 
  not 
  a 
  vector 
  ; 
  

   for 
  two 
  finite 
  rotations 
  cannot 
  be 
  added 
  vectorially. 
  Part 
  II. 
  is 
  

   promised 
  shortly. 
  It 
  will 
  deal 
  with 
  the 
  higher 
  applications 
  to 
  

   conies 
  and 
  other 
  more 
  recondite 
  aspects 
  of 
  modern 
  geometry. 
  An 
  

   interesting 
  feature 
  of 
  Mr. 
  Durell's 
  book 
  is 
  the 
  series 
  of 
  historic 
  

   notes 
  prefixed 
  to 
  most 
  of 
  the 
  chapters. 
  

  

  Mr. 
  Hardy's 
  Course 
  of 
  Pure 
  Mathematics 
  will 
  almost 
  certainly 
  

   l)ecome 
  very 
  useful 
  in 
  University 
  classes. 
  It 
  lays 
  important 
  stress 
  

   on 
  points 
  which 
  are 
  apt 
  to 
  be 
  slurred 
  over 
  when 
  a 
  student 
  passes 
  

   from 
  the 
  ordinary 
  algebra 
  to 
  the 
  study 
  of 
  the 
  calculus. 
  The 
  

   author 
  himself 
  regards 
  the 
  fourth 
  chapter 
  on 
  limits 
  of 
  functions 
  

   of 
  an 
  integral 
  variable 
  as 
  one 
  of 
  the 
  leading 
  features 
  of 
  the 
  book. 
  

   This 
  is 
  followed 
  by 
  a 
  discussion 
  of 
  the 
  limits 
  of 
  functions 
  of 
  a 
  

   continuous 
  variable, 
  and 
  then 
  Chapter 
  VI. 
  on 
  Derivatives 
  and 
  

   Integrals 
  constitutes 
  a 
  " 
  first 
  chapter 
  in 
  the 
  differential 
  and 
  integral 
  

   calculus." 
  Subsequent 
  chapters 
  deal 
  with 
  Taylor's 
  Theorem, 
  with 
  

   definite 
  integrals, 
  with 
  questions 
  of 
  convergency, 
  with 
  infinite 
  

   integrals, 
  and 
  with 
  the 
  theory 
  of 
  logarithmic, 
  exponential 
  and 
  

   circular 
  functions. 
  In 
  introducing 
  the 
  complex 
  variable 
  in 
  

  

  