Notices-  respecting  JSreiv  Bool*.  833 
accurate  if  instead  of  "is  given"  we  read  "  may  be  approximately 
represented,"  and  if  a  and  b  were  characterized  as  quantities  which 
may  be  taken  as  constants  within  limited  ranges.  Both  physically 
and  mathematically  Van  der  Waals'  equation  is  of  the  highest 
importance ;  but  the  bare  statement  quoted  above  is  misleading. 
This  of  course  in  no  way  detracts  from  the  value  of  the  book  as 
an  introduction  to  the  calculus;  and  both  authors  deserve  great 
credit  for.  the  clearness  with  which  they  have  presented  the 
elements  of  a  subject  which  most  students  find  peculiarly  difficult 
at  the  start. 
(Eavres  de  diaries  Hermite.  Publiees  sous  Us  auspices  de  VAcademie 
des  Sciences,  par  Emile  Picakd.  Tome  1.  Paris :  Grauthier- 
Villars.  1905. 
This  is  the  first  of  three  projected  volumes  of  the  mathematical 
memoirs  of  the  great  French  analyst.  Hermite  began  to  write 
matter  worthy  of  publication  at  the  age  of  22,  when  he  was  still  a 
student;  a  year  later  (in  1843)  he  entered  into  correspondence 
with  Jacobi,  and  in  a  few  years  was  recognized  as  one  of  the 
leading  mathematical  minds  of  the  century.  M.  Picard's  preface 
gives  a  delightfully  written  account  of  the  various  lines  of  mathe- 
matical activity  followed  out  by  Hermite  ;  and  the  volume  is 
further  enriched  by  the  reproduction  of  a  crayon  drawing  of  the 
mathematician  at  the  age  of  25.  The  papers  are  arranged  nearly 
chronologically,  and  in  this  volume  come  down  to  1856. 
Vorlesungen  uber  Matliematische  Naliermethoden.  Von  Dr.  Otto 
Biermann.  Braunschweig  :  Eriedrich  Vieweg  und  Sohn.  1905. 
This  is  a  systematic  account  of  various  accredited  methods  of 
carrying  out  approximate  calculations.  When  it  is  remembered 
that  almost  all  scientific  calculations  are  necessarily  approximate, 
not  only  will  the  importance  of  such  a  work  be  recognized,  but 
some  surprise  may  be  felt  that  no  book  of  a  like  nature  has  ever 
before  been  published.  The  successive  chapters  deal  with  ordinary 
arithmetical  operations,  series  and  logarithms,  solution  of  equations 
(arithmetically  and  graphically),  the  several  methods  of  inter- 
polation with  applications  to  quadrature  and  cubature,  and  an 
account  of  certain  mathematical  instruments  including  Amsler's 
Planimeter.  A  brief  appendix  by  Bauschinger  treats  of  the  idea 
underlying  the  method  of  Least  Squares,  which  otherwise  finds  no 
place  in  Dr.  Biermann's  book.  There  are  some  specially 
interesting  sections  in  the  chapter  devoted  to  graphical  solutions 
of  equations. 
The  First  Booh  of  Euclid's  Elements,  ivith  a  Commentary.  Bif 
W.  B.  Eeankland,  M.A.  Cambridge  University  Press.'  1905. 
This  literal  translation  of  the  real  Euclid  with  a  running  commentary 
will  repay  careful  study  by  all  interested  in  geometry.  The 
commentary  is  at  once  historical  and  critical.  It  is  written  in  a, 
fine  style  and  shows  the  author  to  be  familiar  with  all  the  aspects 
of  both  ancient  and  modern  geometry.  He  quotes  largely  from 
Phil.  Mag.  S.  6.  Vol.  11.  No.  GO.  June  1906.         ^  3  I 
