Notices  respecting  New  Books.  377 
K1W1  +  *s+  ...=0, 
v*  +  ua+...=Ot 
v*  +  w*+ua  +  ••  -=0> 
a  similar  process  of  reasoning  shows  that  near  the  origin  the  curve  in 
these  cases  has  two  branches  with  either  distinct  or  coincident  tan- 
gents, or  has  a  conjugate  point  at  the  origin.  It  is  scarcely  neces- 
sary to  observe  that  the  student  who  has  approached  the  subject  in 
this  manner  will  find  that  the  corresponding  parts  of  Treatises  on  the 
Differential  Calculus  offer  no  difficulty,  and  give  in  fact  little  more 
than  rather  obvious  generalizations  of  points  that  have  already  en- 
gaged his  attention. 
The  chief  contents  of  the  Treatise  are  these  : — In  addition  to  pre- 
liminary explanations,  there  is  a  full  discussion  of  the  forms  of  curves 
near  the  origin,  and  of  the  forms  which  they  tend  to  take  at  an  in- 
finite distance,  a  subject  which  includes  the  discussion  of  asymptotes 
both  rectilinear  and  curvilinear.  There  is  also  an  account  of  various 
means  (such  as  division  into  compartments)  by  which  curves 
whose  equations  cannot  be  solved  for  one  of  the  variables  may  be 
attempted  "  before  they  are  given  up  in  despair."  The  properties 
of  the  "  analytical  triangle  "  are  given  at  some  length,  as  well  as 
several  illustrations  of  its  ordinary  use  in  ascertaining  the  form  of 
curves  whose  equations  are  of  a  high  degree.  A  novel  use  of  the 
triangle  is  also  suggested  and  illustrated  by  Mr.  Frost,  viz.  as  an  aid 
in  inferring  the  equation  from  the  form  of  a  traced  curve.  The  work 
is  illustrated  by  seventeen  plates,  which  show  the  forms  of  a  very 
large  number  (about  two  hundred)  of  curves  whose  equations  are 
discussed  in  the  text.  Although  the  book  is  intended  for  those 
whose  mathematical  acquirements  are  limited,  yet  we  presume  it  is 
addressed  to  students  of  more  than  average  capacity.  To  such  it 
will  offer  no  serious  difficulty  ;  they  will  work  through  it  in  a  short 
time,  and  in  doing  so  will  find  much  that  is  instructive,  and  many 
pleasant  exercises  of  their  ingenuity.  To  others,  we  fear,  the  study 
will  prove  a  serious  undertaking. 
Arithmetic  in  Theory  and  Practice.  By  J.  Brook  Smith,  M.A., 
LL.B.  London:  Macmillan  and  Co.  (Crown  8vo,  pp.  426.) 
The  aim  of  the  author  of  the  above  work  is  to  explain  fully  the 
principles  on  which  the  rules  of  Arithmetic  are  based,  and  to  do  this 
by  numbers  only  without  the  aid  of  more  general  symbols.  In  car- 
rying out  this  view,  he  states  most  of  the  principles  of  the  science  in 
the  form  of  propositions,  of  which  he  gives  formal  proof.  The  aim 
and  execution  of  the  work  are  both  excellent ;  but  it  may  be  noted 
that  when  a  general  statement  has  to  be  proved  by  reasoning  on  a 
particular  case,  the  force  of  the  proof  is  apt  not  to  be  felt  by  the 
student  unless  it  is  quite  manifest  that  the  reasoning- which  applies 
to  one  case  will  equally  apply  to  all ;  and  accordingly  in  several 
cases  Mr.  Brook  Smith's  proofs  would  have  gained  in  clearness,  if 
not  in  cogency,  from  the  use  of  general  symbols.     Both  as  speci- 
