Notices  respecting  New  Books.  307 
sea,  and  the  prevalence  or  otherwise  of  clouds  and  rain.  The  land  areas 
are  readily  distinguishable  by  being  shaded.  If  we  are  not  greatly 
mistaken,  the  maps  of  the  barometer  and  wind  will  prove  to  be  the 
most  valuable  portion;  but  we  greatly  regret  that  the  area  over  which 
the  elements  are  given  is  so  small  that  by  our  instruments  we  can  only 
obtain  fragments  of  them  ;  and  unless  the  meteorologist  combines  the 
data  given  in  these  charts  with  other  data  obtainable  on  the  continent 
or  ocean,  we  apprehend  his  work  will  not  be  easy  from  the  charts  before 
us  to  obtain  a  general  view  of  the  affections  of  the  atmosphere  as  they 
exist  in  their  entirety. 
An  Elementary  Treatise  on  the  Differential  Calculus,  containing  the 
Theory  of  Plane  Curves,  with  numerous  Examples.     By  Benjamin 
Williamson,  A.M.,   Fellow  and  Tutor,   Trinity  College,  Dublin. 
London  :  Longmans,  Green,  and  Co.   1872.  (Pp.  343.) 
The   various  English  Treatises    on  the  Differential  Calculus  so 
closely  resemble  each  other  as  to  the  subjects  of  their  successive  chap- 
ters,   that  we  shall  sufficiently  describe  the  scope  of  the  present 
volume  by  saying  that  it  comprises  all  the  parts  of  the  subject  com- 
monly treated.     For  instance,  if  the  table  of  contents  be  compared 
with  that  of  Mr.  Todhunter's  well-known  treatise,  they  will  be  found 
to  differ  simply  in  arrangement.     We  do  not  mean  to  imply  that 
the  resemblance  extends  beyond  this  point.     On  the  contrary,  Mr. 
Williamson's  Treatise  is  as  completely  his  own  as  any  book  on  a 
well-worked  subject  can  be  an  author's  own.   In  fact,  so  far  as  treat- 
ment is  concerned,  these  two  authors  take  a  fundamentally  different 
view.     The  one  says  : — "  My  own  experience  with  pupils  has  been 
decidedly  unfavourable  to  the  system  of  Differentials ;  many  suc- 
cessful teachers  whom  I  have  consulted  have  expressed  a  similar 
opinion  ;   and  I  have  therefore  adopted  exclusively  the   method  of 
Differential  Coefficients"  (Pref.  to  3rd  edit.).     Mr.  Williamson,  on 
the  other  hand,  says  : — "  An  exclusive  adherence  to  the  method  of 
Differential  Coefficients  is  by  no  means  necessary  for  clearness  and 
simplicity  ;  and,  indeed,  I  have  found  by  experience  that  many  fun- 
damental investigations  in  mechanics  and  geometry  are  made  more 
intelligible  to  beginners  by  the  method  of  differentials  than  by  that 
of  differential  coefficients."    We  shall  not  attempt  to  decide  between 
the  authors ;  but  this  at  least  may  be  said,  that  Mr.  Williamson  has 
proved  by  example  that  a  most  instructive  and  useful  book  can  be 
produced,  if  written  from  his  point  of  view.     It  is  perhaps  in  the 
Geometrical  applications  of  the  Calculus  that  the  best  features  of  the 
book  are  to  be  seen  :  e.  g.  the  illustration  of  partial  differentiation 
by  discussing  the  variations  of  the  parts  of  plane  and  spherical  tri- 
angles (pp.  110-118)  is  very  happy,  and  the  chapters  on  Tangents, 
Asymptotes,  Multiple  points,  Inflection,  and  Curvature  are  exceed- 
ingly well  done  ;  they  have  been  written  with  Dr.  Salmon's  work 
on  "Higher  Plane   Curves"   in  view,   and  contain  several  points 
which  are  not  generally  given  in  the  corresponding  chapters  of  other 
elementary  treatises,  such  as  the  notice  of  Inverse  curves,   Pedal 
curves,  the  introductory  notice   on    Multiple  points,   &c.       Very 
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