226  Notices  respecting  New  Books* 
a  failure  of  this  kind  that  Mr.  Todhunter  first  enunciated,  and  in  our 
pages*,  the  cardinal  principle  above  quoted. 
Technical  Arithmetic  and  Mensuration.     By  C.  W.  Merrifield, 
F.R.S.     London :  Longmans  and  Co.      1872.     (Pp.308.) 
This  book  is  intended  for  the  use  of  persons  who  have  already 
learned  arithmetic,  but  who  wish  to  renew  and  extend  their 
acquaintance  with  the  subject.  It  consists  of  a  treatise  on  Arith- 
metic of  1 83  page?,  a  treatise  on  Mensuration  of  64  pages,  and  an 
Appendix  of  60  pages,  comprising  Examination  Papers,  somewhat 
elaborate  Tables  of  Money,  Lengths,  Areas,  &c,  and  Answers  to 
Questions.  The  treatise  on  Arithmetic  gives  clearly  and  concisely 
an  account  of  the  fundamental  operations  of  Arithmetic ;  it  notices 
with  sufficient,  but  not  more  than  sufficient,  fulness  the  points 
which  have  to  be  dwelt  on  in  oral  instruction;  and  it  illustrates 
the  rules  by  a  very  considerable  number  of  examples,  most  of  which 
Mr.  Merrifield  says  he  has  himself  made.  Very  many  of  these  ex- 
amples are  mere  illustrations  of  rules,  such  as  must  of  necessity 
occur  in  every  book  of  Arithmetic  ;  e.g.,  "  If  3|-  of  a  share  cost  £5, 
what  will  5f  cost  ?  "  Many,  however,  occur  here  and  there  which 
illustrate  elementary  scientific  principles  ;  e.  g.,  "  Express  miles  per 
hour  in  metres  per  second,  having  given  a  metre  =39*37  inches." 
The  chief  novelty,  however,  is  the  last  chapter,  which  treats  of  "  the 
applications  of  Arithmetic  to  Machines,  Work,  and  Motion."  This 
is  a  very  useful  chapter ;  it  contains  sufficient  explanations  to  render 
intelligible  the  examples  with  which  it  concludes.  The  number  of 
these  examples  might  easily  have  been  increased  with  great  advan- 
tage. The  objection  to  introducing,  except  with  very  great  judg- 
ment, examples  on  points  of  science  into  a  book  of  Arithmetic,  is 
that  they  presuppose  knowledge  which  the  student  probably  does 
not  possess.  This  objection  does  not,  of  course,  apply  in  the  above 
case  ;  we  are  inclined  to  think  it  does  apply  to  such  examples  as 
(19),  p.  53  ;  (16),  (17),  (25),  (26),  p.  76— not  to  say  that  (25)  is 
inexactly  stated,  and  (26)  clearly  in  the  wrong  place. 
On  pp.  66,  67  there  is  a  statement  with  regard  to  proportion 
which  is  worth  notice.  In  the  text  occur  these  words  : — "Propor- 
tion may  include  the  comparison  of  ratios  which  are  not  definite,  and 
which  are  incapable  of  being  expressed  as  fractions."  And  to  this 
statement  the  following  note  is  added  : — "  I  do  not  think  it  neces- 
sary to  restrict  the  idea  of  proportion  to  numerical  ratio.  In  fact, 
it  appears  to  me  to  be  a  consequence  of 
3  shillings  :  5  shillings  :  :  6  yards  :  10  yards, 
that 
3  shillings  :  6  yards  :  :  5  shillings  :  1 0  yards ; 
*  Phil.  Mag.  June  1866,  vol.  xxxi.  p.  425.  The  question  to  which  Mr. 
Todhunter  applied  the  principle  was  this  : — To  determine  a  solid  of  revolu- 
tion of  given  surface  so  that  it  may  cut  the  axis  of  revolution  at  given 
points  and  have  a  maximum  volume.  A  chapter  of  the  essay  is  devoted  to 
examining  this  question  and  some  cases  arising  out  of  it. 
