Notices  respecting  New  Books.  379 
much  plainer  if  the  preliminary  assumption  had  been 
4 x 
The  reasoning  would  then  have  run  thus : 
Therefore  4  x  y = 5  x  a?. 
As  4  is  prime  to  5,  cc  must  be  divisible  by  4  ;  let  it  equal  4  x  m,  then 
y=5xw. 
Hence  x  and  y  are  equimultiples  of  4  and  5. 
There  is  in  reality  no  objection  to  the  use  of  general  symbols  in  a 
treatise  on  arithmetic.  For  it  must  be  remembered  that  any  student 
who  takes  in  hand  this  or  any  other  complete  treatise  will  already 
be  pretty  familiar  with  the  subject ;  he  will  have  begun  learning  it 
when  a  child,  and  have  acquired  his  knowledge  by  a  daily  practice 
extending  over  some  years.  His  object  will  be  not  to  acquire  know- 
ledge of  a  new  subject,  but  to  extend,  systematize,  and  render  ra- 
tional a  knowledge  which,  as  first  acquired,  was  imperfect  and  em- 
pirical; For  this  purpose  we  know  no  better  treatise  than  Mr. 
Brook  Smith's.  His  explanations  are  clear  and  satisfactory ;  he  has 
gone  with  great  thoroughness,  though  without  prolixity,  into  the 
theory  of  the  subject ;  and  he  has  illustrated  all  the  parts  of  the 
book  by  an  exceedingly  large  number  of  examples.  A  student 
who  works  straight  through  the  book  will  go  through  an  excellent 
preparation  for  passing  any  of  the  numerous  examinations  in  which 
a  thorough  knowledge  of  arithmetic  is  required. 
Kuklos,  an  Experimental  Investigation  into  the  Relationship  of  certain 
Lines.  By  John  Harris.  Part  First.  Montreal:  1870.  4to, 
pp.  35.     Ten  Plates. 
Though  somewhat  in  doubt  whether  it  were  worth' while  to  notice 
this  book,  we  have  on  consideration  determined  to  do  so,  for  two 
reasons  :  in  the  first  place,  it  is  published  in  the  dominion  of  Canada ; 
in  the  next,  although  it  is  undoubtedly  an  attempt  to  square  the 
circle  (and  that  is  a  very  hopeless  sort  of  undertaking),  the  author 
shows  an  acquaintance  with  the  conditions  of  the  question  not  to  be 
generally  found  amongst  his  fellows  ;  for  he  allows  that  the  ratio  of 
the  circumference  to  the  diameter  of  a  circle  is  correctly  expressed 
by  the  number  3*14159  .  .  .  (p.  12).  Moreover  the  problem  which 
he  proposes  to  solve  is  not  intrinsically  absurd  ;  there  must,  of  course, 
be  some  straight  line  intermediate  in  length  to  three  and  four  times 
the  diameter,  which  is  exactly  equal  in  length  to  the  circumference ; 
and,  further,  the  rectangle  under  half  that  line  and  the  radius  equals 
the  area  of  the  circle.  The  problem  would  therefore  be  solved  if  by  a 
direct  geometrical  construction  a  straight  line  could  be  drawn  equal 
in  length  to  a  given  fraction  of  the  circumference.  Mr.  Harris's 
notion  of  the  solution  seems  to  be  this  : — Let  A  B  be  an  arc  of  a 
circle  whose  centre  is  O ;  join  O  A ;  draw  A  T,  a  tangent  to 
the  arc  at  A;  take  A  O^ twice  AO;  with  centre  0„  draw  an 
arc  ABp  join  O,  B,  and  produce  it  to  cut  ABjin  Bj ;  the  length 
of  the  arc  A  BT  is  plainly  the  same  as  that  of  A  B.     If  the  construe- 
