378  Notices  respecting  New  Books, 
mens  of  the  book  and  as  instances  of  what  we  mean,  we  will  cite 
two  propositions. 
"  If  we  interchange  multiplicand  and  multiplier,  the  product  remains 
the  same. 
"  Thus  7  multiplied  by  5  is  the  same  as  1111111 
5  multiplied  by  7.  For  write  down  7  units  1111111 
in  a  horizontal  line,  and  repeat  this  line  51111111 
times;  the  number  of  units  written  down  is  1111111 
7  repeated  5  times  or  7  multiplied  5  times.  1111111 
But  in  each  vertical  line  there  are  5  units, 
and  there  are  7  such  lines ;  therefore  the  number  of  units  written 
down  is  5  repeated  7  times,  or  5  multiplied  by  7 ;  that  is,  7  multi- 
plied by  5  is  the  same  as  5  multiplied  by  7."    (P.  14.) 
Here  the  reasoning  is  obviously  conclusive  ;  what  is  proved  in  the 
case  of  7  and  5  is  obviously  true  of  any  other  two  numbers.  We 
will  now  take  another  proposition. 
"  If  the  numerator  and  denominator  of  a  fraction  be  prime  to  each 
other,  the  numerator  and  denominator  of  any  fraction  of  equal  value 
will  he  equimultiples  of  the  numerator  and  denominator  of  the  given 
fraction." 
"  The  numerator  and  denominator  of  -J  are  prime  to  each  other  ; 
and  suppose  ^  to  be  equal  to  j;\  then  8  and  10  are  equimultiples 
of  4  and  5. 
"For,  multiply  numerator  and  denominator  of  -J  by  10,  and  of  -f$ 
by  5,  these  fractions  will  be  unaltered  in  value ;  therefore 
4xl0_  8X5 
5x10      10x5' 
And  since  the  parts  composing  these  fractions  are  all  equal,  the 
numbers  taken  in  both  cases  must  be  equal,  so  that 
4x10=8x5  or  =5x8. 
And  since  4  divides  4x  10,  it  divides  5x8;  but  4  is  prime  to  5, 
therefore  it  divides  8 ;  let  the  quotient  be  2,  so  that 
8=4x2; 
therefore 
4x10=5x4x2. 
And  dividing  each  of  these  quantities  by  4,  we  have 
10=5x2; 
that  is,  10  is  the  same  multiple  of  5  that  8  is  of  4  ;  or  the  numerator 
and  denominator  of  ^  are  equimultiples  of  the  numerator  and  deno- 
minator of  f."    (P.  63.) 
In  this  case  the  reasoning  is  quite  cogent,  but  is  rendered  obscure 
by  the  use  of  the  second  particular  fraction  -fa,  "We  shrewdly  sus- 
pect that  a  large  number  of  students  will  remark  when  they  come 
to  the  end  of  the  proof,  that  it  was  plain  to  begin  with  that  8  and 
10  are  equimultiples  of  4  and  5  ;  i.  e.  they  will  have  missed  the  point 
of  the  proof.     And  accordingly  the  proof  would  have  been  made 
