It  is  of  the  utmost  importance  that  the  teacher  himself  have
clear  ideas  as  to  rigor  of  proof,  and  these  ideas  are  perhaps
best  acquired  by  a  somewhat  extensive  study  of  mathematics,
even  of  subjects  not  apparently  related  to  those  of  the
elementary  field. ^

Extension  of  proofs.  Generalization  on  insufficient  data
should  be  carefully  avoided.  Thus  when  in  algebra  the  idea
of  number  is  extended  so  as  to  include  negative
number,  it  does  not  follow  that  the  theorems  which  tion  on  insuf-
have  been  proved  for  arithmetical  numbers  will
necessarily  hold  also  for  algebraic  numbers.  The  proofs  were
made  without  thought  of  algebraic  numbers,  and  the  rea-
sons alleged  for  arithmetical  numbers,  and  which  were  true
for  that  class  of  numbers,  may  not  be  true  for  algebraic
numbers.
To  establish  that  a  proof  made  for  certain  definitions  holds
for  more  general  definitions  not  contemplated  when  the  proof
was  made,  it  is  necessary  to  go  over  the  proof  and
ascertain  whether  or  not  all  the  reasons  alleged  are  may  be
valid  also  for  the   more  general  definitions,  even  ^^^^^  ^  '
though  these  were  not  considered  in  making  the  first  proof.
If  so,  the  generalized  proposition  also  holds.     This  is  usually
the  case  when  the  propositions  of  arithmetic  are  generalized
to  cover  algebraic  numbers.      It   is  not  the   case  when   the
binomial  theorem  for  positive  integral  exponents  is  generalized
to  include  fractional  exponents.     The  reasons  alleged  in  the
proof  for  positive   integral  exponents   no    longer   hold    when
applied  to  fractional  exponents  (in  fact  they  then  lose  their
sense),  and  it  is  necessary  to  devise  a  new  proof.
The  conditions  under  which  a  proof  may  be  extended  can
be  expressed  as  follows  :


See  Chapter  IX.,  On  the  preparation  of  teachers.
