It  will  warn  the  teacher  from  any  attempt  to  prove  the  postu-
late, either  for  himself  or  his  pupils  —  since  there  is  no  ques-
tion that  all  such  attempts  must  be  classed  with  attempts  to
"square  the  circle."

Euclid  attempted  to  avoid  definitions  and  proofs  based  on
The  Use  of  motion,  and  his  followers  even  down  to  modern
Signed  Ma^?  days  have  followed  his  example  to  the  detriment
nitudes.  of   geometric    work.      In    recent  times,    however,

there  have  arisen  men  who  boldly  make  free  and  fundamental


^*  "  To  be  sure,  as  Study  has  well  insisted,  for  a  thorough  com-
prehension of  even  the  elementary  parts  of  Euclidean  geometry  the
non-Euclidean  geometries  are  absolutely  essential.  But  the  teacher
is  teaching  the  subject  for  the  benefit  of  the  students,  and  it  must
be  admitted  that  beginners  in  the  study  of  demonstrative  geometry
cannot  appreciate  the  very  delicate  considerations  involved  in  the
thoroughly  abstract  science.  Indeed,  one  may  conjecture  that,  had
it  not  been  for  the  brilliant  success  of  Euclid  in  his  effort  to  organ-
ize into  a  formally  deductive  system  the  geometric  treasures  of  his
times,  the  advent  of  the  reign  of  science  in  the  modern  sense  might
not  have  been  so  Ions;  deferred.  Shall  we  then  hold  that  in  the
schools  the  teaching  of  demonstrative  geometry  should  be  reformed
in  such  a  way  as  to  take  account  of  all  the  wonderful  discoveries
which  have  been  made  —  many  even  recently  —  in  the  domain  of
abstract  geometry  ?  And  should  similar  reforms  be  made  in  the
treatment  of  arithmetic  and  algebra?  To  make  reforms  of  this
kind  :  would  it  not  be  to  repeat  more  gloriously  the  error  of  those
followers  of  Euclid  who  fixed  his  Elements  as  a  text-book  for  ele-
mentary instruction  in  geometry  for  over  two  thousand  years  ?
Every  one  agrees  that  professional  mathematicians  should  cer-
tainly take  account  of  these  great  developments  in  the  technical
foundations  of  mathematics,  and  that  ample  provision  should  be
made  for  instruction  in  these  matters  :  and  on  reflection,  every  one
agrees  further  that  this  provision  should  be  reserved  for  the  later
collegiate  and  university  years."  —  Moore,  Presidential  Address.
