belong　to　line；　odd　and　even；　prime　and　compound；　square　and　oblong；
  to　number；　and　also　the　formula　defining　any　one　of　these　attributes
  contains　its　subject…e。g。　line　or　number　as　the　case　may　be。
  Extending　this　classification　to　all　other　attributes；　I　distinguish
  those　that　answer　the　above　description　as　belonging　essentially　to
  their　respective　subjects；　whereas　attributes　related　in　neither　of
  these　two　ways　to　their　subjects　I　call　accidents　or　'coincidents'；
  e。g。　musical　or　white　is　a　'coincident'　of　animal。
  Further　（a）　that　is　essential　which　is　not　predicated　of　a　subject
  other　than　itself：　e。g。　'the　walking　＇thing＇'　walks　and　is　white　in
  virtue　of　being　something　else　besides；　whereas　substance；　in　the
  sense　of　whatever　signifies　a　'this　somewhat'；　is　not　what　it　is　in
  virtue　of　being　something　else　besides。　Things；　then；　not　predicated
  of　a　subject　I　call　essential；　things　predicated　of　a　subject　I　call
  accidental　or　'coincidental'。
  In　another　sense　again　（b）　a　thing　consequentially　connected　with
  anything　is　essential；　one　not　so　connected　is　'coincidental'。　An
  example　of　the　latter　is　'While　he　was　walking　it　lightened'：　the
  lightning　was　not　due　to　his　walking；　it　was；　we　should　say；　a
  coincidence。　If；　on　the　other　hand；　there　is　a　consequential
  connexion；　the　predication　is　essential；　e。g。　if　a　beast　dies　when　its
  throat　is　being　cut；　then　its　death　is　also　essentially　connected　with
  the　cutting；　because　the　cutting　was　the　cause　of　death；　not　death　a
  'coincident'　of　the　cutting。
  So　far　then　as　concerns　the　sphere　of　connexions　scientifically
  known　in　the　unqualified　sense　of　that　term；　all　attributes　which
  （within　that　sphere）　are　essential　either　in　the　sense　that　their
  subjects　are　contained　in　them；　or　in　the　sense　that　they　are
  contained　in　their　subjects；　are　necessary　as　well　as
  consequentially　connected　with　their　subjects。　For　it　is　impossible
  for　them　not　to　inhere　in　their　subjects　either　simply　or　in　the
  qualified　sense　that　one　or　other　of　a　pair　of　opposites　must　inhere
  in　the　subject；　e。g。　in　line　must　be　either　straightness　or　curvature；
  in　number　either　oddness　or　evenness。　For　within　a　single　identical
  genus　the　contrary　of　a　given　attribute　is　either　its　privative　or　its
  contradictory；　e。g。　within　number　what　is　not　odd　is　even；　inasmuch　as
  within　this　sphere　even　is　a　necessary　consequent　of　not…odd。　So；
  since　any　given　predicate　must　be　either　affirmed　or　denied　of　any
  subject；　essential　attributes　must　inhere　in　their　subjects　of
  necessity。
  Thus；　then；　we　have　established　the　distinction　between　the
  attribute　which　is　'true　in　every　instance'　and　the　'essential'
  attribute。
  I　term　'commensurately　universal'　an　attribute　which　belongs　to
  every　instance　of　its　subject；　and　to　every　instance　essentially　and
  as　such；　from　which　it　clearly　follows　that　all　commensurate
  universals　inhere　necessarily　in　their　subjects。　The　essential
  attribute；　and　the　attribute　that　belongs　to　its　subject　as　such；
  are　identical。　E。g。　point　and　straight　belong　to　line　essentially；　for
  they　belong　to　line　as　such；　and　triangle　as　such　has　two　right
  angles；　for　it　is　essentially　equal　to　two　right　angles。
  An　attribute　belongs　commensurately　and　universally　to　a　subject
  when　it　can　be　shown　to　belong　to　any　random　instance　of　that
  subject　and　when　the　subject　is　the　first　thing　to　which　it　can　be
  shown　to　belong。　Thus；　e。g。　（1）　the　equality　of　its　angles　to　two
  right　angles　is　not　a　commensurately　universal　attribute　of　figure。
  For　though　it　is　possible　to　show　that　a　figure　has　its　angles　equal
  to　two　right　angles；　this　attribute　cannot　be　demonstrated　of　any
  figure　selected　at　haphazard；　nor　in　demonstrating　does　one　take　a
  figure　at　random…a　square　is　a　figure　but　its　angles　are　not　equal
  to　two　right　angles。　On　the　other　hand；　any　isosceles　triangle　has　its
  angles　equal　to　two　right　angles；　yet　isosceles　triangle　is　not　the
  primary　subject　of　this　attribute　but　triangle　is　prior。　So　whatever
  can　be　shown　to　have　its　angles　equal　to　two　right　angles；　or　to
  possess　any　other　attribute；　in　any　random　instance　of　itself　and
  primarily…that　is　the　first　subject　to　which　the　predicate　in　question
  belongs　commensurately　and　universally；　and　the　demonstration；　in
  the　essential　sense；　of　any　predicate　is　the　proof　of　it　as
  belonging　to　this　first　subject　commensurately　and　universally：
  while　the　proof　of　it　as　belonging　to　the　other　subjects　to　which　it
  attaches　is　demonstration　only　in　a　secondary　and　unessential　sense。
  Nor　again　（2）　is　equality　to　two　right　angles　a　commensurately
  universal　attribute　of　isosceles；　it　is　of　wider　application。
  5
  We　must　not　fail　to　observe　that　we　often　fall　into　error　because
  our　conclusion　is　not　in　fact　primary　and　commensurately　universal
  in　the　sense　in　which　we　think　we　prove　it　so。　We　make　this　mistake
  （1）　when　the　subject　is　an　individual　or　individuals　above　which　there
  is　no　universal　to　be　found：　（2）　when　the　subjects　belong　to　different
  species　and　there　is　a　higher　universal；　but　it　has　no　name：　（3）
  when　the　subject　which　the　demonstrator　takes　as　a　whole　is　really
  only　a　part　of　a　larger　whole；　for　then　the　demonstration　will　be　true
  of　the　individual　instances　within　the　part　and　will　hold　in　every
  instance　of　it；　yet　the　demonstration　will　not　be　true　of　this　subject
  primarily　and　commensurately　and　universally。　When　a　demonstration
  is　true　of　a　subject　primarily　and　commensurately　and　universally；
  that　is　to　be　taken　to　mean　that　it　is　true　of　a　given　subject
  primarily　and　as　such。　Case　（3）　may　be　thus　exemplified。　If　a　proof
  were　given　that　perpendiculars　to　the　same　line　are　parallel；　it　might
  be　supposed　that　lines　thus　perpendicular　were　the　proper　subject　of
  the　demonstration　because　being　parallel　is　true　of　every　instance
  of　them。　But　it　is　not　so；　for　the　parallelism　depends　not　on　these
  angles　being　equal　to　one　another　because　each　is　a　right　angle；　but
  simply　on　their　being　equal　to　one　another。　An　example　of　（1）　would　be
  as　follows：　if　isosceles　were　the　only　triangle；　it　would　be　thought
  to　have　its　angles　equal　to　two　right　angles　qua　isosceles。　An
  instance　of　（2）　would　be　the　law　that　proportionals　alternate。
  Alternation　used　to　be　demonstrated　separately　of　numbers；　lines；
  solids；　and　durations；　though　it　could　have　been　proved　of　them　all　by
  a　single　demonstration。　Because　there　was　no　single　name　to　denote
  that　in　which　numbers；　lengths；　durations；　and　solids　are　identical；
  and　because　they　differed　specifically　from　one　another；　this　property
  was　proved　of　each　of　them　separately。　To…day；　however；　the　proof　is
  commensurately　universal；　for　they　do　not　possess　this　attribute　qua
  lines　or　qua　numbers；　but　qua　manifesting　this　generic　character　which
  they　are　postulated　as　possessing　universally。　Hence；　even　if　one
  prove　of　each　kind　of　triangle　that　its　angles　are　equal　to　two
  right　angles；　whether　by　means　of　the　same　or　different　proofs；　still；
  as　long　as　one　treats　separately　equilateral；　scalene；　and
  isosceles；　one　does　not　yet　know；　except　sophistically；　that
  triangle　has　its　angles　equal　to　two　right　angles；　nor　does　one　yet
  know　that　triangle　has　this　property　commensurately　and　universally；
  even　if　there　is　no　other　species　of　triangle　but　these。　For　one
  does　not　know　that　triangle　as　such　has　this　property；　nor　even　that
  'all'　triangles　have　it…unless　'all'　means　'each　taken　singly'：　if
  'all'　means　'as　a　whole　class'；　then；　though　there　be　none　in　which
  one　does　not　recognize　this　property；　one　does　not　know　it　of　'all
  triangles'。
  When；　then；　does　our　knowledge　fail　of　commensurate　universality；
  and　when　it　is　unqualified　knowledge？　If　triangle　be　identical　in
  essence　with　equilateral；　i。e。　with　each　or　all　equilaterals；　then
  clearly　we　have　unqualified　knowledge：　if　on　the　other　hand　it　be　not；
  and　the　attribute　belongs　to　equilateral　qua　triangle；　then　our
  knowledge　fails　of　commensurate　universality。　'But'；　it　will　be　asked；
  'does　this　attribute　belong　to　the　subject　of　which　it　has　been
  demonstrated　qua　triangle　or　qua　isosceles？　What　is　the　point　at　which
  the　subject。　to　which　it　belongs　is　primary？　（i。e。　to　what　subject　can
  it　be　demonstrated　as　belonging　commensurately　and　universally？）'
  Clearly　this　point　is　the　first　term　in　which　it　is　found　to　inhere　as
  the　elimination　of　inferior　differenti