was　bound　to　rise；　and　when　an　exhalation　had　risen　cloud　was　bound　to
  form；　and　from　the　formation　of　cloud　rain　necessarily　resulted　and　by
  the　fall　of　rain　the　earth　was　necessarily　moistened：　but　this　was　the
  starting…point；　so　that　a　circle　is　completed；　for　posit　any　one　of
  the　terms　and　another　follows　from　it；　and　from　that　another；　and　from
  that　again　the　first。
  Some　occurrences　are　universal　（for　they　are；　or　come…to…be　what
  they　are；　always　and　in　ever　case）；　others　again　are　not　always　what
  they　are　but　only　as　a　general　rule：　for　instance；　not　every　man　can
  grow　a　beard；　but　it　is　the　general　rule。　In　the　case　of　such
  connexions　the　middle　term　too　must　be　a　general　rule。　For　if　A　is
  predicated　universally　of　B　and　B　of　C；　A　too　must　be　predicated
  always　and　in　every　instance　of　C；　since　to　hold　in　every　instance　and
  always　is　of　the　nature　of　the　universal。　But　we　have　assumed　a
  connexion　which　is　a　general　rule；　consequently　the　middle　term　B　must
  also　be　a　general　rule。　So　connexions　which　embody　a　general　rule…i。e。
  which　exist　or　come　to　be　as　a　general　rule…will　also　derive　from
  immediate　basic　premisses。
  13
  We　have　already　explained　how　essential　nature　is　set　out　in　the
  terms　of　a　demonstration；　and　the　sense　in　which　it　is　or　is　not
  demonstrable　or　definable；　so　let　us　now　discuss　the　method　to　be
  adopted　in　tracing　the　elements　predicated　as　constituting　the
  definable　form。
  Now　of　the　attributes　which　inhere　always　in　each　several　thing
  there　are　some　which　are　wider　in　extent　than　it　but　not　wider　than
  its　genus　（by　attributes　of　wider　extent　mean　all　such　as　are
  universal　attributes　of　each　several　subject；　but　in　their　application
  are　not　confined　to　that　subject）。　while　an　attribute　may　inhere　in
  every　triad；　yet　also　in　a　subject　not　a　triad…as　being　inheres　in
  triad　but　also　in　subjects　not　numbers　at　all…odd　on　the　other　hand　is
  an　attribute　inhering　in　every　triad　and　of　wider　application
  （inhering　as　it　does　also　in　pentad）；　but　which　does　not　extend　beyond
  the　genus　of　triad；　for　pentad　is　a　number；　but　nothing　outside　number
  is　odd。　It　is　such　attributes　which　we　have　to　select；　up　to　the　exact
  point　at　which　they　are　severally　of　wider　extent　than　the　subject　but
  collectively　coextensive　with　it；　for　this　synthesis　must　be　the
  substance　of　the　thing。　For　example　every　triad　possesses　the
  attributes　number；　odd；　and　prime　in　both　senses；　i。e。　not　only　as
  possessing　no　divisors；　but　also　as　not　being　a　sum　of　numbers。
  This；　then；　is　precisely　what　triad　is；　viz。　a　number；　odd；　and
  prime　in　the　former　and　also　the　latter　sense　of　the　term：　for　these
  attributes　taken　severally　apply；　the　first　two　to　all　odd　numbers；
  the　last　to　the　dyad　also　as　well　as　to　the　triad；　but；　taken
  collectively；　to　no　other　subject。　Now　since　we　have　shown　above'　that
  attributes　predicated　as　belonging　to　the　essential　nature　are
  necessary　and　that　universals　are　necessary；　and　since　the
  attributes　which　we　select　as　inhering　in　triad；　or　in　any　other
  subject　whose　attributes　we　select　in　this　way；　are　predicated　as
  belonging　to　its　essential　nature；　triad　will　thus　possess　these
  attributes　necessarily。　Further；　that　the　synthesis　of　them
  constitutes　the　substance　of　triad　is　shown　by　the　following　argument。
  If　it　is　not　identical　with　the　being　of　triad；　it　must　be　related
  to　triad　as　a　genus　named　or　nameless。　It　will　then　be　of　wider　extent
  than　triad…assuming　that　wider　potential　extent　is　the　character　of
  a　genus。　If　on　the　other　hand　this　synthesis　is　applicable　to　no
  subject　other　than　the　individual　triads；　it　will　be　identical　with
  the　being　of　triad；　because　we　make　the　further　assumption　that　the
  substance　of　each　subject　is　the　predication　of　elements　in　its
  essential　nature　down　to　the　last　differentia　characterizing　the
  individuals。　It　follows　that　any　other　synthesis　thus　exhibited　will
  likewise　be　identical　with　the　being　of　the　subject。
  The　author　of　a　hand…book　on　a　subject　that　is　a　generic　whole
  should　divide　the　genus　into　its　first　infimae　species…number　e。g。
  into　triad　and　dyad…and　then　endeavour　to　seize　their　definitions　by
  the　method　we　have　described…the　definition；　for　example；　of
  straight　line　or　circle　or　right　angle。　After　that；　having　established
  what　the　category　is　to　which　the　subaltern　genus　belongs…quantity
  or　quality；　for　instance…he　should　examine　the　properties　'peculiar'
  to　the　species；　working　through　the　proximate　common　differentiae。
  He　should　proceed　thus　because　the　attributes　of　the　genera　compounded
  of　the　infimae　species　will　be　clearly　given　by　the　definitions　of　the
  species；　since　the　basic　element　of　them　all　is　the　definition；　i。e。
  the　simple　infirma　species；　and　the　attributes　inhere　essentially　in
  the　simple　infimae　species；　in　the　genera　only　in　virtue　of　these。
  Divisions　according　to　differentiae　are　a　useful　accessory　to　this
  method。　What　force　they　have　as　proofs　we　did；　indeed；　explain
  above；　but　that　merely　towards　collecting　the　essential　nature　they
  may　be　of　use　we　will　proceed　to　show。　They　might；　indeed；　seem　to
  be　of　no　use　at　all；　but　rather　to　assume　everything　at　the　start
  and　to　be　no　better　than　an　initial　assumption　made　without
  division。　But；　in　fact；　the　order　in　which　the　attributes　are
  predicated　does　make　a　differenceit　matters　whether　we　say
  animal…tame…biped；　or　biped…animal…tame。　For　if　every　definable
  thing　consists　of　two　elements　and　'animal…tame'　forms　a　unity；　and
  again　out　of　this　and　the　further　differentia　man　（or　whatever　else　is
  the　unity　under　construction）　is　constituted；　then　the　elements　we
  assume　have　necessarily　been　reached　by　division。　Again；　division　is
  the　only　possible　method　of　avoiding　the　omission　of　any　element　of
  the　essential　nature。　Thus；　if　the　primary　genus　is　assumed　and　we
  then　take　one　of　the　lower　divisions；　the　dividendum　will　not　fall
  whole　into　this　division：　e。g。　it　is　not　all　animal　which　is　either
  whole…winged　or　split…winged　but　all　winged　animal；　for　it　is　winged
  animal　to　which　this　differentiation　belongs。　The　primary
  differentiation　of　animal　is　that　within　which　all　animal　falls。　The
  like　is　true　of　every　other　genus；　whether　outside　animal　or　a
  subaltern　genus　of　animal；　e。g。　the　primary　differentiation　of　bird　is
  that　within　which　falls　every　bird；　of　fish　that　within　which　falls
  every　fish。　So；　if　we　proceed　in　this　way；　we　can　be　sure　that　nothing
  has　been　omitted：　by　any　other　method　one　is　bound　to　omit　something
  without　knowing　it。
  To　define　and　divide　one　need　not　know　the　whole　of　existence。　Yet
  some　hold　it　impossible　to　know　the　differentiae　distinguishing　each
  thing　from　every　single　other　thing　without　knowing　every　single　other
  thing；　and　one　cannot；　they　say；　know　each　thing　without　knowing　its
  differentiae；　since　everything　is　identical　with　that　from　which　it
  does　not　differ；　and　other　than　that　from　which　it　differs。　Now
  first　of　all　this　is　a　fallacy：　not　every　differentia　precludes
  identity；　since　many　differentiae　inhere　in　things　specifically
  identical；　though　not　in　the　substance　of　these　nor　essentially。
  Secondly；　when　one　has　taken　one's　differing　pair　of　opposites　and
  assumed　that　the　two　sides　exhaust　the　genus；　and　that　the　subject　one
  seeks　to　define　is　present　in　one　or　other　of　them；　and　one　has
  further　verified　its　presence　in　one　of　them；　then　it　does　not
  matter　whether　or　not　one　knows　all　the　other　subjects　of　which　the
  differentiae　are　also　predicated。　For　it　is　obvious　that　when　by
  this　process　one　reaches　subjects　incapable　of　further　differentiation
  one　will　possess　the　formula　defining　the　substance。　Moreover；　to
  postulate　that　the　division　exhausts　the　genus　is　not　illegitimate
  if　the　opposites　exclude　a　middle；　since　if　it　is　the　differentia　of
  that　genus；　anything　contained　in　the　genus　must　lie　on　one　of　the　two
  sides。
  In　establishing　a　definition　by　division　one　should　keep　three
  objects　in　view：　（1）　the　admission　only　of　elements　in　the　definable
  form；　（2）　the　arrangement　of　these　in　the　right　order；　（3）　the
  omission　of　no　such　elements。　The　first　is　feasible　because　one　can
  establish　genus　and　differentia　through　the　topic　of　the　genus；　just
  as　one　can　conclude　the　inherence　of　an　accident　through　the　topic
  of　the　accident。　The　right　order　will　be　achieved　if　the　right　term　is
  assumed　as　primary；　and　this　will　be　ens