asked…unless；　indeed；　the　terms　can　reciprocate　by　two　different
  modes；　by　accidental　predication　in　one　relation　and　natural
  predication　in　the　other。
  20
  Now；　it　is　clear　that　if　the　predications　terminate　in　both　the
  upward　and　the　downward　direction　（by　'upward'　I　mean　the　ascent　to
  the　more　universal；　by　'downward'　the　descent　to　the　more　particular）；
  the　middle　terms　cannot　be　infinite　in　number。　For　suppose　that　A　is
  predicated　of　F；　and　that　the　intermediates…call　them　BB'B〃。。。…are
  infinite；　then　clearly　you　might　descend　from　and　find　one　term
  predicated　of　another　ad　infinitum；　since　you　have　an　infinity　of
  terms　between　you　and　F；　and　equally；　if　you　ascend　from　F；　there
  are　infinite　terms　between　you　and　A。　It　follows　that　if　these
  processes　are　impossible　there　cannot　be　an　infinity　of
  intermediates　between　A　and　F。　Nor　is　it　of　any　effect　to　urge　that
  some　terms　of　the　series　AB。。。F　are　contiguous　so　as　to　exclude
  intermediates；　while　others　cannot　be　taken　into　the　argument　at
  all：　whichever　terms　of　the　series　B。。。I　take；　the　number　of
  intermediates　in　the　direction　either　of　A　or　of　F　must　be　finite　or
  infinite：　where　the　infinite　series　starts；　whether　from　the　first
  term　or　from　a　later　one；　is　of　no　moment；　for　the　succeeding　terms　in
  any　case　are　infinite　in　number。
  21
  Further；　if　in　affirmative　demonstration　the　series　terminates　in
  both　directions；　clearly　it　will　terminate　too　in　negative
  demonstration。　Let　us　assume　that　we　cannot　proceed　to　infinity　either
  by　ascending　from　the　ultimate　term　（by　'ultimate　term'　I　mean　a
  term　such　as　was；　not　itself　attributable　to　a　subject　but　itself
  the　subject　of　attributes）；　or　by　descending　towards　an　ultimate
  from　the　primary　term　（by　'primary　term'　I　mean　a　term　predicable　of　a
  subject　but　not　itself　a　subject）。　If　this　assumption　is　justified；
  the　series　will　also　terminate　in　the　case　of　negation。　For　a　negative
  conclusion　can　be　proved　in　all　three　figures。　In　the　first　figure
  it　is　proved　thus：　no　B　is　A；　all　C　is　B。　In　packing　the　interval
  B…C　we　must　reach　immediate　propositionsas　is　always　the　case　with
  the　minor　premisssince　B…C　is　affirmative。　As　regards　the　other
  premiss　it　is　plain　that　if　the　major　term　is　denied　of　a　term　D　prior
  to　B；　D　will　have　to　be　predicable　of　all　B；　and　if　the　major　is
  denied　of　yet　another　term　prior　to　D；　this　term　must　be　predicable　of
  all　D。　Consequently；　since　the　ascending　series　is　finite；　the　descent
  will　also　terminate　and　there　will　be　a　subject　of　which　A　is
  primarily　non…predicable。　In　the　second　figure　the　syllogism　is；　all　A
  is　B；　no　C　is　B；。。no　C　is　A。　If　proof　of　this　is　required；　plainly
  it　may　be　shown　either　in　the　first　figure　as　above；　in　the　second
  as　here；　or　in　the　third。　The　first　figure　has　been　discussed；　and
  we　will　proceed　to　display　the　second；　proof　by　which　will　be　as
  follows：　all　B　is　D；　no　C　is　D。。。；　since　it　is　required　that　B
  should　be　a　subject　of　which　a　predicate　is　affirmed。　Next；　since　D　is
  to　be　proved　not　to　belong　to　C；　then　D　has　a　further　predicate
  which　is　denied　of　C。　Therefore；　since　the　succession　of　predicates
  affirmed　of　an　ever　higher　universal　terminates；　the　succession　of
  predicates　denied　terminates　too。
  The　third　figure　shows　it　as　follows：　all　B　is　A；　some　B　is　not　C。
  Therefore　some　A　is　not　C。　This　premiss；　i。e。　C…B；　will　be　proved
  either　in　the　same　figure　or　in　one　of　the　two　figures　discussed
  above。　In　the　first　and　second　figures　the　series　terminates。　If　we
  use　the　third　figure；　we　shall　take　as　premisses；　all　E　is　B；　some　E
  is　not　C；　and　this　premiss　again　will　be　proved　by　a　similar
  prosyllogism。　But　since　it　is　assumed　that　the　series　of　descending
  subjects　also　terminates；　plainly　the　series　of　more　universal
  non…predicables　will　terminate　also。　Even　supposing　that　the　proof
  is　not　confined　to　one　method；　but　employs　them　all　and　is　now　in
  the　first　figure；　now　in　the　second　or　third…even　so　the　regress
  will　terminate；　for　the　methods　are　finite　in　number；　and　if　finite
  things　are　combined　in　a　finite　number　of　ways；　the　result　must　be
  finite。
  Thus　it　is　plain　that　the　regress　of　middles　terminates　in　the
  case　of　negative　demonstration；　if　it　does　so　also　in　the　case　of
  affirmative　demonstration。　That　in　fact　the　regress　terminates　in　both
  these　cases　may　be　made　clear　by　the　following　dialectical
  considerations。
  22
  In　the　case　of　predicates　constituting　the　essential　nature　of　a
  thing；　it　clearly　terminates；　seeing　that　if　definition　is　possible；
  or　in　other　words；　if　essential　form　is　knowable；　and　an　infinite
  series　cannot　be　traversed；　predicates　constituting　a　thing's
  essential　nature　must　be　finite　in　number。　But　as　regards　predicates
  generally　we　have　the　following　prefatory　remarks　to　make。　（1）　We
  can　affirm　without　falsehood　'the　white　（thing）　is　walking'；　and
  that　big　（thing）　is　a　log'；　or　again；　'the　log　is　big'；　and　'the　man
  walks'。　But　the　affirmation　differs　in　the　two　cases。　When　I　affirm
  'the　white　is　a　log'；　I　mean　that　something　which　happens　to　be
  white　is　a　log…not　that　white　is　the　substratum　in　which　log
  inheres；　for　it　was　not　qua　white　or　qua　a　species　of　white　that　the
  white　（thing）　came　to　be　a　log；　and　the　white　（thing）　is
  consequently　not　a　log　except　incidentally。　On　the　other　hand；　when
  I　affirm　'the　log　is　white'；　I　do　not　mean　that　something　else；
  which　happens　also　to　be　a　log；　is　white　（as　I　should　if　I　said　'the
  musician　is　white；'　which　would　mean　'the　man　who　happens　also　to　be　a
  musician　is　white'）；　on　the　contrary；　log　is　here　the　substratum…the
  substratum　which　actually　came　to　be　white；　and　did　so　qua　wood　or　qua
  a　species　of　wood　and　qua　nothing　else。
  If　we　must　lay　down　a　rule；　let　us　entitle　the　latter　kind　of
  statement　predication；　and　the　former　not　predication　at　all；　or　not
  strict　but　accidental　predication。　'White'　and　'log'　will　thus　serve
  as　types　respectively　of　predicate　and　subject。
  We　shall　assume；　then；　that　the　predicate　is　invariably　predicated
  strictly　and　not　accidentally　of　the　subject；　for　on　such
  predication　demonstrations　depend　for　their　force。　It　follows　from
  this　that　when　a　single　attribute　is　predicated　of　a　single　subject；
  the　predicate　must　affirm　of　the　subject　either　some　element
  constituting　its　essential　nature；　or　that　it　is　in　some　way
  qualified；　quantified；　essentially　related；　active；　passive；　placed；
  or　dated。
  （2）　Predicates　which　signify　substance　signify　that　the　subject　is
  identical　with　the　predicate　or　with　a　species　of　the　predicate。
  Predicates　not　signifying　substance　which　are　predicated　of　a
  subject　not　identical　with　themselves　or　with　a　species　of
  themselves　are　accidental　or　coincidental；　e。g。　white　is　a
  coincident　of　man；　seeing　that　man　is　not　identical　with　white　or　a
  species　of　white；　but　rather　with　animal；　since　man　is　identical
  with　a　species　of　animal。　These　predicates　which　do　not　signify
  substance　must　be　predicates　of　some　other　subject；　and　nothing　can　be
  white　which　is　not　also　other　than　white。　The　Forms　we　can　dispense
  with；　for　they　are　mere　sound　without　sense；　and　even　if　there　are
  such　things；　they　are　not　relevant　to　our　discussion；　since
  demonstrations　are　concerned　with　predicates　such　as　we　have　defined。
  （3）　If　A　is　a　quality　of　B；　B　cannot　be　a　quality　of　A…a　quality
  of　a　quality。　Therefore　A　and　B　cannot　be　predicated　reciprocally　of
  one　another　in　strict　predication：　they　can　be　affirmed　without
  falsehood　of　one　another；　but　not　genuinely　predicated　of　each
  other。　For　one　alternative　is　that　they　should　be　substantially
  predicated　of　one　another；　i。e。　B　would　become　the　genus　or
  differentia　of　A…the　predicate　now　become　subject。　But　it　has　been
  shown　that　in　these　substantial　predications　neither　the　ascending
  predicates　nor　the　descending　subjects　form　an　infinite　series；　e。g。
  neither　the　series；　man　is　biped；　biped　is　animal；　&c。；　nor　the　series
  predicating　animal　of　man；　man　of　Callias；　Callias　of　a　further。
  subject　as　an　element　of　its　essential　nature；　is　infinite。　For　all
  such　substance　is　definable；　and　an　infinite　series　cannot　be
  traversed　in　thought：　consequently　neither　the　ascent　nor　the
  descent　is　infinite；　since　a　substance　whose　predicates　were
  infinite　would　not　b