descent　is　infinite；　since　a　substance　whose　predicates　were
  infinite　would　not　be　definable。　Hence　they　will　not　be　predicated
  each　as　the　genus　of　the　other；　for　this　would　equate　a　genus　with　one
  of　its　own　species。　Nor　（the　other　alternative）　can　a　quale　be
  reciprocally　predicated　of　a　quale；　nor　any　term　belonging　to　an
  adjectival　category　of　another　such　term；　except　by　accidental
  predication；　for　all　such　predicates　are　coincidents　and　are
  predicated　of　substances。　On　the　other　hand…in　proof　of　the
  impossibility　of　an　infinite　ascending　series…every　predication
  displays　the　subject　as　somehow　qualified　or　quantified　or　as
  characterized　under　one　of　the　other　adjectival　categories；　or　else　is
  an　element　in　its　substantial　nature：　these　latter　are　limited　in
  number；　and　the　number　of　the　widest　kinds　under　which　predications
  fall　is　also　limited；　for　every　predication　must　exhibit　its　subject
  as　somehow　qualified；　quantified；　essentially　related；　acting　or
  suffering；　or　in　some　place　or　at　some　time。
  I　assume　first　that　predication　implies　a　single　subject　and　a
  single　attribute；　and　secondly　that　predicates　which　are　not
  substantial　are　not　predicated　of　one　another。　We　assume　this
  because　such　predicates　are　all　coincidents；　and　though　some　are
  essential　coincidents；　others　of　a　different　type；　yet　we　maintain
  that　all　of　them　alike　are　predicated　of　some　substratum　and　that　a
  coincident　is　never　a　substratum…since　we　do　not　class　as　a　coincident
  anything　which　does　not　owe　its　designation　to　its　being　something
  other　than　itself；　but　always　hold　that　any　coincident　is　predicated
  of　some　substratum　other　than　itself；　and　that　another　group　of
  coincidents　may　have　a　different　substratum。　Subject　to　these
  assumptions　then；　neither　the　ascending　nor　the　descending　series　of
  predication　in　which　a　single　attribute　is　predicated　of　a　single
  subject　is　infinite。　For　the　subjects　of　which　coincidents　are
  predicated　are　as　many　as　the　constitutive　elements　of　each　individual
  substance；　and　these　we　have　seen　are　not　infinite　in　number；　while　in
  the　ascending　series　are　contained　those　constitutive　elements　with
  their　coincidents…both　of　which　are　finite。　We　conclude　that　there
  is　a　given　subject　（D）　of　which　some　attribute　（C）　is　primarily
  predicable；　that　there　must　be　an　attribute　（B）　primarily　predicable
  of　the　first　attribute；　and　that　the　series　must　end　with　a　term　（A）
  not　predicable　of　any　term　prior　to　the　last　subject　of　which　it　was
  predicated　（B）；　and　of　which　no　term　prior　to　it　is　predicable。
  The　argument　we　have　given　is　one　of　the　so…called　proofs；　an
  alternative　proof　follows。　Predicates　so　related　to　their　subjects
  that　there　are　other　predicates　prior　to　them　predicable　of　those
  subjects　are　demonstrable；　but　of　demonstrable　propositions　one　cannot
  have　something　better　than　knowledge；　nor　can　one　know　them　without
  demonstration。　Secondly；　if　a　consequent　is　only　known　through　an
  antecedent　（viz。　premisses　prior　to　it）　and　we　neither　know　this
  antecedent　nor　have　something　better　than　knowledge　of　it；　then　we
  shall　not　have　scientific　knowledge　of　the　consequent。　Therefore；　if
  it　is　possible　through　demonstration　to　know　anything　without
  qualification　and　not　merely　as　dependent　on　the　acceptance　of　certain
  premisses…i。e。　hypothetically…the　series　of　intermediate
  predications　must　terminate。　If　it　does　not　terminate；　and　beyond
  any　predicate　taken　as　higher　than　another　there　remains　another　still
  higher；　then　every　predicate　is　demonstrable。　Consequently；　since
  these　demonstrable　predicates　are　infinite　in　number　and　therefore
  cannot　be　traversed；　we　shall　not　know　them　by　demonstration。　If；
  therefore；　we　have　not　something　better　than　knowledge　of　them；　we
  cannot　through　demonstration　have　unqualified　but　only　hypothetical
  science　of　anything。
  As　dialectical　proofs　of　our　contention　these　may　carry
  conviction；　but　an　analytic　process　will　show　more　briefly　that
  neither　the　ascent　nor　the　descent　of　predication　can　be　infinite　in
  the　demonstrative　sciences　which　are　the　object　of　our
  investigation。　Demonstration　proves　the　inherence　of　essential
  attributes　in　things。　Now　attributes　may　be　essential　for　two　reasons：
  either　because　they　are　elements　in　the　essential　nature　of　their
  subjects；　or　because　their　subjects　are　elements　in　their　essential
  nature。　An　example　of　the　latter　is　odd　as　an　attribute　of
  number…though　it　is　number's　attribute；　yet　number　itself　is　an
  element　in　the　definition　of　odd；　of　the　former；　multiplicity　or　the
  indivisible；　which　are　elements　in　the　definition　of　number。　In
  neither　kind　of　attribution　can　the　terms　be　infinite。　They　are　not
  infinite　where　each　is　related　to　the　term　below　it　as　odd　is　to
  number；　for　this　would　mean　the　inherence　in　odd　of　another
  attribute　of　odd　in　whose　nature　odd　was　an　essential　element：　but
  then　number　will　be　an　ultimate　subject　of　the　whole　infinite　chain　of
  attributes；　and　be　an　element　in　the　definition　of　each　of　them。
  Hence；　since　an　infinity　of　attributes　such　as　contain　their　subject
  in　their　definition　cannot　inhere　in　a　single　thing；　the　ascending
  series　is　equally　finite。　Note；　moreover；　that　all　such　attributes
  must　so　inhere　in　the　ultimate　subject…e。g。　its　attributes　in　number
  and　number　in　them…as　to　be　commensurate　with　the　subject　and　not　of
  wider　extent。　Attributes　which　are　essential　elements　in　the　nature　of
  their　subjects　are　equally　finite：　otherwise　definition　would　be
  impossible。　Hence；　if　all　the　attributes　predicated　are　essential
  and　these　cannot　be　infinite；　the　ascending　series　will　terminate；　and
  consequently　the　descending　series　too。
  If　this　is　so；　it　follows　that　the　intermediates　between　any　two
  terms　are　also　always　limited　in　number。　An　immediately　obvious
  consequence　of　this　is　that　demonstrations　necessarily　involve　basic
  truths；　and　that　the　contention　of　some…referred　to　at　the　outset…that
  all　truths　are　demonstrable　is　mistaken。　For　if　there　are　basic
  truths；　（a）　not　all　truths　are　demonstrable；　and　（b）　an　infinite
  regress　is　impossible；　since　if　either　（a）　or　（b）　were　not　a　fact；
  it　would　mean　that　no　interval　was　immediate　and　indivisible；　but　that
  all　intervals　were　divisible。　This　is　true　because　a　conclusion　is
  demonstrated　by　the　interposition；　not　the　apposition；　of　a　fresh
  term。　If　such　interposition　could　continue　to　infinity　there　might
  be　an　infinite　number　of　terms　between　any　two　terms；　but　this　is
  impossible　if　both　the　ascending　and　descending　series　of
  predication　terminate；　and　of　this　fact；　which　before　was　shown
  dialectically；　analytic　proof　has　now　been　given。
  23
  It　is　an　evident　corollary　of　these　conclusions　that　if　the　same
  attribute　A　inheres　in　two　terms　C　and　D　predicable　either　not　at　all；
  or　not　of　all　instances；　of　one　another；　it　does　not　always　belong
  to　them　in　virtue　of　a　common　middle　term。　Isosceles　and　scalene
  possess　the　attribute　of　having　their　angles　equal　to　two　right　angles
  in　virtue　of　a　common　middle；　for　they　possess　it　in　so　far　as　they
  are　both　a　certain　kind　of　figure；　and　not　in　so　far　as　they　differ
  from　one　another。　But　this　is　not　always　the　case：　for；　were　it　so；　if
  we　take　B　as　the　common　middle　in　virtue　of　which　A　inheres　in　C　and
  D；　clearly　B　would　inhere　in　C　and　D　through　a　second　common　middle；
  and　this　in　turn　would　inhere　in　C　and　D　through　a　third；　so　that
  between　two　terms　an　infinity　of　intermediates　would　fall…an
  impossibility。　Thus　it　need　not　always　be　in　virtue　of　a　common　middle
  term　that　a　single　attribute　inheres　in　several　subjects；　since
  there　must　be　immediate　intervals。　Yet　if　the　attribute　to　be　proved
  common　to　two　subjects　is　to　be　one　of　their　essential　attributes；　the
  middle　terms　involved　must　be　within　one　subject　genus　and　be
  derived　from　the　same　group　of　immediate　premisses；　for　we　have　seen
  that　processes　of　proof　cannot　pass　from　one　genus　to　another。
  It　is　also　clear　that　when　A　inheres　in　B；　this　can　be
  demonstrated　if　there　is　a　middle　term。　Further；　the　'elements'　of
  such　a　conclusion　are　the　premisses　containing　the　middle　in　question；
  and　they　are　identical　in　number　with　the　middle　terms；　seeing　that
  the　immediate　propositions…or　at　least　such　immediate　propositions
  as　are　universal…are　the　'elements'。　If；　on　the　other　hand；　there　is
  no　middle　term；　demonstration　ceases　to　be　possib