Clearly　this　point　is　the　first　term　in　which　it　is　found　to　inhere　as
  the　elimination　of　inferior　differentiae　proceeds。　Thus　the　angles
  of　a　brazen　isosceles　triangle　are　equal　to　two　right　angles：　but
  eliminate　brazen　and　isosceles　and　the　attribute　remains。　'But'…you
  may　say…'eliminate　figure　or　limit；　and　the　attribute　vanishes。'　True；
  but　figure　and　limit　are　not　the　first　differentiae　whose
  elimination　destroys　the　attribute。　'Then　what　is　the　first？'　If　it　is
  triangle；　it　will　be　in　virtue　of　triangle　that　the　attribute
  belongs　to　all　the　other　subjects　of　which　it　is　predicable；　and
  triangle　is　the　subject　to　which　it　can　be　demonstrated　as　belonging
  commensurately　and　universally。
  6
  Demonstrative　knowledge　must　rest　on　necessary　basic　truths；　for　the
  object　of　scientific　knowledge　cannot　be　other　than　it　is。　Now
  attributes　attaching　essentially　to　their　subjects　attach
  necessarily　to　them：　for　essential　attributes　are　either　elements　in
  the　essential　nature　of　their　subjects；　or　contain　their　subjects　as
  elements　in　their　own　essential　nature。　（The　pairs　of　opposites
  which　the　latter　class　includes　are　necessary　because　one　member　or
  the　other　necessarily　inheres。）　It　follows　from　this　that　premisses　of
  the　demonstrative　syllogism　must　be　connexions　essential　in　the
  sense　explained：　for　all　attributes　must　inhere　essentially　or　else　be
  accidental；　and　accidental　attributes　are　not　necessary　to　their
  subjects。
  We　must　either　state　the　case　thus；　or　else　premise　that　the
  conclusion　of　demonstration　is　necessary　and　that　a　demonstrated
  conclusion　cannot　be　other　than　it　is；　and　then　infer　that　the
  conclusion　must　be　developed　from　necessary　premisses。　For　though
  you　may　reason　from　true　premisses　without　demonstrating；　yet　if
  your　premisses　are　necessary　you　will　assuredly　demonstrate…in　such
  necessity　you　have　at　once　a　distinctive　character　of　demonstration。
  That　demonstration　proceeds　from　necessary　premisses　is　also　indicated
  by　the　fact　that　the　objection　we　raise　against　a　professed
  demonstration　is　that　a　premiss　of　it　is　not　a　necessary　truth…whether
  we　think　it　altogether　devoid　of　necessity；　or　at　any　rate　so　far　as
  our　opponent's　previous　argument　goes。　This　shows　how　naive　it　is　to
  suppose　one's　basic　truths　rightly　chosen　if　one　starts　with　a
  proposition　which　is　（1）　popularly　accepted　and　（2）　true；　such　as
  the　sophists'　assumption　that　to　know　is　the　same　as　to　possess
  knowledge。　For　（1）　popular　acceptance　or　rejection　is　no　criterion
  of　a　basic　truth；　which　can　only　be　the　primary　law　of　the　genus
  constituting　the　subject　matter　of　the　demonstration；　and　（2）　not
  all　truth　is　'appropriate'。
  A　further　proof　that　the　conclusion　must　be　the　development　of
  necessary　premisses　is　as　follows。　Where　demonstration　is　possible；
  one　who　can　give　no　account　which　includes　the　cause　has　no　scientific
  knowledge。　If；　then；　we　suppose　a　syllogism　in　which；　though　A
  necessarily　inheres　in　C；　yet　B；　the　middle　term　of　the　demonstration；
  is　not　necessarily　connected　with　A　and　C；　then　the　man　who　argues
  thus　has　no　reasoned　knowledge　of　the　conclusion；　since　this
  conclusion　does　not　owe　its　necessity　to　the　middle　term；　for　though
  the　conclusion　is　necessary；　the　mediating　link　is　a　contingent
  fact。　Or　again；　if　a　man　is　without　knowledge　now；　though　he　still
  retains　the　steps　of　the　argument；　though　there　is　no　change　in
  himself　or　in　the　fact　and　no　lapse　of　memory　on　his　part；　then
  neither　had　he　knowledge　previously。　But　the　mediating　link；　not　being
  necessary；　may　have　perished　in　the　interval；　and　if　so；　though
  there　be　no　change　in　him　nor　in　the　fact；　and　though　he　will　still
  retain　the　steps　of　the　argument；　yet　he　has　not　knowledge；　and
  therefore　had　not　knowledge　before。　Even　if　the　link　has　not
  actually　perished　but　is　liable　to　perish；　this　situation　is
  possible　and　might　occur。　But　such　a　condition　cannot　be　knowledge。
  When　the　conclusion　is　necessary；　the　middle　through　which　it　was
  proved　may　yet　quite　easily　be　non…necessary。　You　can　in　fact　infer
  the　necessary　even　from　a　non…necessary　premiss；　just　as　you　can　infer
  the　true　from　the　not　true。　On　the　other　hand；　when　the　middle　is
  necessary　the　conclusion　must　be　necessary；　just　as　true　premisses
  always　give　a　true　conclusion。　Thus；　if　A　is　necessarily　predicated　of
  B　and　B　of　C；　then　A　is　necessarily　predicated　of　C。　But　when　the
  conclusion　is　nonnecessary　the　middle　cannot　be　necessary　either。
  Thus：　let　A　be　predicated　non…necessarily　of　C　but　necessarily　of　B；
  and　let　B　be　a　necessary　predicate　of　C；　then　A　too　will　be　a
  necessary　predicate　of　C；　which　by　hypothesis　it　is　not。
  To　sum　up；　then：　demonstrative　knowledge　must　be　knowledge　of　a
  necessary　nexus；　and　therefore　must　clearly　be　obtained　through　a
  necessary　middle　term；　otherwise　its　possessor　will　know　neither　the
  cause　nor　the　fact　that　his　conclusion　is　a　necessary　connexion。
  Either　he　will　mistake　the　non…necessary　for　the　necessary　and　believe
  the　necessity　of　the　conclusion　without　knowing　it；　or　else　he　will
  not　even　believe　it…in　which　case　he　will　be　equally　ignorant；　whether
  he　actually　infers　the　mere　fact　through　middle　terms　or　the
  reasoned　fact　and　from　immediate　premisses。
  Of　accidents　that　are　not　essential　according　to　our　definition　of
  essential　there　is　no　demonstrative　knowledge；　for　since　an
  accident；　in　the　sense　in　which　I　here　speak　of　it；　may　also　not
  inhere；　it　is　impossible　to　prove　its　inherence　as　a　necessary
  conclusion。　A　difficulty；　however；　might　be　raised　as　to　why　in
  dialectic；　if　the　conclusion　is　not　a　necessary　connexion；　such　and
  such　determinate　premisses　should　be　proposed　in　order　to　deal　with
  such　and　such　determinate　problems。　Would　not　the　result　be　the　same
  if　one　asked　any　questions　whatever　and　then　merely　stated　one's
  conclusion？　The　solution　is　that　determinate　questions　have　to　be　put；
  not　because　the　replies　to　them　affirm　facts　which　necessitate　facts
  affirmed　by　the　conclusion；　but　because　these　answers　are　propositions
  which　if　the　answerer　affirm；　he　must　affirm　the　conclusion　and　affirm
  it　with　truth　if　they　are　true。
  Since　it　is　just　those　attributes　within　every　genus　which　are
  essential　and　possessed　by　their　respective　subjects　as　such　that
  are　necessary　it　is　clear　that　both　the　conclusions　and　the
  premisses　of　demonstrations　which　produce　scientific　knowledge　are
  essential。　For　accidents　are　not　necessary：　and；　further；　since
  accidents　are　not　necessary　one　does　not　necessarily　have　reasoned
  knowledge　of　a　conclusion　drawn　from　them　（this　is　so　even　if　the
  accidental　premisses　are　invariable　but　not　essential；　as　in　proofs
  through　signs；　for　though　the　conclusion　be　actually　essential；　one
  will　not　know　it　as　essential　nor　know　its　reason）；　but　to　have
  reasoned　knowledge　of　a　conclusion　is　to　know　it　through　its　cause。　We
  may　conclude　that　the　middle　must　be　consequentially　connected　with
  the　minor；　and　the　major　with　the　middle。
  7
  It　follows　that　we　cannot　in　demonstrating　pass　from　one　genus　to
  another。　We　cannot；　for　instance；　prove　geometrical　truths　by
  arithmetic。　For　there　are　three　elements　in　demonstration：　（1）　what　is
  proved；　the　conclusion…an　attribute　inhering　essentially　in　a　genus；
  （2）　the　axioms；　i。e。　axioms　which　are　premisses　of　demonstration；
  （3）　the　subject…genus　whose　attributes；　i。e。　essential　properties；　are
  revealed　by　the　demonstration。　The　axioms　which　are　premisses　of
  demonstration　may　be　identical　in　two　or　more　sciences：　but　in　the
  case　of　two　different　genera　such　as　arithmetic　and　geometry　you
  cannot　apply　arithmetical　demonstration　to　the　properties　of
  magnitudes　unless　the　magnitudes　in　question　are　numbers。　How　in
  certain　cases　transference　is　possible　I　will　explain　later。
  Arithmetical　demonstration　and　the　other　sciences　likewise
  possess；　each　of　them；　their　own　genera；　so　that　if　the
  demonstration　is　to　pass　from　one　sphere　to　another；　the　genus　must　be
  either　absolutely　or　to　some　extent　the　same。　If　this　is　not　so；
  transference　is　clearly　impossible；　because　the　extreme　and　the　middle
  terms　must　be　drawn　from　the　same　genus：　otherwise；　as　predicated；
  they　will　not　be　essential　and　will　thus　be　accidents。　That　is　why
  it　cannot　be　proved　by　geometry　that　opposites　fall　under　one　science；
  nor　even　that