it　cannot　be　proved　by　geometry　that　opposites　fall　under　one　science；
  nor　even　that　the　product　of　two　cubes　is　a　cube。　Nor　can　the
  theorem　of　any　one　science　be　demonstrated　by　means　of　another
  science；　unless　these　theorems　are　related　as　subordinate　to
  superior　（e。g。　as　optical　theorems　to　geometry　or　harmonic　theorems　to
  arithmetic）。　Geometry　again　cannot　prove　of　lines　any　property　which
  they　do　not　possess　qua　lines；　i。e。　in　virtue　of　the　fundamental
  truths　of　their　peculiar　genus：　it　cannot　show；　for　example；　that
  the　straight　line　is　the　most　beautiful　of　lines　or　the　contrary　of
  the　circle；　for　these　qualities　do　not　belong　to　lines　in　virtue　of
  their　peculiar　genus；　but　through　some　property　which　it　shares　with
  other　genera。
  8
  It　is　also　clear　that　if　the　premisses　from　which　the　syllogism
  proceeds　are　commensurately　universal；　the　conclusion　of　such　i。e。
  in　the　unqualified　sense…must　also　be　eternal。　Therefore　no
  attribute　can　be　demonstrated　nor　known　by　strictly　scientific
  knowledge　to　inhere　in　perishable　things。　The　proof　can　only　be
  accidental；　because　the　attribute's　connexion　with　its　perishable
  subject　is　not　commensurately　universal　but　temporary　and　special。
  If　such　a　demonstration　is　made；　one　premiss　must　be　perishable　and
  not　commensurately　universal　（perishable　because　only　if　it　is
  perishable　will　the　conclusion　be　perishable；　not　commensurately
  universal；　because　the　predicate　will　be　predicable　of　some
  instances　of　the　subject　and　not　of　others）；　so　that　the　conclusion
  can　only　be　that　a　fact　is　true　at　the　moment…not　commensurately　and
  universally。　The　same　is　true　of　definitions；　since　a　definition　is
  either　a　primary　premiss　or　a　conclusion　of　a　demonstration；　or　else
  only　differs　from　a　demonstration　in　the　order　of　its　terms。
  Demonstration　and　science　of　merely　frequent　occurrences…e。g。　of
  eclipse　as　happening　to　the　moon…are；　as　such；　clearly　eternal：
  whereas　so　far　as　they　are　not　eternal　they　are　not　fully
  commensurate。　Other　subjects　too　have　properties　attaching　to　them
  in　the　same　way　as　eclipse　attaches　to　the　moon。
  9
  It　is　clear　that　if　the　conclusion　is　to　show　an　attribute
  inhering　as　such；　nothing　can　be　demonstrated　except　from　its
  'appropriate'　basic　truths。　Consequently　a　proof　even　from　true；
  indemonstrable；　and　immediate　premisses　does　not　constitute　knowledge。
  Such　proofs　are　like　Bryson's　method　of　squaring　the　circle；　for
  they　operate　by　taking　as　their　middle　a　common　character…a　character；
  therefore；　which　the　subject　may　share　with　another…and　consequently
  they　apply　equally　to　subjects　different　in　kind。　They　therefore
  afford　knowledge　of　an　attribute　only　as　inhering　accidentally；　not　as
  belonging　to　its　subject　as　such：　otherwise　they　would　not　have　been
  applicable　to　another　genus。
  Our　knowledge　of　any　attribute's　connexion　with　a　subject　is
  accidental　unless　we　know　that　connexion　through　the　middle　term　in
  virtue　of　which　it　inheres；　and　as　an　inference　from　basic　premisses
  essential　and　'appropriate'　to　the　subject…unless　we　know；　e。g。　the
  property　of　possessing　angles　equal　to　two　right　angles　as　belonging
  to　that　subject　in　which　it　inheres　essentially；　and　as　inferred
  from　basic　premisses　essential　and　'appropriate'　to　that　subject：　so
  that　if　that　middle　term　also　belongs　essentially　to　the　minor；　the
  middle　must　belong　to　the　same　kind　as　the　major　and　minor　terms。
  The　only　exceptions　to　this　rule　are　such　cases　as　theorems　in
  harmonics　which　are　demonstrable　by　arithmetic。　Such　theorems　are
  proved　by　the　same　middle　terms　as　arithmetical　properties；　but　with　a
  qualification…the　fact　falls　under　a　separate　science　（for　the　subject
  genus　is　separate）；　but　the　reasoned　fact　concerns　the　superior
  science；　to　which　the　attributes　essentially　belong。　Thus；　even
  these　apparent　exceptions　show　that　no　attribute　is　strictly
  demonstrable　except　from　its　'appropriate'　basic　truths；　which；
  however；　in　the　case　of　these　sciences　have　the　requisite　identity
  of　character。
  It　is　no　less　evident　that　the　peculiar　basic　truths　of　each
  inhering　attribute　are　indemonstrable；　for　basic　truths　from　which
  they　might　be　deduced　would　be　basic　truths　of　all　that　is；　and　the
  science　to　which　they　belonged　would　possess　universal　sovereignty。
  This　is　so　because　he　knows　better　whose　knowledge　is　deduced　from
  higher　causes；　for　his　knowledge　is　from　prior　premisses　when　it
  derives　from　causes　themselves　uncaused：　hence；　if　he　knows　better
  than　others　or　best　of　all；　his　knowledge　would　be　science　in　a　higher
  or　the　highest　degree。　But；　as　things　are；　demonstration　is　not
  transferable　to　another　genus；　with　such　exceptions　as　we　have
  mentioned　of　the　application　of　geometrical　demonstrations　to　theorems
  in　mechanics　or　optics；　or　of　arithmetical　demonstrations　to　those
  of　harmonics。
  It　is　hard　to　be　sure　whether　one　knows　or　not；　for　it　is　hard　to　be
  sure　whether　one's　knowledge　is　based　on　the　basic　truths
  appropriate　to　each　attribute…the　differentia　of　true　knowledge。　We
  think　we　have　scientific　knowledge　if　we　have　reasoned　from　true　and
  primary　premisses。　But　that　is　not　so：　the　conclusion　must　be
  homogeneous　with　the　basic　facts　of　the　science。
  10
  I　call　the　basic　truths　of　every　genus　those　clements　in　it　the
  existence　of　which　cannot　be　proved。　As　regards　both　these　primary
  truths　and　the　attributes　dependent　on　them　the　meaning　of　the　name　is
  assumed。　The　fact　of　their　existence　as　regards　the　primary　truths
  must　be　assumed；　but　it　has　to　be　proved　of　the　remainder；　the
  attributes。　Thus　we　assume　the　meaning　alike　of　unity；　straight；　and
  triangular；　but　while　as　regards　unity　and　magnitude　we　assume　also
  the　fact　of　their　existence；　in　the　case　of　the　remainder　proof　is
  required。
  Of　the　basic　truths　used　in　the　demonstrative　sciences　some　are
  peculiar　to　each　science；　and　some　are　common；　but　common　only　in
  the　sense　of　analogous；　being　of　use　only　in　so　far　as　they　fall
  within　the　genus　constituting　the　province　of　the　science　in　question。
  Peculiar　truths　are；　e。g。　the　definitions　of　line　and　straight；
  common　truths　are　such　as　'take　equals　from　equals　and　equals　remain'。
  Only　so　much　of　these　common　truths　is　required　as　falls　within　the
  genus　in　question：　for　a　truth　of　this　kind　will　have　the　same　force
  even　if　not　used　generally　but　applied　by　the　geometer　only　to
  magnitudes；　or　by　the　arithmetician　only　to　numbers。　Also　peculiar
  to　a　science　are　the　subjects　the　existence　as　well　as　the　meaning
  of　which　it　assumes；　and　the　essential　attributes　of　which　it
  investigates；　e。g。　in　arithmetic　units；　in　geometry　points　and
  lines。　Both　the　existence　and　the　meaning　of　the　subjects　are
  assumed　by　these　sciences；　but　of　their　essential　attributes　only
  the　meaning　is　assumed。　For　example　arithmetic　assumes　the　meaning
  of　odd　and　even；　square　and　cube；　geometry　that　of　incommensurable；　or
  of　deflection　or　verging　of　lines；　whereas　the　existence　of　these
  attributes　is　demonstrated　by　means　of　the　axioms　and　from　previous
  conclusions　as　premisses。　Astronomy　too　proceeds　in　the　same　way。
  For　indeed　every　demonstrative　science　has　three　elements：　（1）　that
  which　it　posits；　the　subject　genus　whose　essential　attributes　it
  examines；　（2）　the　so…called　axioms；　which　are　primary　premisses　of　its
  demonstration；　（3）　the　attributes；　the　meaning　of　which　it　assumes。
  Yet　some　sciences　may　very　well　pass　over　some　of　these　elements；　e。g。
  we　might　not　expressly　posit　the　existence　of　the　genus　if　its
  existence　were　obvious　（for　instance；　the　existence　of　hot　and　cold　is
  more　evident　than　that　of　number）；　or　we　might　omit　to　assume
  expressly　the　meaning　of　the　attributes　if　it　were　well　understood。　In
  the　way　the　meaning　of　axioms；　such　as　'Take　equals　from　equals　and
  equals　remain'；　is　well　known　and　so　not　expressly　assumed。
  Nevertheless　in　the　nature　of　the　case　the　essential　elements　of
  demonstration　are　three：　the　subject；　the　attributes；　and　the　basic
  premisses。
  That　which　expresses　necessary　self…grounded　fact；　and　which　we　must
  necessarily　believe；　is　distinct　both　from　the　hypotheses　of　a　science
  and　from　illegitimate　postulate…I　say　'must　believe'；　because　all
  syllogism；　and　therefore　a　fortiori　demonstration；　is　addressed　not　to