arguments　formally　illogical　do　sometimes　occur　through　taking　as
  middles　mere　attributes　of　the　major　and　minor　terms。　An　instance　of
  this　is　Caeneus'　proof　that　fire　increases　in　geometrical
  proportion：　'Fire'；　he　argues；　'increases　rapidly；　and　so　does
  geometrical　proportion'。　There　is　no　syllogism　so；　but　there　is　a
  syllogism　if　the　most　rapidly　increasing　proportion　is　geometrical　and
  the　most　rapidly　increasing　proportion　is　attributable　to　fire　in
  its　motion。　Sometimes；　no　doubt；　it　is　impossible　to　reason　from
  premisses　predicating　mere　attributes：　but　sometimes　it　is　possible；
  though　the　possibility　is　overlooked。　If　false　premisses　could　never
  give　true　conclusions　'resolution'　would　be　easy；　for　premisses　and
  conclusion　would　in　that　case　inevitably　reciprocate。　I　might　then
  argue　thus：　let　A　be　an　existing　fact；　let　the　existence　of　A　imply
  such　and　such　facts　actually　known　to　me　to　exist；　which　we　may　call
  B。　I　can　now；　since　they　reciprocate；　infer　A　from　B。
  Reciprocation　of　premisses　and　conclusion　is　more　frequent　in
  mathematics；　because　mathematics　takes　definitions；　but　never　an
  accident；　for　its　premisses…a　second　characteristic　distinguishing
  mathematical　reasoning　from　dialectical　disputations。
  A　science　expands　not　by　the　interposition　of　fresh　middle　terms；
  but　by　the　apposition　of　fresh　extreme　terms。　E。g。　A　is　predicated
  of　B；　B　of　C；　C　of　D；　and　so　indefinitely。　Or　the　expansion　may　be
  lateral：　e。g。　one　major　A；　may　be　proved　of　two　minors；　C　and　E。
  Thus　let　A　represent　number…a　number　or　number　taken
  indeterminately；　B　determinate　odd　number；　C　any　particular　odd
  number。　We　can　then　predicate　A　of　C。　Next　let　D　represent　determinate
  even　number；　and　E　even　number。　Then　A　is　predicable　of　E。
  13
  Knowledge　of　the　fact　differs　from　knowledge　of　the　reasoned　fact。
  To　begin　with；　they　differ　within　the　same　science　and　in　two　ways：
  （1）　when　the　premisses　of　the　syllogism　are　not　immediate　（for　then
  the　proximate　cause　is　not　contained　in　them…a　necessary　condition
  of　knowledge　of　the　reasoned　fact）：　（2）　when　the　premisses　are
  immediate；　but　instead　of　the　cause　the　better　known　of　the　two
  reciprocals　is　taken　as　the　middle；　for　of　two　reciprocally　predicable
  terms　the　one　which　is　not　the　cause　may　quite　easily　be　the　better
  known　and　so　become　the　middle　term　of　the　demonstration。　Thus　（2）　（a）
  you　might　prove　as　follows　that　the　planets　are　near　because　they　do
  not　twinkle：　let　C　be　the　planets；　B　not　twinkling；　A　proximity。
  Then　B　is　predicable　of　C；　for　the　planets　do　not　twinkle。　But　A　is
  also　predicable　of　B；　since　that　which　does　not　twinkle　is　nearwe
  must　take　this　truth　as　having　been　reached　by　induction　or
  sense…perception。　Therefore　A　is　a　necessary　predicate　of　C；　so　that
  we　have　demonstrated　that　the　planets　are　near。　This　syllogism；
  then；　proves　not　the　reasoned　fact　but　only　the　fact；　since　they　are
  not　near　because　they　do　not　twinkle；　but；　because　they　are　near；　do
  not　twinkle。　The　major　and　middle　of　the　proof；　however；　may　be
  reversed；　and　then　the　demonstration　will　be　of　the　reasoned　fact。
  Thus：　let　C　be　the　planets；　B　proximity；　A　not　twinkling。　Then　B　is　an
  attribute　of　C；　and　A…not　twinkling…of　B。　Consequently　A　is　predicable
  of　C；　and　the　syllogism　proves　the　reasoned　fact；　since　its　middle
  term　is　the　proximate　cause。　Another　example　is　the　inference　that　the
  moon　is　spherical　from　its　manner　of　waxing。　Thus：　since　that　which　so
  waxes　is　spherical；　and　since　the　moon　so　waxes；　clearly　the　moon　is
  spherical。　Put　in　this　form；　the　syllogism　turns　out　to　be　proof　of
  the　fact；　but　if　the　middle　and　major　be　reversed　it　is　proof　of　the
  reasoned　fact；　since　the　moon　is　not　spherical　because　it　waxes　in　a
  certain　manner；　but　waxes　in　such　a　manner　because　it　is　spherical。
  （Let　C　be　the　moon；　B　spherical；　and　A　waxing。）　Again　（b）；　in　cases
  where　the　cause　and　the　effect　are　not　reciprocal　and　the　effect　is
  the　better　known；　the　fact　is　demonstrated　but　not　the　reasoned
  fact。　This　also　occurs　（1）　when　the　middle　falls　outside　the　major　and
  minor；　for　here　too　the　strict　cause　is　not　given；　and　so　the
  demonstration　is　of　the　fact；　not　of　the　reasoned　fact。　For　example；
  the　question　'Why　does　not　a　wall　breathe？'　might　be　answered；
  'Because　it　is　not　an　animal'；　but　that　answer　would　not　give　the
  strict　cause；　because　if　not　being　an　animal　causes　the　absence　of
  respiration；　then　being　an　animal　should　be　the　cause　of
  respiration；　according　to　the　rule　that　if　the　negation　of　causes
  the　non…inherence　of　y；　the　affirmation　of　x　causes　the　inherence　of
  y；　e。g。　if　the　disproportion　of　the　hot　and　cold　elements　is　the　cause
  of　ill　health；　their　proportion　is　the　cause　of　health；　and
  conversely；　if　the　assertion　of　x　causes　the　inherence　of　y；　the
  negation　of　x　must　cause　y's　non…inherence。　But　in　the　case　given　this
  consequence　does　not　result；　for　not　every　animal　breathes。　A
  syllogism　with　this　kind　of　cause　takes　place　in　the　second　figure。
  Thus：　let　A　be　animal；　B　respiration；　C　wall。　Then　A　is　predicable
  of　all　B　（for　all　that　breathes　is　animal）；　but　of　no　C；　and
  consequently　B　is　predicable　of　no　C；　that　is；　the　wall　does　not
  breathe。　Such　causes　are　like　far…fetched　explanations；　which
  precisely　consist　in　making　the　cause　too　remote；　as　in　Anacharsis'
  account　of　why　the　Scythians　have　no　flute…players；　namely　because
  they　have　no　vines。
  Thus；　then；　do　the　syllogism　of　the　fact　and　the　syllogism　of　the
  reasoned　fact　differ　within　one　science　and　according　to　the
  position　of　the　middle　terms。　But　there　is　another　way　too　in　which
  the　fact　and　the　reasoned　fact　differ；　and　that　is　when　they　are
  investigated　respectively　by　different　sciences。　This　occurs　in　the
  case　of　problems　related　to　one　another　as　subordinate　and　superior；
  as　when　optical　problems　are　subordinated　to　geometry；　mechanical
  problems　to　stereometry；　harmonic　problems　to　arithmetic；　the　data
  of　observation　to　astronomy。　（Some　of　these　sciences　bear　almost　the
  same　name；　e。g。　mathematical　and　nautical　astronomy；　mathematical
  and　acoustical　harmonics。）　Here　it　is　the　business　of　the　empirical
  observers　to　know　the　fact；　of　the　mathematicians　to　know　the　reasoned
  fact；　for　the　latter　are　in　possession　of　the　demonstrations　giving
  the　causes；　and　are　often　ignorant　of　the　fact：　just　as　we　have
  often　a　clear　insight　into　a　universal；　but　through　lack　of
  observation　are　ignorant　of　some　of　its　particular　instances。　These
  connexions　have　a　perceptible　existence　though　they　are　manifestations
  of　forms。　For　the　mathematical　sciences　concern　forms：　they　do　not
  demonstrate　properties　of　a　substratum；　since；　even　though　the
  geometrical　subjects　are　predicable　as　properties　of　a　perceptible
  substratum；　it　is　not　as　thus　predicable　that　the　mathematician
  demonstrates　properties　of　them。　As　optics　is　related　to　geometry；
  so　another　science　is　related　to　optics；　namely　the　theory　of　the
  rainbow。　Here　knowledge　of　the　fact　is　within　the　province　of　the
  natural　philosopher；　knowledge　of　the　reasoned　fact　within　that　of　the
  optician；　either　qua　optician　or　qua　mathematical　optician。　Many
  sciences　not　standing　in　this　mutual　relation　enter　into　it　at　points；
  e。g。　medicine　and　geometry：　it　is　the　physician's　business　to　know
  that　circular　wounds　heal　more　slowly；　the　geometer's　to　know　the
  reason　why。
  14
  Of　all　the　figures　the　most　scientific　is　the　first。　Thus；　it　is　the
  vehicle　of　the　demonstrations　of　all　the　mathematical　sciences；　such
  as　arithmetic；　geometry；　and　optics；　and　practically　all　of　all
  sciences　that　investigate　causes：　for　the　syllogism　of　the　reasoned
  fact　is　either　exclusively　or　generally　speaking　and　in　most　cases
  in　this　figure…a　second　proof　that　this　figure　is　the　most　scientific；
  for　grasp　of　a　reasoned　conclusion　is　the　primary　condition　of
  knowledge。　Thirdly；　the　first　is　the　only　figure　which　enables　us　to
  pursue　knowledge　of　the　essence　of　a　thing。　In　the　second　figure　no
  affirmative　conclusion　is　possible；　and　knowledge　of　a　thing's　essence
  must　be　affirmative；　while　in　the　third　figure　the　conclusion　can　be
  affirmative；　but　cannot　be　universal；　and　essence　must　have　a
  universal　character：　e。g。　man　is　not　two…footed　animal　in　any
  qualified　sense；　but　universally。　Finally；　the　first　figure　has　no
  need　of　the　others；　while　it　is　by　means