perfectly　fulfils　that　end　is　mathematics。　The　reason　of　that　is　that　quantitative　difference　is　only　the
  difference　which　is　quite　external。　Thus；　in　geometry；　a　triangle　and　a　quadrangle；　figures
  qualitatively　different；　have　this　qualitative　difference　discounted　by　abstraction；　and　are　equalised
  to　one　another　in　magnitude。　It　follows　from　what　has　been　said　formerly　about　mere　Identity　of
  understanding　that；　as　has　also　been　pointed　out　（s。　99）；　neither　philosophy　nor　the　empirical
  sciences　need　envy　this　superiority　of　Mathematics。
  The　story　is　told　that　when　Leibnitz　propounded　the　maxim　of　Variety；　the　cavaliers　and　ladies　of
  the　court；　as　they　walked　round　the　garden；　made　efforts　to　discover　two　leaves　indistinguishable
  from　each　other；　in　order　to　confute　the　law　stated　by　the　philosopher。　Their　device　was
  unquestionably　a　convenient　method　of　dealing　with　metaphysics　…　one　which　has　not　ceased　to　be
  fashionable。　All　the　same；　as　regards　the　principle　of　Leibnitz；　difference　must　be　understood　to
  mean　not　an　external　and　indifferent　diversity　merely；　but　difference　essential。　Hence　the　very
  nature　of　things　implies　that　they　must　be　different。
  Likeness　and　Unlikeness
  §　118
  Likeness　is　an　identity　only　of　those　things　which　are　not　the　same；　not　identical
  with　each　other：　and　Unlikeness　is　a　relation　of　things　alike。　The　two　therefore　do
  not　fall　on　different　aspects　or　points　of　view　in　the　thing；　without　any　mutual
  affinity；　but　one　throws　light　into　the　other。　Variety　thus　comes　to　be　reflexive
  difference　or　difference　（distinction）　implicit　and　essential；　determinate　or
  specific　difference。
  Difference　and　identity　in　natural　science
  §　118n
  While　things　merely　various　show　themselves　unaffected　by　each　other；　likeness　and　unlikeness　on
  the　contrary　are　a　pair　of　characteristics　which　are　in　completely　reciprocal　relation。　This　advance
  from　simple　variety　to　opposition　appears　in　our　common　acts　of　thought　when　we　allow　that
  comparison　has　a　meaning　only　upon　the　hypothesis　of　an　existing　difference；　and　that　on　the　other
  hand　we　can　distinguish　only　on　the　hypothesis　of　existing　similarity。　Hence；　if　the　problem　be　the
  discovery　of　a　difference；　we　attribute　no　great　cleverness　to　the　man　who　only　distinguishes　those
  objects；　of　which　the　difference　is　palpable；　e。g。　a　pen　and　a　camel：　and　similarly　it　implies　no　very
  advanced　faculty　of　comparison　when　the　objects　compared；　e。g。　a　beech　and　an　oak；　a　temple
  and　a　church；　are　near　akin。　In　the　case　of　difference；　in　short；　we　like　to　see　identity；　and　in　the
  case　of　identity；　we　like　to　see　difference。　Within　the　range　of　empirical　sciences；　however；　the　one
  of　these　two　categories　is　often　allowed　to　put　the　other　out　of　sight　and　mind。
  Thus　the　scientific　problem　at　one　time　is　to　reduce　existing　differences　to　identity；　on　another
  occasion；　with　equal　one…sidedness；　to　discover　new　differences。　We　see　this　especially　in
  physical　science。　There　the　problem　consists；　in　the　first　place；　in　the　continual　search　for　new
  'elements'；　new　forces；　new　genera　and　species。　Or；　in　another　direction；　it　seeks　to　show　that　all
  bodies　hitherto　thought　to　be　simple　are　compound：　and　modern　physicists　and　chemists　smile　at
  the　ancients；　who　were　satisfied　with　four　elements；　and　these　not　simple。　Secondly；　and　on　the
  other　hand；　mere　identity　is　made　the　chief　question。　Thus　electricity　and　chemical　affinity　are
  regarded　as　the　same；　and　even　the　organic　processes　of　digestion　and　assimilation　are　looked
  upon　as　a　mere　chemical　operation。　Modern　philosophy　has　often　been　nicknamed　the　Philosophy
  of　Identity。　But；　as　was　already　remarked　（s。　103；　note）　it　is　precisely　philosophy；　and　in　particular
  speculative　logic；　which　lays　bare　the　nothingness　of　the　abstract；　undifferentiated　identity；　known
  to　understanding：　though　it　also　undoubtedly　urges　its　disciples　not　to　rest　at　mere　diversity；　but　to
  ascertain　the　inner　of　all　existence。
  §　119
  Difference　implicit　is　essential　difference；　the　Positive　and　the　negative：　and　that
  is　this　way。　The　Positive　is　the　identical　self…relation　in　such　a　way　as　not　to　be
  the　Negative；　and　the　Negative　is　the　different　by　itself　so　as　not　to　be　the
  Positive。　Thus　either　has　an　existence　of　its　own　in　proportion　as　it　is　not　the
  other。　The　one　is　made　visible　in　the　other；　and　is　only　in　so　far　as　that　other　is。
  Essential　difference　is　therefore　Opposition；　according　to　which　the　different　is
  not　confronted　by　any　other　but　by　its　other。　That　is；　either　of　these　two　（Positive
  and　Negative）　is　stamped　with　a　characteristic　of　its　own　only　in　its　relation　to　the
  other：　the　one　is　only　reflected　into　itself　as　it　is　reflected　into　the　other。　And　so
  with　the　other。　Either　in　this　way　is　the　other's　own　other。
  Difference　implicit　or　essential　gives　the　maxim；　Everything　is　essentially　distinct；
  or；　as　it　has　also　been　expressed；　Of　two　opposite　predicates　the　one　only　can　be
  assigned　to　anything；　and　there　is　no　third　possible。　This　maxim　of　Contrast　or
  Opposition　most　expressly　controverts　the　maxim　of　identity：　the　one　says　a　thing
  should　be　only　self…relation；　the　other　says　it　must　be　an　opposite；　a　relation　to　its
  other。　The　native　unintelligence　of　abstraction　betrays　itself　by　setting　in
  juxtaposition　two　contrary　maxims；　like　these；　as　laws；　without　even　so　much　as
  comparing　them。　The　Maxim　of　Excluded　Middle　is　the　maxim　of　the　definite
  understanding；　which　would　fain　avoid　contradiction；　but　in　so　doing　falls　into　it。
  A　must　be　either　＋A　or　…A；　it　says。　It　virtually　declares　in　these　words　a　third　A
  which　is　neither　＋　nor　…；　and　which　at　the　same　time　is　yet　invested　with　＋　and　…
  characters。　。　。　。
  。　。　。　Identity　and　Opposition　are　themselves　opposed；　and　the　maxim　of
  Opposition　was　taken　even　for　that　of　Identity；　in　the　shape　of　the　principle　of
  Contradiction。　A　notion；　which　possesses　neither　or　both　of　two　mutually
  contradictory　marks；　e。g。　a　quadrangular　circle；　is　held　to　be　logically　false。　Now
  though　a　multiangular　circle　and　a　rectilinear　arc　no　less　contradict　this　maxim；
  geometers　never　hesitate　to　treat　the　circle　as　a　polygon　with　rectilineal　sides。　But
  anything　like　a　circle　（that　is　to　say　its　mere　character　or　nominal　definition）　is　still
  no　notion。　In　the　notion　of　a　circle；　centre　and　circumference　are　equally
  essential；　both　marks　belong　to　it；　and　yet　centre　and　circumference　are　opposite
  and　contradictory　to　each　other。
  The　conception　of　Polarity；　which　is　so　dominant　in　Physics；　contains　by
  implication　the　more　correct　definition　of　Opposition。　But　physics　for　its　theory　of
  the　laws　of　thought　adheres　to　the　ordinary　logic；　it　might　therefore　well　be
  horrified　in　case　it　should　ever　work　out　the　conception　of　Polarity；　and　get　at　the
  thoughts　which　are　implied　in　it。
  §　119n
  （1）　With　the　positive　we　return　to　identity；　but　in　its　higher　truth　as　identical　self…relation；　and　at　the
  same　time　with　the　note　that　it　is　not　the　negative。　The　negative　per　se　is　the　same　as　difference
  itself。　The　identical　as　such　is　primarily　the　yet　uncharacterised：　the　positive　on　the　other　hand　is
  what　is　self…identical；　but　with　the　mark　of　antithesis　to　an　other。　And　the　negative　is　difference　as
  such；　characterised　as　not　identity。　This　is　the　difference　of　difference　within　its　own　self。
  Positive　and　negative　are　supposed　to　express　an　absolute　difference。　The　two　however　are　at
  bottom　the　same：　the　name　of　either　might　be　transferred　to　the　other。　Thus；　for　example；　debts
  and　assets　are　not　two　particular；　self…subsisting　species　of　property。　What　is　negative　to　the
  debtor　is　positive　to　the　creditor。　A　way　to　the　east　is　also　a　way　to　the　west。　Positive　and
  negative　are　therefore　intrinsically　conditioned　by　one　another；　and　are　only　in　relation　to　each
  other。　The　north　pole　of　the　magnet　cannot　be　without　the　south　pole；　and　vice　versa。　If　we　cut　a
  magnet　in　two；　we　have　not　a　north　pole　in　one　piece；　and　a　south　pole　in　the　other。　Similar；　in
  electricity；　the　positive　and　the　negative　are　not　two　diverse　and　independent　fluids。　In　opposition；
  the　different　is　not　confronted　by　an　other；　but　by　its　other。
  Usually　we　regard　different　things　as　unaffected　by　each　other。　Thus　we　say：　I　am　a　human　being；
  and　around　me　are　air；　water；　animals；　and　all　sorts　of　things。　Everything　is　thus　put　outside　of
  every　other。　But　the　aim　of　philosophy　is　to　banish　indifference；　and　to　ascertain　the　necessity　of
  things。　By　that　means　the　other　is　seen　to　stand　over　against　its　other。　Thus；　for　example；　inorganic
  nature　is　not　to　be　considered　merely　something　else　than　organic　nature；　but　the　necessary
  antithesis　of　it。　Both　are　in　essential　relation　to　one　another；　and　the　one　of　the　two　is；　only　in　so　far
  as　it　excludes　the　other　from　it；　and　thus　relates　itself　thereto。　Nature　in　like　manner　is　not　without
  mind；　nor　mind　without　nature。　An　important　step　has　been　taken；　when　we　cease　in　thinking　to
  use　phrases　like：　Of　course　somethin