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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folio 98r, page id 00068195:0195
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intersectiones curvarum explicatae
per tangentes.
si duae curvae A BC, dBE sese secent in B
et ad punctum B ducantur tangentes EBG ad curvam
AC, et  FBH ad curvam DE necesse est ut
EG tangens curvae seaantis;  AC a etiam curvam
secet in B, et cum antea extra eam fuisset B
parte EB nunc intra eamaBco
parte pe neversa. Nam cum tangens currae AB
 cuirae AC portio AB cadat extra curvam DE; et
E tangens ipsius AC nempe EB cadat extra  ipsam ACVE
_ multo magis cadet optia VB. B cadet extra DE.
At BG cadet intra DE nam si cadit extra tunc
tota EG cadet extra DE x ergo EG curvam
E tanget. At At cujus curvae nempe AC
Tangens EBG in padit entra unam ibidem et
in B et curvam aliam DBE taugit ea curva ABC
et ipsam DBC in B tangit. quod est contra
Hypothesin posuimus enim secare.
Restat hoc probandum, quod quarum curvaru
communis est tangens eae se tangt, quod
quidem ex natura directionis manifestum est
Hinc sequitur notabile paradoxun
p
sit linea ABC cujus Tangens EBG nulla potest
linem intra BC et BG linea EB ita duci
ut producta cadat ultra B in D cadat intra
AC, sed necessario cadet extra. Alioqui
secabit curvam, nec tamen Csecabit  erus
tangentem Hinc facile determinatur an
aX  B y ayp, b a y vy unsem d2
curva una intra alteram cadat  et quousue
Ex, his etiam discimus discricme  inter unam curvam, et
agqregatum exduabus; nempe EBD in tali casu non potest esse una
curva, sed aggregatum ex duabus. Excipe nisi curva sit quae in uno
aliquo puncto plares habet tangentes