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# Title: [Problema Austriacum Plus Ultra Quadratura Circuli] Problema Avstriacvm Plvs Vltra Qvadratvra Circvli (shelfmark Leibn. Marg. 230, Stück 1)
# Folio 633, page id 1815201908:0690
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quae
Min.
exiam
mill
ulum
RS
SE PTINA
Varias exhibet geneses; qua tum ex lineis es circulis, tum ex ellipsi
parabola, denique ex ipsa oriuntur hyperbola. Secundo hyperbolas
reducit ad sectiones, equibus orta. Perrioreliquas sectiones ex hyper-
bola educit.
P R O POSITIO CZCII
TEparallelorrammum AB C D, cui adiungatur criangulum BF C:
Drectae vero Dt, ponaneur xquidistances F C; rectangulis aurem
LOF, tanc quadraca C HI aqualia.
Dicopuncta B, H. Es ad eandem esse hyper-
bolam, cuius diamerer est BF.
Demonstratio.
DROducanuE rectE D A, P. B donec conventane in
K. quoniam aequidistane F Cr, rectae D E, vel AB, Sc
AD ips BC; iSICUITVEK B ad R C, ica eAt AB ad CF:
_ve K Bad B C, ica AB, hoc es F P ad ICi; _F VE RG
ad GB, ic2 est F C5 ad C: I. quare rectangulum RGB, ad
RCB, rectangulum cam racionem obcinec, que reperi-
cur incer rectangula FGI ad FGTT, cum ex ijidem ra-
cionibus componaneur: sed ve rectangulum FGI, ad
FOI rectangulum, ita ponicur quadracum GTTad TTC
quadracum; isitur ve es rectargulum R C B ad R C B
rectangulum, ita cA G TI ad e HI quadrarum. Igitur
HIHI sunc ad candem hyperbolam cuius diametererans.
verta eAt XB. Ouod tuic demonstrandums.
Corollarium.
Iinc collixere licee mechodum de-
scribendi heperbolas, non solum
cas, qux asympeoros habene ad rectos an-
culos consticucas, verum eriam alias om-
nes, qux vel obliquissimo selc ansulo de-
cuctanc. hyperbola enim cuius alympeori
selc invicem orchozonalicer dinidune re-
atansula RGB aequalia quadraris. ordi.
nacim applicacirum exhibene: cum vero
rectangula R CB, maiora sunc quadraris
CTT aueminora, cunc alymprocosta-
bene sibi invicem ad angulos obliquos
insistences: unde asempeocorum quond
angulos constirurio, dependet aPropor.
ciove rectangulorum R C B, 2d quadrar2
CFInon aucem aproporrione rectingu.
lorum TC F ad quadeata CFI. nam ve in
Praelenci Ccemare videre est, rectansula
TOF aequalia Cune quadracis FIOs nihi-
lominus asympeociharum hyperbolarum
occurrune sibi ad inaequales angulos,
   a tu li ta  e   t e i  e  se a  t   e t t ate  t a ut   i   e       e   s e te        t  at