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# Title: Leibniz-Handschriften zur Mathematik LH 35, 15, 6
# Folio 33r, page id 00068208:0065
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ad aream infinitam Aesmiptoro AC cuwa 3k versus Asympton AC infini
a -X
ad aream infinitam ok AG3D oK. curva infinili producta (a 3K
versus Asyemptoton AED Asumptoto infinita AEok  ordinata G3K et
Ciogneae utsu ubi notandum est
Jam curvam KK toltam, utriaque productam esse inter duas asuemplotos
siit dischiud
inveniamus, considerandum est esse GK seu EF ad CE ut  radius a seu C3H
G3H erit Trax- XX ergo EF seu Gk erit a: a -X. Frax XX, seu
pro X ponendo 24 2aX - XX sumendo  coco XX: aa seu X faciendo
X aeq pro Xs sumendo 2a: aa +XX fiet EF seu GK aequal. ) aatcoro : 200 aa- 1oco
et proeade ipsorum at darae quaeritur autem ( GF dx. est autem _
4a3c0 dco: () a +oco. Ergo EF dx aequal.  24dco : aa Lcoco
ax aequal
Ergo area infinita supra dicta seu (EFdx aequal -2a) deo: aa-coco
quae facile habetur ex quadratura Hyperbolae seu per logarithmos.
Alia ad huc ratione (eodem tamen reditura) ad eandem quadraturam
sive ad summam se antium arcui applicandorum perveniri potest
sit Triangulum circuli Characteristicum 3HW L. producatur. est versus,
et ex CE educatur normaliter EN. patet triangula NEC, et 3A WL Bsse
similia. Ergo CN CE NC: CE:: 3HL: WL. seu CE in 3HL, hoc est se cans
in elementum arcus aequal. NC iniaatus seu NC in elementum sinus; transferatur
V
arcum, seu areae, ek AG3Kok esse aequalem. Ubi potandum su rectam
per P ductam paralletam ipsi AC. fore currae A9 asyusmptoton. Est auterm
infiniste
AN (vel R∞) radio CA et tangenti AE tertia, proportionalis. Itaque facile
invenietur nempe e erit aAN aequal AE quad AE: ca Est
autem AE  aego G3H. a: GCseu a -X. Ergo et a a sit C fiet ANaequal
si G3Hse vocetur e. erit a -X aequial. raa - ec. ergo fiet AN aequal c aec: aa- ee
si ergo _ G3H vet AR sit C erit AN vel RQ AR (vel G2H sinus) sit e, C
erit  (seu A AN. recta quae ad sinus applic aego a ee: aa - ee.
Ergo ava ARA aequal, a see ac: an ec, quae pendet ex quadratura Hyperbulae.
qua
arta autem AC M  A aequal rectanus. AM + area ARA. et rectanus AM aequal a 8
ergo area ACAA seu summa cantium inarcum aequal ae Acpede aa ee
ut distrnctius jam vstend distincre ad quadraturum Hyperbolae voucamus.
erteisem A aequalis radio) ita pergendum es rec: ea ee poqu.
a = X
ad sinum latitudinis G3H, jam AG vocetur X. erit CE, aa: a -X et
EF seu hK ad PE ut G3H ad BH BH., jam AG vocetur: X, erit CE aequ aa
et G3H aequal. Srax =X Ergo Gk aequal a rax-X
sugitta AG comprehensam
ec
 seaaee
et CNve MV aequal a: an = ee et area ACMVA aequal.  a sd. aa ee
Est autem 2a: aa- ee aequal aa: g- e + aa: a tC. Ergo area AMVA aequat dimid. Ergo
arca neira aequal. demid aasde. ave  dimid aas dei care.