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# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 30
# Folio 99r, page id 00068124:0197
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Nova h
quae tractenus
finstra
In Ellipf Hyperbola si abscissae AB, AC, AD, Suit progressionis Geometricae
etum applicatae DBI BD, CE, DF ejusdem erum progressionis; ergo
et differentiae earum. orro veterum applicatae et abscissae sunt in atione
 reciprota, ergo et differentiae earum curva repraesentatur per
in quibus progsi
Va 7
A
s(X a Hnesi d si sint pot X inter
ns geometricae,
V161
2X
a
ec
n
x erunt
cuurva latera progressionis hujus
64X -42
4 X
erc.
Uende jam transitus habetur
ad hujusmodi progressiones
quae a Geometrias pendent
ambigua in exponcule, ut prugressio hoc loco:
quaesiveram pujus summa invenietur, per curvam hyperbolicam C Et Cumque imohaecens
cantili
Adhiben dellur superficies higurbolica ex data superbolae quadratura unc ducendo in
dulica potest
haec methodus et hnjus serici
etiam ad
obliquas
applicatas.
Oacrt
in A
ino rinutile
la
la
Y(5) 4
poterit simpliciter:
4V
a
28 videtur
=a summa ex data Hyperbolae quad ratura haberi.
Ecce transiturum admirabilem ad calculos in quos exponentes ingrediuntur. loco
demonstrati
Interim Nota in hyperbola rectantulum
1.
scribi etiam
EHD aequari racrangulo
s
FGE posita progressione colinna Geometrica rectarum AB, AC, AD. Hinc jam videtur elegantissime
demonstrari posse aequalitas spatiorum DBZFD et QPAFB. posilis AB, AG, AD, sive positis
BD, ZF, PR progrestionis continuae. Nam i Nam B9 pariter et Q subdi vidatur in infinitum aliais
medias proportionates continne interporendo, ut CE, s
NLR etc. aequales. Trumspatia Diffentitea
V.
erunt ete DCH, DHE, EG. tC itei FHN,
generaliter si BC sint infinte parcae C9. et
sive rectacnta
similes sint infinite parvae, spatiala ipsa infinite carvae lalitudinis, HBCE
qualia fient enim ex ductu
in CX. ponendo aX BHp
F
hec
erunt inters
et BC+ BX si ( sit quantisas
constais infinite parva. Totidem Xgemnando Ergo inter positionem, marifestum est lotidem satiola intem
poni inter B et Q, quot inter 9 et D. si AB, AG, Alotimie proportionales. Habemusergo demonstratione
anam ac facilem
nutcherrimi theorematis Gregori; a S. vincentio quaeque ad longe magora aditum
subtiliora aditum facit. si ordinatae sunt xvl. sive ya emul y in reciprocat duplicuta. Hinc si X
progassioin lometricae erunt et y progressiocis Geome tricae.inc si AB, AC, 14g. progossio ni  Geometricae ergut spatiolam
. Malsesbetur  spalia ordinaria DBEFD. FQPRF fore etiam progressionis Geometricae, adeogo dscinis relipioce
ser