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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 188r, page id 00068196:0371
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Jun. 168)
quadraturan grationalium tractatio, ubi subfine hujus pagina reductces
curaae Coniae
Habeos Methodum Constantem analys ticam pro curvis rationalit
Sed nondum aliquid satis constans habeo pro irrationalibus. Vario tamen
artes atiliter adhibebuntur. Ex. gr. sit
scribemus:
Vb si ordmata curvae
a+ b, vel +  quae singulae separatim forte aliquando
25 a +b
Vatb Natb
utiliter exrminabuntur. Simile quid et in altiocibus
1+ X aego  X + X
SX aequ ~
Hinc si quaeratur r+X
Sitx sX equ. X + X
X+XX aequ X + X. itaque non podit idem.
Sbx
quantur an  a +X, exprimi possit per X + l, et quales nam
futurae Z et V. erit a +X aequ. 12) X I Quod si Z numerus
rationalis quod suppono habebitur aequatio finita. Ja m
quaeritur an sit dabilis valor ipsius c0. semper idem quacunque
posita X. ita arbitior, potest enim exhiberi per seriem infinitam
quod si ita est poterimus etiam facere: Va + X aequ X, et
a + X aequ xe aZ quae aequatio debet essc identica, adeoque talis
102+1
debet esse ae ut sit identica. Quo posito erit. sta + X aego X
sed hoc jam video cur non sucedat, quia quando ipsa X ut c fit coz + 1
ingreditur in exponentem regulae tangentiam locum non habent
2
ni pus
semper habetur.
adx cyctolaeta nir. tat t
AMIXY+CxK
scribatur  X aqu’yy fiet y Sbt cgy
b+ yd  seu cb +9  bb - bus +gy, et quidem: bb -bys +yy
potest aquari quadrato ut bb - bys (y7) aequ y7 + 2Z + 4ZE fiet:
y. aequ : bb - 22. et sp fiet... Sb + bb - 22. seu S2bz. + bb - 23
p
VB+ -x3
2Z+ b
2. + b
Z + b
ag. eb + rebx + exF fiei b - bx + X Xuqu e b+X. et fiat
e aequ 1b + ZH. ergo habenus: b2 bx + XX C aequ b +  EZ+  b+1Z5b +ZZ.
ergo b2 - bx +XX aequ. bb -
X aqu e B) b - bx + XX fiet : b + X squ C. b - bX+ XX
l’+ ilX F XX
ut alibi astensu
ponc X equ. L+ my
quaesi reducatur d Turva ationale, etiam curva conica reducitur in rationale
vslec
 y + by + c fiats: 3y aqu Y te fei  x + 2CX + eC. et haberbimus:
+ b XX + be
faciamus ee + be + c aequ d qu tique semper fieri potest essi tat irrationiles
 imagiasrcas, Eppe in determinata non caremus. ergo fier:
X Lx +2C . jam y que aeqo X  sed non habemus y nisiper tx re
faciamus:
+ b e
substituendo aequemus: radicem ab y2 byyte
 y aequ KX + C
fiet:
cum X + V AN + vx + f
XK+ ax e + xx ec + ae + e + aequ. X +2x + XF + 2VtX +
A
+ bkXX +2bxte + ex
+
t
aequanda cu LelxXth
X +29X + 2hXX
gaeque
et S(O+ bK2  2hk +4Kto2 Erguhabetur h
Let
C
+ bnxe
See. b- g9 ae, ee t
sis seduna argitrarir infeliciter deet nis
el  set  poe s in catu  e
99 n = 2gt a + bi C. in st
Vb
+ X