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# Title: Leibniz-Handschriften zur Mathematik LH 35, 7, 1
# Folio 69v, page id 00068073:0138
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g3 + bXZ + cx Z + ex H . 0. Cjus dsseontialis _Z
AgC
+h
ifferentialis5Z +, bXZ + CX  +
+15. 9X
by + cayy + eaay + hae aeq. 0.
+ dx + f ax y + i aa X
+ gXX y F KaxI
differentialis
+ lx
ubgus +2ca y+e aa F ctus)
+ 2dX
+fax
4gXx
3l X +2ka X + laa +x
+297
+fay
+ dyy
sou C dys
aeg. 0 D dx
aeq. c
seu
d equ
2Z aequ Y 2 bbyy - (7 bb-yy ergo lintum
sit Xy aego (7d. seu Xdy + ydx aequ 2dx
seu 3 aequ. d. Ergo 27 aequ
seu 3. 2 rgo 2. seu 70-Z0 aequ DX
seu 3b y + 2cayy + eaa y _
 2d Xyy +Fax y
v. 
C3lry -3Ca+ y
G.jum
-2477 7
 eaa)
un
ax
oxh
debet
ita explicari
 extrchi possit radix. sit yaequi ec0+ s
debet 26yy - (t bb-gy aequari quadrato
by s2 ( bbayyrgo (bb + 2byZ + 172 firt G en. b s2 - y aeg. 263 + 322
by si = bb + 99 8g. vv + 2bBy + (77) seu y aeq. no + Bb, : b s2 - 2b0
la fax 
faay
+dayy
Z sunt sinus auusalorum, quos
radi aura facit ad tuas
Jitorum sinuum unvenienda relutio
Ad abscissas. Itoc est data cunom
tangentim propriatatem
Et  duta hac sinuum proprietate
quarere curvm est quaerie curva
datae quadratricem
s
ABvel CN sit a. AI
E, y. EG cy
GL +X MZ Z:a:: dy: Ax
Patet spatium ADF aequari divii si pe AB
aequari ipsi DE. seu ipsus DE esse ipsis
spetiis proportinates peu ipsus DF font.
ut differentias ipsius DE
De Methodes invenieudi quadraturasgconates aut earum impossibilitates
praeclaro quaedam Theoremata nemlia al ne aliisque inventa sunt, quorum ope data quavis
Catangen cnve ec tangnis CT, perpendicular BC, oidianti
B vel Ad, abscissa AB  vel DC, rectu constans AE, ex per E ducatur EF pam recto ad
AC angulo tam ordinatis paraleta quiam tangens gese c7 secet in G erot F sumta in
AD ipsa AH aequ. EG erit TH sinue anguli quem curva cc (vec erus tungens TC) facit
ad noscissam seu coordinatam DC  radio vel simi toto existente AE sem GH. V producta
ordinuta CB, usque ad L, ita ut. Bt sit aequalis ipsi siani cutelae, TH, idque ubiu faciendo et per
puncta ec L. ducendo curocm hovam L. Dico figuram quam possis dicere Lineam inclina
tionum prioris curvae, et quidem daca curva CC, semper quadiari potest sti ABL, aequa
tur enia rectangs ulo sub AE (constate ) et sub adinata BC. et proinde sd inatat
c BC cuvae AC, sunt proprationales spatiis AB A Curvae AL pius essum
semper inveniri potest curva AL. qua btusine t quodrabilia
posissemus BL aequalem ipsi Bp. sepe tunc etiam semper A figira ABL A
pro inde data curva AC
ee bys s2 +bgy =ef f... in un ego co
siu sembamus dy aeg. Z fet a 20 + aD 7. 0
+ 297X X
L
Abscissis  ordinatas; radio existente a
A eiusque ordinata a si s
quadrato seu et bys + bb -gy + frbus  bbyu aequ. c
 seu y aeq. ( 12 -2627: 1+Z.
+ eaa
+gr
3LXX_+ 2k aaxt 9 1a3
Co
cucos inveniei potest. figmro ouvilirea inveniri potessalia quadrabilis. Exempli causa si sit.
in figrua
Ad.
ergo bys+ bb-gy. et by bb +yy aequanda
aeque andarationali quodd y. si fiet:
ff by s2 -bb +yy
fiet
3byyZ + 2ca y
2dx
erit quadrabilis. Hine figuram ABCA figurae ABL A aoeo quad rahicem, si idta descripta
haec potest quadrari. Et proinde mnifestum est multos ciusdem figurae quad ratices dari posse Et jacile
quidem est data curva t quadrotice invenire aliam ejus ope quadrabilem, quia data facile
ex conum methodo tongontium jam inventa invevia potest at H vel BL. sed non contra ex oata