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# Catalogue record: 40332 — De ratione ordinatae ad abscissam adhibenda Item de aeqvatione non rationali redacta ad rationalem qvadraturam
# Incipit: Sit y/x aeqv z. fiet y aeqv xz. et dy aeqv xdz + zdx
# Date: 1682-02-25
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=40332&katnr=39613
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 75 (from the shelfmark LH 35, 12, 2 Bl. 75): 2 pages, canvases 149–150, page ids 00068195:0149 to 00068195:0150
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=149
# Lines: 140 recognised on 2 of 2 pages, mean confidence 0.79
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
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# After the first blank line, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 75r — 00068195:0149
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000149.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 107 lines, status machine
5
item de aequatione non rationali, redacta ad rationalem quadraturi
sit F aque Z Hiol BC yrdu
x+d
XX + X dX
destruendis, flet: Vdr Xdyraqu. dZ; seu Zdx - dy aequ. XdZ seun dy aequ
sit Z aequZ. fiet: y aequ XZ. et dy aequ. XdZ + Z dx
differentialis
Data sit jam aequatio servans legem homogeneorum in indeterminatis: x
a aXX +  +byy aequ. dy cujus quaerenda aequatio summatrix:
fiet:  + bZ aqu. +n
xd
 du
XaZ+ Zax seu 2 +bZ .Z aequ X d2. seu 2
dx
oraturas de
Et habebitur. aequ d2 et solutio habetur per aati rationalem
a + bZ. Z
x
vel
vecesse est n
autem 
divendum ergo pu alioe aequationes differentiales componi possint ex aequationibus
differentialibus leges homogeneorum servantibus. Eximpli causus sit.
Ile a xx A d aeqo dx Idem est si ratio fractio homogenectum
2 vel
legem servaris, multiplicata intelligatur per formulam ex una indeterminatarum, ut x
compositam,  quam ita exprimo. X. seu si scribatur: a Xx +b99
et fiet: da X. dx aequ.
X
L
a + bZ-Z
si
0
evanescere x in  seu fieri: x
t.
dy
Hinc sequitur porro si loco X simplex X achibeatu
12 aequ. dx
Hinc sequi videtur aequationem differentialem datam ex pluribus hujusmodi
componi posse ut si scribamus: ac X  b  ax qY (d
a X+ by dx aqu. e sxagy
q +
gxths
ex +
cLXX
a XF b vX squ d 
c X + C v
dx
npcox
er. deo
ax
fiatque dua deo aequ du seu vac aqu y. fiatque v+co aequ. y. seu deadco Equ. dy.
2
habet Ergo ex compositis aequationibus 1 et 2. fiet: ax + b ∞X, Cx +m co  (d vedco sdus
Ax B) de
Cx + eD nXX + pce Y
derum quia duas assum; quaaquatione tollere possumus co ope aequationis
Sed non apparet quomodo possit tolli do. quid si ergo videamus
an unam tantum literam novam Ex his dgabus aequationibus 1 vel 2
alterutram ponamus esse simpliciter solubilem seu inveniri posse relationem
inter co et X. ope aequationis 2. per aequationem communem b. itaque ex aequ. 5 tollem
V ope aequatio 3, et co ope aequationis 6 habebiturue aequatio 7. in qua dye aequabitur
compositae forimulae ex x et y  homogeneorum legem non servanti, et taman per
quadratuas rationales solubili.
Brevius una tantum assumta litera nova. sic procedi potest: sit: d ax + by
Cx + eb
(1) dx
equ
et fiaet fingatur aequatio albitraria inter x. y. v. cujus ope tollatur, habebitur ae quatio nova
differentialis solubilis per quadratur srationales ut: sit vego  mxu
demagnus
fiet du. aequ tampanus quxardus prgnerus ndrendus, et habebimus:
p C (2 p+ quaru
bequanes
st s  b
apX + aq XX taryX + bl = bmx +bnus acu.  m merrus a qu ry. Hem dx
CpX + CgXX= Erxy 4 et t enx t eny
l+mX+nq D- PEgNerus n ex AlemxnPxrndu
Compendii causa loco aequ. 1. scribatur: O
aequ.
47
Ex
 loco aeque. scribatur  aequ. C
fiet: dee eqd vq liqo hb Ergo erit: vdQ + Q dv aequ h db ex aequis. Et in aequ. 6.
no v substituondo, valorem ex aequ. 5. fiet:  hF dq ee pr du aego h Q d B Et eraeqo 3, hb9) d, q Odra qp  C
est autem de aequ. auxerdus et et d8 aequ. m ax +n dy. per aequ. 2. et  comparatas. Ergo
fiet: h5D qdx +h8 Drdy + Odx aequ. hqDmdx + hGondy ex aequ. 8. per 9 et 10. seu ordinando
secundum ex et dy fiet: hq 8D= hm  hD98- hDm + 8O
seu 46- m8 + 20
(13
aequ  dys
ng - ro hD. n rb
ADnO-hDrS
Iu du seu explicando valo
IX
ax faciamus compendu causa BLDi P siet 46. m + B0 (is ay
9 -r F
equ a
+ BnF, ∞8 et Daequ. c9x +eht. 0e
uuibus substitutis in aequ. 12 fint: hgt cx  hgd et B - hmqc Ox - timq ch F + 8 a Px + D bh B.
exteqa
hnqcer = un eh 8 = hrt cqr = het etit
60 qu
aequ
unde tantum opus est valotes Detsum
dasuicalis perus haberunus aquationem ascerdentem ad gradum Vitum seu ad X. is quae hoiim de

## Folio 75v — 00068195:0150
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000150.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 33 lines, status machine
Bermin ita vndeo ad additionem plurium summandarum quantitatum in naum 
lrenc devenni, quod ita melius fat. sit y aego XZ. fiet du aequ Xdque + 2dx. Quaeritur
(X dy. seu quaeritur )xxdZ FX Z dx seu sixXdZe + ydx
ax ∞, bn Q5 - hnc X Q + hocX 9d - time DB + h’ ge d
tncx 8 - hrcX Q5 + Mtene q- h red
(16)
du
aequ
de
Rost aequ. prgxarus. fiet  aequ P EEPG X F INy EGGRXFGrXytyy et Fagu. BEpgt rspag pr  qe
D est aequ lamxanus fiet Saequ. Pxilux + ulnus + max + 2maxy nagus
Q3 est aequ lp + 19 X alr y amgx a mexy + ur yy
imp. +np
Hi valores substituantur in
diffecentialis
et habebitur aequalia resolubilis
per pec aaequs quadraturam ratio.
nalem ope aequ. 1dx positaa
Htyperbolae quadratum
13414y +rY
ix
12
e lapoqurrus
6
lemprny
inpys ingegarg
AmBX imq iurpys
alp nlgx atrus
PP + 2pgX + 2pry + 4X + 29r Xy + rryY
+mX + ny
X appus + +m t Emprs H Angxys tingrres Ahtr 2
mpiX H EmpuX impr. + imqx +imos . +m. 
Elsqe. Elpr. 14. cp. tr.