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# Catalogue record: 60445 — [Aufz. zu Differentialgleichungen]
# Incipit: Problema differentiale primi gradus, id est Tangentium inversum, non d
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60445&katnr=57373
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 43–48 (from the shelfmark LH 35, 13, 1 Bl. 43-48): 12 pages, canvases 85–96, page ids 00068196:0085 to 00068196:0096
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=85
# Lines: 443 recognised on 8 of 12 pages, mean confidence 0.82
# 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 43r — 00068196:0085
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# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000085.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 66 lines, status machine
problema differentiale primi gradus, id est ine Tangentium
inversum, non aeterminat curvam, nisi uno puncto dato.  sunt quidem
data
aliquae ex convitione tangentium deficreates constructiones, quae
unam curvam definiunt a ut cum curva habetur ex concursi aliurum ordinatim
vatarum. Quia tamen a curvae habetur aequatio differentialis, quae
sua nntura ex dato jeno puncto regulariter, construi potest, quantamvis appropin
quando; ideo videndum an non in hec cam generalitas problematis plures alias
contineat curvas. Idque cognoscemus, si ipsas constructiones appropiaquatricus ex hibeamus
Porro ut sublema differentiate primi gradus, seu Tangentium inversum, non determinat
regulariter
curvam nisi uno puncto dato;, ita problema differentiale secundi gradus, seu
eclorum Circulorum osculantium, inversum, non de terminat curvam nisi duobus
punctis sive indefinite vicinis iatis seu non nisi tangente ad punctum illud datum
ata directione curvae, sin tangente, ad punctum illud datum data.
ce probleme bissee vntale tertire geedas Etsi interdum fieris   concuta
Curbarum osculantim  et  ordinatim datarum, concursu sed tangentiali
seu ita, ut se tangant contiane, ceurva quaesita formetur. Quo casu determi
Et problema diffcentiale tertii
nata iterum curva habetur.  u
gradus seu ati curvarum alteragradui osculantium. Inversum, sive osculi secundi
gradus inversum non determinat curvam nisi duobus punctis tribus
indefinite vicinis datis seu non nisi curvatura curo ae teu circuo osculante ad illud
punctum vato. Etsi interdum fieri, possit ut curvarum ordinatim datarum
concursu, sed osculatorio, seu ita ut se osculentur continue curva quaesita forme–
tur quo casu dete varinata iterum curva habetur Et Generaliber problema
differentiale cujuscunque propositi gradus seu curvurum quo tangentiam (aut oscuilam
fium propositi gradus nve nam osculum est factus secundi gradus inversum non
delesminat curvam quaesitam, nisi punctis e tot uno minus quam
indefinite vicinis, tot datis, quot sunt unitates in gradu contactus; nis
Et si interdum fieri possit ut curva quaesita inte determiuetur conculsu
curvarum ordinatis se datarum ordinatim vatarum sese continue sibi
continue occurrentium eo gradu concursus, quibus leujus exponens
contactus.
est in graduarum curvarum i
it unitate minor gradu en exsionente u
Nempe tactus simplex est concursus secundi gradus, seu in duobus punctis. i vi
isculum simplex est factus secundigradus et concursustertii gradus seu in tribus punctis indefinite vicins a
Qcitum pcndigredus est Factus terti gradus et concursus quarti gubus seu in quatuor punctis indefrnite vicims
x
quaevis
si problemam talia solvi possent: invenire curvam, in qua duo puncta, certo interr allo distantia
etsi
porre
vel aliter aex ad se invicem semrelata certa canditione alia adjecta relatimem  absolutam a se invicem
cinversum solvi. Nam  cum duo illa puncta sunt indefnite vicira
prodit hoc problema vngr. si  luron ec) sic ut sumto puncto C, in ipsa, et centro C, ratio
constante descripto circulo secante rectam Axem AT in T, trajectaque dustaque TC, continuata in
acciperens. posset problema tanusenfia
in C(C) intervallo C(C) constante caderet (C) in ipsam curbam propositam; patet, eo
casu quo C ) intervallum, C (C) est infraite parvum, TC esse tangentem, et problema
portio tangentis inter curvam e apem intercepta sit data
datum, sed B(B) ubi quidem curva aliae prodirent quam ante
tunc cum ntervalla essent assignabilia sed cum inefinite parva, desimuntur in eandem qua
fore invenire curvam, silrvat
posset etiam fieri, ut, non 5(C) Cesset
ante curvam, ex tangentium conditione datam:, possupt. per interpo latiopus
sutt habent  aliquid prestantias p amplias praebent quam problerata differertialia. Nam in problematibus
problema uebi duobus punctis is curvae datis interpolatione
cujuae i interpolatorie dsitae, haberi possunt vera puncta, sed in differentialibus non item. etsi approseinquam
haberi semper puncta possint, utcunque vicina veris. d ineet  illuo misimri,s in si ex uno puncto
illad quoque notandum si ex uno puncto, aliud ex vel ex duobus ai se, aut tribus alim sequens definiti possit
non posse curvae ne punta continuitatem satis im exhit impteri, seu inveniri puncta quantumbis ad se ravicem
curva sit in se rediens. ed si veruri si non sequens punctum, sed cmedium ex duobus aut tribus etc. reperit
sic continuitatis, implementum, et tangentum etiam atque osculantiam
ceventia iis potsenser pobt, haberaetur er patura et et u esset cust cencalia posse deduris a linter potit

## Folio 43v — 00068196:0086
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 3 lines, status machine
TFRIED WILHELM LEIBNIZ BIBLIOT
e
NIEDERSALHSISCHE LANDESBIBLIOTHEK

## Folio 44r — 00068196:0087
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 53 lines, status machine
In numeris s, B = bxade pe V, B - 1. Ergo b = 5b (b. C- 1) Erx sb x ad (b= (bdi-issen ad b, ut b -1 a 
Ergo SBx da = BX dx : (b lX -1)
in numeris d, X. B = X lb) + b dx. Ergo Lib dx = fe X db) = X, b - s b dx
ad saqulbus est ad sx  x
 aim s (tdex pragressionis Geomessicae, ad eoque summabilis; tty b . bs a e . (. L 3
Ergo sit
ut pax -1 ad dx Ergo SX bxde sua B) dx
Rursus d Xx, b = (2 xdx + dx dX) bF + XX d(b). Ergo (XX. d) (b) = XX. b - 2 sX b dx = sb du ax = (XX. b 
Et ita habentur ha summae. A)  b - V s  (bx- Ergo b Ergo dei
rgo ent fb SXX. b ex = XX A - 2B H . dx L2
A-52
=(dx: B nosito B- (dX: s ar-, 2
Carsnu
In numeris d, b = Bx+x - bx = b =, b -1. (Ergo b. = sb= bb - 1). Et posita
de constante, erit b (dx: , bar - 1) = SbX dx
d, X. b = X d,B) + b dx seu X d (bY) = dCX. bF) = b dx = X. b, bd -1. quam multipti
aequationem Multiplicando per dx:, bX -1. fet et productum utrinque summando fiet
X. bE dx = X. B dx (, b.  - 1) = S b dxdx(i, b  - d) ac denique per canonem praecedentem
pro sbx ponendo valorem, fiet (x. BB a (Xde X b tx. ()
fet pro sb x ponendo valorem fict, l. X. b dx = X. b (dx:, b = - 1) - B (dx:, b  - 1). (+ be drdn :, bt - 1
+b dxdx.
+b. dedx
d, XX. B = XX llb7  2b. X lx = XX b. (b.= - 1) + 2b: pdxC Empo Xx b = RX. b = (37 - 1)) g 2. X. b (1r: 9b = - ) bs l. 1 2)
di niniurdele
-b dx (ax:, fdu
dex d, X. b =- X db) + 3b. Xx x (= X. b ( (b7 - 1) 43 b  XXdx Ly. X b = (X W(t Y 1) 12 + 3HX. b du P X  sudx
+ b dx. Ergo (x. b dx = X. b (dx :3 b  - 1) = 3 XX. b dx: , b x - 1) 3 + 3. 2. X. b (dx:, b7 - 1)  - 3. 2 b (dx:, bX- 1) 2 +
3 (X dx(dx: bdX - 1)3 = 3 X b= (dx:, b  - 1)  + 3 bX dx (dX: , bX.1)3 = BX Axdx (dX:, bx- 1): eu ordinando
et per compendium dx dx:, bax - 1 appellando B, fiet
(X b dx = X( X3 bs B i 3XX b  + 3. 2 X. BX
(LXdx = b= B
SX. bE x = x. bxB - b BB
IXX. Bdx = XX. BB - 2X. bX BB + 2 X 93
_ axDB
X. b dx = X bF B - 3Xx. H 3B + 3 2 X. b. B3 - 3.2. 76 X. 94
+3  b x3
p. Lxdu: 
dx bY = Xt bB +A X ax6X aplxtaX b  rb 
Ergo sx Vx - X 68. 4X  28B, 43 NX V 84 3. 2x b 197 + 3. 21 fr95
F+. 3. Xb x B3 - 7E1B2
-31X. b  BD + 32b 4- 133 b B.
= b dxdxBD
(x b k = X bB - 4 X b  + 4 3 XrbB3 - 4. 3 LXb Ct + 43 2b95
=6Y X(. a BB H4 X B deb - a (3. b dx 4
+4 X. S ax xb F est B apax B
bx dx BB
aid alaxo.
26XX6 d BB +6. 2XV axB - 6. 2. GX A. B1.
+6 bXdxdx B3
+x6 deBB 24 (Ax B
1b dx D
ita continuando poterit inveniri
regula continuationis

## Folio 44v — 00068196:0088
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 9 lines, status machine
X tlx xrxax
d(=   + deb) - (Xx sed hic haereri necesse est, em cum duo ignola simal occurrant
Be txiy b -1
E db = B C b  = si BX - + (. Ergo et s dxax = b du - b 2e
Xt d
XX +XdX
Ergos (L b) = = B per - = B -   b x + b 
Xt
exxax

## Folio 45r — 00068196:0089
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 83 lines, status machine
Nova ratio differentiandi aliquando utilis, ex hoc fundamento, quod
a - b + b-c =  a -c. itaque interdum in differintiando medium
sia immediata.
interponi potest. sane cum in vulgari differenbiatione a me dudum intro
ducta hoc p revera contingit, perinde ac si medium fuisset assumtum
verb. gr. XL (X) (Y) - xy  ( seu (X+dx) 9+ du). = xy. idem est quod
prodiret faciendo idx) y Xy et (X+dx) (V +dy) - (X + dx) y, et (xraxsy - Xy
sum audo in unum addendo
ita ut variabilium semper alterum varietur. Sed idem transferemus ad
verb. gralia
eos casus, ubi communis differentiatio hoc non dat
vido =X + X unac XFdX - X 
d(X) concipi potest sic:  X+dx
itaqe conjunguntur duae differentiationes jam datae et fit d X (posita X non variata) + d X spot
uti dxy est dxy posita X non variata) + dxus posita mmronvariata. Unde
fiet dX = dv logx X + v X dx. Sed hoc video etiam obtineri nostra via logarithmo
elelbrig rum nam fit V logX = exosite log X. ergo logx do + vdx: x = d. X: X. Erggo
d.X = logx. X de + v. X dx. salte interm si antem leguiento
consiterationem non habuissemus reduxissemus d, X ad db seu infinitasum
Huc applicari pilerit ad curvas alliores sumum seu plus quam Loganthicas
igur logarit hmiiusas ad unam logarithmicam primariam.
si sit A posito P non la esse in direm poteatiae ab X in seipsam ducto sed
producto si invicem ducantur X. X+ dx. Neda X+ 2dx. X+ 3dx, et C.
ut
et
videamus
Nempe
sit ) - 7 B sint duae  sit  X sensu quem dixi, ut v sit index continuationis
inter ed etque Vtdus
y= X. X+ aY X + 2AV. N 3dx ete usque ad X+ V. Videamus differentiam, alemno
Xet xrdx ea est seu quaeramus ly. primum _ N. X et X+dx habent differentiam X;
inter X. Xtde, et inter Xadx X+ dxt dv seu intem
XX+ Xde et XX + XX +Xl (+ dud rxav)
baZ
differentia est X dR. dnter X, X+d0, + Zde CM XFaxt dv XFd X+ Zdv fa) e
XX X+ XX AV F Xx 2dB( XAvdB) .t X2 + XXdx + Xxdv + XX X +cambabus divisis per a
disserentia erit Xxdx + XX Zdv,: a Et differentia inter X, X+ dv, X+2d0 X+2do, _ interg.
aa
sut t. xap. podedes pd
’ext XSaV et
differentia est Xdx. Ergo ginemliter d, X~ = X. dx seu habetur (pga - X = y
si esset X arithmeHcat est X at sit alia quaecunque fiet X (Xe dx X  Zv
Tandem aliquando accedam ad (urvas panscendentes metaloqarthinicas; quae
fiunt ex praeductis continuorum sit  X, index, et   V sit quantitas continue
bracenda  in  facta ex suias anguento, sitque y factus continuorum, et fiet
Y = V, v+ dx, v +idx, vi 3dx ex. usque ad V+ X. et Htdu = vide, v+ Y Fx, v+ dv + 3 dx ea usque ad vtav + Y
par et
inter vord et Y et V+ dv differentia est dv. inter v. et v, vadx et vad
et inter 00 + Vde + vdx + Vde differentia est 2r dicisa per a 2pt 20d0: a
inter v, e+ ax, v+2_x, et vtdv, vidvtdx, Vidot2ax, differentil sseu inter v+vdx + 2VV dX
et v +, VV dv + vvdx + Vvdv F 2WVdx differenti a divisa per ad est 3 vodo: aa  Ergo
generaliter dy = L, ∞X = X. (B: a): do su sive (, X. (V. a)  dv = v~ unde
si sit  X = v fiet  _ XY  AX =  X Si v= xxfiet (2X (xX: aas   Ax= v seu
fiet (3X (X: a) 2X dx = vexiaa si v = X ficit Se. X (X. a): Ix ( seu e sXe (X: a) dx = vex
X  , X= x = y fit X logX = logy. et X de + log X dx - dy:y seu X dx + X log X dx = dy
X21- fit XH) logx = logy se XXHL Ax + log X. X x = dys = x FI X dx + legX. p dx
orratum hic comnissum est, paulounte cum scripsi d. VZ esse X(V: a 
Equidem inter V, V+ dx, et vido, v+dv+ dx disferentia est 2vdo: a sed differentia deberbat quaeni
inter V, vtdx, et + dx vt do, v + ax +dX, vt do selx ergo inter v, t Veto hoc et vide, vedorax
quo diff. est v,  V+ do, v+ du Lx, Vtavgda seu y, vtdvti dx a tme differentiam ham
duaum agpegntur foret disfeentia quae sio: sed postenot pars non placet, nam pon ati vixa
Tentemus ex duoibus punctis invenire tertionurvae, idq vel assumendo Rety vel non
videamus an opus H sit et liceat ut e cojumere reciderem puncta priora verb. gro
ex a et b quaesivimus C, et ex a et et C quaesionnus d, et ex C et bquaesivimus e quaenitur
an endem lege opus sit ex d et e, inveniri C. Hoc vid fieret, si certa observarentur
intervallu in axe, vel aliud tale.
pro sneredes puatis redis
alibi memini ine uisum maximis
+pab d
ly2
lesos
X soyl o
+ax yb =dy
2X. B4
rlu c lagus
dy bg
cincilum
alioque
cerentiae
ut
flosinales
vdebet
eet itn geter

## Folio 45v — 00068196:0090
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# No recognised text on this page (recognized).

## Folio 46r — 00068196:0091
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 99 lines, status machine
venit, novum leu tentamentum in nculem pro resolutione
differentialium. Constentus it dus fassiosue
prirui puncti X, coincidere
es
dua Xdr gdq ydx
V. g. si esset proea aaequatio
fieret en casu du dy. + axdx = dyc Axdys posito Ad, X get 
Dinter se X et dx.
curvam exprimi, continue variando axest, et cujusque axis initialia tantum
puncta consiterando. sit cuva AB ACQ, sit axis AB Apis AH, et alius FH,
et ex puncto curvae Q duae agantur normales una i Zp in AH
altera Eto in FH. sit AH, a; AP, X; QBy. FId, Z
7t, v ex jam ex Z ectae  HF, L) se secent in G
erit AP: pG: AG = LW: WG: Gq C Ergo Hp: PG= Z_: WG
AP: HH= Z: G et TG: GH = 1oG: GZ sed AG + GWF
=G1o + eo et HG = AP= p GV somitamus aequ. 5
quin sequitur ex etr. _HW + WVF= HF, omittuim
initio HF et HA, sex et AP et FWD
10
sit Ap: (A P + pG)= ZW:
(GW + 3W) qu= HH -Hp Z GE et  eX aequ. 1
Hp. pG GH il in ex ag. 2. per
fint, 1 P: pG qU: H, W0, = SHp + pG2, et ex aeque
10 fint Hp: (H p2 F pG2) = L2:, HW + HP + p G2 2Morni
restat pG. tolcenda per aequ. (2 et 13.
lemne 
ex aequ. 12. Ht = BG (ae: HP) + H1 = VHp + pest. seu
Ap G2. (9W0: Hp) + HW  2HW. pG (ZW: HP) = H p. 7 p G.
 4 + pG: H
Ex aequ. 13 = Hp. eu tep G2 Qqu +HW. Hp + Ap +PG. HP - 2H1 is
seu HI. Zu- PGr. quo AH1S. Hp+ Ap L p G. Ap = 2 HV PG QVO H 
+ 2HW. ApV jam ope 14 et 15 tollatur PG
ex i est pGt = H p4 - Hp. H + Z H Wo. ZWo. HP. pG.:, elo- HpE
ex i5 est pGr = e p4 + HW. Hp = Hp. qu + 2 HI qW. Hp. PG, :, QW -  p
et fet 2Hp. H2 = H p e absurdum est erior in caleu  M. Ar
 p0 - F9W + HW + HP E PG - ZHW.  p + p GrA Ho
il
 Ap. HC - pG. Cto :AB. seu H C exp p (Z1G Hp = AP. Hl eS. Q
.tu
pG = VE2. HP P eW HP + H p + HW. Hp 6 3: 2H (B;, P8 - HW. qW. (Ex 3)P i p - H _ : eW - H sol
 fi p p E q- 2 p p B. pG. = ZW. HP + HI. M B + A pd ( BG. H Ph (3AW.) Ap, - 2Hlo Zud PG
Ex, fit AP. HUS - HP SHNL pGr = BG. ∞W. sen AP +P4
Eet h
y vel y
Wd F = HF, A = AH- AP CAPG YGZ = )eZ. sene
jam Gq
HP: pG  = ZW:, HW - Hp + pGi
(e
2+ Ap= H  HG2. ego ex Y  et 8. fit fHW:, F, HP. PQ H1o. 210 - 1, AP. H13: Cd. A
deu A4 sed  axissen iL: LA et fiot PE 6 t 9 et 0  l: Y = HW:, Ap. Pg- H1. quo
seu p9 = oW + Hp + H13, :, 3 AP. PQ = H H a
superest
Ex ) et H est Ap. p Q = (Y. I) MO + H S H t:
1046. l: r = s. Hp - x 1,:, ZW - Hp seu 1H. N = U ZW= H. AP = YrHp- rr HIS. Ergo
seo HP: BG
ut ope 5gg. 8, 9, I1 tollamus HP et PZ,
H p2 ILW= rr HIO,: (l- 1 = (per 8) HW + HS YN
G =(I ZWrrHWD + HWO l HWrD (= 1HI r: U- 1r Cixdevend.
 fit (UNo rNIo= HuUNu et  si Hol 
Wg. v sit Hp: pG = r: l tit X = Z - rV: NFI y - Z+lIV)B. SAM
que est aequ quae aequalio deberet esse identum est erior in calculo
l- 1
in XFIErrLr et  = X se  y Sitr - b: DV. et fit
Y
YrF)  seu v = (y - X) r: Sit rr
H(yrr =Xrr f. Vitrr) seu q = X(F XVD + yir Xrr,: sitrr
seu Pyt = HW = A = qW
paucis res huc redit
H8, Z.
yFitr  ((9-X): SIF Er) seu Z = (+ yer  y, + X: sit
(y-X)Sitrr
AP X. P2By. B
sitem nyllans vides efficiendi ut res lampatium hac modo redueatur ad
Et Z X +
v = (X y) t
Quadraturqs nam duo raeter duas infinile parvas dus, dx duae
Csidendum
coaneseat.
1B373 3βaB, elc reddueret ad 1
ubi pro arcum circuli componendo
ippsas
et Z = X SItEr +y
seu
naneu, Ende lerninalee 2
3 et t
itaque nul juvat, essi velimus
et alla curva obtinea posset, sed tunc opus A pariari, seu A(+) esse focu,
quid sia s , summando componant aliquid
trajectionem, si
scilicet si 11) 2, contipuaretur in 2825 Maut,  2 aequillam et 435 in 39 2A quoritur valo-
3)4)
2γ 3
ipsius E
Malim ergo
tentari a Iaquiri: a
copretus
Ae si feculum de ar ingentim
ex
1. 1

## Folio 46v — 00068196:0092
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## Folio 47r — 00068196:0093
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 4 lines, status machine
1F
TFRIED WILHELM LEIBNIZ B
VIEDERSACHSISCHE LAINOEde
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## Folio 47v — 00068196:0094
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## Folio 48r — 00068196:0095
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## Folio 48v — 00068196:0096
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 126 lines, status machine
Hinc
intendum posset fortisse inveniti solvi problema;  differentiale si series
differentiarum primarum secundarum tertiarum et. esset exponentialis
vel alia tractabilis
sit dy = dx N(a + b X 4). 4a seu dydy = a + b: a) X, det
fic 2 dus Rds sit dx constans, fiet 2dy ddy = 4b: a) X dx 2 aa seu ddy = (4: 2) b: a) x du .
rursus  dydhy + ddy ddy = 2b: 2. 3. (6: a) X dxt: na sed ddy =  a )bb. aa) X dx:
et fiet du d’y + 2bb: aa) X dx = 2. 3. (b: a) xx dx dydy
seu eret:
By = 2. 3. (b: a) x  (a + b: 4) X2)  - 2 1bb: aa) X, dx:, a 1b. a) , atb X
seu d’y - 2.3 (b: a) a . + (2. 3 - 2) bb: aa) X, dx; a + (b: a) X, 1(a + b: a) X2)
in dy dhy = 2. 3 b: a) a Xx + (2. 3 - 2) (bb: 4a) X, dx
3d ydy ady) + dy) dy = 2.3 (b: a) a X + (2. 3 - 2. 3) (bb: aa) X dx
et fiel
A. a)X (a abia X) du et fiet
(2y = 2. 3sb. a) a N+  (2. 3 = 2. 3) bb aa) X - 2. 3 (b: a) da X i a - 1b. a X) 
(a + (b: a) X3) (Y (a +b: 4) p)
ax
Et ita pergendo, uidendum an haberi posset lex. quae si esset retro
continuabiis, posset haberi, y. Et si autem hoc in casu nonsit talis, posset ex cogi–
Et quia in potestate est assumere
relationem inter, X, dx, et dax proarbituo, hinc fortasse faciluis tates leges
tari casos, in quo continuabilis esset
ex cogitari possent.
sit aliqua langentiam inversa, seu differentialis primi gradis aequatio
veluli adue= axs X=+ eax. sumanus X et y ut constantes d et Z, nenoe primas
jam proxima. Erquacdys) = (X + dx + y+ du) (dx) et pro du ponendo valorem, fiet
aldys) = x(dx) + qd Xax) + y A X) +  X dx (d X) + y dx (dx)
X+ c
L(dx) +
seu a(dy) = (X + y) (dx) + a + x + Z dx AX) seu (dy) =
a (dy)) = (X + dx + (dy) + y + dy +(dy)) dx (dxD
a+xts
aa
y) = X (dX)) + d x pd X D) + (dX) pdX) + y fdx) + dy (dx) + fau  (dy) dX)
illeq
ppste
) roZ dx (dxI
Xa Z(dx) l  + d   Ax r  X)
dy) = ( p.. X+ 7 puX) = a + X +y (dx (dXD + (d X) (d X) + a + x 19 dx(.tX fl 7
dx = (1X) = (dx ) fint  dy=
= (X + Axaclx) tXD Fyrdus adus y) dx) (dy)-
ax  ydC +XHLy 1a d, p (d y) =
 L. X Y andx + atxFZ dx + xty apdx
2 a Z dx + ara y
+ dx, a
dyIX, a =
 +dx, a =
atpay
Unde tundem
 dx
dx +
(dy 1
1 d X dx
dx + 2 dxdx +
a
aa
XLy
X + y ex (d- ( 6B
esc
X + 7 dx +
atx tZ dedx
x +y dx + a+ x+7 dxdx +
aa
at 2 ax
axty
a
(y).y
Sed summa omnium d.y+ dx, a est
a+ x +y
etsumma valorum est 
Potuissemus et summare
aa
dx. _ 1+
atxty
1
a + x+y
2
dy
as ++
a
x) a (YL fxea
iaest autam numnus 14 dx X: a a + X +
eaaratim cdx. + idyct
et idede rant
1. dx) in
aal
1 dx + 3dx dx + 3 dx
vi tatio e)
sihed
in finitus qui
stuctus in infinite parvum, facit constantem
1 dx + a dxdx + 6. dx + 4 dx F 1d7
Hinc
ac
a +X + Z dxdx
aa
aa
XtZ dx + 3 a +XFy Axdx +
3
e
cujus valor aliunde haberi potest
Ubi notandum si aliter valorem iave
niremus pure seu sine (Y) ita ut ipsa
(y) haberetur, et poncietur esse
(d =  (1H dx: a) a fore
Si adus .  +ysr. idcoquia dato sus = pA ryB ) dx: C = (1 + (dX: a)). X. a seu fore
aa
 1= a log. a videtur negligi posse dx: a et fieri = 1. sed
y ∞W s dx 2. 10 potio B= Sod
ax lt hoc nonlicet, tam inde videretur absurdum nam toret
+ a) (AXX: a): pax) atia
 C 
x me
 u  intialte pavantordinn  aditamm
xligaetat ast etata  , a. at prt ex sud
um. Uinc st pncii infinite parvam ordinan additam
dy)
itE
+y
+ax, a -
+x+y
= ((d y)) + dx, a
A + X+ y
1xx +y
ait quim dx, 1r   X.: a