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# Catalogue record: 60409 — Investigatio Analytica Tangentialium
# Incipit: Tangentiales voco aequationes differentiales primi gradus
# Date: 1698-09
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60409&katnr=57355
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 20 (from the shelfmark LH 35, 13, 1 Bl. 20): 2 pages, canvases 39–40, page ids 00068196:0039 to 00068196:0040
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=39
# Lines: 199 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.
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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 20r — 00068196:0039
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000039.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 80 lines, status machine
Exe
Investigatio Analytica Tangentialium
Tangenetiales voco aequationes adrs differentiales
primi gradus. Assumo aequationem ordinariam quoad unam
indete ordinatam, cujus coefficientes dentur utcunque ex abscissa, et hujus
hanc differentiactore comparo datae. Vel quod eodem redit et syntheticum
 est, sed si omnia percurras analysin absoboit or dine assamo aequationes
ordinarias ad crdinatam, et video quae differentiates datae per quamque solvi
9+ m = 0 hinc dy + dn = 0 quod hinc ducetur, palet per se
possint.
hytam = hdy + ydh + adm =0 sit data aadus + (By dx + a) dx = o
quae multiplicetur per E: aa fiet prodit hdy + βh ydx: aa + h) dx: a = 0 Et
et fit aadm _ aadh = Bt dx se aa logh = (Bdx et aam = (h) dx
adeoque aequatio quaesita est hy+ sh) dx: a =0 posito logh = (βdx: aa
a) hdyt ydhiel + by + m -0 fit hydy eah dy + aydh + aa dl = 0
ral dy + hydm + at dm
+ab dys
hy+ al + ab
+ ab dm
Sit data aequalio agdys + a Bdys+ Vydx 1ad ex ∞ e, fiet h- a
Sit data squatio aydys ta βdy + Vydx. + aBdx = 0 quae multiplicetur per h: a
fiet hydy + h(Bdy + hVydx: a + hbdx =c. Et comparanda
fiet ex dy fiet h∞ al+ ab , βua al = h B ee βh - an - ab
et denique ex ex fiet a aBdh + ah d(3 - aa dh H(Bh - ah - ab) V dx: a - adh: h)
et ex ydx fiet dm= m = (Vdx: a sadh: h
= hBdx sit b = 0 et fiet aBdh: h + ad (3- aa dh: h + (3 - a)) dx:a  (3 a) adhi. B
-Badh: h + a a dh: h = hAdx. et evanescit h, nihilque agitur
Ergo maneat b, et fet Ca(B- a) dh: h) + a d β + B- a)) dx:a  a (B3. a) dh : h)
Lb) dx: A + aabah: hih = Adx quae aequatio solvi potest
ex Canone proxime praecedente nam bdh: hisi est differentia a - b: h
a Adx = ac dβ + (β- a))) dx - ad ax et sit Lsc Z = su Adx: a
1
20
Si AdX posito logto = s) axiaa
t
tietque
sit b: h = Z. a et aequatio  erit talis
et excanor
posito logW = (Vdx: au seu b =
aleue (hyraltab) + g + m = c. et t = βh. a h b et m= sidxia = dogh
et posito A = aa dβ + (3 - a)) - ab,: a
Data ergo aequatione solvenda aydus + aBdy + Vydx + aD dx = 0. fiet
aloge (hy + al tab) + y +m =0 posito esse - Stonax logro= fdx: aa
U
A = a aadβ + β a) dx abdX,: a dx  l+bBh: a - h -b et
henique M - Sydx. a = alogh, et b manet arbitraria toussans
seu f aloge (hy+ Bh - ah) + y + sy dx: a - alogh = 0
posito
posito  = 1 fuit et W log W = f) dxea posito h = bW: sondx
et legu = (V dx: aa et A = aa d β + (3- a Vdx- abdx: adx
i
repetamus calculum: ashdys +ydh +a dt +94 m 20 Ergo ta hdy F aydhta d +hidy =
tigdm 
im ydx sit
superest x et dabit
t data aequatio Hydus + Baus fVgaX + abi dx nd quae multiplicasa per Ai a dabit hyputhedy+ s ydxia a sd tu) = v. Ergo
(4s p atdB anat  (a Bah at a β-aadhe ((3- a) H) dx: a ((β- a) dh = ABaX. Ergo heva nescit
xtaddx =o
Adtro m= Paxia  log.t ex du fit an + al + ab = hB sit a l+ ab = hB at Bh- ah.
t hypusdy-
kgys dk
Alter si nsNaq Ry  P aaya  + RX y ay + altay +
dk raatam Tal ycus quae multiplicerur per rhy +P fictque
indecoparendo ex udus fit hN+ 2aC - n B seu Al = hB . HH. + xphyam ahgus dus + Ehise ge + 2bst ex taBodus + 2hv rylxia + 2kB yex + apB ax = 
x
K=Fit
et ex dysit lK Tan = C sO ABK - HKK + apt = a (β3Y et ex yudx sit aK = 2hE dX e seu
et exlude it et ex ax ter aa dl + aldm= sp B ux  seu at al aBF a Bah EA ex- = aF d + Bam  PK an = siK. HKK tuas 
supere s ex yax
sit unde ex se pvicem habentum habetur af (B-  K) (t ah +  dM) adeoqe habetur asahih +Mper Quadratura
equai aantin n aleat in ente tacia- aptax  de  atk r uua scy t jusu habesur Sahisi ta e prvinde
caleulus nion succedit
alnil
hunc hac
pagina
aβ est homo-
genea uniteti
ut sax seu
ut loguhuu
ut logX

## Folio 20v — 00068196:0040
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000040.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 119 lines, status machine
Lel
si aequatio vy +hg y +hu et. finita differentiatur, cujus termnus
coeffi ubi ipsius y coefficientes dentur per X, sitque terminus summus sine X, et termnus
infinus sine y, patet aequationem ejus differentialem omnes ejus terminos duplicara
praeter primum et ultimum. Termini autem sunt e+1 itaque termini qui duplicantur
sant C = 1 et fitat 2C - 2 Termini, quibus si addantur duo primu et ut  nempe qui fiunt
ex primo et ultimo differentiatis, patet aequationem differentialem habere tetminis 2
Hinc si sit aequ. y F hy + ly + m, Ubi e = 3,  habebit ejus L fferentialus
ermnrs sex 2e seu 6. nempe yydy ydy, yyax, dy, yax, dm.
Hinc
aequatio. Itaque ep. ut aequatio alta ejus differentiali comparetur alia epa
identificetur alia ejusiem forma opas et terminis
2C assumtitius literis
rC, sed ipsa tantum exibt terminos e, Ergo oportet ut aequalio
differentiatis huic comparanda hujus differentiali multiplicetur per
equati
sit aequatio y, ea quae comparari potest erit multiplicanda per y+ p, et
forautam  terminorum C. Hinc
cum ex y derivetur ydus etc. erit dyoidm 50 huc dyth xh exlta silvenda
Ex p deriontur gyaus, eritus exult ex altaeda idq exallalio facienda
per yy + py+q sed ita Polvenda rarsus esset  dy+hax, et ita porro
sed tatis solvenda nunquam habet opus sobutione
parum praestat
td hE
Ergo hoc methodus
hdx + m dy + _M dx,
yg
Etiyay taldus +Lyycht agal F aado F y+ n = 0 fit AG ylyichir y adtalus a tigus du abs atam
solvi pt haec aequatio
seu nyy de + hydx + m dy
quid Ado pyu raly +aam
+ alydus ast dus regusdi
comporandada sit data aydus +Bsga Vytxa abdx mult per s.
apBau + hygdy F hByduy +. hsyy dx: a + tByex rap Bdx 
a
+ ps dx
(9) +a1) ytr tataus
nou+ a y + aam
aequa
hontantum altormula
aeq
cysitob.duio
ndum  simple tretir
p et ana td
en
arya 1am ut Begla v3 au
Fdamydn
rghyyclus + aglydys Faalrdy +gu ah ta guydl fac gram 
pulo sic uhilitate rans quaer ut tractetur resvelut
pie sic
g
uyds
tfyau rnttusdus
et
Tarsych + faurydl t aerdm
tal pydk + namenk t a mdn
Axyydy +as ly agnotns hystit regodu  aalde
Gytagt (atey naleus ryyda t ayul kacum Ahus tansul
resuma fi am= pB - rl ab ypax fit Asax. a=ddur rash- findrir ob gayfit ap - 2ra t a l- h
Egus ralu+ aam
iquaam = _ 2βrh + aBl - hββ - alY ob ydx fit a Fal+ alda -lnydr:  BhB dx + (pVaX =) =7 rhet Vt iss
estat ex ex et inde fit 2rhBdx + albdx.h BBdx Lardm +a m an (s madr. 1E. = prhndi Faslnartt eer ralus
srhun aB Can -hB(5 du -alian
2Prantar Chartr Erha BAraB l raralaβ C
dne est ad
repreducta est ad agq. O,D, 2
ed verror aplculi prorx later fnciamus m=o res irit simplicur sed credo nondem satis
erit (L B- er BBah- araler- Araral
per laeg. 5 datur
An sic (gytar) Shrayt ydh tadt + an =0 fiet ghydus + ahxalys +gyydh + agydil
an eg, fiet hy yan tadl. Fgyaan ficgyais angdyle +
hus +a
Phay +yan tac  x
gytar
hy+ al
rat
hydc
rydus elguss 
atgdh taarat
sit 9y tar  ∞ hy+al se g-h et Y et fi gyus +2agryt aurr
hiys+ al
hleydus +ahlaus + Ryydhaligal.
ah Gdn +aal du
pandr
aatul = . tollumus yydx et fiet hah + c dh. hde =  sen-dhDC dh + hde
+ anhdue + cyydh Fal ydh dardu
Hyragui C
 + aa ndr
seu2s + = seu c: h = logati
sit aequdala th ydus + tit βdu + hhyydxea xhhbdx
L
+ac Gdr sts hβdh + arth logs F t d(hD +hlogtadr Farah anda-aryde racyar 9
aatal - ar (tB) farat  ad r logh) + e ur hB+ adr arcyhxadral= hnb
si + fyEC yy + fyaldysragotyde sgagA + a gaadm )nygap) Afynay tagndus etyaaptagalp
gaag sifet nesent  fieret pCoppusiri sp glapdu  polysap yalt e gam ax
agregdus fyaada
quiti =Ecgpydy + aglody+ agpydl tanqpam   2cg py dus + 2CG p (dys + 2CGpVy ax: a + ECgpDax =
ande raa ar de
n sent a9 (. 3tali + X aa 
Fh
arydc et fint all + anh = arHlogh = h+ EB (a-B + aployt
-arcas =Qyyde  a
O ayeat
ytas
e
aq = ahgde
+f
fit ab  2cYa al = XC β aadl = 2C) ax quidsitillam dx et Hodus exhac t
fit apa cydys  alaue Fgyydc agay dl rsganam
Am - adp. P
aegulus darta inyae g fiat meento cins aep acpraplm
nygesasenus dal dus 
sit jam opodata cniydus + cgnsay, FC gnispaia + cgnbaxuo,
1-0
e
rget tnygnadnt apadus, et roi dividetur
a
Cgn agnat a sit al- cspax  cb dx -9 aa + ad  =
dg:9+a apip + a dg: g= a ap. p