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# Catalogue record: 1109 — Resolutio fractionum unius literae, respectu rationalium in partes, maxime simplices secundum ipsam Ad usum Quadraturarum
# Incipit: 1/[x+a, x+b] = (1) [1/[x+b] - 1/[x+a]], 1/[a-b]
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=1109&katnr=58659
# Work: https://leibnizlegible.com/work/00068070
# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 25
# Folios: Bl. 1–2 (from the shelfmark LH 35, 6, 25 Bl. 1-2): 4 pages, canvases 1–4, page ids 00068070:0001 to 00068070:0004
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068070&page=1
# Lines: 382 recognised on 4 of 4 pages, mean confidence 0.72
# 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.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
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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 1r — 00068070:0001
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 2v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068070/jpgs/default/00000001.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 93 lines, status machine
eopeem litora
Resolutio rationalium fractionum unius indeserminatae
 respectuetus rationalium in partes, se maxime simplices respect indeter secundum ipsam
le 1 ay.
b xtdC Xia Xrb
eta, xib Xtb xta  a. b
(+) . ..,  =  - L, 1 -.- 
Rta, xe
xta, x+b, xtc
Xta, XtC Mb, XtC
seu fiet xta, xtb, xtC
XtC Xta a-c Xtc Xtb b- c ) a - b
XtC a- c. a- b. bc Xtb a-b, b-c xta a- b, ac
ba-at R + 1, 
Vel ut omnia enuntiemus homoeoptotos
xtar
a-
xta, xtb, Xtc
Xta, XIb, XIC, XFd
e
Xta, bia, aC
Xta,  ac
Xta, a -b, auc, a- d
Mra, aebactsga bud
Xta, Xeb xra, A. b Xeb, b a
Et similiter
X+b, b. d, b-c
Xtb, b-a, bac bad XrC, Ca, C b, C. d XFd, D. ad .b, d. C
a + aa
G22a J. 2a + aa =X  xra
73 - 3aZ + 3a - a = XX + 2aXF aa = a = XX - ax +fau - a
XROR sit X+a = Z fit X = Z- a. et fit XX =
Ac et similiter X = Ei via  a
et similiter X = x - a xx taax -a xra
=3ax3aa Z
Et ita porro.
dy=, smX +a posito V esse rationalam ex X
itaque resolutio est absoluta
Sit jam summanda quantitas de axtap
nat Huc enim
Sit jam summanda quantitas dus= Ax. V smxta. sit  t Cet
bxtap
sit mxtan L
bxtac da
omnes quadratyrae Conicae revocari possunt.
) a(ZZ- ar et dx=
aamX + an = bx 2Z + a(7Z. Ergo X=
Ergo dy = V3
aam - b73
C2ab CZ3 d2) + 2CC MZ  + 2abc Z3d3- 2a bn Za
(2) - b32+ aam
seu dus vZZ 2v7Z aZ, em -b2
lEt b22+ aam
Reslvatur t 
praee
b Faam racsque - ac dn 26
(2- bZZ+ dam
1S. v= aa. X fint dy- 
(7) eb 1X raam L. 2aim
1: 2a tm
AXC BZ aim, Zibraim IZobtom, Esb =aim
(2)Zib- aim b7- aum
1:2am
rarm
posterius ilerum resoloi potest s nam v. aam 2b -aim Zibtarm
1. radm
yb-aam
gobtatm, Zib asm F. gsbratm
Ergo - 1: aaam (- 1: +aamm
1. gaam
b tata
siri Ro bZZ scribatur coco, et pro maa scribatur hh, fiet. I: () coco- fihi
(elbZ2- aan (217 1b -aim b72- aam).
Pscocontin = aht esconti cocatito  corhi
Ergo sit estu. naa =  Etebaam BEE. na   2b r ami
e)  1: + mar
sesoren sebreta
cZZ aan, aam. b2
(A) 1aim - 1. a im. Exquibus prioris summatio datur
14 b7Z aam
(25Zsba num, btaem.
Resolvatur (B2 ma
Jam 
 denique quia
iml + +naa rmua + 1: dma sa
at
o eir
v tti  et coct coth fiet
 1. 8 ma sm
2
1. 4a
bz Z- dans
Ad usum quadraturarum

## Folio 1v — 00068070:0002
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 2r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068070/jpgs/default/00000002.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 105 lines, status machine
Si ergo sit du =   vdx s mxran et fiat nxtan = 2
1(2 2a v22 em n 2
amaa  et
aL
bxtac ac c
fiet ex
+ 1: 2am 1: 2a Im
oram aim Zubtaim Zb - a sm
sit jam du= dx matan, bxrac = d ex bx tac Ax S mx t an =ay fit bX + ac = V = abes . dbn , ac
L5) (abez t abn cubq  a E m
seu V
c em. bn  aebn aem poyt in,  p : , 1. d ratm
Ergo dx Imaran, bxtac =
2m 2 b7Z aam VC A Z.braim Baboam Zebratm Zsb asm
Ubi rursus nova resolutio instituenda est.
.Et quidem ex 6 et 15 in catu aequ
sit _ eesita ag. 2. sit: A  de Atox vac, mxran = Eaa, en bn, ae bnndem q q uy seu dos bprac, naran = 4G2
Quad et sie colligemus: Resolvamus ergo .
orroli c + h l-
fot (3) ocontil
id
cimntin coth
 et s- k 
(3 lb2 aam
hoc modo
LElloco-hh) caur
. 2 3) cocontia scototiti, catr (3). -.  an (et ers. B, oth
ta ico  ed +h
Arfilo scoonth. coth
cothC
1 (3)- 122h
poptu qu a enbri Lelicam
   cotm ast  sadtii hi
cument, utt cort ausit hgu p 1 12) 1.2h
E. h. Coth (2s coco-tih (.) corh, coco-hh
Dcocontihi contr (e) cocontih Ergo L 9 = 1: 2t  1
 et cent
Bel comath, edah () conti er aliti P seoronth
comprrbatio: = 2coco + 2hh +  +CoC rEdh + hh + co  + 2coh + hh = + 4hh
2: 4tik + 1: atit 
ecentin (Tothycdconhh  Pconh, edctih
scoth, a. h Etera, cotrntibi
(2)Coth ocotit  corho cot B) coth
L . Ergo
=1:dk- 1.  p ath 1: 2h
1corh () corh
Eoh consi eo conhit BCcoth  et coldoht soah coth
H 1.B(3 1: 4th, 1.2h
H Hib) 2 1. t X
ath ck
oi scorh, cocon hit p. t 1. 2b
picatui ito
1. abh et li8h
Liscoth Ltcoh 
nto  voit ibti
tui
co-h ro
(t conh 13) conh
Ergo per a49. 10, 15, 16
=1. 8l3 Ergotandem ex 10 per  15, 16, 17 fiet
r: 16t -. d. 1hik 2. 32k
Dcoch 13tcogh
Pcorh coh Coth cont
corh3 conh.  Lt . h 13s coad
1.81 
D) oth ubi quod examinis loco est
1bhd 1. 81
cont. Ch
h et h se habent eodem
(20)  ponatur Jico Eeg. 19, Zso toroco, et a sin foce h.
Atque ita redimus ad colculum superiorem.
superest jam ut summemus Cso coco deo cocodeo, cocoduo cocoduo coco deo corodio
 Co+h
cont scoth jescont 13 corh 13tco-
 sit conh = Z fin eocodeo  (2 - 2lh + hl. 1q =) Zt ehiter tit d2 Ego Scodo deo  L corti - ih cot  +. b S
seu scocoded =
E- 2t E + hitom Ego f ouo =  rt - 2hstes - th, e faoto = v.h. 4. 1.  .  
=h (23)
4. sito-oth +hor a h sto
oth coth ideo coti
-hh
coth 2cot cont 3
conh cohr 2 t
2 cos + 2h coach + hh sunt
-3: 1bh3 oth - M N- t
1og. at
hi o revet
Em oenti 4 Ca b tit
+ c- h + ah San B 
+ Sut -
Qa quad abera deniet a quad atura Hyp.  si a sit quantitas ealis, vel
Snxran, fin festodio
quad. Circuli, si h sit quantitas imaginaria
jam co ponamus Zsb et pro h, ponamus am, et pro Z. a pondamu
oio ara
fint in  
cotti b
= (dx Sbxtac, nX ran
alculuu
quod comprobemus: (co k 3co h t 3bi  9d s co h co + h  phcoco H 2td 4h o ++ humi  8h
112t Ergo
Similiter ergo
Ergo S corodio
a
ocontin Q co+h

## Folio 2r — 00068070:0003
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 1v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068070/jpgs/default/00000003.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 86 lines, status machine
Bllocetin V
18+ 3Co h+rcohZh
o+ 13WB Susi th
= c3L- 3u h - 30h = h
+o 43 (+ rethi tcati
conui cocott Co F reoh X = rcob3 .h
Hh corh conti
 4o titicoth ult  t,B cco + ha Co h - _ 4 hh, d 13) comn
_ cocdtih, coth + 4t  cot A
seu 4c - 8 cont 2= cootmGdatnt vet (- atih. C0   B co (12h C) t+h
trco + iti
F Cot tib saali
(10) - 2. 4hh 1: atib
(E) coco -th (3 coth. c- h
eqe  e 
Ahhco  12h.   2h tco) + 4t
Ergo o
+ 6: 16h 3: 16l+ +7
cco - he la s
2: 1bh 2h: 16h
B) coth 13(Co+ h
+Qco ht cocohht cohh
2 tut 6 cocohl 6co B3) 2 h1
sumtua 6+3 +3+ 2+Eh
beo facoroti b
00 ba (310t
he
rc. I a o tit +
eoto- th
p praev
Calculus ergo est verus
sed summari jam debet
vel quo cocorhiti potes q
((c0co hh (8)
Gvo- 2wtthh v - 2h + hh. Erp  
oitu acoan
)1 + itis + tit
loco
uo. Corte co prte oustulisu p
cintra Comprobatio
hh:, lcota  2H3: (3tco
Eago (tio crinsibes scitit
co d Docohh + h) tti ocd e +ihi
Habemus ergo pro certo: ed
(3Iaco -th 16tih
+9dio frco
Enipsa, est idco + p dco + qdeo + xcocodu rhhdco rcocodeo
S seo do5co tssio  ohn
n oScoscocohtiio  L stddio 41cot de a ct
xcodeo- rhh collco HC Frh cocoded +o shaco- ihdo- n4
Btrontk
cold = 00  -2bn + hh = 1 - 2h
+ht. seu coco 12) 
+3:hC0) 3. h h (+ 3. h h
- 3. h C0) +3. hih
 cdotit tt io
et coco coth + 
Cat frliselertigut grtito
ihitot ( i (2to gati at sio B
 b
coco - 1
(3ICdCo- hh 16s
40ha +h +5teo
tios  rot +Eteju  = onIS
En set t2t Latet a (2
 tosr 1
3h,
coh co Ncor
Abito. m) tsen
sumu . c. tin scdco atiti B seoconth P cococlh
ad conicarum dimensinum apius indigemus
ia peculieri leco prosquerma
superest quae ramus et orontt
( .  B eit
tlita vaeten cath.th:  cot
+3t3qe colutus tgti. sisconh, + 2H5. 1co-h
-Etu: coth +atit Bcoth = 2h5:, 12) coth
: comh +Etet. seh F at. BCo-
Bloo. dui aui  he: cort iote; pu orh As
Carcoh- ih +coco-2coh +_h= qcoh E 
(.3  , d EB) + WoCH = 
PB E  3sib =   pesto  16 ,
2=sti e H  coim
cm conn ibqoconun quona
h  coco rcoh +hn- ldl - 2c h. t-
ticoc

## Folio 2v — 00068070:0004
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 1r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068070/jpgs/default/00000004.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 73, 2026-08-27T15:28:38Z — 98 lines, status machine
r
coit
+sdeo = = 2Cd.
eom
cdlo-tih
(duo =  1. Et Cdo -. Ergo pdao + sdao = est a c    t c0. t
crn o d rti cont 2onh ni 6 accups
hot
Coco-htr
Co).h icd
ecutt scoco) tih
Nota generaliter si eo Crescente decrescat Z, et sit Z explicandar per co, debet d explicuri
per quantitatem debet dz: deo explicari per quantitatem negatinam.
cocodco -
16ht -
BScoco-hh
inter ceto
inter co smipseilco caipus 
,
conh coth L (C0c0)-hh-
Cut
cont S
E inter co cninst e 16h43 - h s + hs  
rootit
et (C0) (majus)
 lusti uatur comprobatio: 
coit
c
o) hh +Co) h
(+ cocot ht.2
LCauot
2cotentaroti
t fe tnotif rot ohi
cortih +cohd ot
reaulariter terminam ubi
(c0) pro 0 seu ubi eo
minus, subintelligo in
calculo
a brocetti le licontnio + a E b ht  Is  ( c hh. recch +6cocoh d . cob
ht qco li- C02- rcocoht +b 60ih + E. P R Co hh Ho ht 2 cop -tih 2c0 duo oh. 34 t et Aoti 1   
(4) co0hh F
sed co+BCocoh-h divis. per cocoatih dat Co + 2gcocot +7 sit
Erque - 2codhh 46coh +2co + d0cot t h
dco titi 
co 21  +24
B) coco H
Alo cdth L17W h=- 7h  ∞t + nacocolh +rht
Fc 1 hh- titi. - th
vo +colo
comprovatio
sib ash ihticocogatt
co corti n Le cd- si
rcoli. c t co h H)
H coco - hh
uhtut etico oticos h
h-- . 1. (9r3olei o ute ( a is co (. rotdrte C
B c 
hh ilocohiha 2h 2t cocoi ocoh
conri corri
Coco -hn2
23 - 213 = 12cd h +4
co-ti3 C0+t3
Coo-th
Secot Fh
2oco2coth +2aecoh- 2cocoh 2 hb.
+ 12 cocoh + 4 Ah
teu 2co th + 12coht + E2h
coco- ph.
attp ratit Eoco - h 6c tih + 2t - 2. 2C E hh + 2Coht = 6  troruhi - Scoloh - . h = dcotit - 8cot 4 L h o 
loco -ht
(coco deo
contludoigitur esse  coco
toto -hh3
15(23 -h faco + hsdu
Nunc similiter quaeramus (Ct dae = ((3hdeo - p3hco A stih,-k
t oco-tit H co-t coth
ut as
coco- th
coch cont
2c+ qui calculus est coprobatus
Coco-ht
ests -gh
datsaca = hit
a coth copsi
Coco-hh2
oce it. bconit au. n vouli o  li = 10 3 1in t 60 h t 2. 2u agec tl B6 eabil. 30 co 1 216L - 20 0 ilt 14 co i, sedt
coco- hti3
C0co- hh =. dco,
itaque (concludendo) comprobatum est esse
(W du 
Dcoc-th 16t4 in
(Codo Lin
Plcoco- hih 16bh
-h sdq
+h Scoth
-
coth
-2hh . . -