# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/59898/text
# Catalogue record: 59898 — [Aufz. zur Reduktion irrationaler Ausdrücke]
# Incipit: √(x,x-b,x-c) formulam continet sub vinculo, quam explicatione aliqua d
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=59898&katnr=57601
# Work: https://leibnizlegible.com/work/00068197
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folios: Bl. 22–23 (from the shelfmark LH 35, 13, 2a Bl. 22-23): 4 pages, canvases 43–46, page ids 00068197:0043 to 00068197:0046
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068197&page=43
# Lines: 264 recognised on 4 of 4 pages, mean confidence 0.78
# 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.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# 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 22r — 00068197:0043
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 23v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000043.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 84 lines, status machine
(X, X - b, X- C) formulam continet subvinculo, quam explicatione aliqua deprimi velim ad formulam rationalem
secundi gradas per quadratum rationale integrum multiplicatam et divisam feci  X. a = qu( hvaaf;, v+Y) sed
hoc non successit. Ergo faciam X: a = qu (hob + alvt aaque,:, suv + arv taar
Fiot X = B = aqu (hvorer) -  qu ( 50v + ete) et X - C = aqu (hvy + etC) = C qu (svo+ etC)
a qu (SWvk.)
a qu (svot etC.)
fine
Unde a X- b, qu (SuVtet.) = ach vv tetC) + Nab (suvretC), in aChov tetC) - tasov + etC)
Et iXub, qu (suotet
aahn b55
Aperi
jute
a, X- C, qu ssuv tete)
aa, atih - c3t
vD + naftar ab V + a3q.t aat iab
ah +5uab
ah + stab
seu compende D = vo Alo ram, 
D = votpetaque
9 = BBAHPtAK
in vy aasartabx agnaatitab - 67
ah- stab ahastab
5= vot potaque
v1 + qut tariaev + a + aat sac in Wv + aaf-a )ac ∞ aeg- aatiae Z
an+sipo  ah +sval
sen comprndio S = Wp + ABtap
d = vv alv+am
C = ve+ NvtaB
ab - 5aC at -sdac
Nomprndio Cx = vv + v + at
2
= e+ L( - Y( l = am), in, vt = l+ + (Z l - am) = 5uq hic pat
= v+ ZLN(ZH-au), in, v+ = 1+ Y(Z AA - an) = BI Hac
= v + = p- d (app - aG), in, v + = p + (C pp - a9) = GB auut
= vA LH - s (LAM-aX) y in v+ L e + A (LAH. aK) = 30 dac
Opus est OSD8 constans ex octo radicibus sV, 2, 3, 7, h, I, d, D, habere s cegales
ita pt tormnta secundi grabus m tres tribus aliis aequales, remanentibus dunbus tantum
radicibus nullam  aequalem habentibus. Ita OF  non continebit sub vinculo
nisi formulam secundi gradus secundum V. Hoc ut quam commodissime efficiamus, tentemus, an
liceat facere 6 et D quadratos, seu B =  et h = 1 et se = S
Ut S sit quadratus seu ut sit Z = F fiet  AA ∞a =0 seu AA = 4 am, ex explicando
fiet seu aaff Nrac F afrrac ipahag tastC tg ah t sac +4495 tac
Rursus ut D sit quadratus seu ut sit h= D fiet pp = aq seu PP = Aaq et explicando
fiet. aaff F er ab - 2afr dac ( )et ahagrt astb - tant lab- 4 ag5 tab
quod si ab aequatione 2 subtea hamus aequationem 1, fiet
+aass + rrab - 2afr 1ao = 4 ahag - 4 astb + Aaht vab + 4 agd Vab =0
=aaf - rrac - 2a fr Vac + gahague +4atC + gaht tac + 4ag5 sac
quae aequatione divisa per sab + hac, firt (XY - gCt) (Tab - tac) - 2a fY + 4aht + 4ag5 β3( = 0
Rursi si addamus inviaem aequs  t et 2 2 et 1. fiet
Iqo col ninus
2aaff gaagh, + (rr - 49t) (ab + ac) +Casr  galib + (afr. + Aaht. + 4ag5) (Y ab -) Vac) = ) - 0
fas aeque. 3 et 4 ad usum commodiores puto quam 1 et 2,
superet
la
rexpentus
1
aqu.
u redisset, si sunsibuns 
 fiet =l-H=l -am) = ZH + IGWI = aK) reprri non licere fatere l =H, nec m = K
quanquam Rorum unum ex altero sequitur, positis scilicet prioribus B = et h= 1. Tentabo
rta
6)
tamen compendium tale, ut fiat =l = S(Z HH - aK) et LI = - M(Zl- am). Unde
Zl = =AH = aK, seu L(H NH) NH - l) = aK et ex posteriore L (HB - U) = - atm
Ergo fiat ak = = am seu m rHC = D K = - M sive K + m = 0. porro
ob   m +K = 0 fiet ag+ t sab, an - sNac t (ag - F sac, ah + 5ta b) ande
siat ags tac, (at bC) = (ag ato + 295 Jab = ast Hb seu
ese aC
 abtrac
e +ant tab
seu agat = ags sac + ahtrab = ast ibc + agah + ag5 tab - altilac- astibe -0
aht (xab + sac) ag5 (tub t dac)  t B
seu raagh + (ag5. + aht) (+ab - tac) - 2 a5t Ibc =  (5) = 0, a qua
aequatione ab sunt Y et f, et res reducitur ad duos valores sitr et t:9, quorum rationum una
per alteram habetur. Superest ad huc explicanda aequatio YH H am l- HN = 4am
seu (a taarrab + 2a frsab, aahih + Spac - 2ah5 sac) (a Hf tuarra c - 2a r Lac, aol h. 4 5sab + 2ahs dlab)
=Ab 4, ag Laat sab, an + susab, aath + scac - 2ahs tac, qua aequatio evoluta ex 30 parsibus
constabit sed pedeo nos posse compendiose procedere, in qu tibus aequationibus quatuor, si seponamus omnes
la borduse un agu.  et 6, ntitus nb a hiber e g9. 1 et. Z, quam Z et 4
int, non possunt
habere divijores
comium
duo  () G
ut, auto reperi

## Folio 22v — 00068197:0044
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 23r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000044.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 95 lines, status machine
itaque compendi causa aftriab af -r sab
vocemus
 aβ
Et ex aq. 1. fiet a) = Aagh + ht dac. 
est 95.
aW
af +r vac
a)
af -r tac
aE
Ixog. 2. fit. L au = agh - ht sab = ag. tiab, cha srab
 ahistac
b5+ ∞95
Ex aeq. 4 fit  aa BC ( qu( ah - 5 aC) - +aa 3 3 (u (ah + 5 ab) = agrt sab, aht s ab, qu ( at - 5 Vac)
Exaequ. 1. fret : aa)
Ex aeq.2 fit
fa
a
= a +d c
h
aa n = a + b  
4ge
paf sac (-aar racrabs= Aaghilam
4 ibc lac + obc sab
Eodemmodo ex aequ. t et 2 quaeremns +
aaW L 4 agh,:  agtu
=aa + 4agh,: Acgh
e
 sac = tad
Uit
+acc rab  Y Aacgh
atb  rb sac
Lacuo -agha): lab. s =
- (2 + 3) tab  E + b
vil _aas ab At  Taturac - 4agh sab
C
=+ 4 abgh
vac
rab
vel fab (aas- 4agli) + Var saatae ragti)
agic lab + b tac
aaf gagh
e
aghrbc
st- aat
Hbc
(af- 49h lab. laD + artbe
= 4 gh, b dac + C sab = - agh, obe, lab + tac + 4gh Hbc
Hec peracto redeanis ad Hunc Hos valores ipsorum  + , et T. d possemus substitutre
 determinatio. Sit ob  l- +(H = 4am) = Ir + (CNN - 4aK), fiet
Il +. _H) - (3ln = H =  (1 -am (+ XV. (D) aK + (2) S(l- aam, AN - 4a X)
agagtac
aa95Yab
ah+Srab
ah -5 sac
Ot + aags vab
EE Caag +at 1ab)
ahu s vac, qu (est tab) art stab, qusel siu.)
a BB (aagah - at ah lac + aa s C) bat  saagah + atahiab + ac
V 55
ta95 Vab
seu fiit qus _t = g5) (tab + sac) = (3 (2aagh - 2 at abC + (aht + ag5) (+ab - Vac)
aags, sac
aa BB= aaf- iafrYab -abr
aaEZ- aaff + rafr tac - aCH
=( (aagh+ a Cst - 4 (ant + ag5) +C) ( poio ht 4 gr L0
si faceremus L. + e = 0, seu ht +9p =0 magnafieret calculi contractio
 jam ex aequ. 1. erat aa B+rrac = 4aagh + Aa C GE Et ifin. Arg. 2. aaque tabur rasrtab = gaaght q abst fit gaagu= qul afgrtac) -qacC
 ohet = (b-c) an +g sr (b sac = C Ab): ga a (b - c)
ficrenim en (lib  + hinet: aque = ghre - stet. aequit AA = 4 am Et aep 2. pp = 4 a9
eragrine
c
al +ssab, ah -5Vac, ah + sb, a h- 5 Vac
eetq (aagati Lat atitab = aats Hbe, + aagah - at ahibac - aa st + bc)
Zaagsic
 aass-afrrac-arribe
natr sab
a(37 ((augratitab, atrus ac) fang- atrae, al t stab)
inde aamm + Aaa KK () 8 aa mk = asam at) ((t IlHa -llrur ibaa m. H. gatull. ganna
est autem l = aa B: alitstab, et ex = aa Z:, an - s taC, et m = acq+at sab et K = aagu - at Hac
H
aequ. 
a
in aeq. G, et habebitur buvalor ipsius gh per r et 5. Inde etium hoec tollentur ex 7. 6.
sed quoniam has contractiones adhibui, tentare volo an non liceat sumplicite
Edhibere N = D sine divulsioneque ita enim minus minor adhibita erit
ah +stab, ah - 5Yac
quslangu t at aniat aastibe f. aagat, + at ab la (L aat. t ibc
 et divisis omnibus per 4, fit qu (M- K) = l_ (mt L) = KI  = M AN
etfirt ((aa g + at Yab, ah - 5 raC) = (a ag - at tac; Mah +5 tab))
qu 
etfiat snd sublalisfractio ib
a
seu
nempe fiet

## Folio 23r — 00068197:0045
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 22v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000045.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 78 lines, status machine
S. feret ht +5g =0. firret ex aequ. 1 agh = aaV- est = (ex rg. 2) + a W - b5 +, et s+ = aa (W -)): 4 (bc
1t agh =  a) = caa (1o. 7 )(: 4 (b. d). sin agt baa) (=C aast caa )V  caa D: 4 (b -c) seu gh = a ()- ci)
Ex nt +5g =0 firret  nertih Euu ses h= posio 3) 1g (onc)5--ht: 9 et -het: g- aaso. . 1sb- )
et oV - cld = Falf +br sac. = (a (Yab - sac) + D  IbC) (Yab + Vac). ut st =  V aaY: A (lab - 1ac)
et oh= af (sab = sac) tas, bE,:, 4(ab  Vac) et fiet
=stgh = Bhutt = 5599 = (a - 10) (b) - c1): qu (b.c afr (tab. Yac) +ar ibe j. 1guffaba tuc
ao  et Veniamus jam ad explicationem aequationis ulticmae paginae praeceventis, quae satis
aghu ( (- 8) st qu (Bract Z Sab) =  a agh tacst, - aa B ( e aorb  matcitib. vac) aatile, a act rasa leb talir
Ldagh- astite, taaBEH
et aas t qu (aptiactaz ab) -- aq aer AH,: 4 (sab - 1ac)
Vab lac) +arribe
2+ hhet=
c qu (tit - 95)
4qu (tab- Vac
Ergo tandem frV (Tab- sac) + rr Ibc + af, (Jac - sab) = f(Vab - lac) taar sbc
compendii causa sit vab - sac = A et IbC = V, et habibimus aequationes sequentes
qu (Tab+ tac)
St = aar: 4_ gh = ab f + a NVY,: 4B
hhHt = sigg = a adfI, apr: b66
et tandem fr vor + e B vrr + afs abss = aaf t aaqvr
5= aa invoniamus S et g, siet
5 = aar: +bt g= abf+ acoY,: 4bh fit =s9 = , a, b5+ aY, : 160Bht = ht
Et = a Hadfr Bar r1): 4D =  -59
+ = as (ab fr + aN rY): ADM et 5= - aa
adh
EBa CaBEr+avr)
 Superest aequatio in qua solae sunt fet Y nempe
= -ahr: Y (adsrtavn)
frDD = r D_ + BD Barr. + adH = aab f + aa VY, seu
y  f + a + PBrr- 2aadY =
at +BOrr -2aaBP - 4 a N + qciV: B) r
4aa
+B- 4a = ab + ac = 2 a sbc - 4 a fbc 2aaD + 4 aV : B = 2aadD + 4 
4
4aa
-2na B + 42  Raa, ab- ac + 2albc + 4a eoc, sed nulla hinc compendia
fiet f =  aa = DY + (a +BBIrY + 4 a(v: b) Y): 2a
=gade
= Z aaor
si si velimus facere f =0 fiet BY = aa seu Y = aa: D et foret g = a V: 4BDh
et  F = aavaN: ADb et 5 = - ah: sac et h manet arbitraria
Rspetau Calculies posito f = 0 et ht + 95 =0 EUam firi arVb- aegale (i antaria a ab
a tib Bst
ob B = 7 fit  AA = gan seu aerre- Aagut 4, gatu
seusarre = 4(a3), gatr (+ ht vad + StC, ob h= B fit pp = 4aq. seusartb = a gahs it lab  Stb
plis aegq. fit Regh= stida = _b, vrat: 4a, ques duos valores aequando sit st = r.
o (P. . Apul P lab e plac)
-ye (aagh+ abst + Cant Fags) iab p. q r abt fa) Btacts saba as t dact
+95 val
E qu af.Y)ab) = 4ab5+
gbYaB
A  g5= -ht s = -ht:g = ghrr = ahhlt ,: Ht sit sthH = gghrr + aghat F, et
et fit h= HH- gorr: 4gH et fiet S= fflt + ggrr, : 4g9t
St = - - AH (+ g9rB,  = = (a rr (C. b)  rasr ( C  la b) 9e (u (C. b)
et on -  ggnr,: (El = da f(b- CD+ rafo (btac - etab) : (ina Cb. C)
(+:gu = q adss Cb arr b) agr loca tab) trY 2aBr (Lctrab) .4(P- 9)
aq: H = 2 afB (bsac = C Nab): aa (C - b) Y) et tuanescentibus F .E, 9, 5, r, fint
d, = 4 aa YbC (ab ad):  a qu Cb - C) et fieret, 1bc =- b +C,
sin fes prasse atrr t ragritaet ast abrr F rasr vab;in qacabs  se ele  rai cherteord gs e B)
qu Cb-c)
go, part  aatislab. tretarobe,- aassatriat
contrahitur ponendo ht+ 95 L0  paagh FabSF  a3Fa
qu (Tab + lac)
4(b-C)
et fit 9 = Ahtt, b - C : aa(V - 10) = a (b) - c1): 4h (b -C). sin IBhhit  Ca V1) (. ad
Et explicundo Zet W0 et) fiet X   V. W= Y(Sact  ab) velb 55gg =
act terrab
Est autem, 3a = r(Nab + 1ac)
Tab- saC)
itaque Et aGh qut B. 15) =- a, afeab - a1) (Nab  tac)  Ibc  (abt actratbi)s: A(tabo tac)
e
 et aβ sac + ac s ab = afiac qbe + af ab Baribe = af (Vab q vac) +ur 
itaque - aaghqu (a β - a Z) = - aa, ex afrr (ab a sac) + ar bc, : 4 (tab - Vac ))
et itue fieret 4bc = bb + c = 2 bc seu bb. + cC + 2bC 
e l) C  inttu ali oetitum succede et si catbulus esset bonu
ee= 8bc +9bb = dbb L bb  erfit a 3b -C = 2b+2. seu
c _V, 3=212, quod mrirater, si res hvccasspenti sucideret
1

## Folio 23v — 00068197:0046
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 22r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000046.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 7 lines, status machine
Repe jamus
Asuuta ht + q5 =0. b3 1X= 4a M. Fiet aa, aats+ Hacyr + rafr iac = ad iaganistiatit vacs
C
1d5t + ag5 tac)
(.b Fursus)  gaque fint aa, aats tabr- ragriab = na, agat (ant rab
abst- ag5 tab 
diique ab se- 3 fit cl-  (a lam9 = Z H + 1(Z AN~ aX). fit qu ( ma X) = LM (m t K) = UX= NFM per piora