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# Catalogue record: 60631 — [Aufz. zur Reduktion irrationaler Ausdrücke]
# Incipit: Proponitur ay = √(a, x+l, x+m, x+n)
# Date: 1714
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60631&katnr=57619
# Work: https://leibnizlegible.com/work/00068197
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2a
# Folios: Bl. 46–47 (from the shelfmark LH 35, 13, 2a Bl. 46-47): 4 pages, canvases 91–94, page ids 00068197:0091 to 00068197:0094
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068197&page=91
# Lines: 237 recognised on 3 of 4 pages, mean confidence 0.81
# 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 46r — 00068197:0091
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 47v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000091.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 76 lines, status machine
Proponitur ay = L(a, XLL, XFm, X + n)
Quaeritur talis explicatio ipsins X, ut ex valore ipsius y radix possit extrali quadratice
Patet esse AB in AC in BC = AABBCC, ideo ((AB. AC. BC) = ABC
itaque si fiati possit ax += AB axtam = AC axt an = BC) et possint haberi A, B,C
seu ax = AB - al = AC = am = BC = an habebitur deside ratum
sit A = ootu + 1 +OD + 12 104 +  tD4
B = 002 + 00
C = 00 131
0o mihi in tatibus significae unitatem
AB, (00. oti + av. 110108 + 00. 11904. + 00. 124 1OY
2000as
= 00. 00 an
+ 11. 320108
+ 11. 335 10q
+ 00. 313 +1E + 00. 31 5N CQ
+12. 339 105)9
200319 731
Ut coinsidentientur hitre, valores ipsius X, sufficit duos reddi tertio coincidentes
Hoc fit per ae quationes identificatorias ex irrationalibus G -Q-  H = R =  I = s - 
eX G = Q queratur O, et rursus ex W = 2. Bi n. hlo
ipsus Caequati dent vulorem 2 per h et 6
H = R .. . . . D . . . - . 9R .. . . . . . .
425 .. . . . . . . . ..
ita habemus valorem ipsius 2 ter primus aqueur
primus aequetur secundo p . S videtur prodire tvalor ipsius h: 6
tertius.... secutido
...    -  --
Sed ne prodeat, oportet valoris utriusque tam numeratorem quam denominatoremfieri = 0.
ita habemus aequationes 4, quan e pro determinandis coefficientibus
Harum ope inveniantur valores ipsorum 11, 13, 31, 357 spe ipsorum aD 12, 32
ocuidum led
tum ut calculi maxima parte possimus supersedere
l. ut 3.
itaque valore uno ejusdem habito habetur alter
ipsius (2 ) fit ex valore sPriore
alet 
Dsvel () terentum
ptinus ex secundo (tertio) permutardo
Tres valores ipsius  siunt scundus ex primo (tertio)
Etertius ex primo (secundo)
Itaque uno calculato, reliqui sine calculo scribi possunt
sexutrous valor ipsius tr: S differt fit ex primo permutando a D et 3
Val ores ipeoru 13 et 11) dif 3 et . 1
(33 et. 31) C differuns permutando
Calores ipsorum), 31 et 11
13 et 2 (it 63 A d)
Tipsius q Cfort 9) Epermutando a2 et c3 scrt . 2 et 1)
quealur adbu resione
02 et. 3 (- 2 et. 1)
0l et 3 C. l et. 2)
in hos valores solae ingrediuntur
residuae 12 et 32
pino
lina
Bid=A N a Y.
in his quae ex se invicem siunt permutando 3. et 1. quodlibet manet permutando 12 et 32
in talibus 12 et 32 pari jure utuntur, et secundum legem justitiae tractantur
Talis observatio  inservit ad examinandam aequationem per se, etsi alteri non comparetur
Venio jam ad identificandos Terminos rationales seu ad F(= P = F
Hic pro O, D, Q, 2t substituantur inventi valores.
Et x aequatio quae fitex( F = B, fit em cgecefet et agu
ft e in
permutandotu et S, ilem l. et 3.
F= t in his aequationibus extunt ex variantibus solae  et d supersunt
fiat a a6 = Pt +ag, et hoc vatore substitutus in utraque aequatione
33. prodibunt binae aequationes hujus formae Btit + aVt +aa d = 0 et Ahq + antu taa A = 0
F+ quae ut coincidentientur inter se, seu habe ntus radix aliqua unius sit identicae radici alicui alterius
fiat en = a= 3 maser e s  e est nrbitiariarum  s et resiua init
aFGLHEito
ap 2tVib2
aT +D  + BB O
syut rab septta
s
erit ag: ah= bf : cf
Ergo et ag
  npo (. Ab. p

## Folio 46v — 00068197:0092
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 47r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000092.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 47 lines, status machine
It calculo generali fiat compendium specialis ponantur duae aequationesi
ad duas literas X et y, verb. gr. 1Lret  14 LX+ 19 Y + 15e xy+ quiy He=
et 24gY Y 29yy + 25py + 2g aX + 2 Bay +249aa 20 Explicetur y per X, faciendo
XXy13e a fient duae aequationes secundum X, quae debent esse coincidentes
vel _a= fi seu habere radicem communem. quae celtirebipe; si  Hae binae aequatiopes com
posit f = -63. 3a pendro expressae sunto (XX + amX + aan= o et exx + agx + aar v. Fiat
1 = m = R. E linabu Unde nascentur binae Aequationes
vilut li p = m: q et n: r = m: q, vel quo alio modo commodissime licebit
per quas determinabuntur ltere F et g. Quod si forte tollantur
 etoet
ambae, orietur relatio inter coefficientes 120, 102, etc.  220, 202, et
quod culaulus docebit. Id ub: generaliter praestiterimus utiliter adhibe-
bitur ad specialem casum, quem dat finis calculi  paginae
praecedentis. Generalis panon poterit etiam servire ad alios casus
et revera s solutio generalis hrarum trium aequatio, 1, et 2, 3
anon
Et atillter loco fet s, od hibebuntur 3, 303, 3Cd. Ita enim perpetuo
inter calculandum observari possunt
A Ita enim possunt
CA  observari permutationes ipsorum 1. et 2.
et permutationes ipsorum 2 et 3, sub quibus comprehenduntur
4 id est squadr. 2) et 19 id est quadr. 3,  At.5, id est A. (+ 3)
eol le a modes aequationetiam q et
dutu seu ex et o permutatis cebent mane
Ergo Calculus ise generalis est ergo pro eo casu, quo bres aequationes
modo
rent ses corum
quales 1, 2, 3.  reducuntur in effectu ad duas, ob coinci
dentiam radicum.
Ut caleuli (anonem is eni commedius inventumus, primum
aeqq. t et 2, reducamus secundum X quasi essent.
54XX F 59y. S2XX + 51aX + sCq et 6Z X X 16I aX + 60 aa - 0
ita eret E g9 et ponendo sepro 19, et Sr pro pro 159+ 12a, et
50aa pro 147y + 1ay + 10 aa.
Compendio ponendo 52 = i9 et 51a = 15G+ p a et se a a = 19y+ nay +1oaa
62 = 29 6ta - 25yt na e 60 aa- 24 gy +2 yt 20aa
ita ex equ tallendo X, fit aequatio per canonem dim generalem
4c
olim jam a me inventum 1(): 1 b = quadr. 150. 
=51. 60) L52. 60
Ubi explicandum 50, si, 52, et 60, 61, 62, habebitur aequatio secundum 
primum ergo in aequ. 1 et 2 tollatur H pr ponendo _=
Habebuntur binae aequationes se secundi gradus secundum ,Y
et 5
+8
rempe
et g- βc. 39) d

## Folio 47r — 00068197:0093
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# No recognised text on this page (recognized).

## Folio 47v — 00068197:0094
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 46r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068197/jpgs/default/00000094.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 114 lines, status machine
B= v0 10 octi . ed. l  
C F 31 16 1321+ 
poAB - auB r P0o a 102+ ac. 109+ e0oi
- 00. 31C
()=AAC = 1. 310
sed hic iluae incommoddea F
occurit: DC  m
rc supersunt literae ad follendu F
00. 31 108
e 3 oq
00. 13∞  =1
yo. 3219
te 8
V1. 32 e sutnuris aufsir tes, t Catsuosima p. A. 32  Sit
IV22. 3( +gia 1b
 uor
(103 +5(  13. 32
quoo quduli asuoa pediade pondiscet q
nisi ipsae L m n datae adhibenaturr:
1 p) )
su irratohalitu
sed supederit in rasionalibu. Ut autem ABC, t non otineat 1000 sin rationalisc 
is tiat  pro 6 D Y. poritur G 
in AC = 0 seu 1. 31BL. + 12, 3291 + 13.3 =0, seu nB1O + 12. 327 + 13. 338 = 0
es non continebit
quia ex assumto 31 = 2. 11 +00 C fi t o+ 1- 31 = -
tis positis eun HC, contincat 0gd.tesimiles, ideo B
 CD et similus et YBDOL
XEA = t +, 0 + ut D AB - h B + 00. 11 1ON
80. HO 00. 12
et  =2.43 et -00 + 1+ 31 =
fit 32: 31 = - 1: 3 et  seu 32= - 2 11 -00,:3
et 12 = 11 + 2.00,:3
An tia d - fi1 NOB + 12 C AB 10. 188 + 0.  sO (Ex mationad  (1i2) = 1GHL A = ve C311
+ M. 3
B = 4 +00 VC +001D
 F 6= 311+3250
pt. B 500 +OG + o0 9C intC. 3. . 3 
 CF 131 +CG + 32 13G +nC- 400u e
lt ne uo ux  re ns
12 = H. 32 00.11 - 90. 31 400. 32- Il. 3 73
tit (23st.   00. 1 3, 400. sit 3
00. 3= 00. 31 5 ( 11. 3131
M fit 31 = 33CF 1. 3 - 00 31
12 = 1.31-, (.3(-00. lB - av. 00. 114 a0. 1. 31
00t qu (00 -31)
00. fl. 31 (- I13L. 32) ~ 00. H. 3l 00 H. 32 (- 11. 31. 3d = 0e ev 31 7 ee 00, 32 4 00. 11. 32
hlb 
fit (32=
Dip. 00. 1531B0 q u - ∞3. H -00. 1 3(. ,. 13 1100. 11. 31
80. H. 31 +00. 01. 31 = 00. 31. 31. Ergo 12 =
sseu 12 -0b. 11, I1 400 +31
00. 31 4 0d. I1-00.00
=. 3E 12 = 1I - an potuus 31 411 = 0V
Y700. l1 = 2. 00. 32
32 = (11 + 12 =31 = 00. 11400. 12 =18. 31 5 i 1Y
11. 12 472 12- 3412 = g0. 1L +00. 12 - (. 31
erto 11a 1412= 31,  H23t
11. (I1 + 1L. 124 1. 31 = 00. 11 400112 - 12. 31
31+I, 12+11 = 31+131+32
D Ea n F 31. 12(+3 1.s)  1. 12 + 1. HI = 31. 3 + 31 32(F 11. 31X 11. 32
60: 114 00. 12 = (11. 9d - 1. 11
00. 11400. 1 = 11 32 F ( 12. 0d - 12. 11
fit 1(2 = 00 t  See (a0 + 31 - 11) 11, =, 11 = 00 + 31
b0 -+ 12 = 4.4 et 12 = 32 -00 = - 11.1
Ergo 12 = = II. et qua AF - 0 fit 32=  31 sed ita A et C habersat cavrcem ratione constante
Ergo faciam, 0c fit 32 = 1  H(00. ) + 3111 F u. 11 4 11. 11 = U. 00 (E l1. 31 = 31. 1100. 3] = 31. 3)
32 = 00 - 31 + 11, 31 ut 12 = 00+31-, 2n 11 = 00, =+ 311
=90 + 11- 31 11+ 31-00
-00 +11+ 31
12 = +00 - H + 31
-00 +H + 31
mon
sit l - n = V
et R- M = A et l - n = n
(90. 31 - 11. 39O + (00. 32-12 32)
 (00. 4 - 1. 3) D H00. 2 - 12. 39)D
 00. 00 Y
00 - 11) 31YC + (00-11) 32fVD = 60-3)1 AO + (00-32) 122D on
sx rstionalibus
debent desrin O et D, fietque (00 -11) 3IV  = (∞ - 31) 11 A ~ et (00 - 11) 32V -(00- 32)121
in posteriore pro 12 et 32 substituantur vatores inventi
ita prodibunt tandem aequ. binae, in quibus solae supererunt
I1 et 31, quae si solvi possunt habetur quaesitum
fin 3C  laor
32 = 00. 12X :, 0OV - IIV + I2X = 00 + 11- 31, 31G :, - 00 +11 4 5, fi 80, 12 A (200 +IIEZ 1) = (00r- AV H LD) (ooru-
D  1A  u e C- v ev 4 00. 11 (+ o8. 31 00 31, n. 31 l31.31) (00-)r (00 ru. 39. H
12 (-00 + 11+ 31) = 11 (+00 - 11 + 31) et fint re X 11 (00 = 11 +31) = (00Y - 11V) (00 + H1- 3931 + 12X (30 + u. 3s)
tit 12 = voX voID A (00- 1 +31) + (11 = 00)Y (0DFU -31) H,: (00 +U-31)BLN. (3 (vv- ut 31): uov FuIL 3)
fit n. 31A (00. 00 = qu (11 - 31) =, 00. 11X (31.31 = qu (00 - 1) )+ (N - 00) (11. 41 - qu (00 - 31))
esset 00 -31 =f 0 sen 31 =00 firit 00(1A (200n 11) +1 - 00) 1 (11) 31, . (11. 71X = (D (2. 00 - 11) : (11)
Jeu 00X (2.00 -11) + 10 (3V (11 - 00) = 8 (200 - 11) sid hoc non licet
 si esset A Nec lient facere 11 = 0D, nam eonnescenst V in Calculo. Nec licet facere H = 3I, nam fircet 32: 31 = 12: 11
(sit 11+ 31 = d seu  31 2 = 11. firet 00 (iA (00 -2. 11) 4 (11 = 00) V (00 4 212)( : (90 + 2. 11.)(1) = - 11 (00 - 2.11):00
Si esset 00. - 11 + 31 = 0 fieret 12 = 0, et  sed hoc qua
Si esset H- 00== 11
si sit 30+11 = 0 parum lucramur, ut et faciendo 00 +31 =0
EZsR.d ita ascenditur ad _ quod nolim.
non licet quia fieret 1-00
1 = 00. et ubique depressio fiet per Mr, lasleed ete 11 et 31
contrahetur calculus tamen haberetui repro si fint H
habetur depressio et fiet 211 =
ascendetque calulus non nisi ad gradum secundum  dutque  31 = 21+ 00 et fint -00 A (2. 00 +11) + (11-
andes casus impossibiles a
firra ut et tint
Ct 00 + 11 = 311
ot Zandem habebimus: -6.00. 00 X - 3v0. 1X + C. 11. 1V FocinY A. 00.4Y =c. voV = 2. II. IIX + +. 00. 11 +00.11X 200.00X
sea (2A = 51) 11. 1 + (Z, 00 X = 3. 00Y) 1 f +: 8. 00. 00X + 3, 000oY  0. et fiet a  2~) Hc-∞) L ( 1to
Sed non video quomodo potuisset deprissioe obtineri usque ad gradum infimum Li posset amen ad huc tenturi fingendo
sdae in intoris t  a piict iberum Encdi in casus impossibiles. Plud tanen adiue habemns
possunt esse affirmativae vulnegativae pro arbitrio
medium contra imposibilitates l valore ipsus i1, quod s ad huc linasuptias iquptio, et id co 3 assumere
VO +is
quod in ABC