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# Piece: https://leibnizlegible.com/api/records/397/text
# Catalogue record: 397 — [Aufz. zur Reduktion irrationaler Ausdrücke]
# Incipit: Quaeritur quadratum ipsius √xz + √yz quod sit av + p√xy erit hoc = xz
# Date: 1676
# Akademie-Ausgabe: AA VII (assigned, not yet published)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=397&katnr=57973
# Work: https://leibnizlegible.com/work/00068199
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folios: Bl. 124–125 (from the shelfmark LH 35, 13, 2c Bl. 124-125): 4 pages, canvases 247–250, page ids 00068199:0247 to 00068199:0250
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068199&page=247
# Lines: 301 recognised on 4 of 4 pages, mean confidence 0.76
# 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 124r — 00068199:0247
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 125v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000247.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 37 lines, status machine
ab
Ho
erit hoc-
quaeritur 
quoD sit
abic ter Vancresa esse quadratum ipsuis Laque + syque. S possit esse a V +pIXy
E erit XZ F yZ + 2Z (py et firt p = 2Z et aB = xZ + yZ
etiam
videamus rursus an a +PX pexu possit etiam esse quadratum ipsius gtru. Esto
trit ap + pxy E~ 49 +hh Xy F rhqexy et av = q9 hhX)-rret p = 2h. 9 = 2Z
aa
AyZg
Ergo ut sit (xZ NyZ = _q + h sxy a erit q- 9Z et aagy 2  Xy = XZ + yZ
th 2
ac
Ih2
tt X
aa.
sad oideo que f Jis fieri rationalem, nam  in valore ipsius y pstist ddividi numerator per nominatorem
Sed hoc est frustraneum nam Numerator potest dividi per nominatorem
et fiet  tsd +g n =
n
9 +2f9
HFmm + ggna+ 2fgmn = (fn F g) Am + gn
nn
XZ + Vay quadrando dabit XZ +ay + 2) a28y quaeritur huius latus
baq Lay pa 2. Tazpy, haius, quod est pp + 49 azxy + 2pqe Tarxis
a
ar
an possit eisse quadratum a
aar
Erit ar = pq et pp + 42aZr9 = XXt ay se P Farxy= px + ay
 24 2 1444
ppq
D= 502 +a
a 11 (00- 31) A. 40 = 3
- 31(.80-11) V

## Folio 124v — 00068199:0248
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 125r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000248.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 69 lines, status machine
X = H~ 10D +, , OD. Ducantur invicem XH, X+m, X+n, fint X + lXX + ImX +
X3 = 100 + 3. 100. 112G 10D + 3. 100. 11290D + 112O P LOD
(Lamin) XX Llamtn), 100. 90 + 2. 100. 112 E LOD + 1120D
lmalu tun, X = latlnema, 100  + 112 f0D
XFl, XIm, X F
Et ordinando fiet, X H Xx + luX Flum =
100. 100. 1004
3. 100. 112. 112 907
+
00100. 100p es 4
00. 112. 112p (9
atC
00. 00. 100a pe
un
Compendio sit lam + n = p
Ut shnando fiet
+1C4 GB
3∞00 +10
cntlntum = a
Quaeritur A(XFC, Xtm,
+ 106. 112G0) 12. 114 +12 (d5
d  irE
+ 0), L3100100   200. 100 mpE
1. 112. 12. 1123
quae formula aequoenda quadrato a ad O + aq
coefficienti
197)
a
Forw. oolmn
Et pars rtionalis valoris ipsius X3+ pXX + ap X + amas partis irrationalis aequare oportebit at
parti vero rationali ad + a8D. Quod fieri poterit explicando C et D) per unam
et 21 = + 123. 11. 3(a + ot. 11293: 00 a
0.00. 
Veninmus jam ad partem rationalem. ibi non potest esse Q, nisi in 40D, Ergo ut destruatur
D debet esse 100r1o + 3. H 108. 00 +3. 112. 10. 30. Ergo fit 100 = 100 112 ed 30 +. 5 ( Y. 31030; 
quos duos valores aequando fit 10. 10. 30. 30g = = 3. 10. 30. 00.00 et 10. 30 = 6 - 27,00.00
superssupersunt aequandi termini  in rationalibus ciefficientes ipsorum e, e et G
ope literarum 10. 30, 15, 31, quae ex Itaque ad ad hoc, et ad tollendum terminum e, opus
est aeqq. quatuor, et e arbitrariae sunt etiam quatuor, 10, 30, 11, 31. Nec videntur
haec nimis prolixa. Scribamus ergo omnia distincte; et primum partem irationationalem
300 100. r +3. 00. 00, 100. 112. 112 ~∞ + 2. 00. 00100. 112. 112 p 4 + 1. 00. 04,00. 00. 112a9
+ 1 a 10.3]2112. U2 W2 a8
+ 1. 11. 31. 112. 112. 112 aa
=1. 00 99, 98, 80, 00 a 8 - 1. 00. 21,90. 0087d
F +1. 1V. 30. 112. 112. 112..
11 30.
( aqu0
1. MF 
fit 100 = - 10. 30112: 3.00 et 20 = 2=. 10. 30. 1 p + 10. 317112 a, : 00t a. Et  X 00. 129 + 11311N a.
octa
In parte rationali habebimus, 100. 00 +3. 112. 10. 30, 9=0 fint 100 = 11, + (- 3. 10.30):00 sed prius erat
2
100 = = 10. 30. 112 : 3.00 quos duos valores aequando fit 10. 10. 30.30. 112. 112 = f - 27. 00.00. 10.30. 112. 112
seu 10.30 = -27. 00.00. Ergo (.+ 100 = 9. 112. Hiac ex aeq. l. fit 27 (11100. 00.27. 11282700.00. 112 =8
et ex aqu. 4 fit 81. 112. - 3. 27. 112 =0, ut oporet.
tar pe g
ru quae rui
communem porrit poterit autem  esse secundi gradus, a son a 00 a t= obi PR +000 aroB naa
et ita equatio ivessi inter coefficientes trrationalium, dabit valores ipsarum 20, 21, 22. Superest ergo
debet autem e esse gradus simplices, et poterit esse 204 + na, ita aequando coefficien
tes irrationalium, habetur 20 et 21 sd et ex irrati habobetur  3
2 = 308)+31a fiat 3. 100. 0d = 112. 1030 = 0 seu 100 = - 112. 10. 30 : 3. 00
112ob:6
in se ipso et
11.
i
extractione
Xy = ab
Fd Xx = ab

## Folio 125r — 00068199:0249
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 124v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000249.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 99 lines, status machine
3 +00 H
1100 8 - 127. 9000. 1dIL. 117
3100.12957 =27. 00. 0,9. 112. 112. 112..- +27.10.312112 112. 12a GQ + 27. 18. 3. 112. 1 A2a 
3r00. 100 pGG-
1. 00. 12 p O) = (r s 12 . 5d
+00. 100 ap E
-0d a 
-00 a4
Erpolabubr
3
t
54 H. 30)
u 431 00. 00, 00. 112. 12 p...
1∞, a0 _. 00. 112. 112p... + 1. 1031. 0B. (12.12pE + 1. 113199, 12. 12a a p
130
11.
 uti nonin semradato 9. 00 00. 00. 12 a p
sus  Lu2 00AOL 2. 12. 12p PG 4 Z, 17. 00 (1 11. 112 1 ap e
1t 3. 2eroro 3
000d 1 1 - 41.  40. 31 112. 12. M a .1.  g.90
quit
simplicu
Tesset aliqua +
in
valit Rihilo
1. 000000 e ombun
an
1o 10204 00. 80. 11249. . .- 1. 1 v0- v000. 112 aaq
quods emo bs pl 2 1. 31. 11242 W aqu. , 19 c. vc, 12 124 c
Bet Vrent con
mtii sequales nihilo porent. quo
oporterot esso
simul Pevalor ipsir
er Aequationem qua destruitur QE. habetur (18. 31712, sine 11 et 31. is substititatur pro eo in aeq.
epotest suppleri
qua destruitur 2. Et iac aequatione posteri qua destinitur e, pro valore ipsa 11. 31, substibuatur
hujus vator ex aequatione qua destuitur absolutus, et, ex aeq. qua destinitur  sit aliqua
in qua reperitur 1+31, simpliciter Ergo quin habe sine a et 31. Quia ergo invenimus
tam _ ∞ 10. 31 + H. 30, quam 11 +31, itaque habetur  1 et 31, simplicile et rationaliter
etsimplit per 10. et 12. Sed nunc non est opus ips ipsis 11, et 31, sufficit quaeri
haberi 10. 31 + 11. 30 et 11+3I, quoram ex  per aequ. ex e habetur et H1. 31, sine 31.
quo valore  et ipsius 11 +31, substituto in aeq. ex absoluto, aberunt 1 et 12, et
supererit aequatio, quam soluum ingredientur 10, 30, et 112. sed porete. 30 = 2700 10
ata omnia reducta erunt ad 10 et 112. possunt autem aequationes efferri compendiosius
 10. 31 + 11. 30 vicetur B131 _ 11. + 31 = 113 et 10+30 = 107
Et ex aqu. qua destinitur FE fints 127. 00 I2a + 5400 p - 18 00. 103. 12, -, 103 131. 112a 
ip aeg. qua destruitur a G fieter. 11. 31 HE 4 00. 131 2p + 9. 00 p ar g. ∞. 118. 113 H2 13. 1311i2 a  18. 9. 1ot p id
v0 
Ex aeq. qua dessimitur absolutus, fir 1211. 31112  aap + 00 lmn - 08. 113, 12 aaq - 113. 112a
ex aeq. 3. et valor ipsius 131 ex aeq. 1. fiet aeq. (4)
Ad has ergo tres aequationes cuncta reducuntur
in qua habebitur valor ipsius 113 His valoribus substitutis in aequ 2. habebitur aeq. 65) in qua
3
Ex aequ. l. habetur Vator ipsius 131.
in aeq. L. substituatur varor ipsius 11. 31. 112
erit valor ipsias I1. 31. sine 11 et 31. jam valores ipsorum H3 131, 113, 11. 31. substitutis in aeq. 3.
ine
(repertis in aeqque 2, 4, et 5) substitutis in aeq. 3, liberabitur ea ab omni H et 31, et habebitur
ditur aatem jam
depressionis ver
aequatia quam ingredicatur i tantum 10130 seu 103, et 112
facilius qaeratur, 103, an 12, et an non posse forte depressionis malioris resolutiors.
commodiis causa cum fructu aequatio resolvi divelli possit in duas. Utcunque sit, modo
arbitrariae 103, et 12 in ea non plane evanescant. sufficit ad nostrum scaepum, neque
enim refert quam difficilibus aequationibus inveniantur. 10, 30, 1, B1,  etc. 2, etc modo
inveniantur. imd etsi ad imaginarias deveniendum esset modo
nihi proveniat
od l +S 13 Vaag f 13. 12a F00. B3I 12 app 1 9.0o app - 18. 00. 1E. N3 app H3 13L ME nap ob. 103. apque to 1. 3 1
absurdum.
lti valores inserantur in aeq. , quae commodiis sic scribet ( ei. 3 (2 7.  a - 103a () + 131 (00. 19p - 113. 12 a) + 9. 00p. 1900. i. 1p. B. 1619
Exae, 3. II. 31. 112 00 ma + 98. 12. p2aag f 1
aet habeb itur aeq. 4. seu valor ipsuis 131 sine H et 31. compendio H. ex 3, erit 11. 31. 112  00 hun L 113. Iaa (0F, et
ert ou octaas, = 06 f = 00q + ATk et ex aeg. 1. 131 a2 13l 69. 00. 013p : 203 ent (023 = 54, 00 - 18. 103. H2
et 203 = 10Z. I - 27.00. ila ex aeg. 2 fel 11. 31. 112  203A F 3~a g + 131. f 112(B0 0B ap - 113 693 ap) - 180d 11., 1BpO3ap
+00h = 0, ubi 9 = 27a -loa A 0o= 90op 1034. itaqe ex aeg. 2 fiet 06 un 2osat 113. 12. 203 fg. + o B 3e pi 13 u, 
19 112. 113, 203 ap F eo 203h =0 seu 113 n ou 203mng eot. B23RapP, : 12. 203 112. 923a p L compendo B: X 113
et ip aeq, fit 11. 31. 112, 2030G + 003. 023 pp - 08. ∞. 112. D: 2 - 18. 112. 203 00 ap D:  +00 203 a h20 Ex aeg. 3 est
11.31. 15 a P = -00 matod p   fB: A, unde duos valores ipsius 1. 31 qui aquab aequati habunt aequatione (6) literam ab H1 et 91, in qua
Jam notae pro compendio assapta sunt explicandae. Explicentur autem primumm et latm quae involonnt 112, ut o2 3 et F et B  in X Asspegit
112 ad spartum usque grad
saudeo tamo ut FD. D
lee
orene
n
2 cb qui s fieut 20y  14,00. 3pu, 103 -5 4 1 1300- 8 00 +0C    te ps
astendere ad gridum stum sed ardeo opimam elont fatd ita tils q et inen tat 12  in .  
trr anque ae se prteratu
Fiat a1 + 3tL
nocete nequit
et Agt E i2M = libo lmn. acp. IX
du
Hacd
27 - 10+ 20, L eX 4P E
27lmn 00.-905 arp t00 . 0+ 50) aagtoro) ed lmn
+0demn = i op 1. fint 11. 12 4 3. Moate lnc inoener iacoghitaru
qui
10
LH eH
o

## Folio 125v — 00068199:0250
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 124r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068199/jpgs/default/00000250.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 96 lines, status machine
y = + (_, X+L, XIm, Xtn). quaeritur extractio.  productum est a X 1XX +m X + lma a quandum quadrato
sit x = ( 100 ( + 112 (O. erit XX = H1OB.vo GG + 2. 100. 12 BD a N -100 + 12  OD, ac
+II2. 112 BD
compendio rmen vocetur p, imtinama vocetur aque
310o. l  +3L 18R..
Quabratum, lum quo aequatio institui debet, sit excitatum a at 2O ta ZD, nempe aa BO + 28 1OD q
+(WE
px 11 1agu   + lmn = = aa  aequ (Grs
troa W0op
l+ 3 100.in9BB ∞
os
400
P0VZPOY
A qui
Eoosntg
+ 05. 100a9
00
1.122OD) _F 1b ag
-
C
a
+ cc8dua
 aa
Q
 20 a8 1()
P. tatos
Ex irrationalibus
aa2
Fiat 9 = 102 +ua
007 = 308131a
O5 = 10. 30 2 9. 10. 3 (e 8 1979
11.30
004-204+4a
300. B0100. 100. 112 ZC + 20 u s ev. 12 p C2 +00. 80. 0002 OVaq ~ _2. 0020  + 2. 00. 1aa
1. 10.30. 12. 112. 112 Q 1. 10HI J. 1. 112 a Q + 11. 31. H12. 12. 12 aa
fit 3. 00. 170 + 10. 30. 115 =0 vel 100 = 112 1(- 10. 30: 3): a  f20 = 2. 00. 100. 112P 208.4 = 00 aq
200
11. 30
in Rationalibus ot evanes cat e3 fit ve 00. 100 43. 12. 10 30. et 100 = 1 (3 0 ) . Wu. 31G ,112 a
Ia 1131. 113 aa
quos duos valoras aequano fit absundam, ha3 Ex Btionalibus Talisergo Hypothesis non succedit,
184058 + 0op 4 + 0. 100 age + eo lmn fat x= A  +4 +On
3. 1V0. MIGOD -15. 100ul 10 to Y + J. 1 3L vort a GQ + 3. u3 s 1B 12 aaq in p halebimus 8 et 3 0 tor
1. 02 M2 8
sicesit in tat Aa latus t + l2 Y(D tytu +2t4 SOD
X = 85 + AVOD asentit ad 56 . 1 ti D sCD
2401
si Latus SVZZ + 7h+OD, ascendil ad 513 775B, (t Y 1GG0D et Si B131OD.
ut ita demum institui potest comparatio
sitonditur ad sextum gradum explirando omnes por eandem; Euso opus in parte rationari 7
7. Cgq. in irrationali ryque 6. summa 13, ad literae autem sunt 1 20, ob 10 variantes simplire
caetem prom et ino liteae ob 66 sunt tres, ob 7Ihes, ob 0D, tres sunt 9, sb 5.73, 4.
ob 73, sunt 3, in tumma 16 ex quibus etiam binae videntur superfluae
deberent. +46 +GVOD, et 46 + +D habere divisorem communem
s
N7 = +95 + qYOD  sZ = 4 + t (OD
+m
5 =. 103 + +07Q. 192 7 - . 107+ . sDE 3 -. 109 +. +94
517 in S 3 in 23 =  sst 7733, 517 = X I 5Z = X + m 7B = X + n
debent aequai. S  - l NZ -m 23 - n Hoc maxiinc succedere, debere videatur
et tamen non successit: ine si N  = NZ -mtl hit _ 7 = Z + 
 
et p et pari jure S Z = =B +H- m fiit s = Y +H m
S - 7 =   (it  addendo s0 -3 - l-  +
m. L
et pari jure 6 +3 = s B = n -m et 2 -5 = n m Ergo FC s - X = lam _n - m - 3F
= lm
Z
  108+ ADQ in a1 +OO + 12. 15Q debet esse, 11 = 31 - 21 et 12 = 2. 32
XL+ IaKA CaX + CNNEX+N =21 +C. 2 194 ita productum fit rationale. Sid hoc fieri deber adhuc semel
DO constans et si O = 10X +M a ut D = 20X +2 a feret 11. 1 - 12. 20 = 0
X taLLI ax FM2a X + AnEXXL (Q ax HL B. . 32a 3(31. 35. 10 = 31. 20 Ergo 10 : 20 = 12: 11 = 31: 31
10X + a f - 1
sed pari pne ficot etium
fiet H 12: 11 = + 31: 32
30Xtta, 20X + 2 a, 30 X131 10X - a f- 1
c
aXX 12210.
I1. 2
10XFIA 20X +21a IOX tUa, 30X +31a
10.2DXX + Z 1WX L L 10 sXa t 1N, 1dXa =na 10 exa +na -10 tx a + 11a
10, 10XX F2, 10. 1a X + 11. 1a4 20 20 Xx +2. 20. 2 ax + 21. 2 a 2 0. 31. y. ( X = + 2. 1511 253IX 411. 4. 3
to Lax La 10X 10. 10- 2040. 30. 30 X 2.10. L1
110. 2021 3. HX + ~191I. et H p 2. X +2.
deona, XX + 2. 2 ax + _ d. aa, _ Ax + 231 ax + g1 21 ai test et 3
u
2010
Or  2. HiloaX+
1.3
3
10.10. 20. 20. 3030 X + 2 1  +2-10 1 21
ex
a
XX+ uax + 12aa
XX+ 2laxtmaa
I1. 2 auX