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# Catalogue record: 60614 — [Aufz. zu Gleichungen 4. Grades, zur Integration irrationaler Integranden]
# Incipit: In fine paginae 3tiae pro <-> usus considerationis similium, seu ejusd
# Date: 1693
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60614&katnr=57530
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 340–341 (from the shelfmark LH 35, 13, 1 Bl. 340-341): 4 pages, canvases 675–678, page ids 00068196:0675 to 00068196:0678
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=675
# Lines: 386 recognised on 4 of 4 pages, mean confidence 0.82
# 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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## Folio 340r — 00068196:0675
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 341v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000675.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 115 lines, status machine
sin fii pagine
olantus usus osides
praevideri potnt caleutum runque non succede
X + aakX + b = quadrato C sit X = sion su Y Y2oVYE + S3oVHI
seu opendio X= I + 13+ 16. Sit fomaula quadhanda 12(hq + B 1E6 + 23a 2,8 + 000B+ NI = C.
tiet es- igo 10. 20V 2. 130. X30B At 24
2. 900, EoV
2301. 20.30. 2,30.296. 3D
00d 600.. 2, 0g 120
120, 10. 21 V.
2.19 210. 20B A 2. 120. 130. 1oV 12E
sit  oi
Sit t = 10v + 1
c = 1t2 + 146.
2. 000, 230V
2 120, 130. 1
2. 000. 23a
2 (2) 20v + 21
6 = 30 V + 31
O = 14 Htrmob 8 = 160  16
Es D = 50 LH8 + 70V+I sit  XX + Aba = 00D
GB ru gotbi ai guat eto XX + A C= D8G: P Ex 8 explicalo fit a+ 22 15  intrdd 24ost.  B
t) = 10rovr ao ebr vut  B
2B = 20 3o vv, 10 11 V. + 231
00- a0 qe trirt e lib
1BD . 43.7. B + 10140 50V
1. Ho 70b 30. sAo. So V
i ng io  sis ode (H6
u s
2. H. 3. 10. 340. 50 p
2. 50290 3 40 G6 30 b (. 2 110340 1 G6
6010. 20VV + b 20V
00. 20. 30..
Geb - 115. abtriptro so (B0. 20. 21 V
2.001. 130
a
seu explicando
(t6 B 1070. 140 vD + 10. 50. 10. v7
30.70. 3901
2103052. 140. 340..
11. 70. 1e 140 v
31.70. 340
10. 71. 140.
30. 71.340
2. 10.34. 50.403)9.
3. 11. 30 50. 140 740.
U1. 71. 145. v.
31. 71. 348.
2. H1. 31.50.190. 340.
2.70. 140.340.
27]. 140. 340. V
ubi comparabimus primum
irrationales, inque eam rem
adhibebimus 140 et 143,
quippe in aeq. 9 explicanda
non recursuras
4110a+190b4100
Reducitur res ad hanc fauam ob tu ab 
et 220 a + 202b + 21 ab + 2104 201 b +200 0. jam ex
sin a, et b siuit 140 - et 340, desunt 110, 101, 210, 201, unde prodit aequatio
+ 100. 100. 230. 220 X
f 1. 120, 120, 200. 200
= 2. l00. 10. 290. 102
jam 120 _ 102
+2.190. 102. 220. 220 bb X  + 1. 102. 102. 230. 220 b (14
+ 2, 120, 120 200, 202
+ 1. 100. 120. 211. 211
+ 1. 111. 11I. 200. 22
- 1. 100. 111. 211, 220
= 1, 111. 120. 200, 211
( 2. 102. 120, 200. +20
2. 100, 120. 292. 220
11 110  10)
Spisinl 1050 n D 30. 50 5 (2. 70
+ 1. 120, 120, 202. 202
=2. 102. 120, 202. 210
4. 1. 11I. 111.. 202. 240
+ 1. 102. 120. 211. 211
= 1. 11I. 102, 211. 220
= 1. 111. 120. 211. 202
220, 202 211 210 201 280
avso 3Iso 1 osstrmatu terminus itta
100. 220 - 120. 200 = - 2. 11 2050b0 + 2. 410. 21.50b0. quod si hoc ponamus =0 fie
oy. 2
11. 20. Sed
et praeserretur
Eoc non placet, quia ita  et _ 2 revera quoad inefinito non se ab o
praestatentare, au licual tollere terminum (t. Nam maxipus etu
ubi et  C
foirt et 102. 420- 470. 20 = IL. H WX CCe - 11I, 11I. 202. 220 102. 120 211. 211 4 111. 102. 211. 220 + 111. 120. 2111. 202
Si abessent 1I et 211, id est si 70 et 71 essent 0. fienet
MI: mH . 1 = n BV1. 211 4 p.11I 2u  Ex aq uafica, D = (s ivomo-MeND + 2100. 1o2. 220. 220 b + (2 (,oL. VOn NOnOE b
+2. 120, 120. 200. 202
 noGA
10229- (26. 262
100. 12
+ in 102. 420, ut. 1H + n. 102., 20. 21. 211 F p. 102120, (11 211
= m 120. 202. 1I. M = n. 120. 22,. 21. 21 = p. 120 202. H. 21 et feiet posset deprimi aequata, fiereta
100-  100. 220 -1
102. 220 = 120. 202
+100. 102. 210. 220 = 102. 120, 300, 22D
+120. 120. 200. 202 = 100.120, 201. 230
Non debet autem s haberi
+100.220 102. td bb = 0 pure rationalite, seu
= 120.200 = o. 202 l-.
xoinciderent B
 se quadratorum (vel sattem quimdato Nane
parationes ad eodem prodeunt put 140
Caeteruum dubito adhuc, an sit consultum omitee
peculiau peculiari his seo culo
qui 
lo opamute aquec ue upentes es
parte rationali, cum Bodlatibus ex rationali. Praterea mimiuns
sed operae pretium ait aequaves efficiendas distuncte describere.
sen 1e0 vel 340 plioqui 100 ut 36

## Folio 340v — 00068196:0676
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 341r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000676.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 115 lines, status machine
ses A irrationalit partem, et  ea v fit 1on1o a 303I8 = 1. 70. 190. 240 +  2. 20b0
propterb V fit 10. 140 + 50. 340 = - 2. 38140. 343. + 2. 20 B0) po d, (P fit n. 140 +31. 390 2 - 2. 31. 1403 40 + 2. 2. b0
in lesimali prute prosic. V fit 1e 1o 130. 70 = - 2 .  190. 340 + bB. 2 1013 4 At V fit 1, 43+ BI. 3 45. 3 -2. 1 H CC Mo. sH +  bo2l. 1r 31 + ba
proper v fet - 2. 704. 140. 3 40. + 2. 20. 71 b0 - 270 31 140. 340 + 2 2t 10i be (, 0. 217
ex aq tetr fit  + 2.10 31 . . . . ( era + =. 1 3 50. . . .
Ex 1013 fit (.  2. 10. 3050 - 2. 78, 190.340 =    20 b0 ror3a - 2. 20.ib
Ex 2 et 4 fit
x 6 et qe lit
190370
d5
2. 11. 31. 50 - 2.71, 140. 340 = 64 + 2169, 1131 - 2. 21. b0
20.50. 10130 b0 = 20. 70. b0 = 2. 10. 30 5050 2. 70. 70
20. 50. b 10+30 bo . 2 20. 70. bo 17 21. 50. 114 31 bo = 2. 21. 71b0
f. 10. 30.50. 50 - 7. 70.70
C fei
7. 11. 31.50, 50- f. 71. 71
e
 20 ((B50. B = 2. 20. 71 - 2. 21. 30b, b0
10) Y 20. 31
140340=
per aeq. 5
21.30
 2. 10. 31.50.50 + 2. 11. 30.50.50 - 4. 70. 71
Htabemus ergo tres valores lipsius 140. 340, adeoque dua aequationes, a quibus abest absunt
140, et  quibus si dertia adhuc accedat, ab his independens, habetimus quae sitam
sufficientes aequationes sine 140, et 340. Et quia ope aequ. 1 et 2 invenimus valorem
ipsarum 1 140 et 340  pagina praecedente. jamque per  s
83 et 4 conjunctas cum 1 et2 ad huc novam aequationem obtinuimus et novam stidem
per aeq. 5. Superest jam ut novam adhuc obtineamus per aeg9. 3 et 4
inter se conjunctas
 +byl C rhc
Cx + 
XyaB) p
e compendum pro 12, 307 ag. 3. et 4. sembamus ax + by = 3 (e rmy =n
 pro 140, 34
talX+ bly = et
alXX- amyL N’a
argep+ bryy = er+ n5 su bt an
l5.. +45..
ar+lS = br + 15 d a - br = m - 15
fit r: 5 = M = l, :, a - b
it yy = 10- a1
 fit tarXX + bY yy + ar xy = er +ns + par
ar + l5
p sun
y etva.
Hoc suopone nocanda
(TY = mC -b
b
seu a Xx = gy + 2 xy = Cr + n S+ ar+ (sBpp
 sit s-1 et Y = ma l 5
et fici XX + 9y + 2xy =
ry -
seo
X = S me-(C + an - bn + rp an. Bl Hm C le + a - bu - 2p am bl
y = Ime - lC + an -bn + 2p amabl = Lme. le an -b 2 - 2 p an bl
et X jignifiell 140. et y fignificet 340. patel
itam el fit xy = LCn amcbl - m - n ab - 40340
i vel 8, vel 9, ve
hic iterum haberi 140 vel 340
=2. 9 70. 1131 50 Xy ( f 10.70, 2, F 3, b0
Un rkab - Cn anebt
 l  
na - 31 q F ffser 10. 540 =
 inubrru o sepliolle. nve t 2
initio hujus paginae,
_ NGC ep. fit C3 = - 2, 10. 30 50Xy F bo. 2 10 430
X=  se  nt) Fugo vdia
ser
seu XX+ ryfrpus = em - cl f an -bn 4 a nc a l + a b1  s
sit eX + 9y + 2Xy = hq et xx tyy -2xy= ht
fet x = 1h9+ iht et y = thq- sht. Ergo
finc si (1.). et ((2
bltam10. 31. v. )
n’a  etfiest
bI - am
XL9 = XX +yy + Exy = em -c + an-bn 4 am- bC 2p
am = bl
n3 = - 2. 11. 31. 50xy + b0. 2t. 11431 C71 
Ny- B sC l.na fibpanXu, = + 2 l. 7 1 1090. 5CXy  - 1.31. 20. 101 5 a b0
bt am
C. tuil t n. a. b + a nil  l. a. tB tip
a. m- l + l. a- b
am= al) + C+ ad - bl
Sam- bl
tam- bl
5.7
am - bl
X. ideo 
non sufficit, sed pretius omissa
extractione scribendum
Ax = 1n umebt- sim - nab 
av. 34
(2) am - 6l~ Bet
et hoc explicato, tandem nova extra.
chore habebitur X G, pova
recg demum erit aequatio ab B et
Vage liberata qui novus
valor cetsi irrationalis, si cum
aliqui ex ipsis 8, 9, 10, conjunga
tur, habebitur aequatio nova.
Nempe ex valore 10 et 8 fil aeg.
prima, ex valore 10 et y aequ. 2da
ex valore et nova inationali
et valore 8 ag. 3tia. Et per
has tres aequades liberas ab, 40
et 340, psa habet praestabitur
oparatio ratabsolvetur.
in ag. XXyy seu 140. 340 = 
explicita pro xxjus potent substitur valqu
 psius ortus exvaloribus desu invicem ductis
ita habebitur x9 rationaliter nova aq
quae esse qui eut varot 1) qui junjendus ipsi
 o dite t a e ete a ea tut lee
Hab

## Folio 341r — 00068196:0677
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 340v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000677.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 80 lines, status machine
ppoliia trausici posse
Hoc loci tamen videntur omnia minus
1. stit (t
(31.2. 20 A2 21
4
Ar  Ex aeg. 3. fit 180. 34d = 20. 1a+30 (3) 2t. 1r1, +ba
2. 10. 30,50
faciamus
110. 2. 1 -72. 20
50, 10. 3( - 11, 30 690= 50, 10. 31 - 11. 30
han
equ. 5. fil. 10. 340 = 10. 21 + 11 20 + 20. 31+ 21.38
2. 50. 10. 31 + 11. 30 40. 30ba
benique ex
Unde ex aq. 9 fit 1031. 2031 + 11. 2030. 31  10. 1. 21. 30 + 10. 21. 30. 31C vel 20l1)1030,. 11F31
Ex 10 et 8. fil 20. 104 30. 10.31Y1130  10.30. 21. 10+ 30 + 10. 30. 20. 11431 fit 20
,. 16. 36 100
10470. 10. 31+ 11. 30 - 10. 30. 114 31
Ex 10 et 9 fit 21. 11431. 10. 31+1130 = 11. 31. 20. IF31 + 11.31. 21 Fo1 50 fit 21 C) 11.31- M F2
possemus jam jungere egq, 13 et 15. qui ambos leges homdeoplotorum observant, 11131. 10. 3) + 130 - 11. 31. 102
ed praestat
tamen alteratiam harum Conjungere cum aeg. 11, nam eo ipso quia sic proditiae quatio
quae si non observat leges homoe oploton, necesse est quaedam in ea evanescere, ut ad homveopu
F homoeoploton redigatur; cum, quae hc proditura est, sam tum sit simplicior
quam quae ex 14  13 et 15 orietur, 2an ex 1 et 13
10+30, 11+31, 1031 +430 = 10. 30. 1143,  = 11. 31. 10F30
fiet
sit 10+ 50. I+ 31 1931 +130 = 10. 30. II431 + 11. 31. 10 + 50 quoe aefontio est honocoptotos
quod si fiugeremus 1 10430 = 11 + 31 ifieret 10. 31 + 11. 30 = 10. 30 + 1. 3) e ex 1 per1s. fere    3
jam quaeramus 140. 340 ex b et 7. fiet 140346 = 2. Y18.31 282] - 10. 14 21. 2 - 3034 30 20 + 11. 30. 20. 21
50. 10. 31 = 11. 30 et fiet
ubi numeratorem explicemus, et primum pro 21 ponamus 20. 11. 31: 10.30 ex aequ. id
10. 3(. 20. 21 = 10. 11. 21.21 - 30. 31. 20. 20 + 11. 30. 20. 21 = 20. 2
10, 30, 11 31, . 31 + 20. 1 = 10 31. m  B. M1. M. 31. L. 10. 30. 3. 3o = . 10. 11. 31.31. 10.3∞ - 1. 15 + 10. 30. 3031. 1111 - 10.30
10. 10. 30. 30
Ergo ex numeratore epharento fit 20. 20. 26. 3,  1. 3130. 30, 10.30- 11. 1 L
10. 10. 30. 3
aliter 10. Z0  H31 -30. 30 + 30. 30, 1. 1- 10 50
aieoque fit 1010 =  10 
31. 20  H1. 1.28 31, 16. 11 26. 31 = 30 30, - 2. 40. 3, 110 30 1 M, 1131 3030
A13. 3B3, 10. 30 - 11 a
10. 31 = 11.3 antequam
a contrahi adhuc amplius calculus in aeg. 21 possit,
eam quadremus, p  lollamus inde prius  et it et 31 quorum valores quaecenus ope ce 9g. 17. et 18.
ex ag. 17. fit _ 31 = 10+ 30 - 11 quo valore substituto in req. 18 fit 10. 10(410.30 -10. 11 H(11. 39 = (0. 3d + 11. 16 13 . 1 1
Contequam   Uet vero apparea
seu 10.10 + 11. 11 - 2. 10.11 = 0 seu 10 -11 = 0 seu 01 1 = 10
 E Et similiter 31= 30 ita in aeqqu. 6 et 7. evaneselrent 140 et 340, nec invenitentur
sed arbilrae sed semi arbitrarcae manerent, facto tantum ex ipsis dato; et
lunde fiet  10. 1t0 - 1) = 11. 10 - 11 
ex aequ. 6. 1t fier (explicata 21) fieret 10.30 - 111. = 0 et 11. 31 - 30. 30 =0, ande ob 22 et 23
feret 30= 10 _ utrobiqus, quod placet, quod ita non prodit nimia determinatio
sed video jam in quo peccaperim, omsomisi perliceti ba, sunt. ex XX + ba
tende falsae aequ 1. 14, 15. itaque rem resumamus
 20, 10+30. 11. 31 = 10. 30. 21. 31+31 + 10. 30b9 et fit
10. 30. . 11+ 31. Et ex 8 et 10 fol 20. 160430. 10. 31 +11 50 = 1033, 10021 + 103. 142, 2
1 sd ergo nunc in aeg. 4 suppleo
sed quia in C3 et n rursus labet 140.340 et fit 21= 20, 10+ 30. 10. 31F11. 50 = 10. 3a. 1+31 3: 10. 20, 10430- Quos duos valores ptru 21 exegg. 11. et 13.
ompe ex
ideo haec aequalio aeque equemue inter se, et fiet: it. 3e 10r 10. 11. 30. 31. 1ot 30 20 - 10. 10. 30. 3c. 101 30 ba = 10 10. 11 31. 10. 3( 1130. 2
1on3o 31.1 0 130 f. 1010,103B, 1131 - o+ 30. 113) 1 t 38 .
- 10. 10.3030. 11+31 29, et fit 20  (8, 10110. 30. 30 1at30 ba
10.11. 30.31. jer30 +. 10. f6. 3630. 1143, = 16.36. 104 30. 1143,. 10314430 10 vel 9 vel 10
Ego per 13 et 1sfit 2(1) 16. 3b ba 00. 31+11. 30 - 10. 30. 112)
31. 20 = 11. 21=10.21 ∞ 30.20
tollendo 140, 340 ex
fiet 140= + 340. Ergo opus est adhuc duab,s literis habemus autem arbitraries ad huc residuas:
Superest tantum una investigenda aequatio
ope a99.  6 et 7. s videamus an non licept facere
sed ita jam prodit ncommodum, quod non praevideram, et cujus remedium in hoc
calculo non apparet, quod nempe ex a (ut aq. 1 reperta est, ita similiter
10, 11, 30, 31 
ficilt 21 = 11. 31. 70 ∞ 10+ 30 = 10. 30 ca : 60. 36. 11+31 unde sequitur absurdum, fore a
Nec video quomodo hoc evitars queat, si Xx + ac et Xx+ ac
ponantur illud = 0∞ 5, hoc 59G Ergo, non licet facere ut O et e similiter se
ba = ca seu b =c
habeant cum inde hoc loco oimodum aequalitas sequatur, secusfuisset, si  sit ecrammpo adrata
 Motandus hic usus considerationis similim. sed factis ergo C et 
valore dissimilibus, perit fructus  divulionis, ut x tac et xxaac separatim aequentur
cateu  alicui

## Folio 341v — 00068196:0678
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 340r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000678.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 76 lines, status machine
quid si sic sint X x+ ab et Xx +  ac aqu. quadralis
x t h 1+ .V 3 = 21+ 20v 8 = 3 lroB
n - (G2. + 148 184 et
XX+ ab = 00 et A O = 1. 40v + 41 + 501h6.
+. 201, 190 = rego d 
20 = 40.50
21 41.50
+ 2. 40. S0
2. 41. 50.
20: 21 = 40:41
0.20 = 50, 10. 30 + 48 ergo 1970. 40.50 - 50.10. 30+40
hinc habetur. 40:50
15291. 50 = 31. 41. 50 + ba =. 11.31. 50 + 4,
hunc habetur 4+. 50
chabetur 41
10+30. 41.50 4 11 +31. 40. 50 = 10. 31. 50 + 1. 30. 58 + 2. 40. 41. hmih 4V
41 = c0. 3F . H35.50 = b, 11451 4850, CH8 1, 50. 10+50 L2. 40
seu
husvalores, ipsius 40 et e substotuendo in aequ
in qua substituendi valores
Restat 4. 41. 50 + 31.4150 + b a = 11. 3150 + 45
per praecedente,
unde lollatur 41
dende 40 per pacedentem, restabit aequalio pro 50
similiter si XX + ac = BG et  = 60v + 61 + 70 st
e similis erit calculus. pero 60,61, 70, ut pro 40, 41, 50.
sit jam  t ab  ag. coto sint y- xrC ergo tabebum
lam ipsum quam quaratum nodn nisi unam cylinet
XXFCX +BCet si sit et yytlly taen. fit Xx +28 + ex fint
= 115. 3 20V 1E
112. 132. 21V
112. 132. 20 = 5
ca non est est p
calione ipsouum
o fue
12. 172. 21 = f. Ergo s2 s. 20 = +: 21
seu 20: 21 =f: t et 112. 12 = 5: 20
I2. 112. 20. 10 + 132.132. 2030 = p se 1 C+ 11276. 104 55, 103
 l 112. 20
seu et BeF. 0. 16. 4+ 132. 26. 30 = p. 20
o ek 20.21 = 5: t. Nam quia 20,21, s
2. 41. 23 +2C+ l. 23
ergo 20+ = 2C +b. Ergo 1= b, ota hypothesia ligores pestangenda ad eum casum
r. 21 et Fu +B2. 51 = r: 21
et f. 1021ruI + 135. 302 - 91
ubi sola r involvit
differentio inter D et 
3E ae jam inventa per 112. sed 12. modudetur per aeu
extitissec
 all aliquam ex his  M. si in aeq. Z et 4 vels,
rursus in absurdum non subtuta I12a habuissemu 12 simperces
sut idem no
 pro 212, annc occo essi 212 et 112
poteren fiet 4D.50 = 60. 70 et  91.50 = 51. 70. quibus prastitis nil ultra superes
melior, ut sit _ a XX + Cx + ab- et 
XX + SX + ag-  quolobet aeq. Clto, et tiat x= 40B + 1+ Hd, ita enim
sed est etiam alia ratio, quae nihi vddetur
et E = 212 ff4 + 232162
2. 40. 23v
12 = 5. M2. 20
sed, non et cun permanus jam emm prodit
sunt cora ipsi C et G etiam + eut coc pdeoque 241. 23 + 2C + B23 =
quo l= b. adeoque revem ad cosum ut gy+ ac ergus+am
-, 40. 400
X763.
sin cquendio se
raru. 11V. 11 F2
et 12. p 25 + 30. 2(11 20 (5] 11.
Porro ad aegu. 3 4, 5, retiacrmus solur ubi 112 o milsa ubi 132,
tc = 1090 W0 n c. pl. 1i
At ex 5q. 3 e A rarsus sut Collendo 112. habetur item aeq. nova ubi jam veniendu ad eplicadem
Ex aeq. 3 et 5 tollumus  112, habelu aequatio nova, quae rusus cois ipsi D et 8 per qua invenietur 20
ipsorum S,t,
valorem, qui fuisset
lamen haec coicident, sed differre possunt sianis.
3 elix 112. simplex