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# Catalogue record: 60462 — [Aufz. zu Differentialgleichungen]
# Incipit: Aequationes habemus 11.310+12d310+21.302+22d302+31.310+32d310+41.301+4
# Date: 1676
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=60462&katnr=57390
# Work: https://leibnizlegible.com/work/00068196
# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folios: Bl. 73–74 (from the shelfmark LH 35, 13, 1 Bl. 73-74): 4 pages, canvases 145–148, page ids 00068196:0145 to 00068196:0148
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068196&page=145
# Lines: 360 recognised on 2 of 4 pages, mean confidence 0.79
# 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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## Folio 73r — 00068196:0145
# Scan: The same scan is also registered as fol. 74v; its text is given once, here.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000145.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 164 lines, status machine
sunto
it aequationes besbinas habemus
31o 31 30] 302
1B.312 20. 31 30 31L 40.312
suli 10, 20, 30, 40, sunt
arbitrarii
ob 312 erit 10. 51 - 11
20. 61 -21
30.91 -31
302
20320 11. 310 + 12d34 + 21. 2 + 22 d 3 + 31. 300 + 32 d 3D0. F 41. 3d + 42 d301, = c
51. 350 + 52 dex +61 3 +62 de + 71. 310 + 72 d3ID + 81. 3)B + 82d301 (70
10. 11. 312 +10. 12 c312
20. 21
30. 31..
40. 41..
g1D. 12..
d20. 22.
d30. 32..
dao. 42.
20- 22
30. 32
40. 42
13
3
10. 51 312 + 10.52 d3120 Hae binae aeque pdnantur edentisicar
20. 6
30. 
40 g1
d10. 52.
d2o. 42..
l30.52
d9u. 62 ..
20.62 Obdy ) Erit 40 = 10(52- 12) + 2c(62- 22) + 30 (72 - 32)
30. 72
40. 82
+ d10. 5et = 0 se 10. 53 + d10. 59 = Hec arpuntio
d2o 64
430. 74
c0. 03
30. 5
d20. 64
d30. 74
multiplicet
per 84 et
- 82
compendio 4  10. 54 + 20. 64 + 30. 749: - 84
et dq=- 8 4F454. 10484 54 d10. _P + d 34 ( 52. 18 3: 84.84
ec
64. d04. 20884. 64 d20
40. 83 440. 84 Explicndo cuiernofirt
34 d74. 30 84.74d30
(7et. 30
. 82. 53. 10 +94. 5951d L 83. 54. 10 + 84. d54.10- de. 5410 f 154. 547di0
84, 63. 20 84. 64 d20 83.64. 20 +84. d6g. 20- 1. 64  84. 64 d 20
84. 73. 3084. 14430) 8. 74. 30 +8+ d7430- 184 3130+84. 7 4d30
superest ergo ut eq. 1o applicetur ad data. Ante omnia videamus an a2- 82 non evanescal, ut haberi possit 40. dende videamus an refficientes ipso
40. 81-41 d40. 84
30 et deo non evanescant.
11
12
22
31 32
41 42
valent eost 11B. 202 00) 12 V0 IBEs -lN2oL 120.210 0
+83. 74
3i 32 61 62 71 72, 81 82 Eyo gr = 84
003 19 202 0o7 1a24 10122 0 na2oI - 1eo. 22
+ 120. 120 202 (201-210)
Bd. 8 fit 42 - 82 = - 84 = 120. 202 73= 71 - 31 = 202 (101 - 110) itaque 30404 74 = 30 f- 120. 202 (1 2 0 d 202 + 202 d120))
81- 41 = 83 = 120 (20(-719 74 = 12. 32 = 120. 202
6C = 62 - 22 = 120. 20]- 120. 210 = 120 (201 - 210) =64
63 = 61 - 21 = 00 7 - 003 =63
4= 52 -12 = 202 (101 -110)
53=9
Ergo tantem ex aequ. 10, fiet , -120 (20- 201) [120. 282. 30
= 202 (110-101
Cadit
+120. 202. 202 (101 - 110) 
(1120. 202 (rodror + 2024120))
120 (10 1. 201)
A. 1B3. 64 (o(2ot. 20) pol vor. ud
d. 20 64 = p) =nonor ano cnom no ao holusaro- drc
463
F12∞ 20200(7- 3)
aguta licccon) eroio   sid
(83. 54 = 120 (201 - 210) 202 (o1 u0)
 82 d 54 - 120. 20 CRO2Cdioi - ducKtd2o2(1oL oB)
323 = 120. 202 0V (7 - 51) - (X. 202 C∞0. 0.) (notrdi r)
s h
Eo 20. 20 ( 12o (u. 1) 1B Zo2. 10  2o lador sq. vorbu 1 robrot en u   Catioinl toB)
sed quia tota aequatio dividi potest per 120 (20-201) + 202 110- 101), fit tandem - 120. 202. 30 + 120. 20, 20 + 110. 202. 10120 vel 30 = 120. 210. 20 + I1c. 202. 10, : 120. 202
22
40 = (120. 202) 202 (101 - fi 9) 10 f120. 202) 120 (20) (=210) 20 + (120. 202) (120. 210+ 29) + 620. 202110. 209. 10),:. 12072
Et ita putehre satis fit 40= 101. 202. 10 + 120. 201.20, ut 30 = 110. 202. 10 + 120. 210. 20 Hi jam valores substituendi sunt in aeq. 3, imiidentificata est aeq. Y
20
200
120. 202
Ex coefficiente ipsius C d2 12 multiplicato per 10. 202, fiet 120. 202. 110. 202. 10 + 120. 202, 120. 210. 20_ = 120. 202. 110.202. 10 - 120. 202. 120. 2020= 0 Epqu. Crefficiens ipsin) 312 evanescit mate
Quaemmus nunc etiam eoefficientem ipsius 312, multiplicutum per 120. 202
tiet 202. 1b. 202. 10  2or. 10. 20 L d1 20 ,1 20. 202. W01 2b. 1DF 20. 202. 202 d 1o. 10)
Vi
ae (F110. 202. 10. 202. 10) (= 12o. 20. 1el. 202. 19
120. 202 =110. 21026 =12V. 120. 210. 201. 2OC1o. 12 0. 2∞9dior 2 ruo. 202. 120d20. 20) (+10. 202. 120. 210. 20) =12O2C. 120. 201. 20)
H
41 20. 2d2. 110. 202. d 10). L trorcoL 1. 21B. d2O 120. 202. 10 d 292. 10- 120. 202. 202 d lio. (0= 120. 202. 110. 202 d10) (̸ u0. 202. 1204 202. 19( 10. 202 202 4 1o. 10)
P2o. 202. 120. 20 d 20)
C120. 202. 120 dio. 20=120. 207210 d12020)= 120. 202. 120. 210 d 20f 120. 20. 120 d20, =0120. 21 6 202 d120 20)
Ergo coefficiens ipsius 312 in aeq 3 vet a, etiam evanescit
postet fai ao aers et 3o uo elanta ad
U ah + e bZ + cD + B dq + fdo sit 5:  eibo fiot abi 4 bg +rv + ce du 4 (aV =  e ta a h + bZ - cv + l.p 1q7 + C 1 = 0
sit bZ1 cV = gW fir ah + guo + e li+ e gdu. sed ita oporti esse constentes, b, c,9
Mitar sit Ha Z F = gt aZ at Z + adv = oduo a codque fist de  gdr + dg W
et
sit qu =  9Z + rv adio = qZ d9 Z + dr V + qu2 a rdio Ergo (W Vdu = (3 aW + Pa duo = Bg + CpV =
Vd Ndr V +) 9Z + Vrdo
qu’od ut coincisal data abzue + ac V +ae dz tafan, fiet β. b et Y  Y af: Y = ae: q
ab = B9 + pdque seu ab = (t aedque et ac = Cr + af dx et fit B = acr - a fdr, : 1Y = abq - aedq, : 99 r - 54. C di- tc. gat
et fiet abq ab - aedq  ab9 - aC dg. 5: 49) = _ aC59 - 4(4 - af - a ss d9 - af 9 d5 ,: 199)
Repetamus. Esto formula bZ + cD + edzue + fdx. videamus an reduci possit ad hanc ) BW + dW.
q E sit aW qq + rV erit adu = at. d9 Z+ de V + gdZ + rdo et aBu +du ext  ent  B9Z + βrV 
(6)
Identificando teminos deb sit 9= af: Y et teminos dzue sit V = aC : q. Ergo r= qF r - 59: C et di . 41o s posta V dgZ + qaV =+ Ga Vrd
Vaeztevtetz efdo
constante quod in arbitrio est
Ob terminos  identificandos, f ope aq.7 fit BY + a fdr. r= ac seu β acr - afdr, : H. Rursus opt ae 4. 8 fit BOb termnos Z, ope ag. 5 sit B9+ aed9 u9 = ab
seu B = abg - aedg,: gq. quos duos ipsius B valores aequando sit acr - afdY, : rr  abq- ae dque: qu et ex 9 et 10, pro Y et dr substituend
1
valores, fiet abq - aedque, : 49 = ac fq: C = a ff dq: e - afqdf: C, :, sf 49: ee, seu  b fgC efdq = ceg ( fe dq - egdf, ergo
tandem fiet in da evanescente q siet in datis  edf = ce  ce - bf. sed haec non est ipo in pte
1op) + updus + 12 pldus an redigi potest ad fC + gde posito () = 20y + 21 dus ubi d0 = droy + 20 dy +2, ddys
a2 du
Gdroy +920 dy +921 ddy  10p y. + up dus + 12 p ddy fit p = 29. 12 = 205 + duo 9: 10 seu _ 10219 = 12. 20f + 2d209
f20y + f 21 dy
superest aequatio 21f + 209 + d 2 g = 11 P, ubi substituantur valoes f et p. Multiphicet hoc aeq. per 12. 20 fint 21. 1. 21 - 21.12420 +12. 20. 20 + 12. 20d21 - 11. 2021
fiet:
fiet f = 9, 10.21 - 12d20, :, 12. 20
Quodsi fiat 21 = 2m 20m: a, fint 20. 20. 10mm (= 12. 20d20) am + 12. 20. 20aa +1 20312. 20. 20dma + . 20 dio and = 11. 20. 20 a m
Es d21 dy
10y + 11 dy + 12 ddus, divitatur in duas partes Bato + = 10 ay + 1 adus +oopdus + 124, ddus prior pars sit differentia ipsius osus, poserae itus gda
qua divisa per 20, habetur ita ve evanescit 20. nec profectum est quicquam
liet   0s df y + oif dus ~ 10au + u aydys siet f. f o. 00 et f  et  a Agdus + aoa gdus ~ vop gdus + 12agibus sit pg = ar
d
vop d
r = dg g = 00g: 12 = df 00 00df: 10 f.  oof q r = 19 - 00 5, . d
119 - 00f 500 fit dg. 9. 10 = 25.5 + 119df = - af5 + 11 dtdf00: 10
200 a + 210Z+ 201V + 200 dz +202 dv =o Ex 1 et 2 fiet ngo 200. 2oa (+ 210. 1207 ) 2o1. noV (220. nodZ +, 1no. 202 dv = 100. 220 a +10 220Z (u0. mo 
c
n4 13
arg
= 10g: 12
ldg = 119 -005, :00
fiet ad prro. 200) + _ 120. 201 dv + d120. 201) V + (120. 20+) ddio + d (120. 202) dv =  (10. 290 - 120. 210 ) dZ = d (10. 20 - 1B. 20)Z andent tolltu Z et aZ, fiet
00. F29
1005
il fdZ
100 a + 10 = + 120 1Z = 0
=20oga = 2019p - 202 9 dv
21947
+ 110 fZ
+21097
F= 120, 210 - 110. 220-2209, : 120
F = d (120. 210 - 110. 210) - 21093: 11C
27

## Folio 73v — 00068196:0146
# Scan: The same scan is also registered as fol. 74r; its text is given once, here.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068196/jpgs/default/00000146.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 196 lines, status machine
Sidua sint aequationes, continentas Z, Aqe, do simplici ratione, possunt inde fieri aliae binae, quarum uni desit d, alteri dpsit dez
Itaque jam tum sic scribamus 100 + 10Z + 120 d 2( = 0 200, + 210ZH
2o rorto f ita in priore millimus 10r dyx. disserentiam dugues note mtir 2, peste tagriloi enq noiti)
posteriore
rior
220 dZ prior dabit
compendio 10 + dno voleituus 130 et 201 + d eor voce mu) 203. Et quatuor istas pric2es equationes e multiplicatas respectiva per l, m, n, p, addamus n unum. fiet:
100l, + 1101Z + 101 lv + 120l. 1Z
200ma +210 m Z + 201 mV
+2o2m do
deo B vid
1202l
3(7) Tentemus an haec aequatio possit rrodi tractabilis seu ec Dentificabilis tractabili
diaonaranonz =+ le1KV+ on dZ = 10In du + 12on ddlk + 202 dlv)
troopa Hlno pZ toloipV + 20 p Aq + 203p do
0o ha +310fZ + 301. fB +320f dZ + 302 f du
)(1)
50+9d0 +00h = 8 = 0 faciendo O = 9 = 310Z + 301V + 320dZ + 302 du. unde dispereatiando
Hyo 9Z. + d 30190 +3109 17 + 301 g dv + 3rogultzu 1302 g dldv =0
fiet d(= d310Z + d301. V + 310dZ. + 301 dv + 320 ddz + 202 dax
d320 d d302 dv
Et hos valores D et do substituendo in aeg. 8. fiet.-
Et aequationes 7 et 1 ientificanda punt inter se, contrahetur autem alcilus facilndo hoco
302=0 fint etiam p=0. fice ob quanesci ergo terminus ildo. Et perinde est as acti
non tantum aequ. 1, 2, 3, multiplicatas respectior per l, m, n, in unam aeg.7. composuissemus
Omisso ergo termino dao, relque ut identificemus aequ. 7Z et 1, aequabimus coefficientes reliquorum termiporum. Ob Terminum habentem ddque, fint n = 14 = 3209: 120
+d3209dZ d3orgdv
Ob terminos et V, erit 110. cuol +210.  n + dy0. 320g   210s, +.d310q et 101  L+201  + Ao1. 3109 = 31 f id 301g quaerum binarum aequatiogum epse invenitutm et m
Ob terminum Y F Ao, cum evanescat p, et fob 202 = o) etiam pruodibit sine litins 101.3r ag orgor , 2o 30cg Ubi feliciter omnis abistdissernisfis.
Ob terminos dze et av, habentur singulatim valoras psarum literarum L et m; et ob aZ firt , nol.  . (30: 120) 320.. + 320 f +3109 +d3209 Et 202M = -(01: 10) 3209 +3019. Quos valches
ipsarum literarum , et m, substitie us in aeg9. pe 15 et 6 prius per 202 multiplicatis , ex eu apis fiet in 2o B. 3og. + 10. vo. 320)- H0 202 3lg + o 202 d qag ti. 1  bo va za g
+ 12 2B2. 32s , 120 202 d30g + 202 12C 3209 hlaq ib fir l 101 202 2C9209 + 101. 262. 520f ,rot 202. 3109+ 101 02 d3209- 120. 20t  Siog to. e1 3019 = honot 101 f 19 1. 202 d301 g
20c
10. 202. 120l
202 320f s
+ 210, 120. 202M
= 10. 202. 130 3209 +  o2 B∞ f F 40 02 yege
+ 110. 202320F
+ 110. 202.
+110. 202 89
Ex aeju. 1 6. 101. 202, 120l
= 101202 Bo 3209
+101. 202. 320f
+101. 202.
3109
F101. 207. 13269
Etexaqe 2d vidi nanss.
Et ex ay 2a ordinundo fit:
+1d. 210. 161 3169
+ 120. 210523019
l. 202m
- 120. 201. 1 3209
+ 120. 201. 3019
-120. 120.502. 310
C-atro or tes 2og
Et exay. 19 firt: porrrb. 40 3olg
Et rpaey. 20 fiet +
3320a
+ 202 d 110. 3209  120. 202. 310 f + 120. 202 d 310g
1202d110. 3209  120. 202. 310f + 120. 202 d3109
+202 dio1. 3209 = 120. 202. 301f + 120. 202 d301 g
+202. dio1. 320 g
(20) 120. 202. 301 f + 120. 202 4301g
siata et  5H. 310
ono. es
a + 110. 120. 202. 320f, + 110. 120. 202 d320
= 110. 130.202.3209
= 120. 210. 101. 3+09 (=5. 30
+120. 2041103209)
1620 (20oC (=L212 dx - 10I. 120. 202. 320f, + 101. 120. 202 d
dibe + 120.120. 201. 3019 = 10on2v. 102 d301 9
=120. 120. 202. 304f
=101130. 202. 3209
= 120. 101. 201. 320924320
invenio autem nullo fructu 350 et 301 reduci ad se invicem +120. 202d01. 3269
seu fieri 301 = 5t. 310, vel contra, sam ita a quando loefficientes ipsorum d31o in diversis aequa-
tionobus tit 51 200 seu 51: 00 = 1, sed  aequando coefficientes ipsorum 310, coanescunt f, et 1
nec possunt inservire ad aequationem. Ergo reducamus 310 ad 320, seu frat 310= 8t, 320
+ seo d. SP 320 a + 110. 120. 202. d3209 +120. 120. 20. 3019
+110. 120, 208.51 3209 = 120, 120, 2025 dg20 G
= 120. 120. 202. 51. 320f
= 120. 120. 20t a 56. 3209
+ b1. 120. 202. d320 9
+ 120120. 201. 3019 = 12o. n0. 2e2 4309 0
= 120. 120. 202. 304f
it bis dabus aequ aequemus harum duarum aequationum termues d320, et 320, Ob dero fiet 51 == 00. 10 -00. 101, :120
Aequenus has bines aequations 3 10]. 120. 202. 5 3209
Terrinos 320 mop aequationibi aequaba. interim ex refiduotum aquatione, fit d301 (5) 2019 - 202f - 2109. Et posito f:g hubetur 301
Jam f. g inveniemus aequando terminos 320, hoc modi X+ 00. 110. Ro. 202f
00. 10. 10. 2020
-00. 120. 210. 101C
+00. 120. 202. d110C
A+0S. 101. 120. 202f d unde fiet 00.1205+202 d101 +120 C2f.  . 5,
or auo
. -00. 101. 130. 2029
.-00. 120. 101. 20f
- +00, 120. 202 d10t
. +101. 120. 202. 51 G
habetitur
fe
+ 210. 101
120. 202 520. 51 = 120. 20r 4
d sedita evunescet f et g ob evanesceatem nom
1I. comparundam cum aeg.7. fietus ob taz, n mrog: 120 et 2) Jorg: 202
tut 1olL ob dq fiet nol  of + 3log. L d3rog - (130: 4120) 3209-(150: 212)302g ob de fiet 202 m ) vorfF 301g +d 3029 (101. 120) 32og - fro 3. 202) 3org
Ejm non licet omittere 302, sed redenndum est nobis ad aequ
Hag secundum Z, fiit: 110. 202. 320 f. + 110. 202. 409. + 110. 202 d320y - 110. 202 (130: 170) 3209- 110. 202 (210. 2023029. + 210. f9 D. 30r5 , 210. 120. 3019 (+. 20 120 d362g. 210, 120 (101: 129) 3rog- 20. 1 20 (203. 202)3029
Ae superest a supersunt aequationes ex coefficientibus ipsorum Z et v, quae multiplicentur per 120. 202
Et ex ap scudmu, fiet. 101. 202. 320 f + 101. 202. 3109 + 111. 202 1329y - 101 202. (130: 12o 13209. - 101. 202 (210202) 32g, + 201120 3ors pro112o. 30191 201. 1o deorg- 201. 1 40 (1o1. 20) og - 2o1. pro (203. 102)9029
+ 120- 202 (a110: 120) 3609 + 120. 202 (d210: 202) 302f 3 19 120. 202. 310f + 120. 202 4310
+ 120. 202fl101 : 120) 3209 + 120. 202 (d 20] - 207)302y (7,
-110. 202. 3205- 110. 202 d 320g + 110.248. 3029  20. 210 dv25
120. 202. 301f + 120. 20243019
Ex aeg. 16 ordinando firt 110. 202. 3109 =120. 202 d 3109  120. 210. 3019
l
E. ex aequa simndo fiel: C
 31
=101.202. 3109
Hiora
12o. 210. 3028
+120. 2102 3C2
= lec. 210 - 610C)
400.8 a
+ 120. 28. d 510
+110. 202 (130 ) posroa
+ 120. 210 (106) 3209
-120 d2103029
216 dirS. 3209
+120202. 301 f + no202 dholg. - 101. 202. 32ofu = 140. 202. 13209 + Br 210. 3org, - 120. 2o) 1302
120. 201-. 3019A
Fiut 2 vs, 310 = 3to 20 1507 302 et 0g. 301 = 1y 20 t60. 302
sit ex aeg. 18 firt - 110. 282. 5109. 320
+ 129 202 510 F
.. --
-120. 201 3027
fint
Exaeg. 15.
a  2, 20 202 hof A1on or 43103
+0 eq oi
2)
lO1, 201. 3209
2 202d10. 32
tas jam bini
XI. 10. 20Y 22g 13oca
+ 10
120d20. 30
aequationes aeque mus inter se pens
sadem ut terminia 302, et termini 302 utrobe
1302. f. t 60p= ob. (201 - 210) + 202. 5017 : 202
30 ut 21. Et ita qu
oi
Dloroni G+a
atipduc
120.4105
I6. 2 hol. 501 ) 400. ror d a  
v1. 202. 5019 +00t 202 la +, 20g def 2019) (0201. v0. 20 + 202 501)
Bt )urtoctsom ngetouohomon
20. 202 d cot
=D orre 2 2um
1209 Morsooo vi eo
501 
Edere
de
Die 29ceetre
 o vorlood
d30 + 10 022
nu tareii ponumm
sai itit
 io aspioiin
tod
=1812022 3209 173104
eliuosimu
(= 101. 120202- 3109
 eo
o no ga
e  por
 +110. 120, 20451
=1 20. 120, 202.5,F
= 120. 120, 202. d510
= 110. 202. 501g. 302 - v0. 10. 202 9d320 -1
 (30)
+ 120. 201611.
302
4 120. 292 5e .. . . 410. cdt 3169. 1. 4+ 128. 202. 302
= 120. 210. 809
+00 I a
+ 129202 d5019.-. -
- 00. 11. 2029320 -00. vio. 2019. d302
in ussoatvroed
A exagitospit
+00. 
de
ja
=1012d2. 5109. 32 - 101. 2c. 50
+(2b. 2006109 111 4 l∞. 202 46 . . ..
devau

## Folio 74r — 00068196:0147
# The same scan as fol. 73v (00068196:0146), where its text is given.

## Folio 74v — 00068196:0148
# The same scan as fol. 73r (00068196:0145), where its text is given.