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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 92r, page id 00068196:0181
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Id agitur resoloptur duae aequationes simplices differentiales duarum literarum quas soc exprimo 100a + u0 Z +120d 3 20 et 200 at 2oZ 
eoro + rocao Y2
ubi in priore omisi prido et in posteriore aroces quid semper tolle possunt. diferventientum dalbunt dioo tdnozr dio. VL 1odZ + 1rd V + Modiqe ( Et drva pdao Zt d 20IP + 2~gdZ 1 201 dv +20 2d dv ) = 0
fiat compendio 110 fel 1 26. L 130, et 201 +d 262 =203. Aespuationes 1, 2, 3, 4, multiplicatae respectire per l, m, n, p addanto
fiet voal + etlZz + 10.lv + 120ldZ
200 am + 2 0M3 + 201 nB + 202 ndv
digan + elion Zt dio1NV F BBonds + 101 n du + 12onddz
qoo ap td 2to Z td 2o p B +OPAZ + 203P dv _C
Htaec aeg.  videamus an possit reddi tractabilis seu identificat.
e
ii. unde fit do +
(7)  posito =9= hroz 1301V + 3 redZ  + 2or do. unde ex do + s fint ag10. x idensificanda aequationi 7
pot2 + a314Z + d301V + a3 310dZ + 2oed +324 dd Z + 302 ddv 2 = 9, Hi ex absolutis h = 100l t 200m + d10on + d200 p
a320dz + d3 2d
rorpddie) ob ldo fit pl 3o2a: 22 1,  n = 12 = 32ua: 120 et ab d 2 M 301a Fdzora -101 3u- 203302a Et eodem
modo ob dz fit 120l  310a + d320a - 178320a - 2(0. 302 a, supersunt aequationes coefficientium Z et V, qrae multilicettur per 120. 282, et in productis pro 120l, Egem, substituantur vilores fiet
110.2023101+ 110. 202d320- 10, 202138320-110. 202. 218. 302 120. 210. 3016+120. 210d502f- 120. 210.701 320-120. 210382302+202d110. 320+1d20. 302 120. 202 de0
101202. 310 + 101. 202d320 - 101. 202 35 320 - 101 202 218 302 + 120. 29. 301 +120. 201302 - 120. 20) 101320 -120. 201 202 302 + 202. dlo 1. 320 + 120 d2o1. 302=120. 202d301
Exosdinpta (10. 202. 310 -10. 202d310- F 128. 210. 301 + X
aequ. I5 fi
Ex ordinata
d301 j- set 202. 190 320 +110. 202 d320, 110. 219, 302 +120. 210d302) (15)
00. 5 7 - 101. 210
+202 d110
007 = 120 210
+120 d210...
ay. 16. firt. por. go2. 310 + X 1310 4120. 201. 301 - 10. 22d301, - 10n 20. . 320 +101. 202 d 329, - 101. 40. 302 + 129. 201 d30270
sed quia plure adfinas aequationes safficerent bini numeri, ad sunt auter
quaterni, nempe 3 10,301, 320, 302, ideo reducamus ad binos explicando, et (23 = 101. 201
+202d01
c7 7 120. 203
+ 120. d
fiat oo 310  sl. 302 et 00. 301 = 52. 302. porro eseffictentes iporum 320, 502, in cgq. 15 et 16 compendiicausa vocentur 06 , 00 2, 008, 00. ita explicando
numenos 310, 301, fiet exirg. 15
o5 5 320 7110. 202 d320 +007. 302 Fa20.210 d302 719) nexae
e bina gequs debent aquari intra s nonaise
12. 202. 61
et, 62
F11, 202. 51 ..
+ 120. 710. 52
Haebinae aequationes ut formint unam tractabilem, aequen su r 2orda.
ex aaquatis Terminis d3o2, fit 5200 00201 e. 210+ 202 st,  02
5120. 202.51 .. . p
00 3. 320 +00. 101. 202d320 + 00 1. 302 +00. 120201d302
=120, 202. 62
t1D1. 202.5].11 = 1∞. 2252.-.
praeteria aequentur utrinque coefficientes terminorum d302
. S Terminorum 302
St + 00 201 - 210, et d52 d51 +00d 2 210 superest jam ut aequemus etiam termin
defficientas terminorum 302 Et pot 00 2 + 10. 202. 51 + 120. 210. s1 + 120. 210 00 7o  e C 10 aor des  0d7 + 101 202. 51  +. 120 2, 1olaco- lol a2d f 710 aiet
quae aequatio non tantum est tractabilis, sed it planc ordinaria, nam evanescente d51, habetur sI absolutr, porro d 21 L25 202. dto1 - 202d e- 0 a 2 + 210 d 2o
Explicando ergo Q et T, et mrultiplicando aequationem 23 per 202, et pro 203 ponendo valorem 201 + d202, fiet
211I0.210. ∞2(,120.210. 20X 120. Hd220D Y 1O. 202 121 110. 202.202. 51 + 120 20. 202 5) (7 120. 210 20  120. 210. 210=00. 101.210. 20291
et Lia ML. 0 C e
seu 5 = 210 100 -101) 202 +u0 =- 205) 12C
notatu dignum
o1. 202. 202
20 201 2o 00.120. 201 de  t 00. 120. 202 d éor
1 + 20. 201. 20L514V. 120, 201. 201 120. 2
 evan escere omnes C00. 120. 20280 1Hbo, 120. 202d 2ff 60. 20 holl 2o = 0b. 120. 2e t202)
51:00 = AO 2D. 202(8, 110. 210. 202 +210. 21. 128,:, tub. 205 E02 +210. 12020
202 G00 - 101) 202 + (210 - 201) 120, id est  se destruentibus mutuo numirutore et denon
natore fit tundem
120.202
CEstaminam
6 2t0, qua foret magna calculi contractio, deou latui
interim difficile est creditu fasum calcurum talem in tam multis consensum produsisse. Erge 52 = 10, 210 + 201 = 210, seu 52= 00, 2 et 310 -  302 et 301 = 2ot. 302
porro ob aequatas binas aequationes lg et 20, evanes destructis utrinque 302 et d302, super Ast equatio 0ct H. 320 + 00. 110. 202 d320 = 00 2320 + 00. 101. 202. d320, vel quod idem est
d320: 32060, B-5V, in 110. 202- 101. 20P EX plicetur 3= si, maltipl prius per 120, fint 101. 2 1o, t∞t 2o2 d120 101. 2 120. + 202 120 dioK C
seu log. 320 + Ser 216. 201)  - log (10 - 101) seu 320 (10= 101)  B 5  2
Eit  32 (01. ru. rormiI: Mt A d320 
Namesplicatis Sipr
L ipto 32b, 140919 fat
mntiplicate per 120. 202. fi
V(o. co2. 10. 218 302
ro. 126: 210 412e
Fuonontncr is o t oos
L et d32033) Bs
10 (30 -202) + 202(d 101 - du0) 210- Z01 vocetur 51
  qu (110- 101) 202
Pit ) B ls 1B. 10
(34)
bL. +, 101. 24 - 202 cu. Invento jam Numero 320, potest et am inveniri numerus 302, per aeq. d vil per ap 20
202 III. 111-
412v. ror. 120. 210
12202. 120 202. 21
HHC pleganter lvenit ut evanescat d302
er 10 i
120 202e
velfietavel
E si Explicetur i5 s far palhpl per sco, firt - 101. 202. 1B 10L 20 2 dexB = 120. 101. 2ot + seo. 202 d10t t 10. 202 10 + 1B. 20L 1120 + 120, (0j. 90 - 120. 292 d1
Explicetur Z - 51 Fet d320. 320  (10 - 101) 202. 139 + (210 - 201) 101. 202 + 20242 (d1v1 - du∞), i, 202, 110. 27 - 101. 2 vel posito 110 - 10  = 111 et 20 - 202 211; fiet
d32o: 320 = (11 (130: 120) + 101. 211~, :, s 11. 202)  = du: 11 seu log. 320+ log. 11) = s(Wt CO) rOt EN S  (130, 120. 202) F) (101 . 2, :, 11. 202)= (gpndio) a: a
et fint  1I. 320 =  05t b: a  ac denique 320 = 00 b: a 2: a : 1II
Hinc jam inveniemus et 302, ope alterutrius, 19 vel 20. Nempe ex aeq. 19
aultiplperror fiet o 00 N 320 + 00. 110. 202 dy 20 H = 10. 26. 102+ 30) 120210. 202) d302 = 0 sed ita eoune scunt 301 non minus quam d30)
quia tali modo fieret et aquius do = N (t epaye 20,  - -00h idem
Sressit suurderet, deberet fieri 10. eb. 51 =, 1h. sed hoc non est ila. sic igitur
i
pothesium quae videntur permissae sedun sunt nolivae
sit numem ueo 3or, et tamem mancat lina apatimes un ageritus solim
(+120. 2026
(=10. 2or. 2013
120. 202 d2
F120. 210 d 202
120. 292. 210)..
et ita 320. d320: 320 haberetur ad huc semel. fieretis
aeqratio inter datas. plisi tam adhibeamus 61 et 62
eadem modo ut 31, 52. Et ita poterimus praestare e aequationem, sed non ita nec 220 nec 302
reperiemus, evanescent enim ambae, et manebunt artitratiae, quod fortasse nut noceret
es
fatimus  videamus an hic locum habeat. Compendio 17c. 320. H72 Ag20 H 3. 3or Ap2 d 30= 0 271320 + 272 d320 + 281302 f 282 d 302=0 add
e trtim E, s fbe fr s o 17L = nrrg. et engon -
intam - 18 - A sit 72 41d1 ataro t et 18 a 28
20, om issis nunc 61, 62, fiet
et ob en   igul tit er2fioI= mridI
Hinc ex novissima hac aego tione multipli pltu
184281
ab. 3u
 184302
12m = 172 + 272
num, fint
 = 17L +27 27 15 - 182 + 282 M = 182 + 2825. 18. fit 
19m = 182 + 282
171+ 172 -
in test = 17. sineli ceni 17 8, fiet ratu aequalilalis
192 +292
arrs
s
oi 1 . Eor + 00. 120. 28 , 120. 202. 51 = 00. 101. 202 LeB, 4 70 ( . 120. 202. SL seu 52 = 51 + 0B. 202 (101 19) 1 00. ro (201 - 201 - 51 - 00 y. 00. r
85  + 00 2. +(0. 202,. 51 +, 20. 210. 52 - 120. 202 451, = 93, 5. + 007 + 101. 202 51 + 120, 201.52 - 120. 202 d5 2
= 23. 1D. 2o2. 1e (- 202. 110. 202 duo ( p0. Dor. 101. 1 + o. Dor Wvrli  = 202. 1oL. 202 10( Zor vf 20 2d1 2.  20 uor 120 p00 2o o iondioV Auo nor es uo a  )
per 2 120. 202, et epplicala sint + = 20. 202. 10. 2(- u. 120. 216. 20) 120 20. 40 2o  3aio2. 12o d 210) = - 00. 1e0. 2o2. 1B1. 2CF a. 1o lb. 20p. 20 L= 0B. 10. 120. 3  20 t 120. 202. 12   21 ao 2o  seluos  
Unde tandem fiet 120. 202]202 (10 - 101)
+120. 202. 1022S A. 202. 120. 210. Ii=00. 120. 120, 210. 21B
42. 120. 202. 110. 20
Unde dividendo utrobique per 210. 110 - 202. 101
seu denique  sL = 110. 202 + 120. 20 et proinde 52 - 101. 202 + 120. - + 20. 216
120.5202
etiam et tam de2o quam d302. Et vercor no etiam tam 320, quam 32, seu
oraio
7120. 202. 101. 2025I =20, 12O. 20101. 1V LI2V. 120. 2V1. 00. 210
+120. 202. 120. 20 51 (F v0. 202. 120. 210. 10D(, 00. 110. 120. 20. 20D. pr. 202. 120. 20151 (F21. 129 201. 01101) (Y 12v. 12D. 201 . 00. 20B
112o Cno. roi) la =  + 202. 202. 110. 110
4, 120. 120. 210, 210
190.YGD. 21 201 = 120, 202, 101. 210
120. 202 10 (Co - 101 ) d202
e raroi mo io flaor (C130 dor o tuo. 01) lere
2 632- (2o2. 10. or. 10
1120. 2013 L120. 210
seu denique
10Vt + 20240
120. 12
sod c vitro eodem opera in aequ n composita qu 19 et 20 ee
120. 202 latus aequationis D, quod per explicationem jam fultam, patit esse verissimum itaque Eac er
hundo non venitur ad valorem ipsorum 2o, 302, sed manent oeterminatae inventis quales posuimus st et52. sit er A-
Fit ergo AI = ~A eHAHEo e2 tM- est tEgto e 182 1Eg2 -0  272~ = 17L et et 271 = 1 et 231 - - 18 et 202 = = 182, et habemus duas aequationes
17f. 320 +172 d3o + 181302 + 192 de 2 =0 et - 171. 320 - 272d320 -181d∞2 - i 130 = qua duae aequationes coincidunt inter se
itaque coincodunt aegg. 19 et 20, si ponendo 61, 62, esse 0, faciamus st: 00 = 10. 202 +12. 20, :120 3t, 52 = 101. 202 f120. 202,, 120. 20L. itaque revera equationes 19 et 40
rexplicatae quis us ae quarum una aequat coefficientes  iptarum Z altera coefficiendes ipsarum V, est una eademque, et proinde alteratia literdium est superflua.
mvde calulo Aberrassa 12 tr2C 192 + 2BI dabit non 3, at scripsimus sed nor 10n1O 2ot +60. 1o1. 202= 00. 120. 210 - nv. 102.5] 4 00 1o a 10. 202. 52
utrumque latus aequationis
fit 152= - 17L e 282 = 27L
unde si fiat 302 = 32d omnia evanescent; posito si modo sit 310 = 38. 51: 00 et 301= 302.52: 0 secuntum valores 51, s2 inventos
 et 181 = - 171 et 281 = le - 271
ad hinc video eadem opera eranescere, utrumque latus aequaribuis 
Est ad huc alius modus redd ex binis aegq. 19 et 20 faciendi unam tractabilem, fint 310 = 61. 3EB et 301 = 52. 302, ita evanescent G1 et 62. Addantur in unumlmbae aequ. 1G et 20, et in aeq. composita destruantur
es
tam  2 qu d3ro, quam d3o2, quod facile est, pro d32o tollendo firt 00. 110. 202 - 12B 201. 61 + 00, 101. 202 =0 seu 6
pro d2g2 tollendo fint. 00. 120. 20400. 129. 201 = 120, 202.52 20 sen fiet 52:00  21
pultistiirmm pid
 2oC
13
at E
or pot on
1. Bri i et 22   
nlo toror. 101. 210( 120. 202. 202
202912d- 120. 202. 101.
xt 202 (10 +). 12. 2 113 = +101, : 128
Iil. C (od N Ao. 202 617, Mo. non d G, +∞o Z. L 1el 2o2 61) 321 (00 3 t ouos tos 1 morosr. navords 1) 30
A6E. 11B. 2B. 1DrIO2. 11 202 10=Z Or. odit 8. e  202 120  e E 207, 1 odi o L. 2oc . diro ) 32
(120. 202. 202 dio t 20r 101. 202. 11 ] 202. 101. 202. 10
20. 2o4oI 0ZCAp D L12O. 2O. L
 st  avoroa e A hovo2 20d 25 + 10 10. v  1. e i 200
prars une pegia versa