Leibniz-Handschriften zur Mathematik LH 35, 13, 1, fol. 92r

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1. Id agitur resoloptur duae aequationes simplices differentiales duarum literarum quas soc exprimo 100a + u0 Z +120d 3 20 et 200 at 2oZ
2. eoro + rocao Y2
3. 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
4. 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
5. fiet voal + etlZz + 10.lv + 120ldZ
6. 200 am + 2 0M3 + 201 nB + 202 ndv
7. digan + elion Zt dio1NV F BBonds + 101 n du + 12onddz
8. qoo ap td 2to Z td 2o p B +OPAZ + 203P dv _C
9. Htaec aeg. videamus an possit reddi tractabilis seu identificat.
10. e
11. ii. unde fit do +
12. (7) posito =9= hroz 1301V + 3 redZ + 2or do. unde ex do + s fint ag10. x idensificanda aequationi 7
13. pot2 + a314Z + d301V + a3 310dZ + 2oed +324 dd Z + 302 ddv 2 = 9, Hi ex absolutis h = 100l t 200m + d10on + d200 p
14. a320dz + d3 2d
15. rorpddie) ob ldo fit pl 3o2a: 22 1, n = 12 = 32ua: 120 et ab d 2 M 301a Fdzora -101 3u- 203302a Et eodem
16. 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
17. 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
18. 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
19. Exosdinpta (10. 202. 310 -10. 202d310- F 128. 210. 301 + X
20. aequ. I5 fi
21. Ex ordinata
22. d301 j- set 202. 190 320 +110. 202 d320, 110. 219, 302 +120. 210d302) (15)
23. 00. 5 7 - 101. 210
24. +202 d110
25. 007 = 120 210
26. +120 d210...
27. 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
28. sed quia plure adfinas aequationes safficerent bini numeri, ad sunt auter
29. quaterni, nempe 3 10,301, 320, 302, ideo reducamus ad binos explicando, et (23 = 101. 201
30. +202d01
31. c7 7 120. 203
32. + 120. d
33. 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
34. numenos 310, 301, fiet exirg. 15
35. o5 5 320 7110. 202 d320 +007. 302 Fa20.210 d302 719) nexae
36. e bina gequs debent aquari intra s nonaise
37. 12. 202. 61
38. et, 62
39. F11, 202. 51 ..
40. + 120. 710. 52
41. Haebinae aequationes ut formint unam tractabilem, aequen su r 2orda.
42. ex aaquatis Terminis d3o2, fit 5200 00201 e. 210+ 202 st, 02
43. 5120. 202.51 .. . p
44. 00 3. 320 +00. 101. 202d320 + 00 1. 302 +00. 120201d302
45. =120, 202. 62
46. t1D1. 202.5].11 = 1∞. 2252.-.
47. praeteria aequentur utrinque coefficientes terminorum d302
48. . S Terminorum 302
49. St + 00 201 - 210, et d52 d51 +00d 2 210 superest jam ut aequemus etiam termin
50. 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
51. 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
52. Explicando ergo Q et T, et mrultiplicando aequationem 23 per 202, et pro 203 ponendo valorem 201 + d202, fiet
53. 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
54. et Lia ML. 0 C e
55. seu 5 = 210 100 -101) 202 +u0 =- 205) 12C
56. notatu dignum
57. o1. 202. 202
58. 20 201 2o 00.120. 201 de t 00. 120. 202 d éor
59. 1 + 20. 201. 20L514V. 120, 201. 201 120. 2
60. evan escere omnes C00. 120. 20280 1Hbo, 120. 202d 2ff 60. 20 holl 2o = 0b. 120. 2e t202)
61. 51:00 = AO 2D. 202(8, 110. 210. 202 +210. 21. 128,:, tub. 205 E02 +210. 12020
62. 202 G00 - 101) 202 + (210 - 201) 120, id est se destruentibus mutuo numirutore et denon
63. natore fit tundem
64. 120.202
65. CEstaminam
66. 6 2t0, qua foret magna calculi contractio, deou latui
67. 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
68. 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
69. 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
70. seu log. 320 + Ser 216. 201) - log (10 - 101) seu 320 (10= 101) B 5 2
71. Eit 32 (01. ru. rormiI: Mt A d320
72. Namesplicatis Sipr
73. L ipto 32b, 140919 fat
74. mntiplicate per 120. 202. fi
75. V(o. co2. 10. 218 302
76. ro. 126: 210 412e
77. Fuonontncr is o t oos
78. L et d32033) Bs
79. 10 (30 -202) + 202(d 101 - du0) 210- Z01 vocetur 51
80. qu (110- 101) 202
81. Pit ) B ls 1B. 10
82. (34)
83. bL. +, 101. 24 - 202 cu. Invento jam Numero 320, potest et am inveniri numerus 302, per aeq. d vil per ap 20
84. 202 III. 111-
85. 412v. ror. 120. 210
86. 12202. 120 202. 21
87. HHC pleganter lvenit ut evanescat d302
88. er 10 i
89. 120 202e
90. velfietavel
91. 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
92. 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
93. 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
94. et fint 1I. 320 = 05t b: a ac denique 320 = 00 b: a 2: a : 1II
95. Hinc jam inveniemus et 302, ope alterutrius, 19 vel 20. Nempe ex aeq. 19
96. 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)
97. quia tali modo fieret et aquius do = N (t epaye 20, - -00h idem
98. Sressit suurderet, deberet fieri 10. eb. 51 =, 1h. sed hoc non est ila. sic igitur
99. i
100. pothesium quae videntur permissae sedun sunt nolivae
101. sit numem ueo 3or, et tamem mancat lina apatimes un ageritus solim
102. (+120. 2026
103. (=10. 2or. 2013
104. 120. 202 d2
105. F120. 210 d 202
106. 120. 292. 210)..
107. et ita 320. d320: 320 haberetur ad huc semel. fieretis
108. aeqratio inter datas. plisi tam adhibeamus 61 et 62
109. eadem modo ut 31, 52. Et ita poterimus praestare e aequationem, sed non ita nec 220 nec 302
110. reperiemus, evanescent enim ambae, et manebunt artitratiae, quod fortasse nut noceret
111. es
112. 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
113. e trtim E, s fbe fr s o 17L = nrrg. et engon -
114. intam - 18 - A sit 72 41d1 ataro t et 18 a 28
115. 20, om issis nunc 61, 62, fiet
116. et ob en igul tit er2fioI= mridI
117. Hinc ex novissima hac aego tione multipli pltu
118. 184281
119. ab. 3u
120. 184302
121. 12m = 172 + 272
122. num, fint
123. = 17L +27 27 15 - 182 + 282 M = 182 + 2825. 18. fit
124. 19m = 182 + 282
125. 171+ 172 -
126. in test = 17. sineli ceni 17 8, fiet ratu aequalilalis
127. 192 +292
128. arrs
129. s
130. 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
131. 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
132. = 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 )
133. 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
134. Unde tandem fiet 120. 202]202 (10 - 101)
135. +120. 202. 1022S A. 202. 120. 210. Ii=00. 120. 120, 210. 21B
136. 42. 120. 202. 110. 20
137. Unde dividendo utrobique per 210. 110 - 202. 101
138. seu denique sL = 110. 202 + 120. 20 et proinde 52 - 101. 202 + 120. - + 20. 216
139. 120.5202
140. etiam et tam de2o quam d302. Et vercor no etiam tam 320, quam 32, seu
141. oraio
142. 7120. 202. 101. 2025I =20, 12O. 20101. 1V LI2V. 120. 2V1. 00. 210
143. +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
144. 112o Cno. roi) la = + 202. 202. 110. 110
145. 4, 120. 120. 210, 210
146. 190.YGD. 21 201 = 120, 202, 101. 210
147. 120. 202 10 (Co - 101 ) d202
148. e raroi mo io flaor (C130 dor o tuo. 01) lere
149. 2 632- (2o2. 10. or. 10
150. 1120. 2013 L120. 210
151. seu denique
152. 10Vt + 20240
153. 120. 12
154. sod c vitro eodem opera in aequ n composita qu 19 et 20 ee
155. 120. 202 latus aequationis D, quod per explicationem jam fultam, patit esse verissimum itaque Eac er
156. hundo non venitur ad valorem ipsorum 2o, 302, sed manent oeterminatae inventis quales posuimus st et52. sit er A-
157. 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
158. 17f. 320 +172 d3o + 181302 + 192 de 2 =0 et - 171. 320 - 272d320 -181d∞2 - i 130 = qua duae aequationes coincidunt inter se
159. 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
160. rexplicatae quis us ae quarum una aequat coefficientes iptarum Z altera coefficiendes ipsarum V, est una eademque, et proinde alteratia literdium est superflua.
161. 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
162. utrumque latus aequationis
163. fit 152= - 17L e 282 = 27L
164. 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
165. et 181 = - 171 et 281 = le - 271
166. ad hinc video eadem opera eranescere, utrumque latus aequaribuis
167. 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
168. es
169. 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
170. pro d2g2 tollendo fint. 00. 120. 20400. 129. 201 = 120, 202.52 20 sen fiet 52:00 21
171. pultistiirmm pid
172. 2oC
173. 13
174. at E
175. or pot on
176. 1. Bri i et 22
177. nlo toror. 101. 210( 120. 202. 202
178. 202912d- 120. 202. 101.
179. xt 202 (10 +). 12. 2 113 = +101, : 128
180. 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
181. A6E. 11B. 2B. 1DrIO2. 11 202 10=Z Or. odit 8. e 202 120 e E 207, 1 odi o L. 2oc . diro ) 32
182. (120. 202. 202 dio t 20r 101. 202. 11 ] 202. 101. 202. 10
183. 20. 2o4oI 0ZCAp D L12O. 2O. L
184. st avoroa e A hovo2 20d 25 + 10 10. v 1. e i 200
185. prars une pegia versa

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Leibniz-Handschriften zur Mathematik LH 35, 13, 1, fol. 92r. Machine transcription, not an edition: Leibniz Legible (model FoNDUE-GD_v2_ft_Leibniz, run 74). https://leibnizlegible.com/page/00068196:0181?run=74