Leibniz-Handschriften zur Mathematik LH 35, 13, 1, fol. 76v

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1. resolvendae
2. 11Y
3. 200 troZ + 20 q a 0
4. dio ie
5. 2
6. 100l + 101Z + 101 lv + n2ol dZ 1 102l dv
7. nodaque 0 per lmin, p addamus in unum fat fit aeq. 5.
8. 202 ddv
9. 200m +2omZ +201mV +22m 7 +202 m dv
10. dioon Fdnoq + dtnV + yon du ton d + 1on ddque + 1o2nda
11. dnon dZ do2n de
12. 42oo pa ducop Z Fdrospo + 210p AZ + 2oip do + 220 p dz F 202p ddr
13. Aerop sqZue idrorp dr
14. t ooah + 310fzue 1301 fV + 320F dZ Y302 fdv
15. =dyogz + 13o1gP+ 310g 4Z 1 201gdv +320gdZ + 302g dav
16. aequatio tractabilis)
17. Assumatur jam fO + gdO tah = (6 - 0
18. et sit = 310Z + 301V + 320 dZ + 302 dv
19. fet de (8 L d3oZ F d3o,. B + 31dz +. 30. de +3rodtqe Bodar
20. fiet eg. 9. tractabilis
21. identificentur jam aequationes C et . Utetiam ipsa N95 fiat tractabilis
22. Habetur h per terminos absolutos aequatos
23. Aequandli daque et ddo coefficientes habemus n etpperus
24. (10) 202. 920 - 220. 202
25. fit p = 3209 -120n, : 220 = 302,g - 201n, : 202, quos duos valores aequando fit 2c2. 329 - 202. 129 = 290. 3 org - 220 0 2, adeoqe N = 20 170- 210107
26. dgrogaqe +d3olgd
27. quo valore ipsius n substituto in prie valore ipsius B, fiet p = (202. 120. 320q 2 1907. 3209(- 202. 320. 1209 + 220. 302. 1209,: (202. 120, 220. 102)20
28. enim _ p = 102. 320 = 120. 302gquod non diffeit a priore, nisi
29. 102. 120 - 120. 20X mutationesignorum umeratoris simul et nocaleris
30. in
31. permutando 10. et 200 ex 1 fier
32. Compendu 1eq. 1B t communis denominator, 120. 202 = 102. 220 sit = 13 = D0. 1002 fit 1 2 L . 3o- 20. 10g et p = noqua tor e9
33. inde eodem
34. modo
35. 110- 0 t -1000Y
36. 120. 202 - 102. 220
37. eodem modo 3015 + hoju -dioin - droip = 17 = 00t et ob Z, fiat nol + 20m = x 18 = oor et ob fiet 101l + 20im = 19
38. quo supra invenimus n et p, invenus l et m, et et el pro 120; 22D, FD 102, 202, ndo 3 20), 302n, B, fit
39. 101. 20
40. 201. 110 - 21010, = t M = 10. 201-101,210
41. et compendio sit 110. 201 - 101. 210 = 22 =00.2001, fiet 1 = 23 = 201 Y - 210t, : 2001 et m = 24 = 110t -101Y, : 2001
42. superest ub aequemus oefficientes ipsius dz, et coefficientes ipsius av. ubi compendii causa fiat 10 +d120 = 25 = 1120 2+ + 220 = 6 - 2120 101 + dp.A. = 27 - 1210 201 4 42od - 18 -240
43. et 30r I +dhon g - 1210n = 2210p = 30 = 00 . Ergo obdue fili2oltiom = atgult2oem = voV
44. 1002
45. paeterea fiat compendio 320f + (310 + 4320) 9 - 1120n - 2120p = 29 - 004
46. Quos duos valores ipsius L ex aeqa 23 er 35, et ipsius M ex aegg. 24 et 36 aequando fit 201. 1002. Y - 210. 1002 t2 202 20018 -220. 2001 ,+u0 100rt - 101.1002 R 2o. 200 p 102 201
47. 120. 202 - 102. 22
48. 1B. 202- 102. 220
49. 110, 290, Y, 101, 201
50. ubi numeri sunt dati sed literae debent explicari Quae libet harum quatuor literalum novissimarum continet n et p, quorum valores supra inventi continent 320 et 302 incognitas
51. Unde rur sus habemus l et m, pro
52. sed non continent 310 et 301. itaque si ad calculum absolvendum sufficerent hae binae, pro duabus 849. 37 et 38, non opus foret devenire ad explicationem num seu valores lite
53. rarum n et B. itaque majoris comprndii causa, in aeqg. 16, 17, 29, 30, faciamus dion + d 2op 00 β, t _oin + d20ip = 00V et 120n + 2120p 00R et 12on + 2410p = 00 p
54. Et ita jam explicando Y, t, 8, I in aequ. norissimis 3738, ope aeqq. 16, 17, 29, 30, et 39, 40, 41, 42, fint ex aq. 37. sequens aequatio 43. et ex aeg. 38 sequens aequatio 4
55. 1002. 201. 310f + 1002. 201 d3109 = 1002. 201. 00B =1002. 210. 301F- 1002. 210. d3019 + 1002. 210. 0072 2001. 208. 3205. +2001202517209- 2001. 28 00-200+2 20. 3 12 f. 2001.20 39 . 00 n
56. 1092. 110. 3015 + 1002. 110. d3o1g = 1002. 118. ∞) - 1002. 101. 310 f = 1002.101 d3199 + 1002. 101. 00 so = 2001. 120. 3025. + 2011 1205, 701 (9-2av1. 129, eo p- 2007 1 1 2. 30 f 2ogi. 12 f. 1. 9 420o1. 104 0 9
57. troons BA
58. Quarum duarum aegg. novissimarum, ope invenire oportet 210 et 301 possunt autem haebines aequaliones ad majus compendii redigi, compendiote comprehue
59. omnia, in quibus abest 310 et 301, si hae binae literae incognitae suffictunt, quod tentabimus. In aequ. 43 fint: 1002. 201 0β3 vor. 210 vol 4 2001. or rof t 3roy). 1001. Wul eot. 2em. la ro
60. Undi ex ceq. 43, fit (1002. 201 f- 2dol. 2029) 310 + 1002 1019 d310 400 1. hrog - 1002. 20 f )301 - 100 2. 219, 1301 p0 ili aivor map Meni te e 2e t. pe a i ut 4 20or 100 t 2 1 a
61. A
62. Unde exaequa fit. (200. 1079 -1002 101F) 310 1002. 1019 d310 + (1002. 10 f - 2001. 1209) 301. 4. 1002. 110y d 30, 007 V
63. Videamus jam an ex his duabas aequationibus una componi possit, in qua absit tam sot, dat tam d301, quam 301, ita ut supersint solum 310 et d310.
64. ita enim habebitur solutio desiderata. Nempe aeq. 4b et aeg. 4 dividatur quaevis per coefficientem ipsius 3ot nempe multiplicetur aequ. 4b per 10 et 48 per 2∞
65. et de inde in unum addantur ita ev anescente d301, per se, superest, ut evanescat 301, fieri X (10, 220. 21I - p20. 410)9 +610.210 - 110. 20)51002 f = 0
66. Itaque reen tentenus aliter coua ex aeggu. 46 et 48 facere tractabilem multiplicvndo ibam per b, hanc per c et dende
67. Sed hoc non licet, quia ita eo unisuit s. g, et urire aeqvatio in datis
68. Comporendo et efficiento ut coefficientes ipsius d310, et 4301 sint aequales, et ipsius 310 et 301 sint etiam inter se aequales iun
69. enim aequationem fier tiae stabilem alibi ostensum 3t. Et fiet (2ag 12 ) b Evrofc
70. (1002) 201 (9b = 1002, 101GC = (1002 2109b + (1002) 11oGC et ubi cum evane seat g, habetur ratio b: c
71. Eiquemus jam et coefficientes ipsorum 310, et 301, fiet
72. 1 corf 1002fab - 101C) + 20019 (102C - 2026) ) 1002 f(110C - 206) + 20019 (220b - 120 c). Ubi loco b et c substituamus
73. Nam fit C: b = et = 210 + 201,: 110 + 104
74. quae ipsis proportione respondent e et fet 1002f (1 29(F 10I. 201 = 101. 210= 101. 201)) + 2009 (02 20 (t o2 roL 10 2o2 (101 20D) = teoz ff2o. 2, 9ruo 20s = 110. B 14. 200,
75. 1019 (110. 220 + 1. 210 - n0. 10 - 1 20201)
76. cesse copstantes, se
77. itaque faciamus non quidem coefficientes ipsorum du dio ut dio
78. eorb oC:, 2t0b + HdC = 13. 1
79. m 2 + 13. 201:, 12. 10 4 12. 101 L et in aequ 50
80. sed ita cum utrotius destinatur f, ehunescit etiam, 9
81. pro C et b, substituamus proportionales fiet 13. 1002f (201 12tC) 1 12. 10079 (102. C - 20 6) = 1. 1002 f (Hoc - 210b) + 13. 20079 (220b - 120C)
82. 3 HC
83. in fiet
84. itrtob
85. ay + Y = 0 quae rebditur
86. 12b iotc
87. tequatio 49 ex 46 et 48 (piost multiplicationem pe respecive per b et C) compositis formata, compendio scribatur fin im 3Id + n 301 + p d310 + 4 3 ee t h = (aB..
88. 2e:
89. ee
90. 3P1 - 181. 1.
91. naaum ntculi instituto inveni enveni prodire et in datis efficiendo mpZ n. mg - np. = gdp - pdque si jam ratio p ad 4 fiat constans, se veluti 9-
92. his positis reperi posse tractabilem, seu aequationem istam aequivalere huic sd + ndB + Y. si fat O = E310 + A 301
93. = 12p dp: B3, seu tandem 12m = 13n C et 13 12 p = 3
94. 2 12p: 13
95. quod epistemcis in
96. 100if (. 110. 201 ( 13. 101. 208 - 12. 210.104 (B 201. 10r)
97. quod tentemus efficere in aequation qs, composita ex aejus 46 et a8 Ept autem m = ex 5eg. 4b multiplicata per b et 48 multiplicata per C, eritu
98. 9- voeg Eferne F. anf 210
99. s et mp. n 12p: (3 = np = 12dp 113 er ( 12 p dp. 13
100. m = 1002f (201 b - 10iC) + 200) (102C - 202b) n = 1002f, (10c - 210b) + 2009 ( 20b - 120C) p = 10029, 201 f. 101c 9- lvorg,
101. -= 12. 201 +1, 210,:, 12. 101 413,110
102. super est jam al faciamus 12M = B3n
103. -1 20. 11
104. 1002. 1f (C (01. 12. 4091 201. 1, 1. D =101. 12 201 -101. 15. v0) + 2 0v1. 129 (102. 12. 201 + 102. 13. 210 - 202. 12. 181 - 202. 13. 10) = 1av2 1f(no. 12 o (+V0. 17 9 210. 12101 f21O. Iu 10)) 12os. 3 (ervo 11 .
105. Ergo si 12p = 139 fiet 12. 201b = 12. 101C = 13. 10C - 13. 210b seu c
106. enimus proportio
107. et in valoribus m et n, pro b et c, substituamus quantitates quas mv
108. Fietque
109. itaquex solis 310 et 3oc rem consequi non licet, sed adhibere postet adhuc 3 320 et 302. Itaque redeundum est ad aequa 43, 44 ibius explicant (), A, er
110. sed video ne hoc quidem prodosse, et similiter eancdestiniae f, unde et evanescit
111. 1do2
112. Et 00B = D d1o1. 202. 320 = d 101. 220. 302 + d201. 120. 302 - a 201, 102.320, : 00. 1002. Compendio sit 120 d 201 - 2e0d 101 2dD, et
113. d0p. 302-20d1.3le: 00. 1002, et 90d = 1129. 202320-1120. 220, 302 +2120. 120. 302 - 2120. 102.320,:00. 1002. compendis sit H20. 202- 21208
114. per 39, 40, _1, e2, et per 14, 15, fint 00 β -7 d10. 202. 320 - di10. 220302 + d2go. 120. 302 - duo. 102. 320, : 00. 100
115. . 2 20. 320 2020. 302: 00. 1002. 5t 00 s=, 1219. 22. 320 -mo. n2o So2 + 2210. 120. 12 - 2210. 102320, : v0 10o2. Compendio til 1210. 202 - 2210. 102=- o2so, et uo tuo. 121a no = 201
116. 001002
117. 29 + 100 2. 201. 10d2. 302 = 1002. 2001. 202. dyrog F1 2v2. 2001. 2w d3 029 = 0
118. et fier 00n O210. 320 = 2010. 30∞: 00. 1002. H. jamvalores inventi ipsorum β8, γ, A, e substituantur in ae q9. 43 et 44
119. 8.
120. et fier s e
121. der 22o = 10d2302,
122. et 103 1701 2o d101 =od1 et fiei Y
123. + 1120. 290 = 2120. 120 2020 et fiet
124. et ex aeq 13. fist 1002 (1o2 201 f - 2001. 2∞ 9) 310 + 100n Vo2 10N 7 1310. + 10ve ( a mrogu 301u ar a E. veg 13a=
125. 410c
126. L200i
127. 2001. 202, 202036.
128. e ex aj. 44 fiil 1002 (2vo1 1029 - 1002. 1015) 310D 1 ver. 10l9 d 310 +1002, 00 2. 105 ,30j u 21uc2. 10, l341 +1 col. 1. 2081. 715) - 1092. 110
129. 2ool. 120
130. 5i0.310 + sol d31o + 52o3B1 +5or. d 3o1 + 5 30. 320 f5(3 d320 +5∞. 2o2 1 504 dzo 63)
131. (10. 310 + 601 43, 0 4+6i0. 301 +602 1701 + 827. 320. 603 432 +640. 302
132. et 301 = 21320122302
133. 30. 90,320 +50300d320 +540. 00. 302 +504.00 dse
134. At in 1501. 12 1.
135. 502. d22.
136. 2
137. +
138. 502. 2.
139. +604 dy 23
140. et in aq G Huimus 510.H.
141. vates, ut haten
142. psisnt
143. Q tor roq ivrdvus -Morar reci tvoiseo
144. 630. 00. 320 +6038320 +640. 8.302 +00t. 00. d3026
145. 610. 1.1
146. 601 di ...
147. 120. 2
148. ot du.
149. quatinae ripnict
150. 4 10
151. 11.
152. 601.11...
153. 602. 4...
154. finni ras coefficientis est
155. lieis olsitns
156. o u o
157. Ex org er iomprdu fint
158. 15964 fint
159. 5ol. di . . 4301. 11
160. L a2
161. 502. 27
162. onanus ce ad ad siat S . E puirae fl 1542
163. pic
164. a susicere d
165. 1 310 p. 320 + 12. 302
166. 6o. oo
167. 210. 3og + 2og1. 20. 2010. 3
168. 1201d8. 3c03
169. o ec aiio
170. 50i deZi 601. _ ... -0
171. 6or 822 . 602.22..
172. multiplicotur aerg. 68 per 60
173. im ab abtera tollendr eomnesce
174. teram aeguatua
175. 43e0 in aeq. prod
176. s 40. 001 510 m2 25odi d d2 C, . A40. v0 + 610. 12H Bo,
177. 604. 00 +bd1 12 ++ 60Z. 22 Py 5oq. 00 +5o1. 12 + 502. 2
178. sini 60r. (526 - 50d= 502 (620 -602)3
179. ant alisele
180. o1. 12
181. C+C
182. ti eo
183. o pir. o
184. en aoormat sond. onr ey) (.
185. ior vornog
186. ner 320 et ag et habebitur ita 320, Ergo et 50ct

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Leibniz-Handschriften zur Mathematik LH 35, 13, 1, fol. 76v. Machine transcription, not an edition: Leibniz Legible (model FoNDUE-GD_v2_ft_Leibniz, run 74). https://leibnizlegible.com/page/00068196:0152?run=74