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Leibniz-Handschriften zur Mathematik LH 35, 13, 1, fol. 88v

Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German. Mean line confidence on this page: 0.80.

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One image of two folio pages side by side, registered by the library under both labels (also fol. 89r); this page is its left half.

The machine reads it as

  1. Esto aequatio data dy: x = XX + gy,: aa, seu
  2. (2)
  3. i3 qaadys +v agdy =d
  4. vi 12, 10, 11 suyt
  5. fet ly :dx= 22aa + Elayt 20gy,:, tiaa F quay ~ 2XX +9y,: aa, et habebimus aequationem
  6. tr. 22a F, 2103y +u2oaayy
  7. 20
  8. in auxx+ qa xy 4 aayys + 28 ay3
  9. vidermus an hic utiliter liceat facere 4) = 11. mar. X1 7X
  10. etfiei 12, 1. 21 a = 11. 20ay + 10.40 yy 120
  11. 10. H ay
  12. dA. D + ac du +dH 20 aydx + 102gt ydx + 1. 2onady + 10. qu.ydy 0
  13. Hanc aeq. 7. differentiando, siet:
  14. U d21 aa do 11 d2o dy dx
  15. 2.1X(12 20 dx A. Et ay ex
  16. 12Xx dqudx 10 dal ay ax
  17. ut (80
  18. 10 d4o gylx 10 4 adus
  19. (ullistinti uite e
  20. eu rindeltiglicata per 60 addita aeg. 2ib
  21. Edinoutst 1 nabru 4 100 obyaue 2240
  22. 00. 6oyydx -00.60 aadu
  23. fier 40F0 sed ita etiam frertet 6020
  24. quid qun biecto itaque vereor, ut liceat tollere temminum absolutum in aeu5. Eo ergo retento
  25. 00. 60 XXdx +
  26. duc du. 21 cayax L dureang X tdlo. 40 ydx + 11. 21 a dy 7 H. 20n ay dy , 10. 40 agy dis
  27. si differentiamus fiet
  28. pde Ta du
  29. d
  30. Bx
  31. Udro
  32. 12 d.. B2XX 140
  33. a10 4
  34. 101a
  35. 10d4d
  36. 12 4 eb 2. 10. 41 aaydy
  37. 140. axedy
  38. comparandum crum aequ. 1. multii l. per 609+61a
  39. bi axplx + bxpydx + biagydu + 60 2 dx -6qa dus - 6causny ce
  40. b=0
  41. iat
  42. fet b2 2. 10 =29 et ex dy fit 2. 1941, 11. 20 = 60, seu 41- 20 4 60
  43. ex joste
  44. Ex gydx fit d40 = (2) 20 + 41 (-20) Ex y, d28 7 d 4) 1)2. 21
  45. d11 =0
  46. 1
  47. denius ex ex sit d21 + 2X 2. 40X =+ X d qu 2.20 + 4, - 20, antitti potest aequatio hac
  48. ultrum qoia sola continet 22 item ag. prim ex is 12, quia sola continet d4
  49. Porro ob sublatum terminum altonum in aeg. 5 Z2= AIXX: aa Ergo ex aeg. 1+ faet
  50. d21. F 2. 40X + XX d40= - 2. axX + 41 - 20 Res ergo redit ad, ce gy. 12. 13, 16
  51. seu 2072521 - 4
  52. d2L ex aeq. 16, d21 + 4sset 4. X() 21 + XX2(2 - - 2. 4 1XX + 41 - 2 s21 + 41 - si Aad per ta set +. EF sat = L XF L a1X + a - 2( it + 41, et differtulando fer.
  53. ponamus 21 esse cognitam pro arbirio assumendam Ex 13 fiet 20+ 41= 2521 Ergo
  54. res redit ad aeqqu. 12) 16, 17. Eit ex 12 fiet d40 = 2 sel Et 40 = 2 s521. superest
  55. et habeber I, atque adeo cum 21 pro arbitio assumi possit habetur quaesitum sea quaesita 4I
  56. la
  57. potest poni 21 constans adeoque (21 = 21x et B2 = 2. U XX
  58. Assumitur alia differentialis ANX A- 4 1xax + 2 a ydx + 20 y ux + N aatus + 40 a y dy = (2) = 0. Et fiet
  59. ex 1; - d g: dx = XXF yy, : - aa = (per 2) - MIXX + 21ay + 20yy, :, 4aa + 4ugy unde prodit aequatio
  60. +40xxA + 41 ay + 40 yy = (4) =0
  61. F 41 ad XX) + 4 cl Xxy + 41 A yy + 40y3. 720 vel +ila + 20 ad
  62. = AI a XX )+ 21 aay + 20 a gy
  63. unde differentiqudo fit 2 4 Xdx + d H ay. + d to yy rodau
  64. ago XXx +d 20 ayx
  65. Haec aequatio coincidetur datae, multiplicatae perbo,
  66. cui addatur assumta 2, multi plicata per b, sart
  67. arlaaix
  68. 60 xxdx
  69. .
  70. + 60 gydx -b aadus
  71. 7 = (9 =0
  72. et fiet .. ...
  73. ∞ AIb xxdx +2by dx + 20b yyax +alabays +40 b ydy s
  74. Iustituta jam cincidentiatione t ob ydu, sit b = 2. Et ob du fit (2)+ 20 = - 60 (+2) 41 Erit 60 = 4 2(7) = 41 - 20
  75. vyarex
  76. ob yydx fit 440 = 60 +2. 20 seu dgo = 41 - 20 +2. 20 seu d90 = 4- (8) 41 + 20 bC f
  77. Ax
  78. ob ydx fit d4l + d20 = 2 t (9) = 2. 21 denique ob dxx fit 2. +oX + ex d G0XX + d2 I = 1) ( 1 - 2C) XX. s
  79. redit ad solvendas aequationes 8, 9, 10. Sit 2 quantitas constans, seu d21 = (11) = 0, f. eth
  80. Et ex 9 fier 41 +20 = K (12) = 2(21. Ergo Ex 8 fit 440 seu A 25i1 et 40 (14) = 2s21. quibus
  81. valoribus ex 11, 13, 14 substitutis in aeg. 10, fiet 2421 + 2XX21 = = 41∞ 2OXX - 2. 4 1XX) -20xp - 21xx
  82. andesit 2X (21 + 2XX) 21 - - 2XX s21 =0 seu B21 + 2Xs21 = d seu (s21) = (2) 521 + 2 1X seu 21+ 2 21 dx + 2X d20
  83. Ad qua 221 =0, hoc non licet, hisi sit 21 = 0. Hoc pacto ex aeq.q fiet d41 +d20 = (17) =0 Et 41 +20 = (18) =c
  84. Ergo et d40 9) = C et 40 = (20) = CX sed ita ex aeq. 10 fiet 2E XX + CXX = (- 41+ 20) = Z Cxx, quod
  85. est atsurdum Ergo nonlicet facere 21 constantem nempe 41 et 20, non constituunt nisi unam quantitatem 21 + 20
  86. itaque in aeq. 10 fiet (2 + 2521 + d21 = - X∞ C 2Xs21. seu 2X (s21 + aex se1 + d210 seu
  87. 25
  88. et posito 131. fiet s21 = dg et 21 = ddgu, et d 21 = d3’g, et fiit 29X + 4XX d + d38 = 0.
  89. si esset d3g d’ =0 fimet 2X = - 2d8 seu log X = logi
  90. ex est conlans non potest ’esse constans, C non litel faicere 2ca = - 41XX ad deprimend a, ey.C
  91. et d = dv et d(() = b lv. et ddlo - b dv acv b
  92. A 8- V Bo sia d log f ag L hg bts v et a V- vB s = do b + dtotr b + de dso. lul epupuirbid qu nus sen foe tile prio sim letepes ater

Cite

Leibniz-Handschriften zur Mathematik LH 35, 13, 1, fol. 88v. Machine transcription, not an edition: Leibniz Legible (model FoNDUE-GD_v2_ft_Leibniz, run 74). https://leibnizlegible.com/page/00068196:0174?run=74