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Leibniz-Handschriften zur Mathematik LH 35, 4, 7, fol. 4v

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: 0.86.

Original at the GWLB · Page image · Annotations (IIIF) · Download text

One image of two folio pages side by side, registered by the library under both labels (also fol. 3r); this page is its left half.

The machine reads it as

  1. 2
  2. by2. b78.
  3. demonstratio generalis de irrationalibus:
  4. sit quantitas
  5. sie aliqua 2b. poterit eadem et sic enuntiari.
  6. b b
  7. E
  8. + 1 - a + b - s - ac Quanquam eim V- ac sit
  9. nons
  10. quantitas inaeginaria, summa tamen ideo, minus est realis; quoniam
  11. imaginarie destruuntur; Dividatur haec formula in duas partes
  12. binomitem et es b+ Y-ac, et residuum - Y-ac, et
  13. utriusque separatiu in vestigetur abus erit ipsius bgr-ac cubus hic:
  14. +b = ac Y- ac et ipsius b - s-ac, cubus ent + b +ac f- ac
  15. 3bac. + b ....
  16. adeoque erit
  17. 16) +b3 = ac Y- ac + +5) +b3 +Vac L- ac
  18. sive
  19. -7bac +2b..
  20. -3bac = 3 62....
  21. ibac B B +Y) bac
  22. btac
  23. b + f- ac 4 + b 6 - - ac
  24. -3bac -3V....
  25. e 2b
  26. N 2b
  27. cum residuo suo conjun
  28. quodsi ergo extali binomio emsmodi semper extrahi posset radio
  29. Cubica, quemadmodum exhoc quidem potest utique semper junctis inter
  30. se binonno et risiduo semper hebebitur tolli posset imaginaria
  31. sed quoniam data ejusmodi expressione:
  32. 16) = +Yr -93 +- Y- 4 qualem exhibent subicae
  33. 4 37
  34. 211 4
  35. 27
  36. quia Cquantillas data
  37. aequationes, non semper extrahi potest, id est non semper (Y. in b.3bac
  38. data
  39. la rec quantitas 93
  40. in duas atres:
  41. 27
  42. aC3 + 6acb- 9btac dispesci potest, a nobis, sine alia
  43. aequatione, aeque ac data, difficili potest; ideo fit ut nalitte
  44. ex quantitatibus realibus. non semper possimus imaginorias eliminite
  45. rationalsibus
  46. Exempla autem in rumeris proponere utile erit
  47. sit
  48. aequatio qua et Atter Altertus Girardus utitur: X X = 13X - 12p0
  49. i cujus radix vera, est 4 constat autem ex regulis
  50. Caud Ex formulis autem scipionis Ferrei sive Cardam erit
  51. XY 15)6 + s- 1225 + (0 +6 - s-mis
  52. 7
  53. videamus ergo
  54. an posito quam expressionem rectam esse et realem, et admissendam
  55. sic demonstrabo: Ponatur X ( 2 + -t + 2 - f-
  56. erit utique X 1, X uti aequatio postulatii videamus nunc an inde derivari
  57. rimu
  58. iu
  59. Cutus 21 Su, erit 18
  60. possit formula cardanica
  61. -37
  62. 2 + Y.
  63. faciendoque
  64. b + Y -ac
  65. b H2. et ac 1 2
  66. habebimus pro cubo ipsius bireac formulam hanc:
  67. 3, 2, 3 2
  68. fa
  69. +A..
  70. sire 6 Nimirum cubando, et Superiorem formulam
  71. + b - s- ac huc asplicant
  72. +8
  73. 46, 2, 4A sive summando 6 + Y -nc
  74. 9. 16, = p 41 296.
  75. Eodem modo cubus a 2 - f. erit 6 - s = 1225
  76. ac proinde
  77. 3) 6+ s = 1225 erit 2 + f - 3. et (36 - s - 1225 erit:
  78. 2 = s.2: ac juigendo binomium residuo erit
  79. X sive 13) 6 + s - 1225 + 10 6 - +
  80. Es idemquo quad 2+ s. t +2 - s. 3
  81. id est erit 4. Ut ostendere propositum erat
  82. vel
  83. Ergo
  84. sive ac H = 13
  85. 8 - 2, 2 ac p 6.
  86. 8 -6 ac.
  87. 486
  88. 29b

Cite

Leibniz-Handschriften zur Mathematik LH 35, 4, 7, fol. 4v. Machine transcription, not an edition: Leibniz Legible (model FoNDUE-GD_v2_ft_Leibniz, run 73). https://leibnizlegible.com/page/00068004:0008?run=73