Leibniz Legible

Machine transcription of the Leibniz Nachlass

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LBr. 57, 1 · Bernoulli, Joh., fol. 119r

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: 0.77.

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. 118v); this page is its right half.

The machine reads it as

  1. ad 10
  2. Addamus adhuc Fractiones (ubicas:
  3. +n +O = A B A~ + , B -8 + A1B, 18
  4. 1
  5. +1
  6. 54 216 =B 9 8. 125 826
  7. A+B = B + ( 1r5 + 316). Ergo A = 8 1B + + 3, 6 et A- B - 5- 2B + 4 1g5 249 =0
  8. Ergo B, = 0 = (64 t 125 F 216) 3:38-2 et _ Ergo AFB = 8C = 8 (647 15 ) + 8 2 (1 t) 8.2
  9. sed Undetandem A+B = 40 = (sit 15t 216) :, 4 -1 seu fiet
  10. us ecis etC)
  11. infinita, fiet: t g + , + 54 + ,25 + 3,6 4( f +t =
  12. +
  13. 1 + + gy Y = ete = 4 (, - + L - etC):, 4-1
  14. (alculus generalis sic institui potesti) + - = = C
  15. et
  16. 3d0) L = + + = + - + = = d et B3tio) + + = A et (4t0) + + = B
  17. 2C
  18. 3. 4. 5.
  19. 6
  20. per set 4 ft1
  21. es2 fit
  22. et ((t0) e + 2 + = L et (610) L + = + = M (7mo) (A - B = 0 (8v0) (+ + B = d
  23. (11m9
  24. Scomeo
  25. 2. ex calculi evidentia
  26. gno) L+ M = D jam D: E- B + C M L A+B, :2 = B + M. 2. Ergo, A = (2. - 1) B + M
  27. Unde ex 7 per 11 fit 2B) (2 -2) B + M = C. seu (13t) B = C - M, :, 2. - 2. Unde ex H. fiet
  28. A= (2-1) O_(2E - 1) M + (2. -2) M, :, 2 -2. se A = (2 DA (2 -1) O - M,:, 2 - 2.
  29. ExErgo ex 13 et 14 A + B = 22O - 2M, :, 2. -2 seu (15to) A + B = 2 C = M, :, 2C. - 1 = -
  30. explicando per 3, 4, 1, 6
  31. itaque sit denique (6t0) e + + + +
  32. 6
  33. c+
  34. 3. 4 5i
  35. jam cum s Se epost ipsius M posterioris partis ipsias D, semper decrescat ratio ad priorem L, ita ut tandem
  36. fiat minor quavis data, si assumatur numerus terminorum in D satis magnus, fiet denique in
  37. ca itaque in casu infiniti M evanescet, atque excepto uno casu, quo e= 1, itaque pro caeteris casibus
  38. in serie infinita M evanescet, et fiet adeo (17mn) + L + te etc ad
  39. 22
  40. (800
  41. ut 2 2 ad 2 -1 seu d: C = 2 :, 2: -1
  42. etc, Rotile
  43. 4
  44. ese
  45. E
  46. dideamus num an O possit inveniri in casu seriei infinitae, ut ita habeatur
  47. etiam D, cujus infinitae seriei a summam potissimum quaerimus. quam in rem ita procedo
  48. quaecitur C = (imo) + L - = ete Sgat 2) X - x + X - etc = y. Ergo si daturY
  49. t
  50. Assumatur.
  51. generaliter, dabitur et C) tanquam casus specialis ipsius y, nam Z C X t ∞ naro (H) D.
  52. nam 3to), O = y sit x = 1. His positis fex 2FB x et C
  53. id e 2 fit (a), dys = X - = + X X erc, dx Ax ax = (6 saleta) ( ax Exx
  54. per tet 6 fit (y = s, ((itX) dxiX id est desideratur area rogarithni quadvatura, figuiae Logarithmicae, quam
  55. =(k)
  56. mo)
  57. a
  58. utique habemus. Ut ergo inspiciatur figura Veniam
  59. vertex _.)
  60. describatur Hyperbola W(IV) cujus centrum C, Asumptotae CAX, ((4) Ia potentiae
  61. seu constantis seu quadrati Aa, latus CA sit a _ abscissa AX sit X, etq
  62. Et X Vordinata, fiet aa = (8v.) a- N +N X etc. per punctum
  63. a at
  64. A describatur logarithimea AZ, cujus asymptota sit ipsa Ca) C, axis autem
  65. atX
  66. sit CA ad quem ordinatae XZ sint logarit hmi ipsorum numerorum CX, subtangens
  67. autem sit a con utique fiat X Z seu X Z seu la+X) saa dx = de (10mo) X - X + X -X ete
  68. mo
  69. at)
  70. a 3a 48
  71. ita nempe erit rectanus. sub AC et Z Z Z aequ areae A VAXWV. Sed hinc jam
  72. orro patet Arcum figurae loginthmicae AXZA aream aequari (12mo) ipsi y (il C per artic 2
  73. per Ca 10 et 2, E, 6, nomfiet at AXZA = ) tCarX) dx = ff(ac du) dX)- f.
  74. Hinc quenitur
  75. ede
  76. Quaeritur (I1FX) ax: X y. ((+X) Ex: X = dus datur ergo (3tio) Sxdu = s, log(1+X) dx
  77. ax 1 -co fit ligx. n. X x +n A_ L(r dx: 14+) facianus
  78. B =L et 14x X. seu Z= C + X + 1, et fiet LX +X + 1 = co et fe LXm n. X L n (+ d)
  79. +M f∞ Z ( x (~m. 1 (+ aud. X, d, L1+ X X = n. X. liex: dx + m. Xu. liF. dx:, 1+X
  80. d, LIFx lux = IX dx(: 1 + X) + l Et X dx: X
  81. fiet: + + + + + L + = + et = 2 (. 4 t 0 25 36 ay 5n)
  82. i
  83. -_t)
  84. lxdx (:, 1 AX) = lxdx, = X + x = B eta f(xdx = xIX C SXIXa = X IX ZeL 19
  85. c
  86. f.ta SX IX x = X IX X Et sp lx ax = 1X. X
  87. hies lin 0 = 1. 1 9 t1s ele - 2( 4 1. 51 3dsen +. 1G ser - E, 1. f 4 4g eut 2
  88. (IXdx - sIX, Z - 2 + x etc
  89. +X
  90. Ergo senique sit
  91. x. H+) Ergo Ex F) = Ex (12X)
  92. unte sepuntur esse ex insinitu opara ipsius A ax:)
  93. - 1, X - X + X etc Eugo sed =- X +x ete = log _ x
  94. stito
  95. Ergo quia liex lx = (1X dx + S LitX cx hic (LiX lX = (LEFX LX) - = + X - X + X esc
  96. a
  97. ita nhil discimus nevi nam itustat esse storede X + L X + X esc = 0

Cite

LBr. 57, 1 · Bernoulli, Joh., fol. 119r. Machine transcription, not an edition: Leibniz Legible (model FoNDUE-GD_v2_ft_Leibniz, run 74). https://leibnizlegible.com/page/DE-611-HS-958803:0239?run=74