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Leibniz-Handschriften zur Mathematik LH 35, 7, 7, fol. 2r

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: 0.78.

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. 1v); this page is its right half.

The machine reads it as

  1. sed quoniam hoc quoque ipsum peculiarem in se habere videtur subtilitatem, et aditum aperit
  2. ad eos casus, ubi coeffirentes potentiarum ab X sunt series repriantes, quae quiae
  3. vel infiitae, vel finitae, sed in altioribus termmnis magis magisque crescentes, videamus an
  4. hiae compendium sese afferat amile succedat haec methodus summam primum
  5. ope ipsius N deinde summam provenientis verso N in n, investigandi. Quod
  6. sane aetificumi imprimis mivum est. Sit Ergo series y - + +
  7. = 2 2: Ergo gy= 4- Mu
  8. +1 8c
  9. altera illa ractione assequi possimus per coninationes ne Biniones. Nempe
  10. n+I
  11. nntp
  12. jam y seu = y =
  13. si tota series ducatur in se ilsam, quilibet Terminus dhicitur in alios omnes. Unde fiet
  14. et terminus unu in alios omnes
  15. erit A inen et facti summa erit in seu verso Nin vuut jum ruo sus hoc
  16. in quolibet instituamus, fiet: 4n- + et et ( + debet aequari y y = 4- 8+G
  17. na nimu
  18. ne Ln
  19. um
  20. a
  21. in incelligendo seriem numericam utcunque productam, seu indeficite finitam, et phemii
  22. quod experimento cum provernus in hoc casu serie tantum es ad duos numeros
  23. 4m + etiam
  24. poum producta sit y = + + = 4 et gy - 16 videamus en s pE
  25. est autem (aex. 4 - e 40 4 + EF0 Videamue
  26. 8+ 4
  27. etiam L Est autem ( _ hic 4 + 2
  28. 1+ 1
  29. n+ 2n +1
  30. n+2n+
  31. quod non procedit vs igitur acmratius investiganda. sit y = a + b + c + d.
  32. yy = a in a +b + CFd + b in a + b + C+ d + C in a +b+ c+ d + d in a + b+C+
  33. e si jam a in a+bt ctd = ay et siterminus quilibet a vocetur X, fiet
  34. gy = sxy poa, b sed jam video cur res non procedat, nam foret
  35. Exy = a ina + b in a+b + C in a + b+C. At yy = y sx = gy nam sx - y
  36. Alia igitur ratiine in compendia inquirendum.
  37. yy - (xy = a in b+ ctd + bin ctd + c in d
  38. sit y- 1 12 +3+4 +X fol y 1, Est atem jam a in b+ c+ d et bin etd
  39. et C in d’exprimi potest, si cogitemus seriem inverso modo sumtam daci in terminum
  40. enci semper in terminum proxime majorem, nempe ultimus seriei ordinalis
  41. sit N, serieique inverso modo expressae prodibit natura, si ubique pro n
  42. scribamus Y = N- n et siet n = N - r. quo et terminus seriei )
  43. numerando scilicet contrart Erit (Y) ponendo ( r esse ordinatem contrarium
  44. sit ergo (n) n sit exempli gratia series quadratorum 1 + 4 + 9 + 16 = y.
  45. cujus quilibet Terminus (n) et L ∞, 4, 4sit1 vel 1 vel 3ec sit n. quaeritur
  46. et secundo: 1 in 4+9+6 + 4 in 9+ 16 +9 in 16 = ((r + 1) in S(r) = in Y y
  47. e
  48. Et ita possumus retinerus (N) ergo yy = ((2) is +) (2) in Y- y. Sed hoc
  49. identicum est, nam nam S(n) in Y -y = ((2) Y - ((2) y et (()Y = Y s(n) = yy
  50. unde fit yy = 5(yy + yy = ()y. At si retineamus Y videamus quio
  51. prodeat: quia (1) = n fit et n = N- Y vel R X (nam N et R coinciduat)
  52. Xr1 Narr
  53. fiet; (Y) = N2Nrar N - 2 Ar = rr
  54. Egps y = (n fiet yy = fri fi + ( rrrr in ) sN acum = fr
  55. n+na
  56. videamus an
  57. 2
  58. Quod est plane mitabile ratociundi genus, et fier genemliter yy= f(Y per
  59. e
  60. sit unus ex terminis NNN
  61. e
  62. endastimutes
  63. ex
  64. 2
  65. In
  66. AEN
  67. Generaliter igtur sit y = ((n) et r- s fiet = Sen) sin) + s rn) pr
  68. ubi tantum (1) eodem modo ontur ex _ I ex H ex N- Y (vel (Y+ 1) ex N- r. 1)
  69. ut (n) oritur ex n
  70. Ergo si y = sSn fier yy S n su +
  71. (
  72. 1+
  73. p
  74. -ENY +rr in (N- 2NrE
  75. +Z Y
  76. Generaliterigitur si y fn ft y = S() et r = N - n fier y = sn = ((2) () + sr+ 1) Sr
  77. Eodem modo licelit, tale theorema comminisci pro cubo, et pro alia potentia quavis
  78. pu si. 0 e ar pterit
  79. 2n +ED: 2
  80. 1+19
  81. AA
  82. norga
  83. +IMFI
  84. sn

Cite

Leibniz-Handschriften zur Mathematik LH 35, 7, 7, fol. 2r. Machine transcription, not an edition: Leibniz Legible (model FoNDUE-GD_v2_ft_Leibniz, run 73). https://leibnizlegible.com/page/00068079:0003?run=73