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Leibniz-Handschriften zur Mathematik LH 35, 13, 1, fol. 386r

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.

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The machine reads it as

  1. XX +X - X 1+ p-1 =
  2. XX + X - XX + 2X - 1 - X + 1 = X.
  3. 2 X dx + d - 2(x - 1) dx + dx. = 2X dx + dx = 2X dx = 2 dx = dx = dx
  4. O KNL +XHI - XXHX = X. Ergo _ i(x + 1) ax +. ax - 2X dx = dx = ex
  5. d (X+ B) + XF1) = A ExEX = dx
  6. Ergo 2X
  7. Ergo d, 2(X tns + 3(x t n) + Xtn, :6
  8. 2. 3 (X+ n) E F 3. 2(x n) ax + dx 8, : 6
  9. unde R(X+ r) + XFn -
  10. t3 XX + 6X 13
  11. a nntX - dx fi
  12. aequ. omn d, XX = aequiomn 2Xdx
  13. E) XX + 23. 2an + 3. 2X F 3. en + 1 :6
  14. 2. 3 XX + 2. 3. 2X, + 2. 3, 4 3. 2X + 3. 2n + 1, :6
  15. aequ
  16. +1X+ 1
  17. omn 2X dx
  18. aequi 2 omnX
  19. ceq
  20. omn X
  21. 2.6+ x 16 Xx 1523 (XF 1) +2.3(+ 1) + 1( 23xA 3X + aeg. 2X + 2 aequ 2, Xt 1
  22. Nam
  23. GXXI OEXX
  24. Hnua
  25. si dip () Alindv = (y) si (X + csisa) = (X sit a) = X + e FD (X + Clis sa)
  26. si des e si st (∞) v
  27. erit (X + nclisa) = fa t csist ) (X ; a) = omn X + C (5i
  28. li diff v e
  29. omn. dx
  30. 2X + 6XX +bX + 245 - 1H( - 3XX -x, es - XXAEX +1
  31. _XEr + XrD - AB (2+G. + XF1 = dx
  32. 2X3 + 3XX + X, :6 = omn XX
  33. aequ.
  34. (Sad) + 12X +6+ = bRx (bX aequ 12X4 - 2x+ 2
  35. quia
  36. seu
  37. s
  38. seu (V) = V H Numericam differentiam notabo
  39. d0 = (V) = p. v
  40. Et sydx sit om
  41. do = (1) =V (∞) v (V) - ∞
  42. dif v nota arithe differentia Arithmetica, et do geometrica.
  43. quoni sucum sit ergo rem cunt complectermur si Eodenmodo (uda = sum geom. et suum ydx est sumattutimet
  44. XFC- X
  45. iste nexte
  46. exxt
  47. l X. XrE XZ OXKIEeX
  48. siff V = ((∞) ()
  49. fiit dooe= summdusm.
  50. si V = summ. G idem quod () vman emin vinax aminim = omn. X
  51. Theorema ergo tale = sit v Z summ
  52. S EF
  53. V. = suum y idem quod supra vulterius insimum oun y ab infima ad
  54. summum assumta
  55. simu
  56. it
  57. dy = (y) - y
  58. Hinc ita demon
  59. 10 = () - et Ergo du - vicinde = (V) - vicin (B) - v + vicino jam v- viein v = 9. Ergo-
  60. y = dy
  61. pen vmovissa vprim sum tunde
  62. c
  63. erit do = suum dus copi
  64. vultra noviss.
  65. =ju aenno davys
  66. ad novissimum,) et conrectitur cum hoc
  67. Demonstratio pendet ab explicatione summus Ce connexa sunt v summus, et
  68. Theorema.
  69. sit V = summ
  70. quo noviss
  71. cummu a primo
  72. dy = (7) - y
  73. Theorema magnum
  74. v- vicin v
  75. Hinc ita ratio cinamur
  76. A(per 3) () = (y) - y per
  77. By) y et (4t)
  78. et dy
  79. et connectitur eum dla) - x viein v) + vicius v
  80. 9 Edo-
  81. it v = 2X1I
  82. et ge X sit B = 2X + 3XX+ X: et y = XX, ubi
  83. Exemplum
  84. XX, do = 6XXdX +6xdx + ax ib et dy = 2xdx
  85. reperietur x (positu dpomstante) dx manente constante (d vi do- vicindo = dy, niam fiet
  86. e
  87. sit = 2
  88. 2X3+BXX+X
  89. sit n X a 3xK
  90. = 2( - 1)3 = 3(X-
  91. 6(X - 1) - 6 (X - 1) - 1
  92. :6 = 2X.
  93. veniendas. Nam quia
  94. t Ergo connexa sunt sux = summ sydx et
  95. vutis nunerice euntet
  96. summas arithmeticas c
  97. Adeoque haec tres aequationes inter se connexae sunt, quin et
  98. videtur via aliqua apparere an
  99. 6XX +6X +
  100. FF + po connexa sunt v = summ. y
  101. in =C: Hinc jam
  102. 2
  103. ulterins novissimo = V er
  104. fit Aua vicindo= (ass victi) n y+ vicin yn Y
  105. etrabitur
  106. est v
  107. per qost de unvicin do
  108. 2
  109. quia v- vpcin = y
  110. V= summ g idem quod
  111. m resse qudq
  112. sim
  113. d AX
  114. 5
  115. a i
  116. et de = summ dy
  117. itaque si
  118. Et ila porro.
  119. quart
  120. uadraturis. Sit series
  121. Ax smum (AX r
  122. indentas vel descendenties numerice summabiles intervententibus a ratur et summ suxt
  123. summ y.
  124. axsvdx = sum faxsyax
  125. 43
  126. demes
  127. + 2 3
  128. i2 5 12 20
  129. ete seu = V
  130. ea sen
  131. et ita porro. sed non deo
  132. sum de= suum sum. dus
  133. avenitur ad summ v sum
  134. num sum t1 SX 4L
  135. u ntala alicest
  136. F EE 2Hinc F summ = summ sum du = ffrum auua
  137. 2tFe
  138. infinete semcal
  139. Xda contranii
  140. BXHI) CI a
  141. daretur sumo
  142. Hinc s or
  143. reciprocat rferentias arthmeticas
  144. datur et summ. du et summ daus, etc. Sc
  145. gressionis harmoniae, darentur et summae potentiarum
  146. noc summatio
  147. non est per
  148. si 2 = 5 fit 271, f. et
  149. dque p + 17 9 sZ Cf. (2, A + (Y 1) (f- 1)
  150. aot s 1+= + 3
  151. oxd a
  152. eries
  153. .+ 4
  154. (Erd flr (Z +d Z +1) f. d R+) fYtdE
  155. = 1ru + 3ad +4 etc
  156. ent differenta
  157. itt = (7+ 1) f = u ptca + (41) f2C Fa2C. Y t 1) 12ta (21 Fs
  158. Z dq + (Z+1) (fd
  159. EtdZ
  160. E
  161. du -
  162. d
  163. (2+1) C
  164. +(7+ 1) (FZt a2 fZ) (= q7 1. f.
  165. q+ dZ)
  166. Hie per summas numericas videntur reperdi posse tales feries summari, et quidem in terminis tam finitis a
  167. enque est
  168. uper
  169. intinitisctitem
  170. 50=q 5, 5 Y -1)
  171. (E ed ( 1, ubi si A Ea
  172. d
  173. cer
  174. iit 2
  175. mimas ronessins
  176. . numerita
  177. edi
  178. lo consti
  179. summam ipsius
  180. +Z(+B- 1)
  181. 20 si do non
  182. i
  183. EF ∞
  184. disfeculiae
  185. FF. EHine si detur summ.
  186. F+
  187. 1 X veri (7+ d
  188. F57B (= dZ
  189. Z + dz +1
  190. eito cum a spostus bil

Cite

Leibniz-Handschriften zur Mathematik LH 35, 13, 1, fol. 386r. Machine transcription, not an edition: Leibniz Legible (model FoNDUE-GD_v2_ft_Leibniz, run 74). https://leibnizlegible.com/page/00068196:0767?run=74