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Leibniz-Handschriften zur Mathematik LH 35, 3 B 6, 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: 0.77.

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The machine reads it as

  1. 20 Decemb. 1699
  2. pag. 1.
  3. pro Fractionibus dyadice exprimendis
  4. 105 (001 seu 100 = e1 + 7, et 1 = 0 ot o101 ete
  5. 14++
  6. u
  7. Nam Ndm = 4B (0∞ 00f s + vel 2010101
  8. eF + 1 orsiaoI
  9. l000000
  10. aetrectea
  11. 100000000
  12. vel 6 01 010101+
  13. ei eb
  14. ex his patet periodum quidem semper descnere in 1. Qualis hoc loco est 01
  15. AXA X,
  16. atet Tibobobo
  17. 166 (00101B7 vel 0. 010101 vel 0. 0101010101 etc
  18. Cat
  19. nu
  20. seu
  21. 101 +
  22. seu
  23. 10101 + L
  24. 100000d
  25. 101010101 et
  26. 1 0000000000 en
  27. patet ex his periodum, ut, hic o OI, semper desinere in 1. et resitrii numeralorem
  28. 10000
  29. ()b) esto antib, ita ut si b sit 1 tunc (b sit 0; et si b sit 0, func (6) sit 1
  30. si divisionemfinias semper esse 1
  31. b Theoremat. b + 1= b (6)
  32. . Nam A 10
  33. 6b1
  34. 1(1) = 4
  35. b + (b) = 1 Nam B si b = 1 erit (6) =0 et fit 1+ 0 = 1
  36. b -bb = B3 etc sed b = 1 peo 4
  37. si si diffi
  38. seu
  39. (6) = 1 - b
  40. b (6) =0
  41. P 4.
  42. 9(es -0s1
  43. sin b =0 ert (6) =1 et fit 0+ 1 =
  44. et b = = b) patet ex praecedenti
  45. Nam si b = 1 erit b(b) = lin 0 -0
  46. si b= 0 eut b(b) = 0 in l20
  47. enempe 0 si b =c et 1si b = (C1
  48. B.) BC20 = b uc
  49. Ci b2c erit b(e) = 0 et (b)C 20
  50. (C) et (6) C= () (B) = (6) = c et
  51. Hic quaelibet t quantitas
  52. si bec ent b∞) Cs t vel 0 si a - 6)
  53. (1b)C)
  54. si molet cec, ert (a) =b
  55. si b = C = 1 erit b-c70
  56. si N b posito = p sit C = (K) et b posito (p) sit C = K fint BC) +d) =
  57. b(C) + (b) C = 0 vel 1 nunquam
  58. + peA _ + = (b) C + P b (p) + C(K) = 1, et (p) = b - bc(K)
  59. semper b + c = c+ b = b(C) + (b)c
  60. bc erit (1) = (bc) = cb) (C) sit t() B)C
  61. dtrosquod sucito ipistma
  62. si b =c fit (b c) = (6) CC
  63. sin (b) =C fit (b d = bc
  64. Nam _ sit, b =c, fit bc = b et (bc) = (b) et (b) (C) = (b) (6) = (b1
  65. Hbc-o
  66. X- 2Fh +nh = X + h = 2Xh+ h
  67. =Xh4h = 0 X = 0 +h
  68. X -X + 2h - 2XF
  69. sin (b) =c fit bc = bb) - 0 et (bc) = 1 at 1b) (C) = c() - 0
  70. pordiseum
  71. ()
  72. er
  73. de de
  74. by
  75. ba
  76. ce ep
  77. u
  78. du dn dta
  79. E s S B
  80. ete
  81. il
  82. bn
  83. n m
  84. 1
  85. an am al
  86. bm
  87. V
  88. V
  89. d vid 8. pag.
  90. nas
  91. si b- a sintque ambo positiva
  92. erit b20. Nam s
  93. si b (c) erit bet c +1) = 6+
  94. ) +
  95. si b= (1) erit b()
  96. P
  97. X, X- 1 =0
  98. a, 4- 1 = 0 = aa - a 22. AXX -2xh-h
  99. xu
  100. 0, 0-1 =0
  101. 1, 1-0 =0
  102. h.
  103. ex
  104. X- 3Xh+3Xh - h)
  105. -x + 2hX = h 20
  106. 2lx ata
  107. seu X = 0+
  108. b1
  109. smu
  110. XX-2Xath
  111. X-h.l
  112. X3 2K CXth
  113. XL+ 2hh h
  114. EX +2xh - h
  115. 3xh tgh Eh
  116. xect. +t2b
  117. =1+3h
  118. nite a bsc =0 sequenti coliadoud . postsequenti 0
  119. al + 1 = 6 150 Ergo n al =1 adeoque a = l = 1
  120. am +bl + i = i + b + 1 = B 150 fit m+ b = 1 seu m = b)
  121. p + b(C) + c(b) + d ∞ 1 = iso mlpo si b = c sit b (C) +b) =0 et p + d + 1 = 1so et prd- 1 et p -Cd)
  122. n +b m + c + 1 = n + c + 1 fit e at = 10 fit, n = (c)
  123. Sin (b) =C fit b (C) + (1b)= b + (6) = 1 et p + 1+ d + 1 = 1s0 vel ipso
  124. , erit p = (d) sin = (b) = P erit p = d
  125. unde b b(C) + p(d) = 1 ret p = , - b (C), : (d) = ad3 b(4)d) - () (() )) (d) (b(1)) = p
  126. 10 et p - d,. itaque si b =c
  127. 1. + b() Cb(0)) 4 Ln) 1 d(b) + e ubi si p = (d) item s seu si P +d = 1 item si p+ d = 0 fit additur 1 in
  128. Ergo p+ d = d l
  129. sequentem columnam. sin p+ d = 10, addetur 10 in sequentem columnam. Aa poterimus hinc aliquid
  130. generale ducere in sequentem columnam addendum 2 id st X. si p+ d = 0 addetur et d + + fiet X = p + d + 1
  131. si p+d_ = 1 fiet X = p+ d sin p+ d = i0 fiet A = p+d Ergo ducantur in se invicem haec tua
  132. quibus ductis
  133. rationatiter, nam X et xx hiac non sequantur
  134. xah L0 vel X-h=0
  135. d per compendium sit p+d = h esto vel X - h - 1 =0 vel
  136. +XXh- H. sed ita qu x potest esse 10 seu 2, toem est de h
  137. ideo sic
  138. sed quia 10 aedi ad columnam sequentem est potius 1 addi columnam post sequentem; = p+ d = 1
  139. it
  140. inficem fit aqualio pro X, nempe X= 22x + Exh-
  141. procedendum ut columnae sequenti addatur 1 si p+d = o et erit X = p Ad + 1
  142. ad fiet X= p + d sed si p+d = 10 erit X = 0 Ergo X = ++d p p- d et Nam tunc p = d = 1 et
  143. ita ducenda in vicem X - 1, X - b - d, X +d - p et fiat XX - 2p X - Dd - pp), X- 1. = X3 - 2 pXX - X (dd PP) + dd-pp
  144. et quia P, d, X, sunt ovil 1 fit +X -d
  145. ita si pa vel
  146. aequatrs. p
  147. Cx + 2pX.
  148. +d- p vit Y pod vel polius x= +p +d. nam
  149. quae aequalio salis facit, nam si d p seu si dep dividatur s
  150. 2P∞ - X(d- p( (+2pX
  151. sigi si de et pe0 utcunque, sed parum apta calculis
  152. -d
  153. s Ansic: si
  154. es
  155. 3 1,d
  156. X -d + p, fit denu X = pd - P
  157. bra
  158. eus erro iam hpli bitu erte
  159. crys cugo jam applisibitum cotibe
  160. ionum ex sedib
  161. t
  162. anta
  163. generaliter
  164. p+ d
  165. et
  166. (0)
  167. conden
  168. Quite linea ben
  169. ticalis,
  170. mini separat
  171. X- k - 1
  172. X = h- 1

Cite

Leibniz-Handschriften zur Mathematik LH 35, 3 B 6, fol. 2r. Machine transcription, not an edition: Leibniz Legible (model FoNDUE-GD_v2_ft_Leibniz, run 74). https://leibnizlegible.com/page/00068256:0003?run=74