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Leibniz-Handschriften zur Mathematik LH 35, 13, 1, fol. 320v

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: 0.88.

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One image of two folio pages side by side, registered by the library under both labels (also fol. 321r); this page is its left half.

The machine reads it as

  1. Ad promovendas Quadraturas magni momenti foret, reducere (do (ovab, i, von ac) ad rationale
  2. quod multis modis tentavio nec tamen satis hactemus successit. Tandem placuit hec via:
  3. avv- aab, a vv =aac aequand. Oto, quod esse poterit 0OY 8 DD, si sit: avv - aab = 008
  4. et a = auc = ZDD. sint l, mn, indeterminatae simplices pendentes ab x, velut. X +..
  5. Faciamus V = 12 Am + 123 Imn fiet v = 112 lm + 2. 112. 123m sIn. sit = 204p + 213In
  6. sit 0 = 310 sal + 303 fan fiet 93 = 315 al + 2. 310. 303 a sIn, et 00G = 204. 310 alpfr 213. 31 al Iln
  7. Ubi notandum Aumerorum fictitiorum notas initiales, 1, 2, 3, esse ad hibitas ad indicandos.
  8. quatuor aequat valores faundamentales assumtos quantitatum v,
  9. 204. 303 arpZ 213. 303 an
  10. 2. 317. 1 9. 3 5a 2. 204. 310. 305 ap..
  11. 204. 405 anp 213. 403 an
  12. 8, 1), D. Et quidem ac ini 2. 13 10. 4ch alm 2. 204. 40ta ap.
  13. 3 ac 4 initiales eodem se modo habere. Porro post initiales potas, eas
  14. quae secuntur indicare 1, 2, literas l, m, n, p. Et 1 et 3 prorsus ut l et n se habere eodem modo, sed 2, interponi prom
  15. cum abest m, loco 2 ponitur 0. id tantum interest inter l et R, adeoque inter + et 3 post initiales quod l’etta praeponitur
  16. ipsi 2 ve m et n, sed n postponitur ipsi L et m; eodemque modo 1 praeponitur iplis 2 et 3, sed 3 poseponitur ipsis 1 et 2.
  17. omplremus jam primum
  18. ob cum valorem ipsias avv- aab cum valore ipsius ODQ, et similiter
  19. Hinc 1, n se habent ut iuc Compare R
  20. Coincidentiemus jand
  21. coincidentiemus jam valorem ipsius aVB-aub cum valore ipsius 0O_ secundumq irrationales
  22. Ex rationalitus ob sarv- aab = a 1D Z piet 1 Im = ab 2 204. 1 6. f ip + 2. 1. 310. 303 In
  23. lavo -aac = ZDD fiet 112 Im = ac = 204. 410. Ip + 2. 217, Mo. 403 In
  24. ope harum duarum aequationum n quaerentur m et p et primum quidem Tollendo (si placet aequationem
  25. ab aequatione 2, tolletur m, et fiet 204 (4102-310) 1p +204 (903 -303)n p 2. 213 (410,403- 310. 303, In + a (c - b) = 0 seu
  26. p. (312. 213 (310. 303- 410.403)17 a lb- c),:, 204(410- 310) l + 204 (405 = 303)n. Addantur jam in unum aegq. 1 et 2. et ex summa
  27. quaeratur m, fiet m L 204 (410 +318) Ip + 204 (403+303)n p + 2. 213 (310.903+ 410. 403) In + a (b+C):, 212l + 13n
  28. avv- aab = O04 fiet 2. 112. 123m = 213. 310 al + 2. 204. 310.303 ap
  29. Exiralioraibis db q avv. - aac = QDD fiet 2. M. 123m = 21. mo pl + 2. 104. 41040) ap
  30. unde ubi rursus inveniri poterunt m et p; et quidem subtrahendo aequationem 5 ab aequ. 6. ut tollatur m, fiet:
  31. p (7) 213 (410 - 310) + 213 (403 -305)n, :, 2. 204 (310. 303- 410. 403), aequando duos valores ipsius p ex a99. 3 et 7, fiet
  32. 213 (410 - 310). Il + 213 (403-303) nn +. 213 (410- 310) (403. 303) 4. 213 (410. 403-310.903) In = 4. (10.903- 903. 310) a (b- 1) =0
  33. ( in jam (418. 403) 4103 303=- 310.403 ft 310. 303)-918. 403 318. 303 + 4. 410. 403. 310. 303 - --(2) (1B. 403 -. 11030) - () (+0. 303- 43. 1 1)
  34. 2. 2132) (210. 403- 310303) In +2 (410. 303 - 403. 310) a (b - c) 0
  35. Ex aequ. 7. cum sit simplicior quam prior ex aequ. 3.
  36. 2. 213 (2) (40.303 -403. 310) Ln
  37. substituemus in aquvalore ipsius m aqu. 4., et firt m = 213 (416 - 308) H -+ 213 (410 4916) (405 - 305) In 4. 2B (410. 403 - 310. 305) In
  38. Porro ut rursus inveniamus m Praeterea novum valorem ipsius
  39. (3 (405- 303) M 213 (405 +305) (410 = 3102) IN
  40. jam 410. 405 (= 410. 3032 4 403. 910)- 318. 303 +
  41. qu. qui 4032. 310 + 410.303)- 3101. 305 = 4. 410. 403 +404. 315. 303 est 2319. 303 -2. 4102 4032,
  42. 2. (410. 403- 310. 303) a (b+C), :, 2. 1 l4 (310. 303- 910. 403) (1 2 l + 1231)
  43. m 213 (410 -3107) l + 213 (4037 - 3057) nn 42. 213 (410. 403 - 310. 305) In =2 (410. 903 - 310. 303) a (b+):
  44. 4(310, 303 -410. 403) (113 l+ 123. n). Rursus ex 5et 6 in unum additis fiet: M = 213 (21543,5) fl + 213 (405 4 3032) an 2. 204(40. 40 3 + 310. 3) B.
  45. : 4 112. 123. Et pro substituendo valorem ex sq. 7. fiet m =(213 (41 + 315) L 4+ 213 (405. + 305 )n B 10. 303. 910 403) 2 3 (a p0 403 + 316. 303 (40. 310
  46. 34 4 ) 213. 46_310 303(213, 910. 10. +o E D + 217. 315. 3a 3o H 2336: 40. L3l
  47. +213. 416. 410. 403) + 213. 410. 319. 30) - 213. 310. 410. 4u3 ( 31 . 213318. 310. 303C F 213. 405. 30. 203 ( 213. 4032. 91. 403 + 213. 30 f. 20. 30113 13. 303. quo tin
  48. F213, 9u. 410. 403 + 213. 4032. 310. 203n - 213. 303, 410. 403f213. 3032. 310. 303n), :, 4. 112. 123 (310. 303-40. 903) vel coutre sive
  49. 4 213 (410. 403 + 310. 303) (403- 303)n,:, 4. 112. 123 (310. 303- 410. 403)
  50. M 2+ ∞, 410 31. 303e 213. 410. 310 (410. 303-403. 310) l + 213. 403. 303 (403. 310- 410. 303)n M 12 (2:, 2. 112. 123(310. 303- 410. 403
  51. Quos duos ipsius m valores ex aegg. 9 et 10. equando fre + 112. 123 (48 - 21 )I+ 112. 123 (403- 303) nn- 112. 123 (4V. 405- 315. 305 )1n - 2. 12 1 9(0 4oh 310 3os seb HC)
  52. =2 2. 1122. 410. 310 (410. 303 - 403 31) l. + 2. 1292. 403. 303 (404. 310- Ho. 303) nn +, 12. 403. 303 (403 310 - 0. 305) In + 2. 125,. 410. 310 (40. 1 0) 40z 310)1 12
  53. de denique (112. 123 (410 -315) -2. 12 ( 410. 30 (mo 303- 40 3. 30)) l + (112. 13 (483 2 90, G - 2. 143. 403, 303 (90 4. 910 - 1o. 103)) MM =, 12 1 3 (40. 403 - 210. 305)
  54. (410. 403 - 310. 303) a (b+C
  55. I 2 coincidentiandae jam sunt duae aequationes
  56. uae aequationes quadraticae ad easdem binapliteras sint coinci
  57. Li mo i eoio to. vom elea
  58. dentiande, absitque utrobius utraque indeterminata simplicis gradus ut hic l, et n hon posse eas habere unam tantum radicem aequalem, sed
  59. as io. 403- 303
  60. secundum l et n nempe 8 et H 12. Ubi notandum si
  61. iicum
  62. seu
  63. Embas necessero coincidere identificari, adeoque terminum termino esse coincidentiam
  64. utrobiu sit Unitas. Sed praeterea opus
  65. tervinum suum ultinum seu alsolutum, itas inmentibilem, ad alives l’et n, cr dudeutur ex se invirem
  66. 610. 603 = ito
  67. tamen ut neque 600, neque 603 sit 0) et fiet (2((410. 403- 3,0.303) A, 13 () (40. 363- 403.310)L (410- 310) (403- 3032)-
  68. 513L ∞,: 2.603. debetque fieri
  69. indeterminatis, ateque fiet sit aequitio 8 per compendrum, Sioll + b03 nn 4613ln +
  70. Cun coincidentie Cum identificentur) e
  71. si + sit +, fiet (210. 403 + s1o. 305, 2 2. 410. 403.310. 303 + 410. 315 + 405: 315 - 2. 410 403. 10. 103 = (410. 4034316. 3055 - 415. 305- 405. 318
  72. seu fiet et n = ( 1(613 I -4. 610. 603l L4, 600, 603aa
  73. et divisis omnibus per 2, fiet 418. 303+ 403. 310 = 2. 410:403. 310. 303 = 0 seu 410. 303 =403. 310 seu 4 403: 303=
  74. seu coinciderent in determinatis. Ergo oportet ess
  75. 2. 410. 403. 310.303 = - 410. 403 =316 7 0 410. 307. 4 40 5. 310
  76. et rursu dividendo per 2 fiet 410. 4032+ 318. 303 2410. 403. 310. 303 =0 seu denique 410. 403 = 310. 303. seu 1410, 403- 310. 303 =0 sede
  77. sed haec aequatio non est licita, ita enim ratio C ad D foret constans
  78. et fiet 410. 403 +318. 303 - 2. 410.403. 310. 303 (4 410.305 + 403, 31
  79. ta alia oriuntur in commoda, nam in aequ. 10. ubi esset valo
  80. e ipsus
  81. son non llue ob laudem
  82. 403 = 303. seu 403 = + 305. Et pari jure qui aequationes dua detisina
  83. deubificantur fiet 410 = ()B = 310 ne scilicet 410 sit ad, 310, ut 403 ad 303, quod prohiberi diximus. Adoque foret hoc modo
  84. iderqu fieret, ex aeg. 9: sentemus de A13. quo facti fter
  85. dentica. Hinc sequitur, ex is solutibnibus aequoe esse duabus esse unam, ut
  86. id ita alia orimntur incommoda, nam ex in aeg. B evacescentibus H et un, restacet In de
  87. b-c . quod esse
  88. vel fitet 410303 403. 30= 0 quod jam rejecimus Ergo n per l rationaliter in ipsis determinatis haberi nequit. Atque edeo
  89. ita jam foret per 8,
  90. 410: 403= - 310. 303 seu 410, 403. + 30303 = Reret 600 = 0 sed hoc mucto minus licet nati fier
  91. cum cistante quod non licet
  92. hac via ad exctum perveniti ae non potest

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Leibniz-Handschriften zur Mathematik LH 35, 13, 1, fol. 320v. Machine transcription, not an edition: Leibniz Legible (model FoNDUE-GD_v2_ft_Leibniz, run 74). https://leibnizlegible.com/page/00068196:0636?run=74