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Leibniz-Handschriften zur Mathematik LH 35, 6, 27, fol. 6v

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: 0.81.

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One image of two folio pages side by side, registered by the library under both labels (also fol. 7r); this page is its left half.

The machine reads it as

  1. quin vero hanc saltem utilitatem hinc ducimus ut transformare liceat rationales aequatione
  2. differentiales in irrationales co generali quadam ratione ideo vidiam, AT,Q, AX, X XY, y
  3. X y C. a et t:a D = x-: dya (3+) de: dt = - -y: a vel Ihii a tratio)olgod
  4. stu y i ade et ex, et 2 (4t) t: a = dx: dy
  5. Haec generalia
  6. et ex 2 et (3+10) e = vu) X - y (dx: dy).
  7. ontra y = (3to) - ade: dt et X = (610) H e = t lv: de)
  8. speciatim sit dy: dx = cla +XX f AX IXX YH _ 1 oC;, Hy 1 S5 gy + bg +CC,:, yy + my enm
  9. = (900) Pa: t Ergo a deded unde trys et bcy+ cet =ayyt amyt ann, seu
  10. hy + bt -am y + cet =ann - 0 seu gy + bt - amys + s2) btuam = = () bluam = cet -ann
  11. +-a
  12. +-a
  13. a
  14. = bb t t - 2 abmt taamm = 4cett + 4 annt + 4acct - 4aann ,: 4, Ht - Eat + aa
  15. -a
  16. lbb -4cc) et (2 ab, - 4ann) C + (mn - qnni) aa) - (1a t - 2m), : 42, t-a
  17. ratonpltr reductibiles,
  18. t- a
  19. sit ergo quaerunque aequalio ad quadraturas dus dx- a dx: du = y: a poterit.
  20. seu : = le: M y=
  21. loco ax: du pom F: a, et loco y: a ponri de: dt, et prodibit alia aequatio disse
  22. vulat vel contra; data Bde: dt = C: a, potest poni loco de: dt poni - y: a
  23. Crcaloroxgav
  24. et loco t: a porie dx: dys
  25. veniamus ad formationem curvam per concul
  26. sit AX, X, XY, y. AZ, Z. ZY, v centrum circuli V, radius VY sit Y
  27. et fit (Z- X) 2 + (y -v) = rr uubi et y quaerendo jam concursum ductum hujus-
  28. modi circulorum, si erint X er y. per un communes, sed ZV, Y variabiles
  29. (Z- X) dZ + (y- v) dv2 rdr. Sed conferendo hunc
  30. calculum cum communi, considerandum est, esse i cum circulus formans tangat Curvan
  31. et differentiando fit
  32. Y(Y) ipsam YY esse perpendicularem ad Y(Y). puducatur ergo dum axi occurrat in P
  33. jam patet sore YX : X p = -dx: dy = (tio )p v,
  34. y seu pro eadem curva
  35. nasic per x’y, a
  36. calculo nam si aequ. 1. differentietur secundum X
  37. sive jam ergo detur dZ dv: dz = per g,F, Ya a
  38. vel dxidy per X, y, a. potent altera ad alteram reduci, sed manebit
  39. arbitraria non ex tibust, D, Y, arbitraria. sed et potest concipi unam exhis tribus
  40. dari ex X, y, a. quo facto in aequ. 2, eta differentialis et quae per eam multiplocant
  41. conrescent. Hinc video utilius esse ut d: ax= p. P. a nonatur aequatio data
  42. seu compendio d: d 2) t: a et Gadius r imgneat datus ex X. y. a seu r= X. y. a
  43. aV. 1Z =(G v 93:, Z X, 7 Z - X, :, v- y. E = (800) et 5) +: R
  44. cursus ot ex 3 est Z Y =. Ex E est vm gua v y Cp
  45. Ex d est Z X Hini ex aeg. 3 fib _ dys. dx a t (gno) +: a. inveniamus jam et Z ope
  46. fint (no) du: dx = Ca. Inveni cmus jam
  47. .a
  48. otis
  49. Z et D, per 1 et 3.. Ex 3 fit y - v = X- Z, dx .dy = (1mo ex 1) A Z, X) 7 N(rr = _ (X - q) 2)
  50. seu per
  51. ag4. I et 3 at possimus tollere in aqui
  52. seu X - Z de dx: dydys = rr - (X - Z) = sive (X- Z= (dxax + dydx) rr dy dy seu 3. X ralys: Sindus tays
  53. Et pari pore vel ex 11, fiit X Z AVy = (Sto) rdx: Faxax + dydus. adeoque tandem erit
  54. 9= (15to) X + Y (dy: xx + dydy)) et v = (s) y + rlax: SC ax dx Faydy)) Qui valores
  55. substituendi in aequ. 10 quam pominus datam. et deinde Y poterit explicari pro arbitrio, ut
  56. tollatur a. verum jam video rem non procedere, nam si ponatur A = 0 cum sit do: dz = dus: dx
  57. necessario fient linere prallea v(V) et Y(Y) paralletae, seu V(Y) sive Y fet constans nec variabilis per
  58. interiu hunc
  59. X et y. Ac pruande in initi seu generaribus non licet ste facere, u=
  60. calcutum tenere possumus si lineae Y () et et V(V) sint paralleli et sit AX, et AV, y;
  61. et AZ, Z, et ZV, 0, fore du:dz= de dy: dx, et 3- x rdy: Saxdxt dypdus et Bay raxisat tely du
  62. sit jam dyo d= in casu aliquo dyidx = X.0 fint duidque py fiet d
  63. per
  64. ubi ∞ significat contrariam relationes
  65. 1,2, est x = 2 et 1 est X ~) +d Z- rev: ravdet d2) et y = V- rd dvelat de a2)
  66. =(gno) dvidzue
  67. X: y ~ 1v. 4u
  68. fiet per 6,7, 8, frat E= r(advi s, dvant an): v- re 1 s evavt at28.)
  69. seu +p.d) quae aequatio comparanda est dataex et s _ ubi do: dz Z. v. ∞r: + seu
  70. lv: dz 5(Z. v. r) et fiet, Za r(∞CZ v.r) A (.v v: 1)F 1),:, v - pl 1: ((~ (. v. r). 1) + 1 = ( C v. B). 1)
  71. adeoque rt
  72. ex,
  73. Z3+ vb,:rr
  74. sit in casu speciasi ~(Z. B. B: r= 22+ v0,: r pyt pe = dv: dZ
  75. V )e15) 2 o, er adeon quaeritur modus qu calculandi seu
  76. fit NZVZZ + v + ir - (2I + VV),: xV LZ2+ 0btrr - X(Z2F0V)1513E+0VC,: rr sed hoc quantum
  77. quod si ergo redeamus ad generalia, seu si r manet variabilis
  78. sitque (ZXZ 4+( b = ) X (Z- X) de 4(V. y) do rar et Eto) vay (:Z. X- ux: dus
  79. fiet
  80. Judico praestari non potestus
  81. et jam in ne data ya dy:dx = (4to) L:a posito t, dam ex F. ya. Ergo
  82. 4dv. riy (dr (V33y-D) unbe
  83. p Z riri natuta
  84. (5to) v-y : ZX = t: a _t Ex 2 et 6si et (Z-X, 2, - V) dx
  85. a dx +Ldb, y- v = =rdr seu y- vE (∞) Ardrocde + d
  86. ex his valores X et y explicalo prius F perxes
  87. 2et
  88. Ex 8
  89. axdr: adZe + + dv L9),-X
  90. v xtyy rar in dZue Hex + yy) da _ (e 1Hamo) seraax D et srl a y) df gyy b) avtXXB
  91. ymo) teri
  92. Q-X =
  93. y, a.+
  94. rxrdi Lyynd et fut. Breous redeam, ad aq
  95. tole
  96. et ful rrdear +(v∞ + yy = Lvy) dydo = 2r(V. y) dudr= _ rrdzb- (∞- V) = de7
  97. Prectius: ala invendimus X et y per aeq. 1. et 2
  98. sit vay per cypendio
  99. 1+rrapdvarar
  100. lii rrdrar + Wlodvdo Zru duar = rrazaz Wlo dede. seu WoW -2rdodea
  101. A12 avartarar)
  102. dodr +Y(e ddz dodv + rd ZdZ drd(rraras duav 1 ddrfrravavarar) R: , dvaytanti
  103. (dZd drar) (avdv +drar) + erdodvdrde,:, (2) (dvdo +drdr) et sit d
  104. v-ys W, seiter Z- X = ravar LYN(aZd dodo +. dvd dedr- drt):, dzaz + arar sit jam in exemplo
  105. e
  106. rum tollantur. X et
  107. x et et
  108. Zet e quaesitum sec
  109. quodsi deinde assupi posset
  110. ite
  111. aeq
  112. Extgy, daa = (per 3) gq X, hucc ope valorum
  113. iut
  114. dr
  115. qu
  116. nam
  117. post
  118. inven
  119. et aliu
  120. Z a2.
  121. aglos substutu
  122. icsilsio
  123. natur aesieratu Leiphe
  124. F vel dv: d)
  125. vet t vel
  126. ar: de
  127. 1d. Eaposito dr-0 ex aque
  128. Hinc duidque n: dis set
  129. Lyrux =V. L

Cite

Leibniz-Handschriften zur Mathematik LH 35, 6, 27, fol. 6v. Machine transcription, not an edition: Leibniz Legible (model FoNDUE-GD_v2_ft_Leibniz, run 73). https://leibnizlegible.com/page/00068072:0012?run=73