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Lexicon Technicum: Or, An Universal English Dictionary Of Arts And Sciences, fol. -

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  1. M A X
  2. M a L
  3. ment Petits, Sect. 3, (90. In the Acta Frud. Lips
  4. 1. D. 1683. P, 122. there is allo a Method o
  5. determinins Maxima C Minima, bY one D. 3
  6. chiesr applicable to the Dramins oF Pangents
  7. to Curres: And in che lame B0Ok, and For che
  8. Xear 1684. P. 467. Vou hare a Method oF che
  9. FaMOUS MIE. LeibnitZ, FoOr che Came Purpole, 2c-
  10. cordins ro his Calculus Differentialis. Sec alle
  11. ihe Serench BOOk OF MI. De La Fires Conick
  12. Sections, in Lacin; and Chap. 7. OF NieumentiitS
  13. Anabsiis Insinitorum. There is also princed in
  14. the First Volume oF Descartes Geometry, 2 MC.
  15. chod oF Fruddes, For Aindins che Maxima (5 MI
  16. mima, P. 107.
  17. The Eollomins Acconne oF this Method, Wa
  18. communicare d ro me by MIr. FTumphry Dit
  19. ton a Peron very Skiltul in chele MaTeIS.
  20. LR OBL F M.
  21. To determine any Flomins Quantity in an F. quation
  22. proposd, to an Extream Value.
  23. Tho chere are varions excellent Mechods Fo
  24. the doins of chis, vet there is nochins chat seems
  25. (0 clear and natural, and is really (0 general, (0
  26. quick and casie, 25 thac wrhich che Doctrine of
  27. Fluxions Furnictres us wwichal.
  28. ToTolre rhe Problem in any Case ThaE Can BC
  29. proposd, is onty co make a just Application of
  30. this General Rule, viz.
  31. FTaving put the E quarion into Fluxions, let the
  32. Fluxion os that Ouantity (mhose Fxtream Value
  33. is souche) be supposed =0; by mhich means all
  34. those Members of the Equation in which it v
  35. found vill vanist, and ehe remainins ones mill
  36. sive the Determination of the Maximum or Mi-
  37. nimum desired.
  38. DENIONSTRATION.
  39. gr = 2rx, vod x = 2; which Value oF X
  40. cubsticured in ihe Equarion ot ihe Curre, sives
  41. =ry, and (0y = 1: = rhe Semi con-
  42. Iugace.
  43. Again, Suppose yy u = d y 2 = c cd - 0
  44. Wrhere y, Z. un are all Florvins Ouanriries; and
  45. y is ro be determind eo an Lxtremum. Nou in
  46. all such Cales, Wvhere che Equation will involve
  47. theFluxions oF (0 many disterent Flowins Ouan-
  48. ciries, e must endearour by convenient substi
  49. curions ro expunse fome ot chem, chac ve may
  50. hare no more than one TOrr OF Fluxions letrro
  51. deal wirh; ot which che Lquation may be
  52. clear d br ordinarr Diriion. And I chink chis
  53. Mechod, which PIl proxose in the Arst place, c0
  54. be a very easie and general one in order eo chis
  55. End. The Equation is y y u - dy Z- ccd =0,
  56. Crom wotence 2y y u -1 yy u = dy Z = dZy
  57. o and purrins y = 0, yy i = dy 2 =0
  58. Nou ro chron oux chele Fluxions in che last F.
  59. quacion, we necd onty substiture che Value ot
  60. cirher ot chem trom che decond Equation, wohere
  61. ve sindt it (ex. er.) = dy 2. F d 7y = 297u
  62. erxo yyu = dy Z = dy 2 ] d 2y = 27y u
  63. dy Z = dy Z - 2 y y 0 =0, er∞0 d Z = 2yu
  64. o Which is che Equarion delired
  65. Suppole y y x = Zy X = m = 0, and y 10
  66. Be decermined c0 an EXtremum. THeN 2yy X
  67. F yy X = 2y X = Zy X = Zy X = 0, ane
  68. matins y = 0, yy X = Z y X = Zy X = 0,
  69. buc = 271 x + 77 r = 27 x = 2) X, ereo
  70. Substimrins ehis in che room oF Z in the FOrmer
  71. Every Maximum or Minium is in ils orn Na. Equarion, Wve tave yy x - 2 y y X = y y X
  72. eure a Stable Ouantity: F0 determine cheretore
  73. any Florvins Ouantier to a Maximum OF Mini-
  74. mum, is eo makre ic (instead ot a Florwing) a
  75. Permanent one; buE the Fluxion OF a PerMA
  76. nent duanties is equal ro Noching. From whence
  77. che Realon ot che Rule is Susticieny clear.
  78. Lerus illustrace chis by Come P. amples. F. X. 81
  79. Suxpose BB x - y y X -] - cy x - d3 = 0
  80. wvhere y and x are Florvins Ouancities, and
  81. 2y ∞ - 2y ∞ = 2y x = 0; That 15, Zy X
  82. = 2 yy x = 0, 2nd Z X = 2 y X-
  83. BHE NIT. Niementiit Furnistres an Expedient for
  84. this Purpose al(o in che 7ch Chap. OF his Analysis
  85. Infinitorum: Ile considers che Flovins Ouanti
  86. ries in the Rquarion, a5 che Ordinares of (0 ma-
  87. ny several Curres, wrhich hare one common
  88. abscissa; chen substicurins che Values oF chC
  89. Fluxions of chese Ordinates (sorren by che sc
  90. neral Properer oF all Curres) he brings aIl chc
  91. is eo be determined t0 an Fxtream Valuc. Then Fluxions EO one Expreltion. LX. Sr. In THCF
  92. Bb X - 2y y X = y y ∞ -) - c y x - (= c X y = 0, Quacion abOvC, y y u = d y Z = ccd = -0, Wvhere
  93. and makins y = 0, bb x - yy x- L cy x = c
  94. and y y = cy = bb, trom wtich Ouadrarick
  95. Equation y mas be derermind.
  96. we had yy i = dy 2 = 0; let y, u, Z BC Con-
  97. ceirdro bethe Ordinares oFchree lereral Curres
  98. wrhose common Abscisse ler = x, and ler the
  99. Subcansent belongins 10 ube = n, and chat for
  100. Equation ot an Fllipte, Wohere che L arus Rectum (Z be = 1; chem v = = and Z = 2 , ereo
  101. =r, che Latus Fransversum == 9; the Abscise
  102. x, and che Ordinare = y; then ve have
  103. 13uX d9Z2
  104. =0, chat 15 17 u = d Zn = 0.
  105. 2yy = 0, Hom irhence che Egnarion For y decermined e0 an Fxtremum.
  106. Fis rrue ehis Equarion appeaes somectiins disse-
  107. La 1 X = - = - 97 = 0, wotich is the

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Lexicon Technicum: Or, An Universal English Dictionary Of Arts And Sciences, fol. - . Machine transcription, not an edition: Leibniz Legible (model FoNDUE-GD_v2_ft_Leibniz, run 74). https://leibnizlegible.com/page/806933011:0499?run=74