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

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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The machine reads it as

  1. R
  2. eR OPOSTTION H
  3. IF Four Ouantities are proporrional, cithex con-
  4. cinualle or dilereetly, che Produet oE che fixtremes
  5. is equalto rhe Product oF the MeanS.
  6. For il an ray rray rera, in Or 2. 4. 8. 16. .
  7. ris plain che Produet oF the Extremes and oF che
  8. Vleans, is rhe same Ouanrity rrraa OF 32.
  9. In Discreer Proporionals, let a: ra:: b: rb;
  10. that 15, 2: 4 : : 15 : 30. The Product OF the FX-
  11. treams ard of che Means can here beno orhet char
  12. ab = 60.
  13. On Wohich Proporrion is grounded che Colden
  14. Rule, Or Rule of Threc in Arithmerick; (0 called,
  15. because havins Threc NTumbers, (25 2. 4. 25.) ic
  16. ceaches hion to find an VInKHOWn FOUrtH PFOPOT-
  17. tional: For altho this Fourth be unknon, yec ic
  18. Produet bV 2, cheFirst Perm, is knomn, becaule ris
  19. the lame wich the Product oF che McanS + and 15.
  20. Wheretore che Rule directo us ro multiply che
  21. Third by che Sccond, chac rou may chereby ob-
  22. tain the Product oF che Extremes; wtrich dirided
  23. by onc oF the Fxrremes, viz. the First, must necds
  24. sire cheorher, that is, the Fourch Toughes
  25. COROLEREES
  26. Etence vis plain, Thar iC. Lvo Products arisine
  27. From che Nulriplication ot any LWo Pairs oF Onan-
  28. ecies, are equal; chose Four Ouantiries will Be ac
  29. cast diserecels proportional.
  30. PROPOSITION HI
  31. IE chere are never somany Continual Proponion-
  32. als, the Produet oF the Lxrremes is equal eo the
  33. Produst oF any Two OF the MeanS Thar arc equa-
  34. Iy distant trom che Farremes, 25 al(0 to ehe Square
  35. ot che Mean or Midale Term, il che. Number ot
  36. the Permsbe odd. le
  37. Thus in etis ecogresion,
  38. 2 4 8 16 32, 64 128.
  39. a. ra. rra. rrras rrra. rrerra. rrerrra. c.
  40. IEs plain che Product dE tHC PXIreMeS, and oE any
  41. Tuo Terms equally distane from them, and che
  42. Square oF ihe Midalo Term, must alwarsibe
  43. 6aa.
  44. As Etie las Tertm bue one, is muliiplied into 2
  45. DeEreC OF POWer OF the Ratio lels By One than chC
  46. last; (0 ehie Second Term is mulciplied into one
  47. more than ehe First: And cheretore che Rectangle
  48. OF thie Fxtremes mus RilI De che Tame Ouantity
  49. wich chat of the Product of any Tvo Terms c.
  50. quallu distant from chat; and chis also equal ro che
  51. Square oF the Midale Lerm, it the Number ot
  52. ehem be odd.
  53. P R O POSF Y LO NmA
  54. Flaviris rhe First and Last Ferms, and ete Rario;
  55. Tofind the Sum os all the Terms in any Ceometrical
  56. Proeression, MI. Ousbtred siveschis Mertod.
  57. Let a beche sirst Perm, andy the last; etien let
  58. all the antecedents, and Z = a = 3Il che
  59. Consequents. Theretore, by Compolition oF Pro-
  60. porcion, ic plainls CoHlows, Thar a. ra : Z 7
  61. LeF F F (OF rb) be a Mean Proporional DC
  62. Eveen the Side oF the Cone B C, (OF rrb;) atic
  63. ihie Radius oF rhe Basc CD, (Or b;) and let
  64. etiere be an equal Number oF Peripheries, rra
  65. ra, and a artiverine 10 un equal Nuiber OF Ra
  66. dii in che lame proxorrion; then Wvill halt che firte
  67. Radius B C, (Ox rr B) mulciplied inco the Peri-
  68. phery a, (thac is = rr a b) be equal ro halF che
  69. eroaut ot uhe midase Line imco chemidale Peri
  70. pherr, chat is (as Betore) halt rrab: The FOF
  71. irier ot wrhich is equal co che surtace oF the Given
  72. Cone, (by the Ach Cons. OF his De. 18.) and ehie
  73. latter ro the Area oF che Circle, Wio Lacius I5
  74. the Mean Proporion EF, (by rhe 2dC09. et tii
  75. Def. 15.) 8. F. D.
  76. Ana Trom hence also Rous naturally chis orhet
  77. Proxotirion, That the Sursace B sc Lo, Bals L lL b,
  78. os s Base hals 2b; ac the Side ef the Cone P F
  79. s to obe Radius es che Pasc b. F0r
  80. Errab: ab:: rrb: b.

Cite

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/806933011:0661?run=74