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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. Thus, luppole
  2. 0. 2. 4: 6. 8. 10. 12. 14. 16. 18, 2a
  3. Thus, in che Collomins Series,
  4. a. a X. a 2. X. a l- 3X. a 1 4X. 5 ( 5X. a1
  5. a -] 7X. a dd. a Gt. d TO.
  6. The Sum of che Fxrremes 2 a = = 1OX, is equal
  7. 10 = a -) 5 X mulciplicd bY 2; thar is, the Double
  8. oF the Midale Perm: AS ic is allo ro the Sum ot
  9. any Trvo oF ehem equallr distane trom boch RX
  10. tremes.
  11. And ehis must erer be, because the last Perm
  12. contains in icphe trst, and also che common Diste
  13. rence super-added as otcen a5 che NTumber oE its
  14. Place is distane from che Arst Perm: Bur, the ArA
  15. Term hach no Aadirion of che Ditterence ac all;
  16. and ac the Sccond Term hach one Difference Or
  17. Rario more than the First, the Third one more
  18. than the Sccond, (5c. Po the LaA lave onc, hach
  19. one lels chan che LaH OE all; che L ast lare tWO,
  20. one lels than che Las Care onc, (96. S0 thac rhe
  21. Sum ot any Trvo oE chele equallr distane from che
  22. Extremes, must be equal ro che Sum ot the Fx-
  23. tremes; because One encreales as much 25 tHC
  24. Ochex decreases.
  25. tence ris plain,
  26. COROL L A R TFS.
  27. 0. X. 2 X. 3X. 4 X. 5X. 6X. 7X 8N. 9X. XOX
  28. 20 che lase Term, mulriplied by er che Number
  29. ot che Terms, gires 220; Thehalt ot irtiict, 11O,
  30. is the sum of all the Terms.
  31. For, By Corollary che First, 20, the Sum ot ehic
  32. Fixrremes, mulciplied bY 5. 5 halt the Number Ot
  33. Terms, produces Tr0, the Sum oE all che TerrS.
  34. 5. Ilence ehis very uletul Theorem may be de
  35. duced, Thac ehie Sum oF all che Pemns in any such
  36. Procression, beginnins From o, is Subduple oF the
  37. Sum ot (omans Terms, all equal co the srearest
  38. For ler che Erst Term beo, andche last x, and che
  39. siven Number oF che Termsn; then (by PF L. 7
  40. Wvill the Sum oF all che Terms oF the Progression
  41. be Znx, Wtich is halF Ox subduple OF n X equal re
  42. che Numiber oF s0 many Terms equal co e
  43. grearest. 8. F. D.
  44. sion, (i. c. aCcendine. bY the Cardinal Numbers
  45. continued trom Vnirs ad Infinitum, is subduple the
  46. Sum oF che same Number oF Terms, each oF
  47. Which is equalrothe oreatest; or, on che contrarr.
  48. this larrer Sum is double of che Former. By pre-
  49. Y. That che Sum oP anx Dumber os Terrris in sixing a Cxpher Detore Vrier, this vould De bur 2
  50. Tuch a Progresion may be trad, it the Sum of the Cale of the last Corollarr, che Sum oE the Pro-
  51. Extremes be multiplied by hals the Number of the
  52. Terms, Or halt chac Sum by che wwhole Numiber ob
  53. rhe Perms.
  54. 2. F0 gain the Sum oE nerer lo many Onanti-
  55. ties in rhis Progression, chere is noching neceslary tO
  56. fression remainine Aillthe lame.
  57. Buc chac ehis is erue in an infnite Series, begin-
  58. mins from Onier, (For in a Finite Or Decerminare
  59. one, the Proxorrion of the sum is alr 2ys less chan
  60. double; tho ic alvays approaches coit, and comes
  61. Po much che nearer by hou much srearer the
  62. be siren bur che Extremes and the Number OF Scries i5) we Mall nou chus demonstrare:
  63. Terms: S0 thac if by having the First Term and
  64. Common Fxcess, one ould sec che Last, cvould
  65. Stranselr disparch Ouestions in Progression.
  66. 3. Wilers sau Lem in Cuch a Progression may
  67. behad caslr, by musciplvins ehe Number oF the
  68. Terms, lessened by one, into the Common PXCelS.
  69. and then 10 thac Produet addine the ArA PerM.
  70. Thus it che last Term, in a Prozression ot 52
  71. Places vere desired, utiere che Common Diterence
  72. 15 3, and che Aest PerM g:
  73. Mulciply 5F by 3, ir produces 133; eo vhich
  74. addino 5 che ArA Ferm, Vou have 158 cqual 1O
  75. he last Perm in such a Prosression.
  76. 4. WVheretore iE che Progression begin wich a Cy
  77. Pher, (wrhich is the most Nacural and Simple Pro-
  78. pression; For when ic begins wich any other Perm,
  79. T0 che Sum oF Three TermS, 1, 2, 3, 1. c. 6,
  80. the Sum oF as man equal in Number co the
  81. Sreatelt, 1. c. 9, hasthelame Proporrion a5 3t02;
  82. but to the Sum OF Six TerS, 1, 2, 3, 4, 5, 6, 1. e
  83. 2T, the Sum of as many equal ro the grearest.
  84. 1. c. 36, has che Tame Proporrion 25 310 1 -] - 3
  85. thav 15 3 FO 2 = 2 the Decreale beins =; bue rc
  86. the Sum oF Tvelre Terms, wrhich mar befound
  87. by Coroll. L. equal 78, che Sum ot s0 mans equal
  88. co che greatest, viz. 244, has che lame Proportion,
  89. (dividing boch des by 48) 45 3 10 128, 1. c. 310
  90. 1 -1 = - =, (For 24 matee in and the Remain-
  91. der =5 is the Came 25 f) Thac i5, 25 3102 - -
  92. theDecrement beins no J. Since cheretore, by
  93. doublins che Number oF Terms. onvards, roull
  94. find the Decrement r0 be =5, and lo onwards in
  95. double Proportion; the Sum ot an Infinire Num-
  96. ber oF such Terms, in Arichmerical ProgresTon.
  97. equal to che grearest, will be tothe Sum of the
  98. Procression, From r ad insinitum; 25 3 202 =
  99. =E, (dc. Thac i6, bY Coroll. 1 and 2, 23
  100. 3 10 2 - =, Thac 15, 25 3EO 5 5, OF 25 2 F0 I.
  101. D. E. D.
  102. O
  103. 6. The Sum oF a Simple Arichmerical Progres.
  104. eis in realier a Compound ot rvo Proeressions; onc
  105. OF EqualS, 25 OF a, a, a, cn (90. aset l Oet 6
  106. Arichnerical ProporrionalS, 25 0. X. 2 X. 3X. 4 X.
  107. CSC.) chen I Cay the Sum of al the Terms will bc
  108. equal ro halF che Produet oF che lac Term muiri-
  109. Plicd by che hole Number ot ette letus. 10b)
  110. rhe First Corollarr, the Su OL H EC LCMIS WII
  111. 2. Xhe Sum oe any Duplieate Aitumeriealero.
  112. Ecito, Corsusca Tiom Onies ad insinitum, ls
  113. lubetisse ot the Sum ot as manr Ferms equat 1b
  114. be equal io ihe dum ol che lSCccies
  115. O halt ihe Dumber Ot Letirs.
  116. EhC ElCaccnus do 1o ehe Leclidet et ECclto. Let
  117. any Tuch linte Losreo IS Sreatet MMaM CHE SHD-
  118. eriple Proportion, bur approaches nearer and near-
  119. et 10 ic concimals, by hou much che laccher clie
  120. Serieo

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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/806933011:0659?run=74