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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: 0.87.

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  1. 3 =- a: LHal 1, IHC FIEH Lerm is Eo the Second )T6Th Place inthis PrOGreNTIOM, SincC 8 1- 8 = 7C
  2. :: 25 che Sum oE all the Anrecedenrs is co rhe Sum Ifind, chac it I multiply 256 by ic PelF, or square
  3. ot allche Consequenis: Wrheretore by mulciplica-
  4. tion oE he Extreams and MeanS, Z4 = aa == r aZ
  5. ray. Wheretore cranstere Za, and ic Will bc
  6. aa
  7. ra
  8. 29. Franspole ray
  9. then Will r ay = a a = r aZ = Za. Divide eact
  10. Pacc byr a = a, and Z will Be equal 1
  11. Thae i5, Multiply the second and Last Terms toce-
  12. sether, and from the Product substract the Square
  13. the First Term; and then divide the Remainder b
  14. the Difference betroeen the First and Second Term;
  15. ond the Ouotient vill bethe sum of al the Terms.
  16. Example in ehis Series 2. 4. 8. 16. 32. 64. 128.
  17. ic, ic Wwill sire me 65536, Which is chersch place
  18. and Cince 15 -)=7 = 23, iC Lmulripls chac Num
  19. ber by 228, ic will produce 8388608, Wrhich isthe
  20. 24MH Place or Nail; (For cho Indices begin witho)
  21. and cheretore char doubled, accordins ro he Con
  22. dirion oF che Ouestion, will gire 16777216 Fac
  23. things, wrtich reduced, i5 174761. 55. 4 d
  24. From ertrich Propositions aboue Geometricat Pro-
  25. porrionals, thele wondertul Corollaries may be de-
  26. duced.
  27. 1. Fhac iis postible by chese Rules co collect an
  28. infinice Series oF Proporrional Perms into one Sum
  29. altho it is impossible co run over aI che Terms le
  30. perately, becaule Intinire. Thus, in a conrinued
  31. Series oF Fractions decreahns in a double ProxOr-
  32. FiOn, 2, 23 2, 23, 32, 9c. ad Infinitum, il vou tate
  33. them backards, ron mas justly recton a Cypher
  34. Or 0; for the First Term; (for berween : and 0
  35. chere mmax be an Intinice Number oF such Ferms
  36. and the Infinire Sum ot chese Terms ill Be pre-
  37. cisets equal r0 Vnier: For subrractine the first c
  38. trom the last:, and che Remainder: beins divi-
  39. ded by che Name ot che Realon sesten d by 1, char
  40. is by i, erhich dirides noching; the Ouoriene : is
  41. The Z oF allthe Lerms burthe last, mar be found che sum ot all che Terms, exceprins chie lat, (b)
  42. very easily chus: From the Lastrake rhe First, and Rule 2.) and (0 che last = Beins added, the Sum oF
  43. diride the Remainder by che Ratio le Nened by Oniey; aIl in chat Series will De T. Now it che last is
  44. the Duocient is ehie Sum oF all the Terms burche ROF =, buc r, che Sum oF all Will necellarily bc2;
  45. lact, a5 trill vers casils appear is rou multiply and
  46. divide Algebraicallr: Which Rule is in some Ca-
  47. ses more reads than the former, and cheretore ris
  48. Tometimes reterred ro in che Collowins Consectaries
  49. under cho Name ot Rule 3.C
  50. P ROPOSTFTONV.
  51. Llarins che Rario ot ene Lertus in ans beolle
  52. trical Prosreition, To find any of the other Terms,
  53. or 10 find an) term sousis. 3
  54. and if 2 bc che last, che Sum oF all wilt be 4: In
  55. a rord, it ill be alvars double the laA Perm.
  56. 2. And snce in this Case che Sunt oF all che pre-
  57. cedent Terms is equal ro che last Perm, the one
  58. beins subtracted From che orher, chere will remain
  59. nochins; 1. c.
  60. 32 (9c. in In-
  61. finitum, is = o; and al(0, x = = = =, (6. OF
  62. 2
  63. = sn 9c. = 0.
  64. 3. In like manner the Sum oE Intinice Fractions
  65. A5For Instance: In Progresion, whose Rario is decrealns in criple Rcalon in an Intinite Series
  66. 2, t0 find rhe 24th Perm; Or rO ansiver expediri-
  67. ousy che Common Ouettion ot chie Price oF an
  68. FPorse Cold at a F’arthins a Nail (oF his Stioe)
  69. and doubled ererrtime.
  70. Besin and double, a5 ris eatie to do, For 6 Or 8
  71. rimes, phus,
  72. 0
  73. 2. 12. 5. 4
  74. 3. 16. 32, 64. 128. 256.
  75. (3-
  76. F = 53 - 51, (9c.) Will bc equal 10
  77. Foril trom che last; (2gain in an inverted Order)
  78. rou luberaet, che tirst o, and the Remainder bC
  79. dirided by che Name ot che Realon lessen d by V.
  80. niry, chat is by 2, che Quoriene? will be che Sum
  81. Ol al che Ancecedent Terms; and addine cortis
  82. lalt5 Or 2, che Sum oC all will De L OF =
  83. Buc had che Fractions decreasd from = in 2
  84. Ouadruple, or From z in a Ouincuple Proporrion,
  85. E5C. the Sum of che whole Series on d hare been
  86. And then Over the Geometrical Dumbers place a laccordinelv = Or; and so any Series ot chis kind
  87. Series oF Ariclimerical ones, besinnins wich 0, 25
  88. is equal ro a Fraction, Whole Denominator is less
  89. rou see: Thele are Indexes or Exponenes ot the by an Onite chan che Denominator of the last Fra
  90. ocher, and Be erery where hou otien he Racio ction in that Series, (suppolins the Series to be
  91. is multiplied into its lel ro produce anr parricular numbred backward, as betore.)
  92. Term. Thus over 64, _ Ia Term, chere sands
  93. che Index or Eexponenr 6; irhich sens chac 64 i5
  94. 4. Generally allo, any Insinite Series os tracti-
  95. the schPorver of cheRario 2, (since the Series be. ons decrealins accordins ro the Proporrion ot the
  96. Sins wich Oniry.) No che Addirion, and Substra- Denominator of che last Term, and havine a com-
  97. etion ot the Inderes antvers ro the Mulciplicarion men Numieraror less by au Vnire phan rhe De-
  98. and Division oF che Wumiberscher Rand oxer: For nominator of che last P et, (25 3- 3 3, 69c.
  99. 8, wrtich ische Index ot 256, the Pro- Or 3 -)
  100. 24 C9C. Or
  101. 190
  102. duct oF. B into 32; ancl 7 = + 4 = 3, Which is che is équal eo Cniry, alter the Parrie vay as tie Se-
  103. Index oF 8, ehC OnorienE OF F28 dirided by 16. ries, Coroll. L. Which may be comprehended under
  104. Thisbeins che admirable Properer and vast Dse oF) ehis kind, and which may be demonstraced in al
  105. these Indices, (wtich is the Reason and Founda ips parricular Cases by che Tame Method ve have
  106. tion ot chelarhole Businels, oE the Losarithms, a5 hicherro made use ot, or allo barelr subsumed trom
  107. roumay see under thar Word) ris very easie by
  108. Coroll. 3. FOr Since
  109. Cheir help to come ac anr Lerm, cho never (o re-( 10
  110. - 3 - 1-
  111. more citien az. For luprotins L moula have che) and l0 in Hhe rest.
  112. (90. is equal
  113. , 13c. Wwill be equal 10 = Or 1:
  114. N
  115. 500 (254 = 254

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/806933011:0662?run=74