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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: 0.89.

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  1. Thole are called Commensurable Surds, urtiiet
  2. are 1o one anocher as Number eo Namber, as onc
  3. Racional Ouanriey ro anorher; Or whiel are, orhen
  4. reduced co cheir lea Verms, nuc Figurace Juan
  5. riries of cheir onn kind.
  6. Todiscorertheretore, Whecher cher are such ot
  7. nOc: IFche Surdsare ot differenekinds, (Or FTetero-
  8. geneat Surds a5 fome calL chem) cher must sirst be re.
  9. duced ro one kind, and then dirided severally by
  10. their grearest common Mcasure, for is then there
  11. vill. Come our Rarional Duorients, che cirst Surds
  12. are Commensurable: buc if che Ouorients are Irra-
  13. nioval, or Surd Numbers or Guanricies, chen che
  14. propoled Surds are Inncommenlurable.
  15. V. er. T0 Examine. Whecher V: 12. andt V 3.
  16. are Commensurable Surds; thes beins Llonose.
  17. neal, L diride chem severally by cheir grearest Com
  18. mon Divitor, Which is V: 3; and the Ouorients
  19. ase v: 4. 2Bd V: c. Thac is 2 and r. Wheretore,
  20. since 2 and r. are Rarional Numbers, L Pay rhaE
  21. V: 12 and v: 3 are Commenlurable Surds; Or
  22. are roone another, 25 2 20 r. Which is very plam:
  23. for no douDr 12. 3:: 25 4. F. and Tis plain chat
  24. as Squares are ro one another, (o are their ROOES)
  25. Wheretore 12. 3. 25 V : 12. V 3. chac 15, 25.
  26. 1: 4. V 3 F. OF 25 2. E0 I.
  27. Wthenerer imo Surds are divided by one com
  28. mon Diritor, (cho noc the sreatest) it their Omori-
  29. rienes come out Rational, or are to one another
  30. as Number eo Number, thole Surds are cettain
  31. ly Commenturable.
  32. TE Fractional Surds ivere giren, noc harine a
  33. common Denominaror, ibler must hirst be reduced
  34. to their sinallest common Denominacor, and then
  35. il cheir Numerators are Commenturable, vou may
  36. conclude the Ars Surd Fractions evere (0.
  37. Burit cicher the NVUICTAEOES OF DenominatOre
  38. et rvo Surds, proper Fractions, or mixe Numbers
  39. in the form oF Fractions (neolectine the Radica
  40. Sien) be Povers oF rhar kind which che Radical
  41. Sien expresses, then chiey will need no Reduction:
  42. Fox ilrheir NIIMCTALOES Or Denominatorsare C’om
  43. menturable, che Whole Surd Fractions propoled
  44. are certainy (0. Thus il it vere enquired ivhe
  45. ihe Radical sien expresses or Denores, omitrins che
  46. HEche Surds proxoled be oF elic same Kinc, Mtul-
  47. ciply chem one by anocher, and presix che common
  48. Because r6 anc 25 are SGnaTCS, OF Su CH PORVerS 25
  49. Sien V: vou necd ouls compareche Numerators
  50. 20 one anortier, 25 10 and consequenels, b7
  51. Mulcipication of Simple Surd Roots.
  52. Kadical sion ro che Produet; bur it che Surds are
  53. Fererogeneat, ox oF ditterent kinds; cher must be
  54. reduced Arst, (accordins ro Rule 2.) 10 Surds har-
  55. es V: and s: are Commenturabie Sucds:
  56. V: 50. and v: 72; Wrhich beins divided by cheir
  57. ercarest common Dirilor, V: 2, che Ouorienrs
  58. Will De 5 and 6 (i. c. V: 25 and 1 : 36) WVhere
  59. tore che siren Surds are Commenlurable, and axe
  60. he precedent Rule may be expressed chus, . S::
  61. ins che lame Radieat Sion.
  62. FhuS eo Mulrixis. V: 7bY Y: 8. che Produst
  63. will bc V: 56.
  64. Fon since in all Mulciplicarion, 25 r. is E0 opie
  65. Factor, (0 is ehie ortier eo the Product; cherefore
  66. here S: P. V: 7. :: V: 8. V: 56. Buc it 4 PCrHS
  67. are Proporcionable, cheir Squares will be 10 100;
  68. WhCretOre T. 7. 3: 8, 56. charis, 56 is rHC FIC
  69. Square ot Y: 56. and V: 56. rhe crye R0Or ot
  70. 7 X 8 = 56.
  71. Octrex Framples.
  72. P. EEV: 8 were r0 BC Mulciplied into V: 4. DC-
  73. cause cher are nor Floinogeneat Surds, etiey must
  74. bereduced io such bY Rule 2. and cen chey vil
  75. stand thus, V: 312 V: 16, wetich beine mudripli
  76. ed inco each ortier, and the common Radical Sien
  77. presix d, Will makie V: 8193; and chusche V: 27
  78. mulcipied by V. 9. uoten reducec, and rixtuts
  79. mulciplica, produces v: 531441.
  80. AT. WVhen 2 Surd is ro be mulriplred by 2 Raric-
  81. nal Ouanriey, chaE Rarional Guantiey OuDHE REs
  82. tobe reduced ro a Surd ot like Nature wich chc
  83. Erue Surd. Bue ris otcentimes conveniene on y ro
  84. connect chem tosether, by prehxine che RationaL
  85. Ouantier co the Letr Fland oF che Surd. AS sup
  86. Pose V 27 vere ro bemulciplied by 6, cheProduct
  87. mlay commodiouss be expressed chus, 6 v: 27
  88. and Coit i: 9 vere co be mulriplied by r0, ic will
  89. Aand cus, 10V: 9.
  90. III. And when rro Bational Quanrities are
  91. thus pretixd eo rro Surds ot the lame kind, rou
  92. nar sind che Produet of chem, by mulciplvins ihe
  93. Racional Parr, by the Rarional, and che Surd Pare
  94. by che Surc chen chose joinedcosether, will bethe
  95. Produet required. Thus, 6 Y: 7 mulciplied b7 5
  96. V: 3 Produces 30 V: 21.
  97. IV. IF anY SUIE LLOBL DE LO DC SAVIIPHCC MOA
  98. Cel, or Involved accordins worhe Index OF iESPrO-
  99. per PoweE, You necd only cast avay che Radica
  100. Sign, and eheri the Ouancier, or Number remain-
  101. ins, is alvars che Square, Cube, or Orhet POIeK
  102. required; and will alivars be Rarional.
  103. Thusche Square ot V: 1I, 15 F1.
  104. The CobC OF V: 30, 15 50.
  105. AIIO2 V: 3 mulciplica by 8 s: 3 = 48. and
  106. 3V: 5. mulriplied bY 2 Y: 5 = 30.
  107. IV. F0 find Whetser tr3. Surd ROOtS siven, at
  108. osa or notia
  109. For an Instance in Species; suppose rhat it wet
  110. enquired whecher v: 27 a a and V : 12 a a Wvet
  111. Commensurable Surds; Diride caca Y che
  112. compound Number srearer chan rwo, and tis re
  113. quiredro square sueh a Surd: Phere necd omly a
  114. Radical Sien, Whose Index is hait che Former, be
  115. pretixd 10 rhe Cuatius, inAtead ot che formet
  116. Componne one, and iis done.
  117. greatest conmtent o V: 3 a a. And cheduo
  118. tienEs V: 9 atid V : 4. Thal 15, 3 al Z. die Raris
  119. ral Numbers; and consequeneiy, che propole
  120. Surds are Commenturablc.
  121. V. And if che Index of che Porver be any even

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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/806933011:0791?run=74