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

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  1. Index oF 100 150; then may che ROOE be casily ) Hnovn Ouantiries, the Ouanciries chemsclres vill
  2. Found by etie folloine Merhod.
  3. AII Equations OF ehis Ranle wwill be in one o
  4. chese chree FOrms.
  5. an =)= a d = R. 2 + Some male four Forms.
  6. a a - - a d = R. but at lons run it comes
  7. a5 = a a = R.tothesame rhins.
  8. En all vohiict Forme, R, rhe abloture Number
  9. giren, is a Rectanele or Produet made out oF rhC
  10. vo Ouanciries or Roors Coushe, a Crearer and:
  11. L eNCF.
  12. OF wtich, inthe First Form, where all is AHir-
  13. matire, the Coefficient d is the Disserence Between
  14. (oon be bnown.
  15. Thus in che Ars FOrM, let a a - d a = R.
  16. Here is given R, the Rectansle OF the ROOES,
  17. a cheir Ditlerence; and ris hnoun thac a repre
  18. Cents the lesser oF them. Lec S Cand Forthe Summ
  19. Eo bc Coughr.
  20. Lec a V c = Sand a = c == d, Then a a -]
  21. 2 a c -) - e e == 55, and a a = 2 a c =) e0 = d
  22. Substract the lacter trom the former, and there re-
  23. mains only 4a c = 4R. Wheretore SS-d d = 4R
  24. Tou may cheretore by Simple Alsebra And cha
  25. 4R == 55= d d, and consequeney chac 4 R -1-
  26. a d == 5S, and cheretore S istnown; and then
  27. those tuvo Ouanticies or Roors; and a is the Lesser havins S and d, a the lesser Roor will be Hnoon t00,
  28. OF chem, a5 is plain is von suppose the EWO ROOES FOF =S = = d =
  29. (25 Oushered doch) ro be a the greater, and c the
  30. Lesser. Porthen lec d = X be che Ditterence Be-
  31. tveen chem: Sochat e - X = a. it then vou
  32. Mulciply cach Parr by e, ic will bC e c -F e
  33. ac; trom whence ic appears allo plainly, chac a c
  34. is equal ro R, the absolure Number siren, oi c
  35. a
  36. Aeain, in che second FOrM. Let aa = a d = R.
  37. Eere d aud R (as betore) che Ditterence and
  38. Rectanse oF The LWO ROOTS are siren; arid a the
  39. crearer ot chem; Wheretore ris easiero tind S che
  40. qual co che Rectansle oF the rvo unknonn ROOrS Summ, and chen =5 Z d = a.
  41. a and c, ot which in this Form, the Coefficiene a
  42. Or dis equal coche Difference betveen them, and e
  43. isthe Lesser os chem.
  44. In rhe Second Form, The Coeficiene d is the Dit.
  45. Ference ot the trvo Roors as betore, Dur a there re
  46. In che chird Form, iotere 5 a = aa = R.
  47. There is siven the Coefficien S = Summ ofche
  48. untnoun Rooes, R che Rectansle berween them;
  49. and a max be cichierrhe bisser or thelesser of them;
  50. prelemts the Creater oFthem, a5 is plain by purtine Llere cheretore ro tind d che Ditterence
  51. (because the Sion is Negatire) a -- x = e, and
  52. Musciptrins each Parr bY a, ic produces a a = a X
  53. a e, che second Form, Where x Or d che C0.
  54. efficiene is the Difference oF the rvo untenovn
  55. Roors; and a represenis the greater oFthem.
  56. Inthe Thira Form, Where che highest Pouer is
  57. Negarive, the Coesficientsis the Sum of the EVO,
  58. Ouanticies ox RoorS Toughr; and a the AfFirmatire
  59. Roor (oughr, may be cirher itre Bigger or theLes-
  60. Because SS- d d = 4R, cheretore SS - ] 4 F
  61. =a d, and consequener d is known; and then
  62. 25- Zd = BrCacer, and ZS = = d = leNer.
  63. II. The Solution os Adfected Dudraricte Equations,
  64. by the Method of Compleatins the Squâre.
  65. Wrhich is BY DIP. Parriot, ehus: Since in every
  66. onc ot che chree Forms of Ouadraricks; one quar-
  67. rer of the Square of the Coefficient will make the
  68. sex oFrhem. For lec (Because chehighest Porver is untenoun Side ofthe Equarion a C ompleat Square,
  69. TFegatire) Z - a = e, Then Mulriplyins Boch by a wtose erue Roor will be a -1 - = d (or Whaterer
  70. it Wvill Z a = a a = a e== R. Or it Z ehad been po
  71. equalro an then it ould hare been 2e = ec=
  72. ac, by Mulciplrins all by c.
  73. So That CIS Method MeS vou IHC OIISIIaL
  74. Consticution ot chese Forms, and the Nature and
  75. Ofice oFcach Member et them.
  76. From all Which may be found chis general Canor
  77. For che Solurion oF Ouadrarick Equarions, accord.
  78. ins eo chis Method.
  79. the Produet add. the Square ot the Coeticient
  80. then Extract che Square Roo OEthat Sum: Whict
  81. RoOF Mallbe the Sum oF the trvo Numbers Touxti
  82. Then e0 ox From the halE OF TaE RROOF, Add anc
  83. Substract hals the Coessicient, and the Sum ane
  84. Remainder are the two Roors required.
  85. For rhe parricular Solurion os Adfected Ouadr
  86. cicks there are chrec WayS.
  87. Lerrer esse be the Coefficient.) Tis plain by chis
  88. means, an Adfected duadratick Equation, may bc
  89. reducedro a simple one.
  90. WVheretore,
  91. In che Arst Form, Wrhere all the Species are AHir-
  92. matire.
  93. be a pertect Square a a - d a- = d d, Whosetrue
  94. ROOE 1S 4 =
  95. a perfect Square Number and Hnowmn: Whose
  96. Square ROOF extracted in Numbers, will be equat
  97. t0a-) = d; and consequener, a will be equal 10
  98. chac Roor, When = d is caten trom ic, and s0 a will
  99. Let aa t a a L
  100. IF = d d be added ro the unkenovn Side, ic will
  101. Add Fhen, = d d EO R, and R - = d d will be
  102. be known.
  103. Musriply che absolure Number By 4, and 10
  104. Lde Llalleat cille 2d latdus
  105. I. That of Oushered, uho proceeds in its
  106. tdet tobco
  107. In all che chree Forms, there is giren cithet che
  108. Foche Absoture Number, add : ot he square
  109. Rectanete ad Summ, Or the Rectansa and Diffe- OE Coefficient, (or the Square of half the Coessici
  110. rence ot the to unknovn Onantiries; Whence ris enF) and Fxrract che ROOE oF the Sum; then FrOM
  111. very eahe eo End cither the Differance in che For- chac ROOr found in PTumbers, Subrract ; che CoeF-
  112. mer, ox the summ in che larter Cafe: And then) ficient, and che Remainder is a, the letser ol che
  113. harius ihte summ aud Dissererce ot auy tuo uEvO ROOO, Oi Values e a.
  114. F Natiole.

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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/806933011:0680?run=74