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  "text": "quae erunt Aequationis datae Radices omnes reales; haec semper ad dextram erunt Radices affirmativae, illae vero ad sinistram Radices negativae. Demonstratio est manifesta ex praecedentibus, habita tantum ratione Parabolae per puncta B, C, c, x, x transiens. Nam positio F foco Parabolae, (cuius distantia à Vertice ait \\( \\frac{1}{2} ON \\)) notum est quod lineae omnes ut FB + BQ, FC + CD, &c., eadem ubique conficiant summam.\n\nAtque ex principiis hic positis proclive erit Instrumentum haud inconcinnum & quantumvis accuratum fabricari, cujus beneficio hujusmodi Aequationum quarumcunque Radices nullo fere negotio inveniri possint, & praecipue oculis exhiberi. Hoc autem quilibet, si id Curae sit, variis modis pro ingenio suo efficere potest, & de his jam satis.\n\n---\n\nIII. Aequationum quarundam Potestatis tertiae, quintae, septimae, nonae, & superiorum, ad infinitum usque pergendo, in terminis finitis, ad instar Regularum pro Cubicis quae vocantur Cardani, Resolutio Analytica.\n\nPer Ab. De Moivre, R. S. S.\n\nSit n Numerus quicunque, y quantitas incognita, sive Aequationis Radix quaesita, sitque a quantitas quævis omnino cognita, sive ut vocant Homogeneum Comparationis: Atque horum inter se relatio exprimatur per Aequationem\n\n\\[ ny + \\frac{nn - 1}{2 \\times 3} ny^3 + \\frac{nn - 1}{2 \\times 3} \\times \\frac{nn - 9}{4 \\times 5} ny^5 + \\frac{nn - 1}{2 \\times 3} \\times \\frac{nn - 9}{4 \\times 5} \\times \\frac{nn - 25}{6 \\times 7} ny^7, \\text{ &c.} = a \\]\n\nEx\nEx hujus seriei natura manifestum est, quod si in summatur numerus aliquis impar (integer scilicet, nec refert utrum sit affirmativus vel negativus) tunc series sponte sua terminabitur, & Æquatio fit una ex supra praesinitis, cujus Radix est\n\n\\[ y = \\frac{1}{2} \\sqrt[n]{\\frac{1}{\\sqrt{n}} + aa + a} - \\frac{1}{2} \\sqrt[n]{\\frac{1}{\\sqrt{n}} + aa + a} \\]\n\nvel (2) \\( y = \\frac{1}{2} \\sqrt[n]{\\frac{1}{\\sqrt{n}} + aa + a} - \\frac{1}{2} \\sqrt[n]{\\frac{1}{\\sqrt{n}} + aa - a} \\)\n\nvel (3) \\( y = \\frac{1}{2} \\sqrt[n]{\\frac{1}{\\sqrt{n}} + aa - a} - \\frac{1}{2} \\sqrt[n]{\\frac{1}{\\sqrt{n}} + aa - a} \\)\n\nvel (4) \\( y = \\frac{1}{2} \\sqrt[n]{\\frac{1}{\\sqrt{n}} + aa - a} - \\frac{1}{2} \\sqrt[n]{\\frac{1}{\\sqrt{n}} + aa - a} \\)\n\nExempli gratia, sit hujus Æquationis potestatis quintae \\( 5y + 20y^3 + 16y^5 = 4 \\) Radix invenienda, quo in ca- fu erit \\( n = 5 \\) & \\( a = 4 \\). Radix juxta formam primam erit \\( y = \\frac{1}{2} \\sqrt{\\frac{1}{\\sqrt{17}} + 4} - \\frac{1}{2} \\), qua in numeris vul- garibus expeditissime explicari potest ad hunc modum. Est \\( \\sqrt{17} + 4 = 8.1231 \\), cujus Logarithmus \\( o.9097164 \\), & hujus pars quinta \\( o.1819433 \\), huic respondens numerus est\n\n\\( 1.5203 = \\sqrt{\\frac{1}{\\sqrt{17}} + 4} \\). Ipsius vero \\( o.1819433 \\) Complementum Arithmeticum est \\( 9.8180567 \\), cui respondet numerus \\( o.6577 = \\frac{1}{\\sqrt{\\frac{1}{\\sqrt{17}} + 4}} \\). Igitur horum numero- rum semidifferentia \\( o.4313 = y \\).\n\n\\[ \\boxed{Q} \\]\nHic venit Observandum quod loco Radicis generalis, non\nincommodè sumeretur \\( y = \\frac{1}{2} \\sqrt[2]{2a - \\frac{1}{n}} \\), si qua-\ndo numerus a respectu unitatis, si satis magnus, ut si\nÆquatio fuerit \\( 5y + 20y^3 + 16y^5 = 682 \\), crit Log.\n\\( 2a = 3.1348143 \\), cujus pars quinta \\( 0.6269628 \\), & huic\nrespondens numerus \\( 4.236 \\). Complementi autem Arith-\nmetici \\( 9.3730372 \\) numerus est \\( 0.236 \\) & horum numero-\nrum semidifferentia \\( 2 = y \\).\n\nAtqui præterea, si in Æquatione praecedenti signa alter-\nnatim sint affirmantia & negantia, vel quod eodem redit,\nsi series obvenerit hujus modi\n\n\\[\nny + \\frac{1 - nn}{2 \\times 3} ny^3 + \\frac{1 - nn}{2 \\times 3} \\times \\frac{9 - nn}{4 \\times 5} ny^5 + \\frac{1 - nn}{2 \\times 3} \\times \\frac{9 - nn}{4 \\times 5} \\times \\frac{25 - nn}{6 \\times 7} ny^7, \\&c. = a\n\\]\n\ncrit hujus Radix\n\n(1) \\( y = \\frac{1}{2} \\sqrt[n]{a + \\sqrt[n]{aa - 1}} + \\frac{r}{2} \\sqrt[n]{a + \\sqrt[n]{aa - 1}} \\)\n\nvel (2) \\( y = \\frac{1}{2} \\sqrt[n]{a + \\sqrt[n]{aa - 1}} + \\frac{1}{2} \\sqrt[n]{a - \\sqrt[n]{aa - 1}} \\)\n\nvel (3) \\( y = \\frac{1}{2} \\sqrt[n]{a - \\sqrt[n]{aa - 1}} + \\frac{1}{2} \\sqrt[n]{a - \\sqrt[n]{aa - 1}} \\)\n\nvel (4) \\( y = \\frac{1}{2} \\sqrt[n]{a - \\sqrt[n]{aa - 1}} + \\frac{1}{2} \\sqrt[n]{a + \\sqrt[n]{aa - 1}} \\)\n\nHic autem Notandum, quod si \\( \\frac{n-1}{2} \\) numerus extiterit\nimpar, Radicis inventæ signum in ei contrarium permu-\ntandum est.\n\nPro-\nProponatur Aequatio $5y - 20y^3 + 16y^5 = 6$, unde $n = 5$ & $a = 6$. Erit Radix $= \\frac{1}{2} \\sqrt{6 + \\sqrt{35}} + \\frac{1}{2} \\sqrt{6 - \\sqrt{35}}$.\n\nVei quoniam $6 + \\sqrt{35} = 11.916$, erit hujus logarithmus $1.0761304$ & ejus pars quinta $0.2152561$, Complementum vero Arithmeticum $9.7847439$. Horum Logarithrorum numeri sunt $1.6415$ & $0.6091$ respective, quorum semisumma $1.1253 = y$.\n\nVerum si acciderit ut a sit minor unitate, tunc Radicis forma secunda, ut quae proposito est magis conveniens, praereliquis feligenda est. Sic si Aequatio fuerit $5y - 20y^3 + 16y^5 = \\frac{61}{64}$, erit $y = \\frac{1}{2} \\sqrt{\\frac{61}{64}} + \\sqrt{\\frac{375}{4096}}$.\n\nEt quidem si Binomialium Radix quintana ullo pacto extrahi queat, prodibit Radix proba & possibilis, et si expressio ipsa impossibilitatem mentiatur. Binomialis vero $\\frac{61}{64} + \\sqrt{\\frac{375}{4096}}$ Radix quintana est $\\frac{1}{2} + \\frac{1}{4} \\sqrt{15}$, & Binomialis $\\frac{61}{64} - \\sqrt{\\frac{375}{4096}}$ Radix itidem quintana est $\\frac{1}{2} - \\frac{1}{4} \\sqrt{15}$, quorum Binomialium semifumma $= \\frac{1}{4} = y$.\n\nSi autem extractio ista vel non peragi poscit, vel etiam difficilior videretur, res ubique consici potest per Tabulam sinuum naturalium ad modum sequentem.\n\nAd Radium $1$ sit $a = \\frac{61}{64} = 0.95112$ sinus arcus cujusdam, qui proinde erit $72^\\circ : 23'$ cujus pars quinta (eo quod $n = 5$) est $14^\\circ : 28'$; hujus sinus $0.24981 = \\frac{1}{4}$ proxime. Nec secus procedendum in Aequationibus graduum superiorum:\n\n$14Q_2$ IV. S",
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    "title": "Aequationum Quarundam Potestatis Tertiae, Quintae, Septimae, Nonae, & Superiorum, ad Infinitum Usque Pergendo, in Terminis Finitis, ad Instar Regularum pro Cubicis Quae Vocantur Cardani, Resolutio Analytica",
    "authors": "Ab. de Moivre",
    "year": 1706,
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    "journal": "Philosophical Transactions (1683-1775)",
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