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Philosophiae naturalis principia mathematica . e cognofcamus : Sicut radices furdae fini-tarum aequationum nec numeris nec quavis arte Analytica ita pofliintexhiberi ut alicujus quantitas a reliquis diffinfta exafte cognofcatur. Denique ad Analyticam merito pertinere cenfeatur cujusbeneficiocurvarum areae & longitudines &c. (id modo fiat) exafte & Geome-trice determinentur. Sed ifi:a narrandi non efl locus, refpicienti duoprae reliquis demonflranda occurrunr. I. Demonflratto quadratura curvarum fmpTictumm Regula prtma. Traparatio pro Regula prima demonjlranda. Sit itaque curvae alicujus AD^ Ba
Philosophiae naturalis principia mathematica . e cognofcamus : Sicut radices furdae fini-tarum aequationum nec numeris nec quavis arte Analytica ita pofliintexhiberi ut alicujus quantitas a reliquis diffinfta exafte cognofcatur. Denique ad Analyticam merito pertinere cenfeatur cujusbeneficiocurvarum areae & longitudines &c. (id modo fiat) exafte & Geome-trice determinentur. Sed ifi:a narrandi non efl locus, refpicienti duoprae reliquis demonflranda occurrunr. I. Demonflratto quadratura curvarum fmpTictumm Regula prtma. Traparatio pro Regula prima demonjlranda. Sit itaque curvae alicujus AD^ Bafis AB^a-, perpendiculariter ap-plicata BD=:jy, & area ABD=:2;, ut fit B/3 = o, BK=i;, & reaangulum B/3HK{ov) aequale fpatio B/3^D. Eft ergo Ai6 = Ar+(7, & K^^—z-\ praemiffis, ex relatione inter a- & 2; ad arbi-trium afiTumpta quaerojy ifto, quem fcquentemvides, modo. Pro lubitu fumatur ^^x\ — z., five ^x^ — «• + 0 (A/3) pro ;v, & ;5 + w (A^/3) pro z fubfiitutis, prodi- C X bit. xo D E A N A L Y S I bit s in x^ + 3a:^(? + 3^0 + 0 = (ex natura curvsB) z + ^zov + o^ fublatis {\x^^zz) fequalibus. reliquisque per (? divifis, reftat | in3A?- +^xo-k0^=%zv + ov^. Si jam fupponamus b/3 ininfinitumdiminui& evanefcere, five 0 efl^e nihil, erunt i; & jy sequales, & termini pertfmultiplicati evanefcent, quare reikbit Ix3^^ = i5;z;, fivel^x(=^y) = f Aij, five .v| (=^) ^y. Quare e contra fi xi=y^ eritf^v^vs, Demonflratlo. : Vel generaliter, fi ;7^ ^a^ ^ = z; five, ponendo —7, = r, & w^ p.+;/=^, fi fA:«=2;, five r^A-frra: tum x + o prOA:, &-s + w (five, quod perinde eft, z + qy) pro z, fubflitutis, prodit fin xf+pox^-\:z + /iojiz~, &c. reliquis nempe terminis, qui tandem eva-nefcerent, omiffis, Jam fublatis c^xp & ^i^aequalibus, reliquisque percx =ny\ vel reftituendo J^ pror, fkm+npro*, hoc eft, m pro/—», & «^ pro />f, fiet ax^^^y. Quare
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Keywords: ., bookauthornewtonisaacsir16421727, booksubj, booksubjectmechanics