Pagina:Principia newton la.djvu/132

per motum continuum, ea lege ut inter movendum crescendo vel decrescendo æquetur semper longitudini LD, describet aream ${\tfrac {LBq.-LAq.}{2}}$, id est, aream SL×AB; quæ subducta de area priore 2SL×AB relinquit aream SL×AB.

Pars autem tertia ${\tfrac {ALB}{LD}}$ ducta itidem per motum localem normaliter in eandem longitudinem, describet aream Hyperbolicam; quæ subducta de area SL×AB relinquet aream quæsitam ABNA. Unde talis emergit Problematis constructio. Ad puncta L, A, B erige perpendicula Ll, Aa, Bb, quorum Aa ipsi LB, & Bb ipsi LA æquetur. Asymptotis Ll, LB, per puncta a, b describatur Hyperbola ab. Et acta chorda ba claudet aream aba areæ quæsitæ ABNA æqualem.

Exempl. 2. Si vis centripeta ad singulas Sphæræ particulas tendens sit reciproce ut cubus distantiæ, vel (quod perinde est) ut cubus ille applicatus ad planum quodvis datum; scribe ${\tfrac {PEcub.}{2ASq.}}$ pro V, dein 2PS×LD pro PEq.; & fiet DN ut ${\tfrac {SL\times ASq.}{PS\times LD}}$−${\tfrac {ASq.}{2PS}}$−${\tfrac {ALB\times ASq.}{2PS\times LDq.}}$ id est (ob continue proportionales PS, AS, SI) ut ${\tfrac {LSI}{LD}}$−${\tfrac {1}{2}}$SI−${\tfrac {ALB\times SI}{2LDq.}}$. Si ducantur hujus partes tres in longitudinem AB, prima ${\tfrac {LSI}{LD}}$ generabit aream Hyperbolicam; secunda ${\tfrac {1}{2}}$SI aream ${\tfrac {1}{2}}$AB×SI; tertia ${\tfrac {ALB\times SI}{2LDq.}}$ aream ${\tfrac {ALB\times SI}{2LA}}$−${\tfrac {ALB\times SI}{2LB}}$, id est ${\tfrac {1}{2}}$AB×SI. De prima subducatur summa secundæ ac tertiæ, & manebit area quæsita ABNA. Unde talis emergit Problematis constructio. Ad puncta L, A, S, B erige perpendic ula Ll, Aa, Ss, Bb, quorum Ss ipsi SI æquetur, perq; punctum s Asymptotis Ll, LB describatur Hyperbola asb occurrens perpendiculis Aa, Bb in a & b; & rectangulum 2ASI subductum de area Hyperbolica AasbB relinquet aream quæsitam ABNA.

Exempl. 3. Si Vis centripeta, ad singulas Sphæræ particulas tendens, decrescit in quadruplicata ratione distantiæ a particulis, scribe ${\tfrac {PE^{4}}{2AS^{3}}}$ pro V, dein √2PS×LD pro PE, & fiet DN ut ${\tfrac {SL\times SI^{\tfrac {3}{2}}}{{\sqrt {2}}\times LD^{\tfrac {3}{2}}}}$−${\tfrac {SI^{\tfrac {3}{2}}}{2{\sqrt {2}}\times LD^{\tfrac {1}{2}}}}$−${\tfrac {ALB\times SI^{\tfrac {3}{2}}}{2{\sqrt {2}}\times LD^{\tfrac {5}{2}}}}$. Cujus tres partes ductæ in longitudinem AB, producunt Areas totidem, viz. ${\tfrac {{\sqrt {2}}\times SL\times SI^{\tfrac {3}{2}}}{LA^{\tfrac {1}{2}}}}$−${\tfrac {{\sqrt {2}}\times SL\times SI^{\tfrac {3}{2}}}{LB^{\tfrac {1}{2}}}}$, ${\tfrac {LB^{\tfrac {1}{2}}\times SI^{\tfrac {3}{2}}-LA^{\tfrac {1}{2}}-SI^{\tfrac {3}{2}}}{\sqrt {2}}}$ & ${\tfrac {ALB\times SI^{\tfrac {3}{2}}}{3{\sqrt {2}}\times LA^{\tfrac {3}{2}}}}$−${\tfrac {ALB\times SI^{\tfrac {3}{2}}}{3{\sqrt {2}}\times LB^{\tfrac {3}{2}}}}$. Et hæ post debitam reductionem, subductis posterioribus de priori, evadunt ${\tfrac {8SIcub.}{3LI}}$. Igitur vis tota, qua corpusculum P in Sphæræ centrum trahitur, est ut ${\tfrac {SIcub.}{PI}}$, id est reciproce ut PScub.×PI. Q.E.I.

Eadem Methodo determinari potest attractio corpusculi siti intra Sphæram, sed expeditius per Theorema sequens.

Prop. LXXXII. Theor. XLI.

In Sphæra centro S intervallo SA descripta, si capiantur SI, SA, SP continue proportionales: dico quod corpusculi intra Sphæram in loco quovis I attractio