Page:Philosophical Transactions - Volume 003.djvu/134

  figuræ & methodo quantum res ferebat accomodaveram) ad principia mea revocatam ab origins repetam. V. Fig. 2.

Ostensum est, in mea Arithmetica Infinitorum, prop. 95.. Spatium Hyperbolium $$AD \beta \beta \gamma$$ (in infinitum continuarum à parte $$\beta \delta$$, sed à parte $$D\beta$$ ubivis terminatum,) Figuram esse quam ex Primanorum Reciprocis conflatam appello, Prop. 88. definitam: Cujus nempe Ordinatim—applicatæ $$d\beta$$, $$d\beta$$, sint Primanis (seu Arithmetice proportionaiibus) $$db$$, $$db$$, (Triangulum complentibus) adeoque ipsis $$dA$$, $$dA$$, (suis à vertice distantiis) Reciproce Proportionales. Hoc est, (posito $$AD=d$$; & rectangulo $$AD\beta = b^2$$; particulisque infinite exiguis $$a$$, $$a$$, &c;) fi à vertice $$A\delta$$ incipias $$\frac{b^2}{0}$$, $$\frac{b^2}{a}$$, $$\frac{b^2}{2a}$$, $$\frac{b^2}{3a}$$ &c. usque ad $$\frac{b^2}{d} = D\delta$$: vel, fi à base $$D\delta$$ incipias, $$\frac{b^2}{d}$$, $$\frac{b^2}{d -a}$$, $$\frac{b^2}{d-2a}$$, $$\frac{b^2}{d-3a}$$, &c. usque ad $$\frac{b^2}{d-d} = A\delta$$ infinitæ, (nempe, fi ad Verticem usque processum continuaveris;) vel, usque ad $$\frac{b^2}{d-A} =C\beta$$, (posito $$DC=A$$,) fi continuaveriis usque ad $$C\beta$$, ubivis intra $$A\delta$$ & $$D\beta$$ sumptam. (Adeoque omnium Aggregatum; $$\frac{b^2}{d} + \frac{b^2}{d-a} + \frac{b^2}{d-2a} + \frac{b^2}{d-3a}$$, &c, est ipsum $$DC\beta\beta$$ planum.)

Manifestum itaque est, (& ibidem pro. 94. ostensum) si intelligantur singulæ $$d\beta$$, in fuas à vertice distantias $$Ad$$, ductæ; hoc est, $$\frac{B^2}{a}$$ in $$a$$, $$\frac{B^2}{2a}$$in $$2a$$, (& sic de reliquis;) crunt omnia rectangula $$A d\beta$$; hoc est, rectarum $$d\beta$$ momenta respectu $$A\delta$$, (intellige, facta ex magnitudine in distantiam ductæ;) seu plana semiquadrantalem Ungulam (cujus acies $$A\delta$$ complentia, (eisdem $$d\beta$$ rectis perpendiculariter insistentia;) invicem æqualia. Quippe singula $${} =b^2$$. (Quorum cum unum sit $$AI V \delta$$ quadratum, erit $$AI=b$$.)

Adeoque Totius $$A D\beta \beta \delta$$ (plani infiniti) seu omnium $$d\beta$$ illud complentium, momentum respectu rectæ $$A\delta$$, (ut axis æquilibrii;) seu Ungula semiquadrntalis eidem $$A D \beta \beta \delta$$ insistens (aciem habens $$A \delta$$;) sunt totidem $$b^2$$; hoc est, $$db^2$$. (Ungula magnitudinis finitæ plano infinitæ magnitudinis insistens.) Ejusque pars plano $$AC\beta \beta \delta$$ insistens (propter $$AC = d-A$$.) similiter ostendetur æqualis ipsi $$d - A$$ in $$b^2$$. ductæ; hoc est $$d b^2 - A b^2$$. Adeoque pars reliqua, ipsi $$D C\beta \beta$$ insistens, æqualis ipsi $$Ab^2$$. Quod itaque est ejusdem $$DC\beta \beta$$ momentum respectu $$A\delta$$. Rh