1Centro S,Aſymptotis SA, Sx,deſcribatur Hyperbola quæ
vis, quæ ſecet perpendicula AH, BI, CK,&c. in a, b, c,&c. ut &
perpendicula ad Aſymptoton Sxdemiſſa Ht, Iu, Kwin h, i, k;
& denſitatum differentiæ tu, uw,&c. erunt üt (AH/SA), (BI/SB),&c. Et
rectangula tuXth, uwXui,&c. ſeu tp, uq,&c. ut (AHXtb/SA),
(BIXui/SB),&c. id eſt, ut Aa, Bb,&c. Eſt enim, ex natura Hyperbolæ,
SAad AHvel St,ut thad Aa,adeoque (AHXth/SA) æquale Aa
173[Figure 173]
Et ſimili argumento eſt (BIXui/SB) æquale Bb,&c. Sunt autem Aa,
Bb, Cc,&c. continue proportionales, & propterea differentiis ſu
is Aa-Bb, Bb-Cc,&c. proportionales; ideoQ.E.D.fferentiis
hiſce proportionalia ſunt rectangula tp, uq,&c. ut & ſummis diffe
rentiarum Aa-Ccvel Aa-Ddſummæ rectangulorum tp+uq
vel tp+uq+wr.Sunto ejuſmodi termini quam plurimi, & ſum
ma omnium differentiarum, puta Aa-Ff,erit ſummæ omnium
rectangulorum, puta zthn,proportionalis. Augeatur numerus
terminorum & minuantur diſtantiæ punctorum A, B, C,&c. in in
nitum, & rectangula illa evadent æqualia areæ Hyperbolicæ zthn,
adeoque huic areæ proportionalis eſt differentia Aa-Ff.Suman-
vis, quæ ſecet perpendicula AH, BI, CK,&c. in a, b, c,&c. ut &
perpendicula ad Aſymptoton Sxdemiſſa Ht, Iu, Kwin h, i, k;
& denſitatum differentiæ tu, uw,&c. erunt üt (AH/SA), (BI/SB),&c. Et
rectangula tuXth, uwXui,&c. ſeu tp, uq,&c. ut (AHXtb/SA),
(BIXui/SB),&c. id eſt, ut Aa, Bb,&c. Eſt enim, ex natura Hyperbolæ,
SAad AHvel St,ut thad Aa,adeoque (AHXth/SA) æquale Aa
173[Figure 173]Et ſimili argumento eſt (BIXui/SB) æquale Bb,&c. Sunt autem Aa,
Bb, Cc,&c. continue proportionales, & propterea differentiis ſu
is Aa-Bb, Bb-Cc,&c. proportionales; ideoQ.E.D.fferentiis
hiſce proportionalia ſunt rectangula tp, uq,&c. ut & ſummis diffe
rentiarum Aa-Ccvel Aa-Ddſummæ rectangulorum tp+uq
vel tp+uq+wr.Sunto ejuſmodi termini quam plurimi, & ſum
ma omnium differentiarum, puta Aa-Ff,erit ſummæ omnium
rectangulorum, puta zthn,proportionalis. Augeatur numerus
terminorum & minuantur diſtantiæ punctorum A, B, C,&c. in in
nitum, & rectangula illa evadent æqualia areæ Hyperbolicæ zthn,
adeoque huic areæ proportionalis eſt differentia Aa-Ff.Suman-