136118
ad ipſam EC parallelæ ducantur rectæ RT, SV;
palàm eſt indeter-
minatum punctum X inter limites T, V conſiſtere (nam extra TV
punctum quodlibet L accipiendo, & indè ducendo LI P ad CE paralle-
lam, erit CL, hoc eft LP, major quàm LI, unde à C ad rectam LI,
nulla duci recta poteſt æqualis ipſi LI). Jam autem dico, quòd
punctum Z ad ellipſin exiſtit, cujus axis TV, focus C. Nam biſe-
cetur TV in K; fiat VD = TC; ducatur KH ad CE parallela;
per H ducatur HN ad CK parallela. Eſtque KH = {TR + VS/2} =
{CT + CV/2} = KT = KV. Et quoniam AV. AT : : (VS.
TR (hoc eſt) : : CV. CT : :) CV. DV; erit per rationis con-
vcrſionem AV. TV : : CV. CD. vel, conſequentes ſubduplando,
AV. KV : : CV. CK. dividendóque AK. KV : : KV. CK; hoc eſt
AK. KH : : KH. CK. hoc eſt HN. NG : : KH. CK. quare
KH x NG = CK x HN = CK x KX. atqui eſt CZq = XGq
= KHq + NGq + 2 KH x NG. & CXq = CKq + KXq
+ 2 CK x KX = CKq + KXq + 2 KH x NG. ergo
KHq + NGq - CKq - KXq = CZq - CXq = XZq.
Ad alteras biſegmenti K partes ſumatur K ξ = KX, ducatúrque ξν ad
KH parallela, ſecans curvam TEZV in ζ, & rectam AH in γ, ac
ipſam NH in ν erit quoque, ſimili ex diſcurſu, ξζq = KHq +
νγq - CKq - Kξq; unde liquet fore ξζ = XZ; connexíſque
proinde rectis Cζ, Dζ, erit Dζ = CZ; & Cζ + CZ = ξγ +
XG = 2 KH = TV. ergò Cζ + Dζ (vel DZ + CZ) = TV.
unde perſpicitur _curvam TζZV eſſe ellipſin_, cujus _axis_ TV; _foci_
C, D.
minatum punctum X inter limites T, V conſiſtere (nam extra TV
punctum quodlibet L accipiendo, & indè ducendo LI P ad CE paralle-
lam, erit CL, hoc eft LP, major quàm LI, unde à C ad rectam LI,
nulla duci recta poteſt æqualis ipſi LI). Jam autem dico, quòd
punctum Z ad ellipſin exiſtit, cujus axis TV, focus C. Nam biſe-
cetur TV in K; fiat VD = TC; ducatur KH ad CE parallela;
per H ducatur HN ad CK parallela. Eſtque KH = {TR + VS/2} =
{CT + CV/2} = KT = KV. Et quoniam AV. AT : : (VS.
TR (hoc eſt) : : CV. CT : :) CV. DV; erit per rationis con-
vcrſionem AV. TV : : CV. CD. vel, conſequentes ſubduplando,
AV. KV : : CV. CK. dividendóque AK. KV : : KV. CK; hoc eſt
AK. KH : : KH. CK. hoc eſt HN. NG : : KH. CK. quare
KH x NG = CK x HN = CK x KX. atqui eſt CZq = XGq
= KHq + NGq + 2 KH x NG. & CXq = CKq + KXq
+ 2 CK x KX = CKq + KXq + 2 KH x NG. ergo
KHq + NGq - CKq - KXq = CZq - CXq = XZq.
Ad alteras biſegmenti K partes ſumatur K ξ = KX, ducatúrque ξν ad
KH parallela, ſecans curvam TEZV in ζ, & rectam AH in γ, ac
ipſam NH in ν erit quoque, ſimili ex diſcurſu, ξζq = KHq +
νγq - CKq - Kξq; unde liquet fore ξζ = XZ; connexíſque
proinde rectis Cζ, Dζ, erit Dζ = CZ; & Cζ + CZ = ξγ +
XG = 2 KH = TV. ergò Cζ + Dζ (vel DZ + CZ) = TV.
unde perſpicitur _curvam TζZV eſſe ellipſin_, cujus _axis_ TV; _foci_
C, D.
III.
Sit autem ſecundò angulus CA E major ſemirecto (vel AC
11Fig. 191.& lt; CE) dico punctum Z ad oppoſitas hyperbolas, conſimili modo
determinabiles, exiſtere. enimverò factis (ad utramque rectæ CA
partem) angulis ſemirectis ACP; & (ab ipſarum CP cum AE
occurſibus) ductis rectis RT, SV ad CE parallelis, punctum X
extra limites TV neceſſariò conſiſtet (etenim ubivis intra TV ductâ
LIP ad CE parallelâ, erit LI & lt; LP, ideóque nulla par ipſi LI
angulo AL I ſubtendi poteſt; id quod extra terminos hoſce nil pro-
hibet fieri) erit jam TV axis, & C focus hyperbolarum. Fiant
enim omnia, quæ in caſu præcedente; erítque rurſus hîc KH =
KV. item ob AV. AT : : CV. DV; & (inversè componendo)
11Fig. 191.& lt; CE) dico punctum Z ad oppoſitas hyperbolas, conſimili modo
determinabiles, exiſtere. enimverò factis (ad utramque rectæ CA
partem) angulis ſemirectis ACP; & (ab ipſarum CP cum AE
occurſibus) ductis rectis RT, SV ad CE parallelis, punctum X
extra limites TV neceſſariò conſiſtet (etenim ubivis intra TV ductâ
LIP ad CE parallelâ, erit LI & lt; LP, ideóque nulla par ipſi LI
angulo AL I ſubtendi poteſt; id quod extra terminos hoſce nil pro-
hibet fieri) erit jam TV axis, & C focus hyperbolarum. Fiant
enim omnia, quæ in caſu præcedente; erítque rurſus hîc KH =
KV. item ob AV. AT : : CV. DV; & (inversè componendo)