329136
Series quarta.
_a_ + {_cc_/_a_} = _n_.
_aa_ + _cc_ = _nn_.
_a_3 + _cca_ = _n_3.
_a_4 + _ccaa_ = _n_4.
Sit recta indefinitè protenſa AH, &
huic perpendicularis AD;
fiat autem angulus RAH ſemirectus; tum utcunque ducatur GZK
11Fig. 211. ad AD parallela; & facto AG. AG: : AC. ZK; per Kintra
angulum DAR deſcribatur _hyperbola_ KXK; ſint denuò curvæ CLL,
AMM, ANN tales, ut inter GZ, GK ſint _media_ GL, _bimedia_
GM, _trimedia_ GN; hæ propoſito deſervient. Nam ſi AG (vel
GZ) dicatur _a_, erit GK = _a_ + {_cc_/_a_}; & GLq = _aa_ + _cc_; &
GMcub = _a_3 + _cca_; & GNqq = _a_4 + _ccaa_.
fiat autem angulus RAH ſemirectus; tum utcunque ducatur GZK
11Fig. 211. ad AD parallela; & facto AG. AG: : AC. ZK; per Kintra
angulum DAR deſcribatur _hyperbola_ KXK; ſint denuò curvæ CLL,
AMM, ANN tales, ut inter GZ, GK ſint _media_ GL, _bimedia_
GM, _trimedia_ GN; hæ propoſito deſervient. Nam ſi AG (vel
GZ) dicatur _a_, erit GK = _a_ + {_cc_/_a_}; & GLq = _aa_ + _cc_; &
GMcub = _a_3 + _cca_; & GNqq = _a_4 + _ccaa_.
Not.
1.
Deſignantur radices, ut in præcedentibus, poſitâ AE = _n_, &
ductâ
EF ad AH parallelâ.
EF ad AH parallelâ.
2.
Si AP = AC, erit PX ad _hyperbolam_ KXK ordinatarum _mi_-
_nima_; unde ſi AE (vel _n_) & lt; PX; nulla dabitur radix in primo
gradu.
_nima_; unde ſi AE (vel _n_) & lt; PX; nulla dabitur radix in primo
gradu.
3.
Curva CLL eſt _hyperbola æquilatera_, cujus _centrum_ A, _ſemi_-
_axis_ AC; quæ & ordinatarum eſt _minima_; alioquin ſi _n_& gt;_ c_, ſem-
per una vera radix habetur, & unica.
_axis_ AC; quæ & ordinatarum eſt _minima_; alioquin ſi _n_& gt;_ c_, ſem-
per una vera radix habetur, & unica.
4.
Reliquæ AMM, ANN ſunt hyperboliformes ad infinitum
excurrentes; unde ſemper una vera radix habetur, neque plures.
excurrentes; unde ſemper una vera radix habetur, neque plures.
5.
Si fuerit Y α = {1/2} YX;
Y β = {1/3}YX;
Y γ = {1/4} YX, &
per
puncta α, β γ, traductæ concipiantur _hpperbola_ (habentes & ipſæ _a_-
_ſymptotos_ DA, AR) α λ, β μ, γ ν; erunt hæ ipſarum curvarum
CLL, AMM, ANN _aſymptoti_. (Similes etiam _aſymptoti_ con-
veniunt lineis poſthac deſcribendis, quanquam de illis conticeamus.)
puncta α, β γ, traductæ concipiantur _hpperbola_ (habentes & ipſæ _a_-
_ſymptotos_ DA, AR) α λ, β μ, γ ν; erunt hæ ipſarum curvarum
CLL, AMM, ANN _aſymptoti_. (Similes etiam _aſymptoti_ con-
veniunt lineis poſthac deſcribendis, quanquam de illis conticeamus.)
6.
Hinc in ſecundo gradu _a_ + {_cc_/2_a_}&
gt;_
n_;
in tertio _a_ + {_cc_/3_a_}&
gt;_
n_;
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