Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

83. 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
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_.

83.1.

Fig. 211.

84. Not.

1. Deſignantur radices, ut in præcedentibus, poſitâ AE = _n_, & ductâ
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.

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.

4. Reliquæ AMM, ANN ſunt hyperboliformes ad infinitum
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.)

6. Hinc in ſecundo gradu _a_ + {_cc_/2_a_}& gt;_ n_; in tertio _a_ + {_cc_/3_a_}& gt;_ n_;

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