Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

59. _Exemp_. II.

Sit ADG _circuli_ quâdrans, & habere debeat TP ad PM ratio-
nem eandem quam PE ad R; eſt ergo PY æqualis _tangenti_ arcûs GE; & ſpat. APYY = R x arc. AE. adeóque PM = arc. AE.

60. _Probl_. III.

Proponatur figura quælibet ADB (cujus _axis_ AD, _baſis_ DB)
reperiatur curva KZL, proprietate talis, ut ductâ rectâ ZPM ad
DB utcunque parallela quæ lineas expoſitas ſecet ut cernis) poſitóque
rectam ZT tangere curvam KZL, ſit intercepta TP æqualis ipſi
PM.

60.1.

Fig. 185.

Hocità perſicietur. Sit curva OYY talis, ut adſumptâ quâdam
R, protractâque PMY, ſit PM. R: : R. PY; tum liberè adſump-
tâ DL (in BD protensâ) ſit DL. R: : R. LE; & _aſymptotis_ DL,
DG per E deſcribatur _Hyperbola_ EXX; tum ſit ſpatium LEXH æ-
quale ſpatio DOYP, & protractæ XH, YP concurrant in Z; erit
Z in curva quæſita; quam ſi tangat ZT, erit TP = PM.

Adnotetur, ſi propoſita ſigura ſit _rectangulum Parallelogrammum_
ADBC, quod curvæ KZL hæc erit proprietas, ut ſit DH eodem
ordine inter DL, DO media _Geometricè_ proportionalis, quo DP
inter DA & θ [?] (ſeu nihilum) eſt media _Aritbmeticè_; quod ſi liberè
juxta proprietatem hanc deſcribatur curva KZL, & _Mechanicè_ re-
periatur tangens ZT, indè quadrabitur _hyperbolicum ſpatium_ LEXH; erit utique hoc æquale _rectangulo_ ex TP, AP.

60.1.

Fig. 186.

Subnotari poſſit fore 1. Spat. ADLK = R x DL - DO. 2. Sum. mam ZPq = R x : {DLq - DOq/2}. & ſummam ZP cub. = R x
{DLcub. - DOcub. /3} & c [?] . 3. Siponatur φ eſſe centrum gr. figu-
ræ ADLK, ducantúrque φψ ad AD, & φξ ad DL perpendicu-
lares, fore φψ = {DL + DO/4}, & φξ = R - {AD x DO/LO}.

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