Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

a [?] quatur ipſi R x RM, vel R x QP. itaque totum ſpatium ADE
quod ab ejuſmodi ſectoribus minimè differt adæquatur toti R x DK. quod erat Propoſitum.

41.1.

(_a_) _Hyp._

XXIII. Iiſdem, quoad cætera, poſitis atque paratis, ducantur KH
ad KT, & MI ad MS perpendiculares; & concipiatur jam curva
AE naturâ talis, ut ſit DE = √ DK x DH; & DF = √ DM x
DI; ac ità perpetuò; erit ſpatium ADE quadrati ex DK ſubqua-
druplum.

41.1.

Fig. 128.

Nam eſt MP. PK: : DK. DH: : DKq. DK x DH: : DKq. DEq. item DP. PM: : DE. EX; hoc eſt DK. PM: : DE. EX. ergò MP x DK. PK x PM: : DKq x DE. DEq x EX. hoc eſt DK PK: : DKq. DE x EX. vel DKq. DK x PK: : DKq. DE x EX. unde DK x PK = DE x EX. Simili ratione DM x MR
(vel DP x PQ) = DF x FY. Verúm omnia DK x PK, DP x
PQ, & c æquantur ſemiſſi quadrati ex DK; & omnia DE x EX,
DF x FY, & c æquantur _duplo ſpatio_ EDA; unde manifeſte con-
ſequitur Propoſitum.

XXIV. Sit curva quæpiam DOK, in qua punctum D; cuique
ſubtendatur recta DK; ſit item curva DZI talis, ut ſumpto in curva
DOK puncto quopiam M, connexâque DM; & ductâ DS ad DM
perpendiculari, & MS curvam DOK tangente; ſumptâ demum
DP = DM, & ductâ PZ ad DK perpendiculari, ſit PZ = DS; erit _ſpatium_ DKI æquale _duplo ſpatio_ DKOD.

41.1.

Fig. 129.

Nam recta KP concipiatur indefinitè parva; & DT ipſi DK per-
pendicularis ſit, & KT curvam DOK tangat. Eſt itaque (ducto
arcu MP) rurſus KP. PM: : KD. DT: : KD. KI. unde KP x
KI = PM x KD. Capiatur alia particula PQ, & centro D per
Q ducatur arcus QN, quem ſecet ſubtenſa DM in R; eſt ergòrur-
ſus MR. RN: : MD. DS; hoc eſt PQ. RN: : MD. PZ qua-
re PQ x PZ = RN x MD; ac ità continuò deinceps. patet igitur
omnia ſimul rectangula KP x KI, PQ x PZ, & c. æquari aggrega-
to omnium PM x KD, RN x MD, & c. hoc eſt ſpatium DKI
duplo ſpatio DKOD æquari.

XXV. Iiſdem quoad cætera poſitis atque paratis, ordinatæ PZ jam
æquales concipiantur ipſis MS reſpectivis; & ad rectam aſſumptam
X_k_, diſtantiáſque X_k_, X_m_, X_n_, & c, æquales ipſis curvæ partibus
DOK, DOM, DON, & c. applicentur rectæ _kd_, _md_, _nd_, & c.

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