TRAVAUX MATHéMATIQUES DIVERS DE 1655 X 1659. 1659. 



375 



Lemma II. 



[Fig- 39]- 



AC diam. AB linea. DFG perp. AC. 

 demonftrandum arcum BD majorem 

 recta DF. 



diicatur fubtenfa DB. arciis AG oo AD, 

 Ergo ^ADG oo ABD. L autem DFB oo 

 00 ADG + DAF. Ergo^DFB |i Z_ABD. 

 Ergo in A^FB latus DB || latere DF. qiiare 

 muko magis arcus BD major reéta DF. 



[Fig. 40.] 



[Theorema.] 



Sit RAQ Cycloides. 

 K datum in ipfapunc- 

 tum in quo tangens fit 

 ducenda. ABC efc cir- 

 culas genitor intra cy- 

 cloidem conftitutus. 



Sit KB parai 1. bafi QC, et jungatur BA, ciii parallela fit NKL. 



dico eam tangere cycloidem in puncto K. 



Sumatur enim in NKL punftum quodvis praster K , ac primo altius ut N, unde 

 agatur NQ parall. BK, quae occurrat cycloidi in O circulo ABC in P. 



Quia ergo PO efl: oo arcui PA ut fuperius dem. s) erit QO oo arc.PA + PQ. 

 hifce autem duobus major ert arc. APB, quia arc.PB major reélâ PQ per lemm. 

 fecund. Eftque arcui APB sequalis BK five QN. Ergo QN major quam QO. 

 Ergo punftum N e(l extra cycloidem. 



deinde infra K fumatur in KL punftum L et ducatur LD parall. KB. occur- 

 rens cycloidi in M, produftaquc AB in F. Ergo quia DM oo arcui DBA. DF 

 aucem major arcu DB per lemma L Erit FM minor arcu BA, hoc ell reétâ BK 

 five FL. Ergo punftum L extra cycloidem. Eft ergo NKL tangens in K. 



5) Voir le § 1 delà Première Partie, p. 347. 



