( 553 )

IV. Theorema.] Si in Cycloide A V D, cujus Basis A D esi Horizonti N*" muc J>

Parallel*,- Vertice V deorsum spettante, ex A ducatur utcunque Rffla A l ! 3f

Cvcloidi occurrens in B, ex quo ducatur Refla B C Curva Cycloidis B D Cycloid thanin B Normalis, ad quam ex A demittatur Perpendicularis Recla A C. Dico

'Tempus quo Grave e Quiete cadens ex A, Vi sua Gravitatis decurrit Reff am n . 215.' />. 424A B, ese ad Tempus quo percurrit Curvam A V B, sicut Recta AB ad Re-ctam A C.

Per B ducatur B L, Parallela Cycloidis Axi VE; & B K, Basi A D Q- ■<?.Parallela, occurrens Axi in G, & Circulo super Diametrum E V descrip-to in F Lc H, 'Cycloidi denique in K. Ducatur Recta E F, quae ex Cy-cloidis natura parallela est Rcctx B C. Unde B M est xqualis E F, &

E M aequalis B F; quae, propter Crcloidcm , aequatur Arcui V F; & pro-inde A M est aequalis Arcui E H V F.

Per Prop. 1 y. Part. II. Horologii Oscillatorii Hugenii , Tempus quo gravec Quiete cadens percurrit A V, est ad Tempus Casiis per E V, ut Semicir-cumferentia ad Diametrum ; & per dictx Partis Prop. Ultimam, Tempusquo Grave percurrit V B, post decuriam A V (nempe aequale Tempori quoGrave percurrit K V, post decursam A K) est ad Tempus Lapsus per AV,

-sicut Arcus V F, ad Semicircumfcrentiam ; adeoque ad Tempus Casus peiE V, sicut Arcus F V ad Diametrum. Quare Tempus quo Grave percur-rit Curvam A V B, est ad Tempus Casiis per E V, sicut Arcus EHVF,ad Diametrum EV. Sed Tempus Casus per E V, est ad Tempus Casus;pcr L B, sive E G, sicut E V ad E F: Igitur ex aequo. Tempus quoGrave percurrit AVB, est ad Tempus Casiis per LB, sicut Arcus E H V Fad subtensum E F; hoc est, ut Recta A M, ad Rectam M B. Rursiis,

Tempus Casus per LB, est ad Tempus Lapsus per AB, ut LB ad ABErgo Ratio Temporis quo Grave percurrit AVB, ad Tempus quo percur-rit AB, componitur ex Ratione AM ad M B, & Ratione LB ad BA;adeoque aequalis est Rationi A M x L B ad M B x B A. Sed A M X L B,est aequale MB x AC, quia utrumque aequatur duplo Trianguli ABM:

Et igitur Tempus quo Grave e Quiete cadens percurrit Curvam CycloidisAVB, est ad Tempus quo percurrit Rectam A B, sicut M B x A C adM B x B A j id est sicut AC ad AB. Q± E. D. Similiterque procedet'Demonstratio, si Punctum B, iit inter A & V.

V. 1. The upper Piate os the iVatch is A B : The Circular BaUance- E>Ci!i For, f :etVbecl CD, of which the Arbrc is EF : The Spring tufned Spirally, Yf, Hugcnf.

G HM, fastned to the Arbre of the BaUance-TVbeel in M, and to the pieccn. 112. f. 272.that is fast tq the VYatch-Plate, in G, all the Spires or Windings ot the F ' J ' lS? ‘Spring being free without touching any thing. N O P Q, is the Cocl^, in

which one of the Pivots of the Ballance-lVbcel turns; R S, is one of thelndentcd iVheels of the Match having a Ballancing Motion, which theBallance-H vccl gives to it. And this l-Vheel R S, Catches, in the Pinion T,which holds on the Arbre of the Ballance , of which by this means the Mo-tion is entertained as much as is neceflary. These Matches are exact for the

VoL I. 13 b b b Pocket,