1id eſt, ratio mutationum momentanearum curvæ AP,rectæ
CP,arcus circularis BP,ac rectæ VP,eadem erit quæ linea
rum PV, PF, PG, PIreſpective. Cum autem VFad CF&
VHad CVperpendiculares ſunt, angulique HVG, VCFprop
terea æquales; & angulus VHG(ob angulos quadrilateri HVEP
ad V& Prectos) angulo CEPæqualis eſt, ſimilia erunt tri
angula VHG, CEP; & inde fiet ut EPad CEita HGad HV
ſeu HP& ita KIad KP,& compoſite vel diviſim ut CBad
CEita PIad PK,& duplicatis conſequentibus ut CBad 2 CE
ita PIad PV,atQ.E.I.a adeo Pqad Pm.Eſt igitur decremen
tum lineæ VP,id eſt, incrementum lineæ BV-VPad incremen
tum lineæ curvæ APin data ratione CBad 2 CE,& prop
terea (per Corol. Lem. IV.) longitudines BV-VP& AP,in
crementis illis genitæ, ſunt in eadem ratione. Sed, exiſtente BVra
dio, eſt VPco-ſinus anguli BVPſeu 1/2 BEP,adeoque BV-VP
ſinus verſus ejuſdem anguli; & propterea in hac Rota, cujus radius
eſt 1/2 BV,erit BV-VPduplus ſinus verſus arcus 1/2 BP.Ergo
APeſt ad duplum ſinum verſum arcus 1/2 BPut 2 CEad CB.
que E. D.
CP,arcus circularis BP,ac rectæ VP,eadem erit quæ linea
rum PV, PF, PG, PIreſpective. Cum autem VFad CF&
VHad CVperpendiculares ſunt, angulique HVG, VCFprop
terea æquales; & angulus VHG(ob angulos quadrilateri HVEP
ad V& Prectos) angulo CEPæqualis eſt, ſimilia erunt tri
angula VHG, CEP; & inde fiet ut EPad CEita HGad HV
ſeu HP& ita KIad KP,& compoſite vel diviſim ut CBad
CEita PIad PK,& duplicatis conſequentibus ut CBad 2 CE
ita PIad PV,atQ.E.I.a adeo Pqad Pm.Eſt igitur decremen
tum lineæ VP,id eſt, incrementum lineæ BV-VPad incremen
tum lineæ curvæ APin data ratione CBad 2 CE,& prop
terea (per Corol. Lem. IV.) longitudines BV-VP& AP,in
crementis illis genitæ, ſunt in eadem ratione. Sed, exiſtente BVra
dio, eſt VPco-ſinus anguli BVPſeu 1/2 BEP,adeoque BV-VP
ſinus verſus ejuſdem anguli; & propterea in hac Rota, cujus radius
eſt 1/2 BV,erit BV-VPduplus ſinus verſus arcus 1/2 BP.Ergo
APeſt ad duplum ſinum verſum arcus 1/2 BPut 2 CEad CB.
que E. D.
LIBER
PRIMUS.
PRIMUS.
Lineam autem APin Propoſitione priore Cycloidem extra
Globum, alteram in poſteriore Cycloidem intra Globum diſtincti
onis gratia nominabimus.
Globum, alteram in poſteriore Cycloidem intra Globum diſtincti
onis gratia nominabimus.
Corol.1. Hinc ſi deſcribatur Cyclois integra ASL& biſecetur
ea in S,erit longitudo partis PSad longitudinem VP(quæ du
plus eſt ſinus anguli VBP,exiſtente EBradio) ut 2 CEad CB,
atque adeo in ratione data.
ea in S,erit longitudo partis PSad longitudinem VP(quæ du
plus eſt ſinus anguli VBP,exiſtente EBradio) ut 2 CEad CB,
atque adeo in ratione data.
Corol.2. Et longitudo ſemiperimetri Cycloidis ASæquabitur
lineæ rectæ quæ eſt ad Rotæ diametrum BV,ut 2 CEad CB.
lineæ rectæ quæ eſt ad Rotæ diametrum BV,ut 2 CEad CB.
PROPOSITIO L. PROBLEMA XXXIII.
Facere ut Corpus pendulum oſcilletur in Cycloide data.
Intra Globum QVS,centro Cdeſcriptum, detur Cyclois QRS
biſecta in R& punctis ſuis extremis Q& Sſuperficiei Globi hinc
inde occurrens. Agatur CRbiſecans arcum QSin O,& produca
tur ea ad A,ut ſit CAad COut COad CR.Centro Cin-
biſecta in R& punctis ſuis extremis Q& Sſuperficiei Globi hinc
inde occurrens. Agatur CRbiſecans arcum QSin O,& produca
tur ea ad A,ut ſit CAad COut COad CR.Centro Cin-
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