Caſ.1. Jam ſi Figura DESCirculus eſt vel Hyperbola, biſece
tur ejus tranſverſa diameter ASin O,& erit
84[Figure 84]
SOdimidium lateris recti. Et quoniam eſt
TCad TDut Ccad Dd,& TDad TSut
CDad SY,erit ex æquo TCad TSut
CDXCcad SYXDd.Sed per Corol. 1. Prop.
XXXIII, eſt TCad TSut ACad AO,puta ſi
in coitu punctorum D, dcapiantur linearum
rationes ultimæ. Ergo ACeſt ad (AOſeu) SK
ut CDXCcad SYXDd.Porro corporis
deſcendentis velocitas in Ceſt ad velocitatem
corporis Circulum intervallo SCcirca cen
trum Sdeſcribentis in ſubduplicata ratione
ACad (AOvel) SK(per Prop. XXXIII.) Et
hæc velocitas ad velocitatem corporis deſcri
bentis Circulum OKkin ſubduplicata ratione
SKad SCper Cor. 6. Prop. IV, & ex æquo velo
citas prima ad ultimam, hoc eſt lineola Ccad
arcum Kkin ſubduplicata ratione ACad SC,
id eſt in ratione ACad CD.Quare eſt CDXCc
æquale ACXKk,& propterea ACad SKut
ACXKkad SYXDd,indeque SKXKkæqua
le SYXDd,& 1/2 SKXKkæquale 1/2 SYXDd,
id eſt area KSkæqualis areæ SDd.Singulis
igitur temporis particulis generantur arearum
duarum particulæ KSk,& SDd,quæ, ſi mag
nitudo earum minuatur & numerus augeatur in infinitum, ratio
nem obtinent æqualitatis, & propterea (per Corollarium Lem
matis IV) areæ totæ ſimul genitæ ſunt ſemper æquales, que E. D.
tur ejus tranſverſa diameter ASin O,& erit
84[Figure 84]SOdimidium lateris recti. Et quoniam eſt
TCad TDut Ccad Dd,& TDad TSut
CDad SY,erit ex æquo TCad TSut
CDXCcad SYXDd.Sed per Corol. 1. Prop.
XXXIII, eſt TCad TSut ACad AO,puta ſi
in coitu punctorum D, dcapiantur linearum
rationes ultimæ. Ergo ACeſt ad (AOſeu) SK
ut CDXCcad SYXDd.Porro corporis
deſcendentis velocitas in Ceſt ad velocitatem
corporis Circulum intervallo SCcirca cen
trum Sdeſcribentis in ſubduplicata ratione
ACad (AOvel) SK(per Prop. XXXIII.) Et
hæc velocitas ad velocitatem corporis deſcri
bentis Circulum OKkin ſubduplicata ratione
SKad SCper Cor. 6. Prop. IV, & ex æquo velo
citas prima ad ultimam, hoc eſt lineola Ccad
arcum Kkin ſubduplicata ratione ACad SC,
id eſt in ratione ACad CD.Quare eſt CDXCc
æquale ACXKk,& propterea ACad SKut
ACXKkad SYXDd,indeque SKXKkæqua
le SYXDd,& 1/2 SKXKkæquale 1/2 SYXDd,
id eſt area KSkæqualis areæ SDd.Singulis
igitur temporis particulis generantur arearum
duarum particulæ KSk,& SDd,quæ, ſi mag
nitudo earum minuatur & numerus augeatur in infinitum, ratio
nem obtinent æqualitatis, & propterea (per Corollarium Lem
matis IV) areæ totæ ſimul genitæ ſunt ſemper æquales, que E. D.
LIBER
PRIMUS.
PRIMUS.
Caſ.2. Quod ſi Figura DESParabola ſit, invenietur eſſe ut ſu
pra CDXCcad SYXDdut TCad TS,hoc eſt ut 2 ad 1, ad
eoque 1/4 CDXCcæquale eſſe 1/2 SYXDd.Sed corporis caden
tis velocitas in Cæqualis eſt velocitati qua Circulus intervallo 1/2 SC
uniformiter deſcribi poſſit (per Prop. XXXIV) Et hæc velocitas ad ve
locitatem qua Circulus radio SKdeſcribi poſſit, hoc eſt, lineola
Ccad arcum Kk(per Corol. 6. Prop. IV) eſt in ſubduplicata ratione
SKad 1/2 SC,id eſt, in ratione SKad 1/2 CD.Quare eſt 1/2 SKXKk
æquale 1/4 CDXCc,adeoque æquale 1/2 SYXDd,hoc eſt, area KSk
æqualis areæ SDd,ut ſupra. que E. D.
pra CDXCcad SYXDdut TCad TS,hoc eſt ut 2 ad 1, ad
eoque 1/4 CDXCcæquale eſſe 1/2 SYXDd.Sed corporis caden
tis velocitas in Cæqualis eſt velocitati qua Circulus intervallo 1/2 SC
uniformiter deſcribi poſſit (per Prop. XXXIV) Et hæc velocitas ad ve
locitatem qua Circulus radio SKdeſcribi poſſit, hoc eſt, lineola
Ccad arcum Kk(per Corol. 6. Prop. IV) eſt in ſubduplicata ratione
SKad 1/2 SC,id eſt, in ratione SKad 1/2 CD.Quare eſt 1/2 SKXKk
æquale 1/4 CDXCc,adeoque æquale 1/2 SYXDd,hoc eſt, area KSk
æqualis areæ SDd,ut ſupra. que E. D.
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