Lemma 10.
Velocitas acquiſita in duabus chordis EIB eſt æqualis acquiſitæ in EB;
quia acquiſita in EI eſt æqualis acquiſitæ in GI; ſunt enim eiuſdem al
titudinis; igitur acquiſita in EIB æqualis acquiſitæ in GB: ſed acqui
ſita in GB eſt æqualis acquiſitæ in EIB; igitur acquiſita in EB eſt æqua
lis acquiſitæ in EIB, itemque acquiſita in ELB acquiſitæ in EB: immò
acquiſita in tribus EILB eſt æqualis acquiſitæ in EB; quia acquiſita in
EIL eſt æqualis acquiſitæ in EL; igitur acquiſita in EILB æqualis
acquiſitæ in ELB: ſed acquiſita in ELB eſt æqualis acquiſitæ in EB; igi
tur acquiſita in EB æqualis acquiſitæ in EILB idem dico de 5. chordis,
6.7. atque ita deinceps.
quia acquiſita in EI eſt æqualis acquiſitæ in GI; ſunt enim eiuſdem al
titudinis; igitur acquiſita in EIB æqualis acquiſitæ in GB: ſed acqui
ſita in GB eſt æqualis acquiſitæ in EIB; igitur acquiſita in EB eſt æqua
lis acquiſitæ in EIB, itemque acquiſita in ELB acquiſitæ in EB: immò
acquiſita in tribus EILB eſt æqualis acquiſitæ in EB; quia acquiſita in
EIL eſt æqualis acquiſitæ in EL; igitur acquiſita in EILB æqualis
acquiſitæ in ELB: ſed acquiſita in ELB eſt æqualis acquiſitæ in EB; igi
tur acquiſita in EB æqualis acquiſitæ in EILB idem dico de 5. chordis,
6.7. atque ita deinceps.
Quod certè mirabile eſt, & quaſi paradoxon;
præſertim cùm duplici
motu acquiratur æqualis velocitas in ſpatiis inæqualibus, quorum mauis
citiùs percurritur; Equidem in AB, EB acquiritur æqualis velocitas,
vel impetus, ſed breuius ſpatium, ſcilicet AB citius percurritur; at verò
in EB, & ELB acquiritur æqualis velocitas; licèt ſpatium longius ELB
percurratur citiùs, quàm EB; ſimiliter EILB velociùs, quam ELB & EB.
motu acquiratur æqualis velocitas in ſpatiis inæqualibus, quorum mauis
citiùs percurritur; Equidem in AB, EB acquiritur æqualis velocitas,
vel impetus, ſed breuius ſpatium, ſcilicet AB citius percurritur; at verò
in EB, & ELB acquiritur æqualis velocitas; licèt ſpatium longius ELB
percurratur citiùs, quàm EB; ſimiliter EILB velociùs, quam ELB & EB.
Hinc ſuprà velocitas acquiſita in perpendiculari ſeu radio quadrantis
non eſt ad velocitatem acquiſitam in toto arcu quadrantis vt quadratum
ſub radio ad ipſum quadrantem, quia ſcilicet velocitas acquiſita per ar
cum ELB eſt æqualis acquiſitæ per omnes chordas facto initio motus
ab E; ſed velocitas acquiſita in 6. chordis. v. g. eſt æqualis acquiſitæ in
5. 4. 3. 2. 1. igitur velocitas acquiſita in EB eſt æqualis acquiſitæ in ar
cu ELB, & in ipſa perpendiculari ER.
non eſt ad velocitatem acquiſitam in toto arcu quadrantis vt quadratum
ſub radio ad ipſum quadrantem, quia ſcilicet velocitas acquiſita per ar
cum ELB eſt æqualis acquiſitæ per omnes chordas facto initio motus
ab E; ſed velocitas acquiſita in 6. chordis. v. g. eſt æqualis acquiſitæ in
5. 4. 3. 2. 1. igitur velocitas acquiſita in EB eſt æqualis acquiſitæ in ar
cu ELB, & in ipſa perpendiculari ER.
Lemma 11.
Hinc Lemma vniuerſaliſſimum ſtatuitur, ſcilicet ab eodem puncto altitudi
nîs ad eandem horizontalem, vel ab eadem horizontali ad idem punctum
deorſum, vel ab eadem horizontali ad aliam horizontalem aquales acquiri
velocitates, ſiue plures ſint lineæ, ſine vnica, ſiue ſimplices, ſiue compoſitæ, ſiue
recta, ſiue curua; quæ omnia ex Lemmate decimo manifeſta redduntur.
nîs ad eandem horizontalem, vel ab eadem horizontali ad idem punctum
deorſum, vel ab eadem horizontali ad aliam horizontalem aquales acquiri
velocitates, ſiue plures ſint lineæ, ſine vnica, ſiue ſimplices, ſiue compoſitæ, ſiue
recta, ſiue curua; quæ omnia ex Lemmate decimo manifeſta redduntur.
Lemma 12.
Velocitas acquiſita in toto arcu quadrantis ELB non debet aſſumi in area
tota quadrantis AEB, ſed in linea recta æquali toti arcui ELB, ductis ſci
licet lineis rectis tranſuerſis, qua ſint ipſis ſinubus rectis æquales, cuius conſtru
ctionis; ſit enim linea AN æqualis arcui quadrantis, & NT radio; igi
tur totum triangulum mixtum ex rectis AN, NT, & curua TQH, eſt
velocitas acquiſita in toto arcu quadrantis; ſit autem A σ æqualis lateri
quadrati inſcripti qua eſt ad AN proximè vt 10. ad 11. eſt enim AB ra
dix quad. 98. ſitque AE ſinus rectus quad. 45. certè rectangulum NE
eſt velocitas acquiſita in chorda A σ, ſed hæc eſt æqualis acquiſitæ in
toto arcu quadrantis AN; igitur rectangulum NE eſt æquale triangulo
mixto NTOA, denique velocitas acquiſita in radio A 4. æquali AF,
eſt vt quadratum 4 F, ſed quadratum 4. F eſt æquale rectangulo BE, vt
conſtat, nam A σ eſt dupla AE; igitur rectangulum eſt ſubduplum qua-
tota quadrantis AEB, ſed in linea recta æquali toti arcui ELB, ductis ſci
licet lineis rectis tranſuerſis, qua ſint ipſis ſinubus rectis æquales, cuius conſtru
ctionis; ſit enim linea AN æqualis arcui quadrantis, & NT radio; igi
tur totum triangulum mixtum ex rectis AN, NT, & curua TQH, eſt
velocitas acquiſita in toto arcu quadrantis; ſit autem A σ æqualis lateri
quadrati inſcripti qua eſt ad AN proximè vt 10. ad 11. eſt enim AB ra
dix quad. 98. ſitque AE ſinus rectus quad. 45. certè rectangulum NE
eſt velocitas acquiſita in chorda A σ, ſed hæc eſt æqualis acquiſitæ in
toto arcu quadrantis AN; igitur rectangulum NE eſt æquale triangulo
mixto NTOA, denique velocitas acquiſita in radio A 4. æquali AF,
eſt vt quadratum 4 F, ſed quadratum 4. F eſt æquale rectangulo BE, vt
conſtat, nam A σ eſt dupla AE; igitur rectangulum eſt ſubduplum qua-
zoom in
zoom out
zoom area
full page
page width
set mark
remove mark
get reference
digilib