Cardano, Girolamo
,
De subtilitate
,
1663
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archimedes
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s.000572
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<
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pagenum
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371
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xlink:href
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016/01/020.jpg
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<
expan
abbr
="
æquilibriũ
">æquilibrium</
expan
>
D cum G coëxtenſo K L, ſi igitur
<
lb
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adderetur æquè graue per totam CK adhuc
<
lb
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faceret æquilibrium, quia in BC & BK æ
<
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quales ſunt, & tunc eſſet proportio ponde
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ris D ad pondus KL, ſicut ponderis LC, ad
<
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pondus CK, ex ſecunda proportione permu
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tata, igitur vt longitudinis LC ad CK, quia
<
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pondus eſt æqualiter diſtributum. </
s
>
<
s
id
="
s.000573
">Sed ſicut
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lb
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ponderis D ad pondus LK, ſic D ad G, quia
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lb
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ſuppoſitum eſt G & LK æqualia eſſe, igitur
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vt LC ad CK, ſic MB ad BC: quare permu
<
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tando vt CK ad CB, ſic LC ad MB: ſed CB
<
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eſt dimidium CK: igitur BM eſt dimidium
<
lb
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LC, quod erat demonſtrandum. </
s
>
<
s
id
="
s.000574
">Quia verò
<
lb
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CB eſt dimidium CK, & BM dimidium
<
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CL, ſequitur vt MK ſit dimidium KL, & ita
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eſt ac ſi ſuſpenſum eſſet in medio loci cui
<
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coextenditur. </
s
>
<
s
id
="
s.000575
">Vnumquodque igitur pondus,
<
lb
/>
iuxta Archimedem, quantumvis inæquale,
<
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/>
<
arrow.to.target
n
="
marg46
"/>
<
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vt triangulum tantum affert grauitatis, coex
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tenſum virgulæ, quantum ſi ſuſpendatur ex
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centro in loco vbi centrum grauitatis ſe
<
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<
arrow.to.target
n
="
marg47
"/>
<
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cundum perpendiculum ſitum eſt. </
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>
<
s
id
="
s.000576
">Hoc au
<
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tem generaliter ſupponit, etiamſi pondus
<
expan
abbr
="
nõ
">non</
expan
>
<
lb
/>
extendatur vſque ad aginam, ſed coex-ten
<
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/>
<
arrow.to.target
n
="
marg48
"/>
<
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datur, gratia exempli per L F, & centrum
<
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eius ſit in directo E, tunc dicit, eſt ac ſi ſuſ
<
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penderetur in ipſo E. </
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>
<
s
id
="
s.000577
">Ex his, vt in Arithme
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ticis docuimus, colligitur ratio conficien
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di ſtateras. </
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</
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<
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type
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margin
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<
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id
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<
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Stateræ ra
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tio.</
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type
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<
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id
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s.000579
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Parabola
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quinta.</
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>
</
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type
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<
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id
="
s.000580
">
<
margin.target
id
="
marg47
"/>
Parabola
<
lb
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octaua.</
s
>
</
p
>
<
p
type
="
margin
">
<
s
id
="
s.000581
">
<
margin.target
id
="
marg48
"/>
Statera
<
lb
/>
quomodo
<
lb
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perfecta eſſe
<
lb
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poſſit.</
s
>
</
p
>
<
p
type
="
main
">
<
s
id
="
s.000582
">Nunc ſolum demonſtrare oportet, quo
<
lb
/>
modo ſtatera perfecta eſſe poſſit: etſi in pre
<
lb
/>
cioſis mercibus mercatores libra vtantur. </
s
>
<
s
id
="
s.000583
">Sit
<
lb
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igitur ſtatera diuiſa tuo modo cum pondere
<
lb
/>
auxiliari, quod mobile eſt G, & æquiponi
<
lb
/>
deret G in F ipſi D, quia itaque D æqui
<
lb
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ponderat G in F, & etiam regulæ KL. </
s
>
<
s
id
="
s.000584
">Po
<
lb
/>
nantur enim N pars D, quæ facit
<
expan
abbr
="
æquilibriũ
">æquilibrium</
expan
>
<
lb
/>
<
expan
abbr
="
cũ
">cum</
expan
>
LK, & O pars reliqua D, quæ facit
<
expan
abbr
="
æqui-libriũ
">æqui
<
lb
/>
librium</
expan
>
cum G. </
s
>
<
s
id
="
s.000585
">Igitur iuxta
<
expan
abbr
="
primã
">primam</
expan
>
<
expan
abbr
="
regulã
">regulam</
expan
>
pro
<
lb
/>
portio O ad G, vt FB ad BC. </
s
>
<
s
id
="
s.000586
">Proportio ve
<
lb
/>
rò N ad LK, & eſt vt LK ſuſpenſa in M, ex
<
lb
/>
tertia regula, vt L C ad CK ex ſecunda re
<
lb
/>
gula. </
s
>
<
s
id
="
s.000587
">Igitur ſtatuemus pondus N primò in
<
lb
/>
directo D, deinde facto æquilibrio additio
<
lb
/>
O ſemper eſt ſecundum proportionem ad G,
<
lb
/>
vt partis LB ad BC. </
s
>
<
s
id
="
s.000588
">Secundum igitur æqua
<
lb
/>
lia incrementa BL creſcet O, ſed N manet
<
lb
/>
idem ſemper: igitur ſecundum æqualia in
<
lb
/>
crementa partium B L creſcet D pondus.
<
lb
/>
</
s
>
<
s
id
="
s.000589
">Æqua igitur ſtatera fieret, ſi ad
<
expan
abbr
="
Capponãtur
">C apponantur</
expan
>
<
lb
/>
pondus faciens
<
expan
abbr
="
equilibrũ
">equilibrum</
expan
>
<
expan
abbr
="
cũ
">cum</
expan
>
LK: inde diuida
<
lb
/>
mus ſpatia ab agina ad L per æqualia. </
s
>
<
s
id
="
s.000590
">Sed
<
lb
/>
quia ipſi
<
expan
abbr
="
nõ
">non</
expan
>
<
expan
abbr
="
apponũt
">apponunt</
expan
>
pondus in C,
<
expan
abbr
="
neceſſariũ
">neceſſarium</
expan
>
<
lb
/>
eſt vt prima nota, puta P, oſtendat etiam
<
lb
/>
pondus LK: vt ſi LK ponderis libras duas,
<
lb
/>
& D pondus duas libras æquet, nota primi
<
lb
/>
ponderis eſſet in K, exempli gratia: ſed quia
<
lb
/>
G poſitum in K, grauaret quantum quatuor
<
lb
/>
libræ, & præter id etiam LK grauitatem
<
lb
/>
efficit duarum librarum, igitur oporteret vt
<
lb
/>
D eſſet librarum ſex, igitur pondus eſſet li
<
lb
/>
brarum ſex, & oſtenderet tantùm quatuor.
<
lb
/>
</
s
>
<
s
id
="
s.000591
">Ob id faciemus primam notam quatuor li
<
lb
/>
brarum in P, nam & ibi C grauitatem effi
<
lb
/>
cit duarum librarum, & LK duarum alia
<
lb
/>
rum, igitur pondus D erit quatuor librarum,
<
lb
/>
quod faciet æquilibrium: igitur nota qua
<
lb
/>
tuor librarum primarum erit in P, & multò
<
lb
/>
minus ab agina diſtans, quàm reliquæ inter
<
lb
/>
ſe. </
s
>
<
s
id
="
s.000592
">At verò reliquæ inter ſe æquidiſtabunt,
<
lb
/>
vt ſi ſecunda fit in Q, tertia erit in H, &
<
lb
/>
quarta in M, & quinta in R, & ſexta in E:
<
lb
/>
ideoque poſito G in E, oſtendet libras vigin
<
lb
/>
tiquatuor. </
s
>
<
s
id
="
s.000593
">Manifeſtum eſt autem ex hoc,
<
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<
arrow.to.target
n
="
marg49
"/>
<
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quod commodum affert non leue, quò G
<
lb
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ſit pondus per ſe notum, id eſt, libra, vel bi
<
lb
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libra, vel trilibra. </
s
>
</
p
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<
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type
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margin
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<
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id
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s.000594
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<
margin.target
id
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"/>
Media pon
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derum quo
<
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modo ha
<
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/>
beantur.</
s
>
</
p
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<
p
type
="
main
">
<
s
id
="
s.000595
">Centra autem ponderum in circulis &
<
lb
/>
rectangulis ſunt in communi ſectione di
<
lb
/>
metientium duarum. </
s
>
<
s
id
="
s.000596
">In omnibus autem
<
lb
/>
figuris æquilateris, quæ circulo poſſunt
<
lb
/>
inſcribi, centrum grauitatis idem eſt cum
<
lb
/>
centro circuli circumſcribentis. </
s
>
<
s
id
="
s.000597
">Supponitur
<
lb
/>
autem in omnibus, quòd ponderoſa hæc æ
<
lb
/>
qualem vbique habeant craſſitudinem, &
<
lb
/>
quòd ex materia quæ vbique æqualem ſor
<
lb
/>
tiatur grauitatem, conſtituantur. </
s
>
<
s
id
="
s.000598
">In trigonis
<
lb
/>
autem omnibus in communi ſectione trium
<
lb
/>
linearum, quarum ſingulæ ad ſingula latera
<
lb
/>
ex angulis oppoſitis venientes, ea per æqua
<
lb
/>
lia diuidunt. </
s
>
<
s
id
="
s.000599
">Has verò in vnum punctum
<
lb
/>
ſecando ſe inuicem concurrere neceſſe eſt:
<
lb
/>
etſi id Archimedes non demonſtrauerit.
<
lb
/>
</
s
>
<
s
id
="
s.000600
">Nos autem in Geometricis elementis illud
<
lb
/>
oſtendimus generaliter, nunc autem pro ne
<
lb
/>
ceſſitate declarabitur. </
s
>
</
p
>
<
p
type
="
main
">
<
s
id
="
s.000601
">Nam diuiſis per æqualia lateribus AB &
<
lb
/>
AC in D & E, & ducta CD, BE, & per
<
lb
/>
communem ſectionem AGH & DE, quæ
<
lb
/>
erit æquidiſtans lateri tertio, vnde BEC &
<
lb
/>
CDB erunt æquales, quia in eadem baſi
<
lb
/>
BC, ſubducto BCG, communi erit CEG æ
<
lb
/>
qualis DBC, ipſi autem ſunt æquales AGE
<
lb
/>
& AGD, quia in æquis baſibus & inter æ
<
lb
/>
quidiſtantes, quare AGE & AGD æquales.
<
lb
/>
<
figure
id
="
id.016.01.020.1.jpg
"
xlink:href
="
016/01/020/1.jpg
"
number
="
19
"/>
<
lb
/>
Quumque ſint ſuper eandem lineam AG,
<
lb
/>
erunt æqualis altitudinis, qui altitudo eſt tri
<
lb
/>
gonorum etiam FGD, & FGE, qui
<
expan
abbr
="
conſiſtũt
">conſiſtunt</
expan
>
<
lb
/>
in eadem baſi FG, igitur etiam ipſi inter ſe
<
lb
/>
ſunt æquales, Quia verò BC æquidiſtat DE,
<
lb
/>
erunt ex 29 primi elementorum & 25. eiuſ
<
lb
/>
dem, DGE & BGC æquianguli, & propor
<
lb
/>
tio BG ad GE, vt CG ad GD. ex eiſdem
<
lb
/>
etiam BGH & GEF ſimiles, itemque GCH
<
lb
/>
& DGF. quare proportio trigoni BGH ad
<
lb
/>
EFG, vt BG ad GE dupla, & CGH ad DFG,
<
lb
/>
vt CG ad GD dupla, eſt autem ( vt dictum
<
lb
/>
fuit ) CG ad GD, vt BG, ad GE, quare BGH
<
lb
/>
ad EFG, yt HCG ad DFG: quare cum DFG
<
lb
/>
& EFG æquales ſint, erunt BGH & CGH
<
lb
/>
æquales: quumque ſint inter æquidiſtantes,
<
lb
/>
erunt in baſibus æqualibus BH & HC, igi
<
lb
/>
tur trigoni omnes ABH, ACH, CDB, CDA,
<
lb
/>
BEC, BEA, erunt, medietas ABC, quare ap
<
lb
/>
penſus trigonus in G, in nullam poterit par
<
lb
/>
tem inclinari. </
s
>
<
s
id
="
s.000602
">Centrum autem ſectionis pa
<
lb
/>
rabolę ſeu coni rectanguli eſt in eius dime
<
lb
/>
tiente, quæ à ſummo ad medium baſis in eo
<
lb
/>
puncto qui à ſummitate coni dimidio plus
<
lb
/>
diſtat quàm à baſi, quæ eſt recta linea ſup-</
s
>
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</
chap
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</
body
>
</
text
>
</
archimedes
>