Cardano, Girolamo
,
De subtilitate
,
1663
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597
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ſpatium ſpiralis ſuæ contentum intra ean
<
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dem cum recta, eſt velut quadrati ſemidia
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metri circuli ad rectangulum ex ſemidia
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metro circuli, in rectam præcedentis ſpira
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lis, cum tertia parte quadrati ſemidiametri
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circuli ambientis primam ſpiram. </
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>
<
s
id
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s.010971
">Quintum
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proportio ſectoris circuli circumſcribentis
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ſpiralem primam aliquam portionem, ad
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ipſam portionem ſpiralem terminatam in
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centro, & angulum habentem eundem
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cum ſectore, eſt veluti quadrati ſemidia
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metri eiuſdem circuli ad rectangulum ex
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rectis ſpiralem ſectorem continentibus, ad
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dita tertia parte quadrati differentiæ earun
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dem linearum. </
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>
<
s
id
="
s.010972
">Sextum, cùm ſectorem mi
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nore circulo abſcideris, conſtantem inter
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duos circulos quorum ſemidiametri ex cir
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cumuolutione aliqua aucti ſint, ſpiralis quæ
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à termino minoris ad maioris lineæ finem
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procedit ſuperficiem diuidens, in duas par
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tes eam diuidit, quarum proportio exterio
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ris ad interiorem, eſt veluti ſemidiametri
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minoris cum duplo tertiæ partis differentiæ
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ſemidiametrorum ad ſemidiametrum mino
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ris cum tertia parte differentiæ ipſorum
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etiam ſemidiametrorum.
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type
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<
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id
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<
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id
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Spiralis li
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neæ priuile
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gia 6.</
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<
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type
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<
s
id
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<
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id
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Rectilinea
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lb
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rum om
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nium figura
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rum priuile
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gium.</
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>
</
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<
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type
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main
">
<
s
id
="
s.010975
">Omnibus figuris rectilineis hoc vnum
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/>
eſt commune, quod protractis ſingulis la
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teribus exteriores omnes anguli pariter ac
<
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cepti, etiamſi mille fuerint, quatuor rectis
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angulis ſunt æquales. </
s
>
<
s
id
="
s.010976
">Hoc autem ex hoc
<
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pendet, quòd omnes qui intrà continen
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tur anguli, tot rectis æquantur, quotus eſt
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numerus duplus laterum, ſeu angulorum,
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/>
quatuor demptis, quòd ex trigonorum ra
<
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/>
tione pendet, in quos figura diuiditur.
<
arrow.to.target
n
="
marg1536
"/>
<
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/>
Nam cuiuſlibet trigoni tres anguli pariter
<
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/>
accepti duobus rectis ſunt æquales, exterior
<
lb
/>
verò angulus duobus interioribus ex aduer
<
lb
/>
ſo poſitis pariter acceptis eſt æqualis. </
s
>
<
s
id
="
s.010977
">Area
<
lb
/>
etiam æqualis eſt producto ex dimidio ag
<
lb
/>
gregati omnium laterum, in differentiam
<
lb
/>
cuiuſlibet lateris, ab eodem dimidio omnia
<
lb
/>
ſimul multiplicando, non iungendo, vt tres
<
lb
/>
fiant multiplicationes.
<
lb
/>
<
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n
="
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</
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</
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<
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type
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<
s
id
="
s.010978
">
<
margin.target
id
="
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Trigonorum
<
lb
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priuilegia.</
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>
</
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>
<
p
type
="
margin
">
<
s
id
="
s.010979
">
<
margin.target
id
="
marg1537
"/>
Quadrati
<
lb
/>
proprietas.</
s
>
</
p
>
<
p
type
="
main
">
<
s
id
="
s.010980
">Quadrati verò proprium eſt, vt latus eius
<
lb
/>
inter aggregatum ex ipſo dimetiente, ac
<
lb
/>
inter differentiam eorundem proportione
<
arrow.to.target
n
="
marg1538
"/>
<
lb
/>
media conſiſtat. </
s
>
<
s
id
="
s.010981
">Hoc autem contingit, quia
<
lb
/>
dimetiens quadrati quadratum duplum effi
<
lb
/>
cit ipſi quadrato, cuius erat dimetiens. </
s
>
<
s
id
="
s.010982
">Æ
<
lb
/>
quilateri vero pentagoni, & æqui anguli
<
lb
/>
latus eſt maior pars lineæ diuiſæ, ſecundum
<
lb
/>
proportionem habentem medium, & duo
<
lb
/>
extrema, in comparatione ad lineam, quæ
<
lb
/>
duobus pentagoni eiuſdem lateribus ſubten
<
lb
/>
ditur. </
s
>
<
s
id
="
s.010983
">Hexagoni verò latus, qui tamen ſit
<
lb
/>
( vt dixi ) æquilaterus atque æquiangulus,
<
lb
/>
æquale eſt ſemidiametro circuli eundem he
<
lb
/>
<
arrow.to.target
n
="
marg1539
"/>
<
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/>
xagonum circumſcribentis. </
s
>
<
s
id
="
s.010984
">Heptagoni ve
<
lb
/>
rò latus, & linea, quæ duobus eiuſdem la
<
lb
/>
teribus, ac linea, quæ tribus pariter ſubten
<
lb
/>
ditur conſtituunt trigonum, ſi fuerit ( vt di
<
lb
/>
xi
<
emph
type
="
italics
"/>
)
<
emph.end
type
="
italics
"/>
æquilaterus ac æquiangulus, cuius pro
<
lb
/>
portio aggregati ex latere & ſubtenſa tri
<
lb
/>
bus ad ſubtenſam duobus; eſt vt ſubtenſæ
<
lb
/>
duobus ad latus eiuſdem, & rurſus late
<
lb
/>
ris, & ſubtenſæ duobus ad ſubtenſam tri
<
lb
/>
bus, eſt vt ſubtenſæ tribus ad lineam ſub
<
lb
/>
tenſam duobus eiuſdem heptagoni late
<
lb
/>
ribus. </
s
>
<
s
id
="
s.010985
">Hoc autem inferiùs demonſtrabitur. </
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>
</
p
>
<
p
type
="
margin
">
<
s
id
="
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">
<
margin.target
id
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"/>
Pentagoni
<
lb
/>
æquilateri,
<
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/>
& æqui an
<
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/>
guli proprie
<
lb
/>
tas.</
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>
</
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>
<
p
type
="
margin
">
<
s
id
="
s.010987
">
<
margin.target
id
="
marg1539
"/>
Hexagonti,
<
lb
/>
& heptagoni
<
lb
/>
conſimilis
<
lb
/>
proprietas.</
s
>
</
p
>
<
p
type
="
main
">
<
s
id
="
s.010988
">Habent & corpora, & ſuperficies planæ,
<
lb
/>
<
arrow.to.target
n
="
marg1540
"/>
<
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/>
obliquæ proprietates: velut Ambiens ſphæ
<
lb
/>
ram, quadrupla eſt illius maximo circulo,
<
lb
/>
ipſa verò ſphæra inter corpora omnium pro
<
lb
/>
<
arrow.to.target
n
="
marg1541
"/>
<
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ambitus ratione capaciſſima. </
s
>
<
s
id
="
s.010989
">Continet au
<
lb
/>
tem, & contineri poteſt à quinque corpori
<
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/>
bus, quæ ſola poſſunt æquas habere omnes
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/>
ſuperficies, æquoſque ſolidos angulos, ac la
<
lb
/>
tera inuicem æqualia. </
s
>
<
s
id
="
s.010990
">Partium verò ſphæ
<
lb
/>
ræ, quæ plano ſuper axem propendiculari
<
lb
/>
diuiduntur, tria ſunt priuilegia. </
s
>
<
s
id
="
s.010991
">Cuiuſcun
<
lb
/>
que partis ſphæræ ſuperficies æqualis eſt
<
lb
/>
circulo, cuius ſemidiameter eſt linea à ver
<
lb
/>
tice portionis ſphæræ ad terminum circuli,
<
lb
/>
qui eſt baſis eiuſdem portionis. </
s
>
<
s
id
="
s.010992
">Ex quo pa
<
lb
/>
tet, quòd proportio ſuperficierum partium
<
lb
/>
ſphærę plano ſeparatarum, eſt veluti partium
<
lb
/>
diametri codem plano diuiſarum, cùm dime
<
lb
/>
tiens tamen ſphæræ ſuper planum perpen
<
lb
/>
dicularis fuerit. </
s
>
<
s
id
="
s.010993
">Proportio partium corpo
<
lb
/>
rearum ſphæræ, quas planum vnum diſtin
<
lb
/>
guit diametrum diuidens ei perpendicula
<
lb
/>
rem, eſt veluti corporis producti ex quadra
<
lb
/>
to maioris portionis axis, in lineam con
<
lb
/>
ſtantem ex minore portione, & dimidio
<
lb
/>
axis, ad corpus, conſtans ex quadrato mino
<
lb
/>
ris portionis in lineam conſtantem ex dimi
<
lb
/>
dio, & maiore axis portione. </
s
>
</
p
>
<
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type
="
margin
">
<
s
id
="
s.010994
">
<
margin.target
id
="
marg1540
"/>
Sphaera pri
<
lb
/>
uilegia 2.</
s
>
</
p
>
<
p
type
="
margin
">
<
s
id
="
s.010995
">
<
margin.target
id
="
marg1541
"/>
Corporum
<
lb
/>
quinque
<
lb
/>
ſphæra con
<
lb
/>
tentorum
<
lb
/>
proprietas.
<
lb
/>
</
s
>
<
s
id
="
s.010996
">Partium
<
lb
/>
ſphæræ pri
<
lb
/>
uilegia 2.</
s
>
</
p
>
<
p
type
="
main
">
<
s
id
="
s.010997
">Hoc autem ex iſto pendet, quòd onus ha
<
lb
/>
<
arrow.to.target
n
="
marg1542
"/>
<
lb
/>
bens baſim eandem cum portione ſphæræ,
<
lb
/>
ſi talem habeat illius altitudo, ad altitudi
<
lb
/>
nem portionis portionem, qualis eſt aggre
<
lb
/>
gati ex altitudine reſiduæ portionis, & dimi
<
lb
/>
dio axis ad altitudinem eiuſdem reſidui por
<
lb
/>
tionis, erit conus ille æqualis portionis. </
s
>
<
s
id
="
s.010998
">Ex
<
lb
/>
hoc patet, quòd quælibet ſphæra eſt qua
<
lb
/>
drupla cono, cuius baſis eſt circulus maior,
<
lb
/>
altitudo verò medietas diametri ſphæræ.
<
lb
/>
</
s
>
<
s
id
="
s.010999
">Quilibet etiam conus æqualis eſt ſectori
<
lb
/>
ſphæræ ( Sectorem autem dico, corpus in
<
lb
/>
centrum ſphæræ terminatum, cuius baſis
<
lb
/>
portio eſt ſuperficiei ſphæræ: manifeſtum
<
lb
/>
eſt etiam illud conſtare portione ſphæræ, &
<
lb
/>
cono baſim habente ſuperficiem planam
<
lb
/>
eiuſdem portionis) cùm coni altitudo fuerit
<
lb
/>
ſemidiameter ſphæræ, & baſis æqualis ſuper
<
lb
/>
ficiei ſectoris, igitur erunt baſes, & altitudi
<
lb
/>
nes tunc æquales. </
s
>
</
p
>
<
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type
="
margin
">
<
s
id
="
s.011000
">
<
margin.target
id
="
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"/>
Coni recti
<
lb
/>
priuilegia
<
lb
/>
tria.</
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>
</
p
>
<
p
type
="
main
">
<
s
id
="
s.011001
">Conoidali corpore rectangulo diuiſo per
<
lb
/>
<
arrow.to.target
n
="
marg1543
"/>
<
lb
/>
planum, portio, quæ ad apicem terminatur,
<
lb
/>
ſeſquialtera eſt coni portioni eandem baſim
<
lb
/>
ac axem habenti: Conoidalium rurſus
<
expan
abbr
="
rectã-gulorum
">rectan
<
lb
/>
gulorum</
expan
>
portiones inuicem proportionem
<
lb
/>
retinent, ſi plano diuiſæ fuerint, quàm axis
<
lb
/>
partium earundem ipſius quadrata. </
s
>
<
s
id
="
s.011002
">Cùm ve
<
lb
/>
<
arrow.to.target
n
="
marg1544
"/>
<
lb
/>
rò Conoidale obtuſiangulum ſecatur plano,
<
lb
/>
erit proportio partis ad verticem termina
<
lb
/>
tæ, ad portionem coni eandem baſim, &
<
lb
/>
axis portionem habenti, qualis lineæ con
<
lb
/>
ſtantis ex axis parte conoidalis portionis
<
expan
abbr
="
cũ
">cum</
expan
>
<
lb
/>
triplo lineæ, quæ ex centro hyperboles, ſeu
<
lb
/>
obliqui formæ lateris ad
<
expan
abbr
="
eandẽ
">eandem</
expan
>
axis
<
expan
abbr
="
portionẽ
">portionem</
expan
>
<
lb
/>
cum duplo
<
expan
abbr
="
eiuſdẽ
">eiuſdem</
expan
>
, quę ex centro hyperboles. </
s
>
</
p
>
<
p
type
="
margin
">
<
s
id
="
s.011003
">
<
margin.target
id
="
marg1543
"/>
Conoida
<
lb
/>
lium rectan
<
lb
/>
gulum pri
<
lb
/>
uilegia 2.</
s
>
</
p
>
<
p
type
="
margin
">
<
s
id
="
s.011004
">
<
margin.target
id
="
marg1544
"/>
Conoida
<
lb
/>
lium obtuſi
<
lb
/>
ſi angulorum
<
lb
/>
priuilegium</
s
>
</
p
>
<
p
type
="
main
">
<
s
id
="
s.011005
">Sed
<
expan
abbr
="
ſphæroidaliũ
">ſphæroidalium</
expan
>
priuilegia quatuor ſunt
<
lb
/>
<
arrow.to.target
n
="
marg1545
"/>
<
lb
/>
cùm enim plano diuiditur, per centrum, per
<
lb
/>
æqualia diuiditur, eritque quælibet portio
<
lb
/>
dupla cono baſim, & axem æquales portioni
<
lb
/>
ipſius ſphæroidis habenti. </
s
>
<
s
id
="
s.011006
">Si præter cen
<
lb
/>
ſphæroides ſecetur, quomodolibet proportio </
s
>
</
p
>
</
chap
>
</
body
>
</
text
>
</
archimedes
>