Benedetti, Giovanni Battista de
,
Io. Baptistae Benedicti ... Diversarvm specvlationvm mathematicarum, et physicarum liber : quarum seriem sequens pagina indicabit ; [annotated and critiqued by Guidobaldo Del Monte]
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151 - 163
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151 - 163
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332
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IO. BAPT. BENED.
"
n
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344
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file
="
0344
"
xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0344
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lineas
<
var
>.b.q.</
var
>
et
<
var
>.b.n.</
var
>
ſimul ſumptas longiores eſſe omnibus alijs lineis exeuntibus ab ip
<
lb
/>
ſis punctis
<
var
>.q.n.</
var
>
quæ in aliquo puncto dictæ circunferentiæ ſimul concurrant.</
s
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<
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<
s
xml:id
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xml:space
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">Sint igitur aliæ duæ
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var
>.q.o.</
var
>
et
<
var
>.n.o.</
var
>
quas probare volo ſimul ſumptas, eſſe minores dua
<
lb
/>
bus ſimul ſumptis
<
var
>.q.b.</
var
>
et
<
var
>.n.b</
var
>
. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Nam ex .20. tertij Eucli. cognoſcimus angulos
<
var
>.q.b.n.</
var
>
<
lb
/>
et
<
var
>.q.o.n.</
var
>
inuicem æquales eſſe, & ſimiliter angulos
<
var
>.b.n.o.</
var
>
et
<
var
>.b.q.o</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4021
"
xml:space
="
preserve
">deinde ex .15. pri
<
lb
/>
mi eiuſdem habemus angulos contra ſe poſitos,
<
lb
/>
circa
<
var
>.a.</
var
>
eſſe etiam inuicem ęquales. </
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>
<
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xml:id
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xml:space
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">Vnde ex .4
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fig-0344-01a
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number
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368
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file
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0344-01
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0344-01
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</
figure
>
ſexti, habebimus proportionem
<
var
>.a.b.</
var
>
ad .a
<
lb
/>
o. eandem eſſe, quæ
<
var
>.a.n.</
var
>
ad
<
var
>.a.q.</
var
>
& ſic .b
<
lb
/>
n. ad
<
var
>.o.q</
var
>
. </
s
>
<
s
xml:id
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xml:space
="
preserve
">Quare ita erit
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var
>.a.b.n.</
var
>
ad
<
var
>.a.o.q.</
var
>
vt
<
var
>.a.n</
var
>
<
lb
/>
ad
<
var
>.a.q.</
var
>
ſed cum
<
var
>.a.n.</
var
>
maior ſit
<
var
>.q.a.</
var
>
ex .18. primi,
<
lb
/>
eo quod angulus
<
var
>.b.q.n.</
var
>
(qui æqualis eſt angulo
<
var
>.
<
lb
/>
b.n.q.</
var
>
ex .5. eiuſdem) maior eſt angulo
<
var
>.a.n.q.</
var
>
<
lb
/>
qui pars eſt ipſius
<
var
>.b.n.q.</
var
>
ergo latera ſimul ſum-
<
lb
/>
pta
<
var
>.a.b.n.</
var
>
maiora erunt lateribus
<
var
>.a.o.q.</
var
>
ſed ex
<
num
value
="
20
">.
<
lb
/>
20.</
num
>
primi
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var
>.a.b.n.</
var
>
<
reg
norm
="
etiam
"
type
="
context
">etiã</
reg
>
maior erit
<
var
>.a.n.</
var
>
vnde ex .25.
<
lb
/>
quinti
<
var
>.q.a.b.n.</
var
>
maior erit
<
var
>.n.a.o.q</
var
>
. </
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>
<
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xml:id
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xml:space
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">quare ſequi-
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tur verum eſſe propofitum.</
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>
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>
<
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>
<
s
xml:id
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xml:space
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">Sed ſi oculus eſſet in
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var
>.u.</
var
>
quemadmodum in ſubſcripta hic
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reg
norm
="
ſecunda
"
type
="
context
">ſecũda</
reg
>
figura videre eſt,
<
lb
/>
res autem viſibilis in
<
var
>.n.</
var
>
ambo extra dictum circulum, eſto etiam primum
<
var
>.b.u.</
var
>
æqua-
<
lb
/>
lis
<
var
>.b.n.</
var
>
probabo ſimiliter
<
var
>.u.b.n.</
var
>
maiores eſſe
<
var
>.u.o.n</
var
>
. </
s
>
<
s
xml:id
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xml:space
="
preserve
">Nam angulus
<
var
>.o.</
var
>
maior eſt angu-
<
lb
/>
lo
<
var
>.b.</
var
>
eo quod ſi circulum
<
var
>.u.b.n.</
var
>
cogitemus circunſcribere triangulum
<
var
>.u.b.n.</
var
>
ducen-
<
lb
/>
do vſque ad ſuam circunferentiam
<
var
>.o.n.</
var
>
in puncto
<
var
>.s.</
var
>
deinde ducendo
<
var
>.u.s.</
var
>
habebimus
<
lb
/>
ex .20. tertij angulum
<
var
>.u.s.n.</
var
>
<
reg
norm
="
æqualem
"
type
="
context
">æqualẽ</
reg
>
angulo
<
var
>.u.b.n.</
var
>
ſed
<
reg
norm
="
cum
"
type
="
context
">cũ</
reg
>
angulus
<
var
>.u.o.n.</
var
>
exterior trian
<
lb
/>
guli
<
var
>.u.o.s.</
var
>
exiſtat, ipſe maior erit angulo
<
var
>.s.</
var
>
ex .16. primi. </
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>
<
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xml:space
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">duco poſtea
<
var
>.o.q.</
var
>
parallelam
<
lb
/>
ad
<
var
>.u.s.</
var
>
quæ ſecabit
<
var
>.a.u.</
var
>
in puncto
<
var
>.q.</
var
>
& habebimus angulum
<
var
>.a.o.q.</
var
>
ęqualem angulo
<
var
>.
<
lb
/>
n.s.u.</
var
>
ex .29. eiuſdem, hoc eſt angulo
<
var
>.n.b.u.</
var
>
fed ex ſu-
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lb
/>
<
figure
xlink:label
="
fig-0344-02
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xlink:href
="
fig-0344-02a
"
number
="
369
">
<
image
file
="
0344-02
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0344-02
"/>
</
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>
pradictis rationibus, lineæ
<
var
>.q.b.n.</
var
>
ſimul ſumptæ maio-
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lb
/>
rem efficient longitudinem, quam
<
var
>.q.o.n</
var
>
. </
s
>
<
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xml:id
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xml:space
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">Nunc cum
<
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/>
ipſi
<
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>.q.b.</
var
>
addita fuerit
<
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>.u.q.</
var
>
& vice
<
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>.q.o.</
var
>
ſumpta fuerit ali-
<
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/>
qua linea minor ipſa
<
var
>.u.q.o.</
var
>
eo amplius
<
var
>.u.q.b.n.</
var
>
maior
<
lb
/>
erit, quod quidem hoc modo faciendum. </
s
>
<
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xml:id
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xml:space
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">Acci-
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/>
piatur
<
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>.o.u.</
var
>
vt comes
<
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>.o.n.</
var
>
quæ minor eſt ambabus
<
var
>.o.
<
lb
/>
q.</
var
>
et
<
var
>.q.u.</
var
>
ex .20. primi, ita enim habebimus
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.
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</
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<
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xml:id
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xml:space
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">ſed breuiori modo hoc ipſum videbis ex pręcedenti,
<
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& ex .21. primi Euclid. </
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>
<
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xml:space
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">Nam ex præcedenti
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>.u.b.n.</
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>
lon-
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gior eſt ipſa
<
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>.u.s.n.</
var
>
ex .21. autem primi
<
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>.u.s.n.</
var
>
longior eſt
<
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/>
ipſa
<
var
>.u.o.n.</
var
>
ergo verum eſt propoſitum.</
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>
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<
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370
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<
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0344-03
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xlink:href
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</
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>
<
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>
<
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xml:id
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xml:space
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">Si verò radius incidentiæ
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norm
="
non
"
type
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">nõ</
reg
>
fuerit æqualis radio
<
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/>
reflexionis, ſit vt in hac ſubſcripta tertia figura vide
<
lb
/>
re eſt
<
var
>.u.b.p</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Cum autem probauerim longitudinem
<
var
>.u.b.n.</
var
>
ma
<
lb
/>
iorem eſſe longitudine
<
var
>.u.o.n.</
var
>
coniungatur
<
var
>.n.p.</
var
>
cum
<
lb
/>
<
var
>u.b.n</
var
>
. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">deinde. ab
<
var
>.o.</
var
>
ad
<
var
>.p.</
var
>
ducatur
<
var
>.o.p.</
var
>
quæ minor
<
lb
/>
erit longitudine
<
var
>.o.n.p.</
var
>
ex .20. primi, & illicò
<
lb
/>
manifeſtabitur verum eſſe propoſitum, etiam hoc
<
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/>
tertio modo.</
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