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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IO. BAPT. BENED.
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<
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xml:space
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">Demonstrationes quarundam propoſitionum de quibus agit
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Cardanus capite primo libro .16. de
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ſubtilitate.</
head
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<
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xml:space
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<
s
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xml:space
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">EA quæ Cardanus in primo cap. lib. 16. de ſubtilitate ita ſcribit, quod ſi diame-
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tros producatur extra quantumlibet, alia verò diametro in centro ſecetur ad
<
lb
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rectos, ex huius fine
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&c.
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type
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>
quæ quidem ſecundum illum eſt vndecima proprietas cir
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culi, quoniam te id non intelligere ſcribis,
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reg
norm
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idemque
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type
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simple
">idemq́;</
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>
dicis etiam de duodecima, & ſi-
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militer de tribus illis paſſionibus, quas ipſæ communes facit circulo, defectioni, ſeu
<
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ellipſi, & hyperboli, tibi breuiter reſpondebo.</
s
>
</
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<
p
>
<
s
xml:id
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xml:space
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">Circa vndecimam proprietatem circuli verum dicit. </
s
>
<
s
xml:id
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xml:space
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preserve
">Imaginemur circulum
<
var
>.p.
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lb
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d.q.</
var
>
à duabus diametris, inuicem ad angulos rectos coniunctis, diuiſum
<
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>.p.d.</
var
>
et
<
var
>.d.g.</
var
>
di
<
lb
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uidatur enim quarta
<
var
>.q.d.</
var
>
per quot partes æquales volueris, mediantibus punctis
<
var
>.b.a.
<
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/>
o.</
var
>
<
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norm
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ducanturque
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type
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ab ijſdem punctis tot perpendiculares diametro
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var
>.d.g.</
var
>
quæ ſint
<
var
>.b.m.a.n.</
var
>
<
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/>
et
<
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>.o.s.</
var
>
quæ quidem erunt parallelæ diametro
<
var
>.q.p.</
var
>
coniungatur deinde extremitas
<
var
>.d.</
var
>
<
lb
/>
diametri
<
var
>.d.g.</
var
>
cum primo puncto
<
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>.b.</
var
>
& protrahatur
<
var
>.d.b.</
var
>
vſque ad concurſum cum diz
<
lb
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metro
<
var
>.p.q.</
var
>
protracto in puncto, h. </
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<
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xml:space
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quæ adiacet diametro
<
var
>.q.p.</
var
>
æqua-
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/>
lem eſſe omnibus dictis perpendicularibus, quapropter coniungantur puncta
<
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>.m.a</
var
>
:
<
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/>
<
var
>n.o.</
var
>
et
<
var
>.s.q.</
var
>
& producantur vſque ad adiacentem diametro
<
var
>.q.p.</
var
>
in punctis
<
var
>.c.</
var
>
et
<
var
>.e.</
var
>
vn
<
lb
/>
de habebimus angulos
<
var
>.b.a.o.q.</
var
>
inuicem æquales ex .26. tertij, cum verò
<
var
>.o.s.a.n.</
var
>
et
<
lb
/>
<
var
>b.m.</
var
>
parallelæ ſint ipſi
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var
>.p.h</
var
>
. </
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>
<
s
xml:id
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xml:space
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preserve
">tunc anguli
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>.b.h.c</
var
>
:
<
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>a.c.e</
var
>
: et
<
var
>.o.e.q.</
var
>
æquales erunt angulis
<
var
>.d.
<
lb
/>
b.m</
var
>
:
<
var
>m.a.n.</
var
>
et
<
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>.n.o.s.</
var
>
ex .29. primi: </
s
>
<
s
xml:id
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xml:space
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preserve
">quare anguli
<
var
>.h.c.e.q.</
var
>
erunt inuicem æquales, vnde
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/>
ex .28. eiuſdem
<
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>.b.h</
var
>
:
<
var
>m.c</
var
>
:
<
var
>n.e.</
var
>
et
<
var
>.s.q.</
var
>
erunt
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reg
norm
="
inuicem
"
type
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">inuicẽ</
reg
>
parallelę, & ex .34.
<
var
>e.q.</
var
>
æqualis erit
<
var
>.
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o.s.</
var
>
et
<
var
>.e.c.</
var
>
æqualis
<
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>.n.a.</
var
>
et
<
var
>.m.b.</
var
>
æqualis
<
var
>.c.h.</
var
>
verum eſt igitur propoſitum.</
s
>
</
p
>
<
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>
<
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xml:space
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">Duodecima vero
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eſt, ut ſi fuerit circulus
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>.a.b.e.q.</
var
>
cuius duo diametriad
<
lb
/>
rectos coniuncti ſint
<
var
>.a.e.</
var
>
et
<
var
>.q.b.</
var
>
& diameter
<
var
>.a.e.</
var
>
protractus indeterminatè ad partem
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/>
e. </
s
>
<
s
xml:id
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xml:space
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">tunc ſi ab extremo
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var
>.b.</
var
>
diametri
<
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>.q.b.</
var
>
ducta fuerit
<
var
>.b.n.u.</
var
>
extra circulum, ſeu
<
var
>.b.u.n.</
var
>
in
<
lb
/>
tra circulum, vt in ſubiecta figura patet, ita vt ſecta ſit à circunferentia circuli in
<
reg
norm
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pum
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type
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">pũ</
reg
>
<
lb
/>
cto
<
var
>.n.</
var
>
vel à diametro in puncto
<
var
>.u.</
var
>
ſemper id quod fit ex
<
var
>.u.b.</
var
>
in
<
var
>.b.n.</
var
>
æquale erit qua-
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lb
/>
drato inſcriptibili in dicto circulo, hoc autem diuerſimodè cognoſci poteſt, tribus
<
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/>
enim modis ego inueni, quorum primus ita ſe habet. </
s
>
<
s
xml:id
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xml:space
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preserve
">Nam ſi punctus
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var
>.u.</
var
>
fuerit ex-
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/>
tra circulum, ducantur
<
var
>.b.e.</
var
>
et
<
var
>.e.n.</
var
>
& habebimus duos triangulos
<
var
>.b.n.e.</
var
>
et
<
var
>.b.e.u.</
var
>
ſimi
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lb
/>
les inuicem, eo, quod angulus
<
var
>.b.</
var
>
communis ambobus exiſtit, & angulus
<
var
>.b.n.e.</
var
>
æqua
<
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/>
lis eſt angulo
<
var
>.b.e.u.</
var
>
quod ita probatur, nam angulus
<
var
>.b.n.e.</
var
>
cum angulo
<
var
>.b.a.e.</
var
>
(ducta
<
lb
/>
cum fuerit
<
var
>.b.a.</
var
>
) æquatur duobus rectis ex .21. tertij, ſed ex quinta primi angulus
<
var
>.b.
<
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/>
e.a.</
var
>
ęqualis eſt angulo
<
var
>.b.a.e</
var
>
: </
s
>
<
s
xml:id
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"
xml:space
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preserve
">quare angulus
<
var
>.b.n.e.</
var
>
cum angulo
<
var
>.b.e.a.</
var
>
ęquatur duobus
<
lb
/>
rectis, ſed ex .13. eiuſdem angulus
<
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>.b.n.e.</
var
>
cum angulo etiam
<
var
>.e.n.u.</
var
>
æquatur duobus re
<
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ctis, ergo angulus
<
var
>.e.n.u.</
var
>
æquatur angulo
<
var
>.b.e.a</
var
>
. </
s
>
<
s
xml:id
="
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xml:space
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preserve
">quare angulus
<
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>.b.n.e.</
var
>
æquatur
<
reg
norm
="
etiam
"
type
="
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">etiã</
reg
>
an-
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/>
gulo
<
var
>.b.e.u.</
var
>
vnde ex .32. eiuſdem reliquus angulus
<
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>.b.u.e.</
var
>
æqualis erit reliquo angulo
<
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/>
<
var
>b.e.n.</
var
>
latera igitur erunt proportionalia ex .4. ſexti, vnde ita ſe habebit
<
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>.u.b.</
var
>
ad
<
var
>.b.
<
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/>
e.</
var
>
vt
<
var
>.b.e.</
var
>
ad
<
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>.b.n.</
var
>
ex .16. ſexti igitur
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erit propoſitum.</
s
>
</
p
>
<
p
>
<
s
xml:id
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"
xml:space
="
preserve
">Sed ſi punctus
<
var
>.u.</
var
>
intra circulum fuerit, triangulus
<
var
>.b.e.n.</
var
>
ſimilis erit triangulo
<
var
>.b.u.
<
lb
/>
e.</
var
>
nam angulus
<
var
>.b.</
var
>
ambobus communis erit. </
s
>
<
s
xml:id
="
echoid-s3183
"
xml:space
="
preserve
">Angulus vero
<
var
>.b.n.e.</
var
>
ęqualis eſt angulo
<
var
>.
<
lb
/>
b.e.u.</
var
>
ex .26. tertij, </
s
>
<
s
xml:id
="
echoid-s3184
"
xml:space
="
preserve
">quare ex .32. primi reliquus angulus
<
var
>.b.e.n.</
var
>
æqualis erit reliquo </
s
>
</
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>
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