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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<
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392
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rhead
="
IO. BAPT. BENED.
"
n
="
404
"
file
="
0404
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0404
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<
p
>
<
s
xml:id
="
echoid-s4640
"
xml:space
="
preserve
">In vltima verò propoſitione ſecundi lib. de ponderibus Archi. hoc modo intelli
<
lb
/>
gendus eſt, vt ſi diceret,
<
lb
/>
Sit paraboles
<
var
>.a.</
var
>
cuius baſis ſit
<
var
>.a.c.</
var
>
<
reg
norm
="
ſitque
"
type
="
simple
">ſitq́;</
reg
>
<
var
>.d.e.</
var
>
recta parallela dictæ baſi
<
var
>.a.c.</
var
>
<
reg
norm
="
diameterque
"
type
="
simple
">diameterq́;</
reg
>
<
lb
/>
<
var
>b.f</
var
>
.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s4641
"
xml:space
="
preserve
">Inquit deinde quod linea contingens in
<
var
>.b.</
var
>
parallela erit ipſi
<
var
>.a.c.</
var
>
et
<
var
>.e.d.</
var
>
quod proba
<
lb
/>
bimus hoc modo.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s4642
"
xml:space
="
preserve
">Cum
<
var
>.b.f.</
var
>
diameter ſit et
<
var
>.a.c.</
var
>
baſis, clarum erit ex definitione quod
<
var
>.b.f.</
var
>
diuidet
<
var
>.a.c.</
var
>
<
lb
/>
per æqualia in
<
var
>.g</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4643
"
xml:space
="
preserve
">Vnde ex .7. vel etiam ex .46. primi Pergei
<
var
>.d.e.</
var
>
diuiſa erit per æqua
<
lb
/>
lia à diametro
<
var
>.b.f</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4644
"
xml:space
="
preserve
">Quare verum dicit ex quinta ſecundi ipſius Pergei hoc eſt quod
<
lb
/>
dicta contingens in puncto. b parallela erit ambobus
<
var
>.a.c.</
var
>
et
<
var
>.e.d</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4645
"
xml:space
="
preserve
">Inquit poſtea quod diuiſa cum fuerit pars diametri quę inter
<
var
>.d.e.</
var
>
et
<
var
>.a.c.</
var
>
poſita eſt
<
lb
/>
(hoc eſt
<
var
>.g.f.</
var
>
) per quinque partes æquales,
<
reg
norm
="
quarum
"
type
="
context
">quarũ</
reg
>
partium media ſit
<
var
>.h.k.</
var
>
diuiſa etiam
<
lb
/>
imaginatione ſit in puncto
<
var
>.i.</
var
>
ita quod proportio ipſius
<
var
>.h.i.</
var
>
ad
<
var
>.i.K.</
var
>
eadem ſit quæ in-
<
lb
/>
ter duo ſolida quorum vnum (illud ſcilicet à quo relatio incipit, hoc eſt antecedens)
<
lb
/>
pro ſua baſi teneat quadratum ipſius
<
var
>.a.f.</
var
>
cuius etiam ſolidi altitudo compoſita ſit ex
<
lb
/>
<
note
xlink:label
="
note-0404-01
"
xlink:href
="
note-0404-01a
"
position
="
left
"
xml:space
="
preserve
">R</
note
>
duplo ipſius
<
var
>.d.g.</
var
>
cum ſimplo
<
var
>.a.f</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4646
"
xml:space
="
preserve
">Aliud verò ſolidum habeat pro ſua baſi quadra-
<
lb
/>
tum ipſius
<
var
>.d.g.</
var
>
eius verò altitudo compoſita ſit ex duplo ipſius
<
var
>.a.f.</
var
>
cum ſimplo
<
var
>.d.g</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4647
"
xml:space
="
preserve
">Inquit nunc Archi. quod cum ita factum fuerit, oſtendet punctum
<
var
>.i.</
var
>
centrum eſſe
<
lb
/>
portionis abſciſſę à tota ſectione, quod
<
reg
norm
="
fruſtum
"
type
="
context
">fruſtũ</
reg
>
<
reg
norm
="
nominatur
"
type
="
simple
">nominat̃</
reg
>
<
reg
norm
="
ſignatum
"
type
="
context
">ſignatũ</
reg
>
characteribus
<
var
>.a.d.e.c</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4648
"
xml:space
="
preserve
">Sit igitur num@.
<
var
>m.n.</
var
>
inquit, æqualis diametro
<
var
>.b.f.</
var
>
et
<
var
>.n.o.</
var
>
æqualis
<
var
>.b.g.</
var
>
<
reg
norm
="
ſitque
"
type
="
simple
">ſitq́;</
reg
>
<
var
>.x.n.</
var
>
me
<
lb
/>
dia proportionalis inter
<
var
>.n.m.</
var
>
et
<
var
>.n.o.</
var
>
et
<
var
>.t.n.</
var
>
in continua proportionalitate poſt
<
var
>.o.n.</
var
>
<
lb
/>
hoc eſt quod ea proportio quæ eſt ipſius
<
var
>.o.n.</
var
>
ad
<
var
>.n.t.</
var
>
eadem ſit ipſius
<
var
>.x.n.</
var
>
ad
<
var
>.n.o</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4649
"
xml:space
="
preserve
">Hinc
<
lb
/>
habebimus .4. lineas in continua proportionalitate ſibi inuicem coniunctas
<
var
>.m.n</
var
>
:
<
var
>x.
<
lb
/>
n</
var
>
:
<
var
>o.n.</
var
>
et
<
var
>.t.n</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4650
"
xml:space
="
preserve
">Vult etiam quod à linea
<
var
>.i.b.</
var
>
incipiens ab
<
var
>.i.</
var
>
verſus
<
var
>.g.</
var
>
alia linea abſciſſa ſit, cui li-
<
lb
/>
<
note
xlink:label
="
note-0404-02
"
xlink:href
="
note-0404-02a
"
position
="
left
"
xml:space
="
preserve
">A</
note
>
neæ, ita proportionata ſit
<
var
>.f.h.</
var
>
vt
<
var
>.t.m.</
var
>
eſt ad
<
var
>.t.n.</
var
>
quæ quidem linea ſignata ſit
<
var
>.i.r</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4651
"
xml:space
="
preserve
">Dicit poſtea quod diameter
<
var
>.b.f.</
var
>
erit fortaſſe a xis vel aliqua reliquarum diame-
<
lb
/>
trorum, quod quidem in .46. primi Pergei videre eſt, cum omnes diametri ſint in-
<
lb
/>
uicem paralleli ipſi axi.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4652
"
xml:space
="
preserve
">Cum poſtea dicit, quod
<
var
>.a.f.</
var
>
et
<
var
>.d.g.</
var
>
ſunt intentæ ductæq́ue, ibi vult id em infer-
<
lb
/>
re, quod Pergeus vocat ordinatè, vt ex .11. et .49. primi ipſius Pergei videre li-
<
lb
/>
cet, vnde ex .20. eiuſdem proportio
<
var
>.b.f.</
var
>
ad
<
var
>.b.g.</
var
>
erit vt quadrati
<
var
>.a.f.</
var
>
ad quadratum
<
lb
/>
ipſius
<
var
>.d.g.</
var
>
vt ipſe dicit.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4653
"
xml:space
="
preserve
">Sed ita erit quadrati
<
var
>.m.n.</
var
>
ad qua
<
reg
norm
="
dratum
"
type
="
context
">dratũ</
reg
>
<
var
>.x.n.</
var
>
ex .18. ſexti Eucli. </
s
>
<
s
xml:id
="
echoid-s4654
"
xml:space
="
preserve
">Quare ex .11. quin-
<
lb
/>
<
note
xlink:label
="
note-0404-03
"
xlink:href
="
note-0404-03a
"
position
="
left
"
xml:space
="
preserve
">α</
note
>
ti quadratum ipſius
<
var
>.m.n.</
var
>
ad quadratum ipſius
<
var
>.n.x.</
var
>
eandem habebit proportionem,
<
lb
/>
quam quadratum ipſius
<
var
>.a.f.</
var
>
ad quadratum ipſius
<
var
>.d.g</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4655
"
xml:space
="
preserve
">Vnde ex .18. & ex communi
<
lb
/>
<
reg
norm
="
ſcientia
"
type
="
context
">ſciẽtia</
reg
>
, eadem proportio erit ipſius
<
var
>.m.n.</
var
>
ad
<
var
>.n.x.</
var
>
quę ipſius
<
var
>.a.f.</
var
>
ad
<
var
>.d.g.</
var
>
vt inquit Arch.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4656
"
xml:space
="
preserve
">Quaptopter proportio cubi ipſius
<
var
>.m.n.</
var
>
ad cubum ipſius
<
var
>.n.x.</
var
>
erit vt cubi ipſius
<
var
>.a.
<
lb
/>
f.</
var
>
ad cubum ipſius
<
var
>.d.g.</
var
>
vt etiam dicit ex communi ſcientia, nec non ex .36. vndecimi.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4657
"
xml:space
="
preserve
">Inquit poſtea quod proportio totius ſectionis
<
var
>.a.b.c.</
var
>
ad portionem
<
var
>.d.b.e.</
var
>
eadem
<
lb
/>
eſt quæ cubi ipſius
<
var
>.a.f.</
var
>
ad cubum ipſius
<
var
>.d.g.</
var
>
quod verum eſt, vt aliàs tibi monſtraui in
<
lb
/>
diuiſione parabolæ ſecundum aliquam propoſitam proportionem.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4658
"
xml:space
="
preserve
">Quando autem dicit quod proportio cubi ipſius
<
var
>.m.n.</
var
>
ad cubum ipſius
<
var
>.n.x.</
var
>
eadem
<
lb
/>
<
note
xlink:label
="
note-0404-04
"
xlink:href
="
note-0404-04a
"
position
="
left
"
xml:space
="
preserve
">β</
note
>
eſt quæ ipſius
<
var
>.m.n.</
var
>
ad
<
var
>.n.t.</
var
>
verum dicit ex .36. vndecimi. </
s
>
<
s
xml:id
="
echoid-s4659
"
xml:space
="
preserve
">Vnde ex .11. quinti ita ſe
<
lb
/>
habebit totalis ſectio
<
var
>.a.b.c.</
var
>
ad portionem
<
var
>.d.b.c.</
var
>
vt
<
var
>.m.n.</
var
>
ad
<
var
>.n.t.</
var
>
& ex .17. eiuſdem ita
<
lb
/>
erit ipſius
<
var
>.m.t.</
var
>
ad
<
var
>.t.n.</
var
>
vt fruſti
<
var
>.a.d.e.c.</
var
>
ad ſectionem
<
var
>.d.b.e.</
var
>
quemadmodum ipſe di-
<
lb
/>
cit. </
s
>
<
s
xml:id
="
echoid-s4660
"
xml:space
="
preserve
">Sed quia ſuperius, vbi
<
var
>.A.</
var
>
ipſa
<
var
>.f.h.</
var
>
(quæ eſt tres quintæ ipſius
<
var
>.f.g.</
var
>
) ad
<
var
>.i.r.</
var
>
ita rela- </
s
>
</
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