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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<text type="book" xml:lang="la">
<div xml:id="echoid-div7" type="body" level="1" n="1">
<div xml:id="echoid-div477" type="chapter" level="2" n="6">
<div xml:id="echoid-div737" type="section" level="3" n="42">
<div xml:id="echoid-div737" type="letter" level="4" n="1">
<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>
<var>.i.K.</var>
<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/>
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>
<div xml:id="echoid-div748" type="float" level="5" n="12">
</div>
<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>
<var>.n.t.</var>
<var>.x.n.</var>
<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/>
neæ, ita proportionata ſit
<var>.f.h.</var>
vt
<var>.t.m.</var>
<var>.t.n.</var>
quæ quidem linea ſignata ſit
<var>.i.r</var>
.</s>
</p>
<div xml:id="echoid-div749" type="float" level="5" n="13">
</div>
<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>
<var>.b.g.</var>
<var>.a.f.</var>
<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>
<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/>
<var>.m.n.</var>
<var>.n.x.</var>
eandem habebit proportionem,
<lb/>
<var>.a.f.</var>
<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>
<var>.m.n.</var>
<var>.n.x.</var>
quę ipſius
<var>.a.f.</var>
<var>.d.g.</var>
vt inquit Arch.</s>
</p>
<div xml:id="echoid-div750" type="float" level="5" n="14">
</div>
<p>
<s xml:id="echoid-s4656" xml:space="preserve">Quaptopter proportio cubi ipſius
<var>.m.n.</var>
<var>.n.x.</var>
erit vt cubi ipſius
<var>.a.
<lb/>
f.</var>
<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>
<var>.d.b.e.</var>
<lb/>
eſt quæ cubi ipſius
<var>.a.f.</var>
<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>
<var>.n.x.</var>
<lb/>
eſt quæ ipſius
<var>.m.n.</var>
<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>
<var>.d.b.c.</var>
vt
<var>.m.n.</var>
<var>.n.t.</var>
& ex .17. eiuſdem ita
<lb/>
erit ipſius
<var>.m.t.</var>
<var>.t.n.</var>
vt fruſti
<var>.a.d.e.c.</var>
<var>.d.b.e.</var>
<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>