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. BABPT. BENED.
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278
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ſis verò eadem quæ eſt portionis, cuius diameter eſt
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ex .9. 12. Eucli. & ex .42. id-
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eſt vltima primi Archimedis de ſphæra, & cyllindro.</
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<
s
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echoid-s3321
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xml:space
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">Nunc autem ex hoc aggregato iam vltimo dicto detrahatur conus, cuius
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eſt
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axis et
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diameter baſis, qui quidem conus nobis cognitus eſt, cum
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ſemidia-
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meter eius baſis, nobis cognita ſit ex .34. 3. Eucli. </
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<
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xml:space
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">& ſic quantitas eius baſis, & ita ter-
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tia pars
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eius axis, quę multiplicata cum dicta baſi, cuius
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>.n.u.</
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>
eſt diameter, produ
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cit dictum conum, qui quidem conus, vt diximus, demptus cum fuerit ex dicto ag-
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gre gato, relinquet nobis ſoliditatem portionis
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>.n.e.u.</
var
>
vnde cognoſcemus proportio
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nem iſtius portionis ad totam ſphæram propoſitam.</
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<
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echoid-s3323
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xml:space
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">Sed cum nobis propoſita ſit proportio portionis
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ad portionem
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>.i.e.t.</
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>
cogno
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ſcemus etiam ſoliditatem huius ſecundę portionis
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>.i.e.t.</
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>
& ſimiliter
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reg
norm
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proportionem
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type
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>
hu-
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ius ad totam ſphęram, & ad
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type
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<
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norm
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etiam
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type
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">etiã</
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ipſius ſphęrę hoc eſt portioni
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var
>.i.c.t</
var
>
.</
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<
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<
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xml:id
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xml:space
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">Protrahatur nunc diameter
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>.c.e.</
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>
à parte
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>.e.</
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>
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norm
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type
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">vſq;</
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>
quo
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>.e.f.</
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>
æqualis ſit
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>.e.o.</
var
>
ſemidiame
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tro ſphęrę, quæ quidem
<
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>.f.e.</
var
>
diuidatur in puncto
<
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>.h.</
var
>
ita vt proportio
<
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>.f.h.</
var
>
ad
<
var
>.h.e.</
var
>
æqua-
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lis ſit proportioni portionis
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>.i.c.t.</
var
>
ad portionem
<
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>.i.e.t.</
var
>
quod quidem hoc modo efficie
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tur. </
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<
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xml:space
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">applicabimus lineam
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>
(indeterminatam) cum
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ad quemuis angulum in
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pun- cto
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type
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cto</
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>
<
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>.f.</
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in qua accipiemus duas lineas
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>.f.p.</
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>
et
<
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>p.q.</
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>
inuicem ita relatas, vt ſe habent in pro
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portione duæ iam dictæ portiones, hoc eſt, vt
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portio ad portionem
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var
>
ducen
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do poſtea
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>.q.e.</
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>
et
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>.p.h.</
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parallelam ad ipſam
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>.q.e.</
var
>
diuiſam habebimus
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>
in eadem pro
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portione vt dictum eſt ex .2. ſexti, & .11 quinti Euclidis, vnde
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>.c.e</
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:
<
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>e.f.</
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et
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>
nobis co
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gnitę erunt.</
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<
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<
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xml:space
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">Oportebit nos nunc cognoſcere quantitatem
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hoc modo, videlicet, quęramus
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quadratum, cuius
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>.c.x.</
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eius ſit radix, cui quadratum lineę
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>.c.e.</
var
>
cognitum, ita ſit propor-
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tionatum, vt eſt linea
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>.x.f.</
var
>
ad lineam
<
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>.f.h.</
var
>
quę nobis cognita eſt, quod rectè factum erit
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ex eo, quod ſcripſit Archimedes in .4. ſecundi de ſphęra, & cyllindro.</
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<
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xml:space
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">Sed quia Archimedes eo in loco ſupponit id, quod necipſe, nec alius adhuc inue
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nit, niſi via naturali, hoc eſt tres partes ęquales ex proportione data effici, non erit in
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conueniens etiam nobis hac via, circa hoc aliquid dicere.</
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<
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xml:space
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">Accipiemus igitur diametrum
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cum addita
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>.e.f.</
var
>
eius ſemidiametro, diuidemus
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q́ue
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>.f.e.</
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>
in puncto
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>.h.</
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>
vt ſupra factum fuit, applicabimus poſtea
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>.c.m.</
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>
indeterminatam
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angulariter ad
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>.c.e.</
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>
à qua
<
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>.c.m.</
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accipiemus
<
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>.c.g.</
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>
æqualem
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>.f.h.</
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>
quęremus deinde natu-
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rali via punctum
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>.b.</
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>
ita ut protrahendo à puncto
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>.e.</
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>
(altero extremo diametri)
<
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>e.m.</
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>
pa
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rallelam ad
<
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>.b.g.</
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>
ductam, erigendo
<
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>.b.d.</
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>
perpendicularem ad
<
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>.c.e.</
var
>
in puncto
<
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>.b.</
var
>
protra
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<
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ctaque
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type
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">ctaq́;</
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>
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quæ à diametro
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>.e.c.</
var
>
deducta ab
<
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>.c.</
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>
incohando vſque ad
<
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>.x.</
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>
relinquat nobis
<
var
>.
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x.f.</
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ęqualem
<
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>.c.m</
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>
.</
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<
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xml:space
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">Cuius rei ratio eſt, quia quadratum
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ſe habet ad quadratum
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>.c.d.</
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vt
<
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>.c.e.</
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>
ad
<
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>.c.
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b.</
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ex .4. et .18. ſexti Eucl. </
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<
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xml:space
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">ſed ex .4. ita ſe habet
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ad
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>.c.g.</
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>
vt
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>.e.c.</
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>
ad
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>.b.c.</
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>
& cum ſit
<
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>.c.
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g.</
var
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ęq alis
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>.f.h.</
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>
ſi
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>.c.m.</
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>
ęqualis fuerit
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>.f.x.</
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>
habebimus propoſitum. </
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<
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">Quod ſi quis per di-
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ſcretum vel et hoc facere, ita ei agendum erit.</
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</
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<
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<
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xml:space
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">Ponamus exempli gratia totum diametrum
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>
propoſitæ ſphæræ eſſe ut decem,
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<
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="
proportionemque
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type
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">proportionemq́;</
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reſiduę portionis
<
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>.i.c.t.</
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>
ad ſecundam
<
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>.i.e.t.</
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>
hoc eſt
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>.f.h.</
var
>
ad
<
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>.h.e.</
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>
ſeſqui-
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alteram eſſe, vnde
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>.e.h.</
var
>
bis tertia erit ìpſius
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>.f.h.</
var
>
<
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="
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type
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">totaq́;</
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linea
<
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>.c.f.</
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>
erit .15. et
<
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>.f.h.</
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>
erit .3.
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& quadratum lineæ
<
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>.c.e.</
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>
erit .100.</
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<
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<
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xml:space
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">Quærendo poſtea quadratum lineæ
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>.c.x.</
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>
cui quadratum
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>.c.e.</
var
>
hoc eſt .100. ita pro-
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portionatum ſit vt
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>.f.x.</
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>
ad
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>.f.h.</
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>
hoc eſt ad .3. ſi autem cogitauerimus
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>.c.x.</
var
>
eſſe nouem
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partium talium qualium
<
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>.c.e.</
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>
eſt decem, eius quadratum erit .81. et
<
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>.x.f.</
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>
erit .6. par-
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tium talium qualium
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>.c.f.</
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>
eſt .15. dicendo poſtea ſi .100. dat .81. (ex regula de tribus) </
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