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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382
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IO. BAPT. BENED.
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394
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file
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0394
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0394
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<
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<
s
xml:id
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xml:space
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">In eo quod à me petis, mittendo te ad Eutotium, tibi non ſatisfacerem, cum Eu-
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totius citet ſextum librum Pergei, quem nunquam vidimus,
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ea, quæ nec
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ipſe nec alius vnquam quod ſcimus probauit.</
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</
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<
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<
s
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xml:space
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">Deſideras enim demonſtrationem illius quod Archimedes dicit inter primam,
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& ſecundam propoſitionem ſecundi libri, vbi tractat de centris grauium, propte-
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rea quod illud ſupponit pro manifeſto.</
s
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</
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<
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<
s
xml:id
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echoid-s4525
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xml:space
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preserve
">Sit enim figura hic ſubſcripta, ferè ſimilis parabolæ poſitæ in .2. propoſitione di
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cti libri, vt in impreſſione Baſileenſi habetur,
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diuiſæ duæ
<
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et
<
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>.b.c.</
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per æqua
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lia à punctis
<
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>.x.</
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et
<
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>.u.</
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protractisque
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type
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<
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et
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>.u.i.</
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ad
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>.b.d.</
var
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quæ inuicem etiam erunt parallelę
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ex .30. primi Eucli. </
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<
s
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xml:space
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">vnde ipſæ etiam, diametri erunt ipſarum portionum: </
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>
<
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xml:space
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">vt ex eo col
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ligere eſt, quod in .49. primi lib. Pergei probatur. </
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<
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xml:space
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">Imaginando poſtea ad puncta
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>.b.
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f.</
var
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er
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unsure
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<
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tres contingentes, manifeſtum erit punctum
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>.b.</
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>
illud eſſe quod terminat alti-
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tudinem huiuſmodi portionis, et
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et
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terminantia altitudines partialium, ex .5. ſe
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cundi ipſius Pergei, eo quod dictæ contingentes paralellæ erunt ipſis baſibus, vnde
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trianguli inſcripti, eaſdem habebunt altitudines, quas portiones ipſæ, quod erit ex
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mente Archimedis. </
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>
<
s
xml:id
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xml:space
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preserve
">Et ſic deinceps poteris multiplicare angulos ſiguræ rectilineæ
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in parabola, quæ deſignata erit vt deſiderat Archimedes, qui quidem dicit, quod
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protractæ cum fuerint aliæ deinceps poſt
<
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>.f.i.</
var
>
ipſæ inuicem ęquidiſtantes
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erunt
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type
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">erũt</
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, diuiſę-
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q́ue peræqualia ab
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quod
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ſit,
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type
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ab Eutotio non ſatis
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norm
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type
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">demõſtratũ</
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>
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eſt, cum ſupponat
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>.a.f.b.</
var
>
æqualem eſſe ipſi
<
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>.b.i.c.</
var
>
probare volens eius diametros æqua
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/>
les eſſe abſque aliqua citata ratione, quæ quidem ratio eſſet conuerſum .4. propoſi-
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tionis libri de conoidalibus. </
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>
<
s
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xml:space
="
preserve
">Sed oporteret nos
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norm
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type
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">etiã</
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>
videre .6. librum ipſius Pergei,
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& propterea tibi non ſatisfacerem.</
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>
</
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<
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<
s
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xml:space
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preserve
">Eſto igitur, ut inuenta ſit linea
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>.K.</
var
>
cuius productum in
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>.u.i.</
var
>
æquale ſit qua drato ip
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ſius
<
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>.u.c.</
var
>
inuenta etiam ſit linea
<
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>.h.</
var
>
cuius productum cum
<
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>.f.x.</
var
>
æquale ſit quadrato ip-
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ſius
<
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>.a.x.</
var
>
vnde ex conuerſo .49. primi ipſius Pergei, proportio ipſius
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>.K.</
var
>
ad
<
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>.b.c.</
var
>
erit ut
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ipſius
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>.b.c.</
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ad
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>.b.d.</
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& ipſius
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>.h.</
var
>
ad
<
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>.a.b.</
var
>
vt ipſius
<
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>.a.b.</
var
>
ad
<
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>.b.d</
var
>
. </
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>
<
s
xml:id
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xml:space
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preserve
">Erit igitur ex .16. ſexti Eucl.
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quadratum
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var
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æquale producto ipſius
<
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>.K.</
var
>
in
<
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>.b.d.</
var
>
& quadratum
<
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>.a.b.</
var
>
æquale produ-
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cto ipſius
<
var
>.h.</
var
>
in
<
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>.b.d.</
var
>
& ex prima ſexti, ita erit ipſius
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>.K.</
var
>
ad
<
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>.h.</
var
>
vt producti quod fit ex
<
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>.K.</
var
>
<
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/>
in
<
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>.b.d.</
var
>
ad productum ipſius
<
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>.h.</
var
>
in
<
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>.b.d.</
var
>
hoc eſt vt quadrati ipſius
<
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>.b.c.</
var
>
ad quadratum ip
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ſius
<
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>.b.a.</
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>
ex .16. et .11. quinti, hoc eſt vt quadrati ipſius
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>.u.c.</
var
>
ad quadratum ipſius
<
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>.a.x.</
var
>
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/>
hoc eſt ut productum ipſius
<
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>.k.</
var
>
in
<
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>.u.i.</
var
>
ad productnm ipſius
<
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>.h.</
var
>
in
<
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>.x.f</
var
>
. </
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<
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xml:space
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">Nunc ſi ipſius
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ad
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c
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ſt vt producti ipſius
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>.K.</
var
>
in
<
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>.u.i.</
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>
ad productum ipſius
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>.h.</
var
>
in
<
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>.f.x.</
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>
ergo ex .24. ſexti,
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& communi conceptu, proportio ipſius
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>.k.</
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>
ad
<
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>.h.</
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>
compoſita erit ex ea quæ ipſius
<
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>.u.i.</
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>
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ad
<
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>.f.x.</
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& ex ea quæ ipſius
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>.k.</
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>
ad
<
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>.h</
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>
. </
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>
<
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xml:space
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">Cum ergo dempta fuerit proportio ipſius
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>.k.</
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ad
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>.h.</
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>
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(vt ſimplex) à proportione ipſius
<
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>.k.</
var
>
ad
<
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>.h.</
var
>
(vt compoſita) reliquum nihil erit. </
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<
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xml:space
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">Qua-
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re
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æqualis erit ipſi
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>.u.i</
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>
.</
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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 quod
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æqualis ſit ipſi
<
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>.m.i</
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>
. </
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>
<
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xml:space
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">Videto in Eutotio, quia hoc ſatis ſui natura
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facile eſt.</
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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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preserve
">Sed accipe alium modum breuiorem ad probandum
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>.f.x.</
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>
eſſe æqualem ipſi
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>.u.i</
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>
.</
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>
</
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<
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<
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xml:space
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">Finge lineam
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>.e.b.g.</
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>
conting entem in puncto
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>.b.</
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>
prolungatisq́ue diametris
<
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>f.
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x.</
var
>
et
<
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>.u.i.</
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>
vſque ad contingentem ipſam, habebis
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>.f.e.</
var
>
æqualem ipſi
<
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>.f.x.</
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>
et
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>.g.i.</
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ipſi
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>
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Ex .35. primi Pergei, producta poſtea
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>
habeb is ex .2. ſexti Eucli
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>.x.u.</
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>
parallelam
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ipſi
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>.a.c.</
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>
ſed
<
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>.e.g.</
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>
parallela eſt ipſimet
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>.a.c.</
var
>
ex quinta ſecundi ipſius Pergei, </
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>
<
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xml:space
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">quare ex .30
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primi Euclid
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>
parallela erit ipſi
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>.u.x.</
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>
& ex .34. eiuſdem æqualis erit
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>
ipſi
<
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>.u.g.</
var
>
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vnde
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>.f.x.</
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>
etiam æqualis erit
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>.u.i.</
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>
ex communi conceptu.</
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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 ne quid deſideres probabo
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>
æqualem eſſe ipſi
<
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>.m.i</
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>
. </
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<
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xml:space
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