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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42
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file
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0042
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xlink:href
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rimus, ſi ſumma vnius dictorum prouenientium cum vnitate dat primum numerum,
<
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quid ipſa eadem vnitas dabit? </
s
>
<
s
xml:id
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echoid-s397
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xml:space
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preserve
">ex quo propoſitum oriatur.</
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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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">Exempli gratia, proponuntur tres numeri, primus .20. ſecundus .34. tertius .8.
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Iam quærimus diuidere primum .20. in duas partes quæ mutuò diuiſæ prębeant duo
<
lb
/>
prouenientia, quorum ſumma tanta ſit vt per eam diuiſo .34. proueniat numerus
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æqualis tertio numero .8. </
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<
s
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xml:space
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">Quod vt præſtemus iubet regula ſecundum .34. per
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type
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<
num
value
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8
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8.</
num
>
diuidi, vnde proueniet .4. cum vna quarta parte, quod proueniens erit ſumma pro
<
lb
/>
uenientium ex diuiſione duarum partium quæſitarum, quæ ſi diſtinguere volueri-
<
lb
/>
mus, præcedentis theorematis methodum ſequemur, vnitate ſuperficiali pro ſecun
<
lb
/>
do numero propoſito ſumpta, ac ſi diceremus, diuidatur .4. cum vna quarta parte
<
lb
/>
in duas eiuſmodi partes, vt productum vnius in alteram ſit vnitas ſuperficialis, cer-
<
lb
/>
tè fractis integris cum quarta parte coniungendis, darentur vnitatis decemſeptem
<
lb
/>
quartæ lineares, verum cum neceſſe ſit, ex præcedenti theoremate, dimidium in
<
lb
/>
ſeipſum multiplicare,
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eſſetque
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type
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dimidium .8. quartarum partium cum octaua, com-
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modius totum conſtituetur .34. octauarum, quarum dimidium, nempe decemſep-
<
lb
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tem octauæ, in ſeipſum multiplicatum erunt .289. ſexageſimæ quartæ vnius integri
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ſuperficialis, quandoquidem
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ſuperficiale, cuius vnitas linearis in .8. partes
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diuiditur eſt .64. vt ex primo theoremate huius libri depræhendi poteſt. </
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xml:space
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">Nunc vni-
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tate hac ſuperficiali, nempe .64. ex .289. detracta, ſupererit .225. cuius radix qua-
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lb
/>
drata, ſcilicet .15. coniuncta dimidio dictorum prouenientium, nempe .17. dabit
<
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/>
maius proueniens .32.
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type
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ex altero dimidio, dabit proueniens minus .2. hoc
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lb
/>
eſt pro maiore proueniente .32. octauas, & pro minore duas, quatuor ſcilicet inte-
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gros pro maiore, & quartam partem vnius integri pro minore. </
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>
<
s
xml:id
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xml:space
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">Nunc ſi ex regula
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de tribus dixerimus, ſi .4. iuncta vni, nempe .5. dant .20. primum numerum, quid
<
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/>
dabunt .4. integra (proueniens inquam maius)
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dabunt
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type
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certè .16. partem maiorem.
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</
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<
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xml:space
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">Tum ſi dixerimus, ſi quarta pars coniuncta vnitati dat .20: </
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<
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xml:space
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">quid dabit quarta illa
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pars (hoc eſt proueniens minus) dabit
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type
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quatuor ſcilicet
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minorem
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type
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partem, quod
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ab antiquis certè ignoratum fuit, qui, inuentis prouenientibus quieuerunt, ne-
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/>
ſcientes ijs vti ad inueniendas duas primi numeri partes.</
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>
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<
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<
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xml:space
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">Cuius ſpeculationis gratia, demus primum numerum ſignificari linea
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>
cuius
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partes
<
var
>.e.a.</
var
>
&
<
var
>a.u.</
var
>
ſint quæ quæruntur, alter verò numerus ſignificetur linea
<
var
>.b.
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d.</
var
>
tertius linea
<
var
>.g.f.</
var
>
proueniens
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type
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diuiſionis
<
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>.e.a.</
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per
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var
>
ſit
<
var
>.n.t.</
var
>
diuiſionis
<
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norm
="
autem
"
type
="
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">aũt</
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>
<
var
>.a.u.</
var
>
<
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/>
per
<
var
>.a.e.</
var
>
ſit
<
var
>.t.o.</
var
>
ſumma erit
<
var
>.n.t.o.</
var
>
vnitas verò
<
var
>.n.i.</
var
>
et
<
var
>.o.i</
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>
. </
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>
<
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xml:id
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xml:space
="
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">Iam ſi numerus
<
var
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var
>
tertiò
<
lb
/>
propoſitus ex diuiſione ſecundi per
<
var
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var
>
proferri debet. </
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>
<
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xml:space
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preserve
">Ex .13. theoremate patet,
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lb
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quòd ſi
<
var
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var
>
per
<
var
>.g.f.</
var
>
diuiſerimus, proferetur
<
var
>.o.t.n.</
var
>
qui cum fuerit inuentus,
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type
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eſſe oportet
<
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duorum
"
type
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">duorũ</
reg
>
<
reg
norm
="
prouenientium
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type
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">prouenientiũ</
reg
>
, ex diuiſione mutua
<
reg
norm
="
duorum
"
type
="
context
">duorũ</
reg
>
numerorum, nempe
<
var
>.
<
lb
/>
a.e.</
var
>
per
<
var
>.a.u.</
var
>
et
<
var
>.a.u.</
var
>
per
<
var
>.a.e.</
var
>
deinde manifeſtum eſt ex .24. aut .25. theoremate
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type
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<
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productum (multiplicatis prouenientibus adinuicem) vnitatem ſuperficialem futu
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ram eſſe. </
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<
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xml:space
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">Hactenus igitur, totum
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>
ex doctrina præcedentis theorematis diui-
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lb
/>
ditur in puncto
<
var
>.t.</
var
>
ita vt productum
<
var
>.o.t.</
var
>
in
<
var
>.t.n.</
var
>
<
lb
/>
ſolam vnitatem ſuperficialem
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, quo
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58
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xlink:href
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facto, ſi, vt antedictum eſt, cogitauerimus
<
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>.n.
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t.</
var
>
<
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proueniens
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type
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eſſe ex diuiſione
<
var
>.e.a.</
var
>
per
<
var
>.a.u.</
var
>
et
<
var
>.
<
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/>
t.o.</
var
>
proueniens ex diuiſione
<
var
>.a.u.</
var
>
per
<
var
>.a.e.</
var
>
pa-
<
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/>
tebit ex definitione diuiſionis, quod eadem
<
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/>
erit proportio
<
var
>.a.e.</
var
>
ad
<
var
>.n.t.</
var
>
quæ eſt
<
var
>.a.u.</
var
>
ad vni-
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/>
tatem
<
var
>.n.i.</
var
>
et
<
var
>.a.u.</
var
>
ad
<
var
>.o.t.</
var
>
eadem quæ eſt
<
var
>.e.a.</
var
>
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