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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<text type="book" xml:lang="la">
<div xml:id="echoid-div7" type="body" level="1" n="1">
<div xml:id="echoid-div7" type="chapter" level="2" n="1">
<div xml:id="echoid-div71" type="math:theorem" level="3" n="32">
<p>
<s xml:id="echoid-s299" xml:space="preserve">
retur .20. ſcilicet et .4. certè .24. perſingulas partes diuiſo, daretur vnum proue-
<lb/>
niens ſex integra, & alterum vnum & quinta pars, quorum ſumma eſſet ſeptem in-
<lb/>
tegra cum quinta parte, tum altera parte per alteram diuiſa, daretur vnum proue-
<lb/>
niens quinque integrorum & alterum vnius quinti tantum, quorum ſumma eſſet
<lb/>
quinque integra, & vna quinta pars, minor prima reliquorum duorum prouenien-
<lb/>
tium per binarium.</s>
</p>
<p>
<s xml:id="echoid-s300" xml:space="preserve">Cuius conſiderationis cauſa, propoſitus numerus linea
<var>.q.p.</var>
ſignificetur, eius duę
<lb/>
partes lineis
<var>.q.x.</var>
et
<var>.x.p.</var>
<reg norm="tum" type="context">tũ</reg>
<var>.q.f.</var>
ſit proueniens ex diuiſione totius
<var>.q.p.</var>
per
<var>.x.p.</var>
et
<var>.
<lb/>
q.i.</var>
ſit proueniens ex diuiſione eiuſdem
<var>.q.p.</var>
per
<var>.q.x.</var>
<var>.h.m.</var>
ſit proueniens,
<lb/>
ex diuiſione
<var>.q.x.</var>
per
<var>x.p.</var>
et
<var>.h.k.</var>
proue-
<lb/>
niensex diuiſione
<var>.p.x.</var>
per
<var>.q.x.</var>
patet igi-
<lb/>
</figure>
tur ex .22. theoremate huiuslibri proue-
<lb/>
niés.h.m. minus eſſe proueniente
<var>.q.f.</var>
per
<lb/>
vnitaté, & proueniens
<var>.h.k.</var>
minus proue-
<lb/>
niente
<var>.q.i.</var>
per alteram vnitatem. </s>
<s xml:id="echoid-s301" xml:space="preserve">Itaque
<var>.
<lb/>
f.q.i.</var>
maior erit
<var>.m.h.k.</var>
per numerum binarium, quoderat propoſitum.</s>
</p>
</div>
<div xml:id="echoid-div73" type="math:theorem" level="3" n="33">
<num value="33">XXXIII</num>
<p>
<s xml:id="echoid-s302" xml:space="preserve">
<emph style="sc">QVilibet</emph>
numerus, medius eſt
<lb/>
proportionalis inter numerum
<lb/>
</figure>
</p>
<p>
<s xml:id="echoid-s303" xml:space="preserve">Detur enim numerus propoſitus,
<lb/>
qui linea
<var>.a.u.</var>
ſignificetur, cuiusqua-
<lb/>
dratum ſit
<var>.u.n.</var>
vnitas linearis ſit
<var>.i.a.</var>
<lb/>
et ſuperficialis
<var>.o.</var>
patebit ex .18. ſexti
<lb/>
aut 11. octaui proportionem
<var>.u.n.</var>
<var>.
<lb/>
o.</var>
futuram duplam proportioni
<var>.u.a.</var>
<lb/>
<var>.i.a.</var>
ſed
<var>.i.a.</var>
e
<unsure/>
<lb/>
res
<reg norm="sunt" type="context">sũt</reg>
, tanta ſcilicet
<var>.a.i.</var>
quanta
<var>.o.</var>
vni
<lb/>
</figure>
tas eſt, Itaque proportio numeri
<var>.u.n.</var>
<lb/>
<var>.u.a.</var>
æqualis erit proportioni
<var>.u.a.</var>
<lb/>
<var>.i.a</var>
. </s>
<s xml:id="echoid-s304" xml:space="preserve">Quare numerus
<var>.u.a.</var>
inter nu-
<lb/>
merum
<var>.u.n.</var>
& vnitatem, medius erit
<lb/>
proportionalis.</s>
</p>
</div>
<div xml:id="echoid-div76" type="math:theorem" level="3" n="34">
<num value="34">XXXIIII</num>
<p>
<s xml:id="echoid-s305" xml:space="preserve">
<emph style="sc">HOc</emph>
ipſum quod diximus & alia ratione ſpeculari licebit.</s>
</p>
<p>
<s xml:id="echoid-s306" xml:space="preserve">Propoſitus numerus, nunc etiam per
<var>.a.u.</var>
<var>.
<lb/>
u.n.</var>
vnitas linearis per
<var>.a.i.</var>
<reg norm="productumque" type="simple">productumq́;</reg>
<var>.a.u.</var>
in
<var>.a.i.</var>
terminetur,
<reg norm="ſitque" type="simple">ſitq́;</reg>
<var>.n.i</var>
. </s>
<s xml:id="echoid-s307" xml:space="preserve">quare
<lb/>
<var>n.i.</var>
conſtabit numero íuperficiali æquali numero lineari
<var>.a.u.</var>
& ex prima fexti aut .
<lb/>
18. vel .19. ſeptimi, eadem erit proportio
<var>.u.n.</var>
<var>.i.n.</var>
quæ eſt
<var>.a.u.</var>
<var>.a.i.</var>
ſed nu-
<lb/>
merus
<var>.a.u.</var>
cum numero
<var>.n.i.</var>
idem ſpecie eſt. </s>
<s xml:id="echoid-s308" xml:space="preserve">Itaque medius eſt proportiona-
<lb/>
lis inter
<var>.u.n.</var>
& vnitatem.</s>
</p>
</div>
</div>
</div>
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