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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THEOR. ARITH.
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<div xml:id="echoid-div7" type="chapter" level="2" n="1">
<div xml:id="echoid-div106" type="math:theorem" level="3" n="50">
<p>
<s xml:id="echoid-s444" xml:space="preserve">
numero, verbi gratia .92. præcepit regula detrahi primum numerum ex ſecundo,
<lb/>
nempe .20. ex .92. cuius reſiduum, ſcilicet .72. conſeruetur, tum detrahi iubet bi
<lb/>
narium ex primo, ſic in propoſito exemplo remanebunt .18. huius autem .18. dimi
<lb/>
dium in ſeipſum multiplicari iubet, quod cum ſit .9. datur numerus .81. ex quo .81.
<lb/>
primum numerum conſeruatum, nempe .72. vult regula detrahi, ſic remanebit .9.
<lb/>
tum huius .9. quadrata radix detrahenda eſt ex dimidio ipſius .18. quod fuit ante qua
<lb/>
dratum, ſic ſupererit .6. hoc eſt .9. excepta radice quadrata, qui .6. erit minor pars
<lb/>
quæſita, maior verò .14. quarum productum .84. coniunctum cum partium differen
<lb/>
tia præbet exactè .92.</s>
</p>
<p>
<s xml:id="echoid-s445" xml:space="preserve">Cuius rei hæc eſt ſpeculatio. </s>
<s xml:id="echoid-s446" xml:space="preserve">Primus numerus minor, qui proponitur diuiſibilis
<lb/>
ſignificetur linea
<var>.q.g.</var>
maior vero linea
<var>.x.</var>
tum cogitemus
<var>.q.g.</var>
diuiſam, cuius maior
<lb/>
pars ſit
<var>.q.o.</var>
minor
<var>.o.g.</var>
differentia
<var>.q.p.</var>
ex quo
<var>.p.o.</var>
æqualis erit
<var>.o.g.</var>
ſit autem produ-
<lb/>
ctum
<var>.b.o</var>
. </s>
<s xml:id="echoid-s447" xml:space="preserve">Oportet igitur, ut
<var>.b.o.</var>
ſimul cum differentia
<var>.q.p.</var>
æquale ſit numero
<var>.x.</var>
ſe-
<lb/>
cundò propoſito, qui notus eſt, </s>
<s xml:id="echoid-s448" xml:space="preserve">quare etiam ſumma producti
<var>.b.o.</var>
cum differentia
<lb/>
<var>q.p.</var>
cognita erit, ex qua detracto primo numero
<var>.q.g.</var>
reſiduum cognitum erit, nunc
<lb/>
igitur quodnam erit hoc reſiduum? </s>
<s xml:id="echoid-s449" xml:space="preserve">attendamus qua ratione ex ſumma
<var>.b.o.</var>
et
<var>.q.p.</var>
<lb/>
detrahenda ſit
<var>.q.g</var>
. </s>
<s xml:id="echoid-s450" xml:space="preserve">In primis ſi ſubtraxerimus ex dicta ſumma
<var>.q.p.</var>
quę pars eſt
<var>.q.g.</var>
<lb/>
ſupererit detrahenda
<var>.p.g.</var>
ex
<var>.b.o.</var>
pars inquam ipſius
<var>.q.g.</var>
quod fiet quotieſcunque
<lb/>
cogitauerimus
<var>.q.o.</var>
duabus vnitatibus diminutam, et per
<var>.o.g.</var>
multiplicatam, ſit au-
<lb/>
tem productum
<var>.b.e.</var>
nam cum
<var>.o.g.</var>
toties
<var>.b.o.</var>
ingrediatur, quot ſunt in
<var>.q.o.</var>
vnitates
<lb/>
ex prima ſexti aut .18. vel .19. ſeptimi,
<reg norm="detrahendaque" type="simple">detrahendaq́;</reg>
ſit
<var>.p.g.</var>
ex
<var>.b.o.</var>
quæ
<var>.p.g.</var>
dupla
<lb/>
eſt
<var>.o.g.</var>
patebit
<var>.o.c.</var>
æqualem eſſe
<var>.p.g.</var>
fu-
<lb/>
pererit ita que
<var>.b.e.</var>
productum
<var>.q.e.</var>
in
<var>.e.</var>
<lb/>
i. cognitum, erutis autem ex
<var>.q.g.</var>
ijſdem
<lb/>
duabus vnitatibus, remanebit
<var>.q.i.</var>
nobis
<lb/>
nota, ex quo
<var>.e.i.</var>
æqualis erit
<var>.e.c</var>
. </s>
<s xml:id="echoid-s451" xml:space="preserve">Cum
<lb/>
igitur productum
<var>.q.e.</var>
in
<var>.e.i.</var>
cognoſcamus
<lb/>
ſimul cum
<var>.q.i</var>
: Sivoluerimus partes
<var>.q.e.</var>
<lb/>
et
<var>.e.i.</var>
cognoſcere, vtemur .45. theorema-
<lb/>
te huius libri, & propoſitum obtinebimus, nam cognoſcemus
<var>.e.i.</var>
& ex conſequen-
<lb/>
ti
<var>.o.g.</var>
eius æqualem.</s>
</p>
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<div xml:id="echoid-div108" type="math:theorem" level="3" n="51">
<num value="51">LI</num>
<p>
<s xml:id="echoid-s452" xml:space="preserve">
<emph style="sc">DIvidere</emph>
numerum in duas eiuſmodi partes, quæ pro medio proportionali
<lb/>
alterum numerum propoſitum recipiant, primi dimidio minorem, aliud ni
<lb/>
hil eſt, quàm binas primi numeri partes inuenire, quæ inter ſe multiplicatæ quadra
<lb/>
to ſecundi numeri numerum æqualem proferant, ex .16. ſexti aut .20. ſeptimi, quod
<lb/>
tamen .45. theoremate fuit à nobis ſpeculatum.</s>
</p>
</div>
<div xml:id="echoid-div109" type="math:theorem" level="3" n="52">
<num value="52">LII</num>
<p>
<s xml:id="echoid-s453" xml:space="preserve">CVR pro poſitis tribus numeris quibuſcunque, ſi productum primi in ſecun-
<lb/>
dum per tertium multiplicetur, atque ſecundum hoc productum
<reg norm="corporeum" type="context">corporeũ</reg>
,
<lb/>
per primum numerum diuidatur, proueniens erit numerus æqualis producto ſe-
<lb/>
cundi in tertium.</s>
</p>
<p>
<s xml:id="echoid-s454" xml:space="preserve">Exempli cauſa, proponantur hi tres numeri .10. 11. 12.
<reg norm="multiplicenturque" type="simple">multiplicenturq́;</reg>
.10.
<reg norm="cum" type="context">cũ</reg>
.</s>
</p>
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