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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<div xml:id="echoid-div130" type="math:theorem" level="3" n="64">
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<div xml:id="echoid-div131" type="math:theorem" level="3" n="65">
<num value="65">LXV</num>
<p>
<s xml:id="echoid-s557" xml:space="preserve">CVR propoſito numero in tres qualeſcunque partes diuiſo, ſi prima in
<lb/>
tertiam multiplicetur, & huic producto, ſecundæ in primam productum
<lb/>
coniungatur,
<reg norm="itemque" type="simple">itemq́;</reg>
ſecundæ in tertiam, hæc ſumma duplicata æqualis ſit ſummæ
<lb/>
productorum ſingularum in cæteras duas.</s>
</p>
<p>
<s xml:id="echoid-s558" xml:space="preserve">Exempli gratia, ſi proponatur .20. diuiſus in tres partes nempe .12. 5. 3. multipli-
<lb/>
cato primo .12. per .3. tertiam partem dabitur .36. ſecunda verò multiplicata per re
<lb/>
liquas duas, hoc eſt .5. per .12. et .3. in primis dabitur .60. poſtea .15.
<reg norm="quorum" type="context">quorũ</reg>
<reg norm="trium" type="context">triũ</reg>
pro
<lb/>
ductorum ſumma erit .111. quæ duplicata dabit .222. qui numerus æqualis eſſe di-
<lb/>
citur ſummæ productorum ſingularum partium in reliquas duas, nempe ſummæ .60.
<lb/>
36. 60. 15. 36. 15. hoc eſt ipſis .222.</s>
</p>
<p>
<s xml:id="echoid-s559" xml:space="preserve">Cuius rei per ſe patet ſpeculatio, cum in his ſex vltimis productis, ſingula tria
<lb/>
prima duplicentur.</s>
</p>
</div>
<div xml:id="echoid-div132" type="math:theorem" level="3" n="66">
<num value="66">LXVI</num>
<p>
<s xml:id="echoid-s560" xml:space="preserve">CVR propoſito numero in .3. qualeſcunque partes diuiſo, ſi in reliquas duas ſin-
<lb/>
gulæ multiplicentur, & hæc producta cum ſumma ſuorum quadratorum con-
<lb/>
iungantur, tota ſumma hæc vltima æqualis erit quadrato totali propoſiti numeri.</s>
</p>
<p>
<s xml:id="echoid-s561" xml:space="preserve">Exempli gratia, ſi fuerit idem numerus .20. in .3. partes diuiſus .12. 5. 3. </s>
<s xml:id="echoid-s562" xml:space="preserve">Si .12. in
<lb/>
5. et .3. producatur, ſumma productorum erit .96. at .5. in .12. et .3. erit .75. poſtmo-
<lb/>
dum .3. in .12. et .5. erit .51. nempe in vniuerſum .222. quadratorum porrò ſumma
<lb/>
erit .178 quæ coniuncta .222. dabit .400. quadratum ipſius .20.</s>
</p>
<p>
<s xml:id="echoid-s563" xml:space="preserve">Erit autem huiuſce rei facillima ſpeculatio, ſi ſequentem figuram mente conce-
<lb/>
perimus, in qua
<var>.a.b.</var>
propoſitum numerum ſignificet, cuius partes diſtinctæ ſint me-
<lb/>
dio
<var>.e.</var>
et
<var>.c</var>
. </s>
<s xml:id="echoid-s564" xml:space="preserve">Ip ſum autem
<var>.q.b.</var>
<lb/>
totale parallelis
<var>.e.s.</var>
et
<var>.c.x.</var>
diuiſum, quæ qua
<lb/>
dratum in triarectangula diuident, quorum
<lb/>
primum erit
<var>.q.e.</var>
compoſitum ex producto
<var>.a.
<lb/>
e.</var>
<var>.o.e.</var>
&
<lb/>
ex producto eiuſdem
<var>.a.e.</var>
in
<var>.e.b.</var>
quod erit re
<lb/>
ctangulum
<var>.o.s.</var>
ex quo tria rectangula
<var>.o.s.</var>
et
<var>.
<lb/>
n.x.</var>
et
<var>.t.u.</var>
tria producta erunt ſingularum par
<lb/>
tium in cæteras duas, et
<var>.e.o</var>
:
<var>c.n</var>
:
<var>b.t.</var>
tria qua-
<lb/>
drata erunt: </s>
<s xml:id="echoid-s565" xml:space="preserve">quibus ſex quantitatibus quadra
<lb/>
tum totale
<var>.q.b.</var>
completur.</s>
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</figure>
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<div xml:id="echoid-div134" type="math:theorem" level="3" n="67">
<num value="67">LXVII</num>
<p>
<s xml:id="echoid-s566" xml:space="preserve">
<emph style="sc">VEteres</emph>
aliud quoque problema indefinitum propoſuerunt, quod tamen à
<lb/>
nobis determinabitur.</s>
</p>
<p>
<s xml:id="echoid-s567" xml:space="preserve">Cur diuiſuri propoſitum numerum in duas eiuſmodi partes, vt mutuò diuiſis, &
<lb/>
per ſummam prouenientium diuiſa ſumma qua dratorum partium, oriatur proue-
<lb/>
niens alter numerus propoſitus.</s>
</p>
<p>
<s xml:id="echoid-s568" xml:space="preserve">Propoſito deinde tertio quolibet numero diuidendo per ſingulas partes primi, </s>
</p>
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