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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<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-div175" type="math:theorem" level="3" n="89">
<p>
<s xml:id="echoid-s763" xml:space="preserve">Exempli gratia, caſu ſeſe offerunt hi quatuor numeri .8. 5. 3. 2. multiplicato .8.
<lb/>
per .5. & hoc .40. per .3. rurſus hoc .120. per .2. vltimum productum eſſet .240. æqua
<lb/>
le producto .15. (quod ex .5. in .3. oritur) in productum .16. quod ex .8. in .2. pro-
<lb/>
fertur.</s>
</p>
<p>
<s xml:id="echoid-s764" xml:space="preserve">Cuius ſpeculationis gratia, cogitemus quatuor numeros quatuor lineis
<var>.a.e.i.o.</var>
<lb/>
ſignifi cari, productum autem
<var>.e.</var>
in
<var>.i.</var>
eſſe
<var>.m.f.</var>
et
<var>.r.s.</var>
ſimiliter & productum
<var>.a.</var>
in
<var>.o.</var>
eſ-
<lb/>
ſe
<var>.m.z</var>
: et
<var>.z.f.</var>
productum eſſe
<var>.m.f.</var>
in
<var>.m.z.</var>
cui productum
<var>.a.</var>
in
<var>.e.</var>
multiplicatum per
<lb/>
i. & hoc tandem per
<var>.o.</var>
æquari debet.</s>
</p>
<p>
<s xml:id="echoid-s765" xml:space="preserve">Sit itaque
<var>.u.y.</var>
productum
<var>.a.</var>
in
<var>.e.</var>
quod
<var>.u.y.</var>
per
<var>.i.</var>
multiplicatum proferat
<var>.u.s.</var>
<lb/>
hocq́ue
<var>.u.s.</var>
multiplicatum per
<var>.o</var>
. </s>
<s xml:id="echoid-s766" xml:space="preserve">Dico quod dabit numerum æqualem numero
<var>.f.z.</var>
<lb/>
Quamobrem
<var>.r.s.</var>
aut
<var>.m.f.</var>
quod idem eſt, in figura præcedentis theore matis ſigni-
<lb/>
ficetur linea
<var>.n.u.</var>
& linea
<var>.r.u.</var>
hu-
<lb/>
ius, nempe
<var>.a.</var>
ſignificetur per
<var>.u.t.</var>
<lb/>
</figure>
præcedentis, ex quo numerus pro
<lb/>
ducti
<var>.u.s.</var>
præſentis, in præcedenti
<lb/>
ſignificabitur producto
<var>.n.t.</var>
quod
<lb/>
<reg norm="productum" type="simple context">ꝓductũ</reg>
<var>.u.s.</var>
<reg norm="pręsens" type="context">pręsẽs</reg>
<reg norm="per" type="simple">ꝑ</reg>
<reg norm="præsens" type="context">præsẽs</reg>
<var>.o.</var>
mul­
<lb/>
tiplicatum, quod erat in præceden
<lb/>
ti
<var>.u.c.</var>
ſignificabitur per
<var>.d.u.</var>
præce
<lb/>
dentis, quod non modo ex multi-
<lb/>
plicatione
<var>.n.t.</var>
præcedentis, nempe
<var>.u.s.</var>
præſentis. in
<var>.u.c.</var>
præcedentis æquali
<var>.o.</var>
præ-
<lb/>
ſentis oritur, ſed etiam ex
<var>.c.t.</var>
præcedentis æquali
<var>.m.z.</var>
præſentis in
<var>.n.u.</var>
præceden
<lb/>
tis æquali
<var>.m.f.</var>
præſentis. </s>
<s xml:id="echoid-s767" xml:space="preserve">Itaque verum eſt propoſitum.</s>
</p>
</div>
<div xml:id="echoid-div177" type="math:theorem" level="3" n="90">
<num value="90">XC</num>
<p>
<s xml:id="echoid-s768" xml:space="preserve">CVR quibuſlibet & quantiſuis numeris in ſummam collectis, ſi ab vnitate in ſe-
<lb/>
cunda ſpecie progreſſionis arithmeticę imparium numerorum progreſſi fue-
<lb/>
rimus, eiuſmodi ſumma ſemper eſt quadratus numerus.</s>
</p>
<p>
<s xml:id="echoid-s769" xml:space="preserve">Exempli gratia, ſi horum quatuor diſparium numerorum
<reg norm="ſummam" type="context">ſummã</reg>
, in dicta pro-
<lb/>
greſſione arithmetica quis ſumat, principio ab vnitate ſumpto, nempe .1. 3. 5. 7. ſum-
<lb/>
ma erit .16. numerus quadratus inquam. </s>
<s xml:id="echoid-s770" xml:space="preserve">Idem de cæteris.</s>
</p>
<p>
<s xml:id="echoid-s771" xml:space="preserve">Quamobrem animaduertendum eſt, vnitatem, tam ſumi pro ſui ipſius radicem,
<lb/>
quam pro quadrato, cubo, cenſo cenſi, primo relato, & alia quauis dignitate.
<lb/>
</s>
<s xml:id="echoid-s772" xml:space="preserve">Nunc autem pro quadrato ſumamus per
<var>.o.</var>
ſignificato,
<reg norm="cogitemusque" type="simple">cogitemusq́</reg>
<var>.o.</var>
<lb/>
includi quadrato vnitatem ſequenti, quod, vt patet, eſt quatuor vnitatum, ac pro-
<lb/>
priè primum quadratum numerorum, ex quo etiam nomen accepit, vnde ex ſimi-
<lb/>
litudine quam cætera quadrata cum hoc primo retinent, ex quaternario denomina-
<lb/>
tionem acceperunt. </s>
<s xml:id="echoid-s773" xml:space="preserve">
<reg norm="Hocitaque" type="simple">Hocitaq;</reg>
ſit
<var>.o.u.c.e.</var>
<var>.o.</var>
iun-
<lb/>
gitur gnomon
<var>.e.c.u.</var>
conſtans tribus vnitatibus, quare primus gnomon, numero im-
<lb/>
pari conſtat. </s>
<s xml:id="echoid-s774" xml:space="preserve">Scimus etiam ex additione numeri binarij ad imparem, numeris di-
<lb/>
ſparibus ſummam excreſcere, cum propius accedere
<reg norm="quam" type="context">quã</reg>
binario nequeant, ex quo
<lb/>
medio binario, ſibi inuicem ſuccedunt. </s>
<s xml:id="echoid-s775" xml:space="preserve">Dico igitur quòd quinario ternarium ſub
<lb/>
<var>.o.u.c.e.</var>
profertur quadratum, quod in numeris, bi-
<lb/>
<reg norm="eritque" type="simple">eritq́;</reg>
ternarij,
<reg norm="quodque" type="simple">quodq́;</reg>
ſignificetur per
<var>.o.f.</var>
patet enim pri
<lb/>
mo non differre ab
<var>.o.c.</var>
præter quam gnomone
<var>.b.f.d.</var>
<var>.o.
<lb/>
c.</var>
quique duabus vnitatibus maior eſt
<var>.e.c.u</var>
. </s>
<s xml:id="echoid-s776" xml:space="preserve">
<reg norm="Iam" type="context">Iã</reg>
ſcimus gnomonem
<var>.e.o.u.</var>
æqualem </s>
</p>
</div>
</div>
</div>
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