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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THEOREM. AR IT.
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<p>
<s xml:id="echoid-s530" xml:space="preserve">
<pb o="39" rhead="THEOREM. AR IT." n="51" file="0051" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0051"/>
trahemus,
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numerus .16. cuius dimidium ſcilicet .8. in ſeipſum multipli-
<lb/>
cabimus,
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numerus .64. qui cum ex quadrato dimidij primi detractus fue-
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rit, nempe ex .100. & reſiduo .36. radix quadrata nempe .6. coniuncta denario, di-
<lb/>
midio primi, dabit .16. partem maiorem, & ex denario detracta, partem minorem.</s>
</p>
<p>
<s xml:id="echoid-s531" xml:space="preserve">Cuius ſpeculationis cauſa, primus numerus
<lb/>
propoſitus ſigniſicetur linea
<var>.x.y.</var>
pro voto diui-
<lb/>
<anchor type="figure" xlink:label="fig-0051-01a" xlink:href="fig-0051-01"/>
ſa in puncto
<var>.c.</var>
et
<var>.x.t.</var>
productum ſit ipſius
<var>.x.
<lb/>
c.</var>
in
<var>.c.y.</var>
pariter etiam
<var>.q.p.</var>
ſit ſumma radicum
<lb/>
quadratarum, nempe
<var>.q.g.</var>
ipſius
<var>.t.c.</var>
et
<var>.g.p.</var>
ip-
<lb/>
ſius
<var>.c.y</var>
. </s>
<s xml:id="echoid-s532" xml:space="preserve">Tum ſuper
<var>.q.p.</var>
extruatur & diuidatur
<lb/>
quadratum
<var>.q.u.</var>
ea ratione qua .41. theoremate
<lb/>
aut .29. diuiſimus, in quo ſanè quadrato, quadra
<lb/>
tum ipſius
<var>.q.i.</var>
cernemus datæ differentiæ, & in
<lb/>
eo collocata quadrata
<var>.x.c.</var>
et
<var>.c.y.</var>
ita etiam &
<lb/>
rationem, qua cognoſcimus productum
<var>.g.r.</var>
(vſi
<lb/>
modo .29. theorematis) cuius quidem
<var>.g.r.</var>
qua-
<lb/>
dratum, ex .19. theoremate æquale erit produ-
<lb/>
cto
<var>.x.t.</var>
ideo etiam
<reg norm="cognitum" type="context">cognitũ</reg>
, ac proinde cum no
<lb/>
uerimus
<var>.x.y.</var>
ſi rationem ſequemur .45. theore
<lb/>
mate cognoſcemus non ſolum ratione .41. theoremate allata hocrectè perfici, ſed
<lb/>
hac etiam alia ratione.</s>
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<figure xlink:label="fig-0051-01" xlink:href="fig-0051-01a">
<image file="0051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0051-01"/>
</figure>
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<head xml:id="echoid-head78" xml:space="preserve">THEOREMA
<num value="62">LXII</num>
.</head>
<p>
<s xml:id="echoid-s533" xml:space="preserve">CVR propoſitum numerum diuiſuri in duas eiuſmodi partes, vt differentia
<lb/>
<reg norm="ſuarum" type="context">ſuarũ</reg>
<reg norm="radicum" type="context">radicũ</reg>
<reg norm="quadratarum" type="context">quadratarũ</reg>
æqualis ſit alteri numero propoſito. </s>
<s xml:id="echoid-s534" xml:space="preserve">Cuius
<reg norm="tamen" type="wordlist">tamẽ</reg>
<reg norm="qua- dratum" type="context">qua-
<lb/>
dratũ</reg>
maius non ſit quadrato medietatis ipſius primi propoſiti numeri. </s>
<s xml:id="echoid-s535" xml:space="preserve">Rectè
<reg norm="etiam" type="context">etiã</reg>
<lb/>
<reg norm="quadratum" type="context">quadratũ</reg>
dimidij ſecundi numeri ex dimidio primi
<reg norm="detrahunt" type="context">detrahũt</reg>
,
<reg norm="reſiduique" type="simple">reſiduiq́;</reg>
radicem per
<lb/>
ſecundum multiplicant, & productum ex dimidio primi detrahunt, vt reſiduum
<lb/>
pars quæſita minor ſit, & illud alterum totius reſiduum, pars maior.</s>
</p>
<p>
<s xml:id="echoid-s536" xml:space="preserve">Exempli gratia, ſi numerus .50. in
<lb/>
prædictas duas partes diuidendus pro-
<lb/>
<anchor type="figure" xlink:label="fig-0051-02a" xlink:href="fig-0051-02"/>
poneretur, & alter etiam .6. quadratum
<lb/>
dimidij ſecundi numeri eſſet .9. eo detra
<lb/>
cto ex dimidio primi, remaneret .16. cu
<lb/>
ius radix .4. ſcilicet per totum ſecundum
<lb/>
nempe .6. multiplicata, proferet .24.
<lb/>
quo producto ex dimidio primi detra-
<lb/>
cto, nempe .25. dabitur .1. pars minor,
<lb/>
maior
<reg norm="autem" type="context">autẽ</reg>
erit
<reg norm="reſidum" type="context">reſidũ</reg>
.50. hoc eſt .49.
<lb/>
radices autem erunt .1. et .7. differentes
<lb/>
inter ſe, numero ſenario.</s>
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<figure xlink:label="fig-0051-02" xlink:href="fig-0051-02a">
<image file="0051-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0051-02"/>
</figure>
</div>
<p>
<s xml:id="echoid-s537" xml:space="preserve">Hocvt ſciamus, duo numeri lineis
<reg norm="ſi- gnificentur" type="context">ſi-
<lb/>
gnificẽtur</reg>
, primus linea .b:
<reg norm="ſecundus" type="context">ſecũdus</reg>
linea
<var>.
<lb/>
c.</var>
duæ autem partes
<var>.b.</var>
duobus quadra-
<lb/>
tis
<var>.q.i.</var>
et
<var>.i.d.</var>
notentur, eorum verò radi-
<lb/>
ces lineis
<var>.a.g.</var>
et
<var>.g.d.</var>
differentia porrò ip
<lb/>
ſi
<var>.c.</var>
æqualis & co gnita ſit
<var>.a.h.</var>
ex quo
<var>.h.</var>
</s>
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