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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39
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rhead
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THEOREM. AR IT.
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n
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51
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file
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0051
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0051
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trahemus,
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numerus .16. cuius dimidium ſcilicet .8. in ſeipſum multipli-
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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-
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midio primi, dabit .16. partem maiorem, & ex denario detracta, partem minorem.</
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<
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xml:space
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">Cuius ſpeculationis cauſa, primus numerus
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propoſitus ſigniſicetur linea
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>.x.y.</
var
>
pro voto diui-
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<
figure
xlink:label
="
fig-0051-01
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xlink:href
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fig-0051-01a
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number
="
69
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<
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file
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0051-01
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0051-01
"/>
</
figure
>
ſa in puncto
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et
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>.x.t.</
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>
productum ſit ipſius
<
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>.x.
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c.</
var
>
in
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>.c.y.</
var
>
pariter etiam
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>.q.p.</
var
>
ſit ſumma radicum
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quadratarum, nempe
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>.q.g.</
var
>
ipſius
<
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>.t.c.</
var
>
et
<
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>.g.p.</
var
>
ip-
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ſius
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>.c.y</
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>
. </
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<
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xml:space
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">Tum ſuper
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extruatur & diuidatur
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quadratum
<
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>.q.u.</
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>
ea ratione qua .41. theoremate
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aut .29. diuiſimus, in quo ſanè quadrato, quadra
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tum ipſius
<
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>.q.i.</
var
>
cernemus datæ differentiæ, & in
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eo collocata quadrata
<
var
>.x.c.</
var
>
et
<
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>.c.y.</
var
>
ita etiam &
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/>
rationem, qua cognoſcimus productum
<
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>.g.r.</
var
>
(vſi
<
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/>
modo .29. theorematis) cuius quidem
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>.g.r.</
var
>
qua-
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dratum, ex .19. theoremate æquale erit produ-
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cto
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>.x.t.</
var
>
ideo etiam
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, ac proinde cum no
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uerimus
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>.x.y.</
var
>
ſi rationem ſequemur .45. theore
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mate cognoſcemus non ſolum ratione .41. theoremate allata hocrectè perfici, ſed
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hac etiam alia ratione.</
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<
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xml:space
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">THEOREMA
<
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value
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62
">LXII</
num
>
.</
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>
<
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<
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xml:space
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">CVR propoſitum numerum diuiſuri in duas eiuſmodi partes, vt differentia
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<
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norm
="
ſuarum
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type
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">ſuarũ</
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<
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radicum
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type
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">radicũ</
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>
<
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="
quadratarum
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type
="
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">quadratarũ</
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>
æqualis ſit alteri numero propoſito. </
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>
<
s
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xml:space
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">Cuius
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tamen
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type
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">tamẽ</
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>
<
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norm
="
qua- dratum
"
type
="
context
">qua-
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dratũ</
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>
maius non ſit quadrato medietatis ipſius primi propoſiti numeri. </
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>
<
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xml:id
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xml:space
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">Rectè
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norm
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etiam
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">etiã</
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>
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/>
<
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norm
="
quadratum
"
type
="
context
">quadratũ</
reg
>
dimidij ſecundi numeri ex dimidio primi
<
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norm
="
detrahunt
"
type
="
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">detrahũt</
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>
,
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="
reſiduique
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type
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">reſiduiq́;</
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>
radicem per
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/>
ſecundum multiplicant, & productum ex dimidio primi detrahunt, vt reſiduum
<
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pars quæſita minor ſit, & illud alterum totius reſiduum, pars maior.</
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<
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<
s
xml:id
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xml:space
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preserve
">Exempli gratia, ſi numerus .50. in
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prædictas duas partes diuidendus pro-
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<
figure
xlink:label
="
fig-0051-02
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xlink:href
="
fig-0051-02a
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number
="
70
">
<
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file
="
0051-02
"
xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0051-02
"/>
</
figure
>
poneretur, & alter etiam .6. quadratum
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dimidij ſecundi numeri eſſet .9. eo detra
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cto ex dimidio primi, remaneret .16. cu
<
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ius radix .4. ſcilicet per totum ſecundum
<
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nempe .6. multiplicata, proferet .24.
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quo producto ex dimidio primi detra-
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cto, nempe .25. dabitur .1. pars minor,
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maior
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erit
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.50. hoc eſt .49.
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radices autem erunt .1. et .7. differentes
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inter ſe, numero ſenario.</
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>
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<
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<
s
xml:id
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xml:space
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">Hocvt ſciamus, duo numeri lineis
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ſi- gnificentur
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gnificẽtur</
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>
, primus linea .b:
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linea
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c.</
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>
duæ autem partes
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>.b.</
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>
duobus quadra-
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tis
<
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>.q.i.</
var
>
et
<
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>.i.d.</
var
>
notentur, eorum verò radi-
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ces lineis
<
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>.a.g.</
var
>
et
<
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>.g.d.</
var
>
differentia porrò ip
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ſi
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>.c.</
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>
æqualis & co gnita ſit
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>.a.h.</
var
>
ex quo
<
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>.h.</
var
>
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