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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27
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THEOREM. ARIT.
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39
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file
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0039
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0039
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<
s
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xml:space
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">Quod vt ſpeculemus, conſideremus ſubſcriptam figuram, vigefiminoni theore-
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matis figuræ ſimilem, in qua numeri quæſiti duabus
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lineis directè coniunctis
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et
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fignificentur, ho
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0039-01
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rum
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quadrata
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erunt
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<
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>.r.c.</
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et
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>.g.s.</
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<
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iterum
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type
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propo
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nitur, quare etiam cognita. </
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<
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Differentia
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type
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">Differẽtia</
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autem
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duorum
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type
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numerorum primo propofita fit
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>.q.i.</
var
>
eius verò qua-
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dratum
<
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>.m.e.</
var
>
quod cognitum eſt ex ſua radice
<
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>.q.i</
var
>
.
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</
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<
s
xml:id
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xml:space
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">quare gnomon
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>.e.n.m.</
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ſimul cum quadrato minori
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>.
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g.s.</
var
>
cognitus erit, quæ ſumma æqualis eſt duplo
<
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>.g.r.</
var
>
<
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producto datorum numerorum. </
s
>
<
s
xml:id
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xml:space
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preserve
">Itaque & ipſa
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>.g.
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r.</
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cognoſcetur, nunc ſi præcedentis theorematis ſpe-
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culationem in reliquis conſuluerimus propoſitum
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conſequemur.</
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>
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<
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n
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<
head
xml:id
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xml:space
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">THEOREMA
<
num
value
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42
">XLII</
num
>
.</
head
>
<
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<
s
xml:id
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xml:space
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">ADhuc etiam & alia ratione idipſum conſequi poſſemus, non conſulto qua-
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drageſimo theoremate. </
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<
s
xml:id
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xml:space
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">Nam ſubtracto quadrato differentiæ, numeri primi
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(
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) propoſiti, ex
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duorum quadratorum, nempe ex ſecundo numero pro-
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poſito colligendum eſſet reſiduum in ſummam cum prædicto ſecundo numero, &
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ex ſumma hac deſumenda quadrata radix, quæ duorum numerorum ſumma erit,
<
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de qua detracto primo numero, remanebit duplum minoris numeri quæſiti, cuius
<
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/>
dimidio addito primo numero propoſito, aut detracto minore inuento ex radice
<
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poſtremo inuenta, dabitur numerus maior, qui quæritur.</
s
>
</
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<
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<
s
xml:id
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xml:space
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">Exempli gratia, cum ſuperfuerint .128. hæc ſi cum ſecundo numero
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.272.
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iunxerimus, dabunt .400. quorum radix erit .20. de quo numero detracto primo
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propoſito, nempe .12. ſupererunt .8. quorum
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erit .4. quo ex .20. detracto
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aut coniuncto .12. maior numerus orietur.</
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</
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<
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<
s
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xml:space
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">Cuius rei contemplatio, præcedenti figura aperitur. </
s
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<
s
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xml:space
="
preserve
">Nam reſiduum detractionis
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quadrati
<
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>.m.e.</
var
>
ex ſumma
<
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quadratorum
<
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>.r.c.</
var
>
et
<
var
>.g.s.</
var
>
numerum præbet æqua-
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lem duobus ſupplementis
<
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>.q.n.</
var
>
et
<
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>.n.u.</
var
>
ex .8. ſecundi Euclidis. qui coniunctus duo-
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bus quadratis (quorum ſumma ſecundo propoſita fuit) cognitionem profert qua-
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drati
<
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>.q.u.</
var
>
& eius radicis
<
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>.q.p.</
var
>
de qua, detracto primo dato numero, ſcilicet
<
var
>.q.i.</
var
>
ſu-
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pereſt
<
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>.i.p.</
var
>
cuius dimidium nempe
<
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>.g.p.</
var
>
minor eſt numerus qui quęritur; </
s
>
<
s
xml:id
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xml:space
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">reſiduum
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verò totius
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>.g.q.</
var
>
maior ſcilicet.</
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>
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<
head
xml:id
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xml:space
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preserve
">THEOREMA
<
num
value
="
43
">XLIII</
num
>
.</
head
>
<
p
>
<
s
xml:id
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"
xml:space
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preserve
">CVR ij, qui volunt duos numeros inuenire, quorum ſumma æqualis propo-
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fito alicui numero futura ſit, & ſumma quadratorum maior eorum produ-
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cto per quantitatem alterius propoſiti numeri, rectè dimidium primi dati numeri in
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ſeipſum multiplicant, quod quadratum ex
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ſecundo
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dato numero detrahunt, ſumunt
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q́ue tertię partis refidui quadratam radicem, quam dimidio primi numeri coniun-
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gunt, ex quo maior numerus
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<
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datur, quo ex toto primo detracto, ſu-
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pererit minor.</
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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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">Exempli gratia, propoſito numero .20. cui æquanda eſt ſumma duorum nume-
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rorum quæſitorum,
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ſecundo numero .208. qui ſemper maior eſſe debet </
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