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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49
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THEOREM. ARIT.
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61
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0061
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0061
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Secundus tertiusq́ue terminus reperiuntur, eſt
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enim ſecundus
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tertius
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et
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quando-
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quidem ex præſuppoſito
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æqualis eſt
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et
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<
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æqualis
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et
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cum ſit æqualis
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cui
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pariter æqualis eſt
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ex quo
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æqualis
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eſt
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. </
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<
s
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xml:space
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">Itaque illud ſequitur
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ipſi
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æqualem eſſe.</
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<
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xml:space
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">CVR ſumma duorum terminorum extremorum imparium arithmeticæ pro-
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portionalitatis ſemper duplo medij termini æqualis eſt.</
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<
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<
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xml:space
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">Exempli gratia, ſunt hitres termini proportionalitatis arithmeticæ .20. 15. 10
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ſumma duorum extremorum erit .30. quæ duplum eſt medij termini .15.</
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<
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">Quod vt ſpeculemur, tres termini, tribus lineis
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:
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et
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norm
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ſignificentur
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type
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">ſignificẽtur</
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. </
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<
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xml:space
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">Di-
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co nunc quòd ſumma
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cum
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nempe
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h.d.</
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ſemper duplo
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ſcilicet
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>
æqualis
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<
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fig-0061-02
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xlink:href
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fig-0061-02a
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number
="
84
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file
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0061-02
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erit. </
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ad
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ſit
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quæ
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æqualis erit
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differentiæ inter
<
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>n.u.</
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>
et
<
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>.q.p.</
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>
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patet enim in linea
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>.h.d</
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>
:
<
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>b.c.</
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>
æqualem eſſe
<
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>.n.
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u.</
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>
ſed
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>
ex
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componitur æquali
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>.q.p.</
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>
et
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ex
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>.e.u.</
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æquali
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>.c.d.</
var
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cum
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norm
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itaque
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type
="
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">itaq;</
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in
<
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var
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partem
<
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h.b.</
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>
reperiamus æqualem
<
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var
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gratia
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>
&
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partem
<
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var
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æquale
<
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>m.e.u.</
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manifeſtum erit
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<
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>h.d.</
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æqualem eſſe
<
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>.g.u</
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>
.</
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">BINA PROBLEMAT A EX DVOBVS PRAEDICTIS
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THEOREMATIBVS DEPENDENTIA.</
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xml:space
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">EX duobus prædictis theorematibus duo problemata oriuntur,
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primum
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eſt. </
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xml:space
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">Datis tribus quantitatibus cognitis, ſi quis quartam inuenire voluerit,
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quæ eiuſmodi ſit reſpectu tertiæ, qualis eſt ſecunda reſpectu primæ, ſecunda cum
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tertia in ſummam colligenda erit, ex qua detracta prima, ſupererit quarta.</
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<
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xml:space
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">Exempli gratia, cognitis tribus quantitatibus .20. 17. 9. ſi quartam inuenire vo
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luerimus eiuſmodi proportionem cum tertia arithmeticè ſeruantem, quam ſecunda
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cum prima, ſecundam cum tertia in ſummam colligemus,
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ſumma .26. ex
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qua detracta prima quantitate, quarta relinquetur nempe .6. quod ex .74. theore-
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mate dependet.</
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<
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xml:space
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">Idipſum tamen proueniret ſi quis ex tertio termino differentiam primi atque ſe-
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cundi detraheret; </
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<
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">hæc tamen via non tam vniuerſalis eſtqu àm illa. </
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<
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xml:space
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">N ſi quartus ter
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minus incognitus tertio maior eſſe deberet, dictam differentiam cum tertio termi-
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mino in ſummam colligere oporteret.</
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<
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<
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xml:space
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">Alterum problema eſt, quòd inuentis duobus terminis, ſi tertius requiratur, ſe-
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cundus duplicandus erit, ex qua ſumma detracto primo, ſtatim tertius proferetur,
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quod problema ex præcedenti theoremate dependet.</
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