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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IO. BAPT. BENED.
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68
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0068
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poſitorum .3. et .2. multiplicato .3. per .10.
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.6. cum .4. dantur .30. quod pro-
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ductum æquale erit producto .2. per .15. nempe per ſummam 9. et .6.</
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titates continuæ proportionales ſint
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proportione
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productum
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autem
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in ſummam
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cum
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ſit
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& productum
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in ſummam
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ſit
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et
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ſit æqualis
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et
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æqua
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lis
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& ita etiam
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eidem
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et
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æqualis
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et
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ipſi
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et
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ipſi
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.
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</
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<
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xml:id
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xml:space
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">quare ita ſe habebit
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>.K.n.</
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ad
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>.n.o.</
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ſicut
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<
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ad
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& componendo
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var
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ad
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n.o.</
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vt
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ad
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& permutando
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o.</
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ad
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vt
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var
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hoc eſt
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ad
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&
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pariter
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ad
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vt
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ad
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. </
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>
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que ſicut
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>.k.o.</
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>
ad
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>.o.t.</
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>
ex quo ex .15. ſexti aut .20. ſeptimi
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>.K.h.</
var
>
æqualis erit
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>.f.t</
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>
.</
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>
</
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</
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<
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xml:id
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<
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xml:id
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xml:space
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">THEOREMA
<
num
value
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">LXXXVI</
num
>
.</
head
>
<
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<
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xml:space
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">CVR multiplicatis ſingulis tribus quantitatibus continuis proportionalibus in
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reliquas duas, ſex producta æqualia ſint producto dupli ſummæ ipſarum trium
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in mediam proportionalem.</
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<
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<
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xml:id
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xml:space
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">Exempli gratia, proponuntur hitres termini continui proportionales .9. 6. 4. pro
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ductum .9. in .6. erit .54. at .9. in .4. erit .36. et .6. in .9: 54. et .6. in .4: 24. et .4. in .9: 36. et
<
num
value
="
4
">.
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4.</
num
>
in .6: 24. quæ producta ſimul collecta efficiunt numerum .228 ſed
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type
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eſt pro-
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ductum dupli ſummæ trium terminorum in ſecundum nempe .38 in .6.</
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<
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xml:space
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">Cuius
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type
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>
cauſa, tres termini
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continui
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type
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">cõtinui</
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>
proportionales ſignificentur linea
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>.
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b.e.</
var
>
nempe
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>.b.d</
var
>
:
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var
>
:
<
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var
>
cuius duplum ſit
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>.u.e.</
var
>
et
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>.b.f.</
var
>
æqualis ſit
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>.b.d.</
var
>
et
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>.f.n</
var
>
:
<
var
>d.c.</
var
>
et
<
var
>.n.u</
var
>
:
<
lb
/>
c. e productum verò
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>.u.e.</
var
>
in
<
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>.d.c.</
var
>
ſit
<
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>.u.s.</
var
>
cui dico æqualem eſſe ſummam productorum
<
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/>
ſingulorum trium terminorum in reliquos duos. </
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>
<
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xml:space
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">Quamobrem ducantur perpendi-
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culares
<
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>.c.g</
var
>
:
<
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>d.o</
var
>
:
<
var
>b.i</
var
>
:
<
var
>f.a.</
var
>
et
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var
>.n.p.</
var
>
inter
<
var
>.u.e.</
var
>
et
<
var
>.q.s.</
var
>
ex quo pro producto
<
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>.c.e.</
var
>
in
<
var
>.c.d.</
var
>
ha-
<
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bebimus rectangulum
<
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>.c.s.</
var
>
& rectan-
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xlink:href
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</
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>
gulum
<
var
>.d.g.</
var
>
pro producto
<
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>.c.e.</
var
>
in
<
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>.d.b.</
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>
<
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/>
ex .16. ſexti aut .20. ſeptimi itemq́ue
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rectangulum
<
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>.q.n.</
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>
pro producto
<
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>.d.c.</
var
>
<
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/>
in
<
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>.c.e.</
var
>
& rectangulum
<
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>.b.o.</
var
>
ex
<
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>.d.c.</
var
>
in
<
var
>.
<
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/>
b.d.</
var
>
& rectangulum
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>.b.a.</
var
>
ex
<
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>.b.d.</
var
>
in
<
var
>.d.
<
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/>
c.</
var
>
et
<
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>.p.f.</
var
>
ex
<
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>.d.b.</
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>
in
<
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>.c.e.</
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>
ex .16. aut .20.
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prędictas. </
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<
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xml:space
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">Quare ſex producta æquantur inter ſe,
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productum
<
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>.u.s.</
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>
ex quo
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verum eſt propoſitum.</
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</
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</
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>
<
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<
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xml:id
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xml:space
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">THEOREMA
<
num
value
="
87
">LXXXVII</
num
>
.</
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>
<
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<
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eſſe proportionem ſummæ quatuor quan-
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titatum continuarum proportionalium ad ſummam ſecundæ & tertiæ, ean-
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dem eſſe, quæ ſummæ primæ & tertiæ ad ſecundam ſimplicem.</
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<
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xml:space
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">Exempli gratia, ſi inue nirentur hæ quatuor quantitates continuæ proportiona-
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es .16. 8. 4. 2. earum ſumma erit .30. ſunima verò ſecundæ & tertiæ .12. tum ſumma </
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