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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<
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xml:space
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<
num
value
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82
">LXXXII</
num
>
.</
head
>
<
p
>
<
s
xml:id
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xml:space
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">CVR quantitate aliqua in quatuor partes
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norm
="
continuas
"
type
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context
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reg
>
proportionales ſecta per-
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lb
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q́ue ſingulas diuiſa, ſumma quatuor prouenientium æqualis ſit producto ſe-
<
lb
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cundi in tertium.</
s
>
</
p
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<
p
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<
s
xml:id
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xml:space
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preserve
">Exempli gratia, ſi triginta in quatuor partes proportionales ſecetur, hoc eſt.
<
lb
/>
16. 8. 4. 2.
<
reg
norm
="
perque
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type
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simple
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reg
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harum ſingulas idem numerus .30. diuidatur, primum proueniens
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erit .1. cum ſeptem octauis partibus. </
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<
s
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xml:space
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preserve
">Secundum .3. cum tribus quartis, tertium .7.
<
lb
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cum dimidio, quartum .15. integri, quorum ſumma erit .28. cum octaua parte, tan
<
lb
/>
<
reg
norm
="
tumque
"
type
="
simple
">tumq́;</
reg
>
erit productum ſecundi prouenientis in tertium.</
s
>
</
p
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<
p
>
<
s
xml:id
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echoid-s720
"
xml:space
="
preserve
">Quod vt ſciamus, quantitas
<
var
>.n.c.</
var
>
in partes continuas proportionales quatuor ſe-
<
lb
/>
cetur
<
var
>.n.a</
var
>
:
<
var
>a.t</
var
>
:
<
var
>t.e.</
var
>
et
<
var
>.e.c.</
var
>
<
reg
norm
="
rurſusque
"
type
="
simple
">rurſusq́;</
reg
>
per ſingulas partes illa ipſa diuiſa, prouenientia
<
lb
/>
ſint
<
var
>.i.d</
var
>
:
<
var
>d.x</
var
>
:
<
var
>x.u</
var
>
:
<
var
>u.o.</
var
>
<
reg
norm
="
quorum
"
type
="
context
">quorũ</
reg
>
ſumma ſit
<
var
>.i.o.</
var
>
hanc
<
reg
norm
="
ſummam
"
type
="
context
">ſummã</
reg
>
dicimus æqualem eſſe nume-
<
lb
/>
ro producti
<
var
>.d.x.</
var
>
in
<
var
>.x.u</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
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xml:space
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preserve
">Quod hac ratione probo, cogito productam eſſe lineam
<
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>.i.o.</
var
>
<
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norm
="
quousque
"
type
="
simple
">quousq́;</
reg
>
<
var
>.o.p.</
var
>
æqua
<
lb
/>
lis ſit
<
var
>.o.u.</
var
>
<
reg
norm
="
erectamque
"
type
="
simple
">erectamq́;</
reg
>
<
var
>.m.o.</
var
>
æqualem
<
var
>.i.d.</
var
>
perpendiculariter
<
var
>.o.p.</
var
>
& productam donec
<
var
>.
<
lb
/>
o.q.</
var
>
vnitati ſit æqualis. </
s
>
<
s
xml:id
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xml:space
="
preserve
">Iam terminatis rectangulis
<
var
>.m.p.</
var
>
et
<
var
>.i.q.</
var
>
patebit ex .15. ſexti
<
lb
/>
aut .20. ſeptimi, productum
<
var
>.m.p.</
var
>
producto
<
var
>.d.x.</
var
>
in
<
var
>.x.u.</
var
>
æquale eſſe. </
s
>
<
s
xml:id
="
echoid-s723
"
xml:space
="
preserve
">Ita quòd ſi pro-
<
lb
/>
bauero productum
<
var
>.i.q.</
var
>
producto
<
var
>.m.p.</
var
>
æquale eſſe, facile patebit propoſitum. </
s
>
<
s
xml:id
="
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xml:space
="
preserve
">Cuius
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gratia, ſequuti præcedentis theorematis ordinem, primum ex
<
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norm
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type
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diuiſionis,
<
lb
/>
eadem proportio erit
<
var
>.n.c.</
var
>
ad
<
var
>.i.d.</
var
>
quæ
<
var
>.n.a.</
var
>
ad
<
var
>.o.q.</
var
>
ex quo permutando
<
var
>.n.c.</
var
>
ad
<
var
>.n.a.</
var
>
ſic
<
lb
/>
ſe habebit vt
<
var
>.i.d.</
var
>
hoc eſt
<
var
>.m.o.</
var
>
ad
<
var
>.o.q.</
var
>
& ſi progrediamur eodem ordine, quo præ-
<
lb
/>
cedenti theoremate, ſumpto principio ab
<
var
>.i.d.</
var
>
et
<
var
>.e.c.</
var
>
verſus
<
var
>.d.x.</
var
>
et
<
var
>.e.t.</
var
>
gradatimq́ue
<
lb
/>
permutando ac coniungendo, inue-
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niemus eandem proportionem eſſe
<
lb
/>
<
var
>c.n.</
var
>
ad
<
var
>.n.a.</
var
>
quæ
<
var
>.i.o.</
var
>
ad
<
var
>.o.u.</
var
>
nempe
<
var
>.
<
lb
/>
o.p.</
var
>
ex quo ex .11 quinti, ita ſe habe
<
lb
/>
bit
<
var
>.i.o.</
var
>
ad
<
var
>.o.p.</
var
>
vt
<
var
>.m.o.</
var
>
ad
<
var
>.o.q.</
var
>
</
s
>
<
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xml:id
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xml:space
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">quare
<
lb
/>
ex .15. ſextiaut .20. ſeptimi
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type
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ctũ</
reg
>
<
var
>.i.q.</
var
>
erit producto
<
unsure
/>
<
var
>.m.p.</
var
>
æquale,
<
lb
/>
ex quo etiam æquale erit producto
<
var
>.
<
lb
/>
d.x.</
var
>
in
<
var
>.x.u</
var
>
. </
s
>
<
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xml:id
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xml:space
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">Idem ordo in qualibet
<
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quantitate in quantaſuis partes diuiſa ſeruari poterit, cum huiuſmodi
<
reg
norm
="
ſcientia
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type
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in vni
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uerſum pateat.</
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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
="
83
">LXXXIII</
num
>
.</
head
>
<
p
>
<
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xml:id
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xml:space
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">CVR termini medij cubus, trium continuè proportionalium, ſemper producto
<
lb
/>
rectanguli compræhenſi à maximo & medio in minimo termino æqualis ſit.</
s
>
</
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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, datis his tribus terminis continuis proportionalibus .9. 6. 4. ſi
<
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ſumpſerimus productum maximi in medium nempe .54. quod per
<
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.4. multi-
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plicemus, dabitur numerus .216. cubo medij .6. æqualis.</
s
>
</
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<
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>
<
s
xml:id
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xml:space
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preserve
">In cuius gratiam tres numeri continui proportionales tribus lineis
<
var
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>
<
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ſignifi- centur
"
type
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cẽtur</
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, cubus autem
<
var
>.e.</
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>
ſignificetur figura
<
var
>.d.n.</
var
>
<
reg
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productumque
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type
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reg
>
<
var
>.a.</
var
>
in
<
var
>.e.</
var
>
ſit
<
var
>.b.n.</
var
>
ipſius
<
reg
norm
="
au- temmet
"
type
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">au-
<
lb
/>
tẽmet</
reg
>
in
<
var
>.i.</
var
>
ſit
<
var
>.p.o.</
var
>
ita quod
<
var
>.q.p.</
var
>
aut
<
var
>.b.o.</
var
>
cum ſint
<
reg
norm
="
eiuſdem
"
type
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context
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>
ſpeciei, æqualis erit .a: et
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var
>.o.n.</
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>
</
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>
</
p
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