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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rhead
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I O. BAPT. BENED.
"
n
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32
"
file
="
0032
"
xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0032
"/>
biturum, ſicut
<
var
>.u.x.</
var
>
ad
<
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>.n.x.</
var
>
ex prima ſexti aut .18. vel .19. ſeptimi, </
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<
s
xml:id
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xml:space
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preserve
">quare ex 11.
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quinti ita ſe habebit
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>.o.x.</
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ad
<
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>.e.x.</
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>
ſicut
<
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>.s.x.</
var
>
ad vnitatem; </
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<
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xml:id
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xml:space
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preserve
">ſed ſicut ſe habet
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ad.
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vnitatem, ita ſe habet pariter
<
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>.o.x.</
var
>
ad
<
var
>.m</
var
>
. </
s
>
<
s
xml:id
="
echoid-s288
"
xml:space
="
preserve
">vnde ex .11. prædicta ita ſe habebit
<
var
>.o.
<
lb
/>
x.</
var
>
ad
<
var
>.m.</
var
>
ſicut idipſum
<
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>.o.x.</
var
>
ad
<
var
>.e.x.</
var
>
itaq́ue ex .9. prædicti quinti
<
var
>.m.</
var
>
æqualis erit
<
var
>.o.x</
var
>
.</
s
>
</
p
>
</
div
>
<
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xml:id
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type
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"
level
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3
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n
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31
">
<
head
xml:id
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echoid-head47
"
xml:space
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preserve
">THEOREMA
<
num
value
="
31
">XXXI</
num
>
.</
head
>
<
p
>
<
s
xml:id
="
echoid-s289
"
xml:space
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preserve
">CVR propoſito aliquo numero in duas partes inæquales diuiſo, ſi rurſus per
<
lb
/>
quamlibet ipſarum diuidatur, prouenientia tantumdem coniuncta quantum
<
lb
/>
multiplicata efficiant.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s290
"
xml:space
="
preserve
">Exempli gratia, ſit denarius prop oſitus numerus, per binarium & octonarium
<
lb
/>
diuiſus, prouenientia erunt quinque & vnum cum quarta parte, quæ coniuncta
<
lb
/>
crunt .6. cum quarta parte lineari, quæ ſi mul multiplicata, pariter erunt .6. cum
<
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/>
quarta parte ſuperficiali.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s291
"
xml:space
="
preserve
">Cuius ſpeculationis cauſa, totalis numerns, linea
<
var
>.q.p.</
var
>
ſignificetur, eius duæ
<
lb
/>
partes, per
<
var
>.k.</
var
>
maiorem et
<
var
>.u.</
var
>
minorem, ipſa vnitas per .t: proueniens ex diuiſio-
<
lb
/>
ne
<
var
>.q.p.</
var
>
per
<
var
>.k.</
var
>
ſit
<
var
>.q.i.</
var
>
proueniens autem ipſius
<
var
>.q.p.</
var
>
per
<
var
>.u.</
var
>
ſit
<
var
>.q.f.</
var
>
</
s
>
<
s
xml:id
="
echoid-s292
"
xml:space
="
preserve
">quare ex defini-
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/>
tione diuiſionis ita ſe habebit
<
var
>.q.p.</
var
>
ad
<
var
>.q.i.</
var
>
ſicut
<
var
>.k.</
var
>
ad
<
var
>.t.</
var
>
et
<
var
>.q.p.</
var
>
ad
<
var
>.q.f.</
var
>
ſicut
<
var
>.u.</
var
>
ad
<
var
>.t.</
var
>
<
lb
/>
hoc eſt
<
var
>.q.f.</
var
>
ad
<
var
>.q.p.</
var
>
ſicut
<
var
>.t.</
var
>
ad
<
var
>.u.</
var
>
vnde ex æqualitate
<
reg
norm
="
proportionum
"
type
="
context
">proportionũ</
reg
>
ſic ſe habebit
<
var
>.q.f.</
var
>
<
lb
/>
ad
<
var
>.q.i.</
var
>
ſicut
<
var
>.k.</
var
>
ad
<
var
>.u.</
var
>
et conuerſim. </
s
>
<
s
xml:id
="
echoid-s293
"
xml:space
="
preserve
">Ad hæc in linea
<
var
>.q.p.</
var
>
vnitas, per lineam
<
var
>.q.o.</
var
>
ſigni-
<
lb
/>
ficetur, quo facto, dicamus, ſi
<
var
>.q.p.</
var
>
ad
<
var
>.q.i.</
var
>
ſic ſe habet vt
<
var
>.k.</
var
>
ad
<
var
>.q.o.</
var
>
itaque permu-
<
lb
/>
tando, ſic ſe habebit
<
var
>.q.p.</
var
>
ad
<
var
>.k.</
var
>
ſicut
<
var
>.q.i.</
var
>
ad
<
var
>.q.o.</
var
>
hoc eſt
<
var
>.k.u.</
var
>
ad
<
var
>.k.</
var
>
ſicut
<
var
>.i.q.f.</
var
>
ad
<
var
>.
<
lb
/>
q.f.</
var
>
(nam
<
var
>.k.u.</
var
>
partes ſunt integrales totius
<
var
>.q.p.</
var
>
et
<
var
>.k.u.</
var
>
ad
<
var
>.k.</
var
>
eſt ſicut
<
var
>.i.q.f.</
var
>
ad
<
var
>.q.f.</
var
>
<
lb
/>
ex .18. quinti) </
s
>
<
s
xml:id
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echoid-s294
"
xml:space
="
preserve
">Quare ita erit
<
var
>.i.q.f.</
var
>
ad
<
var
>.q.f.</
var
>
ſicut
<
var
>.q.i.</
var
>
ad vnitatem
<
var
>.q.o.</
var
>
ex .11. quinti
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/>
Addatur deinde
<
var
>.q.i.</
var
>
ad
<
var
>.q.f.</
var
>
et
<
var
>.q.i.</
var
>
per
<
var
>.
<
lb
/>
q.f.</
var
>
multiplicetur, cuius multiplicatio-
<
lb
/>
<
figure
xlink:label
="
fig-0032-01
"
xlink:href
="
fig-0032-01a
"
number
="
43
">
<
image
file
="
0032-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0032-01
"/>
</
figure
>
nis productum, ſit
<
var
>.x.f.</
var
>
quod probabo
<
lb
/>
æquale eſſe ſummæ
<
var
>.f.q.</
var
>
cum
<
var
>.q.i</
var
>
. </
s
>
<
s
xml:id
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echoid-s295
"
xml:space
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preserve
">Sece-
<
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tur enim linea
<
var
>.q.x.</
var
>
in puncto
<
var
>.s.</
var
>
ita. vt
<
var
>.
<
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/>
q.s.</
var
>
æqualis ſit
<
var
>.q.o.</
var
>
ſigneturq́ue pro-
<
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ductum
<
var
>.s.f.</
var
>
</
s
>
<
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xml:id
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xml:space
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preserve
">quare
<
reg
norm
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eadem
"
type
="
context
">eadẽ</
reg
>
erit propor-
<
lb
/>
tio quantitatis
<
var
>.x.f.</
var
>
ad
<
var
>.s.f.</
var
>
quæ eſt
<
var
>.q.x.</
var
>
<
lb
/>
ad
<
var
>.q.s.</
var
>
ex prima ſexti, aut .18. vel 19.
<
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/>
ſeptimi, hoc eſt, ſicut
<
var
>.q.i.</
var
>
ad
<
var
>.q.o.</
var
>
et
<
lb
/>
ex .11. quinti (vt dictum eſt) ſicut
<
var
>.i.q.
<
lb
/>
f.</
var
>
ad
<
var
>.q.f.</
var
>
ſed numerus
<
var
>.s.f.</
var
>
fuperficia-
<
lb
/>
lis tantus eſt, quantus linearis
<
var
>.q.f</
var
>
.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s297
"
xml:space
="
preserve
">quare ex .9. quinti tantus erit (ſu-
<
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/>
perficialiter) numerus
<
var
>.x.f.</
var
>
quantus
<
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/>
(lineariter).
<
var
>f.q.i.</
var
>
quod erat pro-
<
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/>
poſitum.</
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>
</
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>
</
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<
div
xml:id
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type
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level
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n
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<
head
xml:id
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echoid-head48
"
xml:space
="
preserve
">THEOREMA.
<
num
value
="
32
">XXXII</
num
>
.</
head
>
<
p
>
<
s
xml:id
="
echoid-s298
"
xml:space
="
preserve
">CVR numero aliquo in duas partes inæquales diuiſo, ſi rurſus diuidatur per
<
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/>
ſingulas partes, ſumma duorum prouenientium per binarium, ſemper ma-
<
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/>
ior ſit ſumma prouenientium ex diuiſione vnius partis per alteram.</
s
>
</
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>
<
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>
<
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xml:space
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<
reg
norm
="
Exempli
"
type
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context
">Exẽpli</
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>
gratia, ſi proponeretur numerus .24. qui in duas partes inæquales diuide </
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>
</
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