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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56
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0056
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0056
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<
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xml:space
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">THEOREMA
<
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68
">LXVIII</
num
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.</
head
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<
s
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xml:space
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">CVR numero per numerum diuiſo,
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norm
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productoque
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type
="
simple
">productoq́;</
reg
>
duorum numerorum per pro-
<
lb
/>
ueniens multiplicato, quod vltimò productum eſt, diuiſi numeri ſemper qua
<
lb
/>
dratum exiſtat.</
s
>
</
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<
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<
s
xml:id
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xml:space
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">Exempli gratia, ſi diuidamus .10. per .2. proueniens erit .5. quo producto ex duo
<
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bus numeris multiplicato, nempe .20. habe
<
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bimus .100. quadratum numeri diuiſi.</
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<
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<
s
xml:id
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xml:space
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preserve
">Cuius gratia duo numeri ſint
<
var
>.a.</
var
>
et
<
var
>.e.</
var
>
por
<
lb
/>
rò
<
var
>.a.</
var
>
per
<
var
>.e.</
var
>
diuiſo detur
<
var
>.u.</
var
>
tum
<
var
>.o.</
var
>
produ-
<
lb
/>
ctum
<
var
>.a.</
var
>
in
<
var
>.e.</
var
>
eſſe conſtituatur, quo per
<
var
>.u.</
var
>
<
lb
/>
multiplicato dabitur
<
var
>.x.</
var
>
quadratum
<
var
>.a.</
var
>
pro-
<
lb
/>
ptereà quòd
<
var
>.a.</
var
>
medium eſt proportionale
<
lb
/>
inter
<
var
>.o.</
var
>
et
<
var
>.u.</
var
>
ex .35. theoremate. </
s
>
<
s
xml:id
="
echoid-s585
"
xml:space
="
preserve
">itaque
<
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/>
ex .16. ſexti aut .20. ſeptimi, propoſiti veri-
<
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/>
tas eluceſcet.</
s
>
</
p
>
</
div
>
<
div
xml:id
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type
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math:theorem
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level
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n
="
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">
<
head
xml:id
="
echoid-head85
"
xml:space
="
preserve
">THEOREMA
<
num
value
="
69
">LXIX</
num
>
.</
head
>
<
p
>
<
s
xml:id
="
echoid-s586
"
xml:space
="
preserve
">CVR numero aliquo per duos alios multiplicato & diuiſo, ſi per horum duo-
<
lb
/>
rum productum, ſumma duorum primorum productorum diuiſa fuerit, vl-
<
lb
/>
timum proueniens, ſummæ duorum primorum prouenientium æquale ſit.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Exempli gratia, proponitur numerus .24. per .8. et .6. multiplicandus & diuiden
<
lb
/>
dus ſumma productorum crit .336. prouenientium autem .7. ſi igitur ſummam .336.
<
lb
/>
productorum per productum duorum ſecundorum numerorum nempe .48. diuiſe-
<
lb
/>
rimus, proueniens pariter erit .7.</
s
>
</
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<
p
>
<
s
xml:id
="
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"
xml:space
="
preserve
">In cuius
<
reg
norm
="
gratiam
"
type
="
context
">gratiã</
reg
>
primus numerus ſignificetur linea
<
var
>.q.b.</
var
>
multiplicandus & diuiden-
<
lb
/>
dus numeris deſignatis per
<
var
>.k.m.</
var
>
et
<
var
>.y.m.</
var
>
productorum ſumma ſit
<
var
>.k.z.</
var
>
prouenien-
<
lb
/>
tium autem
<
var
>.a.e</
var
>
: et
<
var
>.a.o.</
var
>
ex
<
var
>.k.m.</
var
>
et
<
var
>.o.e.</
var
>
ex
<
var
>.y.m</
var
>
: tum productum
<
var
>.k.m.</
var
>
in
<
var
>.m.y.</
var
>
ſit
<
var
>.f.
<
lb
/>
m</
var
>
. </
s
>
<
s
xml:id
="
echoid-s589
"
xml:space
="
preserve
">Dico quòd ſi
<
var
>.k.z.</
var
>
per
<
var
>.f.m.</
var
>
diuiſerimus proueni et
<
var
>.a.e</
var
>
. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Quod cum ſic fuerit, erit
<
lb
/>
quoque verum quòd diuiſa
<
var
>.k.z.</
var
>
per
<
var
>.a.e.</
var
>
proueniet
<
var
>.f.m.</
var
>
numerus ſcilicet æqualis
<
lb
/>
numero
<
var
>.f.m.</
var
>
ex .13. theoremate huius. </
s
>
<
s
xml:id
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"
xml:space
="
preserve
">Itaque quotieſcunque probauero quòd di-
<
lb
/>
uiſa
<
var
>.k.z.</
var
>
per
<
var
>.a.e.</
var
>
proueniat numerus æqualis ipſi
<
var
>.f.m.</
var
>
propoſitum verum eſſe con
<
lb
/>
ſequetur. ex .13. theoremate. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Quòd ſi proueniens ex diuiſione
<
var
>.k.z.</
var
>
per
<
var
>.a.e.</
var
>
æqua
<
lb
/>
le fuerit
<
var
>.f.m.</
var
>
patet ex .7. quinti quòd
<
reg
norm
="
eadem
"
type
="
context
">eadẽ</
reg
>
erit proportio numeri
<
var
>.k.m.y.</
var
>
ad ipſum
<
lb
/>
proueniens, quæ ad numerum
<
var
>.f.m</
var
>
. </
s
>
<
s
xml:id
="
echoid-s593
"
xml:space
="
preserve
">Cogitemus
<
reg
norm
="
itaque
"
type
="
simple
">itaq;</
reg
>
<
var
>.k.u.</
var
>
æqualem
<
var
>.a.e.</
var
>
ſuper quam
<
lb
/>
mente concipiamus rectangulum
<
var
>.u.p.</
var
>
æqualem
<
var
>.k.z.</
var
>
ex quo eadem erit proportio
<
var
>.
<
lb
/>
k.p.</
var
>
ad
<
var
>.k.y.</
var
>
quæ
<
var
>.g.k.</
var
>
ad
<
var
>.k.u.</
var
>
ex .15. ſexti, aut, 20. ſeptimi, numerus autem
<
var
>.k.p.</
var
>
erit
<
lb
/>
proueniens, quod probandum eſt æquale eſſe
<
var
>.f.m</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s594
"
xml:space
="
preserve
">Probabitur autem ſic, ex .9. quinti, nempe demonſtrato quòd numerus
<
var
>.k.p.</
var
>
ean
<
lb
/>
dem proportionem habeat ad numerum
<
var
>.k.y.</
var
>
quam habet numerus
<
var
>.f.m.</
var
>
ad eundem
<
lb
/>
<
var
>k.y</
var
>
. </
s
>
<
s
xml:id
="
echoid-s595
"
xml:space
="
preserve
">Sed probatum eſt ſic ſe habere
<
var
>.k.g.</
var
>
ad
<
var
>.k.u.</
var
>
ſicut
<
var
>.k.p.</
var
>
ad
<
var
>.k.y.</
var
>
ſufficiet igitur pro-
<
lb
/>
bare ſic ſe habere
<
var
>.k.g.</
var
>
ad
<
var
>.k.u.</
var
>
ſicut
<
var
>.f.m.</
var
>
ad
<
var
>.k.y</
var
>
. </
s
>
<
s
xml:id
="
echoid-s596
"
xml:space
="
preserve
">Sed
<
var
>.k.g.</
var
>
dicitur æqualis eſſe
<
var
>.q.b</
var
>
: et
<
var
>.k.</
var
>
<
lb
/>
u; </
s
>
<
s
xml:id
="
echoid-s597
"
xml:space
="
preserve
">a.e. ſatis erit igitur probare ita ſe habere
<
var
>.q.b.</
var
>
ad
<
var
>.a.e.</
var
>
ſicut
<
var
>.f.m.</
var
>
ad
<
var
>.k.y</
var
>
. </
s
>
<
s
xml:id
="
echoid-s598
"
xml:space
="
preserve
">Scimus au-
<
lb
/>
tem quòd eadem eſt proportio
<
var
>.q.b.</
var
>
ad
<
var
>.a.o.</
var
>
quæ
<
var
>.m.k.</
var
>
ad vnitatem, quæ ſit
<
var
>.x.</
var
>
& quod
<
lb
/>
proportio
<
var
>.o.e.</
var
>
ad
<
var
>.q.b.</
var
>
eadem eſt, quæ
<
var
>.x.</
var
>
ad
<
var
>.m.y.</
var
>
ex definitione diuiſionis. </
s
>
<
s
xml:id
="
echoid-s599
"
xml:space
="
preserve
">Quare
<
lb
/>
ex æqualitate proportionum eadem erit proportio
<
var
>.k.m.</
var
>
ad
<
var
>.m.y.</
var
>
quæ
<
var
>.e.o.</
var
>
ad
<
var
>.o.a.</
var
>
& </
s
>
</
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>
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