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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19
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rhead
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THEOREM. ARIT.
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n
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31
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file
="
0031
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0031
"/>
bit quadratum
<
var
>.e.d.</
var
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cognitum, cuius radix æqualis erit
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>.c.t.</
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qua coniuncta dimi-
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dio
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ex quinta ſecundi Eucli. dabit quod propoſitum erat.</
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<
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xml:space
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">THEOREMA
<
num
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29
">XXIX</
num
>
.</
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xml:space
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<
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style
="
sc
">QVid</
emph
>
cauſæ eſt, cur ſubtracto duplo producti duorum numerorum ad inui-
<
lb
/>
cem
<
reg
norm
="
multiplicatorum
"
type
="
context
">multiplicatorũ</
reg
>
ex ſumma ſuorum quadratorum, ſemper quod ſuper
<
lb
/>
eſt duorum numerorum quadratum differentiæ ſit?</
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>
</
p
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<
p
>
<
s
xml:id
="
echoid-s277
"
xml:space
="
preserve
">Exempli gratia ſi proponerentur duo numeri .16. et .4. duplum producti eorum
<
lb
/>
eſſet .128. quò detracto ex ſumma ſuorum quadratorum, nempè ex .272. rema-
<
lb
/>
neret .144. cuius quadrati radix eſſet .12. tanquam differentia inter .4. et .16.</
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</
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<
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<
s
xml:id
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echoid-s278
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xml:space
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preserve
">Id vtſciamus, duo numeri propoſiti, duabus lineis ſignificentur, maiore
<
var
>.q.g.</
var
>
<
lb
/>
et minore
<
var
>.g.p.</
var
>
directè coniunctis, ſuper quas, totale quadratum extruatur
<
var
>.a.p.</
var
>
<
lb
/>
in quo cogitetur diameter
<
var
>.a.p.</
var
>
et à puncto
<
var
>.g.</
var
>
ducatur parallela
<
var
>.g.n.c.</
var
>
et à pun-
<
lb
/>
cto
<
var
>.n.</
var
>
parallela
<
var
>.n.s.r.</
var
>
ex quo duo producta
<
reg
norm
="
dabuntur
"
type
="
context
">dabũtur</
reg
>
<
var
>.q.n.</
var
>
et
<
var
>.n.u.</
var
>
ſingula æqualia pro-
<
lb
/>
ducto
<
var
>.q.g.</
var
>
in
<
var
>g.p.</
var
>
et
<
var
>.a.n.</
var
>
et
<
var
>.n.p.</
var
>
duo quadrata dictorum numerorum propoſi-
<
lb
/>
torum, quod ſatis
<
reg
norm
="
ſuperque
"
type
="
simple
">ſuperq́</
reg
>
, probatur quarta ſecundi Eucli. </
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>
<
s
xml:id
="
echoid-s279
"
xml:space
="
preserve
">Cogitemus deinde
<
var
>.n.
<
lb
/>
o.</
var
>
æqualem
<
var
>.n.p.</
var
>
et à puncto
<
var
>.o.</
var
>
ducatur
<
var
>.o.m.t.</
var
>
parallela
<
var
>.r.s.</
var
>
et
<
var
>.o.e.</
var
>
ad
<
var
>.n.
<
lb
/>
c</
var
>
. </
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>
<
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xml:id
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echoid-s280
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xml:space
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preserve
">quare ex allatis ab Eucli. octaua ſecundi, dabi-
<
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/>
tur quantitas
<
var
>.m.n.</
var
>
æqualis
<
var
>.q.n.</
var
>
producto
<
var
>.q.g.</
var
>
in
<
lb
/>
<
figure
xlink:label
="
fig-0031-01
"
xlink:href
="
fig-0031-01a
"
number
="
41
">
<
image
file
="
0031-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0031-01
"/>
</
figure
>
<
var
>g.p.</
var
>
et quantitas
<
var
>.o.c.</
var
>
minor ipſo producto, ex
<
lb
/>
quantitate quadrati
<
var
>.n.p.</
var
>
ex quo quantitas
<
var
>.m.n.e.</
var
>
<
lb
/>
vna cum quadrato
<
var
>.n.p.</
var
>
æqualis erit duplo produ-
<
lb
/>
cti
<
var
>.q.g.</
var
>
in
<
var
>.g.p.</
var
>
ſed hæ duæ quantitates, ſunt par-
<
lb
/>
tes duorum quadratorum dictorum, & quæ ſuper
<
lb
/>
eſt
<
var
>.m.e.</
var
>
quadratum differentiæ vnius numeri pro-
<
lb
/>
poſiti ab altero, prout in ſubſcripta figura licebit cui
<
lb
/>
libet conſiderare. </
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>
<
s
xml:id
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xml:space
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preserve
">Itaque veritas hæc manifeſta
<
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erit.</
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</
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<
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n
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<
head
xml:id
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xml:space
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">THEOREMA
<
num
value
="
30
">XXX</
num
>
.</
head
>
<
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>
<
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xml:id
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<
emph
style
="
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">CVr</
emph
>
ij qui ex duobus numeris propoſitis maiorem per minorem diuidunt, ſi
<
lb
/>
proueniens per maiorem numerum multiplicauerint, productum æquale
<
lb
/>
erit prouenienti ex diuiſione quadrati maioris numeri per minorem?</
s
>
</
p
>
<
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>
<
s
xml:id
="
echoid-s283
"
xml:space
="
preserve
">Exempli gratia ſi proponantur duo numeri .20. et .4.
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type
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>
.20. per .4. diui-
<
lb
/>
datur, dabit quinque, tum .400. quadrato .20. diuiſo per prioré .4. dabit .100.
<
lb
/>
quod proueniens, producto ex .20. in .5. primo prouenienti adæquatur.</
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>
</
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<
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>
<
s
xml:id
="
echoid-s284
"
xml:space
="
preserve
">Cuius ſpeculationis cauſa, ſint duo numeri, qui lineis
<
var
>.x.u.</
var
>
et
<
var
>.x.s.</
var
>
maiore
<
reg
norm
="
atque
"
type
="
simple
">atq;</
reg
>
mi-
<
lb
/>
nore ſignificétur, tum
<
var
>.u.x.</
var
>
numerus per
<
var
>.s.x.</
var
>
di-
<
lb
/>
uidatur, ſitq́ue proueniens
<
var
>.x.n.</
var
>
poſtmodum qua-
<
lb
/>
<
figure
xlink:label
="
fig-0031-02
"
xlink:href
="
fig-0031-02a
"
number
="
42
">
<
image
file
="
0031-02
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0031-02
"/>
</
figure
>
dratum
<
var
>.u.x.</
var
>
ſit
<
var
>.x.o.</
var
>
et productum ex
<
var
>.n.x.</
var
>
in
<
var
>.u.
<
lb
/>
x.</
var
>
ſit
<
var
>.x.e.</
var
>
quod æquale eſſe dico prouenienti ex
<
lb
/>
diuiſione quadrati
<
var
>.o.x.</
var
>
per
<
var
>.s.x.</
var
>
quod ſit
<
var
>.m</
var
>
. </
s
>
<
s
xml:id
="
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"
xml:space
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preserve
">Patet
<
lb
/>
enim ex definitione diuiſionis, talem futuram pro-
<
lb
/>
portionem
<
var
>.u.x.</
var
>
ad
<
var
>.n.x.</
var
>
qualis eſt
<
var
>.s.x.</
var
>
ad vnitatem,
<
lb
/>
& quadratum
<
var
>.o.x.</
var
>
ad rectangulum
<
var
>.e.x.</
var
>
ita ſe ha- </
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>
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