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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<
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58
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rhead
="
IO. BAPT. BENED.
"
n
="
70
"
file
="
0070
"
xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0070
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<
p
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<
s
xml:id
="
echoid-s763
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xml:space
="
preserve
">Exempli gratia, caſu ſeſe offerunt hi quatuor numeri .8. 5. 3. 2. multiplicato .8.
<
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/>
per .5. & hoc .40. per .3. rurſus hoc .120. per .2. vltimum productum eſſet .240. æqua
<
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/>
le producto .15. (quod ex .5. in .3. oritur) in productum .16. quod ex .8. in .2. pro-
<
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/>
fertur.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s764
"
xml:space
="
preserve
">Cuius ſpeculationis gratia, cogitemus quatuor numeros quatuor lineis
<
var
>.a.e.i.o.</
var
>
<
lb
/>
ſignifi cari, productum autem
<
var
>.e.</
var
>
in
<
var
>.i.</
var
>
eſſe
<
var
>.m.f.</
var
>
et
<
var
>.r.s.</
var
>
ſimiliter & productum
<
var
>.a.</
var
>
in
<
var
>.o.</
var
>
eſ-
<
lb
/>
ſe
<
var
>.m.z</
var
>
: et
<
var
>.z.f.</
var
>
productum eſſe
<
var
>.m.f.</
var
>
in
<
var
>.m.z.</
var
>
cui productum
<
var
>.a.</
var
>
in
<
var
>.e.</
var
>
multiplicatum per
<
lb
/>
i. & hoc tandem per
<
var
>.o.</
var
>
æquari debet.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s765
"
xml:space
="
preserve
">Sit itaque
<
var
>.u.y.</
var
>
productum
<
var
>.a.</
var
>
in
<
var
>.e.</
var
>
quod
<
var
>.u.y.</
var
>
per
<
var
>.i.</
var
>
multiplicatum proferat
<
var
>.u.s.</
var
>
<
lb
/>
hocq́ue
<
var
>.u.s.</
var
>
multiplicatum per
<
var
>.o</
var
>
. </
s
>
<
s
xml:id
="
echoid-s766
"
xml:space
="
preserve
">Dico quod dabit numerum æqualem numero
<
var
>.f.z.</
var
>
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/>
Quamobrem
<
var
>.r.s.</
var
>
aut
<
var
>.m.f.</
var
>
quod idem eſt, in figura præcedentis theore matis ſigni-
<
lb
/>
ficetur linea
<
var
>.n.u.</
var
>
& linea
<
var
>.r.u.</
var
>
hu-
<
lb
/>
ius, nempe
<
var
>.a.</
var
>
ſignificetur per
<
var
>.u.t.</
var
>
<
lb
/>
<
figure
xlink:label
="
fig-0070-01
"
xlink:href
="
fig-0070-01a
"
number
="
98
">
<
image
file
="
0070-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0070-01
"/>
</
figure
>
præcedentis, ex quo numerus pro
<
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/>
ducti
<
var
>.u.s.</
var
>
præſentis, in præcedenti
<
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/>
ſignificabitur producto
<
var
>.n.t.</
var
>
quod
<
lb
/>
<
reg
norm
="
productum
"
type
="
simple context
">ꝓductũ</
reg
>
<
var
>.u.s.</
var
>
<
reg
norm
="
pręsens
"
type
="
context
">pręsẽs</
reg
>
<
reg
norm
="
per
"
type
="
simple
">ꝑ</
reg
>
<
reg
norm
="
præsens
"
type
="
context
">præsẽs</
reg
>
<
var
>.o.</
var
>
mul
<
lb
/>
tiplicatum, quod erat in præceden
<
lb
/>
ti
<
var
>.u.c.</
var
>
ſignificabitur per
<
var
>.d.u.</
var
>
præce
<
lb
/>
dentis, quod non modo ex multi-
<
lb
/>
plicatione
<
var
>.n.t.</
var
>
præcedentis, nempe
<
var
>.u.s.</
var
>
præſentis. in
<
var
>.u.c.</
var
>
præcedentis æquali
<
var
>.o.</
var
>
præ-
<
lb
/>
ſentis oritur, ſed etiam ex
<
var
>.c.t.</
var
>
præcedentis æquali
<
var
>.m.z.</
var
>
præſentis in
<
var
>.n.u.</
var
>
præceden
<
lb
/>
tis æquali
<
var
>.m.f.</
var
>
præſentis. </
s
>
<
s
xml:id
="
echoid-s767
"
xml:space
="
preserve
">Itaque verum eſt propoſitum.</
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>
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</
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<
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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
="
90
">
<
head
xml:id
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xml:space
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preserve
">THEOREMA
<
num
value
="
90
">XC</
num
>
.</
head
>
<
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>
<
s
xml:id
="
echoid-s768
"
xml:space
="
preserve
">CVR quibuſlibet & quantiſuis numeris in ſummam collectis, ſi ab vnitate in ſe-
<
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/>
cunda ſpecie progreſſionis arithmeticę imparium numerorum progreſſi fue-
<
lb
/>
rimus, eiuſmodi ſumma ſemper eſt quadratus numerus.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s769
"
xml:space
="
preserve
">Exempli gratia, ſi horum quatuor diſparium numerorum
<
reg
norm
="
ſummam
"
type
="
context
">ſummã</
reg
>
, in dicta pro-
<
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/>
greſſione arithmetica quis ſumat, principio ab vnitate ſumpto, nempe .1. 3. 5. 7. ſum-
<
lb
/>
ma erit .16. numerus quadratus inquam. </
s
>
<
s
xml:id
="
echoid-s770
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xml:space
="
preserve
">Idem de cæteris.</
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>
</
p
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<
p
>
<
s
xml:id
="
echoid-s771
"
xml:space
="
preserve
">Quamobrem animaduertendum eſt, vnitatem, tam ſumi pro ſui ipſius radicem,
<
lb
/>
quam pro quadrato, cubo, cenſo cenſi, primo relato, & alia quauis dignitate.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s772
"
xml:space
="
preserve
">Nunc autem pro quadrato ſumamus per
<
var
>.o.</
var
>
ſignificato,
<
reg
norm
="
cogitemusque
"
type
="
simple
">cogitemusq́</
reg
>
quadratum
<
var
>.o.</
var
>
<
lb
/>
includi quadrato vnitatem ſequenti, quod, vt patet, eſt quatuor vnitatum, ac pro-
<
lb
/>
priè primum quadratum numerorum, ex quo etiam nomen accepit, vnde ex ſimi-
<
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/>
litudine quam cætera quadrata cum hoc primo retinent, ex quaternario denomina-
<
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tionem acceperunt. </
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>
<
s
xml:id
="
echoid-s773
"
xml:space
="
preserve
">
<
reg
norm
="
Hocitaque
"
type
="
simple
">Hocitaq;</
reg
>
ſit
<
var
>.o.u.c.e.</
var
>
ita ex communi ſcientia quadrato
<
var
>.o.</
var
>
iun-
<
lb
/>
gitur gnomon
<
var
>.e.c.u.</
var
>
conſtans tribus vnitatibus, quare primus gnomon, numero im-
<
lb
/>
pari conſtat. </
s
>
<
s
xml:id
="
echoid-s774
"
xml:space
="
preserve
">Scimus etiam ex additione numeri binarij ad imparem, numeris di-
<
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/>
ſparibus ſummam excreſcere, cum propius accedere
<
reg
norm
="
quam
"
type
="
context
">quã</
reg
>
binario nequeant, ex quo
<
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/>
medio binario, ſibi inuicem ſuccedunt. </
s
>
<
s
xml:id
="
echoid-s775
"
xml:space
="
preserve
">Dico igitur quòd quinario ternarium ſub
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ſequente, coniuncto quadrato
<
var
>.o.u.c.e.</
var
>
profertur quadratum, quod in numeris, bi-
<
lb
/>
narij quadratum ſequitur,
<
reg
norm
="
eritque
"
type
="
simple
">eritq́;</
reg
>
ternarij,
<
reg
norm
="
quodque
"
type
="
simple
">quodq́;</
reg
>
ſignificetur per
<
var
>.o.f.</
var
>
patet enim pri
<
lb
/>
mo non differre ab
<
var
>.o.c.</
var
>
præter quam gnomone
<
var
>.b.f.d.</
var
>
qui coniungitur quadrato
<
var
>.o.
<
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/>
c.</
var
>
quique duabus vnitatibus maior eſt
<
var
>.e.c.u</
var
>
. </
s
>
<
s
xml:id
="
echoid-s776
"
xml:space
="
preserve
">
<
reg
norm
="
Iam
"
type
="
context
">Iã</
reg
>
ſcimus gnomonem
<
var
>.e.o.u.</
var
>
æqualem </
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>
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