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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43
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28
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IO. BAPT. BENED.
"
n
="
40
"
file
="
0040
"
xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0040
"/>
quadrato dimidij, prout ex ſpeculatione huiuſmodi operis cognoſcetur,
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eſt
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differentia
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type
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context
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>
inter
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norm
="
ſummam
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type
="
context
">ſummã</
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>
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reg
norm
="
quadratorum
"
type
="
context
">quadratorũ</
reg
>
<
reg
norm
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duorum
"
type
="
context
">duorũ</
reg
>
qui
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reg
norm
="
quæruntur
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type
="
context
">quærũtur</
reg
>
<
reg
norm
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numerorum
"
type
="
context
">numerorũ</
reg
>
, ſimul
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reg
norm
="
cum
"
type
="
context
">cũ</
reg
>
pro
<
lb
/>
ducto
<
reg
norm
="
eorum
"
type
="
context
">eorũ</
reg
>
radicum. </
s
>
<
s
xml:id
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xml:space
="
preserve
">Dimidium numeri .20. in ſeipſum multiplicandum eſſet, qua-
<
lb
/>
<
reg
norm
="
dratumque
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type
="
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>
detrahendum ex .208. vtremanerent .108. quorum .108. tertiæ partis qua
<
lb
/>
drata radix eſſet .6. quæ ſi iuncta fuerit dimidio .20. nempe .10. daretur maior nu-
<
lb
/>
merus quæſitus .16. quo detracto è .20. darentur .4.</
s
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<
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<
s
xml:id
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xml:space
="
preserve
">Cuius ſpeculationis cauſa, datus primus numerus ſignificetur linea
<
var
>.g.h.</
var
>
in qua
<
lb
/>
maior numerus incognitus ſit
<
var
>.g.h.</
var
>
minor verò
<
var
>.b.h.</
var
>
quorum quadrata ſint
<
var
>.y.t.</
var
>
et
<
var
>.
<
lb
/>
b.l.</
var
>
in quadrato maximo
<
var
>.g.p.</
var
>
tum productum
<
var
>.g.b.</
var
>
in
<
var
>.b.h.</
var
>
ſit
<
var
>.g.c.</
var
>
<
reg
norm
="
cogitenturque
"
type
="
simple
">cogitenturq́;</
reg
>
duo
<
lb
/>
diametri
<
var
>.q.h.</
var
>
et
<
var
>.g.p.</
var
>
diuiſi per medium in puncto
<
var
>.o.</
var
>
per quod duę lineæ ducan-
<
lb
/>
tur
<
var
>.f.d.</
var
>
et
<
var
>.k.m.</
var
>
parallelæ lateribus maximi quadrati. </
s
>
<
s
xml:id
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xml:space
="
preserve
">Hæ dictum quadratum in
<
lb
/>
quatuor quadrata æqualia diuident, quorum
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norm
="
vnumquodque
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type
="
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">vnumquodq́;</
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>
, æquale erit quadrato
<
var
>.
<
lb
/>
g.f.</
var
>
dimidij ipſius
<
var
>.g.h.</
var
>
datę, </
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>
<
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xml:id
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xml:space
="
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">quare eorum
<
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vnumquodque
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type
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>
cognitum erit. </
s
>
<
s
xml:id
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xml:space
="
preserve
">Iterum co
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/>
gitemus
<
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>.s.x.</
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>
per
<
var
>.e.</
var
>
<
reg
norm
="
parallelam
"
type
="
context
">parallelã</
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>
<
var
>.g.k.</
var
>
tantum diſtan-
<
lb
/>
tem à
<
var
>.g.k.</
var
>
quantum
<
var
>.y.l.</
var
>
ab
<
var
>.g.h.</
var
>
diſtare inueni-
<
lb
/>
<
figure
xlink:label
="
fig-0040-01
"
xlink:href
="
fig-0040-01a
"
number
="
55
">
<
image
file
="
0040-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0040-01
"/>
</
figure
>
tur. </
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>
<
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xml:space
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preserve
">Cogitetur pariter
<
var
>.z.i.a.</
var
>
per punctum
<
var
>.i.</
var
>
<
lb
/>
parallela
<
var
>.d.p.</
var
>
</
s
>
<
s
xml:id
="
echoid-s382
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xml:space
="
preserve
">quare
<
var
>.a.t.</
var
>
æqualis erit
<
var
>.f.c.</
var
>
et
<
var
>.y.x.</
var
>
<
lb
/>
æqualis
<
var
>.f.e.</
var
>
et
<
var
>.y.s</
var
>
:
<
var
>b.l.</
var
>
æqualis. </
s
>
<
s
xml:id
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xml:space
="
preserve
">Ita ſubtractis è
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lb
/>
duobus quadratis ſuperius dictis
<
var
>.a.t.y.x.</
var
>
et
<
var
>.b.l.</
var
>
<
lb
/>
producto
<
var
>.y.b.</
var
>
æqualibus, ſupererunt
<
var
>.k.d.</
var
>
et
<
var
>.a.c.
<
lb
/>
x.</
var
>
cognita, tanquam æqualia dato ſecundo nu-
<
lb
/>
mero, ſed
<
var
>.k.d.</
var
>
quadratum eſt medietatis
<
var
>.g.f.</
var
>
<
lb
/>
cognitæ, cognoſcetur igitur reſiduum
<
var
>.a.c.x.</
var
>
vnà
<
lb
/>
etiam ſingulæ tertiæ partes nempe quadrata
<
var
>.o.
<
lb
/>
i.o.c.</
var
>
et
<
var
>.o.e.</
var
>
& radix
<
var
>.b.f.</
var
>
vel
<
var
>.f.s.</
var
>
ſingularum,
<
lb
/>
qua coniuncta dimidio
<
var
>.g.f.</
var
>
<
reg
norm
="
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type
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">rurfusq́;</
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>
ab
<
reg
norm
="
eodem
"
type
="
context
">eodẽ</
reg
>
de-
<
lb
/>
tracta, propoſitum conſequemur.</
s
>
</
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</
div
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xml:id
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="
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<
head
xml:id
="
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xml:space
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preserve
">THEOREMA
<
num
value
="
44
">XLIIII</
num
>
.</
head
>
<
p
>
<
s
xml:id
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xml:space
="
preserve
">CVR ſi quis cupiat numerum propoſitum in duas eiuſmodi partes diuidere, vt
<
lb
/>
quadratum maioris, quadratum minoris ſuperet quantitate alterius numeri
<
lb
/>
propoſiti, rectè primum numerum in ſeipſum multiplicabit, & ab eodem ſecun-
<
lb
/>
dum numerum detrahet, reſiduum verò per duplum primi diuidet, ex quo proue-
<
lb
/>
niens primi pars minor erit, quæ ex illo primo detracta, partem maiorem
<
lb
/>
proferet.</
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>
</
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>
<
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>
<
s
xml:id
="
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"
xml:space
="
preserve
">Exempli gratia, ſi proponantur .20. diuiſa in duas eiuſmodi partes, vt
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quadratum
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type
="
context
">quadratũ</
reg
>
<
lb
/>
maioris ſuperet quadratum minoris numero æquali ipſi .240. oportebit primum
<
lb
/>
numerum, qui quadratus cum fuerit, erit .400. in ſeipſum multiplicare, & ex hoc
<
lb
/>
quadrato ſecundum numerum nempe .240. detrahere, </
s
>
<
s
xml:id
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"
xml:space
="
preserve
">tunc remanebunt .160. quę
<
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/>
diuiſa per .40.
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numerum
"
type
="
context
">numerũ</
reg
>
<
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="
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"
type
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primo, dabuntur quatuor pro minori numero, à reſi-
<
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/>
duo verò .20. detractis quatuor, erunt .16. pro maiorinumero.</
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>
</
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>
<
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>
<
s
xml:id
="
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"
xml:space
="
preserve
">Quod vt exactè conſideremus, primus numerus propoſitus ſignificetur linea
<
var
>.q.
<
lb
/>
h.</
var
>
diuidendus in duas partes
<
var
>.q.p.</
var
>
et
<
var
>.p.h.</
var
>
tales quales quærimus. </
s
>
<
s
xml:id
="
echoid-s388
"
xml:space
="
preserve
">Poſtmodum eriga
<
lb
/>
<
gap
extent
="
2
"/>
r quadratum
<
var
>.q.e.</
var
>
diuiſum diametro
<
var
>.f.h.</
var
>
<
reg
norm
="
ductisque
"
type
="
simple
">ductisq́;</
reg
>
<
var
>.p.o.t.</
var
>
et
<
var
>.a.o.c.</
var
>
parallelis lateri-
<
lb
/>
bus quadrati, dabuntur imaginaria quadrata
<
var
>.c.t.</
var
>
et
<
var
>.p.a.</
var
>
duarum partium
<
var
>.q.p.</
var
>
et
<
var
>.p.
<
lb
/>
h.</
var
>
incognitarum. </
s
>
<
s
xml:id
="
echoid-s389
"
xml:space
="
preserve
">Ad hæc cogitemus quadratum
<
var
>.u.n.</
var
>
æquale quadrato
<
var
>.p.a.</
var
>
è quadra </
s
>
</
p
>
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