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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0060
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0060
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<
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<
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xml:space
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">THEOREMA
<
num
value
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73
">LXXIII</
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>
.</
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>
<
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>
<
s
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xml:space
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preserve
">HOC etiam problema à me inuentum eſt, nempe ſi duæ radices quadratæ in
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ſummam collectæ fuerint, & ex dimidio eiuſmodi ſummæ detracta fuerit mi
<
lb
/>
nor radix,
<
reg
norm
="
reſiduique
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type
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>
quadratum duplicatum
<
reg
norm
="
eique
"
type
="
simple
">eiq́;</
reg
>
ſummæ coniungatur du-
<
lb
/>
plum producti ipſius reſidui in dimidium ſummæ radicum, atque huic ſummæ du-
<
lb
/>
plum producti eiuſdem reſidui in radicem minorem coniunctum fuerit; </
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>
<
s
xml:id
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"
xml:space
="
preserve
">vltima hæc
<
lb
/>
ſumma differentia erit duorum quadratorum propoſitorum.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s632
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xml:space
="
preserve
">Exempli gratia duæ radices quadraræ ſint .5. et .11. harum ſumma erit .16. & dimi
<
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/>
dium .8. differentia minoris ab ipſo dimidio erit .3: duplum quadrati huius differen
<
lb
/>
tiæ erit .18: </
s
>
<
s
xml:id
="
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xml:space
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preserve
">duplum producti huius differentię in dimidium ſummę radicum erit .48.
<
lb
/>
item & huius differentiæ duplum in minorem radicem erit .30. quarum omnium
<
lb
/>
ſumma erit .96. tantaq́ue erit differentia ſuorum quadratorum, quorum vnum
<
lb
/>
erit .25. alterum verò .121.</
s
>
</
p
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<
p
>
<
s
xml:id
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xml:space
="
preserve
">Pro cuius rei ſcientia, duæ quadratæ radices ſint
<
var
>.h.o.</
var
>
et
<
var
>.o.d.</
var
>
directæ inter ſe con-
<
lb
/>
iunctæ, quæ ſumma per medium in puncto
<
var
>.e.</
var
>
diuidatur, tum cogitetur
<
var
>.e.b.</
var
>
æqualis
<
lb
/>
<
var
>o.e.</
var
>
perpendicularis
<
var
>.h.d.</
var
>
<
reg
norm
="
ducanturque
"
type
="
simple
">ducanturq́;</
reg
>
lineæ
<
var
>.b.h</
var
>
:
<
var
>b.o.</
var
>
et
<
var
>.b.d</
var
>
. </
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>
<
s
xml:id
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xml:space
="
preserve
">Iam ex .4. primi
<
var
>.b.h.</
var
>
æqua
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lis erit
<
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>.b.d.</
var
>
& quadratum
<
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>.b.h.</
var
>
æquale quadrato
<
var
>.h.o.</
var
>
& quadrato
<
var
>.o.b.</
var
>
ſimul cum du
<
lb
/>
plo producti
<
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>.o.e.</
var
>
in
<
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>.o.h.</
var
>
ex .12. ſecundi Eucli. </
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>
<
s
xml:id
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xml:space
="
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">Sed ex .13.
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reg
norm
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eiuſdem
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>
quadratum
<
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>.b.d.</
var
>
<
lb
/>
minus eſt quadrato
<
var
>.o.d.</
var
>
cum quadrato
<
var
>.o.b.</
var
>
ex duplo producti
<
var
>.o.e.</
var
>
in
<
var
>.o.d.</
var
>
at duplum
<
lb
/>
eiuſmodi producti æquale eſt duplo qua-
<
lb
/>
drati
<
var
>.o.e.</
var
>
& duplo producti
<
var
>.o.e.</
var
>
in
<
var
>.e.d.</
var
>
ex
<
lb
/>
<
figure
xlink:label
="
fig-0060-01
"
xlink:href
="
fig-0060-01a
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number
="
82
">
<
image
file
="
0060-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0060-01
"/>
</
figure
>
tertia eiuſdem, itaque duo quadrata ſcili-
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/>
cet
<
var
>.o.b.</
var
>
et
<
var
>.o.d.</
var
>
maiora erunt duobus qua-
<
lb
/>
dratis, nempe
<
var
>.o.b.</
var
>
et
<
var
>.o.h.</
var
>
collectis cum du
<
lb
/>
plo producti
<
var
>.o.e.</
var
>
in
<
var
>.o.h.</
var
>
ex duplo quadrati
<
lb
/>
<
var
>o.e.</
var
>
vna
<
reg
norm
="
cum
"
type
="
context
">cũ</
reg
>
duplo producti
<
var
>.o.e.</
var
>
in
<
var
>.e.d</
var
>
. </
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>
<
s
xml:id
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xml:space
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preserve
">Qua
<
lb
/>
re
<
reg
norm
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differentia
"
type
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">differẽtia</
reg
>
ſummæ duorum quadratorum
<
lb
/>
<
var
>o.b.</
var
>
et
<
var
>.o.d.</
var
>
à ſumma duorum
<
var
>o.b.</
var
>
et
<
var
>.o.h.</
var
>
du
<
lb
/>
plum erit quadrati
<
var
>.o.e.</
var
>
cum duplo produ-
<
lb
/>
cti
<
var
>.o.e.</
var
>
in
<
var
>.e.d.</
var
>
& duplo producti
<
var
>.o.e.</
var
>
in
<
var
>.o.h.</
var
>
<
lb
/>
Quòd ſi ex ſingulis duabus ſummis quadratorum demptum fuerit quadratum
<
var
>.o.b.</
var
>
<
lb
/>
eadem producta & quadrata ipſius
<
var
>.o.e.</
var
>
remanebunt, tanquam differentia duorum
<
lb
/>
quadratorum
<
var
>.o.u.</
var
>
et
<
var
>.h.c</
var
>
.</
s
>
</
p
>
</
div
>
<
div
xml:id
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type
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"
level
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3
"
n
="
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">
<
head
xml:id
="
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"
xml:space
="
preserve
">THEOREMA
<
num
value
="
74
">LXXIIII</
num
>
.</
head
>
<
p
>
<
s
xml:id
="
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"
xml:space
="
preserve
">CVR ſumma duorum
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extremorum
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type
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">extremorũ</
reg
>
quatuor terminorum
<
reg
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"
type
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">proportionaliũ</
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>
arith-
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meticè, æqualis eſt ſummæ duorum mediorum, vbi nota hac in re neceſſa-
<
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/>
rium non eſſe proportionalitatem continuam exiſtere.</
s
>
</
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>
<
p
>
<
s
xml:id
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xml:space
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preserve
">Exempli gratia, ſi darentur hi quatuor termini .20. 17. 9. 6. quorum proportio ea
<
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dem eſſet primi ad ſecundum quæ tertij ad quartum, ſumma primi cum quarto eſſet
<
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/>
26.
<
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ſecundi cum tertio.</
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>
</
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<
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>
<
s
xml:id
="
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"
xml:space
="
preserve
">Cuius ſpeculationis cauſa, primus
<
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type
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>
numerus ſignificetur linea
<
var
>.e.o.</
var
>
ſecun-
<
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/>
dus
<
var
>.s.q.</
var
>
tertius
<
var
>.u.c.</
var
>
quartus
<
var
>.g.t.</
var
>
differentia porrò inter
<
var
>.e.o.</
var
>
et
<
var
>.s.q.</
var
>
ſit
<
var
>.i.o.</
var
>
quæ æqualis
<
lb
/>
erit differentiæ
<
var
>.r.c.</
var
>
qua quartus à tertio ſuperatur ex hypotheſi. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Itaque aſſero ſum
<
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/>
mam
<
var
>.e.o.</
var
>
cum
<
var
>.g.t.</
var
>
nempe
<
var
>.a.o.</
var
>
æqualem eſſe ſummę
<
var
>.q.s.</
var
>
et
<
var
>.u.c.</
var
>
<
reg
norm
="
ſitque
"
type
="
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">ſitq́;</
reg
>
<
var
>.q.p</
var
>
. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Nam in
<
var
>.a.o.</
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>
</
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>
</
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>
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