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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130
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<
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<
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<
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88
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rhead
="
IO. BAPT. BENED.
"
n
="
100
"
file
="
0100
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0100
"/>
ducto .2. g. in
<
var
>.d.p.</
var
>
ex .20. ſeptimi, proptereà quòd proportio
<
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>.q.o.</
var
>
ad
<
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>.o.p.</
var
>
hoc eſt ad
<
var
>.
<
lb
/>
d.p.</
var
>
eſt vt
<
var
>.a.g.</
var
>
ad
<
var
>.g.n.</
var
>
coniunctim cum diſiunctim it a ſit
<
var
>.q.p.</
var
>
ad
<
var
>.p.o.</
var
>
vt
<
var
>.a.n.</
var
>
ad
<
var
>.n.g.</
var
>
<
lb
/>
<
reg
norm
="
permutando
"
type
="
context
">permutãdo</
reg
>
eo quòd
<
var
>.q.p.</
var
>
ad
<
var
>.a.n.</
var
>
(ideſt ad
<
var
>.e.c.</
var
>
) ita ſe
<
reg
norm
="
hent
"
type
="
context
">hẽt</
reg
>
ut
<
var
>.p.o.</
var
>
(hoc eſt
<
var
>.d.p.</
var
>
) ad
<
var
>.n.g.</
var
>
<
lb
/>
ex
<
reg
norm
="
conditionibus
"
type
="
context
">cõditionibus</
reg
>
armonicæ proportio nalitatis. </
s
>
<
s
xml:id
="
echoid-s1144
"
xml:space
="
preserve
">Deinde ſi detraxerimus
<
var
>.n.g.</
var
>
ex
<
var
>.a.g.</
var
>
<
lb
/>
remanebit
<
var
>.e.c.</
var
>
minor terminus.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s1145
"
xml:space
="
preserve
">Sed ſi
<
var
>.e.c.</
var
>
tertius terminus nobis propoſitus eſſet ſimul cum
<
var
>.a.g.</
var
>
medio, & volue
<
lb
/>
rimus maiorem inuenire
<
var
>.q.p.</
var
>
ſcilicet, oportebit
<
var
>.e.c.</
var
>
ex
<
var
>.a.g.</
var
>
detrahere, differentiam
<
lb
/>
verò
<
var
>.n.g.</
var
>
ſimiliter demeremus
<
lb
/>
ex
<
var
>.e.c.</
var
>
unde remaneret nobis
<
var
>.e.t.</
var
>
<
lb
/>
<
figure
xlink:label
="
fig-0100-01
"
xlink:href
="
fig-0100-01a
"
number
="
136
">
<
image
file
="
0100-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0100-01
"/>
</
figure
>
cognitum, quo reſiduo
<
var
>.c.t.</
var
>
me-
<
lb
/>
diante diuidemus productum,
<
reg
norm
="
quod
"
type
="
simple
">ꝙ</
reg
>
<
lb
/>
furgit ex
<
var
>.a.g.</
var
>
in
<
var
>.t.c.</
var
>
& prouentus
<
var
>.
<
lb
/>
d.p.</
var
>
erit differentia maior, eo
<
reg
norm
="
quod
"
type
="
simple
">ꝙ</
reg
>
<
lb
/>
<
reg
norm
="
productum
"
type
="
context
">productũ</
reg
>
quod ſit ex
<
var
>.e.t.</
var
>
in
<
var
>.d.p.</
var
>
<
lb
/>
æquale eſt producto quòd fit ex
<
var
>.a.g.</
var
>
in
<
var
>.t.c.</
var
>
per 20. ſeptimi Eucli. eo quòd
<
var
>.a.g.</
var
>
(id-
<
lb
/>
eſt
<
var
>.q.d.</
var
>
) ad
<
var
>.d.p.</
var
>
eſt ut
<
var
>.e.t.</
var
>
ad
<
var
>.t.c.</
var
>
diſiunctim, cum coniunctim ita ſit
<
var
>.q.p.</
var
>
ad
<
var
>.d.p.</
var
>
vt
<
var
>.e.
<
lb
/>
c.</
var
>
ad
<
var
>.t.c.</
var
>
permutando, quia
<
var
>.q.p.</
var
>
ad
<
var
>.e.c.</
var
>
eſt vt
<
var
>.d.p.</
var
>
ad
<
var
>.t.c.</
var
>
hoc eſt ad
<
var
>.n.g.</
var
>
ex legibus
<
lb
/>
dictis.</
s
>
</
p
>
</
div
>
<
div
xml:id
="
echoid-div250
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type
="
math:theorem
"
level
="
3
"
n
="
131
">
<
head
xml:id
="
echoid-head149
"
xml:space
="
preserve
">THEOREMA
<
num
value
="
131
">CXXXI</
num
>
.</
head
>
<
p
>
<
s
xml:id
="
echoid-s1146
"
xml:space
="
preserve
">ALIA etiam methodo hoc perfici poſſe comperi. </
s
>
<
s
xml:id
="
echoid-s1147
"
xml:space
="
preserve
">Propoſiti enim cum nobis fue
<
lb
/>
rint duo termini
<
var
>.c.e.</
var
>
minimus et
<
var
>.g.a.</
var
>
medius, maximus verò quærendus ſit, de
<
lb
/>
trahatur differentia
<
var
>.g.n.</
var
>
ex
<
var
>.e.c.</
var
>
& per reſiduum
<
var
>.e.t.</
var
>
diuidatur productum
<
reg
norm
="
quod
"
type
="
simple
">ꝙ</
reg
>
fit ex
<
var
>.a.
<
lb
/>
g.</
var
>
in
<
var
>.e.c.</
var
>
prouentus quæ erit
<
var
>.q.p.</
var
>
terminus quæſitus.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s1148
"
xml:space
="
preserve
">Pro cuius ratione, ponamus in eſſe terminum
<
var
>.q.p.</
var
>
</
s
>
<
s
xml:id
="
echoid-s1149
"
xml:space
="
preserve
">tunc ex forma huius proportio
<
lb
/>
nalitatis nulli dubium erit quin
<
var
>.q.p.</
var
>
ad
<
var
>.e.c.</
var
>
fit vt
<
var
>.d.p.</
var
>
ad
<
var
>.n.g.</
var
>
hoc eft ad
<
var
>.t.c.</
var
>
vnde ex
<
lb
/>
19. quinti vel .12. ſeptimi ita eſſet
<
var
>.q.d.</
var
>
ad
<
var
>.e.t.</
var
>
vt
<
var
>.q.p.</
var
>
ad
<
var
>.e.c.</
var
>
</
s
>
<
s
xml:id
="
echoid-s1150
"
xml:space
="
preserve
">quare ex .20. @cptimi pro
<
lb
/>
ductum
<
reg
norm
="
quod
"
type
="
simple
">ꝙ</
reg
>
naſcitur ex
<
var
>.p.d.</
var
>
(hoc eſt
<
var
>.a.g.</
var
>
) in
<
var
>.e.c.</
var
>
æquale eric producto
<
var
>.e.t.</
var
>
in
<
var
>.q.p.</
var
>
qua-
<
lb
/>
propter ſi diuiſerimus id per
<
var
>.e.t.</
var
>
proueniet nobis
<
var
>.q.p</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s1151
"
xml:space
="
preserve
">Sed
<
reg
norm
="
cum
"
type
="
context
">cũ</
reg
>
nobis propoſiti fuerint duo termini
<
var
>.q.p.</
var
>
maximus, et
<
var
>.a.g.</
var
>
medius, ſi
<
reg
norm
="
mini- mum
"
type
="
context
">mini-
<
lb
/>
mũ</
reg
>
<
var
>.e.c.</
var
>
<
reg
norm
="
voluerimus
"
type
="
simple
">voluerimꝰ</
reg
>
inuenire. </
s
>
<
s
xml:id
="
echoid-s1152
"
xml:space
="
preserve
">Termino
<
var
>.q.p.</
var
>
maximo,
<
reg
norm
="
iungatur
"
type
="
context simple
">iũgat̃</
reg
>
.
<
var
>p.o.</
var
>
ęqualis,
<
var
>p.d.</
var
>
<
reg
norm
="
differentię
"
type
="
context
">differẽtię</
reg
>
<
lb
/>
propoſitæ, diuidatur poſtea productum
<
reg
norm
="
quod
"
type
="
simple
">ꝙ</
reg
>
ex
<
var
>.q.p.</
var
>
in
<
var
>.a.g.</
var
>
generatur per
<
var
>.q.o.</
var
>
prouen
<
lb
/>
tus autem ſit
<
var
>.e.c.</
var
>
qui quidem erit terminus quæſitus.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s1153
"
xml:space
="
preserve
">Cuius operationis ſpeculutio hæc erit, ſupponatur terminum
<
var
>.e.c.</
var
>
inuentum eſſe
<
lb
/>
vnde
<
var
>.n.g.</
var
>
differentia ſit inter
<
var
>.e.c.</
var
>
<
lb
/>
et
<
var
>.a.g.</
var
>
ex forma igitur armonicæ
<
lb
/>
<
figure
xlink:label
="
fig-0100-02
"
xlink:href
="
fig-0100-02a
"
number
="
137
">
<
image
file
="
0100-02
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0100-02
"/>
</
figure
>
proportionalitis ita erit
<
var
>.q.p.</
var
>
ad
<
var
>.a.
<
lb
/>
n.</
var
>
vt
<
var
>.p.o.</
var
>
ad
<
var
>.n.g.</
var
>
vnde ex .13. quin-
<
lb
/>
ti. </
s
>
<
s
xml:id
="
echoid-s1154
"
xml:space
="
preserve
">Ita erit
<
var
>.q.o.</
var
>
ad
<
var
>.a.g.</
var
>
vt
<
var
>.q.p.</
var
>
ad
<
var
>.a.
<
lb
/>
n.</
var
>
ergo
<
reg
norm
="
productum
"
type
="
context
">productũ</
reg
>
quòd fit ex
<
var
>.a.g.</
var
>
<
lb
/>
in
<
var
>.q.p.</
var
>
(ex .20. ſeptimi) æquale erit
<
lb
/>
producto
<
var
>.q.o.</
var
>
in
<
var
>.a.n</
var
>
. </
s
>
<
s
xml:id
="
echoid-s1155
"
xml:space
="
preserve
">Quare ſi diuiſum fuerit tale productum per
<
var
>.q.o.</
var
>
proueniet no-
<
lb
/>
bis
<
var
>.e.c.</
var
>
quòd querebamus.</
s
>
</
p
>
</
div
>
</
div
>
</
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>
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