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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EPISTOL AE.
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0353
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0353
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æquales. </
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xml:space
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et
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ab iiſdem punctis
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ad aliud punctum,
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quod volueris ipſius lineæ
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quas probabo
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(ſimul ſumptas) eſſe priori-
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bus. </
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<
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xml:space
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et
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>.q.r.a.</
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punctis
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r.</
var
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ad
<
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<
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norm
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type
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simple
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ſit linea
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in puncto
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>.x.</
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ita quod
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var
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æqualis ſit ipſi
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>.b.a.</
var
>
quod
<
lb
/>
nulli dubium erit poſſe effici, cum
<
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>.o.b.</
var
>
<
reg
norm
="
longiot
"
type
="
context
">lõgiot</
reg
>
ſit
<
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>.b.a.</
var
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co quod opponatur angulo ob-
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lb
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tuſo ipſius trianguli
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>.b.a.o.</
var
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quę
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ſimiliter protrahatur vſque ad
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ita quod
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æqualis ſit
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>
</
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<
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xml:id
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xml:space
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>.o.i.</
var
>
quouſque
<
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>.i.h.</
var
>
æqualis ſit
<
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>.a.i</
var
>
. </
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<
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xml:id
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xml:space
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preserve
">In alia parte po-
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/>
ſtea idem faciendum eſt ſecando
<
var
>.a.r.</
var
>
in puncto
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var
>.u.</
var
>
ita quod
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>.u.r.</
var
>
æqualis ſit
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var
>.r.o.</
var
>
efficien
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lb
/>
do
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var
>.r.s.</
var
>
æqualem
<
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>.r.u.</
var
>
et
<
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>.q.p.</
var
>
æquale
<
var
>.q.o.</
var
>
vnde habebimus
<
reg
norm
="
productum
"
type
="
context
">productũ</
reg
>
<
var
>.o.d.</
var
>
in
<
var
>.o.x.</
var
>
æqua
<
lb
/>
le producto
<
var
>.o.h.</
var
>
in
<
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>.o.a.</
var
>
& productum
<
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>.a.s.</
var
>
in
<
var
>.a.u.</
var
>
æquale producto
<
var
>.a.p.</
var
>
in
<
var
>.a.o.</
var
>
exiſtis
<
lb
/>
rationibus. </
s
>
<
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xml:id
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xml:space
="
preserve
">Nam cum quadratum ipſius
<
var
>.o.b.</
var
>
æquale ſit duobus quadratis
<
var
>.o.i.</
var
>
et
<
var
>.i.
<
lb
/>
b.</
var
>
ex penultima primi Eucli. ipſa quadrata
<
var
>.o.i.</
var
>
et
<
var
>.i.b.</
var
>
æqualia erunt producto
<
var
>.o.d.</
var
>
in
<
lb
/>
<
var
>o.x.</
var
>
ſimul ſumpto cum quadrato
<
var
>.b.x.</
var
>
ex .6. ſecundi, hoc eſt ipſi producto ſimul ſum-
<
lb
/>
pto cum quadrato
<
var
>.b.a.</
var
>
hoc eſt ipſi producto ſimul ſumpto cum duobus quadratis
<
var
>.a.
<
lb
/>
i.</
var
>
et
<
var
>.i.b.</
var
>
ſed quia productum
<
var
>.o.h.</
var
>
in
<
var
>.o.a.</
var
>
ſimul ſumpto cum quadrato
<
var
>.a.i.</
var
>
ęquatur qua
<
lb
/>
drato
<
var
>.o.i.</
var
>
ideo productum
<
var
>.o.h.</
var
>
in
<
var
>.o.a.</
var
>
ſimul ſumptum cum quadrato
<
var
>.a.i.</
var
>
& cum qua-
<
lb
/>
drato
<
var
>.i.b.</
var
>
æquale erit producto
<
var
>.o.d.</
var
>
in
<
var
>.o.x.</
var
>
ſimul ſumpto
<
reg
norm
="
cum
"
type
="
context
">cũ</
reg
>
duobus quadratis dictis
<
lb
/>
hoc eſt ipſius
<
var
>.a.i.</
var
>
et
<
var
>.i.b.</
var
>
quę quadrata dempta cum fuerint ab vtraque parte, tunc cer
<
lb
/>
ti erimus producta eſſe inuicem æqualia. </
s
>
<
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xml:id
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xml:space
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s
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<
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xml:id
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xml:space
="
preserve
">Nunc
<
lb
/>
imaginemur protractam eſſc
<
var
>.a.e.</
var
>
parallelam ipſi
<
var
>.o.b.</
var
>
& habebimus proportionem
<
lb
/>
ipſius
<
var
>.a.b.</
var
>
ad
<
var
>.a.i.</
var
>
maiorem eſſe ea quæ eſt ipſius
<
var
>.a.e.</
var
>
ad eandem
<
var
>.a.i.</
var
>
cum
<
var
>.a.b.</
var
>
maior
<
lb
/>
ſit ipſa
<
var
>.a.e.</
var
>
vt oppoſita angulo obtuſo, quapropter proportio
<
var
>.x.b.</
var
>
ad
<
var
>.a.i.</
var
>
maior erit
<
lb
/>
ea quæ eſt
<
var
>.o.b.</
var
>
ad
<
var
>.o.i</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4144
"
xml:space
="
preserve
">Iam enim ſcis proportionem
<
var
>.o.b.</
var
>
ad
<
var
>.o.i.</
var
>
eſſe, vt
<
var
>.a.e.</
var
>
ad
<
var
>.a.i.</
var
>
ex
<
lb
/>
ſimilitudine triangulorum. </
s
>
<
s
xml:id
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"
xml:space
="
preserve
">quare proportio
<
var
>.b.d.</
var
>
ad
<
var
>.i.h.</
var
>
maior erit proportione
<
var
>.o.b.</
var
>
<
lb
/>
ad
<
var
>.o.i.</
var
>
<
reg
norm
="
tunc
"
type
="
context
">tũc</
reg
>
ex .27. quinti
<
reg
norm
="
permutando
"
type
="
simple context
">ꝑmutãdo</
reg
>
<
reg
norm
="
proportio
"
type
="
simple
">ꝓportio</
reg
>
<
var
>.b.d.</
var
>
ad
<
var
>.b.o.</
var
>
maior erit proportione
<
var
>.i.h.</
var
>
<
lb
/>
ad
<
var
>.i.o.</
var
>
& ex .26.
<
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eiuſdem
"
type
="
context
">eiuſdẽ</
reg
>
<
reg
norm
="
componendo
"
type
="
context context
">cõponẽdo</
reg
>
maior
<
reg
norm
="
proportio
"
type
="
simple
">ꝓportio</
reg
>
erit
<
var
>.o.d.</
var
>
ad
<
var
>.o.b.</
var
>
ea quę eſt
<
var
>.o.h.</
var
>
ad. o
<
lb
/>
i. &
<
reg
norm
="
permutando
"
type
="
context
">permutãdo</
reg
>
maior ipſius
<
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>.o.d.</
var
>
ad
<
var
>.o.h.</
var
>
ea quæ
<
var
>.o.b.</
var
>
ad
<
var
>.o.i.</
var
>
& ex .33. maior ipſius
<
var
>.b.
<
lb
/>
d.</
var
>
ad
<
var
>.i.h.</
var
>
ea quæ
<
var
>.o.d.</
var
>
ad
<
var
>.o.h</
var
>
. </
s
>
<
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xml:id
="
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"
xml:space
="
preserve
">Sed vt
<
var
>.b.a.</
var
>
ad
<
var
>.a.i.</
var
>
ita eſt
<
var
>.a.r.</
var
>
ad
<
var
>.a.q.</
var
>
ex ſimilitudine
<
reg
norm
="
triam
"
type
="
context
">triã</
reg
>
<
lb
/>
gulorum. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Erit igitur
<
var
>.a.r.</
var
>
ad
<
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>.a.q.</
var
>
maior proportio, ea quæ eſt
<
var
>.o.b.</
var
>
ad
<
var
>.o.i.</
var
>
& exijſdem
<
lb
/>
ſupradictis rationibus maior erit proportio ipſius
<
var
>.s.a.</
var
>
ad
<
var
>.p.a.</
var
>
ea quæ eſt
<
var
>.a.r.</
var
>
ad
<
var
>.a.q.</
var
>
<
lb
/>
ſed cum iam probatum fuit proportio
<
lb
/>
<
figure
xlink:label
="
fig-0353-01
"
xlink:href
="
fig-0353-01a
"
number
="
384
">
<
image
file
="
0353-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0353-01
"/>
</
figure
>
nem
<
var
>.b.d.</
var
>
ad
<
var
>.i.h.</
var
>
hoc eſt
<
var
>.a.b.</
var
>
ad
<
var
>.a.i.</
var
>
ma
<
lb
/>
iorem eſſe
<
var
>.o.d.</
var
>
ad
<
var
>.o.h.</
var
>
ergo eo ma-
<
lb
/>
gis maior erit proportio ipſius
<
var
>.a.s.</
var
>
ad
<
lb
/>
<
var
>a.p.</
var
>
ca quæ
<
var
>.o.d.</
var
>
ad
<
var
>.o.h.</
var
>
ſed cum ex .15
<
lb
/>
ſexti, eadem ſit proportio
<
var
>.o.d.</
var
>
ad
<
var
>.o.a.</
var
>
<
lb
/>
quæ
<
var
>.o.h.</
var
>
ad
<
var
>.o.x.</
var
>
et
<
var
>.s.a.</
var
>
ad
<
var
>.o.a.</
var
>
quę
<
var
>a.p.</
var
>
<
lb
/>
ad
<
var
>.a.u.</
var
>
</
s
>
<
s
xml:id
="
echoid-s4148
"
xml:space
="
preserve
">tunc erit
<
reg
norm
="
permutando
"
type
="
context
">permutãdo</
reg
>
eadem
<
lb
/>
proportio ipſius
<
var
>.o.d.</
var
>
ad
<
var
>.o.h.</
var
>
quæ
<
var
>.o.a.</
var
>
<
lb
/>
ad
<
var
>.o.x.</
var
>
& ipſius
<
var
>.a.o.</
var
>
ad
<
var
>.a.u.</
var
>
quemad-
<
lb
/>
modum ipſius
<
var
>.a.s.</
var
>
ad
<
var
>.a.p</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4149
"
xml:space
="
preserve
">Quare maior proportio erit ipſius
<
var
>.a.o.</
var
>
ad
<
var
>.a.u.</
var
>
quam
<
var
>.a.</
var
>
o
<
unsure
/>
.
<
lb
/>
ad
<
var
>.o.x</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4150
"
xml:space
="
preserve
">Vnde ſequitur
<
var
>.o.x.</
var
>
maiorem eſſe
<
var
>.a.u.</
var
>
ex .8. quinti, ergo
<
var
>.b.x.o.r.</
var
>
longior erit
<
lb
/>
ipſa
<
var
>.b.a.u.r</
var
>
. </
s
>
<
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xml:id
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xml:space
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preserve
">Quod eſt propoſitum.</
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>
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