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. BABPT. BENED.
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0400
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conuerſa proportionalitate in .19. quinti, cum
<
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>.a.d.</
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ad
<
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>.d.o.</
var
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iam probatum fuit (vbi
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B.) ita eſſe ut
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>.A.</
var
>
ad
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.</
s
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</
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<
p
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<
s
xml:id
="
echoid-s4600
"
xml:space
="
preserve
">Sed in principio huius ſpeculationis probatum iam fuit ita eſſe ipſius
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ad
<
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>.d.e.</
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vt ipſius
<
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>.D.</
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ad
<
var
>.B.</
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vbi notatum eſt
<
var
>.M</
var
>
. </
s
>
<
s
xml:id
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echoid-s4601
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xml:space
="
preserve
">quare ex .23. quinti, Archimedes verum dicit,
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qu od
<
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>.d.o.</
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>
ad
<
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>.d.e.</
var
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erit vt
<
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>.D.</
var
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ad
<
var
>.A</
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>
.</
s
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</
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<
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<
s
xml:id
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xml:space
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">Sed cum
<
var
>.d.o.</
var
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ad
<
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>.d.e.</
var
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ſe habeat ut
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>.D.</
var
>
ad
<
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>.A.</
var
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erit ex conuerſa proportionalitate iam
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dicta
<
var
>.d.e.</
var
>
ad
<
var
>.d.o.</
var
>
vt
<
var
>.A.</
var
>
ad
<
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>.D.</
var
>
per euerſam vero erit
<
var
>.d.e.</
var
>
ad
<
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>.a.o.</
var
>
vt
<
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>.A.</
var
>
ad ſuum reſi-
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">λ</
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duum. </
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>
<
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xml:space
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">quod reſiduum componitur ex ſimplo
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>.b.c.</
var
>
cum triplo
<
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>.b.</
var
>
cum duplo
<
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>.b.o.</
var
>
quod
<
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/>
à te ipſo videre poteris detrahendo numeros ipſarum quantitatum quæ in
<
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>.D.</
var
>
<
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/>
reperiuntur, ex numeris earundem, quæ in
<
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>.A.</
var
>
quod quidem reſiduum ſigniſicetur
<
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/>
à charactere
<
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>.E</
var
>
. </
s
>
<
s
xml:id
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xml:space
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">Vnde ex conuerſa proportionalitate verum dicit Archime. hoc eſt
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quod ita ſe hab ebit
<
var
>.o.e.</
var
>
ad
<
var
>.d.e.</
var
>
vt
<
var
>.E.</
var
>
ad
<
var
>.A</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
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xml:space
="
preserve
">Cum autem ſit
<
var
>.a.b.</
var
>
ad
<
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>.c.b.</
var
>
vt
<
var
>.c.b.</
var
>
ad
<
var
>.d.b.</
var
>
& ita
<
var
>.d.b.</
var
>
ad
<
var
>.e.b.</
var
>
ex ſuppoſito, ideo ex
<
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17. quinti verum dicit Archim. </
s
>
<
s
xml:id
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xml:space
="
preserve
">hoc eſt quod ita erit ipſius
<
var
>.d.e.</
var
>
ad
<
var
>.e.b.</
var
>
vt
<
var
>.a.c.</
var
>
ad
<
var
>.c.b.</
var
>
<
lb
/>
& vt
<
var
>.c.d.</
var
>
ad
<
var
>.d.b.</
var
>
& ex .13. eiuſdem eadem proportio erit tripli ipſius
<
var
>.c.d.</
var
>
ad triplum
<
lb
/>
ipſius
<
var
>.d.b.</
var
>
quæ dupli ipſius
<
var
>.d.e.</
var
>
ad. duplum ipſius
<
var
>.e.b.</
var
>
vt inquit Archi.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4607
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xml:space
="
preserve
">Ex qua .13. compoſitum ex
<
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>.a.c.</
var
>
cum triplo ipſius
<
var
>.c.d.</
var
>
cum duplo ipſius
<
var
>.d.e.</
var
>
ean-
<
lb
/>
dem proportionem habebit ad
<
reg
norm
="
compoſitum
"
type
="
context
">compoſitũ</
reg
>
ipſius
<
var
>.c.b.</
var
>
cum triplo
<
reg
norm
="
ipſius
"
type
="
simple
">ipſiꝰ</
reg
>
<
var
>.d.b.</
var
>
cum duplo
<
lb
/>
ipſius
<
var
>.e.b.</
var
>
quam ipſius
<
var
>.d.e.</
var
>
ad
<
var
>.e.b</
var
>
. </
s
>
<
s
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="
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xml:space
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preserve
">Sed horum compoſitorum primum ſignificetur
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per
<
var
>.H.</
var
>
ſecundum verò ſignificatum fuit per
<
var
>.E.</
var
>
vnde
<
var
>.H.</
var
>
ad
<
var
>.E.</
var
>
ſe habebit vt
<
var
>.d.e.</
var
>
ad
<
lb
/>
<
var
>e.b.</
var
>
ſed
<
var
>.E.</
var
>
ad
<
var
>.A.</
var
>
iam dictum eſt eſſe vt
<
var
>.o.e.</
var
>
ad
<
var
>.d.e.</
var
>
vbi ſignatum eſt
<
var
>.λ</
var
>
. </
s
>
<
s
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="
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xml:space
="
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">quare ex .23.
<
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quinti eadem proportio erit ipſius
<
var
>.o.e.</
var
>
ad
<
var
>.e.b.</
var
>
quæ
<
var
>.H.</
var
>
ad
<
var
>.A.</
var
>
vt ipſe inquit.</
s
>
</
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<
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<
s
xml:id
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xml:space
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">Ex .18. poſtea eiuſdem ita erit
<
var
>.o.b.</
var
>
ad
<
var
>.e.b.</
var
>
vt
<
var
>.H.A.</
var
>
ad
<
var
>.A</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
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xml:space
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">Notandum etiam eſt quod ſi collectæ fuerint omnes partes compoſiti
<
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>.H.A.</
var
>
hoc
<
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/>
eſt duplum
<
var
>.a.b.</
var
>
cum duplo
<
var
>.b.e.</
var
>
cum quadruplo
<
var
>.b.c.</
var
>
cum quadruplo
<
var
>.b.d.</
var
>
cum ſimplo
<
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/>
<
var
>a.c.</
var
>
cum triplo
<
var
>.c.d.</
var
>
cum duplo
<
var
>.d.e.</
var
>
habebitur triplum
<
var
>.a.b.</
var
>
triplum
<
var
>.b.d.</
var
>
& ſexcuplum
<
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/>
<
var
>b.c.</
var
>
vt ipſe dixit. </
s
>
<
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xml:space
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">Quod autem hoc verum ſit, cum diſtinctæ fuerint omnes partes,
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vt in ſubſcriptis his lineis videre eſt, videbis quod ſi ex
<
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>.H.</
var
>
detracta fuerit ſimplex
<
var
>.a.
<
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/>
c.</
var
>
quæ quidem poſtea iuncta vni ex partibus quadrupli
<
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>.b.c.</
var
>
ipſius
<
var
>.A.</
var
>
reſultabit nobis
<
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/>
vna inte gra
<
var
>.a.b</
var
>
. </
s
>
<
s
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="
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"
xml:space
="
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">Vnde habebimus triplum ipſius
<
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>.a.b.</
var
>
& in
<
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>.A.</
var
>
remanebit triplum ip
<
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/>
ſius
<
var
>.c.b</
var
>
. </
s
>
<
s
xml:id
="
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xml:space
="
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">Deinde ſi ex
<
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>.H.</
var
>
auferatur triplum ipſius
<
var
>.c.d.</
var
>
& ipſum addatur tribus parti-
<
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/>
bus quadrupli
<
var
>.b.d.</
var
>
ipſius
<
var
>.A.</
var
>
habebimus tres vices
<
var
>.b.c.</
var
>
quæ ſi iungantur tribus, quæ
<
lb
/>
remanebant in
<
var
>.A.</
var
>
vt dixi, habebimus ſexcuplum ipſius
<
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>.b.c.</
var
>
& in
<
var
>.A.</
var
>
remanebit ſim
<
lb
/>
plum
<
var
>.b.d.</
var
>
cum duplo ipſius
<
var
>.b.e</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4615
"
xml:space
="
preserve
">Vnde ſi ex
<
var
>.H.</
var
>
demptum fuerit duplum ipſius
<
var
>.d.
<
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/>
e.</
var
>
quod quidem iungatur cum duplo ipſius
<
var
>.b.e.</
var
>
habebimus duplum ipſius
<
var
>.b.d.</
var
>
quod
<
lb
/>
coniunctum cum ſimplo
<
var
>.b.d.</
var
>
quod in
<
var
>.A.</
var
>
relictum fuerat, habebimus triplum ipſius
<
lb
/>
<
var
>d.b</
var
>
. </
s
>
<
s
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"
xml:space
="
preserve
">Verum igitur eſt quod inquit Archimedes, hoc eſt, quod
<
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>.H.A.</
var
>
eſt triplum ip-
<
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/>
ſius
<
var
>.a.b.</
var
>
ſexcuplum ipſius
<
var
>.b.c.</
var
>
& triplum ipſius
<
var
>.b.d</
var
>
.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4617
"
xml:space
="
preserve
">Verum etiam dicit ex eo (vt ſupra probatum eſt) quod
<
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>.a.c</
var
>
:
<
var
>c.d</
var
>
: et
<
var
>.d.e.</
var
>
ſe
<
reg
norm
="
habebant
"
type
="
context
">habebãt</
reg
>
<
lb
/>
in continua proportionalitate, </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">quare ex conuerſa proportionalitate erunt ſibi inui-
<
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/>
cem continuæ proportionales.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s4619
"
xml:space
="
preserve
">Nunc autem cum
<
var
>.a.c</
var
>
:
<
var
>c.d.</
var
>
et
<
var
>.d.e.</
var
>
ſint continuæ proportionales in ea proportione
<
lb
/>
in qua ſunt
<
var
>.a.b</
var
>
:
<
var
>c.b</
var
>
:
<
var
>d.b</
var
>
: et
<
var
>.e.b.</
var
>
vt in principio diximus, erit ex .22. quinti
<
var
>.a.c.</
var
>
ad
<
var
>.d.
<
lb
/>
e.</
var
>
vt
<
var
>.a.b.</
var
>
ad
<
var
>.d.b.</
var
>
& ſic etiam
<
var
>.c.b.</
var
>
ad
<
var
>.e.b</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4620
"
xml:space
="
preserve
">Vnde ex .24. eiuſdem
<
var
>.a.d.</
var
>
ad
<
var
>.d.e.</
var
>
erit vt
<
var
>.a.
<
lb
/>
b.</
var
>
cum
<
var
>.b.e.</
var
>
ad
<
var
>.d.b.</
var
>
& vt
<
var
>.c.b.</
var
>
cum
<
var
>.b.d.</
var
>
ad
<
var
>.e.b.</
var
>
& ex .13. dicti vt
<
var
>.a.b.</
var
>
cum
<
var
>.b.e.</
var
>
bis
<
lb
/>
ſumpto, & cum
<
var
>.b.d.</
var
>
ad
<
var
>.e.b</
var
>
. </
s
>
<
s
xml:id
="
echoid-s4621
"
xml:space
="
preserve
">Quare ex conuerſa proportionalitate, vt ſe habet
<
var
>.e.d.</
var
>
<
lb
/>
ad
<
var
>.d.a.</
var
>
ita ſe habebit
<
var
>.e.b.</
var
>
<
reg
norm
="
cum
"
type
="
context
">cũ</
reg
>
<
var
>.d.b.</
var
>
ad
<
var
>d.b.</
var
>
<
reg
norm
="
cum
"
type
="
context
">cũ</
reg
>
<
var
>.b.c.</
var
>
duplicato &
<
reg
norm
="
cum
"
type
="
context
">cũ</
reg
>
<
var
>.b.a.</
var
>
vt inquit Archi
<
lb
/>
medes. </
s
>
<
s
xml:id
="
echoid-s4622
"
xml:space
="
preserve
">Nunc antecedens vocetur
<
var
>.M.</
var
>
hoc eſt
<
var
>.b.e.</
var
>
cum
<
var
>.d.b.</
var
>
conſequens verò, hoc </
s
>
</
p
>
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