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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n
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24
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file
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0024
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0024
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<
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16
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<
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xml:space
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">THEOREMA
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value
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.</
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<
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<
s
xml:id
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echoid-s179
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xml:space
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">INuenire autem cupienti cuius numeri, duæ tertiæ, ſint quatuor quintę partes, mul
<
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tiplicandę eſſent duæ tertiæ per denominantem communem, & productum diui-
<
lb
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dendum per quatuor quintas ipſius de-
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nominantis. </
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<
s
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xml:space
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">Ac ſi quis diceret ſi
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var
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dat
<
var
>.
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<
figure
xlink:label
="
fig-0024-01
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xlink:href
="
fig-0024-01a
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number
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23
">
<
image
file
="
0024-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0024-01
"/>
</
figure
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o.</
var
>
quid dabit
<
var
>.a</
var
>
? </
s
>
<
s
xml:id
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echoid-s181
"
xml:space
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preserve
">nempe dabit
<
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>.u.</
var
>
nam in
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propoſito exemplo, terminus
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>.a.</
var
>
loco
<
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>.e.</
var
>
<
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/>
duos ſortietur denominantes, cognitum
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videlicet
<
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>.o.</
var
>
et
<
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>.u.</
var
>
incognitum quod po-
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ſtea cognitum oritur ex regula de tribus, vt dictum eſt.</
s
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</
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<
div
xml:id
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type
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math:theorem
"
level
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n
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17
">
<
head
xml:id
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xml:space
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">THEOREMA
<
num
value
="
17
">XVII</
num
>
.</
head
>
<
p
>
<
s
xml:id
="
echoid-s182
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xml:space
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preserve
">QVA ratione cognoſci poterit proportionem quantitatis cenſicæ cenſicæ ad
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ſimilem quantitatem quadruplam eſſe ad eam, quæ eſt ſuarum radicum; </
s
>
<
s
xml:id
="
echoid-s183
"
xml:space
="
preserve
">pro-
<
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portionem
<
reg
norm
="
autem
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type
="
context
">autẽ</
reg
>
primarum relatarum eſſe quintuplam,
<
reg
norm
="
atque
"
type
="
simple
">atq;</
reg
>
ita deinceps?</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s184
"
xml:space
="
preserve
">Cuiusrei gratia,
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norm
="
ſciendus
"
type
="
context
">ſciẽdus</
reg
>
eſt modus
<
reg
norm
="
productionis
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type
="
simple
">ꝓductionis</
reg
>
<
reg
norm
="
harum
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type
="
context
">harũ</
reg
>
<
reg
norm
="
dignitatum
"
type
="
context
">dignitatũ</
reg
>
qui oritur ex produ-
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/>
ctione primæ radicis in ſeipſam, prout qui
<
reg
norm
="
cubum
"
type
="
context
">cubũ</
reg
>
requirit, ducat radicé in ſuo quadra-
<
lb
/>
to, & orietur cubus, hæc poſtea ducta in cubum,
<
reg
norm
="
quantitatem
"
type
="
context
">quantitatẽ</
reg
>
cenſicam
<
reg
norm
="
cenſicam
"
type
="
context
">cenſicã</
reg
>
, et in
<
lb
/>
hanc, prædictam radicem, dabit quantitatem primam relatam. </
s
>
<
s
xml:id
="
echoid-s185
"
xml:space
="
preserve
">Quod vbi ſciueri-
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mus, meminiſſe oportet Euclidem decimaoctaua ſexti aut .11. octaui docere, pro-
<
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/>
portionem quadrati ad
<
reg
norm
="
quadratum
"
type
="
context
">quadratũ</
reg
>
, duplam eſſe proportioni ſuarum radicum, & .36.
<
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/>
vndecimi aut .11. octaui, cubi ad
<
reg
norm
="
cubum
"
type
="
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">cubũ</
reg
>
triplam eſſe, ego verò nunc aſſero, cenſici cen
<
lb
/>
ſici ad radicum proportionem quadruplam eſſe, primi verò relati ad primum re-
<
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/>
latum quintuplam
<
reg
norm
="
atque
"
type
="
simple
">atq;</
reg
>
ita gradatim.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s186
"
xml:space
="
preserve
">Cuius ſpeculationis gratia, detur linea
<
var
>.d.</
var
>
quæ cubum maiorem ſignificet. et
<
var
>.b.</
var
>
<
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/>
minorem
<
var
>.c.</
var
>
verò ſit radixipſius
<
var
>.d.</
var
>
et
<
var
>.e.</
var
>
ipſius
<
var
>.b.</
var
>
ita ordinate adinuicem, vt in ſub-
<
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ſcripta figura cernitur. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Iam
<
var
>.c.</
var
>
cum
<
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>.d.</
var
>
producatur
<
reg
norm
="
proueniatque
"
type
="
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">proueniatq́;</
reg
>
<
var
>.q.</
var
>
cenſicum cenſi-
<
lb
/>
cum, tum producatur
<
var
>.e.</
var
>
cum
<
var
>.b.</
var
>
et dabitur
<
var
>.p.</
var
>
alterum cenſicum cenſicum. </
s
>
<
s
xml:id
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xml:space
="
preserve
">Dico
<
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/>
igitur proportionem
<
var
>.q.</
var
>
ad
<
var
>.p.</
var
>
quadruplam eſſe proportioni
<
var
>.c.</
var
>
ad
<
var
>.e.</
var
>
hac de
<
lb
/>
cauſa quòd proportio
<
var
>.q.</
var
>
ad
<
var
>.p.</
var
>
compo-
<
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/>
natur ex proportione
<
var
>.d.</
var
>
ad
<
var
>.b.</
var
>
et
<
var
>.c.</
var
>
ad
<
var
>.e.</
var
>
<
lb
/>
<
figure
xlink:label
="
fig-0024-02
"
xlink:href
="
fig-0024-02a
"
number
="
24
">
<
image
file
="
0024-02
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0024-02
"/>
</
figure
>
prout facile ex .24. ſexti, aut quinta octaui
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depręhenditur. </
s
>
<
s
xml:id
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xml:space
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preserve
">Quare
<
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norm
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cum
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type
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">cũ</
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proportio
<
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>.d.</
var
>
ad
<
var
>.
<
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b.</
var
>
proportioni
<
var
>.c.</
var
>
ad
<
var
>.e.</
var
>
tripla ſit, patet pro-
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portionem
<
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>.q.</
var
>
ad
<
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>.p.</
var
>
quadruplam eſſe pro-
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/>
portioni
<
var
>.c.</
var
>
ad
<
var
>.e</
var
>
. </
s
>
<
s
xml:id
="
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xml:space
="
preserve
">Idem de cæteris dignitati
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bus dico, ſumptis ſemper
<
var
>.d</
var
>
et
<
var
>.b.</
var
>
pro duo-
<
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/>
bus cenſibus cenſuum, aut duobus primis relatis, aut alio quouis axiomate.</
s
>
</
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</
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<
div
xml:id
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type
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n
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<
head
xml:id
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xml:space
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">THEOREMA.
<
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value
="
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num
>
.</
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<
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<
s
xml:id
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xml:space
="
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">CVR diuidentibus nobis dignitatem, per dignitatem, radix prouenientis: </
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>
<
s
xml:id
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xml:space
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">pro
<
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ueniens ſit diuiſionis vnius radicis per alteram?</
s
>
</
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>
<
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>
<
s
xml:id
="
echoid-s193
"
xml:space
="
preserve
">Sint exempli gratia duę lineæ
<
var
>.b.q.</
var
>
et
<
var
>.f.g.</
var
>
quæ ſignificent duas radices cuiuſuis
<
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/>
dignitatis; </
s
>
<
s
xml:id
="
echoid-s194
"
xml:space
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">
<
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norm
="
demusque
"
type
="
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">demusq́;</
reg
>
eſſe radices duorum quadratorum,
<
reg
norm
="
quadratumque
"
type
="
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">quadratumq́;</
reg
>
ipſius
<
var
>b.q.</
var
>
<
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/>
per quadratum ipſius
<
var
>.f.g.</
var
>
diuidatur; </
s
>
<
s
xml:id
="
echoid-s195
"
xml:space
="
preserve
">quadrataq́ue radix prouenientis ſit
<
var
>.d.q.</
var
>
<
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/>
vnitas verò linearis ſit
<
var
>.i.g</
var
>
. </
s
>
<
s
xml:id
="
echoid-s196
"
xml:space
="
preserve
">Dico ipſam
<
var
>.d.q.</
var
>
eſſe proueniens ex diuiſione
<
var
>.b.q.</
var
>
<
lb
/>
per
<
var
>.f.g</
var
>
. </
s
>
<
s
xml:id
="
echoid-s197
"
xml:space
="
preserve
">Patet enim ex definitione diuiſionis nono theoremate tradita quadra- </
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>
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