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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57
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THEOREM. ARITH.
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69
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file
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0069
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0069
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primæ cum tertia .20. ex quo ſicſe habet .20. ad .8. nempe ad ſecundam, vt .30.
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ad .12.</
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<
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<
s
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xml:space
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">Quod vt ſciamus, quatuor prædictæ quantitates ſignificentur linea
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pro-
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babo ita ſe habere
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ad
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>.e.i.</
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vt
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ad
<
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>.e</
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>
. </
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<
s
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xml:space
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preserve
">Nam cum ſic ſe habeat
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ad
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>.e.</
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ut
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>.e.</
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>
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ad
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>.i.</
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& vt
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>.i.</
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ad .o: ex æqualitate proportionum vel permutando ita ſe habebit
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>.a.</
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ad
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>.i.</
var
>
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vt
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>.e.</
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ad
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>.o.</
var
>
& è conuerſo ita
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>.o.</
var
>
ad
<
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>.e.</
var
>
vt
<
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>.i.</
var
>
ad
<
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>.a.</
var
>
&
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reg
norm
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componendo
"
type
="
context
">cõponendo</
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>
ita
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>.o.e.</
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ad e. vt
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>.i.a.</
var
>
ad
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>.a.</
var
>
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/>
<
reg
norm
="
permutandoque
"
type
="
simple
">permutandoq́</
reg
>
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>.o.e.</
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ad
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>.i.a.</
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>
vt
<
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>.e.</
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ad
<
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>.a.</
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nempe
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>.i.</
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>
ad
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>.e.</
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>
& componendo ita
<
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>.o.i.e.a.</
var
>
ad
<
var
>.
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i.a.</
var
>
vt
<
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>.i.e.</
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>
ad
<
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>.e.</
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>
& permutando ita
<
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>.o.i.e.a.</
var
>
ad
<
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>.i.e.</
var
>
vt
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>.i.a.</
var
>
ad
<
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>.e.</
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>
quod erat propoſitum.
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</
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<
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xml:space
="
preserve
">Ex quo patet error antiquorum quiidipſum, accidere arbitrati ſunt in quantitatibus
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diſcretæ proportionalitatis, quod tamen falſum eſt.</
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<
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<
s
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xml:space
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preserve
">Exempli gratia, ſi proponantur .12. 6. 4. 2. proportio .12. ad .6. eadem eſt quæ .4.
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ad .2. </
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<
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xml:space
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preserve
">Sed à proportione .6. ad .4. frangitur, cum non ſit eadem quæ .12. ad .6. harum
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autem ſumma erit .24. & ſumma ſecundæ cum tertia .10. ſed primæ cum tertia erit
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16. ex quo .16. ad .6. non ſic ſe habebit vt .24. ad .10.
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At in ſpeculatione quatuor quantitatum
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>.a.</
var
>
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<
figure
xlink:label
="
fig-0069-01
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xlink:href
="
fig-0069-01a
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number
="
96
">
<
image
file
="
0069-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0069-01
"/>
</
figure
>
<
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>e.i.o.</
var
>
ſi proportio
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>.e.</
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>
ad
<
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>.i.</
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>
non eſſet eadem
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quæ
<
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>.a.</
var
>
ad
<
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>.e.</
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>
minimè licuiſſet dicere ita ſe
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/>
habere
<
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>.i.</
var
>
ad
<
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>.e.</
var
>
vt
<
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>.e.</
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>
ad
<
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>.a</
var
>
.</
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>
</
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</
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>
<
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xml:id
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type
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level
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n
="
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">
<
head
xml:id
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xml:space
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preserve
">THEOREMA
<
num
value
="
88
">LXXXVIII</
num
>
.</
head
>
<
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<
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xml:space
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">CVR extribus quantitatibus quibuſlibet, productum duarum in tertiam, vna
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ſemper
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norm
="
eademque
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type
="
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">eademq́;</
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>
ſit quantitas.</
s
>
</
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<
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<
s
xml:id
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xml:space
="
preserve
">Exempli gratia, proponuntur .15. 8. 2. ſi multiplicauerimus .15. per .8. tum produ
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ctum per .2. tantum erit quantum ſi quis multiplicaret .8. per .2. & hoc per .15. et .15.
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per .2.
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rurſusque
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type
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per .8.</
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</
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<
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<
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xml:id
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xml:space
="
preserve
">Quod ut pateat, tres quantitates tri-
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<
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fig-0069-02
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xlink:href
="
fig-0069-02a
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number
="
97
">
<
image
file
="
0069-02
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0069-02
"/>
</
figure
>
bus lineis ſignificentur
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>.m.f</
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>
: a. et
<
var
>.o</
var
>
. </
s
>
<
s
xml:id
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xml:space
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">Dico
<
lb
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productum
<
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>.m.f.</
var
>
in
<
var
>.a.</
var
>
multiplicatum.
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/>
</
s
>
<
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xml:id
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xml:space
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">per
<
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>.o.</
var
>
æquale eſſe producto
<
var
>.a.</
var
>
in
<
var
>.o.</
var
>
mul-
<
lb
/>
tiplicato per
<
var
>.m.f.</
var
>
aut producto
<
var
>.m.f.</
var
>
in
<
var
>.
<
lb
/>
o.</
var
>
multiplicato per
<
var
>.a</
var
>
. </
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>
<
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xml:id
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xml:space
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">Sit enim corpus
<
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>.d.
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u.</
var
>
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rectangulum
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type
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">rectãgulum</
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, cuius latus
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>.n.u.</
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>
ſit æquale
<
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/>
<
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>m.f.</
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>
et
<
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>.u.t</
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>
: a: et
<
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>.u.c</
var
>
: o. patebit manifeſtè
<
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/>
<
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>n.t.</
var
>
eſſe productum
<
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>.m.f.</
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>
in
<
var
>.a.</
var
>
quod
<
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>.n.t.</
var
>
<
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/>
multiplicatum in
<
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>.u.c.</
var
>
æquali
<
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>.o.</
var
>
producit
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/>
corpus
<
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>.d.u.</
var
>
ſed idipſum corpus
<
var
>.d.u.</
var
>
ex
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/>
multiplicatione producti
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>.c.t.</
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>
in latus
<
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>.n.
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/>
u.</
var
>
æquale
<
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>.m.f.</
var
>
oritur, & idipſum
<
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>.d.u.</
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>
ex
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multiplicatione
<
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>.n.c.</
var
>
in latus
<
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>.u.t.</
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>
æquale
<
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>.a.</
var
>
profertur.</
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>
</
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</
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<
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xml:id
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type
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<
head
xml:id
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"
xml:space
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preserve
">THEOREMA
<
num
value
="
89
">LXXXIX</
num
>
.</
head
>
<
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>
<
s
xml:id
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"
xml:space
="
preserve
">CVR quarumcunque quatuor quantitatum, ſi prima in ſecundam multiplice-
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tur & hoc productum in tertiam,
<
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norm
="
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type
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">rurſusq́</
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>
hoc alterum in quartam, vltimum
<
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productum æquale ſit producto producti ſecundæ in tertiam, in productum primæ
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in quartam.</
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