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]
Text
Text Image
Image
XML
Thumbnail overview
Document information
None
Concordance
Notes
Handwritten
Figures
Content
Thumbnails
Table of handwritten notes
<
1 - 22
[out of range]
>
<
1 - 22
[out of range]
>
page
|<
<
(50)
of 445
>
>|
<
echo
version
="
1.0
">
<
text
type
="
book
"
xml:lang
="
la
">
<
div
xml:id
="
echoid-div7
"
type
="
body
"
level
="
1
"
n
="
1
">
<
div
xml:id
="
echoid-div7
"
type
="
chapter
"
level
="
2
"
n
="
1
">
<
div
xml:id
="
echoid-div148
"
type
="
math:theorem
"
level
="
3
"
n
="
75
">
<
pb
o
="
50
"
rhead
="
IO. BAPT. BENED.
"
n
="
62
"
file
="
0062
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0062
"/>
<
p
>
<
s
xml:id
="
echoid-s656
"
xml:space
="
preserve
">Progredi nihilominus etiam hac in re poſſemus per differentiam primi & ſecun-
<
lb
/>
di termini, eam detrahendo aut in ſummam cum ſecunda colligendo, attamen prior
<
lb
/>
ratio magis latè patet, ideſt vniuerſalior eſt.</
s
>
</
p
>
</
div
>
<
div
xml:id
="
echoid-div150
"
type
="
math:theorem
"
level
="
3
"
n
="
76
">
<
head
xml:id
="
echoid-head93
"
xml:space
="
preserve
">THEOREMA
<
num
value
="
76
">LXXVI</
num
>
.</
head
>
<
p
>
<
s
xml:id
="
echoid-s657
"
xml:space
="
preserve
">CVR ſi quis cupiat ſecundum terminum inuenire, quatuor terminorum arith-
<
lb
/>
meticè proportionalis continuæ, quorum nobis duo extrema proponantur.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s658
"
xml:space
="
preserve
">Rectè primum duplicabit
<
reg
norm
="
coniungetque
"
type
="
simple
">coniungetq́;</
reg
>
vltimo termino, nempe quarto, ex qua ſum-
<
lb
/>
ma tertiam partem deſumet, quæ erit ſecundus terminus quęſitus.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s659
"
xml:space
="
preserve
">Exempli gratia, ſi horum quatuor terminorum .12. 9. 6. 3. duo nobis extrema
<
lb
/>
proponantur. </
s
>
<
s
xml:id
="
echoid-s660
"
xml:space
="
preserve
">nempe .12. et .3. quorum ſecundus inueniendus ſit, ſumpto quolibet
<
lb
/>
pro primo, ſit autem .3. primus numerus, quartus verò .12. </
s
>
<
s
xml:id
="
echoid-s661
"
xml:space
="
preserve
">quare duplicato 3. vtpo
<
lb
/>
tè primo, & coniuncto .12. quarto, ſumma erit .18. cuius eſt tertia pars .6. ſecundus
<
lb
/>
numerus ſcilicet ſumpto principio à minimo. </
s
>
<
s
xml:id
="
echoid-s662
"
xml:space
="
preserve
">Idipſum euenit ſumpto principio à
<
lb
/>
maximo. </
s
>
<
s
xml:id
="
echoid-s663
"
xml:space
="
preserve
">Nam ſi datur ſecundus à minimo aut à maximo, illico tertius datur diffe-
<
lb
/>
rentia inter hunc & primum, ſecundo coniuncta, aut ex eodem detracta.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s664
"
xml:space
="
preserve
">Cuius ratio ſic demonſtratur, quatuor termini quatuor lineis
<
var
>.m.g</
var
>
:
<
var
>q.p</
var
>
:
<
var
>u.n</
var
>
:
<
var
>c.t.</
var
>
<
lb
/>
ſignificentur, quorum
<
var
>.m.g.</
var
>
et
<
var
>.c.t.</
var
>
tantummodo cognoſcantur. </
s
>
<
s
xml:id
="
echoid-s665
"
xml:space
="
preserve
">
<
reg
norm
="
ſitque
"
type
="
simple
">ſitq́;</
reg
>
<
var
>.m.g.</
var
>
primus ac
<
lb
/>
maior terminus: </
s
>
<
s
xml:id
="
echoid-s666
"
xml:space
="
preserve
">k.g. verò ſit duplum primi
<
var
>.m.g</
var
>
: cui coniungatur
<
var
>.b.k.</
var
>
æqualis
<
var
>.c.t.</
var
>
<
lb
/>
Dico tertiam partem
<
var
>.b.g.</
var
>
quæ ſumma totalis eſt, æqualem eſſe
<
var
>.q.p</
var
>
. </
s
>
<
s
xml:id
="
echoid-s667
"
xml:space
="
preserve
">In primis enim
<
lb
/>
certi ſumus
<
var
>.m.f.</
var
>
in
<
var
>.m.g.</
var
>
reperiri æqualem
<
var
>.q.p.</
var
>
<
reg
norm
="
ſupereſtque
"
type
="
simple
">ſupereſtq́;</
reg
>
<
var
>.f.g.</
var
>
differentia inter
<
var
>.m.g.</
var
>
<
lb
/>
et
<
var
>.q.p.</
var
>
æqualis
<
var
>.e.p.</
var
>
differentiæ inter
<
var
>.q.p.</
var
>
et
<
var
>.u.n.</
var
>
& æqualis
<
var
>.o.n.</
var
>
differen-
<
lb
/>
tiæ inter
<
var
>.u.n.</
var
>
et
<
var
>.c.t</
var
>
: ſimul etiam in
<
var
>.k.m.</
var
>
habemus
<
var
>.d.m.</
var
>
æqualem
<
var
>.m.f.</
var
>
</
s
>
<
s
xml:id
="
echoid-s668
"
xml:space
="
preserve
">quare etiam
<
var
>.q.
<
lb
/>
p.</
var
>
et
<
var
>.k.d.</
var
>
æqualem
<
var
>.f.g.</
var
>
nempe
<
var
>.e.p.</
var
>
aut
<
var
>.o.n</
var
>
: Hactenus in
<
var
>.k.g.</
var
>
reperimus duplum
<
var
>.q.
<
lb
/>
p.</
var
>
ſimul cum
<
var
>.f.g.</
var
>
et
<
var
>.k.d.</
var
>
æqualibus
<
var
>.e.p.</
var
>
et
<
var
>.o.n.</
var
>
& quia
<
var
>.b.K.</
var
>
æqualis
<
var
>.c.t.</
var
>
fuit coniuncta.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s669
"
xml:space
="
preserve
">conſiderandum eſt an hætres quantitates
<
var
>.f.g</
var
>
:
<
var
>K.d.</
var
>
et
<
var
>.b.K.</
var
>
ſimul æquales ſint
<
var
>.q.p.</
var
>
<
lb
/>
quod tamen per ſe manifeſtum eſt. </
s
>
<
s
xml:id
="
echoid-s670
"
xml:space
="
preserve
">nam
<
var
>.q.p.</
var
>
ſuperat
<
var
>.u.n.</
var
>
per
<
var
>.e.p.</
var
>
et
<
var
>.u.n.</
var
>
ex-
<
lb
/>
cedit
<
var
>.c.t.</
var
>
per
<
var
>.o.n.</
var
>
æqualem
<
var
>.e.p.</
var
>
</
s
>
<
s
xml:id
="
echoid-s671
"
xml:space
="
preserve
">quare
<
var
>.q.p.</
var
>
per duplum differentię
<
var
>.f.g.</
var
>
ſuperat
<
var
>.c.t.</
var
>
ita
<
lb
/>
que
<
var
>.f.g</
var
>
:
<
var
>k.d.</
var
>
et
<
var
>.K.b.</
var
>
ipſi
<
var
>.q.p.</
var
>
ſunt
<
reg
norm
="
ae- quales
"
type
="
simple
">ę-
<
lb
/>
quales</
reg
>
, ex quo ſequitur
<
var
>.q.p.</
var
>
<
reg
norm
="
tertiam
"
type
="
context
">tertiã</
reg
>
<
lb
/>
<
figure
xlink:label
="
fig-0062-01
"
xlink:href
="
fig-0062-01a
"
number
="
85
">
<
image
file
="
0062-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0062-01
"/>
</
figure
>
partem eſſe
<
var
>.b.g.</
var
>
Hæc quæ hacte-
<
lb
/>
nus dicta fuerunt, in genere maio-
<
lb
/>
ris inæqualitatis probata fuerunt.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s672
"
xml:space
="
preserve
">At in genere minoris, ſumpto or-
<
lb
/>
dinis principio à minimo termino
<
lb
/>
rum, duplicetur
<
var
>.c.t.</
var
>
<
reg
norm
="
ſitque
"
type
="
simple
">ſitq́;</
reg
>
duplum
<
lb
/>
hoc
<
var
>.K.t.</
var
>
cui
<
var
>.k.b.</
var
>
æqualis
<
var
>.m.g.</
var
>
con-
<
lb
/>
iungatur, quæſumma ſit
<
var
>.b.t</
var
>
. </
s
>
<
s
xml:id
="
echoid-s673
"
xml:space
="
preserve
">Di-
<
lb
/>
co
<
var
>.u.n.</
var
>
tertiam eſſe partem ipſius.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s674
"
xml:space
="
preserve
">Nam in primis in
<
var
>.b.t.</
var
>
datur termi
<
lb
/>
nus
<
var
>.b.K.</
var
>
æqualis vltimo
<
var
>.m.g.</
var
>
in
<
lb
/>
quo ſemel reperitur
<
var
>.u.n.</
var
>
vnà cum
<
lb
/>
duabus differentijs, nempe
<
var
>.i.g.</
var
>
in
<
lb
/>
ipſa autem
<
var
>.b.t</
var
>
:
<
var
>u.n.</
var
>
ſignificetur pri
<
lb
/>
mo loco per
<
var
>.r.K.</
var
>
ex quo ſupererit
<
var
>.b.r.</
var
>
duabus differentijs prædictis æqualis, ſed ex
<
lb
/>
præſuppoſito
<
var
>.u.n.</
var
>
componitur ex
<
var
>.o.u.</
var
>
æquali
<
var
>.c.t.</
var
>
et
<
var
>.o.n.</
var
>
ęquali vni differentiæ. </
s
>
<
s
xml:id
="
echoid-s675
"
xml:space
="
preserve
">
<
reg
norm
="
Itaque
"
type
="
simple
">Itaq;</
reg
>
</
s
>
</
p
>
</
div
>
</
div
>
</
div
>
</
text
>
</
echo
>