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 Notes
<
1 - 22
>
[Note]
Page: 398
[Note]
Page: 398
[Note]
Page: 398
[Note]
Page: 398
[Note]
Page: 400
[Note]
Page: 400
[Note]
Page: 402
[Note]
Page: 404
[Note]
Page: 404
[Note]
Page: 404
[Note]
Page: 404
[Note]
Page: 406
[Note]
Page: 406
<
1 - 22
>
page
|<
<
(43)
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-div134
"
type
="
math:theorem
"
level
="
3
"
n
="
67
">
<
p
>
<
s
xml:id
="
echoid-s568
"
xml:space
="
preserve
">
<
pb
o
="
43
"
rhead
="
THEOREM. ARIT.
"
n
="
55
"
file
="
0055
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0055
"/>
ita vt ſimul prouenientibus in ſummam collectis huius fummæ ad primum nume-
<
lb
/>
rum propoſitum proportio futura ſit ea quæ eſt tertij ad ſecundum. </
s
>
<
s
xml:id
="
echoid-s569
"
xml:space
="
preserve
">Rectè dimidium
<
lb
/>
primi numeri in ſeipſum multiplicant, ex quo quadrato ſecundum numerum detra
<
lb
/>
hunt, tum reſidui radicem ſumunt, quam iungentes, & detrahentes ex dimidio
<
lb
/>
primi, partes quæſitas habent, cætera ex neceſsitate ſubſequuntur, prout nunc a
<
lb
/>
me docebitur.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s570
"
xml:space
="
preserve
">Exempli gratia, proponitur numerus .20. in duas partes diuidendus, quibus po
<
lb
/>
ſtea mutuò diuiſis, & per ſummam prouenientium diuiſa ſumma quadratorum,
<
lb
/>
dent
<
reg
norm
="
ſecundum
"
type
="
context
">ſecundũ</
reg
>
numerum propoſitum .36. nam reliqua conſequuntur. </
s
>
<
s
xml:id
="
echoid-s571
"
xml:space
="
preserve
">Itaque .10.
<
lb
/>
dimidium primi in ſeipſum multiplicatur, & ex quadrato .100. eruitur numerus .36.
<
lb
/>
nempe ſecundus propoſitus reſidui porrò .64. quadrata radix .8. fumitur, quam con
<
lb
/>
iungimus & detrahimus ex dimidio primi ſcilicet .10. ex quo partes quæſitæ dabun
<
lb
/>
tur .18. et .2. quæ mutuo diuiſæ dabunt ſuorum prouenientium ſummam .9. cum no-
<
lb
/>
na parte, per quam diuidentes .328. ſummam quadratorum ipſarum partium,
<
lb
/>
exactè dabitur numerus .36. qui fuit ſecundò propoſitus. </
s
>
<
s
xml:id
="
echoid-s572
"
xml:space
="
preserve
">Tum ſi per ſingu-
<
lb
/>
las iam inuentas partes quilibet numerus diuiſus fuerit, verbi gratia .72. ſumma pro
<
lb
/>
uenientium erit .40. qui num@rus eandem proportionem cum primo nempe .20. ſer
<
lb
/>
uabit, quam tertius propoſitus .72. cum ſecundo .36.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s573
"
xml:space
="
preserve
">Quod vt ſpeculemur, primus numerus ſignificetur linea
<
var
>.n.e.</
var
>
ita diuidendus à
<
lb
/>
puncto
<
var
>.o.</
var
>
vt diuiſa parte
<
var
>.n.o.</
var
>
per
<
var
>.o.e.</
var
>
et
<
var
>.o.e.</
var
>
per
<
var
>.n.o.</
var
>
& per ſummam prouenien-
<
lb
/>
tium diuiſa ſumma quadratorum
<
var
>.n.o.</
var
>
et
<
var
>.o.e.</
var
>
detur ſecundus numerus notatus linea
<
var
type
="
line
">.
<
lb
/>
q.K</
var
>
. </
s
>
<
s
xml:id
="
echoid-s574
"
xml:space
="
preserve
">Porrò meminiſſe oportet quòd .26. theoremate probatum fuit vltimum hoc
<
lb
/>
proueniens æquale producto partium inter ſe futurum, nempe producto
<
var
>.n.o.</
var
>
in
<
var
>.o.
<
lb
/>
e.</
var
>
quod ſignificetur rectangulo
<
var
>.n.e</
var
>
. </
s
>
<
s
xml:id
="
echoid-s575
"
xml:space
="
preserve
">Itaque datis
<
var
>.n.e.</
var
>
et
<
var
>.q.K.</
var
>
ſi .45. theorema conſu-
<
lb
/>
luerimus, partes
<
var
>.n.o.</
var
>
et
<
var
>.o.e.</
var
>
cognoſcemus.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s576
"
xml:space
="
preserve
">Proponitur deinde tertius quilibetnumerus, verbi gratia
<
var
>.x.</
var
>
diuidendus per
<
var
>.o.e.</
var
>
<
lb
/>
et
<
var
>.o.n.</
var
>
qui ſi diuidatur per
<
var
>.o.e.</
var
>
dabit pro
<
lb
/>
ueniens
<
var
>.b.o</
var
>
. </
s
>
<
s
xml:id
="
echoid-s577
"
xml:space
="
preserve
">Si verò per
<
var
>.n.o.</
var
>
proueniens
<
lb
/>
<
figure
xlink:label
="
fig-0055-01
"
xlink:href
="
fig-0055-01a
"
number
="
75
">
<
image
file
="
0055-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0055-01
"/>
</
figure
>
erit
<
var
>.d.n.</
var
>
nunc aſſerimus
<
reg
norm
="
ſummam
"
type
="
context
">ſummã</
reg
>
duorum
<
lb
/>
horum prouenientium, ſic primo nume-
<
lb
/>
ro
<
var
>.n.e.</
var
>
dato proportionatam eſſe, ſicut
<
lb
/>
tertius
<
var
>.x.</
var
>
<
reg
norm
="
ſecundo
"
type
="
context
">ſecũdo</
reg
>
<
var
>.q.K</
var
>
. </
s
>
<
s
xml:id
="
echoid-s578
"
xml:space
="
preserve
">Producatur enim li-
<
lb
/>
nea
<
var
>.d.n.</
var
>
donec
<
var
>.n.q.</
var
>
æqualis ſit
<
var
>.o.b.</
var
>
ex
<
lb
/>
quo
<
var
>.q.d.</
var
>
erit ſumma vltimò prouenien-
<
lb
/>
tium: </
s
>
<
s
xml:id
="
echoid-s579
"
xml:space
="
preserve
">item producatur
<
var
>.e.n.</
var
>
donec
<
var
>.n.u.</
var
>
æ-
<
lb
/>
qualis ſit
<
var
>.o.e.</
var
>
<
reg
norm
="
termineturque
"
type
="
simple
">termineturq́</
reg
>
rectangulum
<
var
>.
<
lb
/>
q.u.</
var
>
quod tertio numero propoſito
<
var
>.x.</
var
>
vt
<
lb
/>
patet, æquale erit, </
s
>
<
s
xml:id
="
echoid-s580
"
xml:space
="
preserve
">quare ex .15. ſexti aut .
<
lb
/>
20. ſeptimi eadem erit proportio
<
var
>.d.n.</
var
>
ad
<
lb
/>
<
var
>n.q.</
var
>
quæ
<
var
>.u.n.</
var
>
nempe
<
var
>.o.e.</
var
>
ad
<
var
>.o.n.</
var
>
& com-
<
lb
/>
ponendo
<
var
>.d.q.</
var
>
ad
<
var
>.q.n.</
var
>
ſicut
<
var
>.e.n.</
var
>
ad
<
var
>.n.o.</
var
>
&
<
lb
/>
permutando
<
var
>.d.q.</
var
>
ad
<
var
>.e.n.</
var
>
quæ
<
var
>.q.n.</
var
>
hoc eſt
<
var
>.
<
lb
/>
b.o.</
var
>
ad
<
var
>.o.n.</
var
>
nempe ſicut
<
var
>.b.e.</
var
>
ad
<
var
>.e.n.</
var
>
ſuperficialem, ex prima ſexti aut .18. vel .19.
<
lb
/>
ſeptimi, ſed rectangulum
<
var
>.e.n.</
var
>
conſtitutum fuit æquale numero
<
var
>.q.K</
var
>
. </
s
>
<
s
xml:id
="
echoid-s581
"
xml:space
="
preserve
">itaque verum
<
lb
/>
eſt propoſitum.</
s
>
</
p
>
</
div
>
</
div
>
</
div
>
</
text
>
</
echo
>