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
List of thumbnails
<
0 - 9
10 - 19
20 - 29
30 - 39
40 - 49
50 - 59
60 - 69
70 - 79
80 - 89
90 - 99
100 - 109
110 - 119
120 - 129
130 - 139
140 - 149
150 - 159
160 - 169
170 - 179
180 - 189
190 - 199
200 - 209
210 - 219
220 - 229
230 - 239
240 - 249
250 - 259
260 - 269
270 - 279
280 - 289
290 - 299
300 - 309
310 - 319
320 - 329
330 - 339
340 - 349
350 - 359
360 - 369
370 - 379
380 - 389
390 - 399
400 - 409
410 - 419
420 - 429
430 - 439
440 - 445
>
61
(49)
62
(50)
63
(51)
64
(52)
65
(53)
66
(54)
67
(55)
68
(56)
69
(57)
70
(58)
<
0 - 9
10 - 19
20 - 29
30 - 39
40 - 49
50 - 59
60 - 69
70 - 79
80 - 89
90 - 99
100 - 109
110 - 119
120 - 129
130 - 139
140 - 149
150 - 159
160 - 169
170 - 179
180 - 189
190 - 199
200 - 209
210 - 219
220 - 229
230 - 239
240 - 249
250 - 259
260 - 269
270 - 279
280 - 289
290 - 299
300 - 309
310 - 319
320 - 329
330 - 339
340 - 349
350 - 359
360 - 369
370 - 379
380 - 389
390 - 399
400 - 409
410 - 419
420 - 429
430 - 439
440 - 445
>
page
|<
<
(66)
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-div194
"
type
="
math:theorem
"
level
="
3
"
n
="
103
">
<
p
>
<
s
xml:id
="
echoid-s885
"
xml:space
="
preserve
">
<
pb
o
="
66
"
rhead
="
IO. BAPT. BENED.
"
n
="
78
"
file
="
0078
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0078
"/>
affirmabant. </
s
>
<
s
xml:id
="
echoid-s886
"
xml:space
="
preserve
">Quæ ſanè regula, non ſemper, etſi interdum vera ſit.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s887
"
xml:space
="
preserve
">Sumebant hi exemplum progreſſionis, quæ ab vnitate incohata creſcit per bina
<
lb
/>
rium, in qua per accidens euenit vt numerus dimidium vltimi termini proximè ſe-
<
lb
/>
quens, nempe è duabus partibus vltimi termini maior, æqualis ſit numero termino
<
lb
/>
rum, qui per ſe vnus è producentibus, ex ijs que .94. theoremate diximus, eſſe debet;
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s888
"
xml:space
="
preserve
">alter vero producens, qui per ſe dimidium ſummæ primi & vltimi eſſe debet, per
<
lb
/>
accidens pars maior eſt duarum vltimi termini, & alteri producenti æqualis.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s889
"
xml:space
="
preserve
">Aut alio modo ratiocinemur, dicentes, in huiuſmodi progreſſione dimidium
<
lb
/>
ſummæ vltimi termini cum primo, ſemper medium proportionale eſt inter eam
<
lb
/>
ſummam & dimidium numeri terminorum, etenim huiuſmodi ſumma numero ter-
<
lb
/>
minorum ſemper dupla eſt, prout .94. theoremate tradimus. </
s
>
<
s
xml:id
="
echoid-s890
"
xml:space
="
preserve
">Itaque ex .20. ſeptimi,
<
lb
/>
quadratum partis maioris, producto ſummæ dictæ in numerum dimidij
<
reg
norm
="
terminorum
"
type
="
context
">terminorũ</
reg
>
<
lb
/>
æquale erit, quod productum per ſe ſummæ progreſſionis eſt æquale. </
s
>
<
s
xml:id
="
echoid-s891
"
xml:space
="
preserve
">At in cæte-
<
lb
/>
ris eiuſmodi progreſſionibus fallit regula, vt ex ſupradictis facilè demonſtratur.</
s
>
</
p
>
</
div
>
<
div
xml:id
="
echoid-div195
"
type
="
math:theorem
"
level
="
3
"
n
="
104
">
<
head
xml:id
="
echoid-head121
"
xml:space
="
preserve
">THEOREMA
<
num
value
="
104
">CIIII</
num
>
.</
head
>
<
p
>
<
s
xml:id
="
echoid-s892
"
xml:space
="
preserve
">PErmultis terminis ad libitum propoſitis, diſpoſitis nihilominus progreſſio-
<
lb
/>
ne, aut proportionalitate geometrica continua, ſi minimus ex maximo & exfe-
<
lb
/>
quenti minimum detrahatur, reſiduum maximi, eam proportionem ad fum-
<
lb
/>
mam reliquorum omnium terminorum retinebit, quam reſiduum ſecundi ad pri-
<
lb
/>
mum.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s893
"
xml:space
="
preserve
">Proponuntur, exempli gratia, quatuor termini .3. 12. 48. 192. continui geome-
<
lb
/>
tricè proportionales, ſi primum, hoc eſt minimum, ex ſecundo, & maximo detra
<
lb
/>
has, exſecundo ſupererit .9. ex maximo .189. quod ſi minimum per reſiduum maxi
<
lb
/>
mi multiplicaueris, hoc eſt .189. orietur .567. tum ſi huiuſmodi productum per .9.
<
lb
/>
( refiduum ſecundi ) diuiſeris, proueniet .63. quod proueniens æquale erit ſummæ
<
lb
/>
reliquorum omnium terminorum, maximo excepto. </
s
>
<
s
xml:id
="
echoid-s894
"
xml:space
="
preserve
">Ex quo inferre licet ex .20. ſe
<
lb
/>
ptimi eandem proportionem eſſe .189. ad .63. quæ .9. ad .3. aut ſi reſiduum ſecundi
<
lb
/>
per ſummam dictorum terminorum multiplicaueris produceturidem .567. </
s
>
<
s
xml:id
="
echoid-s895
"
xml:space
="
preserve
">quare
<
lb
/>
ex .20. ſeptimi & cætera.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s896
"
xml:space
="
preserve
">Quod vt
<
reg
norm
="
ſcientificè
"
type
="
context
">ſciẽtificè</
reg
>
poſſimus, & in vniuerſum ſpeculari. </
s
>
<
s
xml:id
="
echoid-s897
"
xml:space
="
preserve
">Quatuor termini propo-
<
lb
/>
ſiti, quatuor ſubſcriptis lineis
<
reg
norm
="
ſignificentur
"
type
="
context
">ſignificẽtur</
reg
>
<
var
>.b.i</
var
>
:
<
var
>c.a</
var
>
:
<
var
>f.r</
var
>
:
<
var
>m.s.</
var
>
(quod
<
reg
norm
="
autem
"
type
="
wordlist
">aũt</
reg
>
de his quatuor di
<
lb
/>
co de
<
reg
norm
="
centummillibus
"
type
="
context
">centũmillibus</
reg
>
, & eo amplius dicere poſſum.) </
s
>
<
s
xml:id
="
echoid-s898
"
xml:space
="
preserve
">Nunc minimus terminus
<
var
>.m.s.</
var
>
ex
<
lb
/>
maximo
<
var
>.b.i.</
var
>
detrahatur,
<
reg
norm
="
ſuperſitque
"
type
="
simple
">ſuperſitq́;</
reg
>
<
var
>.n.i.</
var
>
<
reg
norm
="
idemque
"
type
="
simple
">idemq́;</
reg
>
<
var
>.m.s.</
var
>
ex ſecundo termino
<
var
>.f.r.</
var
>
ſubtra-
<
lb
/>
hatur,
<
reg
norm
="
ſuperſitque
"
type
="
simple
">ſuperſitq́;</
reg
>
<
var
>.o.r</
var
>
. </
s
>
<
s
xml:id
="
echoid-s899
"
xml:space
="
preserve
">Dico proportionem
<
var
>.n.i.</
var
>
ad ſummam reliquorum omnium ter-
<
lb
/>
minorum
<
var
>.c.a</
var
>
:
<
var
>f.r</
var
>
:
<
var
>m.s.</
var
>
eandem effe, quæ
<
var
>.o.r.</
var
>
ad
<
var
>.m.s</
var
>
. </
s
>
<
s
xml:id
="
echoid-s900
"
xml:space
="
preserve
">Quamobrem ex tertio & quar-
<
lb
/>
to ſecundus
<
var
>.f.r.</
var
>
<
reg
norm
="
detrahatur
"
type
="
simple
">detrahat̃</
reg
>
, ex
<
reg
norm
="
tertioque
"
type
="
simple
">tertioq́;</
reg
>
ſuperſit
<
var
>.t.a.</
var
>
& ex quarto
<
var
>.e.i.</
var
>
ita etiam tertius
<
var
>.
<
lb
/>
c.a.</
var
>
ex quarto
<
var
>.b.i.</
var
>
<
reg
norm
="
ſuperſitque
"
type
="
simple
">ſuperſitq́;</
reg
>
<
var
>.d.i.</
var
>
ſanè
<
lb
/>
<
figure
xlink:label
="
fig-0078-01
"
xlink:href
="
fig-0078-01a
"
number
="
106
">
<
image
file
="
0078-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0078-01
"/>
</
figure
>
ſic ſe habebit
<
var
>.c.a.</
var
>
ad
<
var
>.f.r.</
var
>
vt
<
var
>.c.t.</
var
>
ad
<
var
>.f.o.</
var
>
<
lb
/>
vt
<
reg
norm
="
quisque
"
type
="
simple
">quisq;</
reg
>
per ſe ſcire poteſt. </
s
>
<
s
xml:id
="
echoid-s901
"
xml:space
="
preserve
">Quare ex
<
lb
/>
19. quinti ſic ſe habebit
<
var
>.a.t.</
var
>
ad
<
var
>.r.o.</
var
>
vt
<
var
>.
<
lb
/>
c.a.</
var
>
ad
<
var
>.f.r.</
var
>
& permutando ita
<
var
>.a.t.</
var
>
ad
<
var
>.a.
<
lb
/>
c.</
var
>
vt
<
var
>.o.r.</
var
>
ad
<
var
>.r.f.</
var
>
& ſeparando ſic
<
var
>.a.t.</
var
>
ad
<
var
>.
<
lb
/>
a.c.</
var
>
(hoc eſt
<
var
>.f.r.</
var
>
) vt
<
var
>.r.o.</
var
>
ad
<
var
>.o.f.</
var
>
vide-
<
lb
/>
licet
<
var
>.m.s</
var
>
. </
s
>
<
s
xml:id
="
echoid-s902
"
xml:space
="
preserve
">
<
reg
norm
="
Idem
"
type
="
context
">Idẽ</
reg
>
dico de
<
var
>.d.i.</
var
>
ad
<
var
>.a.c.</
var
>
nem-
<
lb
/>
pe ſic ſe habebit
<
var
>.d.i.</
var
>
ad
<
var
>.a.c.</
var
>
vt
<
var
>.a.t.</
var
>
ad
<
var
>. </
var
>
</
s
>
</
p
>
</
div
>
</
div
>
</
div
>
</
text
>
</
echo
>