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
[out of range]
>
<
1 - 22
[out of range]
>
page
|<
<
(419)
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-div477
"
type
="
chapter
"
level
="
2
"
n
="
6
">
<
div
xml:id
="
echoid-div776
"
type
="
section
"
level
="
3
"
n
="
50
">
<
div
xml:id
="
echoid-div778
"
type
="
letter
"
level
="
4
"
n
="
3
">
<
p
>
<
s
xml:id
="
echoid-s5071
"
xml:space
="
preserve
">
<
pb
o
="
419
"
rhead
="
EPISTOLAE.
"
n
="
431
"
file
="
0431
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0431
"/>
gi, methodo etiam qua vtebar dum in iſtisrebus me aliquo modo exercebam.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s5072
"
xml:space
="
preserve
">Quotieſcunque igitur ſcire volueris quantitatem corpulentiæ
<
reg
norm
="
cuiuſque
"
type
="
simple
">cuiuſq;</
reg
>
<
reg
norm
="
quinque
"
type
="
simple
">quinq;</
reg
>
cor-
<
lb
/>
porum regularium ab vna
<
reg
norm
="
eademque
"
type
="
simple
">eademq́;</
reg
>
ſphæra terminatorum ſeu
<
reg
norm
="
circunſcriptibilium
"
type
="
context
">circunſcriptibiliũ</
reg
>
cu-
<
lb
/>
rabis primum, cognoſcere quantitatem lateris
<
reg
norm
="
cuiusque
"
type
="
simple
">cuiusq́;</
reg
>
eorum, talium partium, qua-
<
lb
/>
lium ſemidiameter dictæ ſphæræ ſit .100000. extabulis ſinuum Nicolai Copernici.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s5073
"
xml:space
="
preserve
">Propone igitur tibiante oculos figuram ſemicircularem vltimæ propoſitionis .13.
<
lb
/>
lib. Eucli. & inuenies
<
var
>.c.d.</
var
>
tertiam partem ſemidiametri
<
var
>.d.b.</
var
>
eſſe partium .33333. æ-
<
lb
/>
qualem ſinui arcus
<
var
>.f.e.</
var
>
graduum .19. mi .28. qui quidem arcus
<
reg
norm
="
demptus
"
type
="
context
">dẽptus</
reg
>
<
reg
norm
="
cum
"
type
="
context
">cũ</
reg
>
fuerit à tota
<
lb
/>
quarta
<
var
>.b.f.</
var
>
remanebitarcus
<
var
>.e.b.</
var
>
gra .70. mi .32. cuius corda erit latus exaedri, quod
<
unsure
/>
<
lb
/>
latus ita cognoſces, ſumendo ſcilicet ſinum medietatis
<
var
>.b.e.</
var
>
hoc eſt ſinum gra .35. mi
<
num
value
="
16
">.
<
lb
/>
16.</
num
>
qui erit partium .57738. cuius duplum erit partium .115476. pro latere cubi.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s5074
"
xml:space
="
preserve
">Dempto poſtea quadrato lateris exaedri, & quadrato totius diametri
<
var
>.a.b.</
var
>
reſi-
<
lb
/>
dui radix quadrata, erit
<
var
>.a.e.</
var
>
latus Tetraedri. </
s
>
<
s
xml:id
="
echoid-s5075
"
xml:space
="
preserve
">Vel ſi duplicaueris ſinum dimidij ar-
<
lb
/>
cus
<
var
>.a.e.</
var
>
qui quidem arcus, componitur ex quarta
<
var
>.a.f.</
var
>
& ex arcu
<
var
>.f.e.</
var
>
iam inuento, ſiue,
<
lb
/>
vt reſiduus totius dimidij circuli, dempto
<
var
>.b.e.</
var
>
iam ſupra inuento, habebimus idem
<
lb
/>
latus
<
var
>.a.e.</
var
>
partium .163294.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s5076
"
xml:space
="
preserve
">Pro latere verò Octaedri accipere potes radicem quadratam dupli quadrati ip-
<
lb
/>
ſius
<
var
>.d.b.</
var
>
& habebis
<
var
>.f.b.</
var
>
latus quæſitum. </
s
>
<
s
xml:id
="
echoid-s5077
"
xml:space
="
preserve
">Vel ſi malis accipe duplum ſinus medietatis
<
lb
/>
arcus
<
var
>.b.f.</
var
>
quod duplum erit
<
var
>.f.b.</
var
>
partium .14142.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s5078
"
xml:space
="
preserve
">Pro latere verò Duodecaedri, diuide latus Exaedri ex methodo .11. ſecundi
<
lb
/>
Eucli. cuius maior pars erit latus quæſitum, partium .71368.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s5079
"
xml:space
="
preserve
">Sed pro latere Icoſaedri, te primum oportebit inuenire quantitatem anguli
<
var
>g.d.
<
lb
/>
a.</
var
>
hoc eſt ipſius arcus
<
var
>.b.n.</
var
>
qui tali angulo ſubiacet, quod cum pluribus modis inue-
<
lb
/>
niri poſſit, nihilominus, hunc ſeruabis, inuenies primò quantitatem
<
var
>.d.g.</
var
>
quæ eſt ra
<
lb
/>
dix quadrata ſummæ duorum quadratorum hoc eſt
<
var
>.d.a.</
var
>
et
<
var
>.a.g.</
var
>
quæ
<
var
>.a.g.</
var
>
æqualis eſt
<
lb
/>
diametro
<
var
>.a.b.</
var
>
vt ſcis, dices poſtea, ſi
<
var
>.d.g.</
var
>
correſpondet ipſi
<
var
>.g.a.</
var
>
cui correſpondet
<
var
>.d.
<
lb
/>
h.</
var
>
ſemidiametro ſphæræ? </
s
>
<
s
xml:id
="
echoid-s5080
"
xml:space
="
preserve
">tibi veniet
<
var
>.h.k.</
var
>
ſinus arcus
<
var
>.a.h.</
var
>
hoc eſt
<
var
>.b.n.</
var
>
graduum .63 -
<
lb
/>
min .26. cuius medietas gra .31. mi .43. pro ſinu ſuo habet partes .52571. cuius ſinus du
<
lb
/>
plum eſt partium .105142. pro latere Icoſaedri.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s5081
"
xml:space
="
preserve
">Incipiendo nunc à Tetraedro, ſcire debes, quod pars
<
var
>.a.c.</
var
>
totius diametri
<
var
>.a.b.</
var
>
æ-
<
lb
/>
qualis eſt axi ipſius Tetraedri, quæ quidem
<
var
>.a.c.</
var
>
vt ſubſeſquialtera ipſius
<
var
>.a.b.</
var
>
erit par
<
lb
/>
tium .13333.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s5082
"
xml:space
="
preserve
">Quæres poſtea quantitatem ſuperficialem vnius faciei ipſius Tetraedri, hac me-
<
lb
/>
thodo, inueniendo primum radicem quadratam trium quartarum quadrati
<
lb
/>
ipſius
<
var
>.a.e.</
var
>
lateris Tetraedri, eo quod latus hoc, ſeſquitertium in potentia eſt ipſi per
<
lb
/>
pendiculari terminatę ab vno angulorum trianguli æquilateris & à latere ei oppoſi-
<
lb
/>
to ex .11. tertijdecimi ipſius Eucli. quę quidem perpendicularis, erit
<
reg
norm
="
partium
"
type
="
context
">partiũ</
reg
>
.141416.
<
lb
/>
& hæc multiplicata cum medietate lateris trianguli, hoc eſt cum .81647. tibi dabit
<
lb
/>
ſuperficiem quæſitam, hoc eſt baſim Tetraedri
<
reg
norm
="
partium
"
type
="
context
">partiũ</
reg
>
<
reg
norm
="
ſuperficialium
"
type
="
context
">ſuperficialiũ</
reg
>
.11546192152.
<
lb
/>
<
reg
norm
="
Hanc
"
type
="
context
">Hãc</
reg
>
demum baſim multiplicando cum tertia parte axis Tetraedri habebis corpu-
<
lb
/>
lentiam totius Tetraedri, quæ erit .513158964003488.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s5083
"
xml:space
="
preserve
">Neque tibi hoc loco occultare volo quandam meam animaduerſionem, quæ eſt,
<
lb
/>
quod diameter ſeu perpendicularis (ſupradicta) faciei ipſius Tetraedri, ſemper æ-
<
lb
/>
qualis eſt lateri ipſius Octaedri circunſcriptibilis ab eadem ſphæra, hoc eſt ipſi
<
var
>.b.f.</
var
>
<
lb
/>
quapropter quotieſcunque ipſam perpendicularem habere voluerimus accipiendo
<
lb
/>
<
var
>b.f.</
var
>
habebimus intentum. </
s
>
<
s
xml:id
="
echoid-s5084
"
xml:space
="
preserve
">Et quod hoc verum ſit poſſumus ita demonſtrare.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s5085
"
xml:space
="
preserve
">Primum, notum nobis eſt, ipſam perpendicularem, triplam eſſe eius parti, quæ </
s
>
</
p
>
</
div
>
</
div
>
</
div
>
</
div
>
</
text
>
</
echo
>