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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EPISTOLAE.
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431
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0431
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0431
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gi, methodo etiam qua vtebar dum in iſtisrebus me aliquo modo exercebam.</
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<
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xml:space
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">Quotieſcunque igitur ſcire volueris quantitatem corpulentiæ
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cor-
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porum regularium ab vna
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type
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ſphæra terminatorum ſeu
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circunſcriptibilium
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type
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cu-
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rabis primum, cognoſcere quantitatem lateris
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eorum, talium partium, qua-
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lium ſemidiameter dictæ ſphæræ ſit .100000. extabulis ſinuum Nicolai Copernici.
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</
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<
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xml:space
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">Propone igitur tibiante oculos figuram ſemicircularem vltimæ propoſitionis .13.
<
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/>
lib. Eucli. & inuenies
<
var
>.c.d.</
var
>
tertiam partem ſemidiametri
<
var
>.d.b.</
var
>
eſſe partium .33333. æ-
<
lb
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qualem ſinui arcus
<
var
>.f.e.</
var
>
graduum .19. mi .28. qui quidem arcus
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type
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<
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type
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fuerit à tota
<
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quarta
<
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>.b.f.</
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>
remanebitarcus
<
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>.e.b.</
var
>
gra .70. mi .32. cuius corda erit latus exaedri, quod
<
unsure
/>
<
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/>
latus ita cognoſces, ſumendo ſcilicet ſinum medietatis
<
var
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var
>
hoc eſt ſinum gra .35. mi
<
num
value
="
16
">.
<
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16.</
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>
qui erit partium .57738. cuius duplum erit partium .115476. pro latere cubi.</
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<
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<
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xml:space
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">Dempto poſtea quadrato lateris exaedri, & quadrato totius diametri
<
var
>.a.b.</
var
>
reſi-
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/>
dui radix quadrata, erit
<
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>.a.e.</
var
>
latus Tetraedri. </
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xml:space
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cus
<
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>
qui quidem arcus, componitur ex quarta
<
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>.a.f.</
var
>
& ex arcu
<
var
>.f.e.</
var
>
iam inuento, ſiue,
<
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/>
vt reſiduus totius dimidij circuli, dempto
<
var
>.b.e.</
var
>
iam ſupra inuento, habebimus idem
<
lb
/>
latus
<
var
>.a.e.</
var
>
partium .163294.</
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</
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<
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<
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xml:id
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xml:space
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">Pro latere verò Octaedri accipere potes radicem quadratam dupli quadrati ip-
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/>
ſius
<
var
>.d.b.</
var
>
& habebis
<
var
>.f.b.</
var
>
latus quæſitum. </
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xml:space
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arcus
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>
quod duplum erit
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var
>.f.b.</
var
>
partium .14142.</
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<
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<
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xml:id
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xml:space
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">Pro latere verò Duodecaedri, diuide latus Exaedri ex methodo .11. ſecundi
<
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Eucli. cuius maior pars erit latus quæſitum, partium .71368.</
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<
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<
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xml:space
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">Sed pro latere Icoſaedri, te primum oportebit inuenire quantitatem anguli
<
var
>g.d.
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a.</
var
>
hoc eſt ipſius arcus
<
var
>.b.n.</
var
>
qui tali angulo ſubiacet, quod cum pluribus modis inue-
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/>
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.
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h.</
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>
ſemidiametro ſphæræ? </
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xml:space
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ſinus arcus
<
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>
hoc eſt
<
var
>.b.n.</
var
>
graduum .63 -
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min .26. cuius medietas gra .31. mi .43. pro ſinu ſuo habet partes .52571. cuius ſinus du
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plum eſt partium .105142. pro latere Icoſaedri.</
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<
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xml:space
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">Incipiendo nunc à Tetraedro, ſcire debes, quod pars
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>
totius diametri
<
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>.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
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tium .13333.</
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<
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xml:space
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">Quæres poſtea quantitatem ſuperficialem vnius faciei ipſius Tetraedri, hac me-
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thodo, inueniendo primum radicem quadratam trium quartarum quadrati
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/>
ipſius
<
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var
>
lateris Tetraedri, eo quod latus hoc, ſeſquitertium in potentia eſt ipſi per
<
lb
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pendiculari terminatę ab vno angulorum trianguli æquilateris & à latere ei oppoſi-
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to ex .11. tertijdecimi ipſius Eucli. quę quidem perpendicularis, erit
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.141416.
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& hæc multiplicata cum medietate lateris trianguli, hoc eſt cum .81647. tibi dabit
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ſuperficiem quæſitam, hoc eſt baſim Tetraedri
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.11546192152.
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demum baſim multiplicando cum tertia parte axis Tetraedri habebis corpu-
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lentiam totius Tetraedri, quæ erit .513158964003488.</
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<
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xml:space
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">Neque tibi hoc loco occultare volo quandam meam animaduerſionem, quæ eſt,
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quod diameter ſeu perpendicularis (ſupradicta) faciei ipſius Tetraedri, ſemper æ-
<
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qualis eſt lateri ipſius Octaedri circunſcriptibilis ab eadem ſphæra, hoc eſt ipſi
<
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var
>
<
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quapropter quotieſcunque ipſam perpendicularem habere voluerimus accipiendo
<
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<
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>
habebimus intentum. </
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<
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xml:space
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<
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<
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xml:space
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