206

In Mysterium Cosmographicum

Latus dodecaödri prodit proportione : 2) a : x = x : a — x_ x 2 -j- ax = a 2

=-f+V

5 a 2

T

= 2 (-1 + V5)

Ergo ex 1) et 2) Latus dodecaedri = 5 (V 15 — V 3).

Pro icosaedro: Sit iterum D axis, r radius circuli in quem inscribitur quinquangu-Ium, cujus latus = a.

Erit 1) D 2 = 5r 2

2 )

,=.v

5 + V510

... c+fi)

\ / 2Quare a = D \

49) p. 150. Franziscus Flussates Candalla (Foix, comes Candallensis, episcopusAirensis, mort. a. 1594) edidit anno 1566 „Euclidis Elementa, libris XV ad germanam geo-metriae intelligentiam &c., restituta &c.“ Parisiis (recus. a. 1578).

His quae Keplerus Candalla duce proponit, haec addimus:

Sit tetraedron FGHN , MN diameter sphaerae circumscriptae (axis) , 0 centrum cor-poris, I centrum trianguli FGH, NI altitudo corporis, 01 radius sphaerae inscriptae; latustetraedri = a.

1) IH = — V 3 ; IH 2 = , ergo a 2 = 3IH 2 .

3a 2

2) ON = a V 3 / 8 ; ON 2 — —- = % IH 2 ; NM 2 = •/, IH 2 = 3 /* a 2 .

o

3) IN 2 = NH 2 — IH 2 = a 2 — -|'= 2 / 3 a 2 = 2 IH 2 .

4) Cum sint puncta M, N et H in peripheria circuli, cujus centrum O, et IH _j_ NM,

• xttt 2 ,,, T . . ,^T __t NH 2 a 2 a 2 SON 2

erit IsH 2 = M1S X IN, quare IN == —— = = —— = ■ •

MN MN 2 ON 60N

4 ON

5) 01 = V OH 2 —IH 2 = V ON 2 —IH 2 = VoN 2 — % ON 2 = ^ = —- ; 01 2 =

ON 2 _ MN^ _ a^ 3 b

9 — 36 ~ 24’

6) IP 2 = IH 2 — PH 2 == = 4*-

' 3 4 12

7) NP 2 = NH 2 — PH 2 = a 2 — V* a 2 = 7* a 2 .

Comparatis his numeris, positoque 01 = 1, prodeunt, quos Keplerus in secunda editioneaddit, numeri: