History of Science Department University of Aarhus
History of Science Department University of Aarhus
History of Science Department University of Aarhus
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24 1. PTOLEMAIOS' PLANETMODELLER I ALMAGESTEN<br />
approksimation 2e0 til 2e. I den approksimerede situation, Fig. 1.17 ses positionerne<br />
E 0<br />
i pa equantcirklen under vinklerne 0<br />
i = i fra O og fra E 0<br />
ses de som f r under vinklerne i.<br />
(approksimationen til E)<br />
E<br />
X<br />
A<br />
00 11<br />
00 11<br />
00 11<br />
01<br />
01<br />
δ1<br />
00 11 0<br />
00 11<br />
00<br />
00 11<br />
00 11<br />
00 11<br />
B 00 11<br />
00 11<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
000000<br />
111111<br />
01<br />
000000<br />
111111<br />
01<br />
000000<br />
111111<br />
000000<br />
111111<br />
1 R 000000<br />
111111<br />
0000000000<br />
1111111111<br />
000000<br />
111111<br />
0000000000<br />
1111111111<br />
000000<br />
111111<br />
0000000000<br />
1111111111<br />
000000<br />
111111<br />
0000000000<br />
1111111111<br />
000000<br />
111111<br />
0000000000<br />
1111111111<br />
000000<br />
111111<br />
0000000000<br />
1111111111<br />
000000<br />
111111<br />
0000000000<br />
1111111111<br />
000000<br />
111111<br />
0000000000<br />
1111111111<br />
000000<br />
111111 N<br />
0000000000<br />
1111111111<br />
000000<br />
111111<br />
01<br />
0000000000<br />
1111111111<br />
000000<br />
111111<br />
01<br />
0000000000<br />
1111111111 00 1101<br />
E<br />
000000<br />
111111<br />
0000000000<br />
1111111111 00 11<br />
000000<br />
111111<br />
0000000000<br />
1111111111<br />
000000<br />
111111 2e<br />
0000000000<br />
1111111111<br />
000000<br />
111111 00 11<br />
0000000000<br />
1111111111<br />
00 110000<br />
1111 O 00 110000<br />
1111<br />
00 110000<br />
1111<br />
00 110000<br />
1111<br />
01<br />
00 110000<br />
1111<br />
01<br />
00 110000<br />
1111<br />
00 110000<br />
1111<br />
00 110000<br />
1111 δ2<br />
00 110000<br />
1111<br />
00 110000<br />
1111<br />
00 110000<br />
1111<br />
00 110000<br />
1111<br />
00 110000<br />
1111<br />
1<br />
00 110000<br />
1111<br />
00 11<br />
00 11 1<br />
00 110000<br />
1111<br />
00 11<br />
00 11 1<br />
00 11<br />
00 11<br />
00 11 00 11<br />
00 11 00 11<br />
Figur 1.17.<br />
E Π E<br />
Equantcirklen<br />
2 3<br />
E<br />
B<br />
Equantcirklen<br />
01<br />
01<br />
01<br />
01<br />
01<br />
00 11 0<br />
00 11<br />
0 1 1 0 0 1 1<br />
01<br />
00000<br />
11111 0000 1111 000000000<br />
111111111<br />
01<br />
000000<br />
111111<br />
0000 1111<br />
00000<br />
11111 0000 1111 000000000<br />
111111111<br />
000000<br />
111111<br />
0000 1111<br />
00000<br />
11111 0000 1111 000000000<br />
111111111<br />
000000<br />
111111<br />
0000 1111<br />
00000<br />
11111 0000 1111 000000000<br />
111111111<br />
000000<br />
111111<br />
0000 1111 01<br />
G<br />
00000<br />
11111 01<br />
0000 1111 000000000<br />
111111111<br />
000000<br />
111111<br />
0000 1111 01<br />
000 111<br />
00000<br />
11111 01<br />
0000 1111<br />
01<br />
00 11<br />
000000<br />
111111<br />
0000 1111 000 111<br />
00000<br />
11111 0000 1111<br />
00 11<br />
000000<br />
111111<br />
0000 1111<br />
1<br />
D<br />
000 111<br />
00000<br />
11111 0000 1111<br />
0000000000<br />
1111111111<br />
00 11<br />
000000<br />
111111<br />
0000 1111 000 111<br />
00000<br />
11111 0000 1111000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111<br />
01<br />
0000 1111000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111<br />
01<br />
F 000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111 000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111 000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111 000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111 000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111 000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111 000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111 000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111 000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111 000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111 000000<br />
111111<br />
0000 1111<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111 000000<br />
111111<br />
0000 111100<br />
11<br />
0000000000<br />
1111111111<br />
000 111<br />
00000000<br />
11111111 0000 1111 δ 00 110000<br />
1111<br />
00000000<br />
11111111 0000 1111 1 O 00 110000<br />
1111<br />
00000000<br />
11111111 0000 111100<br />
110000<br />
1111<br />
00000000<br />
11111111 0000 111100<br />
110000<br />
1111<br />
00000000<br />
11111111 0000 111100<br />
110000<br />
1111<br />
00000000<br />
11111111 0000 111100<br />
110000<br />
1111<br />
00000000<br />
11111111 0000 111100<br />
110000<br />
1111<br />
00000000<br />
11111111 0000 111100<br />
11δ0000<br />
1111<br />
00000000<br />
11111111 0000 1111 2 00 110000<br />
1111<br />
00000000<br />
11111111 0000 111100<br />
110000<br />
1111<br />
00000000<br />
11111111 0000 111100<br />
110000<br />
1111<br />
00000000<br />
11111111 0000 111100<br />
110000<br />
1111<br />
00000000<br />
11111111 0000 111100<br />
110000<br />
1111<br />
00000000<br />
11111111 0000 111100<br />
110000<br />
1111<br />
00000000<br />
11111111 0000 111100<br />
110000<br />
1111<br />
00000000<br />
11111111 0000 1111<br />
00 11 00 11<br />
00 11 00 11<br />
00 11 00 11<br />
Figur 1.18.<br />
E E<br />
2 3<br />
Da R's virkelige v rdi ikke kan males, s ttes R =60 p . Den excentricitet der<br />
udregnes er saledes heller ikke den absolutte v rdi, men blot et udtryk der fort ller<br />
om st rrelsesforholdet mellem 2e 0 og R. For at nde 2e 0 udnyttes g. identitet:<br />
BO OE 0<br />
3 =(R +2e0 ) (R ; 2e 0 ) (1.5.3)<br />
Der kr ves altsa en bestemmelse af BO og OE 0<br />
3 i enheder af equantcirklen, hvor<br />
R =60 p .<br />
Betragt Fig. 1.18<br />
BE 0<br />
1, i enheden hvor BO = 120p : I ekliptika haves<br />
^E 0<br />
1E 0<br />
3 = 0<br />
1 + 0<br />
2 = 141 12 ) \E 0<br />
1OE 0<br />
3 =14112<br />
+<br />
\E 0<br />
1OB = 180 ; 141 12 = 38 48 .<br />
I 4OBF hvor BO = 120 p :<br />
^BF = 77 36 1:1:1<br />
) BF = 75 12 p : (1.5.4)<br />
Ptolemaios bruger her, at en vinkel i en trekant er halvt sa stor som den tilh rende<br />
buevinkel i den omskrevne cirkel. Herved er han i stand til at ga direkte fra trekanten<br />
til cirklen og derfra til kordetabellen, idet radius v lges til 60 p (i en retvinklet trekant<br />
er hypotenusen, diameter, i den omskrevne cirkel).<br />
I equantcirklen:<br />
+<br />
^E 0<br />
1E 0<br />
2E 0<br />
3 = 1 + 2 =13321 ) \E 0<br />
1BE 0<br />
3 =6640 30<br />
\BE 0<br />
1F = 180 ; 38 48 ; 66 40 30 =743130 .
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