dionysios's calendar

8/13/2019 Dionysios's calendar

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Gr eek stronomi cal CalendarsI lL The Calendar of D ionysiosB L VAN DER WAERDEN

1 Introduction

In PTOLEMY'S Almagest we find eight observations of the planets Mercury,Mars and Jupiter dated in the calendar o f DIoNYSIOS. As an example I shall quotethe first observation of Mercury (Almagest IX 9, p. 264 HEmER~):

In the 3 rd year of the era of DIONYSIOS, n the morning of the 21st day of Hydroni .e. in a part of the year corresponding to the zodiacal sign Aquarius) Mercurystood 3 lunar diameters north of the brilliant star (6) in the tail of Capricorn .. .The date was in the year 486 since NABONASSARon Egyptian Choiak 17/18 early inthe morning (-261 February 12).Note. Some manuscripts have the day number 29, but the codices G and D have21. For good reasons LEPSIUSand BOECKHas well as the German translator MANITIUSprefer the reading 21. GERHARD of Cremona, who translated from an Arabic manu-script, also has 21.In all cases PTOLEMYpresents double dates: first a date in the era of DIoNYslos,

and next a date in the Egyptian calendar. The latter dates can easily be reducedto Julian dates.

In Chapter 13 of his bo ok Sonnenkreise '1 AUGUST BOECKH has r econstruc tedthe calendar of DIONYSIOS, basing himself on the eight double dates of PTOLEMY.According to BOECKH, the first year of the era of DIONYSIOS began in the Julianyear -2 84 ( = 285 B.C.) on June 26, about the time o f the summer solstice. Theyears with numbers 4n, 4n q- 1 and 4 n + 2 had 365 days each, and the years4n -b 3 had 366 days. The year was divided into 12 parts named after the zodiacalsigns, beginning with Cancer. The first eleven signs had 30 clays each, and the lastsign Gemini had 35 or 36 days.

1 A. BOECKH: Ueber die vierj/ihrigen Sonnenkreise der Alten (Berlin 1863), pp. 286-340.


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1 2 6 B . L . V A N D ER W A E RD E NOTTO NEUGEBAUER criticize s BOECKH S r e c o n s t r u c t i o n a s f o l l o w s :

M u c h s p e c u l a t i o n h a s b e e n s p e n t i n a t t e m p t s t o r e c o n s t r u c t t h e c o m p l e t e D i o -nys i an ca l enda r , a r a t h e r va lue l e ss en t e rp r i s e a s l ong a s we hav e no ho pe o f check ingthese h igh ly hypo the t i c a l r e cons t ruc t i ons on add i t i ona l ma t e r i a l .I n m y o p i n i o n , t h i s h a r s h j u d g m e n t i s n o t j u s ti f ie d . I n w h a t f o l l o w s i t w i ll b e

s h o w n t h a t B OE CK H'S r e c o n s t r u c t i o n i s n o t b a s e d o n h i g h l y h y p o t h e t i c a l r e c o n -s t r u c t i o n s , b u t o n f a c t s . I f w e a c c e p t P TO LE M Y'S d o u b l e d a t e s , w e h a v e n o o t h e rc h o i c e t h a n t o a g r e e w i t h B O EC K H'S c o n c l u s i o n s .

2 . P t o l e m y s D o u b l e D a t e sF o l l o w i n g B O EC K H , I s h a l l d i v i d e t h e d o u b l e d a t e s r e c o r d e d b y P TO LE M Y i n t o

3 g r o u p s :A . Y e a r n u m b e r s 4 n -k 1: y e a r s 1 3, 2 1 a n d 4 5 .C . Y e a r n u m b e r s 4 n + 3 : y e a r 2 3.D . Y e a r n u m b e r s 4 n : y e a r s 2 4 a n d 2 8 .I n B O E CK H 'S g r o u p B ( y e a r s 4 n + 2 ) th e r e h a p p e n t o b e n o d a t e s .T h e d o u b l e d a t e s in g r o u p A a r e :

Y e a r 1 3: C a p r i c o r n 2 5 = A t h y r 2 0 /2 1 m o r n i n g (HEIBERG p . 352 )= - 2 7 1 J a n u a r y 1 8

Y e a r 2 1 : S c o r p i o 2 2 = T h o t h 1 8/1 9 m o r n i n g (HEIBERG p. 288 )- 2 6 4 N o v e m b e r 15

Y e a r 2 1 : S c o r p i o 2 6 = T h o t h 2 2 /2 3 m o r n i n g (HEIBERG p . 289 )= - 2 6 4 N o v e m b e r 19

Y e a r 4 5 : V i r g o 1 0 = E p i p h i 1 7 /1 8 m o r n i n g (HEIBERG p . 386 )----- - 2 40 Se p t em be r 4

I n g r o u p C w e h a v e t w o e n t r i e sY e a r 2 3 : A q u a r i u s 2 1 = C h o i a k 1 7 /1 8 m o r n i n g ( HE m ER G p . 2 6 4)

= --2 61 F e b r u a r y 1 2Y e a r 2 3 : T a u r u s 4 = M e c h i r 3 0 / P h a m e n o t h 1 e v e n i n g ( HE IB E RG p . 2 6 5 )

= - -261 A pr i l 25I n g r o u p D w e h a v e

Y e a r 2 4 : L e o 2 8 = P a y n i 3 0 e v e n i n g ( H EI BE R G p . 2 6 7 )= --2 61 A u g u s t 2 32 0 . NEUGEBAUER: A Hi s to ry o f An c i en t M a them a t i ca l A s t ron om y (Sp r inge r -Ver lag 1975) , Par t Three , p . 1067.


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G r e e k A s t r o n o m i c a l C a l e n d ar s . I I I 1 27Y e a r 2 8 : G e m i n i 7 ---- P h a r m u t h i 5 / 6 e v e n i n g (H E IB E R G p . 2 6 6 )

---- - 2 56 M ay 28F r o m t h e se d o u b l e d a t e s w e s e e t h a t t h e y e a r s o f D I o N v s I o s w e r e d i v id e di n t o 1 2 p a r t s n a m e d a f t e r t h e z o d i a c a l s i g n s . I n t h i s r e s p e c t D IO N Y SIO S f o l l o w e d

t h e t r a d i t i o n o f E UC TE MO N a n d CALLIPPOS w h o u s e d t r o p i c a l y e a r s b e g i n n i n gw i t h t h e s u m m e r s o l st ic e o n t h e d a y C a n c e r 1.

T h e f ir st y e a r o f t h e E r a o f D I o y v s i o s b e g a n i n t h e J u l i a n y e a r -2 8 4 . T h i sc a n b e v e r i f ie d b y c a l c u l a t i n g t h e d i f fe r e n c es b e t w e e n t h e y e a r n u m b e r s o f D I oN Y -s l o s a n d t h e J u l i a n y e a r n u m b e r s . I n t h e f i rs t h a l f o f t h e J u l i a n y e a r s , b e f o r e t h es u m m e r s o l s t i c e , t h e d i f f e r e n c e s a r e

13 - - ( - 2 7 1 ) = 2 8 4 i n J a n u a r y2 3 - - ( - 2 6 1 ) ---- 2 8 4 in F e b r u a r y a n d A p r i l2 8 - - ( -2 5 6 ) = 2 8 4 i n M a y

b u t a f t e r t h e s o ls t ic e t h e y a r e2 4 - - ( - 2 6 1 ) = 2 8 5 i n A u g u s t4 5 - - ( - 2 4 0 ) = 2 8 5 i n S e p t e m b e r2 1 - - ( -2 6 4 ) = 2 8 5 i n N o v e m b e r .

S i n c e t h e y e a r s o f DIO NY SlO S b e g a n a n d e n d e d w i t h t h e s u m m e r s o ls ti c e, t h e yc a n n o t h a v e b e e n E g y p t i a n y e a r s o f 36 5 d a y s . I t i s e a s y t o c h e c k t h is c o n c l u s i o nb y c o m p a r i n g d a t e s i n y e a r s f a r a p a r t . F o r i n s t a n c e w e m a y s t a r t w i t h t h e l a s td o u b l e d a t e o f g r o u p A :( 1) Y e a r 4 5 : V i r g o 1 0 m o r n i n g = S e p t e m b e r 4 .

A d d i n g 7 6 d a y s a n d s u p p o s i n g t h a t t h e s i g n s V i r g o a n d L i b r a h a v e 3 0 d a y se a c h , a s i n t h e p a r a p e g m a t a o f E UC TE MO N a n d CALLIPPOS one o b t a i n s :(2 ) Y e a r 4 5 : S c o r p io 2 6 m o r n i n g = N o v e m b e r 19w i t h a p o s s i b l e e r r o r o f o n e o f tw o d a y s , b e c a u s e t h e s ig n s V i r g o a n d L i b r a m i g h th a v e 2 9 o r 3 1 d a y s .T w e n t y - f o u r y e a r s e a r l i e r w e h a v e t h e d o u b l e d a t e( 3) Y e a r 2 1 : S c o r p i o 2 6 m o r n i n g = N o v e m b e r 19.T h e t i m e b e t w e e n t h e d a t e s ( 2) a n d ( 3) i s 2 4 J u l ia n y e a r s , o r 24 E g y p t i a n y e a r s a n d6 d a y s , b u t t h e D i o n y s i a n d a y n u m b e r s i n ( 2) a n d ( 3) a r e e x a c t l y t h e s a m e . I tf o l l o w s t h a t D IO N Y SIO S d i d u s e y e a r s o f 3 6 5 ~ d a y s , a n d i t a l s o f o l l o w s t h a t t h es i g n e V i r g o a n d L i b r a h a d e x a c t l y 3 0 d a y s e a c h .

I n g r o u p C t h e d i f f er e n c e b e t w e e n t h e t w o n i g h t s in t h e E g y p t i a n c a l e n d a r is7 3 d a y s, a n d t h e d i s t a n c e b e t w e e n t h e t w o D i o n y s i a n d a t e s i s 3 z o d i a c a l s i g n sm i n u s 1 7 d a y s . I f w e a s s u m e , t e n t a t iv e l y , t h a t t h e s i g n s A q u a r i u s , P i sc e s a n d A r i e sh a v e 3 0 d a y s e a c h , a n d i f w e a s s u m e ( w i th B O E CK H ) t h a t t h e D i o n y s i a n d a y d i dn o t b e g i n a t m i d n i g h t , b u t a t s u n r i s e o r n o o n o r s u n s e t, t h e a g r e e m e n t b e t w e e n7 3 d a y s a n d 3 s i g n s m i n u s 1 7 d a y s i s p e r f e c t .


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128 B . L . VAN DERWAERDENS t i l l m o r e s i g n s c a n b e s h o w n t o h a v e 3 0 d a y s o n l y . I n g r o u p D w e h a v e t h ee q u a t i o n

( 4) Y e a r 2 4 : L e o 2 8 e v e n i n g = A u g u s t 2 3 .A d d i n g 2 7 9 d a y s a n d s u p p o s in g t h a t t h e s ig ns L e o t h r o u g h T a u r u s h a v e

3 0 d a y s e a c h , o n e o b t a i n s( 5) Y e a r 2 4 : G e m i n i 7 e v e n i n g = M a y 2 8 .

F o u r y e a r s l a t e r o n e h a s f r o m t h e t e x t o f PTO LEM Y( 6 ) Y e a r 2 8 : G e m i n i 7 e v e n i n g ---- M a y 2 8 .

T h e s e t w o e q u a t i o n s a r e i n p e r f e c t a c c o r d a n c e , s i n c e 4 J u l i a n y e a r s a r e e q u a lt o 4 D i o n y s i a n y e a r s , a s w e h a v e s e e n . I t f o l l o w s t h a t t h e t e n s i g n s f r o m L e o t oT a u r u s h a v e t o g e t h e r e x a c t l y 3 0 0 d a y s .

T h i s i s a v e r y r e m a r k a b l e r e s u lt . I n t h e p a r a p e g m a o f EV CT EM ON t h e s e 1 0 s ig n sh a v e t o g e t h e r 3 0 4 d a y s , a n d i n t h e p a r a p e g m a o f C A LL IP PO S, w h i c h a c c o r d s w e l lw i t h m o d e r n t h e o r y , t h e y h a v e t o g e t h e r 3 0 2 d a y s . I f 1 0 s i g n s h a v e t o g e t h e r 3 0 0d a y s , t h e re m a i n i n g t w o s ig n s G e m i n i a n d C a n c e r h a v e to g e t h e r 6 5 o r 6 6 d a y s :a n i m p o s s i b i li t y i n a n y r e a s o n a b l e t h e o r y o f t h e a n n u a l m o t i o n o f t h e s u n . A sN EU GE BA UE R ( n o t e 2 , p . 6 2 9 ) r i g h t l y n o t e s , t h e D i o n y s i a n z o d i a c a l d a t e s , c o n s i d -e r e d a s s o l ar lo n g i t u d e s , d e v i a t e in a n i r r e g u l a r f a s h i o n b o t h f r o m t r u e a n d f r o mm e a n s o l a r l o n g i t u d e s .

T h u s w e a r e b o u n d t o c o n c l u d e ( w i t h B O EC K H) t h a t D IO NY SIO S u s e d a d i v i s i o no f t h e y e a r in t o 11 p a r t s ( C a n c e r t h r o u g h T a u r u s ) o f 3 0 d a y s e a c h , a n d o n e p a r t( G e m i n i ) o f 35 o f 3 6 d a y s . A s i n th e E g y p t i a n a n d A l e x a n d r i a n c a l e n d a r s t h e 5o r 6 e p a g o m e n a l d a y s w e r e p l a c e d a t t h e e n d o f t h e y e a r , w h i c h is v e r y c o n -v e n i e n t f o r c a l e n d a r i c c a l c u l a t i o n s .

I f t h is is a s s u m e d , w e c a n g o b a c k f r o m e a c h o f th e e i g h t d o u b l e d a t e s a n dc a l c u l a te t h e b e g i n n i n g o f t h e D i o n y s i a n y e a r in e a c h c a se . T h i s h a s b e e n d o n e b yBOECKH w i t h t h e f o l l o w i n g r e s u l t ( S o n n e n k r e i s e , p . 3 1 7 ) :

T h e y e a r s A , B , C , w h o s e D i o n y s i a n y e a r n u m b e r s a r e n o t d M s i b l e b y 4 , b e g i no n J u n e 2 6 a n d t h e y e a r s D b e g i n o n J u n e 2 7 .

I t f o l l o w s t h a t t h e y e a r s A , B , D h a v e 3 6 5 d a y s e a c h , a n d t h e y e a r s C 3 6 6 d a y s .I f th i s is a s s u m e d , a ll d o u b l e d a t e s a r e c o r r e c t l y r e d u c e d t o t h e E g y p t i a n c a l e n -

d a r i n P TO LE M Y'S t e x t , w i t h t h e e x c e p t i o n o f t h e f ir s t, w h i c h s h o u l d r e a dY e a r 1 3 : C a p r i c o r n 2 6 = A t h y r 2 0 /2 1 m o r n i n g = - 2 7 1 J a n u a r y 18 .

T h e c a l e n d a r o f D IO NY SIO S i s n o t a n a s t r o n o m i c a l c a l e n d a r i n t h e s t r ic t s e n s e ,b e c a u s e i t i s n o t b a s e d o n a t h e o r y a b o u t t h e s u n ' s c o u r s e i n t h e z o d i a c . I t i s ac o n v e n i e n t c o m p r o m i s e b e t w e e n t h e E g y p t i a n c a le n d a r w i th i ts 1 2 m o n t h s o f3 0 d a y s a n d 5 e p a g o m e n a l d a y s , a n d t h e m o r e s o p h i s t i c a t e d z o d i a c a l c a l e n d a r so f EUCTEMON an d CALLIPPOS.


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1 3 0 B L V A N D ER W A E R DE N

a n d r i a f r o m t h e c l a s si c a l p e r i o d t o t h e t i m e o f PT OL EM Y . W e n o w m a y a s k :W h a t t y p e o f t h e o r y d i d ARISTARCHOS TIMOCHARISA R IS TY L LO S a n d o u r A n o n y -m o u s a i m a t ?

F o r A R IS TA R CH O S t h e c a s e i s c l e a r . H e w a s t h e a u t h o r o f t h e h e l i o c e n t r i ct h e o r y , i n w h i c h t h e p la n e t s ( i n c lu d i n g t h e e a r th ) w e r e s u p p o s e d t o r o t a t e a r o u n dt h e s u n . I f h e w a n t e d t o d e t e r m i n e t h e c o n s t a n t s o f t h i s t h e o r y , h e f ir s t h a d t os t u d y t h e a p p a r e n t m o t i o n o f t h e s u n. H e c o u l d t a k e t h e d u r a ti o n s o f t h e a s t ro -n o m i c a l s e a s o n s f r o m t h e p a r a p e g m a o f CALLIPPOS b u t h e h a d t o o b s e r v e a tl e a s t o n e s o l s t i t i u m o r e q u i n o c t i u m f o r h i s o w n t i m e , a n d t h i s h e d i d .

A s f o r DIONYSIOSh is c a l e n d a r f o l lo w e d t h e t r a d i ti o n o f t h e G r e e k p a r a p e g m a -f i s ts E U CT EM O N a n d C A L LIP PO S . H i s d i v i s i o n o f t h e z o d i a c i n t o 1 2 si g n s w a s a t r o -p i c a l d i v is i o n : t h e s u n w a s s u p p o s e d t o e n t e r C a n c e r a t t h e t i m e o f t h e s u m m e rs o l st ic e , w h e r e a s i n t h e B a b y l o n i a n s y s t e m A t h e s a m e s o l s ti c e w a s l o c a t e d a tC a n c e r 1 0 , a n d i n s y s t e m B a t 8 . S o t h e c a l e n d a r o f D IO N YS IO S b e l o n g e d t o t h eG r e e k a n d n o t t o t h e B a b y l o n i a n t r a d i t i o n .

T h e u n k n o w n a u t h o r o f t he o b s e r v a t io n s o f M e r c u r y , M a r s a n d J u p i te r l i ve da t t h e t i m e o f A R IS TA R CH O S a n d u s e d t h e c a l e n d a r o f D IO N Y SIO S . H e u s e d , l i k eD Io NY S IO S, t h e G r e e k d i v i s i o n o f th e z o d i a c , i n w h i c h t h e s i g n C a n c e r b e g i n s w i t ht h e s u m m e r s o ls t ic e . L a t e r E g y p t i a n p l a n e t a r y t e x t s a l l u s e d t h e B a b y l o n i a n s i d e r ia ld i v i s i o n o f t h e z o d i a c , a n d a s i d e re a l y e a r l o n g e r t h a n 3 6 5 ff d a y s . F o r t h e s e r e a s o n s ,i t i s n o t l i k e l y t h a t t h e p l a n e t a r y o b s e r v a t i o n s w e r e m a d e i n o r d e r t o d e t e r m i n et h e c o n s t a n t s o f a B a b y l o n i a n t y p e t h e o ry .

W h a t g e o m e t r i c a l p l a n e t a r y t h e o r i es w e r e a v a il a b l e b et w e e n - 2 7 1 a n d - 2 4 0 ,w h e n t h e s e o b s e r v a t i o n s w e r e m a d e ?

T h e p r i m i t i v e ep i cy c le t h e o r y d i s cu s s e d i n m y p a p e r s , w h i c h w a s p r o b a b l yi n v e n t e d b y t h e P y t h a g o r e a n s , c a n n o t b e u s e d t o e x p l a i n t h e m o t i o n o f M a r s ,a s AABOE6 h a s s h o w n . T h e s a m e h o l d s f o r t h e h o m o c e n t r i c s p h e r e s o f EU DO XO Sand CALLIPPOS, as SCHIAPARELLI 7 p r o v e d . T h u s , i f o u r o b s e r v e r s i n A l e x a n d r i ah a d a d e f in i te g e o m e t r i c a l t h e o r y i n m i n d , t h e m o s t p r o b a b l e c a n d i d a t e i s t h eh e l i o c e n t r i c t h e o r y o f A R IS TA R CH O S o f S a m o s .

H e n c e m y h y p o t h e s is : The purpose of the observation of ARISTARCHOSTIMOCHARIS, ARISTYLLOSand our Anonymous was to determine the constants inthe heliocentric theory of ARISTARCHOS.

Wies l i a che r 5CH-8053 ZiJ r ichReceived February 2, 1983)

5 VA N DER WAERDEN: T h e M o t i o n o f V e n u s, M e r c u r y a n d t h e S u n i n E a r l y G r e e kAs t ro nom y , Arch ive fo r H i s to ry o f Exac t Sc ie r.c es 26 , pp . 99 -113 (1982) .6 A . AABOE: On a G reek Qua l i t a t i ve M ode l o f t he Ep icyc l e Va r i e ty , Cen t au rus 9 ,pp . 1-10 (1963) .

G V SCHIAPARELLI: Scr i t t i su l la s tor ia de l la a s t ro no m ia an t ic a I I , pp . 3-112.G e r m a n t r a n s l a t i o n : A b h a n d l u n g e n z u r G e s c h i c h t e d e r M a t h . I ( 1 8 7 7 ) , S u p p l e m e n t t oZei t schr i f t f i J r Math . und Phys . 22 , pp . 101-198.

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